Subgroups — C₂₃×#3493 #15671112

Order 2944.

1040 conjugacy classes of subgroups.

The subgroup lattice has 290 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	4	yes	maximal, complemented
2	C₂₃×#555	C₂	1	4	yes	maximal, complemented
3	C₂₃×#553	C₂	1	4	yes	maximal, complemented
4	C₂₃×#552	C₂	1	4	yes	maximal, complemented
5	C₂₃×#544	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#538	C₂	1	4	yes	maximal, complemented
7	C₂₃×#537	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#529	C₂	1	4	yes	maximal, complemented
9	C₂₃×#525	C₂	1	2	yes	maximal, complemented
10	C₂₃×#520	C₂	1	2	yes	maximal, complemented
11	C₂×C₂₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×C₄⋊Q₈	C₂²	1	4	yes	complemented
15	C₂₃×#126	C₂²	1	8	yes	—
16	C₂₃×#126	C₂²	1	8	yes	complemented
17	C₂₃×#125	C₂²	1	4	yes	complemented
18	C₂₃×#125	C₂²	1	4	yes	complemented
19	C₂₃×#124	C₂²	1	8	yes	—
20	C₂₃×#124	C₂²	1	4	yes	complemented
21	C₂₃×#123	C₂²	1	8	yes	—
22	C₂₃×#123	C₂²	1	4	yes	—
23	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
25	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	—
26	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	complemented
27	C₂₃×#121	C₂²	1	2	yes	—
28	C₂₃×#121	C₂²	1	8	yes	—
29	C₂₃×#121	C₂²	1	2	yes	—
30	C₂₃×#121	C₂²	1	4	yes	—
31	C₂₃×#121	C₂²	1	4	yes	complemented
32	C₂₃×#120	C₂²	1	4	yes	—
33	C₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
34	C₄×Q₈×C₂₃	C₂²	1	4	yes	complemented
35	C₄×D₄×C₂₃	C₂²	1	4	yes	—
36	C₄×D₄×C₂₃	C₂²	1	4	yes	—
37	C₄×D₄×C₂₃	C₂²	1	8	yes	—
38	C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
39	C₄×D₄×C₂₃	C₂²	1	4	yes	complemented
40	C₂₃×#117	C₂²	1	4	yes	—
41	C₂₃×#117	C₂²	1	4	yes	—
42	C₂₃×#117	C₂²	1	2	yes	—
43	C₂₃×#117	C₂²	1	8	yes	complemented
44	C₂₃×#117	C₂²	1	4	yes	complemented
45	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₂₃	C₂²	1	4	yes	complemented
48	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
49	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
50	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
51	#41×C₂₃	—	2	8	no	—
52	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
53	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
54	C₂×D₄×C₂₃	C₂³	1	4	yes	—
55	C₂×D₄×C₂₃	C₂³	1	2	yes	—
56	C₂×D₄×C₂₃	C₂³	1	4	yes	—
57	C₂×D₄×C₂₃	C₂³	1	4	yes	—
58	C₂×D₄×C₂₃	C₂³	1	4	yes	—
59	C₂²×C₄×C₂₃	—	2	4	no	—
60	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
62	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
63	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
64	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
65	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
66	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
67	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
69	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
70	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
71	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
72	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
73	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
74	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
75	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
76	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
77	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
78	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
79	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
80	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
81	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
82	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
83	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
84	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
85	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
86	C₄²×C₂₃	C₂³	1	4	yes	—
87	C₄²×C₂₃	C₂³	1	4	yes	—
88	C₄²×C₂₃	C₂³	1	4	yes	—
89	C₄²×C₂₃	C₂³	1	8	yes	—
90	C₂³×C₂₃	—	2	2	no	—
91	C₂³×C₂₃	—	4	1	no	—
92	C₂³×C₂₃	—	4	1	no	—
93	C₂³×C₂₃	C₂⁴	1	2	yes	—
94	C₂³×C₂₃	C₂⁴	1	4	yes	—
95	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
96	Q₈×C₂₃	—	2	4	no	—
97	Q₈×C₂₃	—	4	4	no	—
98	D₄×C₂₃	—	2	8	no	—
99	D₄×C₂₃	—	2	4	no	—
100	D₄×C₂₃	—	4	4	no	—
101	D₄×C₂₃	—	4	4	no	—
