Subgroups — C₂₃×#3294 #15670913

Order 2944.

494 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄²×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#480	C₂	1	2	yes	maximal, complemented
5	C₂₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#466	C₂	1	2	yes	maximal, complemented
7	C₂₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#447	C₂	1	2	yes	maximal, complemented
9	C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#102	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₂₃	—	2	1	no	—
13	C₂²×D₄×C₂₃	—	2	1	no	—
14	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
15	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₂₃	C₂²	1	2	yes	complemented
17	C₂²×C₈×C₂₃	C₂²	1	2	yes	complemented
18	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₂₃×#127	C₂²	1	1	yes	characteristic
20	C₂₃×#123	—	2	2	no	—
21	C₂₃×C₂²⋊Q₈	—	2	1	no	—
22	C₂₃×#121	—	2	1	no	—
23	C₂₃×#121	C₂²	1	2	yes	—
24	C₂₃×#121	C₂²	1	2	yes	complemented
25	C₂₃×#120	—	2	2	no	—
26	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
30	C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
31	C₂₃×#107	C₂²	1	2	yes	complemented
32	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	C₂₃×#102	C₂²	1	1	yes	characteristic
34	C₂₃×#102	C₂²	1	2	yes	—
35	C₂₃×#102	C₂²	1	2	yes	complemented
36	C₂₃×#102	C₂²	1	2	yes	complemented
37	C₂₃×#102	C₂²	1	1	yes	characteristic, complemented
38	C₂₃×#102	C₂²	1	2	yes	complemented
39	C₂₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
40	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
41	C₂⁴×C₂₃	—	2	2	no	—
42	#41×C₂₃	—	4	4	no	—
43	C₂×Q₈×C₂₃	—	2	1	no	—
44	C₂×D₄×C₂₃	—	2	4	no	—
45	C₂×D₄×C₂₃	—	2	2	no	—
46	C₂×D₄×C₂₃	—	2	2	no	—
47	C₂×D₄×C₂₃	—	2	2	no	—
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	—	2	1	no	—
50	C₂×D₄×C₂₃	—	4	2	no	—
51	C₂×D₄×C₂₃	—	4	4	no	—
52	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
53	C₂×D₄×C₂₃	C₂³	1	2	yes	—
54	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	—	2	1	no	—
58	C₂²×C₄×C₂₃	—	2	1	no	—
59	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
60	D₈×C₂₃	—	2	2	no	—
61	D₈×C₂₃	—	4	2	no	—
62	C₂×C₈×C₂₃	—	4	2	no	—
63	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₂₃	C₂³	1	2	yes	—
65	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	—	2	2	no	—
67	C₄⋊C₄×C₂₃	—	2	1	no	—
68	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
71	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₂²⋊C₄×C₂₃	—	2	1	no	—
74	C₂²⋊C₄×C₂₃	—	2	2	no	—
75	C₂²⋊C₄×C₂₃	—	2	2	no	—
76	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
77	C₄²×C₂₃	C₂³	1	1	yes	characteristic
78	C₂³×C₂₃	—	2	1	no	—
79	C₂³×C₂₃	—	2	1	no	—
80	C₂³×C₂₃	—	2	2	no	—
81	C₂³×C₂₃	—	4	4	no	—
82	C₂³×C₂₃	—	4	4	no	—
83	C₂³×C₂₃	—	8	2	no	—
84	C₂³×C₂₃	C₂×D₄	1	2	yes	—
85	Q₈×C₂₃	—	4	2	no	—
86	D₄×C₂₃	—	2	1	no	—
87	D₄×C₂₃	—	2	2	no	—
88	D₄×C₂₃	—	2	1	no	—
89	D₄×C₂₃	—	2	1	no	—
90	D₄×C₂₃	—	2	2	no	—
91	D₄×C₂₃	—	4	4	no	—
92	D₄×C₂₃	—	4	4	no	—
93	D₄×C₂₃	—	4	4	no	—
94	D₄×C₂₃	—	4	2	no	—
95	D₄×C₂₃	—	8	1	no	—
96	D₄×C₂₃	D₈	1	4	yes	complemented
97	D₄×C₂₃	#41	1	2	yes	complemented
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	2	2	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	1	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	—	4	2	no	—
105	C₂×C₄×C₂₃	—	4	2	no	—
106	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
