Subgroups — C₂₃×#3564 #15671183

Order 2944.

936 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#563	C₂	1	2	yes	maximal, complemented
2	C₂₃×#563	C₂	1	2	yes	maximal, complemented
3	C₂₃×#556	C₂	1	2	yes	maximal, complemented
4	C₂₃×#554	C₂	1	2	yes	maximal, complemented
5	C₂₃×#552	C₂	1	2	yes	maximal, complemented
6	C₂₃×#552	C₂	1	2	yes	maximal, complemented
7	C₂₃×#551	C₂	1	2	yes	maximal, complemented
8	C₂₃×#551	C₂	1	2	yes	maximal, complemented
9	C₂₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#543	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#542	C₂	1	2	yes	maximal, complemented
12	C₂₃×#541	C₂	1	2	yes	maximal, complemented
13	C₂₃×#541	C₂	1	2	yes	maximal, complemented
14	C₂₃×#539	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#539	C₂	1	2	yes	maximal, complemented
16	C₂₃×#538	C₂	1	1	yes	maximal, characteristic
17	C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
19	C₂×C₂₃×#126	C₂	1	1	yes	maximal, characteristic, complemented
20	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
21	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
22	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
23	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
25	C₂₃×#126	C₂²	1	2	yes	—
26	C₂₃×#126	C₂²	1	2	yes	—
27	C₂₃×#126	C₂²	1	2	yes	—
28	C₂₃×#126	C₂²	1	2	yes	—
29	C₂₃×#126	C₂²	1	2	yes	—
30	C₂₃×#126	C₂²	1	2	yes	—
31	C₂₃×#126	C₂²	1	2	yes	—
32	C₂₃×#126	C₂²	1	2	yes	—
33	C₂₃×#126	C₂²	1	2	yes	—
34	C₂₃×#126	C₂²	1	2	yes	complemented
35	C₂₃×#126	C₂²	1	2	yes	complemented
36	C₂₃×#126	C₂²	1	2	yes	complemented
37	C₂₃×#126	C₂²	1	2	yes	complemented
38	C₂₃×#125	C₂²	1	2	yes	complemented
39	C₂₃×#125	C₂²	1	2	yes	complemented
40	C₂₃×#125	C₂²	1	2	yes	complemented
41	C₂₃×#125	C₂²	1	2	yes	complemented
42	C₂₃×#125	C₂²	1	2	yes	complemented
43	C₂₃×#125	C₂²	1	2	yes	complemented
44	C₂₃×#125	C₂²	1	1	yes	characteristic, complemented
45	C₂₃×#125	C₂²	1	2	yes	complemented
46	C₂₃×#125	C₂²	1	2	yes	complemented
47	C₂₃×#125	C₂²	1	2	yes	complemented
48	C₂₃×#124	C₂²	1	2	yes	—
49	C₂₃×#124	C₂²	1	2	yes	—
50	C₂₃×#124	C₂²	1	2	yes	—
51	C₂₃×#124	C₂²	1	1	yes	characteristic, complemented
52	C₂₃×#123	C₂²	1	2	yes	—
53	C₂₃×#123	C₂²	1	2	yes	—
54	C₂₃×#123	C₂²	1	2	yes	—
55	C₂₃×#123	C₂²	1	2	yes	—
56	C₂₃×#123	C₂²	1	2	yes	—
57	C₂₃×#123	C₂²	1	2	yes	—
58	C₂₃×#123	C₂²	1	2	yes	—
59	C₂₃×#123	C₂²	1	2	yes	complemented
60	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
61	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
62	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
63	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
64	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
65	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
66	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
67	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
68	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
69	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
70	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
71	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
72	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
73	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
74	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
75	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
76	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
77	C₂₃×#121	C₂²	1	1	yes	characteristic
78	C₂₃×#121	C₂²	1	2	yes	—
79	C₂₃×#120	C₂²	1	2	yes	—
80	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
81	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
82	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
83	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
84	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
85	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
86	C₄×D₄×C₂₃	C₂²	1	2	yes	—
87	C₄×D₄×C₂₃	C₂²	1	2	yes	—
88	C₄×D₄×C₂₃	C₂²	1	2	yes	—
89	C₄×D₄×C₂₃	C₂²	1	2	yes	—
90	C₄×D₄×C₂₃	C₂²	1	2	yes	—
91	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
92	C₂₃×#117	C₂²	1	2	yes	—
93	C₂₃×#117	C₂²	1	2	yes	—
94	C₂₃×#117	C₂²	1	2	yes	—
95	C₂₃×#117	C₂²	1	2	yes	—
96	C₂₃×#117	C₂²	1	2	yes	complemented
97	C₂₃×#117	C₂²	1	2	yes	complemented
98	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
102	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
105	C₂×C₂²⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
106	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
107	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
