Subgroups — C₂₃×#3295 #15670914

Order 2944.

462 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄²×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#481	C₂	1	2	yes	maximal, complemented
5	C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#465	C₂	1	2	yes	maximal, complemented
7	C₂₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#450	C₂	1	2	yes	maximal, complemented
9	C₄×SD₁₆×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	C₂²×D₄×C₂₃	—	2	1	no	—
13	C₂²×D₄×C₂₃	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
16	C₂²×C₈×C₂₃	C₂²	1	2	yes	complemented
17	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₂₃×#127	C₂²	1	1	yes	characteristic
19	C₂₃×#125	—	2	1	no	—
20	C₂₃×C₂²⋊Q₈	—	2	2	no	—
21	C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	complemented
22	C₂₃×#121	—	2	1	no	—
23	C₂₃×#121	C₂²	1	2	yes	—
24	C₂₃×#120	—	2	2	no	—
25	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
26	C₄×D₄×C₂₃	—	2	1	no	—
27	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	complemented
30	C₂₃×#106	C₂²	1	1	yes	characteristic, complemented
31	C₂₃×#106	C₂²	1	2	yes	complemented
32	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
33	C₂₃×#103	C₂²	1	1	yes	characteristic, complemented
34	C₂₃×#102	C₂²	1	1	yes	characteristic
35	C₂₃×#102	C₂²	1	2	yes	—
36	C₂₃×#102	C₂²	1	2	yes	complemented
37	C₂₃×#102	C₂²	1	2	yes	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	C₂×Q₈×C₂₃	—	2	1	no	—
43	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
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	1	no	—
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	—	4	2	no	—
50	C₂×D₄×C₂₃	—	4	4	no	—
51	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
52	C₂×D₄×C₂₃	C₂³	1	2	yes	—
53	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	2	1	no	—
56	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
57	SD₁₆×C₂₃	—	2	2	no	—
58	SD₁₆×C₂₃	—	4	2	no	—
59	C₂×C₈×C₂₃	—	4	2	no	—
60	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₂₃	C₂³	1	2	yes	—
62	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₂₃	—	2	2	no	—
64	C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₄⋊C₄×C₂₃	—	2	1	no	—
66	C₄⋊C₄×C₂₃	—	2	2	no	—
67	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
70	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₂₃	—	2	2	no	—
72	C₂²⋊C₄×C₂₃	—	2	1	no	—
73	C₂²⋊C₄×C₂₃	—	2	2	no	—
74	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
75	C₄²×C₂₃	—	2	1	no	—
76	C₄²×C₂₃	C₂³	1	1	yes	characteristic
77	C₂³×C₂₃	—	2	1	no	—
78	C₂³×C₂₃	—	2	2	no	—
79	C₂³×C₂₃	—	4	4	no	—
80	C₂³×C₂₃	—	4	4	no	—
81	C₂³×C₂₃	—	8	2	no	—
82	C₂³×C₂₃	C₂×D₄	1	2	yes	—
83	Q₈×C₂₃	—	2	1	no	—
84	Q₈×C₂₃	—	2	1	no	—
85	Q₈×C₂₃	—	8	1	no	—
86	D₄×C₂₃	—	2	2	no	—
87	D₄×C₂₃	—	2	1	no	—
88	D₄×C₂₃	—	2	2	no	—
89	D₄×C₂₃	—	4	4	no	—
90	D₄×C₂₃	—	4	4	no	—
91	D₄×C₂₃	—	4	2	no	—
92	D₄×C₂₃	—	8	1	no	—
93	D₄×C₂₃	SD₁₆	1	4	yes	complemented
94	D₄×C₂₃	#41	1	2	yes	—
95	C₂×C₄×C₂₃	—	2	2	no	—
96	C₂×C₄×C₂₃	—	2	2	no	—
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	2	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	—	8	2	no	—
103	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
107	C₈×C₂₃	—	2	1	no	—
108	C₈×C₂₃	—	2	1	no	—
109	C₈×C₂₃	—	2	2	no	—
