Subgroups — C₂₃×#3495 #15671114

Order 2944.

1022 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#556	C₂	1	2	yes	maximal, complemented
2	C₂₃×#554	C₂	1	2	yes	maximal, complemented
3	C₂₃×#552	C₂	1	2	yes	maximal, complemented
4	C₂₃×#552	C₂	1	4	yes	maximal, complemented
5	C₂₃×#552	C₂	1	2	yes	maximal, complemented
6	C₂₃×#551	C₂	1	4	yes	maximal, complemented
7	C₂₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#542	C₂	1	2	yes	maximal, complemented
9	C₂₃×#541	C₂	1	2	yes	maximal, complemented
10	C₂₃×#536	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#533	C₂	1	2	yes	maximal, complemented
12	C₂₃×#529	C₂	1	2	yes	maximal, complemented
13	C₂₃×#525	C₂	1	1	yes	maximal, characteristic
14	C₂×C₂₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
20	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
21	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
22	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
23	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	complemented
24	C₂₃×#126	C₂²	1	4	yes	—
25	C₂₃×#126	C₂²	1	4	yes	—
26	C₂₃×#126	C₂²	1	4	yes	complemented
27	C₂₃×#126	C₂²	1	4	yes	complemented
28	C₂₃×#125	C₂²	1	4	yes	complemented
29	C₂₃×#125	C₂²	1	4	yes	complemented
30	C₂₃×#125	C₂²	1	2	yes	complemented
31	C₂₃×#125	C₂²	1	4	yes	complemented
32	C₂₃×#125	C₂²	1	2	yes	complemented
33	C₂₃×#124	C₂²	1	4	yes	—
34	C₂₃×#124	C₂²	1	2	yes	complemented
35	C₂₃×#123	C₂²	1	4	yes	—
36	C₂₃×#123	C₂²	1	4	yes	—
37	C₂₃×#123	C₂²	1	4	yes	—
38	C₂₃×#123	C₂²	1	2	yes	—
39	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
40	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
41	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
42	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
43	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
44	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
45	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
46	C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
47	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
48	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	complemented
49	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
50	C₂₃×#121	C₂²	1	1	yes	characteristic
51	C₂₃×#121	C₂²	1	1	yes	characteristic
52	C₂₃×#120	C₂²	1	2	yes	—
53	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
54	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
55	C₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
56	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
57	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
58	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
59	C₄×D₄×C₂₃	C₂²	1	4	yes	—
60	C₄×D₄×C₂₃	C₂²	1	4	yes	—
61	C₄×D₄×C₂₃	C₂²	1	2	yes	—
62	C₄×D₄×C₂₃	C₂²	1	2	yes	—
63	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
64	C₂₃×#117	C₂²	1	4	yes	—
65	C₂₃×#117	C₂²	1	4	yes	—
66	C₂₃×#117	C₂²	1	1	yes	characteristic
67	C₂₃×#117	C₂²	1	2	yes	—
68	C₂₃×#117	C₂²	1	1	yes	characteristic
69	C₂₃×#117	C₂²	1	1	yes	characteristic
70	C₂₃×#117	C₂²	1	1	yes	characteristic
71	C₂₃×#117	C₂²	1	2	yes	—
72	C₂₃×#117	C₂²	1	1	yes	characteristic
73	C₂₃×#117	C₂²	1	4	yes	complemented
74	C₂₃×#117	C₂²	1	2	yes	complemented
75	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
77	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
80	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
81	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
82	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
83	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
84	#41×C₂₃	—	2	4	no	—
85	C₂×Q₈×C₂₃	—	2	4	no	—
86	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
87	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
88	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
89	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
90	C₂×D₄×C₂₃	C₂³	1	2	yes	—
91	C₂×D₄×C₂₃	C₂³	1	2	yes	—
92	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₂₃	C₂³	1	2	yes	—
94	C₂²×C₄×C₂₃	—	2	4	no	—
95	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
97	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
98	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
101	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
104	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
105	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
