Subgroups — C₂₃×#3324 #15670943

Order 2944.

448 conjugacy classes of subgroups.

The subgroup lattice has 272 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	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×#482	C₂	1	2	yes	maximal, complemented
5	C₂₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#468	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	C₂₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#418	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	C₂×M₁₆×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×#126	—	2	2	no	—
20	C₂₃×#124	—	2	1	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	—	2	1	no	—
26	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
27	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
29	C₂₃×#117	C₂²	1	2	yes	complemented
30	C₂₃×#107	C₂²	1	2	yes	complemented
31	C₂₃×#106	C₂²	1	1	yes	characteristic, 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	1	no	—
55	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
56	SD₁₆×C₂₃	—	2	2	no	—
57	SD₁₆×C₂₃	—	4	2	no	—
58	M₁₆×C₂₃	—	4	2	no	—
59	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₂₃	C₂³	1	2	yes	—
61	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₄⋊C₄×C₂₃	—	2	2	no	—
63	C₄⋊C₄×C₂₃	—	2	1	no	—
64	C₄⋊C₄×C₂₃	—	2	1	no	—
65	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
66	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
68	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²⋊C₄×C₂₃	—	2	2	no	—
70	C₂²⋊C₄×C₂₃	—	2	2	no	—
71	C₂²⋊C₄×C₂₃	—	2	1	no	—
72	C₂²⋊C₄×C₂₃	—	2	2	no	—
73	C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
74	C₄²×C₂₃	—	2	2	no	—
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	—	4	2	no	—
86	D₄×C₂₃	—	2	2	no	—
87	D₄×C₂₃	—	2	2	no	—
88	D₄×C₂₃	—	2	1	no	—
89	D₄×C₂₃	—	2	2	no	—
90	D₄×C₂₃	—	4	4	no	—
91	D₄×C₂₃	—	4	4	no	—
92	D₄×C₂₃	—	4	2	no	—
93	D₄×C₂₃	—	8	1	no	—
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	2	no	—
100	C₂×C₄×C₂₃	—	2	1	no	—
101	C₂×C₄×C₂₃	—	4	2	no	—
102	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
106	C₈×C₂₃	—	2	1	no	—
107	C₈×C₂₃	—	2	1	no	—
108	C₈×C₂₃	—	2	2	no	—
109	#3324	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₂₃	—	2	2	no	—
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	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₂₃	—	2	2	no	—
122	C₄×C₂₃	—	2	1	no	—
123	C₄×C₂₃	—	4	1	no	—
124	C₄×C₂₃	—	4	1	no	—
125	C₄×C₂₃	—	4	2	no	—
126	C₄×C₂₃	—	4	1	no	—
127	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
128	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
129	C₄×C₂₃	H(5,𝔽₂)	1	1	yes	characteristic
130	#556	C₂×C₂₃	1	1	yes	characteristic, complemented
131	D₄²	C₂×C₂₃	1	1	yes	characteristic
132	#486	C₂×C₂₃	1	1	yes	characteristic, complemented
133	#482	C₂×C₂₃	1	2	yes	complemented
134	#474	C₂×C₂₃	1	1	yes	characteristic, complemented
135	#468	C₂×C₂₃	1	2	yes	complemented
136	#460	C₂×C₂₃	1	1	yes	characteristic, complemented
137	#450	C₂×C₂₃	1	2	yes	complemented
138	C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
139	#436	C₂×C₂₃	1	1	yes	characteristic, complemented
140	#418	C₂×C₂₃	1	2	yes	complemented
141	C₂×C₂₃	—	4	2	no	—
142	C₂×C₂₃	—	4	2	no	—
143	C₂×C₂₃	—	8	2	no	—
144	C₂×C₂₃	#546	1	1	yes	characteristic
145	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₂₃	#577	1	1	yes	characteristic
147	C₂²×D₄	—	2	1	no	—
148	C₂²×D₄	C₂²×C₂₃	1	2	yes	—
149	C₂×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
150	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
151	C₂×M₁₆	C₂²×C₂₃	1	2	yes	complemented
152	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
153	#127	C₂²×C₂₃	1	1	yes	characteristic
154	#126	—	2	2	no	—
155	#124	—	2	1	no	—
156	C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	complemented
157	#121	—	2	1	no	—
158	#121	C₂²×C₂₃	1	2	yes	—
159	#120	—	2	2	no	—
160	C₄×Q₈	—	2	1	no	—
161	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
162	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
163	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
164	#117	C₂²×C₂₃	1	2	yes	complemented
165	#107	C₂²×C₂₃	1	2	yes	complemented