108	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
110	C₈×C₂₃	—	2	1	no	—
111	C₈×C₂₃	—	2	1	no	—
112	C₈×C₂₃	—	2	2	no	—
113	#3294	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₂₃	—	2	2	no	—
115	C₂²×C₂₃	—	2	2	no	—
116	C₂²×C₂₃	—	2	2	no	—
117	C₂²×C₂₃	—	4	1	no	—
118	C₂²×C₂₃	—	4	1	no	—
119	C₂²×C₂₃	—	4	2	no	—
120	C₂²×C₂₃	—	4	1	no	—
121	C₂²×C₂₃	—	4	1	no	—
122	C₂²×C₂₃	—	4	2	no	—
123	C₂²×C₂₃	—	4	2	no	—
124	C₂²×C₂₃	—	4	1	no	—
125	C₂²×C₂₃	—	8	8	no	—
126	C₂²×C₂₃	C₂×D₈	1	4	yes	—
127	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₂₃	—	2	2	no	—
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	—	4	1	no	—
131	C₄×C₂₃	—	4	1	no	—
132	C₄×C₂₃	—	4	2	no	—
133	C₄×C₂₃	C₂×D₈	1	2	yes	—
134	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
135	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
136	#547	C₂×C₂₃	1	1	yes	characteristic, complemented
137	D₄²	C₂×C₂₃	1	1	yes	characteristic, complemented
138	#486	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#480	C₂×C₂₃	1	2	yes	complemented
140	#474	C₂×C₂₃	1	1	yes	characteristic, complemented
141	#466	C₂×C₂₃	1	2	yes	complemented
142	#459	C₂×C₂₃	1	1	yes	characteristic, complemented
143	#447	C₂×C₂₃	1	2	yes	complemented
144	C₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
145	C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
146	C₂×#102	C₂×C₂₃	1	2	yes	complemented
147	C₂×C₂₃	—	2	4	no	—
148	C₂×C₂₃	—	4	2	no	—
149	C₂×C₂₃	—	8	1	no	—
150	C₂×C₂₃	—	8	2	no	—
151	C₂×C₂₃	#546	1	1	yes	characteristic
152	C₂×C₂₃	C₂²×D₈	1	1	yes	characteristic
153	C₂×C₂₃	#577	1	1	yes	characteristic
154	C₂×#41	—	2	1	no	—
155	C₂²×D₄	—	2	1	no	—
156	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
157	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×D₈	C₂²×C₂₃	1	2	yes	complemented
159	C₂²×C₈	C₂²×C₂₃	1	2	yes	complemented
160	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
161	#127	C₂²×C₂₃	1	1	yes	characteristic
162	#123	—	2	2	no	—
163	C₂²⋊Q₈	—	2	1	no	—
164	#121	—	2	1	no	—
165	#121	C₂²×C₂₃	1	2	yes	—
166	#121	C₂²×C₂₃	1	2	yes	complemented
167	#120	—	2	2	no	—
168	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
169	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
170	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
171	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
172	#107	C₂²×C₂₃	1	1	yes	characteristic, complemented
173	#107	C₂²×C₂₃	1	2	yes	complemented
174	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
175	#102	C₂²×C₂₃	1	1	yes	characteristic
176	#102	C₂²×C₂₃	1	2	yes	—
177	#102	C₂²×C₂₃	1	2	yes	complemented
178	#102	C₂²×C₂₃	1	2	yes	complemented
179	#102	C₂²×C₂₃	1	1	yes	characteristic, complemented
180	#102	C₂²×C₂₃	1	2	yes	complemented
181	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	complemented
182	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
183	C₂₃	#3294	1	1	yes	characteristic, direct product factor, Sylow
184	C₂⁴	—	2	2	no	—
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	1	no	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
196	C₂×D₄	C₂³×C₂₃	1	2	yes	—
197	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
198	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
203	D₈	—	2	2	no	—
204	D₈	—	4	2	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
207	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
208	C₂×C₈	C₂³×C₂₃	1	2	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