102	D₄×C₂₃	—	4	4	no	—
103	C₂×C₄×C₂₃	—	2	2	no	—
104	C₂×C₄×C₂₃	—	2	4	no	—
105	C₂×C₄×C₂₃	—	2	4	no	—
106	C₂×C₄×C₂₃	—	2	8	no	—
107	C₂×C₄×C₂₃	—	4	2	no	—
108	C₂×C₄×C₂₃	—	4	8	no	—
109	C₂×C₄×C₂₃	—	4	2	no	—
110	C₂×C₄×C₂₃	#41	1	4	yes	—
111	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
114	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
116	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
117	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
118	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
119	#3493	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	—	2	1	no	—
122	C₂²×C₂₃	—	2	2	no	—
123	C₂²×C₂₃	—	2	4	no	—
124	C₂²×C₂₃	—	2	4	no	—
125	C₂²×C₂₃	—	2	4	no	—
126	C₂²×C₂₃	—	2	2	no	—
127	C₂²×C₂₃	—	2	2	no	—
128	C₂²×C₂₃	—	4	4	no	—
129	C₂²×C₂₃	C₂×#41	1	2	yes	—
130	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	2	4	no	—
134	C₄×C₂₃	—	2	4	no	—
135	C₄×C₂₃	—	2	4	no	—
136	C₄×C₂₃	—	2	4	no	—
137	C₄×C₂₃	—	2	4	no	—
138	C₄×C₂₃	C₂×#41	1	4	yes	—
139	#556	C₂×C₂₃	1	4	yes	complemented
140	#555	C₂×C₂₃	1	4	yes	complemented
141	#553	C₂×C₂₃	1	4	yes	complemented
142	#552	C₂×C₂₃	1	4	yes	complemented
143	#544	C₂×C₂₃	1	1	yes	characteristic, complemented
144	#538	C₂×C₂₃	1	4	yes	complemented
145	#537	C₂×C₂₃	1	1	yes	characteristic, complemented
146	#529	C₂×C₂₃	1	4	yes	complemented
147	#525	C₂×C₂₃	1	2	yes	complemented
148	#520	C₂×C₂₃	1	2	yes	complemented
149	C₂×#117	C₂×C₂₃	1	1	yes	characteristic, complemented
150	C₂×C₂₃	—	2	2	no	—
151	C₂×C₂₃	—	4	4	no	—
152	C₂×C₂₃	—	4	2	no	—
153	C₂×C₂₃	C₂²×#41	1	1	yes	characteristic
154	C₂×C₂₃	#585	1	1	yes	characteristic
155	C₂×C₂₃	#585	1	1	yes	characteristic
156	C₂×#41	C₂²×C₂₃	1	2	yes	—
157	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
158	C₄⋊Q₈	C₂²×C₂₃	1	4	yes	complemented
159	#126	C₂²×C₂₃	1	8	yes	—
160	#126	C₂²×C₂₃	1	8	yes	complemented
161	#125	C₂²×C₂₃	1	4	yes	complemented
162	#125	C₂²×C₂₃	1	4	yes	complemented
163	#124	C₂²×C₂₃	1	8	yes	—
164	#124	C₂²×C₂₃	1	4	yes	complemented
165	#123	C₂²×C₂₃	1	4	yes	—
166	#123	C₂²×C₂₃	1	8	yes	—
167	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
168	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
169	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	—
170	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	complemented
171	#121	C₂²×C₂₃	1	2	yes	—
172	#121	C₂²×C₂₃	1	2	yes	—
173	#121	C₂²×C₂₃	1	4	yes	—
174	#121	C₂²×C₂₃	1	8	yes	—
175	#121	C₂²×C₂₃	1	4	yes	complemented
176	#120	C₂²×C₂₃	1	4	yes	—
177	C₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
178	C₄×Q₈	C₂²×C₂₃	1	4	yes	complemented
179	C₄×D₄	C₂²×C₂₃	1	8	yes	—
180	C₄×D₄	C₂²×C₂₃	1	4	yes	—
181	C₄×D₄	C₂²×C₂₃	1	4	yes	—
182	C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
183	C₄×D₄	C₂²×C₂₃	1	4	yes	complemented
184	#117	C₂²×C₂₃	1	4	yes	—
185	#117	C₂²×C₂₃	1	4	yes	—
186	#117	C₂²×C₂₃	1	2	yes	—
187	#117	C₂²×C₂₃	1	4	yes	complemented
188	#117	C₂²×C₂₃	1	8	yes	complemented
189	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
190	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂²×C₂₃	1	4	yes	complemented
192	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
193	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
194	C₂₃	#3493	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
196	#41	—	2	8	no	—
197	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
198	C₂×Q₈	C₂³×C₂₃	1	4	yes	—
199	C₂×D₄	C₂³×C₂₃	1	4	yes	—
200	C₂×D₄	C₂³×C₂₃	1	2	yes	—
201	C₂×D₄	C₂³×C₂₃	1	4	yes	—
202	C₂×D₄	C₂³×C₂₃	1	4	yes	—
203	C₂×D₄	C₂³×C₂₃	1	4	yes	—
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
206	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
207	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
208	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
209	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
210	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
211	C₂²×C₄	C₂³×C₂₃	1	8	yes	—
212	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
213	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
214	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
215	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
216	C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
