Subgroups — D₂₃×#542 #15686767

Order 2944.

1132 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13093041	C₂	1	1	yes	maximal, characteristic, complemented
2	#13093040	C₂	1	1	yes	maximal, characteristic, complemented
3	#13093039	C₂	1	1	yes	maximal, characteristic, complemented
4	#13093038	C₂	1	1	yes	maximal, characteristic, complemented
5	#13093037	C₂	1	2	yes	maximal, complemented
6	#13093036	C₂	1	1	yes	maximal, characteristic, complemented
7	#13093035	C₂	1	2	yes	maximal, complemented
8	#13093034	C₂	1	1	yes	maximal, characteristic, complemented
9	#13093033	C₂	1	2	yes	maximal, complemented
10	#13093032	C₂	1	1	yes	maximal, characteristic, complemented
11	#13093031	C₂	1	2	yes	maximal, complemented
12	C₄⋊Q₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
13	#126×D₂₃	C₂	1	2	yes	maximal, complemented
14	#124×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#124×D₂₃	C₂	1	2	yes	maximal, complemented
16	#123×D₂₃	C₂	1	2	yes	maximal, complemented
17	C₂²⋊Q₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂²⋊Q₈×D₂₃	C₂	1	2	yes	maximal, complemented
19	#121×D₂₃	C₂	1	1	yes	maximal, characteristic
20	C₄×Q₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
21	C₄×D₄×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
22	#117×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
23	C₂₃×#542	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×Q₈×D₂₃	C₂²	1	1	yes	characteristic
25	C₂×Q₈×D₂₃	C₂²	1	2	yes	—
26	C₂×D₄×D₂₃	C₂²	1	1	yes	characteristic
27	C₂×D₄×D₂₃	C₂²	1	2	yes	—
28	C₂²×C₄×D₂₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₂₃	C₂²	1	2	yes	—
30	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
31	C₂₃×#126	C₂²	1	2	yes	complemented
32	C₂₃×#124	C₂²	1	2	yes	complemented
33	C₂₃×#124	C₂²	1	1	yes	characteristic, complemented
34	C₂₃×#123	C₂²	1	2	yes	complemented
35	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
36	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
37	C₂₃×#121	C₂²	1	1	yes	characteristic
38	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
39	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
40	C₂₃×#117	C₂²	1	1	yes	characteristic, complemented
41	#10625118	C₂²	1	2	yes	complemented
42	#10625118	C₂²	1	1	yes	characteristic, complemented
43	#10625117	C₂²	1	2	yes	complemented
44	#10625117	C₂²	1	1	yes	characteristic, complemented
45	Q₈×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
46	#10625115	C₂²	1	2	yes	complemented
47	#10625110	C₂²	1	2	yes	—
48	#10625109	C₂²	1	1	yes	characteristic
49	#10625107	C₂²	1	1	yes	characteristic, complemented
50	#10625107	C₂²	1	2	yes	complemented
51	#10625106	C₂²	1	2	yes	complemented
52	D₄×Dic₂₃	C₂²	1	1	yes	characteristic, complemented
53	#10625101	C₂²	1	1	yes	characteristic
54	#10625100	C₂²	1	2	yes	complemented
55	C₄×C₂²⋊D₂₃	C₂²	1	2	yes	complemented
56	#10625097	C₂²	1	1	yes	characteristic, complemented
57	#10625095	C₂²	1	2	yes	complemented
58	#10625095	C₂²	1	1	yes	characteristic, complemented
59	#10625069	C₂²	1	2	yes	complemented
60	#10625069	C₂²	1	2	yes	complemented
61	#10625069	C₂²	1	2	yes	complemented
62	#10625068	C₂²	1	1	yes	characteristic, complemented
63	#10625068	C₂²	1	2	yes	complemented