110	#3295	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂²×C₂₃	—	2	2	no	—
112	C₂²×C₂₃	—	2	2	no	—
113	C₂²×C₂₃	—	2	2	no	—
114	C₂²×C₂₃	—	4	1	no	—
115	C₂²×C₂₃	—	4	1	no	—
116	C₂²×C₂₃	—	4	2	no	—
117	C₂²×C₂₃	—	4	2	no	—
118	C₂²×C₂₃	—	4	2	no	—
119	C₂²×C₂₃	—	8	8	no	—
120	C₂²×C₂₃	C₂×SD₁₆	1	4	yes	—
121	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
122	C₄×C₂₃	—	2	2	no	—
123	C₄×C₂₃	—	2	1	no	—
124	C₄×C₂₃	—	4	1	no	—
125	C₄×C₂₃	—	4	1	no	—
126	C₄×C₂₃	—	4	1	no	—
127	C₄×C₂₃	—	4	2	no	—
128	C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
129	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
130	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
131	#554	C₂×C₂₃	1	1	yes	characteristic, complemented
132	D₄²	C₂×C₂₃	1	1	yes	characteristic
133	#486	C₂×C₂₃	1	1	yes	characteristic, complemented
134	#481	C₂×C₂₃	1	2	yes	complemented
135	#476	C₂×C₂₃	1	1	yes	characteristic, complemented
136	#465	C₂×C₂₃	1	2	yes	complemented
137	#460	C₂×C₂₃	1	1	yes	characteristic, complemented
138	#450	C₂×C₂₃	1	2	yes	complemented
139	C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
140	C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
141	C₂×#102	C₂×C₂₃	1	2	yes	complemented
142	C₂×C₂₃	—	2	4	no	—
143	C₂×C₂₃	—	4	2	no	—
144	C₂×C₂₃	—	8	2	no	—
145	C₂×C₂₃	#546	1	1	yes	characteristic
146	C₂×C₂₃	C₂²×SD₁₆	1	1	yes	characteristic
147	C₂×C₂₃	#576	1	1	yes	characteristic
148	C₂²×D₄	—	2	1	no	—
149	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
150	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
151	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
152	C₂²×C₈	C₂²×C₂₃	1	2	yes	complemented
153	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
154	#127	C₂²×C₂₃	1	1	yes	characteristic
155	#125	—	2	1	no	—
156	C₂²⋊Q₈	—	2	2	no	—
157	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
158	#121	—	2	1	no	—
159	#121	C₂²×C₂₃	1	2	yes	—
160	#120	—	2	2	no	—
161	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
162	C₄×D₄	—	2	1	no	—
163	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
164	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	complemented
166	#106	C₂²×C₂₃	1	1	yes	characteristic, complemented
167	#106	C₂²×C₂₃	1	2	yes	complemented
168	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
169	#103	C₂²×C₂₃	1	1	yes	characteristic, complemented
170	#102	C₂²×C₂₃	1	1	yes	characteristic
171	#102	C₂²×C₂₃	1	2	yes	—
172	#102	C₂²×C₂₃	1	2	yes	complemented
173	#102	C₂²×C₂₃	1	2	yes	complemented
174	#102	C₂²×C₂₃	1	2	yes	complemented
175	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	complemented
176	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
177	C₂₃	#3295	1	1	yes	characteristic, direct product factor, Sylow
178	C₂⁴	—	2	2	no	—
179	C₂×Q₈	—	2	1	no	—
180	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	4	no	—
188	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
189	C₂×D₄	C₂³×C₂₃	1	2	yes	—
190	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
194	SD₁₆	—	2	2	no	—
195	SD₁₆	—	4	2	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
198	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
199	C₂×C₈	C₂³×C₂₃	1	2	yes	—
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