109	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
110	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
111	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
112	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
113	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
114	C₂×D₄×C₂₃	C₂³	1	2	yes	—
115	C₂×D₄×C₂₃	C₂³	1	2	yes	—
116	C₂×D₄×C₂₃	C₂³	1	2	yes	—
117	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
118	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
119	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
120	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
121	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
122	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
123	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
124	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
125	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
126	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
127	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
128	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
129	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
130	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
131	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
132	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
133	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
134	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
135	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
136	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
137	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
138	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
139	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
140	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
141	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
142	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
143	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
144	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
145	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
146	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
147	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
148	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
149	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
150	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
151	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
152	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
153	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
154	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
155	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
156	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
157	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
158	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
159	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
160	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
161	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
162	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
163	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
164	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
165	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
166	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
167	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
168	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
169	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
170	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
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	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
177	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
178	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
179	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
180	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
181	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
182	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
183	C₄²×C₂₃	C₂³	1	1	yes	characteristic
184	C₄²×C₂₃	C₂³	1	1	yes	characteristic
185	C₄²×C₂₃	C₂³	1	1	yes	characteristic
186	C₄²×C₂₃	C₂³	1	1	yes	characteristic
187	C₄²×C₂₃	C₂³	1	2	yes	—
188	C₄²×C₂₃	C₂³	1	2	yes	—
189	C₄²×C₂₃	C₂³	1	2	yes	—
190	C₄²×C₂₃	C₂³	1	2	yes	—
191	C₄²×C₂₃	C₂³	1	2	yes	—
192	C₄²×C₂₃	C₂³	1	2	yes	—
193	C₄²×C₂₃	C₂³	1	2	yes	—
194	C₂³×C₂₃	—	2	2	no	—
195	C₂³×C₂₃	—	4	1	no	—
