Subgroups — C₂₃×#3485 #15671104

Order 2944.

1560 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#41×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#547	C₂	1	8	yes	maximal, complemented
4	C₂₃×#546	C₂	1	8	yes	maximal, complemented
5	C₂₃×#536	C₂	1	2	yes	maximal, complemented
6	C₂₃×#534	C₂	1	4	yes	maximal, complemented
7	C₂₃×#525	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#123	C₂	1	4	yes	maximal, complemented
9	C₂₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×H(5,𝔽₂)	—	2	8	no	—
11	C₂×#41×C₂₃	—	2	8	no	—
12	C₂×#41×C₂₃	C₂²	1	2	yes	—
13	C₂×#41×C₂₃	C₂²	1	4	yes	complemented
14	C₂×#41×C₂₃	C₂²	1	8	yes	complemented
15	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
16	C₂²×D₄×C₂₃	C₂²	1	4	yes	—
17	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×D₄×C₂₃	C₂²	1	2	yes	complemented
19	C₂²×D₄×C₂₃	C₂²	1	4	yes	complemented
20	C₂³×C₄×C₂₃	C₂²	1	2	yes	complemented
21	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
22	C₂₃×C₄⋊Q₈	C₂²	1	4	yes	complemented
23	C₂₃×#124	C₂²	1	4	yes	complemented
24	C₂₃×#123	C₂²	1	8	yes	complemented
25	C₂₃×#123	C₂²	1	16	yes	complemented
26	C₂₃×#123	C₂²	1	4	yes	complemented
27	C₂₃×#123	C₂²	1	8	yes	complemented
28	C₂₃×C₂²⋊Q₈	C₂²	1	4	yes	complemented
29	C₂₃×C₂²⋊Q₈	C₂²	1	8	yes	complemented
30	C₂₃×#121	C₂²	1	16	yes	complemented
31	C₂₃×#121	C₂²	1	4	yes	complemented
32	C₂₃×#120	C₂²	1	4	yes	complemented
33	C₂₃×#120	C₂²	1	8	yes	complemented
34	C₄×D₄×C₂₃	C₂²	1	8	yes	complemented
35	C₄×D₄×C₂₃	C₂²	1	8	yes	complemented
36	C₂₃×#117	C₂²	1	2	yes	complemented
37	C₂×C₄⋊C₄×C₂₃	C₂²	1	8	yes	complemented
38	C₂×C₂²⋊C₄×C₂₃	C₂²	1	4	yes	complemented
39	C₂×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	complemented
40	C₂⁴×C₂₃	C₂³	1	4	yes	—
41	C₂⁴×C₂₃	C₂³	1	1	yes	characteristic
42	C₂⁴×C₂₃	C₂³	1	2	yes	—
43	#41×C₂₃	—	2	32	no	—
44	#41×C₂₃	—	2	16	no	—
45	#41×C₂₃	—	2	8	no	—
46	C₂×Q₈×C₂₃	—	2	4	no	—
47	C₂×Q₈×C₂₃	C₂³	1	4	yes	—
48	C₂×Q₈×C₂₃	C₂³	1	2	yes	—
49	C₂×D₄×C₂₃	—	2	16	no	—
50	C₂×D₄×C₂₃	—	2	8	no	—
51	C₂×D₄×C₂₃	—	2	16	no	—
52	C₂×D₄×C₂₃	D₄	1	8	yes	complemented
53	C₂×D₄×C₂₃	C₂³	1	8	yes	—
54	C₂×D₄×C₂₃	C₂³	1	4	yes	—
55	C₂×D₄×C₂₃	C₂³	1	8	yes	—
56	C₂×D₄×C₂₃	C₂³	1	2	yes	—
57	C₂×D₄×C₂₃	C₂³	1	8	yes	—
58	C₂×D₄×C₂₃	C₂³	1	4	yes	—
59	C₂²×C₄×C₂₃	—	2	8	no	—
60	C₂²×C₄×C₂₃	—	2	4	no	—
61	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
62	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
63	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
64	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
65	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
67	C₂²×C₄×C₂₃	C₂³	1	8	yes	—
68	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	C₂³	1	8	yes	—
70	C₄⋊C₄×C₂₃	C₂³	1	16	yes	—
71	C₄⋊C₄×C₂₃	C₂³	1	4	yes	—
72	C₂²⋊C₄×C₂₃	C₂³	1	4	yes	—
73	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