64	#10625067	C₂²	1	2	yes	complemented
65	#10625067	C₂²	1	2	yes	complemented
66	#10625067	C₂²	1	2	yes	complemented
67	#10625066	C₂²	1	1	yes	characteristic
68	#10625065	C₂²	1	2	yes	complemented
69	#10625065	C₂²	1	2	yes	complemented
70	#10625064	C₂²	1	1	yes	characteristic, complemented
71	#10625063	C₂²	1	1	yes	characteristic, complemented
72	#10625063	C₂²	1	1	yes	characteristic, complemented
73	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
74	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
75	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
76	C₄⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
77	C₄⋊C₄×D₂₃	C₂²	1	2	yes	—
78	C₄⋊C₄×D₂₃	C₂²	1	2	yes	—
79	C₄⋊C₄×D₂₃	C₂²	1	2	yes	—
80	#10625059	C₂²	1	1	yes	characteristic, complemented
81	#10625059	C₂²	1	2	yes	complemented
82	#10625059	C₂²	1	1	yes	characteristic, complemented
83	#10625058	C₂²	1	1	yes	characteristic, complemented
84	#10625058	C₂²	1	2	yes	complemented
85	#10625057	C₂²	1	2	yes	complemented
86	#10625056	C₂²	1	2	yes	complemented
87	#10625056	C₂²	1	2	yes	complemented
88	#10625056	C₂²	1	2	yes	complemented
89	#10625056	C₂²	1	2	yes	complemented
90	#10625056	C₂²	1	2	yes	complemented
91	#10625056	C₂²	1	2	yes	complemented
92	#10625055	C₂²	1	2	yes	—
93	#10625054	C₂²	1	2	yes	complemented
94	#10625054	C₂²	1	2	yes	complemented
95	(C₂²⋊D₂₃)⋊C₄	C₂²	1	2	yes	complemented
96	C₂²⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
97	C₂²⋊C₄×D₂₃	C₂²	1	1	yes	characteristic
98	C₂²⋊C₄×D₂₃	C₂²	1	2	yes	—
99	C₂²⋊C₄×D₂₃	C₂²	1	2	yes	—
100	C₂²⋊C₄×D₂₃	C₂²	1	2	yes	—
101	C₂²⋊C₄×D₂₃	C₂²	1	2	yes	—
102	C₂²⋊C₄×D₂₃	C₂²	1	2	yes	—
103	#10625050	C₂²	1	2	yes	complemented
104	#10625050	C₂²	1	2	yes	complemented
105	#10625050	C₂²	1	2	yes	complemented
106	#10625049	C₂²	1	2	yes	complemented
107	#10625049	C₂²	1	2	yes	complemented
108	#10625049	C₂²	1	2	yes	complemented
109	#10625048	C₂²	1	1	yes	characteristic, complemented
110	#10625048	C₂²	1	1	yes	characteristic, complemented
111	#10625047	C₂²	1	2	yes	complemented
112	#10625046	C₂²	1	2	yes	complemented
113	#10625046	C₂²	1	1	yes	characteristic, complemented
114	C₄×D₉₂	C₂²	1	1	yes	characteristic, complemented
115	#10625043	C₂²	1	1	yes	characteristic, complemented
116	#10625043	C₂²	1	1	yes	characteristic, complemented
117	C₄²×D₂₃	C₂²	1	1	yes	characteristic
118	C₄²×D₂₃	C₂²	1	1	yes	characteristic
119	C₄²×D₂₃	C₂²	1	2	yes	—
120	#10625040	C₂²	1	1	yes	characteristic, complemented
121	C₄×Dic₄₆	C₂²	1	2	yes	complemented
122	C₄×Dic₄₆	C₂²	1	1	yes	characteristic, complemented
123	C₂³×D₂₃	C₂³	1	1	yes	characteristic
124	C₂³×D₂₃	C₂³	1	2	yes	—
125	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
126	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
127	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
128	C₂×D₄×C₂₃	C₂³	1	2	yes	—
129	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
130	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
131	C₂×C₂²⋊D₂₃	C₂³	1	2	yes	—
132	C₂×C₂²⋊D₂₃	C₂³	1	2	yes	—
133	C₂×C₂²⋊D₂₃	C₂³	1	2	yes	—
134	C₂²×Dic₂₃	C₂³	1	1	yes	characteristic
135	C₂²×Dic₂₃	C₂³	1	2	yes	—
136	Q₈×D₂₃	—	2	4	no	—
137	Q₈×D₂₃	—	4	4	no	—
138	D₄×D₂₃	—	2	4	no	—
139	D₄×D₂₃	—	4	4	no	—
140	C₂×D₉₂	C₂³	1	1	yes	characteristic