196	C₂³×C₂₃	—	4	1	no	—
197	C₂³×C₂₃	C₂⁴	1	2	yes	—
198	C₂³×C₂₃	C₂⁴	1	2	yes	—
199	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
200	Q₈×C₂₃	—	4	1	no	—
201	Q₈×C₂₃	—	4	2	no	—
202	Q₈×C₂₃	—	4	2	no	—
203	Q₈×C₂₃	—	4	1	no	—
204	Q₈×C₂₃	—	4	2	no	—
205	Q₈×C₂₃	—	4	1	no	—
206	D₄×C₂₃	—	2	4	no	—
207	D₄×C₂₃	—	4	2	no	—
208	D₄×C₂₃	—	4	1	no	—
209	D₄×C₂₃	—	4	2	no	—
210	C₂×C₄×C₂₃	—	2	2	no	—
211	C₂×C₄×C₂₃	—	2	2	no	—
212	C₂×C₄×C₂₃	—	4	2	no	—
213	C₂×C₄×C₂₃	—	4	1	no	—
214	C₂×C₄×C₂₃	—	4	1	no	—
215	C₂×C₄×C₂₃	—	4	1	no	—
216	C₂×C₄×C₂₃	—	4	2	no	—
217	C₂×C₄×C₂₃	—	4	2	no	—
218	C₂×C₄×C₂₃	—	4	2	no	—
219	C₂×C₄×C₂₃	—	4	2	no	—
220	C₂×C₄×C₂₃	—	4	1	no	—
221	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
222	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
223	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
224	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
225	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
226	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
227	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
228	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
229	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
230	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
231	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
232	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
233	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
234	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
235	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
236	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
237	#3564	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
238	C₂²×C₂₃	—	2	1	no	—
239	C₂²×C₂₃	—	2	1	no	—
240	C₂²×C₂₃	—	2	2	no	—
241	C₂²×C₂₃	—	2	2	no	—
242	C₂²×C₂₃	—	2	2	no	—
243	C₂²×C₂₃	—	2	2	no	—
244	C₂²×C₂₃	—	2	2	no	—
245	C₂²×C₂₃	—	2	2	no	—
246	C₂²×C₂₃	—	8	2	no	—
247	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
248	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
249	C₄×C₂₃	—	2	1	no	—
250	C₄×C₂₃	—	2	1	no	—
251	C₄×C₂₃	—	2	2	no	—
252	C₄×C₂₃	—	2	2	no	—
253	C₄×C₂₃	—	2	1	no	—
254	C₄×C₂₃	—	2	2	no	—
255	C₄×C₂₃	—	2	2	no	—
256	C₄×C₂₃	—	2	2	no	—
257	C₄×C₂₃	—	2	2	no	—
258	C₄×C₂₃	—	2	2	no	—
259	C₄×C₂₃	—	2	1	no	—
260	C₄×C₂₃	—	2	1	no	—
261	C₄×C₂₃	—	2	2	no	—
262	C₄×C₂₃	—	2	2	no	—
263	C₄×C₂₃	—	2	1	no	—
264	C₄×C₂₃	—	2	2	no	—
265	#563	C₂×C₂₃	1	2	yes	complemented
266	#563	C₂×C₂₃	1	2	yes	complemented
267	#556	C₂×C₂₃	1	2	yes	complemented
268	#554	C₂×C₂₃	1	2	yes	complemented
269	#552	C₂×C₂₃	1	2	yes	complemented
270	#552	C₂×C₂₃	1	2	yes	complemented
271	#551	C₂×C₂₃	1	2	yes	complemented
272	#551	C₂×C₂₃	1	2	yes	complemented
273	#544	C₂×C₂₃	1	1	yes	characteristic, complemented
274	#543	C₂×C₂₃	1	1	yes	characteristic, complemented
275	#542	C₂×C₂₃	1	2	yes	complemented
276	#541	C₂×C₂₃	1	2	yes	complemented
277	#541	C₂×C₂₃	1	2	yes	complemented
278	#539	C₂×C₂₃	1	1	yes	characteristic, complemented
279	#539	C₂×C₂₃	1	2	yes	complemented
280	#538	C₂×C₂₃	1	1	yes	characteristic
281	#533	C₂×C₂₃	1	1	yes	characteristic, complemented
282	#529	C₂×C₂₃	1	1	yes	characteristic, complemented
283	C₂×#126	C₂×C₂₃	1	1	yes	characteristic, complemented
284	C₂×C₂₃	—	2	2	no	—
285	C₂×C₂₃	—	4	2	no	—
286	C₂×C₂₃	—	4	2	no	—
287	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
288	C₂×C₂₃	#585	1	1	yes	characteristic
289	C₂×C₂₃	#585	1	1	yes	characteristic
290	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
291	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
292	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
293	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
294	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
295	#126	C₂²×C₂₃	1	2	yes	—
296	#126	C₂²×C₂₃	1	2	yes	—
297	#126	C₂²×C₂₃	1	2	yes	—
298	#126	C₂²×C₂₃	1	2	yes	—
299	#126	C₂²×C₂₃	1	2	yes	—
300	#126	C₂²×C₂₃	1	2	yes	—
301	#126	C₂²×C₂₃	1	2	yes	—
302	#126	C₂²×C₂₃	1	2	yes	—
303	#126	C₂²×C₂₃	1	2	yes	—
304	#126	C₂²×C₂₃	1	2	yes	complemented
305	#126	C₂²×C₂₃	1	2	yes	complemented
306	#126	C₂²×C₂₃	1	2	yes	complemented
307	#126	C₂²×C₂₃	1	2	yes	complemented
308	#125	C₂²×C₂₃	1	1	yes	characteristic, complemented
309	#125	C₂²×C₂₃	1	2	yes	complemented
310	#125	C₂²×C₂₃	1	2	yes	complemented
311	#125	C₂²×C₂₃	1	2	yes	complemented
312	#125	C₂²×C₂₃	1	2	yes	complemented
313	#125	C₂²×C₂₃	1	2	yes	complemented
314	#125	C₂²×C₂₃	1	2	yes	complemented
315	#125	C₂²×C₂₃	1	2	yes	complemented
316	#125	C₂²×C₂₃	1	2	yes	complemented