74	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
75	C₂²⋊C₄×C₂₃	C₂³	1	16	yes	—
76	C₂²⋊C₄×C₂₃	C₂³	1	8	yes	—
77	C₄²×C₂₃	C₂³	1	4	yes	—
78	C₂³×C₂₃	—	2	2	no	—
79	C₂³×C₂₃	—	2	8	no	—
80	C₂³×C₂₃	—	4	4	no	—
81	C₂³×C₂₃	—	4	2	no	—
82	C₂³×C₂₃	—	4	8	no	—
83	C₂³×C₂₃	C₂×D₄	1	8	yes	—
84	C₂³×C₂₃	C₂⁴	1	4	yes	—
85	C₂³×C₂₃	C₂⁴	1	2	yes	—
86	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
87	C₂³×C₂₃	C₂⁴	1	4	yes	—
88	Q₈×C₂₃	—	2	8	no	—
89	Q₈×C₂₃	—	2	4	no	—
90	D₄×C₂₃	—	2	8	no	—
91	D₄×C₂₃	—	2	16	no	—
92	D₄×C₂₃	—	2	16	no	—
93	D₄×C₂₃	—	2	4	no	—
94	D₄×C₂₃	—	4	8	no	—
95	D₄×C₂₃	C₂×D₄	1	16	yes	—
96	C₂×C₄×C₂₃	—	2	2	no	—
97	C₂×C₄×C₂₃	—	2	4	no	—
98	C₂×C₄×C₂₃	—	2	16	no	—
99	C₂×C₄×C₂₃	—	2	8	no	—
100	C₂×C₄×C₂₃	—	2	8	no	—
101	C₂×C₄×C₂₃	—	2	8	no	—
102	C₂×C₄×C₂₃	—	4	8	no	—
103	C₂×C₄×C₂₃	C₂×D₄	1	4	yes	—
104	C₂×C₄×C₂₃	C₂⁴	1	8	yes	—
105	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
106	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
107	C₂×C₄×C₂₃	C₂⁴	1	4	yes	—
108	C₂×C₄×C₂₃	C₂⁴	1	2	yes	—
109	#3485	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₂₃	—	2	8	no	—
111	C₂²×C₂₃	—	2	2	no	—
112	C₂²×C₂₃	—	2	2	no	—
113	C₂²×C₂₃	—	2	4	no	—
114	C₂²×C₂₃	—	2	4	no	—
115	C₂²×C₂₃	—	2	8	no	—
116	C₂²×C₂₃	—	4	8	no	—
117	C₂²×C₂₃	—	4	4	no	—
118	C₂²×C₂₃	—	8	8	no	—
119	C₂²×C₂₃	C₂²×D₄	1	2	yes	—
120	C₂²×C₂₃	C₂²×D₄	1	8	yes	—
121	C₂²×C₂₃	C₂⁵	1	1	yes	characteristic
122	C₄×C₂₃	—	2	2	no	—
123	C₄×C₂₃	—	2	4	no	—
124	C₄×C₂₃	—	2	4	no	—
125	C₄×C₂₃	—	2	8	no	—
126	C₄×C₂₃	C₂²×D₄	1	4	yes	—
127	C₂×H(5,𝔽₂)	C₂×C₂₃	1	1	yes	characteristic, complemented
128	C₂²×#41	C₂×C₂₃	1	1	yes	characteristic, complemented
129	#547	C₂×C₂₃	1	8	yes	complemented
130	#546	C₂×C₂₃	1	8	yes	complemented
131	#536	C₂×C₂₃	1	2	yes	complemented
132	#534	C₂×C₂₃	1	4	yes	complemented
133	#525	C₂×C₂₃	1	2	yes	complemented
134	C₂×#123	C₂×C₂₃	1	4	yes	complemented
135	#518	C₂×C₂₃	1	1	yes	characteristic, complemented
136	C₂×C₂₃	—	2	2	no	—
137	C₂×C₂₃	—	2	8	no	—
138	C₂×C₂₃	—	4	2	no	—
139	C₂×C₂₃	—	4	4	no	—
140	C₂×C₂₃	C₂³×D₄	1	1	yes	characteristic
141	C₂×C₂₃	#585	1	2	yes	—
142	H(5,𝔽₂)	—	2	8	no	—
143	C₂×#41	—	2	8	no	—
144	C₂×#41	C₂²×C₂₃	1	2	yes	—
145	C₂×#41	C₂²×C₂₃	1	8	yes	complemented
146	C₂×#41	C₂²×C₂₃	1	4	yes	complemented
147	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
148	C₂²×D₄	C₂²×C₂₃	1	4	yes	—
149	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
150	C₂²×D₄	C₂²×C₂₃	1	2	yes	complemented
151	C₂²×D₄	C₂²×C₂₃	1	4	yes	complemented
152	C₂³×C₄	C₂²×C₂₃	1	2	yes	complemented
153	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
154	C₄⋊Q₈	C₂²×C₂₃	1	4	yes	complemented
155	#124	C₂²×C₂₃	1	4	yes	complemented
156	#123	C₂²×C₂₃	1	4	yes	complemented
157	#123	C₂²×C₂₃	1	8	yes	complemented
158	#123	C₂²×C₂₃	1	16	yes	complemented
159	#123	C₂²×C₂₃	1	8	yes	complemented
160	C₂²⋊Q₈	C₂²×C₂₃	1	4	yes	complemented