106	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
109	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
110	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
111	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
112	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
113	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
114	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
115	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
116	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
117	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
118	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
119	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
120	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
121	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
122	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
123	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
124	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
125	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
126	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
127	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
128	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
129	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
130	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
131	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
132	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
133	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
134	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
135	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
136	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
137	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
138	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
139	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
140	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
141	C₂²⋊C₄×C₂₃	C₂³	1	2	yes	—
142	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
143	C₄²×C₂₃	C₂³	1	2	yes	—
144	C₄²×C₂₃	C₂³	1	2	yes	—
145	C₄²×C₂₃	C₂³	1	2	yes	—
146	C₄²×C₂₃	C₂³	1	2	yes	—
147	C₄²×C₂₃	C₂³	1	4	yes	—
148	C₄²×C₂₃	C₂³	1	4	yes	—
149	C₄²×C₂₃	C₂³	1	4	yes	—
150	C₄²×C₂₃	C₂³	1	2	yes	—
151	C₂³×C₂₃	—	2	2	no	—
152	C₂³×C₂₃	—	2	2	no	—
153	C₂³×C₂₃	—	4	1	no	—
154	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
155	C₂³×C₂₃	C₂⁴	1	2	yes	—
156	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
157	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
158	Q₈×C₂₃	—	2	4	no	—
159	Q₈×C₂₃	—	2	4	no	—
160	Q₈×C₂₃	—	2	2	no	—
161	Q₈×C₂₃	—	4	4	no	—
162	D₄×C₂₃	—	2	4	no	—
163	D₄×C₂₃	—	2	4	no	—
164	D₄×C₂₃	—	2	2	no	—
165	D₄×C₂₃	—	4	2	no	—
166	C₂×C₄×C₂₃	—	2	2	no	—
167	C₂×C₄×C₂₃	—	2	4	no	—
168	C₂×C₄×C₂₃	—	2	4	no	—
169	C₂×C₄×C₂₃	—	2	2	no	—
170	C₂×C₄×C₂₃	—	2	2	no	—
171	C₂×C₄×C₂₃	—	2	2	no	—
172	C₂×C₄×C₂₃	—	2	2	no	—
173	C₂×C₄×C₂₃	—	2	4	no	—
174	C₂×C₄×C₂₃	—	4	4	no	—
175	C₂×C₄×C₂₃	—	4	2	no	—
176	C₂×C₄×C₂₃	—	4	2	no	—
177	C₂×C₄×C₂₃	#41	1	4	yes	—
178	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
179	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
180	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
181	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
182	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
183	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
184	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
185	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
186	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
187	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
188	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
189	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
190	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
191	#3495	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂²×C₂₃	—	2	1	no	—
193	C₂²×C₂₃	—	2	1	no	—
194	C₂²×C₂₃	—	2	1	no	—
195	C₂²×C₂₃	—	2	1	no	—
196	C₂²×C₂₃	—	2	1	no	—
197	C₂²×C₂₃	—	2	2	no	—
198	C₂²×C₂₃	—	2	2	no	—
199	C₂²×C₂₃	—	2	1	no	—
200	C₂²×C₂₃	—	2	2	no	—
201	C₂²×C₂₃	—	2	1	no	—
202	C₂²×C₂₃	—	4	4	no	—
203	C₂²×C₂₃	C₂×#41	1	2	yes	—
204	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
205	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