317	#125	C₂²×C₂₃	1	2	yes	complemented
318	#124	C₂²×C₂₃	1	2	yes	—
319	#124	C₂²×C₂₃	1	2	yes	—
320	#124	C₂²×C₂₃	1	2	yes	—
321	#124	C₂²×C₂₃	1	1	yes	characteristic, complemented
322	#123	C₂²×C₂₃	1	2	yes	—
323	#123	C₂²×C₂₃	1	2	yes	—
324	#123	C₂²×C₂₃	1	2	yes	—
325	#123	C₂²×C₂₃	1	2	yes	—
326	#123	C₂²×C₂₃	1	2	yes	—
327	#123	C₂²×C₂₃	1	2	yes	—
328	#123	C₂²×C₂₃	1	2	yes	—
329	#123	C₂²×C₂₃	1	2	yes	complemented
330	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
331	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
332	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
333	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
334	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
335	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
336	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
337	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
338	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
339	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
340	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
341	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
342	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
343	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
344	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
345	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
346	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
347	#121	C₂²×C₂₃	1	1	yes	characteristic
348	#121	C₂²×C₂₃	1	2	yes	—
349	#120	C₂²×C₂₃	1	2	yes	—
350	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
351	C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
352	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
353	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
354	C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
355	C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
356	C₄×D₄	C₂²×C₂₃	1	2	yes	—
357	C₄×D₄	C₂²×C₂₃	1	2	yes	—
358	C₄×D₄	C₂²×C₂₃	1	2	yes	—
359	C₄×D₄	C₂²×C₂₃	1	2	yes	—
360	C₄×D₄	C₂²×C₂₃	1	2	yes	—
361	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
362	#117	C₂²×C₂₃	1	2	yes	—
363	#117	C₂²×C₂₃	1	2	yes	—
364	#117	C₂²×C₂₃	1	2	yes	—
365	#117	C₂²×C₂₃	1	2	yes	—
366	#117	C₂²×C₂₃	1	2	yes	complemented
367	#117	C₂²×C₂₃	1	2	yes	complemented
368	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
369	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
372	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
373	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
374	C₂×C₂²⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
375	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
376	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
377	C₂₃	#3564	1	1	yes	characteristic, direct product factor, Sylow
378	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
379	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
380	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
381	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
382	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
383	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
384	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
385	C₂×D₄	C₂³×C₂₃	1	2	yes	—
386	C₂×D₄	C₂³×C₂₃	1	2	yes	—
387	C₂×D₄	C₂³×C₂₃	1	2	yes	—
388	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
389	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
390	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
391	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
392	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
393	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
394	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
395	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
396	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
397	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
398	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
399	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
400	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
401	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
402	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
403	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
404	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
405	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
406	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
407	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
408	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
