Subgroups — C₂₃×#3068 #15670687

Order 2944.

466 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂₃×#575	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#472	C₂	1	4	yes	maximal, complemented
6	C₂₃×#471	C₂	1	4	yes	maximal, complemented
7	C₂₃×#430	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂₃×#413	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#100	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
11	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂₃×#137	—	2	1	no	—
14	C₂₃×#137	C₂²	1	2	yes	complemented
15	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
16	C₂₃×AGL(1,ℤ/8)	—	2	1	no	—
17	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
18	C₂₃×AGL(1,ℤ/8)	—	2	2	no	—
19	C₂₃×AGL(1,ℤ/8)	C₂²	1	4	yes	—
20	C₂₃×AGL(1,ℤ/8)	C₂²	1	2	yes	complemented
21	C₂₃×#135	—	2	2	no	—
22	C₂×SD₁₆×C₂₃	—	2	1	no	—
23	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
24	C₂×D₈×C₂₃	—	2	1	no	—
25	C₂×D₈×C₂₃	C₂²	1	2	yes	complemented
26	C₂₃×#131	—	2	2	no	—
27	C₂₃×#131	C₂²	1	4	yes	complemented
28	C₂×M₁₆×C₂₃	—	2	2	no	—
29	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
30	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
32	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#108	C₂²	1	4	yes	complemented
34	C₂₃×#101	C₂²	1	2	yes	complemented
35	C₂₃×#100	C₂²	1	4	yes	complemented
36	C₂₃×#100	C₂²	1	2	yes	complemented
37	C₂⁴×C₂₃	—	2	1	no	—
38	#41×C₂₃	—	2	1	no	—
39	#41×C₂₃	—	2	1	no	—
40	#41×C₂₃	—	2	1	no	—
41	#41×C₂₃	—	2	1	no	—
42	#41×C₂₃	—	2	1	no	—
43	#41×C₂₃	—	2	1	no	—
44	#41×C₂₃	D₄	1	2	yes	—
45	#41×C₂₃	C₂³	1	2	yes	—
46	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
47	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₂₃	—	2	1	no	—
49	C₂×D₄×C₂₃	—	2	1	no	—
50	C₂×D₄×C₂₃	—	2	1	no	—
51	C₂×D₄×C₂₃	—	2	1	no	—
52	C₂×D₄×C₂₃	—	2	1	no	—
53	C₂×D₄×C₂₃	—	2	1	no	—
54	C₂×D₄×C₂₃	—	2	1	no	—
55	C₂×D₄×C₂₃	D₄	1	1	yes	characteristic
56	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
57	C₂×D₄×C₂₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₂₃	—	2	1	no	—
59	C₂²×C₄×C₂₃	—	2	1	no	—
60	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	Q₁₆×C₂₃	—	2	2	no	—
62	SD₁₆×C₂₃	—	2	4	no	—
63	SD₁₆×C₂₃	—	2	2	no	—
64	SD₁₆×C₂₃	—	2	2	no	—
65	SD₁₆×C₂₃	—	2	4	no	—
66	D₈×C₂₃	—	2	4	no	—
67	D₈×C₂₃	—	2	2	no	—
68	D₈×C₂₃	—	2	4	no	—
69	M₁₆×C₂₃	—	2	1	no	—
70	M₁₆×C₂₃	—	2	1	no	—
71	M₁₆×C₂₃	—	2	1	no	—
72	M₁₆×C₂₃	—	2	4	no	—
73	M₁₆×C₂₃	D₄	1	2	yes	—
74	M₁₆×C₂₃	D₄	1	2	yes	—
75	M₁₆×C₂₃	C₂³	1	2	yes	—
76	M₁₆×C₂₃	C₂³	1	2	yes	—
77	M₁₆×C₂₃	C₂³	1	2	yes	—
78	C₂×C₈×C₂₃	—	2	1	no	—
79	C₂×C₈×C₂₃	—	2	1	no	—
80	C₂×C₈×C₂₃	—	2	1	no	—
81	C₂×C₈×C₂₃	—	2	4	no	—
82	C₂×C₈×C₂₃	C₂³	1	2	yes	—
83	C₂³×C₂₃	—	2	1	no	—
84	C₂³×C₂₃	—	2	1	no	—
85	C₂³×C₂₃	—	2	2	no	—
86	C₂³×C₂₃	—	4	1	no	—
87	C₂³×C₂₃	—	4	1	no	—
88	C₂³×C₂₃	—	8	2	no	—
89	C₂³×C₂₃	#41	1	1	yes	characteristic
90	Q₈×C₂₃	—	2	1	no	—
91	Q₈×C₂₃	—	2	1	no	—
92	Q₈×C₂₃	—	2	1	no	—
93	Q₈×C₂₃	C₂×D₄	1	2	yes	—
94	D₄×C₂₃	—	2	2	no	—
95	D₄×C₂₃	—	2	2	no	—
96	D₄×C₂₃	—	2	1	no	—
97	D₄×C₂₃	—	2	1	no	—
98	D₄×C₂₃	—	2	2	no	—
99	D₄×C₂₃	—	2	2	no	—
100	D₄×C₂₃	—	2	1	no	—
101	D₄×C₂₃	—	4	1	no	—
102	D₄×C₂₃	—	4	1	no	—
103	D₄×C₂₃	—	4	1	no	—
104	D₄×C₂₃	—	4	1	no	—
105	D₄×C₂₃	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₂₃	—	2	1	no	—
107	C₂×C₄×C₂₃	—	2	2	no	—
108	C₂×C₄×C₂₃	—	2	2	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	4	1	no	—
111	C₂×C₄×C₂₃	—	4	1	no	—