141	C₂×D₉₂	C₂³	1	2	yes	—
142	C₂×C₄×D₂₃	—	2	4	no	—
143	C₂×C₄×D₂₃	—	4	4	no	—
144	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
145	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
146	C₂×C₄×D₂₃	C₂³	1	2	yes	—
147	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
148	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
149	C₂×C₄×D₂₃	C₂³	1	2	yes	—
150	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
151	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
152	C₂×C₄×D₂₃	C₂³	1	2	yes	—
153	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
154	C₂×C₄×D₂₃	C₂³	1	2	yes	—
155	C₂×C₄×D₂₃	C₂³	1	2	yes	—
156	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
157	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
158	C₂×Dic₄₆	C₂³	1	2	yes	—
159	C₂×Dic₄₆	C₂³	1	2	yes	—
160	C₂×Dic₄₆	C₂³	1	2	yes	—
161	C₂×Dic₄₆	C₂³	1	2	yes	—
162	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
163	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
164	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
165	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
166	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
167	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
168	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
169	C₂²⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
170	C₂²⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
171	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
172	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
173	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
174	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
175	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
176	C₄²×C₂₃	C₂³	1	1	yes	characteristic
177	C₄²×C₂₃	C₂³	1	2	yes	—
178	C₄²×C₂₃	C₂³	1	1	yes	characteristic
179	C₂²⋊Dic₂₃	C₂³	1	1	yes	characteristic
180	C₂²⋊Dic₂₃	C₂³	1	1	yes	characteristic
181	C₂²⋊Dic₂₃	C₂³	1	2	yes	—
182	C₂²⋊Dic₂₃	C₂³	1	2	yes	—
183	C₂²⋊Dic₂₃	C₂³	1	2	yes	—
184	C₂²⋊Dic₂₃	C₂³	1	2	yes	—
185	C₂²⋊Dic₂₃	C₂³	1	2	yes	—
186	(C₂×D₂₃)⋊C₄	C₂³	1	1	yes	characteristic
187	(C₂×D₂₃)⋊C₄	C₂³	1	1	yes	characteristic
188	(C₂×D₂₃)⋊C₄	C₂³	1	1	yes	characteristic
189	(C₂×D₂₃)⋊C₄	C₂³	1	1	yes	characteristic
190	(C₂×D₂₃)⋊C₄	C₂³	1	2	yes	—
191	(C₂×D₂₃)⋊C₄	C₂³	1	2	yes	—
192	(C₂×D₂₃)⋊C₄	C₂³	1	2	yes	—
193	(C₂×D₂₃)⋊C₄	C₂³	1	2	yes	—
194	(C₂×D₂₃)⋊C₄	C₂³	1	2	yes	—
195	(C₂×D₂₃)⋊C₄	C₂³	1	2	yes	—
196	(C₂×D₂₃)⋊C₄	C₂³	1	2	yes	—
197	(C₂×D₂₃)⋊C₄	C₂³	1	2	yes	—
198	(C₂×D₂₃)⋊C₄	C₂³	1	2	yes	—
199	(C₂×D₂₃)⋊C₄	C₂³	1	2	yes	—
200	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
201	C₄⋊Dic₂₃	C₂³	1	2	yes	—
202	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
203	C₄⋊Dic₂₃	C₂³	1	2	yes	—
204	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
205	C₄⋊Dic₂₃	C₂³	1	2	yes	—
206	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
207	Dic₂₃⋊C₄	C₂³	1	2	yes	—
208	Dic₂₃⋊C₄	C₂³	1	2	yes	—
209	Dic₂₃⋊C₄	C₂³	1	2	yes	—
210	Dic₂₃⋊C₄	C₂³	1	1	yes	characteristic
211	Dic₂₃⋊C₄	C₂³	1	1	yes	characteristic
212	Dic₂₃⋊C₄	C₂³	1	1	yes	characteristic
213	Dic₂₃⋊C₄	C₂³	1	1	yes	characteristic
214	Dic₂₃⋊C₄	C₂³	1	2	yes	—
215	Dic₂₃⋊C₄	C₂³	1	2	yes	—
216	Dic₂₃⋊C₄	C₂³	1	2	yes	—
217	Dic₂₃⋊C₄	C₂³	1	2	yes	—