409	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
410	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
411	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
412	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
413	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
414	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
415	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
416	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
417	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
418	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
419	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
420	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
421	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
422	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
423	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
424	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
425	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
426	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
427	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
428	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
429	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
430	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
431	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
432	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
433	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
434	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
435	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
436	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
437	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
438	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
439	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
440	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
441	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
442	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
443	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
444	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
445	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
446	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
447	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
448	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
449	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
450	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
451	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
452	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
453	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
454	C₄²	C₂³×C₂₃	1	1	yes	characteristic
455	C₄²	C₂³×C₂₃	1	1	yes	characteristic
456	C₄²	C₂³×C₂₃	1	1	yes	characteristic
457	C₄²	C₂³×C₂₃	1	1	yes	characteristic
458	C₄²	C₂³×C₂₃	1	2	yes	—
459	C₄²	C₂³×C₂₃	1	2	yes	—
460	C₄²	C₂³×C₂₃	1	2	yes	—
461	C₄²	C₂³×C₂₃	1	2	yes	—
462	C₄²	C₂³×C₂₃	1	2	yes	—
463	C₄²	C₂³×C₂₃	1	2	yes	—
464	C₄²	C₂³×C₂₃	1	2	yes	—
465	C₂³	—	2	2	no	—
466	C₂³	—	4	1	no	—
467	C₂³	—	4	1	no	—
468	C₂³	C₂⁴×C₂₃	1	2	yes	—
469	C₂³	C₂⁴×C₂₃	1	2	yes	—
470	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
471	Q₈	—	4	1	no	—
472	Q₈	—	4	2	no	—
473	Q₈	—	4	2	no	—
474	Q₈	—	4	1	no	—
475	Q₈	—	4	2	no	—
476	Q₈	—	4	1	no	—
477	D₄	—	2	4	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	1	no	—
480	D₄	—	4	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	1	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	2	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
493	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
494	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
495	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
496	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
497	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
498	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
499	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
500	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
501	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
502	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
503	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
504	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
505	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
506	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
507	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