112	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
117	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
118	C₈×C₂₃	—	2	2	no	—
119	C₈×C₂₃	—	2	2	no	—
120	C₈×C₂₃	—	2	2	no	—
121	C₈×C₂₃	C₂×D₄	1	4	yes	—
122	#3068	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₂₃	—	2	2	no	—
124	C₂²×C₂₃	—	4	1	no	—
125	C₂²×C₂₃	—	4	1	no	—
126	C₂²×C₂₃	—	4	2	no	—
127	C₂²×C₂₃	—	8	1	no	—
128	C₂²×C₂₃	—	8	1	no	—
129	C₂²×C₂₃	—	8	2	no	—
130	C₂²×C₂₃	—	8	2	no	—
131	C₂²×C₂₃	—	8	2	no	—
132	C₂²×C₂₃	#121	1	1	yes	characteristic
133	C₂²×C₂₃	#121	1	1	yes	characteristic
134	C₂²×C₂₃	C₂×#41	1	1	yes	characteristic
135	C₄×C₂₃	—	2	2	no	—
136	C₄×C₂₃	—	4	1	no	—
137	C₄×C₂₃	#121	1	1	yes	characteristic
138	C₄×C₂₃	#121	1	1	yes	characteristic
139	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
141	#575	C₂×C₂₃	1	1	yes	characteristic, complemented
142	C₂×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic, complemented
143	C₂×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic, complemented
144	#568	C₂×C₂₃	1	1	yes	characteristic, complemented
145	#472	C₂×C₂₃	1	4	yes	complemented
146	#471	C₂×C₂₃	1	4	yes	complemented
147	#430	C₂×C₂₃	1	1	yes	characteristic, complemented
148	#413	C₂×C₂₃	1	1	yes	characteristic, complemented
149	C₂×#100	C₂×C₂₃	1	1	yes	characteristic, complemented
150	C₂×C₂₃	—	2	1	no	—
151	C₂×C₂₃	—	2	1	no	—
152	C₂×C₂₃	—	2	1	no	—
153	C₂×C₂₃	—	4	2	no	—
154	C₂×C₂₃	—	8	1	no	—
155	C₂×C₂₃	—	8	2	no	—
156	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
157	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
158	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
159	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
160	#137	—	2	1	no	—
161	#137	C₂²×C₂₃	1	2	yes	complemented
162	AGL(1,ℤ/8)	—	2	2	no	—
163	AGL(1,ℤ/8)	—	2	1	no	—
164	AGL(1,ℤ/8)	—	2	2	no	—
165	AGL(1,ℤ/8)	—	2	2	no	—
166	AGL(1,ℤ/8)	C₂²×C₂₃	1	4	yes	—
167	AGL(1,ℤ/8)	C₂²×C₂₃	1	2	yes	complemented
168	#135	—	2	2	no	—
169	C₂×SD₁₆	—	2	1	no	—
170	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
171	C₂×D₈	—	2	1	no	—
172	C₂×D₈	C₂²×C₂₃	1	2	yes	complemented
173	#131	—	2	2	no	—
174	#131	C₂²×C₂₃	1	4	yes	complemented
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
177	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
178	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
179	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
180	#108	C₂²×C₂₃	1	4	yes	complemented
181	#101	C₂²×C₂₃	1	2	yes	complemented
182	#100	C₂²×C₂₃	1	4	yes	complemented
183	#100	C₂²×C₂₃	1	2	yes	complemented
184	C₂₃	#3068	1	1	yes	characteristic, direct product factor, Sylow
185	C₂⁴	—	2	1	no	—
186	#41	—	2	1	no	—
187	#41	—	2	1	no	—
188	#41	—	2	1	no	—
189	#41	—	2	1	no	—
190	#41	—	2	1	no	—
191	#41	—	2	1	no	—
192	#41	D₄×C₂₃	1	2	yes	—
193	#41	C₂³×C₂₃	1	2	yes	—
194	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
195	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	—	2	1	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	—	2	1	no	—
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	2	1	no	—
203	C₂×D₄	D₄×C₂₃	1	1	yes	characteristic
204	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
205	C₂×D₄	C₂³×C₂₃	1	2	yes	—
206	C₂²×C₄	—	2	1	no	—
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
209	Q₁₆	—	2	2	no	—
210	SD₁₆	—	2	4	no	—
211	SD₁₆	—	2	2	no	—
212	SD₁₆	—	2	2	no	—
213	SD₁₆	—	2	4	no	—
214	D₈	—	2	4	no	—