206	C₄×C₂₃	—	2	1	no	—
207	C₄×C₂₃	—	2	1	no	—
208	C₄×C₂₃	—	2	2	no	—
209	C₄×C₂₃	—	2	4	no	—
210	C₄×C₂₃	—	2	2	no	—
211	C₄×C₂₃	—	2	2	no	—
212	C₄×C₂₃	—	2	2	no	—
213	C₄×C₂₃	—	2	2	no	—
214	C₄×C₂₃	—	2	4	no	—
215	C₄×C₂₃	—	2	2	no	—
216	C₄×C₂₃	—	2	2	no	—
217	C₄×C₂₃	C₂×#41	1	2	yes	—
218	C₄×C₂₃	C₂×#41	1	2	yes	—
219	#556	C₂×C₂₃	1	2	yes	complemented
220	#554	C₂×C₂₃	1	2	yes	complemented
221	#552	C₂×C₂₃	1	2	yes	complemented
222	#552	C₂×C₂₃	1	4	yes	complemented
223	#552	C₂×C₂₃	1	2	yes	complemented
224	#551	C₂×C₂₃	1	4	yes	complemented
225	#544	C₂×C₂₃	1	1	yes	characteristic, complemented
226	#542	C₂×C₂₃	1	2	yes	complemented
227	#541	C₂×C₂₃	1	2	yes	complemented
228	#536	C₂×C₂₃	1	1	yes	characteristic, complemented
229	#533	C₂×C₂₃	1	2	yes	complemented
230	#529	C₂×C₂₃	1	2	yes	complemented
231	#525	C₂×C₂₃	1	1	yes	characteristic
232	C₂×C₂²⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
233	#520	C₂×C₂₃	1	1	yes	characteristic, complemented
234	#519	C₂×C₂₃	1	1	yes	characteristic, complemented
235	C₂×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
236	C₂×C₂₃	—	2	2	no	—
237	C₂×C₂₃	—	2	2	no	—
238	C₂×C₂₃	—	4	2	no	—
239	C₂×C₂₃	—	4	1	no	—
240	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
241	C₂×C₂₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
242	C₂×C₂₃	#585	1	1	yes	characteristic
243	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
244	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
245	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
246	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
247	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
248	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
249	#126	C₂²×C₂₃	1	4	yes	—
250	#126	C₂²×C₂₃	1	4	yes	—
251	#126	C₂²×C₂₃	1	4	yes	complemented
252	#126	C₂²×C₂₃	1	4	yes	complemented
253	#125	C₂²×C₂₃	1	2	yes	complemented
254	#125	C₂²×C₂₃	1	4	yes	complemented
255	#125	C₂²×C₂₃	1	4	yes	complemented
256	#125	C₂²×C₂₃	1	4	yes	complemented
257	#125	C₂²×C₂₃	1	2	yes	complemented
258	#124	C₂²×C₂₃	1	4	yes	—
259	#124	C₂²×C₂₃	1	2	yes	complemented
260	#123	C₂²×C₂₃	1	2	yes	—
261	#123	C₂²×C₂₃	1	4	yes	—
262	#123	C₂²×C₂₃	1	4	yes	—
263	#123	C₂²×C₂₃	1	4	yes	—
264	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
265	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
266	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
267	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
268	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
269	C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
270	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
271	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
272	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
273	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
274	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	complemented
275	#121	C₂²×C₂₃	1	1	yes	characteristic
276	#121	C₂²×C₂₃	1	1	yes	characteristic
277	#120	C₂²×C₂₃	1	2	yes	—
278	C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
279	C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
280	C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
281	C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
282	C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
283	C₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
284	C₄×D₄	C₂²×C₂₃	1	4	yes	—
285	C₄×D₄	C₂²×C₂₃	1	2	yes	—
286	C₄×D₄	C₂²×C₂₃	1	2	yes	—
287	C₄×D₄	C₂²×C₂₃	1	4	yes	—
288	C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
289	#117	C₂²×C₂₃	1	1	yes	characteristic
290	#117	C₂²×C₂₃	1	1	yes	characteristic
291	#117	C₂²×C₂₃	1	2	yes	—
292	#117	C₂²×C₂₃	1	1	yes	characteristic
293	#117	C₂²×C₂₃	1	2	yes	—
294	#117	C₂²×C₂₃	1	1	yes	characteristic
295	#117	C₂²×C₂₃	1	1	yes	characteristic
296	#117	C₂²×C₂₃	1	4	yes	—
297	#117	C₂²×C₂₃	1	4	yes	—
298	#117	C₂²×C₂₃	1	2	yes	complemented
299	#117	C₂²×C₂₃	1	4	yes	complemented
300	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
301	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
303	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
304	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
305	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
306	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
307	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