508	C₂²	—	2	1	no	—
509	C₂²	—	2	1	no	—
510	C₂²	—	2	2	no	—
511	C₂²	—	2	2	no	—
512	C₂²	—	2	2	no	—
513	C₂²	—	2	2	no	—
514	C₂²	—	2	2	no	—
515	C₂²	—	2	2	no	—
516	C₂²	—	8	2	no	—
517	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
518	C₂²	C₂⁵×C₂₃	1	1	yes	characteristic
519	C₄	—	2	1	no	—
520	C₄	—	2	1	no	—
521	C₄	—	2	2	no	—
522	C₄	—	2	2	no	—
523	C₄	—	2	1	no	—
524	C₄	—	2	2	no	—
525	C₄	—	2	2	no	—
526	C₄	—	2	2	no	—
527	C₄	—	2	2	no	—
528	C₄	—	2	2	no	—
529	C₄	—	2	1	no	—
530	C₄	—	2	1	no	—
531	C₄	—	2	2	no	—
532	C₄	—	2	2	no	—
533	C₄	—	2	1	no	—
534	C₄	—	2	2	no	—
535	C₂	—	2	2	no	—
536	C₂	—	4	2	no	—
537	C₂	—	4	2	no	—
538	C₂	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
539	C₂	C₂₃×#585	1	1	yes	characteristic
540	C₂	C₂₃×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₃
prime radical	C₂₃×#538	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#563	C₂	semidirect
2. C₂₃×#563	C₂	semidirect
3. C₂₃×#556	C₂	semidirect
4. C₂₃×#554	C₂	semidirect
5. C₂₃×#552	C₂	semidirect
6. C₂₃×#552	C₂	semidirect
7. C₂₃×#551	C₂	semidirect
8. C₂₃×#551	C₂	semidirect
9. C₂₃×#544	C₂	semidirect
10. C₂₃×#543	C₂	semidirect
11. C₂₃×#542	C₂	semidirect
12. C₂₃×#541	C₂	semidirect
13. C₂₃×#541	C₂	semidirect
14. C₂₃×#539	C₂	semidirect
15. C₂₃×#539	C₂	semidirect
17. C₂₃×#533	C₂	semidirect
18. C₂₃×#529	C₂	semidirect
19. C₂×C₂₃×#126	C₂	semidirect
20. C₂₃×C₄⋊Q₈	C₂²	semidirect
21. C₂₃×C₄⋊Q₈	C₂²	semidirect
22. C₂₃×C₄⋊Q₈	C₂²	semidirect
23. C₂₃×C₄⋊Q₈	C₂²	semidirect
24. C₂₃×C₄⋊Q₈	C₂²	semidirect
34. C₂₃×#126	C₂²	semidirect
35. C₂₃×#126	C₂²	semidirect
36. C₂₃×#126	C₂²	semidirect
37. C₂₃×#126	C₂²	semidirect
38. C₂₃×#125	C₂²	semidirect
39. C₂₃×#125	C₂²	semidirect
40. C₂₃×#125	C₂²	semidirect
41. C₂₃×#125	C₂²	semidirect
42. C₂₃×#125	C₂²	semidirect
43. C₂₃×#125	C₂²	semidirect
44. C₂₃×#125	C₂²	semidirect
45. C₂₃×#125	C₂²	semidirect
46. C₂₃×#125	C₂²	semidirect
47. C₂₃×#125	C₂²	semidirect
51. C₂₃×#124	C₂²	semidirect
59. C₂₃×#123	C₂²	semidirect
72. C₂₃×C₂²⋊Q₈	C₂²	semidirect
73. C₂₃×C₂²⋊Q₈	C₂²	semidirect
74. C₂₃×C₂²⋊Q₈	C₂²	semidirect
75. C₂₃×C₂²⋊Q₈	C₂²	semidirect
76. C₂₃×C₂²⋊Q₈	C₂²	semidirect
80. C₄×Q₈×C₂₃	C₂²	semidirect
81. C₄×Q₈×C₂₃	C₂²	semidirect
82. C₄×Q₈×C₂₃	C₂²	semidirect
83. C₄×Q₈×C₂₃	C₂²	semidirect
84. C₄×Q₈×C₂₃	C₂²	semidirect
85. C₄×Q₈×C₂₃	C₂²	semidirect
91. C₄×D₄×C₂₃	C₂²	semidirect
96. C₂₃×#117	C₂²	semidirect
97. C₂₃×#117	C₂²	semidirect
101. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
237. #3564	C₂₃	direct (N × H)
265. #563	C₂×C₂₃	semidirect
266. #563	C₂×C₂₃	semidirect
267. #556	C₂×C₂₃	semidirect
268. #554	C₂×C₂₃	semidirect
269. #552	C₂×C₂₃	semidirect
270. #552	C₂×C₂₃	semidirect
271. #551	C₂×C₂₃	semidirect
272. #551	C₂×C₂₃	semidirect
273. #544	C₂×C₂₃	semidirect
274. #543	C₂×C₂₃	semidirect
275. #542	C₂×C₂₃	semidirect
276. #541	C₂×C₂₃	semidirect
277. #541	C₂×C₂₃	semidirect
278. #539	C₂×C₂₃	semidirect
279. #539	C₂×C₂₃	semidirect
281. #533	C₂×C₂₃	semidirect
282. #529	C₂×C₂₃	semidirect
283. C₂×#126	C₂×C₂₃	semidirect
290. C₄⋊Q₈	C₂²×C₂₃	semidirect
291. C₄⋊Q₈	C₂²×C₂₃	semidirect
292. C₄⋊Q₈	C₂²×C₂₃	semidirect
293. C₄⋊Q₈	C₂²×C₂₃	semidirect
294. C₄⋊Q₈	C₂²×C₂₃	semidirect
304. #126	C₂²×C₂₃	semidirect
305. #126	C₂²×C₂₃	semidirect
306. #126	C₂²×C₂₃	semidirect
307. #126	C₂²×C₂₃	semidirect
308. #125	C₂²×C₂₃	semidirect
309. #125	C₂²×C₂₃	semidirect
310. #125	C₂²×C₂₃	semidirect
311. #125	C₂²×C₂₃	semidirect
312. #125	C₂²×C₂₃	semidirect
313. #125	C₂²×C₂₃	semidirect
314. #125	C₂²×C₂₃	semidirect
315. #125	C₂²×C₂₃	semidirect
316. #125	C₂²×C₂₃	semidirect
317. #125	C₂²×C₂₃	semidirect
321. #124	C₂²×C₂₃	semidirect
329. #123	C₂²×C₂₃	semidirect
342. C₂²⋊Q₈	C₂²×C₂₃	semidirect
343. C₂²⋊Q₈	C₂²×C₂₃	semidirect
344. C₂²⋊Q₈	C₂²×C₂₃	semidirect
345. C₂²⋊Q₈	C₂²×C₂₃	semidirect
346. C₂²⋊Q₈	C₂²×C₂₃	semidirect
350. C₄×Q₈	C₂²×C₂₃	semidirect
351. C₄×Q₈	C₂²×C₂₃	semidirect
352. C₄×Q₈	C₂²×C₂₃	semidirect
353. C₄×Q₈	C₂²×C₂₃	semidirect
354. C₄×Q₈	C₂²×C₂₃	semidirect
355. C₄×Q₈	C₂²×C₂₃	semidirect
361. C₄×D₄	C₂²×C₂₃	semidirect
366. #117	C₂²×C₂₃	semidirect
367. #117	C₂²×C₂₃	semidirect
371. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
377. C₂₃	#3564	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂₃×#563	2
C₂₃×#556	2
C₂₃×#554	2
C₂₃×#552	2
C₂₃×#551	2
C₂₃×#544	2
C₂₃×#543	2
C₂₃×#542	2
C₂₃×#541	2
C₂₃×#539	2
C₂₃×#538	2
C₂₃×#533	2
C₂₃×#529	2
C₂×C₂₃×#126	2
#3564	23

Subgroups — C23×#3564 #15671183

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3564 #15671183