166	#106	C₂²×C₂₃	1	1	yes	characteristic, complemented
167	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
168	#103	C₂²×C₂₃	1	1	yes	characteristic, complemented
169	#102	C₂²×C₂₃	1	1	yes	characteristic
170	#102	C₂²×C₂₃	1	2	yes	—
171	#102	C₂²×C₂₃	1	2	yes	complemented
172	#102	C₂²×C₂₃	1	2	yes	complemented
173	#102	C₂²×C₂₃	1	2	yes	complemented
174	C₂²⋊C₈	C₂²×C₂₃	1	2	yes	complemented
175	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
176	C₂₃	#3324	1	1	yes	characteristic, direct product factor, Sylow
177	C₂⁴	—	2	2	no	—
178	C₂×Q₈	—	2	1	no	—
179	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
180	C₂×D₄	—	2	4	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	—	2	2	no	—
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂×D₄	—	4	4	no	—
187	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
188	C₂×D₄	C₂³×C₂₃	1	2	yes	—
189	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
192	SD₁₆	—	2	2	no	—
193	SD₁₆	—	4	2	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
196	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
197	C₂×C₈	C₂³×C₂₃	1	2	yes	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
203	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
204	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	2	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
210	C₄²	—	2	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	C₂³×C₂₃	1	1	yes	characteristic
213	C₂³	—	2	1	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	4	4	no	—
216	C₂³	—	4	4	no	—
217	C₂³	—	8	2	no	—
218	C₂³	C₂×D₄×C₂₃	1	2	yes	—
219	Q₈	—	2	1	no	—
220	Q₈	—	2	1	no	—
221	Q₈	—	4	2	no	—
222	D₄	—	2	2	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	#41×C₂₃	1	2	yes	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
239	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
240	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
241	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	1	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	8	8	no	—
255	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	4	2	no	—
260	C₄	—	4	1	no	—
261	C₄	—	4	1	no	—
262	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
263	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
264	C₄	C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
265	C₂	—	4	2	no	—
266	C₂	—	4	2	no	—
267	C₂	—	8	2	no	—
268	C₂	C₂₃×#546	1	1	yes	characteristic
269	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
270	C₂	C₂₃×#577	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×#556	C₂	semidirect
3. C₂₃×#486	C₂	semidirect
4. C₂₃×#482	C₂	semidirect
5. C₂₃×#474	C₂	semidirect
6. C₂₃×#468	C₂	semidirect
7. C₂₃×#460	C₂	semidirect
8. C₂₃×#450	C₂	semidirect
9. C₄×SD₁₆×C₂₃	C₂	semidirect
10. C₂₃×#436	C₂	semidirect
11. C₂₃×#418	C₂	semidirect
15. C₂×SD₁₆×C₂₃	C₂²	semidirect
16. C₂×M₁₆×C₂₃	C₂²	semidirect
17. C₂₃×C₄⋊Q₈	C₂²	semidirect
21. C₂₃×C₂²⋊Q₈	C₂²	semidirect
26. C₄×Q₈×C₂₃	C₂²	semidirect
29. C₂₃×#117	C₂²	semidirect
30. C₂₃×#107	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
109. #3324	C₂₃	direct (N × H)
130. #556	C₂×C₂₃	semidirect
132. #486	C₂×C₂₃	semidirect
133. #482	C₂×C₂₃	semidirect
134. #474	C₂×C₂₃	semidirect
135. #468	C₂×C₂₃	semidirect
136. #460	C₂×C₂₃	semidirect
137. #450	C₂×C₂₃	semidirect
138. C₄×SD₁₆	C₂×C₂₃	semidirect
139. #436	C₂×C₂₃	semidirect
140. #418	C₂×C₂₃	semidirect
150. C₂×SD₁₆	C₂²×C₂₃	semidirect
151. C₂×M₁₆	C₂²×C₂₃	semidirect
152. C₄⋊Q₈	C₂²×C₂₃	semidirect
156. C₂²⋊Q₈	C₂²×C₂₃	semidirect
161. C₄×Q₈	C₂²×C₂₃	semidirect
164. #117	C₂²×C₂₃	semidirect
165. #107	C₂²×C₂₃	semidirect
166. #106	C₂²×C₂₃	semidirect
167. C₄⋊C₈	C₂²×C₂₃	semidirect
168. #103	C₂²×C₂₃	semidirect
171. #102	C₂²×C₂₃	semidirect
172. #102	C₂²×C₂₃	semidirect
173. #102	C₂²×C₂₃	semidirect
174. C₂²⋊C₈	C₂²×C₂₃	semidirect
175. C₄×C₈	C₂²×C₂₃	semidirect
176. C₂₃	#3324	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
D₄²×C₂₃	2
C₂₃×#486	2
C₂₃×#482	2
C₂₃×#474	2
C₂₃×#468	2
C₂₃×#460	2
C₂₃×#450	2
C₄×SD₁₆×C₂₃	2
C₂₃×#436	2
C₂₃×#418	2
#3324	23

Subgroups — C23×#3324 #15670943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3324 #15670943