Subgroups — C₂₃×#3280 #15670899

Order 2944.

526 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#546	C₂	1	2	yes	maximal, complemented
4	C₂₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#466	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#463	C₂	1	2	yes	maximal, complemented
8	C₂₃×#450	C₂	1	2	yes	maximal, complemented
9	C₂₃×#449	C₂	1	2	yes	maximal, complemented
10	C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	2	yes	complemented
13	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
14	C₂₃×#135	—	2	4	no	—
15	C₂×Q₁₆×C₂₃	C₂²	1	1	yes	characteristic, complemented
16	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
17	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
18	C₂×D₈×C₂₃	—	2	4	no	—
19	C₂×D₈×C₂₃	C₂²	1	2	yes	—
20	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
24	C₂₃×#124	C₂²	1	2	yes	complemented
25	C₂₃×#123	—	2	2	no	—
26	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂₃×#121	—	2	2	no	—
28	C₂₃×#121	C₂²	1	2	yes	complemented
29	C₂₃×#120	—	2	2	no	—
30	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
32	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
33	C₂₃×#107	C₂²	1	1	yes	characteristic, complemented
34	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
35	C₂₃×#103	C₂²	1	2	yes	complemented
36	C₂₃×#102	C₂²	1	2	yes	complemented
37	C₂₃×#102	C₂²	1	2	yes	complemented
38	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
41	C₂⁴×C₂₃	—	2	2	no	—
42	#41×C₂₃	—	2	4	no	—
43	#41×C₂₃	—	4	2	no	—
44	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
45	C₂×D₄×C₂₃	—	2	4	no	—
46	C₂×D₄×C₂₃	—	2	2	no	—
47	C₂×D₄×C₂₃	—	2	2	no	—
48	C₂×D₄×C₂₃	—	2	2	no	—
49	C₂×D₄×C₂₃	—	4	2	no	—
50	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
51	C₂×D₄×C₂₃	C₂³	1	2	yes	—
52	C₂×D₄×C₂₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₂₃	—	2	2	no	—
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
57	Q₁₆×C₂₃	—	2	2	no	—
58	SD₁₆×C₂₃	—	2	4	no	—
59	SD₁₆×C₂₃	—	4	2	no	—
60	D₈×C₂₃	—	2	4	no	—
61	D₈×C₂₃	—	2	2	no	—
62	D₈×C₂₃	D₄	1	4	yes	complemented
63	C₂×C₈×C₂₃	—	2	2	no	—
64	C₂×C₈×C₂₃	—	2	2	no	—
65	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	—	2	2	no	—
70	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
71	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
72	C₂²⋊C₄×C₂₃	—	2	2	no	—
73	C₂²⋊C₄×C₂₃	—	2	2	no	—
74	C₂²⋊C₄×C₂₃	—	2	2	no	—
75	C₂²⋊C₄×C₂₃	—	2	2	no	—
76	C₂²⋊C₄×C₂₃	—	2	2	no	—
77	C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₂₃	D₄	1	1	yes	characteristic
79	C₄²×C₂₃	C₂³	1	1	yes	characteristic
80	C₂³×C₂₃	—	2	2	no	—
81	C₂³×C₂₃	—	2	2	no	—
82	C₂³×C₂₃	—	4	4	no	—
83	C₂³×C₂₃	—	4	4	no	—
84	C₂³×C₂₃	—	8	2	no	—
85	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
87	Q₈×C₂₃	—	2	2	no	—
88	Q₈×C₂₃	—	2	2	no	—
89	D₄×C₂₃	—	2	2	no	—
90	D₄×C₂₃	—	2	2	no	—
91	D₄×C₂₃	—	2	2	no	—
92	D₄×C₂₃	—	2	2	no	—
93	D₄×C₂₃	—	4	4	no	—
94	D₄×C₂₃	—	4	4	no	—
95	D₄×C₂₃	—	8	2	no	—
96	D₄×C₂₃	C₂×D₄	1	4	yes	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	1	no	—
99	C₂×C₄×C₂₃	—	2	1	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	2	2	no	—
102	C₂×C₄×C₂₃	—	2	2	no	—
103	C₂×C₄×C₂₃	—	4	4	no	—
104	C₂×C₄×C₂₃	—	8	2	no	—
105	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
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	1	no	—
113	C₈×C₂₃	C₂×D₄	1	2	yes	—
114	#3280	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂²×C₂₃	—	2	1	no	—