218	Dic₂₃⋊C₄	C₂³	1	2	yes	—
219	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
220	C₄×Dic₂₃	C₂³	1	2	yes	—
221	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
222	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
223	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
224	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
225	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
226	C₄×Dic₂₃	C₂³	1	2	yes	—
227	C₄×Dic₂₃	C₂³	1	2	yes	—
228	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
229	C₂³×C₂₃	C₂⁴	1	2	yes	—
230	C₂²×D₂₃	—	2	2	no	—
231	C₂²×D₂₃	—	2	4	no	—
232	C₂²×D₂₃	—	2	4	no	—
233	C₂²×D₂₃	—	2	2	no	—
234	C₂²×D₂₃	—	2	2	no	—
235	C₂²×D₂₃	—	2	4	no	—
236	C₂²×D₂₃	C₂⁴	1	1	yes	characteristic
237	C₂²×D₂₃	C₂⁴	1	2	yes	—
238	C₂²×D₂₃	C₂⁴	1	1	yes	characteristic
239	Q₈×C₂₃	—	2	2	no	—
240	Q₈×C₂₃	—	4	2	no	—
241	D₄×C₂₃	—	2	2	no	—
242	D₄×C₂₃	—	4	2	no	—
243	C₂×C₄×C₂₃	—	2	2	no	—
244	C₂×C₄×C₂₃	—	4	2	no	—
245	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
246	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
247	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
248	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
249	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
250	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
251	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
252	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
253	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
254	C₂²⋊D₂₃	—	2	4	no	—
255	C₂²⋊D₂₃	—	4	2	no	—
256	C₂²⋊D₂₃	—	4	2	no	—
257	C₂×Dic₂₃	—	2	2	no	—
258	C₂×Dic₂₃	—	4	2	no	—
259	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
260	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
261	C₂×Dic₂₃	C₂⁴	1	2	yes	—
262	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
263	C₂×Dic₂₃	C₂⁴	1	2	yes	—
264	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
265	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
266	C₂×Dic₂₃	C₂⁴	1	2	yes	—
267	C₂×Dic₂₃	C₂⁴	1	1	yes	characteristic
268	D₉₂	—	2	2	no	—
269	D₉₂	—	4	2	no	—
270	C₄×D₂₃	—	2	2	no	—
271	C₄×D₂₃	—	2	4	no	—
272	C₄×D₂₃	—	2	2	no	—
273	C₄×D₂₃	—	2	4	no	—
274	C₄×D₂₃	—	2	4	no	—
275	C₄×D₂₃	—	2	2	no	—
276	C₄×D₂₃	—	2	2	no	—
277	C₄×D₂₃	—	2	2	no	—
278	C₄×D₂₃	—	2	2	no	—
279	C₄×D₂₃	—	2	2	no	—
280	C₄×D₂₃	—	4	2	no	—
281	C₄×D₂₃	—	4	2	no	—
282	C₄×D₂₃	#41	1	4	yes	—
283	Dic₄₆	—	2	2	no	—
284	Dic₄₆	—	2	4	no	—
285	Dic₄₆	—	4	2	no	—
286	Dic₄₆	—	4	2	no	—
287	Dic₄₆	—	4	2	no	—
288	C₂×#542	—	23	1	no	maximal, Sylow
289	C₂²×C₂₃	—	2	1	no	—
290	C₂²×C₂₃	—	2	2	no	—
291	C₂²×C₂₃	—	2	2	no	—
292	C₂²×C₂₃	—	2	1	no	—
293	C₂²×C₂₃	—	2	1	no	—
294	C₂²×C₂₃	—	2	2	no	—
295	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
296	C₂×D₂₃	—	2	1	no	—
297	C₂×D₂₃	—	2	2	no	—
298	C₂×D₂₃	—	2	2	no	—
299	C₂×D₂₃	—	2	1	no	—
300	C₂×D₂₃	—	2	1	no	—
301	C₂×D₂₃	—	2	2	no	—
302	C₂×D₂₃	—	4	4	no	—
303	C₂×D₂₃	—	4	8	no	—
304	C₂×D₂₃	C₂×#41	1	2	yes	—
305	C₂×D₂₃	H(5,𝔽₂)	1	2	yes	—
306	C₂×D₂₃	2₋¹⁺⁴	1	2	yes	—
307	C₄×C₂₃	—	2	2	no	—
308	C₄×C₂₃	—	2	1	no	—
309	C₄×C₂₃	—	2	1	no	—
310	C₄×C₂₃	—	2	2	no	—
311	C₄×C₂₃	—	2	2	no	—
312	C₄×C₂₃	—	2	1	no	—
313	C₄×C₂₃	—	2	1	no	—
314	C₄×C₂₃	—	2	1	no	—
315	C₄×C₂₃	C₂×#41	1	2	yes	—
316	Dic₂₃	—	2	2	no	—
317	Dic₂₃	—	2	1	no	—