215	D₈	—	2	2	no	—
216	D₈	—	2	4	no	—
217	M₁₆	—	2	1	no	—
218	M₁₆	—	2	1	no	—
219	M₁₆	—	2	1	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	D₄×C₂₃	1	2	yes	—
222	M₁₆	D₄×C₂₃	1	2	yes	—
223	M₁₆	C₂³×C₂₃	1	2	yes	—
224	M₁₆	C₂³×C₂₃	1	2	yes	—
225	M₁₆	C₂³×C₂₃	1	2	yes	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	C₂³×C₂₃	1	2	yes	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	4	1	no	—
235	C₂³	—	4	1	no	—
236	C₂³	—	8	2	no	—
237	C₂³	#41×C₂₃	1	1	yes	characteristic
238	Q₈	—	2	1	no	—
239	Q₈	—	2	1	no	—
240	Q₈	—	2	1	no	—
241	Q₈	C₂×D₄×C₂₃	1	2	yes	—
242	D₄	—	2	2	no	—
243	D₄	—	2	2	no	—
244	D₄	—	2	1	no	—
245	D₄	—	2	1	no	—
246	D₄	—	2	2	no	—
247	D₄	—	2	2	no	—
248	D₄	—	2	1	no	—
249	D₄	—	4	1	no	—
250	D₄	—	4	1	no	—
251	D₄	—	4	1	no	—
252	D₄	—	4	1	no	—
253	D₄	C₂×D₄×C₂₃	1	2	yes	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
261	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
263	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
264	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
265	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
266	C₈	—	2	2	no	—
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₈	C₂×D₄×C₂₃	1	4	yes	—
270	C₂²	—	2	2	no	—
271	C₂²	—	4	1	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	8	1	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	1	no	—
277	C₂²	—	8	2	no	—
278	C₂²	—	8	2	no	—
279	C₂²	C₂₃×#121	1	1	yes	characteristic
280	C₂²	C₂₃×#121	1	1	yes	characteristic
281	C₂²	C₂×#41×C₂₃	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	4	1	no	—
284	C₄	C₂₃×#121	1	1	yes	characteristic
285	C₄	C₂₃×#121	1	1	yes	characteristic
286	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
287	C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	4	2	no	—
292	C₂	—	8	2	no	—
293	C₂	—	8	1	no	—
294	C₂	C₂×C₂₃×#121	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃	C₂×#121
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#575	C₂	semidirect
2. C₂×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
3. C₂×C₂₃×AGL(1,ℤ/8)	C₂	semidirect
4. C₂₃×#568	C₂	semidirect
5. C₂₃×#472	C₂	semidirect
6. C₂₃×#471	C₂	semidirect
7. C₂₃×#430	C₂	semidirect
8. C₂₃×#413	C₂	semidirect
9. C₂×C₂₃×#100	C₂	semidirect
14. C₂₃×#137	C₂²	semidirect
20. C₂₃×AGL(1,ℤ/8)	C₂²	semidirect
23. C₂×SD₁₆×C₂₃	C₂²	semidirect
25. C₂×D₈×C₂₃	C₂²	semidirect
27. C₂₃×#131	C₂²	semidirect
33. C₂₃×#108	C₂²	semidirect
34. C₂₃×#101	C₂²	semidirect
35. C₂₃×#100	C₂²	semidirect
36. C₂₃×#100	C₂²	semidirect
122. #3068	C₂₃	direct (N × H)
141. #575	C₂×C₂₃	semidirect
142. C₂×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
143. C₂×AGL(1,ℤ/8)	C₂×C₂₃	semidirect
144. #568	C₂×C₂₃	semidirect
145. #472	C₂×C₂₃	semidirect
146. #471	C₂×C₂₃	semidirect
147. #430	C₂×C₂₃	semidirect
148. #413	C₂×C₂₃	semidirect
149. C₂×#100	C₂×C₂₃	semidirect
161. #137	C₂²×C₂₃	semidirect
167. AGL(1,ℤ/8)	C₂²×C₂₃	semidirect
170. C₂×SD₁₆	C₂²×C₂₃	semidirect
172. C₂×D₈	C₂²×C₂₃	semidirect
174. #131	C₂²×C₂₃	semidirect
180. #108	C₂²×C₂₃	semidirect
181. #101	C₂²×C₂₃	semidirect
182. #100	C₂²×C₂₃	semidirect
183. #100	C₂²×C₂₃	semidirect
184. C₂₃	#3068	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×#575	2
C₂×C₂₃×AGL(1,ℤ/8)	2
C₂₃×#568	2
C₂₃×#472	2
C₂₃×#471	2
C₂₃×#430	2
C₂₃×#413	2
C₂×C₂₃×#100	2
#3068	23

Subgroups — C23×#3068 #15670687

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3068 #15670687