308	C₂₃	#3495	1	1	yes	characteristic, direct product factor, Sylow
309	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
310	#41	—	2	4	no	—
311	C₂×Q₈	—	2	4	no	—
312	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
313	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
314	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
315	C₂×Q₈	C₂³×C₂₃	1	4	yes	—
316	C₂×D₄	C₂³×C₂₃	1	2	yes	—
317	C₂×D₄	C₂³×C₂₃	1	2	yes	—
318	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
319	C₂×D₄	C₂³×C₂₃	1	2	yes	—
320	C₂²×C₄	—	2	4	no	—
321	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
322	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
323	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
324	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
325	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
326	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
327	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
328	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
329	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
330	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
331	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
332	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
333	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
334	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
335	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
336	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
337	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
338	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
339	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
340	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
341	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
342	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
343	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
344	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
345	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
346	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
347	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
348	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
349	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
350	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
351	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
352	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
353	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
354	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
355	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
356	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
357	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
358	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
359	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
360	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
361	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
362	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
363	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
364	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
365	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
366	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
367	C₂²⋊C₄	C₂³×C₂₃	1	2	yes	—
368	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
369	C₄²	C₂³×C₂₃	1	2	yes	—
370	C₄²	C₂³×C₂₃	1	2	yes	—
371	C₄²	C₂³×C₂₃	1	2	yes	—
372	C₄²	C₂³×C₂₃	1	2	yes	—
373	C₄²	C₂³×C₂₃	1	4	yes	—
374	C₄²	C₂³×C₂₃	1	4	yes	—
375	C₄²	C₂³×C₂₃	1	4	yes	—
376	C₄²	C₂³×C₂₃	1	2	yes	—
377	C₂³	—	2	2	no	—
378	C₂³	—	2	2	no	—
379	C₂³	—	4	1	no	—
380	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
381	C₂³	C₂⁴×C₂₃	1	2	yes	—
382	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
383	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
384	Q₈	—	2	4	no	—
385	Q₈	—	2	4	no	—
386	Q₈	—	2	2	no	—
387	Q₈	—	4	4	no	—
388	D₄	—	2	4	no	—
389	D₄	—	2	4	no	—
390	D₄	—	2	2	no	—
391	D₄	—	4	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	4	4	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	#41×C₂₃	1	4	yes	—
404	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
405	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
406	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