217	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
218	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
219	C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
220	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
221	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
222	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
223	C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
224	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
225	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
226	C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
227	C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
228	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
229	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
230	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
231	C₄²	C₂³×C₂₃	1	4	yes	—
232	C₄²	C₂³×C₂₃	1	4	yes	—
233	C₄²	C₂³×C₂₃	1	4	yes	—
234	C₄²	C₂³×C₂₃	1	8	yes	—
235	C₂³	—	2	2	no	—
236	C₂³	—	4	1	no	—
237	C₂³	—	4	1	no	—
238	C₂³	C₂⁴×C₂₃	1	2	yes	—
239	C₂³	C₂⁴×C₂₃	1	4	yes	—
240	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	4	4	no	—
243	D₄	—	2	8	no	—
244	D₄	—	2	4	no	—
245	D₄	—	4	4	no	—
246	D₄	—	4	4	no	—
247	D₄	—	4	4	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	8	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	#41×C₂₃	1	4	yes	—
256	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
257	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
258	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
259	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
260	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
261	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
262	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
263	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	4	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	4	no	—
273	C₂²	C₂×#41×C₂₃	1	2	yes	—
274	C₂²	C₂⁵×C₂₃	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	4	no	—
279	C₄	—	2	4	no	—
280	C₄	—	2	4	no	—
281	C₄	—	2	4	no	—
282	C₄	C₂×#41×C₂₃	1	4	yes	—
283	C₂	—	2	2	no	—
284	C₂	—	4	2	no	—
285	C₂	—	4	4	no	—
286	C₂	C₂²×#41×C₂₃	1	1	yes	characteristic
287	C₂	C₂₃×#585	1	1	yes	characteristic
288	C₂	C₂₃×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₃
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×#555	C₂	semidirect
3. C₂₃×#553	C₂	semidirect
4. C₂₃×#552	C₂	semidirect
5. C₂₃×#544	C₂	semidirect
6. C₂₃×#538	C₂	semidirect
7. C₂₃×#537	C₂	semidirect
8. C₂₃×#529	C₂	semidirect
9. C₂₃×#525	C₂	semidirect
10. C₂₃×#520	C₂	semidirect
11. C₂×C₂₃×#117	C₂	semidirect
14. C₂₃×C₄⋊Q₈	C₂²	semidirect
16. C₂₃×#126	C₂²	semidirect
17. C₂₃×#125	C₂²	semidirect
18. C₂₃×#125	C₂²	semidirect
20. C₂₃×#124	C₂²	semidirect
26. C₂₃×C₂²⋊Q₈	C₂²	semidirect
31. C₂₃×#121	C₂²	semidirect
33. C₄×Q₈×C₂₃	C₂²	semidirect
34. C₄×Q₈×C₂₃	C₂²	semidirect
38. C₄×D₄×C₂₃	C₂²	semidirect
39. C₄×D₄×C₂₃	C₂²	semidirect
43. C₂₃×#117	C₂²	semidirect
44. C₂₃×#117	C₂²	semidirect
47. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
119. #3493	C₂₃	direct (N × H)
139. #556	C₂×C₂₃	semidirect
140. #555	C₂×C₂₃	semidirect
141. #553	C₂×C₂₃	semidirect
142. #552	C₂×C₂₃	semidirect
143. #544	C₂×C₂₃	semidirect
144. #538	C₂×C₂₃	semidirect
145. #537	C₂×C₂₃	semidirect
146. #529	C₂×C₂₃	semidirect
147. #525	C₂×C₂₃	semidirect
148. #520	C₂×C₂₃	semidirect
149. C₂×#117	C₂×C₂₃	semidirect
158. C₄⋊Q₈	C₂²×C₂₃	semidirect
160. #126	C₂²×C₂₃	semidirect
161. #125	C₂²×C₂₃	semidirect
162. #125	C₂²×C₂₃	semidirect
164. #124	C₂²×C₂₃	semidirect
170. C₂²⋊Q₈	C₂²×C₂₃	semidirect
175. #121	C₂²×C₂₃	semidirect
177. C₄×Q₈	C₂²×C₂₃	semidirect
178. C₄×Q₈	C₂²×C₂₃	semidirect
182. C₄×D₄	C₂²×C₂₃	semidirect
183. C₄×D₄	C₂²×C₂₃	semidirect
187. #117	C₂²×C₂₃	semidirect
188. #117	C₂²×C₂₃	semidirect
191. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
194. C₂₃	#3493	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×#555	2
C₂₃×#553	2
C₂₃×#552	2
C₂₃×#544	2
C₂₃×#538	2
C₂₃×#537	2
C₂₃×#529	2
C₂₃×#525	2
C₂₃×#520	2
C₂×C₂₃×#117	2
#3493	23

Subgroups — C23×#3493 #15671112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3493 #15671112