161	C₂²⋊Q₈	C₂²×C₂₃	1	8	yes	complemented
162	#121	C₂²×C₂₃	1	4	yes	complemented
163	#121	C₂²×C₂₃	1	16	yes	complemented
164	#120	C₂²×C₂₃	1	4	yes	complemented
165	#120	C₂²×C₂₃	1	8	yes	complemented
166	C₄×D₄	C₂²×C₂₃	1	8	yes	complemented
167	C₄×D₄	C₂²×C₂₃	1	8	yes	complemented
168	#117	C₂²×C₂₃	1	2	yes	complemented
169	C₂×C₄⋊C₄	C₂²×C₂₃	1	8	yes	complemented
170	C₂×C₂²⋊C₄	C₂²×C₂₃	1	4	yes	complemented
171	C₂×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	complemented
172	C₂₃	#3485	1	1	yes	characteristic, direct product factor, Sylow
173	C₂⁴	C₂³×C₂₃	1	4	yes	—
174	C₂⁴	C₂³×C₂₃	1	1	yes	characteristic
175	C₂⁴	C₂³×C₂₃	1	2	yes	—
176	#41	—	2	32	no	—
177	#41	—	2	16	no	—
178	#41	—	2	8	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	C₂³×C₂₃	1	4	yes	—
181	C₂×Q₈	C₂³×C₂₃	1	2	yes	—
182	C₂×D₄	—	2	16	no	—
183	C₂×D₄	—	2	8	no	—
184	C₂×D₄	—	2	16	no	—
185	C₂×D₄	D₄×C₂₃	1	8	yes	complemented
186	C₂×D₄	C₂³×C₂₃	1	8	yes	—
187	C₂×D₄	C₂³×C₂₃	1	4	yes	—
188	C₂×D₄	C₂³×C₂₃	1	8	yes	—
189	C₂×D₄	C₂³×C₂₃	1	2	yes	—
190	C₂×D₄	C₂³×C₂₃	1	8	yes	—
191	C₂×D₄	C₂³×C₂₃	1	4	yes	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	C₂³×C₂₃	1	8	yes	—
195	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
196	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
197	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
198	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
199	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
200	C₂²×C₄	C₂³×C₂₃	1	8	yes	—
201	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₂₃	1	8	yes	—
203	C₄⋊C₄	C₂³×C₂₃	1	16	yes	—
204	C₄⋊C₄	C₂³×C₂₃	1	4	yes	—
205	C₂²⋊C₄	C₂³×C₂₃	1	4	yes	—
206	C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
207	C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
208	C₂²⋊C₄	C₂³×C₂₃	1	16	yes	—
209	C₂²⋊C₄	C₂³×C₂₃	1	8	yes	—
210	C₄²	C₂³×C₂₃	1	4	yes	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	8	no	—
213	C₂³	—	4	4	no	—
214	C₂³	—	4	2	no	—
215	C₂³	—	4	8	no	—
216	C₂³	C₂×D₄×C₂₃	1	8	yes	—
217	C₂³	C₂⁴×C₂₃	1	4	yes	—
218	C₂³	C₂⁴×C₂₃	1	2	yes	—
219	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
220	C₂³	C₂⁴×C₂₃	1	4	yes	—
221	Q₈	—	2	8	no	—
222	Q₈	—	2	4	no	—
223	D₄	—	2	8	no	—
224	D₄	—	2	16	no	—
225	D₄	—	2	16	no	—
226	D₄	—	2	4	no	—
227	D₄	—	4	8	no	—
228	D₄	C₂×D₄×C₂₃	1	16	yes	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	4	8	no	—
236	C₂×C₄	C₂×D₄×C₂₃	1	4	yes	—
237	C₂×C₄	C₂⁴×C₂₃	1	8	yes	—
238	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
239	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
240	C₂×C₄	C₂⁴×C₂₃	1	4	yes	—
241	C₂×C₄	C₂⁴×C₂₃	1	2	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	4	no	—
247	C₂²	—	2	8	no	—
248	C₂²	—	4	8	no	—
249	C₂²	—	4	4	no	—
250	C₂²	—	8	8	no	—
251	C₂²	C₂²×D₄×C₂₃	1	2	yes	—
252	C₂²	C₂²×D₄×C₂₃	1	8	yes	—