212	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
213	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
214	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
220	C₄²	C₂³×C₂₃	1	1	yes	characteristic
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	4	no	—
226	C₂³	—	8	2	no	—
227	C₂³	C₂×D₄×C₂₃	1	2	yes	—
228	Q₈	—	4	2	no	—
229	D₄	—	2	1	no	—
230	D₄	—	2	2	no	—
231	D₄	—	2	1	no	—
232	D₄	—	2	1	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	4	no	—
237	D₄	—	4	2	no	—
238	D₄	—	8	1	no	—
239	D₄	D₈×C₂₃	1	4	yes	complemented
240	D₄	#41×C₂₃	1	2	yes	complemented
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
251	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
253	C₈	—	2	1	no	—
254	C₈	—	2	1	no	—
255	C₈	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	8	8	no	—
268	C₂²	C₂×D₈×C₂₃	1	4	yes	—
269	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	2	no	—
275	C₄	C₂×D₈×C₂₃	1	2	yes	—
276	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
277	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
278	C₂	—	2	4	no	—
279	C₂	—	4	2	no	—
280	C₂	—	8	2	no	—
281	C₂	—	8	1	no	—
282	C₂	C₂₃×#546	1	1	yes	characteristic
283	C₂	C₂²×D₈×C₂₃	1	1	yes	characteristic
284	C₂	C₂₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×D₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#547	C₂	semidirect
2. D₄²×C₂₃	C₂	semidirect
3. C₂₃×#486	C₂	semidirect
4. C₂₃×#480	C₂	semidirect
5. C₂₃×#474	C₂	semidirect
6. C₂₃×#466	C₂	semidirect
7. C₂₃×#459	C₂	semidirect
8. C₂₃×#447	C₂	semidirect
9. C₄×D₈×C₂₃	C₂	semidirect
10. C₈×D₄×C₂₃	C₂	semidirect
11. C₂×C₂₃×#102	C₂	semidirect
16. C₂×D₈×C₂₃	C₂²	semidirect
17. C₂²×C₈×C₂₃	C₂²	semidirect
18. C₂₃×C₄⋊Q₈	C₂²	semidirect
24. C₂₃×#121	C₂²	semidirect
28. C₄×D₄×C₂₃	C₂²	semidirect
29. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
30. C₂₃×#107	C₂²	semidirect
31. C₂₃×#107	C₂²	semidirect
32. C₂₃×C₄⋊C₈	C₂²	semidirect
35. C₂₃×#102	C₂²	semidirect
36. C₂₃×#102	C₂²	semidirect
37. C₂₃×#102	C₂²	semidirect
38. C₂₃×#102	C₂²	semidirect
39. C₂₃×C₂²⋊C₈	C₂²	semidirect
40. C₄×C₈×C₂₃	C₂²	semidirect
96. D₄×C₂₃	D₈	semidirect
97. D₄×C₂₃	#41	semidirect
113. #3294	C₂₃	direct (N × H)
136. #547	C₂×C₂₃	semidirect
137. D₄²	C₂×C₂₃	semidirect
138. #486	C₂×C₂₃	semidirect
139. #480	C₂×C₂₃	semidirect
140. #474	C₂×C₂₃	semidirect
141. #466	C₂×C₂₃	semidirect
142. #459	C₂×C₂₃	semidirect
143. #447	C₂×C₂₃	semidirect
144. C₄×D₈	C₂×C₂₃	semidirect
145. C₈×D₄	C₂×C₂₃	semidirect
146. C₂×#102	C₂×C₂₃	semidirect
158. C₂×D₈	C₂²×C₂₃	semidirect
159. C₂²×C₈	C₂²×C₂₃	semidirect
160. C₄⋊Q₈	C₂²×C₂₃	semidirect
166. #121	C₂²×C₂₃	semidirect
170. C₄×D₄	C₂²×C₂₃	semidirect
171. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
172. #107	C₂²×C₂₃	semidirect
173. #107	C₂²×C₂₃	semidirect
174. C₄⋊C₈	C₂²×C₂₃	semidirect
177. #102	C₂²×C₂₃	semidirect
178. #102	C₂²×C₂₃	semidirect
179. #102	C₂²×C₂₃	semidirect
180. #102	C₂²×C₂₃	semidirect
181. C₂²⋊C₈	C₂²×C₂₃	semidirect
182. C₄×C₈	C₂²×C₂₃	semidirect
183. C₂₃	#3294	direct (N × H)
239. D₄	D₈×C₂₃	semidirect
240. D₄	#41×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#547	2
D₄²×C₂₃	2
C₂₃×#486	2
C₂₃×#480	2
C₂₃×#474	2
C₂₃×#466	2
C₂₃×#459	2
C₂₃×#447	2
C₄×D₈×C₂₃	2
C₈×D₄×C₂₃	2
C₂×C₂₃×#102	2
#3294	23

Subgroups — C23×#3294 #15670913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3294 #15670913