318	Dic₂₃	—	2	1	no	—
319	Dic₂₃	—	2	2	no	—
320	Dic₂₃	—	2	2	no	—
321	Dic₂₃	—	2	1	no	—
322	Dic₂₃	—	2	1	no	—
323	Dic₂₃	—	2	1	no	—
324	Dic₂₃	C₂×#41	1	2	yes	—
325	#542	—	23	1	no	—
326	#542	—	23	2	no	—
327	#542	—	23	2	no	—
328	#542	—	23	1	no	—
329	#542	—	23	2	no	—
330	#542	—	23	2	no	—
331	#542	—	23	1	no	—
332	#542	—	23	1	no	—
333	#542	—	23	1	no	—
334	#542	—	23	1	no	—
335	#542	—	23	1	no	—
336	#542	D₂₃	1	1	yes	characteristic, direct product factor
337	C₂×C₄⋊Q₈	—	23	1	no	—
338	C₂×#126	—	23	2	no	—
339	C₂×#124	—	23	1	no	—
340	C₂×#124	—	23	2	no	—
341	C₂×#123	—	23	2	no	—
342	C₂×C₂²⋊Q₈	—	23	2	no	—
343	C₂×C₂²⋊Q₈	—	23	1	no	—
344	C₂×#121	—	23	1	no	—
345	C₂×C₄×Q₈	—	23	1	no	—
346	C₂×C₄×D₄	—	23	1	no	—
347	C₂×#117	—	23	1	no	—
348	C₂×C₂₃	—	4	1	no	—
349	C₂×C₂₃	—	4	2	no	—
350	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
351	C₂×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
352	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
353	D₂₃	—	4	1	no	—
354	D₂₃	—	4	2	no	—
355	D₂₃	#542	1	4	yes	direct product factor
356	C₂²×Q₈	—	23	1	no	—
357	C₂²×Q₈	—	23	2	no	—
358	C₂²×D₄	—	23	1	no	—
359	C₂²×D₄	—	23	2	no	—
360	C₂³×C₄	—	23	1	no	—
361	C₂³×C₄	—	23	2	no	—
362	C₄⋊Q₈	—	23	2	no	—
363	C₄⋊Q₈	—	23	2	no	—
364	C₄⋊Q₈	—	23	1	no	—
365	C₄⋊Q₈	—	23	1	no	—
366	C₄⋊Q₈	—	23	1	no	—
367	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
368	#126	—	23	2	no	—
369	#126	—	23	2	no	—
370	#126	—	23	2	no	—
371	#126	—	23	2	no	—
372	#126	—	23	2	no	—
373	#126	—	23	2	no	—
374	#126	—	23	2	no	—
375	#126	C₂×D₂₃	1	2	yes	complemented
376	#124	—	23	1	no	—
377	#124	—	23	2	no	—
378	#124	—	23	1	no	—
379	#124	—	23	2	no	—
380	#124	—	23	2	no	—
381	#124	—	23	2	no	—
382	#124	—	23	2	no	—
383	#124	—	23	2	no	—
384	#124	—	23	2	no	—
385	#124	—	23	2	no	—
386	#124	—	23	1	no	—
387	#124	—	23	2	no	—
388	#124	C₂×D₂₃	1	1	yes	characteristic, complemented
389	#124	C₂×D₂₃	1	2	yes	complemented
390	#123	—	23	2	no	—
391	#123	—	23	2	no	—
392	#123	—	23	2	no	—
393	#123	—	23	2	no	—
394	#123	—	23	2	no	—
395	#123	—	23	2	no	—
396	#123	—	23	2	no	—
397	#123	C₂×D₂₃	1	2	yes	complemented
398	C₂²⋊Q₈	—	23	2	no	—
399	C₂²⋊Q₈	—	23	1	no	—
400	C₂²⋊Q₈	—	23	2	no	—
401	C₂²⋊Q₈	—	23	1	no	—
402	C₂²⋊Q₈	—	23	2	no	—
403	C₂²⋊Q₈	—	23	2	no	—
404	C₂²⋊Q₈	—	23	2	no	—
405	C₂²⋊Q₈	—	23	2	no	—
406	C₂²⋊Q₈	—	23	1	no	—
407	C₂²⋊Q₈	—	23	2	no	—
408	C₂²⋊Q₈	—	23	2	no	—
409	C₂²⋊Q₈	—	23	2	no	—
410	C₂²⋊Q₈	C₂×D₂₃	1	2	yes	complemented
411	C₂²⋊Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
412	#121	—	23	1	no	—
413	#121	—	23	2	no	—
414	#121	—	23	1	no	—
415	#121	—	23	1	no	—
416	#121	—	23	2	no	—
417	#121	C₂×D₂₃	1	1	yes	characteristic
418	C₄×Q₈	—	23	1	no	—
419	C₄×Q₈	—	23	2	no	—
420	C₄×Q₈	—	23	1	no	—
421	C₄×Q₈	—	23	2	no	—
422	C₄×Q₈	—	23	1	no	—
423	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic, complemented
424	C₄×D₄	—	23	1	no	—
425	C₄×D₄	—	23	2	no	—
426	C₄×D₄	—	23	1	no	—
427	C₄×D₄	—	23	2	no	—
428	C₄×D₄	—	23	1	no	—
429	C₄×D₄	C₂×D₂₃	1	1	yes	characteristic, complemented
430	#117	—	23	1	no	—
431	#117	—	23	1	no	—
432	#117	—	23	1	no	—
433	#117	—	23	1	no	—
434	#117	—	23	1	no	—
435	#117	—	23	1	no	—
436	#117	—	23	1	no	—
437	#117	C₂×D₂₃	1	1	yes	characteristic, complemented
438	C₂×C₄⋊C₄	—	23	1	no	—
439	C₂×C₄⋊C₄	—	23	1	no	—