253	C₂²	C₂⁵×C₂₃	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	4	no	—
256	C₄	—	2	4	no	—
257	C₄	—	2	8	no	—
258	C₄	C₂²×D₄×C₂₃	1	4	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	2	8	no	—
261	C₂	—	4	2	no	—
262	C₂	—	4	4	no	—
263	C₂	C₂³×D₄×C₂₃	1	1	yes	characteristic
264	C₂	C₂₃×#585	1	2	yes	—

	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. C₂×C₂₃×H(5,𝔽₂)	C₂	semidirect
2. C₂²×#41×C₂₃	C₂	semidirect
3. C₂₃×#547	C₂	semidirect
4. C₂₃×#546	C₂	semidirect
5. C₂₃×#536	C₂	semidirect
6. C₂₃×#534	C₂	semidirect
7. C₂₃×#525	C₂	semidirect
8. C₂×C₂₃×#123	C₂	semidirect
9. C₂₃×#518	C₂	semidirect
13. C₂×#41×C₂₃	C₂²	semidirect
14. C₂×#41×C₂₃	C₂²	semidirect
15. C₂²×Q₈×C₂₃	C₂²	semidirect
18. C₂²×D₄×C₂₃	C₂²	semidirect
19. C₂²×D₄×C₂₃	C₂²	semidirect
20. C₂³×C₄×C₂₃	C₂²	semidirect
21. C₂³×C₄×C₂₃	C₂²	semidirect
22. C₂₃×C₄⋊Q₈	C₂²	semidirect
23. C₂₃×#124	C₂²	semidirect
24. C₂₃×#123	C₂²	semidirect
25. C₂₃×#123	C₂²	semidirect
26. C₂₃×#123	C₂²	semidirect
27. C₂₃×#123	C₂²	semidirect
28. C₂₃×C₂²⋊Q₈	C₂²	semidirect
29. C₂₃×C₂²⋊Q₈	C₂²	semidirect
30. C₂₃×#121	C₂²	semidirect
31. C₂₃×#121	C₂²	semidirect
32. C₂₃×#120	C₂²	semidirect
33. C₂₃×#120	C₂²	semidirect
34. C₄×D₄×C₂₃	C₂²	semidirect
35. C₄×D₄×C₂₃	C₂²	semidirect
36. C₂₃×#117	C₂²	semidirect
37. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
38. C₂×C₂²⋊C₄×C₂₃	C₂²	semidirect
39. C₂×C₂²⋊C₄×C₂₃	C₂²	semidirect
52. C₂×D₄×C₂₃	D₄	semidirect
109. #3485	C₂₃	direct (N × H)
127. C₂×H(5,𝔽₂)	C₂×C₂₃	semidirect
128. C₂²×#41	C₂×C₂₃	semidirect
129. #547	C₂×C₂₃	semidirect
130. #546	C₂×C₂₃	semidirect
131. #536	C₂×C₂₃	semidirect
132. #534	C₂×C₂₃	semidirect
133. #525	C₂×C₂₃	semidirect
134. C₂×#123	C₂×C₂₃	semidirect
135. #518	C₂×C₂₃	semidirect
145. C₂×#41	C₂²×C₂₃	semidirect
146. C₂×#41	C₂²×C₂₃	semidirect
147. C₂²×Q₈	C₂²×C₂₃	semidirect
150. C₂²×D₄	C₂²×C₂₃	semidirect
151. C₂²×D₄	C₂²×C₂₃	semidirect
152. C₂³×C₄	C₂²×C₂₃	semidirect
153. C₂³×C₄	C₂²×C₂₃	semidirect
154. C₄⋊Q₈	C₂²×C₂₃	semidirect
155. #124	C₂²×C₂₃	semidirect
156. #123	C₂²×C₂₃	semidirect
157. #123	C₂²×C₂₃	semidirect
158. #123	C₂²×C₂₃	semidirect
159. #123	C₂²×C₂₃	semidirect
160. C₂²⋊Q₈	C₂²×C₂₃	semidirect
161. C₂²⋊Q₈	C₂²×C₂₃	semidirect
162. #121	C₂²×C₂₃	semidirect
163. #121	C₂²×C₂₃	semidirect
164. #120	C₂²×C₂₃	semidirect
165. #120	C₂²×C₂₃	semidirect
166. C₄×D₄	C₂²×C₂₃	semidirect
167. C₄×D₄	C₂²×C₂₃	semidirect
168. #117	C₂²×C₂₃	semidirect
169. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
170. C₂×C₂²⋊C₄	C₂²×C₂₃	semidirect
171. C₂×C₂²⋊C₄	C₂²×C₂₃	semidirect
172. C₂₃	#3485	direct (N × H)
185. C₂×D₄	D₄×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂₃×H(5,𝔽₂)	2
C₂²×#41×C₂₃	2
C₂₃×#547	2
C₂₃×#546	2
C₂₃×#536	2
C₂₃×#534	2
C₂₃×#525	2
C₂×C₂₃×#123	2
C₂₃×#518	2
#3485	23

Subgroups — C23×#3485 #15671104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3485 #15671104