407	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
408	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
409	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
410	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
411	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
412	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
413	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
414	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
415	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
416	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
417	C₂²	—	2	1	no	—
418	C₂²	—	2	1	no	—
419	C₂²	—	2	1	no	—
420	C₂²	—	2	1	no	—
421	C₂²	—	2	1	no	—
422	C₂²	—	2	2	no	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	1	no	—
427	C₂²	—	4	4	no	—
428	C₂²	C₂×#41×C₂₃	1	2	yes	—
429	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
430	C₂²	C₂⁵×C₂₃	1	1	yes	characteristic
431	C₄	—	2	1	no	—
432	C₄	—	2	1	no	—
433	C₄	—	2	2	no	—
434	C₄	—	2	4	no	—
435	C₄	—	2	2	no	—
436	C₄	—	2	2	no	—
437	C₄	—	2	2	no	—
438	C₄	—	2	2	no	—
439	C₄	—	2	4	no	—
440	C₄	—	2	2	no	—
441	C₄	—	2	2	no	—
442	C₄	C₂×#41×C₂₃	1	2	yes	—
443	C₄	C₂×#41×C₂₃	1	2	yes	—
444	C₂	—	2	2	no	—
445	C₂	—	2	2	no	—
446	C₂	—	4	1	no	—
447	C₂	—	4	2	no	—
448	C₂	C₂²×#41×C₂₃	1	1	yes	characteristic
449	C₂	C₂×C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
450	C₂	C₂₃×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₃
prime radical	C₂₃×#525	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×#556	C₂	semidirect
2. C₂₃×#554	C₂	semidirect
3. C₂₃×#552	C₂	semidirect
4. C₂₃×#552	C₂	semidirect
5. C₂₃×#552	C₂	semidirect
6. C₂₃×#551	C₂	semidirect
7. C₂₃×#544	C₂	semidirect
8. C₂₃×#542	C₂	semidirect
9. C₂₃×#541	C₂	semidirect
10. C₂₃×#536	C₂	semidirect
11. C₂₃×#533	C₂	semidirect
12. C₂₃×#529	C₂	semidirect
14. C₂×C₂₃×C₂²⋊Q₈	C₂	semidirect
15. C₂₃×#520	C₂	semidirect
16. C₂₃×#519	C₂	semidirect
17. C₂×C₂₃×#117	C₂	semidirect
21. C₂₃×C₄⋊Q₈	C₂²	semidirect
22. C₂₃×C₄⋊Q₈	C₂²	semidirect
23. C₂₃×C₄⋊Q₈	C₂²	semidirect
26. C₂₃×#126	C₂²	semidirect
27. C₂₃×#126	C₂²	semidirect
28. C₂₃×#125	C₂²	semidirect
29. C₂₃×#125	C₂²	semidirect
30. C₂₃×#125	C₂²	semidirect
31. C₂₃×#125	C₂²	semidirect
32. C₂₃×#125	C₂²	semidirect
34. C₂₃×#124	C₂²	semidirect
47. C₂₃×C₂²⋊Q₈	C₂²	semidirect
48. C₂₃×C₂²⋊Q₈	C₂²	semidirect
49. C₂₃×C₂²⋊Q₈	C₂²	semidirect
53. C₄×Q₈×C₂₃	C₂²	semidirect
54. C₄×Q₈×C₂₃	C₂²	semidirect
55. C₄×Q₈×C₂₃	C₂²	semidirect
56. C₄×Q₈×C₂₃	C₂²	semidirect
57. C₄×Q₈×C₂₃	C₂²	semidirect
58. C₄×Q₈×C₂₃	C₂²	semidirect
63. C₄×D₄×C₂₃	C₂²	semidirect
73. C₂₃×#117	C₂²	semidirect
74. C₂₃×#117	C₂²	semidirect
79. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
191. #3495	C₂₃	direct (N × H)
219. #556	C₂×C₂₃	semidirect
220. #554	C₂×C₂₃	semidirect
221. #552	C₂×C₂₃	semidirect
222. #552	C₂×C₂₃	semidirect
223. #552	C₂×C₂₃	semidirect
224. #551	C₂×C₂₃	semidirect
225. #544	C₂×C₂₃	semidirect
226. #542	C₂×C₂₃	semidirect
227. #541	C₂×C₂₃	semidirect
228. #536	C₂×C₂₃	semidirect
229. #533	C₂×C₂₃	semidirect
230. #529	C₂×C₂₃	semidirect
232. C₂×C₂²⋊Q₈	C₂×C₂₃	semidirect
233. #520	C₂×C₂₃	semidirect
234. #519	C₂×C₂₃	semidirect
235. C₂×#117	C₂×C₂₃	semidirect
246. C₄⋊Q₈	C₂²×C₂₃	semidirect
247. C₄⋊Q₈	C₂²×C₂₃	semidirect
248. C₄⋊Q₈	C₂²×C₂₃	semidirect
251. #126	C₂²×C₂₃	semidirect
252. #126	C₂²×C₂₃	semidirect
253. #125	C₂²×C₂₃	semidirect
254. #125	C₂²×C₂₃	semidirect
255. #125	C₂²×C₂₃	semidirect
256. #125	C₂²×C₂₃	semidirect
257. #125	C₂²×C₂₃	semidirect
259. #124	C₂²×C₂₃	semidirect
272. C₂²⋊Q₈	C₂²×C₂₃	semidirect
273. C₂²⋊Q₈	C₂²×C₂₃	semidirect
274. C₂²⋊Q₈	C₂²×C₂₃	semidirect
278. C₄×Q₈	C₂²×C₂₃	semidirect
279. C₄×Q₈	C₂²×C₂₃	semidirect
280. C₄×Q₈	C₂²×C₂₃	semidirect
281. C₄×Q₈	C₂²×C₂₃	semidirect
282. C₄×Q₈	C₂²×C₂₃	semidirect
283. C₄×Q₈	C₂²×C₂₃	semidirect
288. C₄×D₄	C₂²×C₂₃	semidirect
298. #117	C₂²×C₂₃	semidirect
299. #117	C₂²×C₂₃	semidirect
304. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
308. C₂₃	#3495	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×#556	2
C₂₃×#554	2
C₂₃×#552	2
C₂₃×#551	2
C₂₃×#544	2
C₂₃×#542	2
C₂₃×#541	2
C₂₃×#536	2
C₂₃×#533	2
C₂₃×#529	2
C₂₃×#525	2
C₂×C₂₃×C₂²⋊Q₈	2
C₂₃×#520	2
C₂₃×#519	2
C₂×C₂₃×#117	2
#3495	23

Subgroups — C23×#3495 #15671114

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3495 #15671114