440	C₂×C₄⋊C₄	—	23	1	no	—
441	C₂×C₄⋊C₄	—	23	1	no	—
442	C₂×C₄⋊C₄	—	23	2	no	—
443	C₂×C₄⋊C₄	—	23	2	no	—
444	C₂×C₄⋊C₄	—	23	2	no	—
445	C₂×C₂²⋊C₄	—	23	1	no	—
446	C₂×C₂²⋊C₄	—	23	1	no	—
447	C₂×C₂²⋊C₄	—	23	2	no	—
448	C₂×C₂²⋊C₄	—	23	2	no	—
449	C₂×C₂²⋊C₄	—	23	2	no	—
450	C₂×C₂²⋊C₄	—	23	2	no	—
451	C₂×C₂²⋊C₄	—	23	2	no	—
452	C₂×C₄²	—	23	1	no	—
453	C₂×C₄²	—	23	1	no	—
454	C₂×C₄²	—	23	2	no	—
455	C₂₃	C₂×#542	1	1	yes	characteristic, complemented, Sylow
456	C₂⁴	—	23	1	no	—
457	C₂⁴	—	23	2	no	—
458	C₂×Q₈	—	23	2	no	—
459	C₂×Q₈	—	23	1	no	—
460	C₂×Q₈	—	23	2	no	—
461	C₂×Q₈	—	23	2	no	—
462	C₂×Q₈	—	23	2	no	—
463	C₂×Q₈	—	46	4	no	—
464	C₂×Q₈	—	92	4	no	—
465	C₂×Q₈	C₂²×D₂₃	1	1	yes	characteristic
466	C₂×Q₈	C₂²×D₂₃	1	2	yes	—
467	C₂×D₄	—	23	2	no	—
468	C₂×D₄	—	23	2	no	—
469	C₂×D₄	—	23	2	no	—
470	C₂×D₄	—	23	1	no	—
471	C₂×D₄	—	23	2	no	—
472	C₂×D₄	—	46	4	no	—
473	C₂×D₄	—	92	4	no	—
474	C₂×D₄	C₂²×D₂₃	1	1	yes	characteristic
475	C₂×D₄	C₂²×D₂₃	1	2	yes	—
476	C₂²×C₄	—	23	1	no	—
477	C₂²×C₄	—	23	1	no	—
478	C₂²×C₄	—	23	2	no	—
479	C₂²×C₄	—	23	1	no	—
480	C₂²×C₄	—	23	1	no	—
481	C₂²×C₄	—	23	2	no	—
482	C₂²×C₄	—	23	1	no	—
483	C₂²×C₄	—	23	1	no	—
484	C₂²×C₄	—	23	2	no	—
485	C₂²×C₄	—	23	1	no	—
486	C₂²×C₄	—	23	2	no	—
487	C₂²×C₄	—	23	1	no	—
488	C₂²×C₄	—	23	1	no	—
489	C₂²×C₄	—	23	2	no	—
490	C₂²×C₄	—	23	2	no	—
491	C₂²×C₄	—	46	4	no	—
492	C₂²×C₄	—	92	4	no	—
493	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
494	C₂²×C₄	C₂²×D₂₃	1	2	yes	—
495	C₄⋊C₄	—	23	1	no	—
496	C₄⋊C₄	—	23	1	no	—
497	C₄⋊C₄	—	23	1	no	—
498	C₄⋊C₄	—	23	1	no	—
499	C₄⋊C₄	—	23	1	no	—
500	C₄⋊C₄	—	23	1	no	—
501	C₄⋊C₄	—	23	2	no	—
502	C₄⋊C₄	—	23	1	no	—
503	C₄⋊C₄	—	23	2	no	—
504	C₄⋊C₄	—	23	1	no	—
505	C₄⋊C₄	—	23	2	no	—
506	C₄⋊C₄	—	23	2	no	—
507	C₄⋊C₄	—	23	2	no	—
508	C₄⋊C₄	—	23	2	no	—
509	C₄⋊C₄	—	23	2	no	—
510	C₄⋊C₄	—	23	2	no	—
511	C₄⋊C₄	—	23	2	no	—
512	C₄⋊C₄	—	23	2	no	—
513	C₄⋊C₄	—	23	2	no	—
514	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
515	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
516	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
517	C₄⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
518	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
519	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
520	C₄⋊C₄	C₂²×D₂₃	1	2	yes	—
521	C₂²⋊C₄	—	23	1	no	—
522	C₂²⋊C₄	—	23	1	no	—
523	C₂²⋊C₄	—	23	2	no	—
524	C₂²⋊C₄	—	23	2	no	—
525	C₂²⋊C₄	—	23	2	no	—
526	C₂²⋊C₄	—	23	2	no	—
527	C₂²⋊C₄	—	23	2	no	—
528	C₂²⋊C₄	—	23	1	no	—
529	C₂²⋊C₄	—	23	1	no	—
530	C₂²⋊C₄	—	23	1	no	—
531	C₂²⋊C₄	—	23	1	no	—
532	C₂²⋊C₄	—	23	2	no	—
533	C₂²⋊C₄	—	23	2	no	—
534	C₂²⋊C₄	—	23	2	no	—
535	C₂²⋊C₄	—	23	2	no	—
536	C₂²⋊C₄	—	23	2	no	—
537	C₂²⋊C₄	—	23	2	no	—
538	C₂²⋊C₄	—	23	2	no	—
539	C₂²⋊C₄	—	23	2	no	—
540	C₂²⋊C₄	—	23	2	no	—
541	C₂²⋊C₄	—	23	2	no	—
542	C₂²⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
543	C₂²⋊C₄	C₂²×D₂₃	1	1	yes	characteristic
544	C₂²⋊C₄	C₂²×D₂₃	1	2	yes	—
545	C₂²⋊C₄	C₂²×D₂₃	1	2	yes	—
546	C₂²⋊C₄	C₂²×D₂₃	1	2	yes	—
547	C₂²⋊C₄	C₂²×D₂₃	1	2	yes	—
548	C₂²⋊C₄	C₂²×D₂₃	1	2	yes	—
549	C₄²	—	23	1	no	—
550	C₄²	—	23	1	no	—
551	C₄²	—	23	1	no	—
552	C₄²	—	23	1	no	—
553	C₄²	—	23	1	no	—
554	C₄²	—	23	1	no	—
555	C₄²	—	23	2	no	—
556	C₄²	—	23	2	no	—
557	C₄²	—	23	2	no	—