116	C₂²×C₂₃	—	2	1	no	—
117	C₂²×C₂₃	—	2	1	no	—
118	C₂²×C₂₃	—	2	4	no	—
119	C₂²×C₂₃	—	2	1	no	—
120	C₂²×C₂₃	—	2	1	no	—
121	C₂²×C₂₃	—	4	2	no	—
122	C₂²×C₂₃	—	4	2	no	—
123	C₂²×C₂₃	—	4	2	no	—
124	C₂²×C₂₃	—	4	2	no	—
125	C₂²×C₂₃	—	4	2	no	—
126	C₂²×C₂₃	—	8	8	no	—
127	C₂²×C₂₃	#135	1	2	yes	complemented
128	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
129	C₄×C₂₃	—	2	1	no	—
130	C₄×C₂₃	—	2	1	no	—
131	C₄×C₂₃	—	2	1	no	—
132	C₄×C₂₃	—	2	1	no	—
133	C₄×C₂₃	—	4	2	no	—
134	C₄×C₂₃	—	4	2	no	—
135	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
137	C₂×#135	C₂×C₂₃	1	1	yes	characteristic, complemented
138	C₂²×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#546	C₂×C₂₃	1	2	yes	complemented
140	#495	C₂×C₂₃	1	1	yes	characteristic, complemented
141	#467	C₂×C₂₃	1	1	yes	characteristic, complemented
142	#466	C₂×C₂₃	1	1	yes	characteristic, complemented
143	#463	C₂×C₂₃	1	2	yes	complemented
144	#450	C₂×C₂₃	1	2	yes	complemented
145	#449	C₂×C₂₃	1	2	yes	complemented
146	C₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
147	C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
148	C₂×C₂₃	—	2	2	no	—
149	C₂×C₂₃	—	4	1	no	—
150	C₂×C₂₃	—	4	4	no	—
151	C₂×C₂₃	—	8	2	no	—
152	C₂×C₂₃	D₄²	1	1	yes	characteristic
153	C₂×C₂₃	C₂×#135	1	1	yes	characteristic
154	C₂×C₂₃	#576	1	1	yes	characteristic
155	C₂×#41	C₂²×C₂₃	1	2	yes	complemented
156	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
157	#135	—	2	4	no	—
158	C₂×Q₁₆	C₂²×C₂₃	1	1	yes	characteristic, complemented
159	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
160	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
161	C₂×D₈	—	2	4	no	—
162	C₂×D₈	C₂²×C₂₃	1	2	yes	—
163	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
164	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
166	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
167	#124	C₂²×C₂₃	1	2	yes	complemented
168	#123	—	2	2	no	—
169	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
170	#121	—	2	2	no	—
171	#121	C₂²×C₂₃	1	2	yes	complemented
172	#120	—	2	2	no	—
173	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
174	C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
175	C₂×C₂²⋊C₄	—	2	2	no	—
176	#107	C₂²×C₂₃	1	1	yes	characteristic, complemented
177	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
178	#103	C₂²×C₂₃	1	2	yes	complemented
179	#102	C₂²×C₂₃	1	2	yes	complemented
180	#102	C₂²×C₂₃	1	2	yes	complemented
181	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
182	C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
183	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
184	C₂₃	#3280	1	1	yes	characteristic, direct product factor, Sylow
185	C₂⁴	—	2	2	no	—
186	#41	—	2	4	no	—
187	#41	—	4	2	no	—
188	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
195	C₂×D₄	C₂³×C₂₃	1	2	yes	—
196	C₂×D₄	C₂³×C₂₃	1	2	yes	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
201	Q₁₆	—	2	2	no	—
202	SD₁₆	—	2	4	no	—
203	SD₁₆	—	4	2	no	—
204	D₈	—	2	4	no	—
205	D₈	—	2	2	no	—
206	D₈	D₄×C₂₃	1	4	yes	complemented
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
210	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
211	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
212	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