205	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
206	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
207	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
212	C₄²	—	2	1	no	—
213	C₄²	C₂³×C₂₃	1	1	yes	characteristic
214	C₂³	—	2	1	no	—
215	C₂³	—	2	2	no	—
216	C₂³	—	4	4	no	—
217	C₂³	—	4	4	no	—
218	C₂³	—	8	2	no	—
219	C₂³	C₂×D₄×C₂₃	1	2	yes	—
220	Q₈	—	2	1	no	—
221	Q₈	—	2	1	no	—
222	Q₈	—	8	1	no	—
223	D₄	—	2	2	no	—
224	D₄	—	2	1	no	—
225	D₄	—	2	2	no	—
226	D₄	—	4	4	no	—
227	D₄	—	4	4	no	—
228	D₄	—	4	2	no	—
229	D₄	—	8	1	no	—
230	D₄	SD₁₆×C₂₃	1	4	yes	complemented
231	D₄	#41×C₂₃	1	2	yes	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
242	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
243	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	1	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	8	8	no	—
256	C₂²	C₂×SD₁₆×C₂₃	1	4	yes	—
257	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	1	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	1	no	—
264	C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
265	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
266	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
267	C₂	—	2	4	no	—
268	C₂	—	4	2	no	—
269	C₂	—	8	2	no	—
270	C₂	C₂₃×#546	1	1	yes	characteristic
271	C₂	C₂²×SD₁₆×C₂₃	1	1	yes	characteristic
272	C₂	C₂₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	D₄²×C₂₃	C₂
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×#554	C₂	semidirect
3. C₂₃×#486	C₂	semidirect
4. C₂₃×#481	C₂	semidirect
5. C₂₃×#476	C₂	semidirect
6. C₂₃×#465	C₂	semidirect
7. C₂₃×#460	C₂	semidirect
8. C₂₃×#450	C₂	semidirect
9. C₄×SD₁₆×C₂₃	C₂	semidirect
10. C₈×D₄×C₂₃	C₂	semidirect
11. C₂×C₂₃×#102	C₂	semidirect
15. C₂×SD₁₆×C₂₃	C₂²	semidirect
16. C₂²×C₈×C₂₃	C₂²	semidirect
17. C₂₃×C₄⋊Q₈	C₂²	semidirect
21. C₂₃×C₂²⋊Q₈	C₂²	semidirect
25. C₄×Q₈×C₂₃	C₂²	semidirect
29. C₂×C₄⋊C₄×C₂₃	C₂²	semidirect
30. C₂₃×#106	C₂²	semidirect
31. C₂₃×#106	C₂²	semidirect
32. C₂₃×C₄⋊C₈	C₂²	semidirect
33. C₂₃×#103	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
93. D₄×C₂₃	SD₁₆	semidirect
110. #3295	C₂₃	direct (N × H)
131. #554	C₂×C₂₃	semidirect
133. #486	C₂×C₂₃	semidirect
134. #481	C₂×C₂₃	semidirect
135. #476	C₂×C₂₃	semidirect
136. #465	C₂×C₂₃	semidirect
137. #460	C₂×C₂₃	semidirect
138. #450	C₂×C₂₃	semidirect
139. C₄×SD₁₆	C₂×C₂₃	semidirect
140. C₈×D₄	C₂×C₂₃	semidirect
141. C₂×#102	C₂×C₂₃	semidirect
151. C₂×SD₁₆	C₂²×C₂₃	semidirect
152. C₂²×C₈	C₂²×C₂₃	semidirect
153. C₄⋊Q₈	C₂²×C₂₃	semidirect
157. C₂²⋊Q₈	C₂²×C₂₃	semidirect
161. C₄×Q₈	C₂²×C₂₃	semidirect
165. C₂×C₄⋊C₄	C₂²×C₂₃	semidirect
166. #106	C₂²×C₂₃	semidirect
167. #106	C₂²×C₂₃	semidirect
168. C₄⋊C₈	C₂²×C₂₃	semidirect
169. #103	C₂²×C₂₃	semidirect
172. #102	C₂²×C₂₃	semidirect
173. #102	C₂²×C₂₃	semidirect
174. #102	C₂²×C₂₃	semidirect
175. C₂²⋊C₈	C₂²×C₂₃	semidirect
176. C₄×C₈	C₂²×C₂₃	semidirect
177. C₂₃	#3295	direct (N × H)
230. D₄	SD₁₆×C₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂₃×#554	2
D₄²×C₂₃	2
C₂₃×#486	2
C₂₃×#481	2
C₂₃×#476	2
C₂₃×#465	2
C₂₃×#460	2
C₂₃×#450	2
C₄×SD₁₆×C₂₃	2
C₈×D₄×C₂₃	2
C₂×C₂₃×#102	2
#3295	23

Subgroups — C23×#3295 #15670914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3295 #15670914