558	C₄²	C₂²×D₂₃	1	1	yes	characteristic
559	C₄²	C₂²×D₂₃	1	1	yes	characteristic
560	C₄²	C₂²×D₂₃	1	2	yes	—
561	C₂³	—	23	1	no	—
562	C₂³	—	23	1	no	—
563	C₂³	—	23	2	no	—
564	C₂³	—	46	2	no	—
565	C₂³	—	46	4	no	—
566	C₂³	—	46	4	no	—
567	C₂³	—	46	2	no	—
568	C₂³	—	46	2	no	—
569	C₂³	—	46	4	no	—
570	C₂³	C₂³×D₂₃	1	1	yes	characteristic
571	C₂³	C₂³×D₂₃	1	2	yes	—
572	Q₈	—	2	2	no	—
573	Q₈	—	4	2	no	—
574	Q₈	—	46	4	no	—
575	Q₈	—	46	2	no	—
576	Q₈	—	92	2	no	—
577	Q₈	—	92	2	no	—
578	Q₈	—	92	2	no	—
579	D₄	—	2	2	no	—
580	D₄	—	4	2	no	—
581	D₄	—	46	4	no	—
582	D₄	—	46	2	no	—
583	D₄	—	92	2	no	—
584	D₄	—	92	2	no	—
585	D₄	—	92	2	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	4	2	no	—
588	C₂×C₄	—	23	1	no	—
589	C₂×C₄	—	23	4	no	—
590	C₂×C₄	—	23	1	no	—
591	C₂×C₄	—	23	1	no	—
592	C₂×C₄	—	23	2	no	—
593	C₂×C₄	—	23	1	no	—
594	C₂×C₄	—	23	1	no	—
595	C₂×C₄	—	23	2	no	—
596	C₂×C₄	—	23	1	no	—
597	C₂×C₄	—	23	2	no	—
598	C₂×C₄	—	46	2	no	—
599	C₂×C₄	—	46	2	no	—
600	C₂×C₄	—	46	2	no	—
601	C₂×C₄	—	46	4	no	—
602	C₂×C₄	—	46	4	no	—
603	C₂×C₄	—	46	4	no	—
604	C₂×C₄	—	46	2	no	—
605	C₂×C₄	—	46	2	no	—
606	C₂×C₄	—	46	2	no	—
607	C₂×C₄	—	46	2	no	—
608	C₂×C₄	—	46	2	no	—
609	C₂×C₄	—	92	2	no	—
610	C₂×C₄	—	92	2	no	—
611	C₂×C₄	—	92	2	no	—
612	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
613	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
614	C₂×C₄	C₂³×D₂₃	1	2	yes	—
615	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
616	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
617	C₂×C₄	C₂³×D₂₃	1	2	yes	—
618	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
619	C₂×C₄	C₂³×D₂₃	1	2	yes	—
620	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
621	C₂²	—	2	1	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	2	2	no	—
624	C₂²	—	2	1	no	—
625	C₂²	—	2	1	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	23	2	no	—
628	C₂²	—	23	2	no	—
629	C₂²	—	23	2	no	—
630	C₂²	—	46	1	no	—
631	C₂²	—	46	2	no	—
632	C₂²	—	46	2	no	—
633	C₂²	—	46	1	no	—
634	C₂²	—	46	1	no	—
635	C₂²	—	46	2	no	—
636	C₂²	—	92	4	no	—
637	C₂²	—	92	8	no	—
638	C₂²	C₂⁴×D₂₃	1	1	yes	characteristic
639	C₄	—	2	1	no	—
640	C₄	—	2	2	no	—
641	C₄	—	2	1	no	—
642	C₄	—	2	2	no	—
643	C₄	—	2	2	no	—
644	C₄	—	2	1	no	—
645	C₄	—	2	1	no	—
646	C₄	—	2	1	no	—
647	C₄	—	23	2	no	—
648	C₄	—	46	1	no	—
649	C₄	—	46	2	no	—
650	C₄	—	46	1	no	—
651	C₄	—	46	2	no	—
652	C₄	—	46	2	no	—
653	C₄	—	46	1	no	—
654	C₄	—	46	1	no	—
655	C₄	—	46	1	no	—
656	C₄	#41×D₂₃	1	2	yes	—
657	C₂	—	4	1	no	—
658	C₂	—	4	2	no	—
659	C₂	—	23	4	no	—
660	C₂	—	92	1	no	—
661	C₂	—	92	2	no	—
662	C₂	C₂×#41×D₂₃	1	1	yes	characteristic
663	C₂	H(5,𝔽₂)×D₂₃	1	1	yes	characteristic
664	C₂	2₋¹⁺⁴×D₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₂₃
commutator, socle, cosocle	C₂²×C₂₃	C₂⁵
Fitting	C₂₃×#542	C₂
prime radical	#121×D₂₃	C₂
hypercenter	#542	D₂₃
hypocenter	C₂₃	C₂×#542
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ #542

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #13093041	C₂	semidirect
2. #13093040	C₂	semidirect
3. #13093039	C₂	semidirect
4. #13093038	C₂	semidirect
5. #13093037	C₂	semidirect
6. #13093036	C₂	semidirect
7. #13093035	C₂	semidirect
8. #13093034	C₂	semidirect
9. #13093033	C₂	semidirect
10. #13093032	C₂	semidirect