215	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	2	no	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
222	C₂²⋊C₄	D₄×C₂₃	1	1	yes	characteristic
223	C₄²	C₂³×C₂₃	1	1	yes	characteristic
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	4	4	no	—
227	C₂³	—	4	4	no	—
228	C₂³	—	8	2	no	—
229	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
230	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
231	Q₈	—	2	2	no	—
232	Q₈	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	2	2	no	—
237	D₄	—	4	4	no	—
238	D₄	—	4	4	no	—
239	D₄	—	8	2	no	—
240	D₄	C₂×D₄×C₂₃	1	4	yes	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	8	2	no	—
249	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
251	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
252	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
253	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	C₂×D₄×C₂₃	1	2	yes	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	8	8	no	—
270	C₂²	C₂₃×#135	1	2	yes	complemented
271	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	2	no	—
278	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
279	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
280	C₂	—	2	2	no	—
281	C₂	—	4	1	no	—
282	C₂	—	4	4	no	—
283	C₂	—	8	2	no	—
284	C₂	D₄²×C₂₃	1	1	yes	characteristic
285	C₂	C₂×C₂₃×#135	1	1	yes	characteristic
286	C₂	C₂₃×#576	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. C₂×C₂₃×#135	C₂	semidirect
2. C₂²×D₈×C₂₃	C₂	semidirect
3. C₂₃×#546	C₂	semidirect
4. C₂₃×#495	C₂	semidirect
5. C₂₃×#467	C₂	semidirect
6. C₂₃×#466	C₂	semidirect
7. C₂₃×#463	C₂	semidirect
8. C₂₃×#450	C₂	semidirect
9. C₂₃×#449	C₂	semidirect
10. C₄×D₈×C₂₃	C₂	semidirect
11. C₈×D₄×C₂₃	C₂	semidirect
12. C₂×#41×C₂₃	C₂²	semidirect
15. C₂×Q₁₆×C₂₃	C₂²	semidirect
16. C₂×SD₁₆×C₂₃	C₂²	semidirect
17. C₂×SD₁₆×C₂₃	C₂²	semidirect
23. C₂²×C₈×C₂₃	C₂²	semidirect
24. C₂₃×#124	C₂²	semidirect
28. C₂₃×#121	C₂²	semidirect
31. C₄×D₄×C₂₃	C₂²	semidirect
33. C₂₃×#107	C₂²	semidirect
34. C₂₃×C₄⋊C₈	C₂²	semidirect
35. C₂₃×#103	C₂²	semidirect
36. C₂₃×#102	C₂²	semidirect
37. C₂₃×#102	C₂²	semidirect
39. C₂₃×C₂²⋊C₈	C₂²	semidirect
40. C₄×C₈×C₂₃	C₂²	semidirect
62. D₈×C₂₃	D₄	semidirect
114. #3280	C₂₃	direct (N × H)
127. C₂²×C₂₃	#135	semidirect
137. C₂×#135	C₂×C₂₃	semidirect
138. C₂²×D₈	C₂×C₂₃	semidirect
139. #546	C₂×C₂₃	semidirect
140. #495	C₂×C₂₃	semidirect
141. #467	C₂×C₂₃	semidirect
142. #466	C₂×C₂₃	semidirect
143. #463	C₂×C₂₃	semidirect
144. #450	C₂×C₂₃	semidirect
145. #449	C₂×C₂₃	semidirect
146. C₄×D₈	C₂×C₂₃	semidirect
147. C₈×D₄	C₂×C₂₃	semidirect
155. C₂×#41	C₂²×C₂₃	semidirect
158. C₂×Q₁₆	C₂²×C₂₃	semidirect
159. C₂×SD₁₆	C₂²×C₂₃	semidirect
160. C₂×SD₁₆	C₂²×C₂₃	semidirect
166. C₂²×C₈	C₂²×C₂₃	semidirect
167. #124	C₂²×C₂₃	semidirect
171. #121	C₂²×C₂₃	semidirect
174. C₄×D₄	C₂²×C₂₃	semidirect
176. #107	C₂²×C₂₃	semidirect
177. C₄⋊C₈	C₂²×C₂₃	semidirect
178. #103	C₂²×C₂₃	semidirect
179. #102	C₂²×C₂₃	semidirect
180. #102	C₂²×C₂₃	semidirect
182. C₂²⋊C₈	C₂²×C₂₃	semidirect
183. C₄×C₈	C₂²×C₂₃	semidirect
184. C₂₃	#3280	direct (N × H)
206. D₈	D₄×C₂₃	semidirect
270. C₂²	C₂₃×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×#135	2
C₂²×D₈×C₂₃	2
C₂₃×#546	2
C₂₃×#495	2
C₂₃×#467	2
C₂₃×#466	2
C₂₃×#463	2
C₂₃×#450	2
C₂₃×#449	2
C₄×D₈×C₂₃	2
C₈×D₄×C₂₃	2
#3280	23

Subgroups — C23×#3280 #15670899

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3280 #15670899