11. #13093031	C₂	semidirect
12. C₄⋊Q₈×D₂₃	C₂	semidirect
13. #126×D₂₃	C₂	semidirect
14. #124×D₂₃	C₂	semidirect
15. #124×D₂₃	C₂	semidirect
16. #123×D₂₃	C₂	semidirect
17. C₂²⋊Q₈×D₂₃	C₂	semidirect
18. C₂²⋊Q₈×D₂₃	C₂	semidirect
20. C₄×Q₈×D₂₃	C₂	semidirect
21. C₄×D₄×D₂₃	C₂	semidirect
22. #117×D₂₃	C₂	semidirect
23. C₂₃×#542	C₂	semidirect
30. C₂₃×C₄⋊Q₈	C₂²	semidirect
31. C₂₃×#126	C₂²	semidirect
32. C₂₃×#124	C₂²	semidirect
33. C₂₃×#124	C₂²	semidirect
34. C₂₃×#123	C₂²	semidirect
35. C₂₃×C₂²⋊Q₈	C₂²	semidirect
36. C₂₃×C₂²⋊Q₈	C₂²	semidirect
38. C₄×Q₈×C₂₃	C₂²	semidirect
39. C₄×D₄×C₂₃	C₂²	semidirect
40. C₂₃×#117	C₂²	semidirect
41. #10625118	C₂²	semidirect
42. #10625118	C₂²	semidirect
43. #10625117	C₂²	semidirect
44. #10625117	C₂²	semidirect
45. Q₈×Dic₂₃	C₂²	semidirect
46. #10625115	C₂²	semidirect
49. #10625107	C₂²	semidirect
50. #10625107	C₂²	semidirect
51. #10625106	C₂²	semidirect
52. D₄×Dic₂₃	C₂²	semidirect
54. #10625100	C₂²	semidirect
55. C₄×C₂²⋊D₂₃	C₂²	semidirect
56. #10625097	C₂²	semidirect
57. #10625095	C₂²	semidirect
58. #10625095	C₂²	semidirect
59. #10625069	C₂²	semidirect
60. #10625069	C₂²	semidirect
61. #10625069	C₂²	semidirect
62. #10625068	C₂²	semidirect
63. #10625068	C₂²	semidirect
64. #10625067	C₂²	semidirect
65. #10625067	C₂²	semidirect
66. #10625067	C₂²	semidirect
68. #10625065	C₂²	semidirect
69. #10625065	C₂²	semidirect
70. #10625064	C₂²	semidirect
71. #10625063	C₂²	semidirect
72. #10625063	C₂²	semidirect
80. #10625059	C₂²	semidirect
81. #10625059	C₂²	semidirect
82. #10625059	C₂²	semidirect
83. #10625058	C₂²	semidirect
84. #10625058	C₂²	semidirect
85. #10625057	C₂²	semidirect
86. #10625056	C₂²	semidirect
87. #10625056	C₂²	semidirect
88. #10625056	C₂²	semidirect
89. #10625056	C₂²	semidirect
90. #10625056	C₂²	semidirect
91. #10625056	C₂²	semidirect
93. #10625054	C₂²	semidirect
94. #10625054	C₂²	semidirect
95. (C₂²⋊D₂₃)⋊C₄	C₂²	semidirect
103. #10625050	C₂²	semidirect
104. #10625050	C₂²	semidirect
105. #10625050	C₂²	semidirect
106. #10625049	C₂²	semidirect
107. #10625049	C₂²	semidirect
108. #10625049	C₂²	semidirect
109. #10625048	C₂²	semidirect
110. #10625048	C₂²	semidirect
111. #10625047	C₂²	semidirect
112. #10625046	C₂²	semidirect
113. #10625046	C₂²	semidirect
114. C₄×D₉₂	C₂²	semidirect
115. #10625043	C₂²	semidirect
116. #10625043	C₂²	semidirect
120. #10625040	C₂²	semidirect
121. C₄×Dic₄₆	C₂²	semidirect
122. C₄×Dic₄₆	C₂²	semidirect
336. #542	D₂₃	direct (N × H)
355. D₂₃	#542	direct (N × H)
367. C₄⋊Q₈	C₂×D₂₃	semidirect
375. #126	C₂×D₂₃	semidirect
388. #124	C₂×D₂₃	semidirect
389. #124	C₂×D₂₃	semidirect
397. #123	C₂×D₂₃	semidirect
410. C₂²⋊Q₈	C₂×D₂₃	semidirect
411. C₂²⋊Q₈	C₂×D₂₃	semidirect
423. C₄×Q₈	C₂×D₂₃	semidirect
429. C₄×D₄	C₂×D₂₃	semidirect
437. #117	C₂×D₂₃	semidirect
455. C₂₃	C₂×#542	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13093041	2
#13093040	2
#13093039	2
#13093038	2
#13093037	2
#13093036	2
#13093035	2
#13093034	2
#13093033	2
#13093032	2
#13093031	2
C₄⋊Q₈×D₂₃	2
#126×D₂₃	2
#124×D₂₃	2
#123×D₂₃	2
C₂²⋊Q₈×D₂₃	2
#121×D₂₃	2
C₄×Q₈×D₂₃	2
C₄×D₄×D₂₃	2
#117×D₂₃	2
C₂₃×#542	2
C₂×#542	23

Subgroups — D23×#542 #15686767

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₂₃×#542 #15686767