Subgroups — C₂₃×#2946 #15670565

Order 2944.

506 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×AGL(1,ℤ/8)	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#442	C₂	1	2	yes	maximal, complemented
5	C₂₃×#440	C₂	1	2	yes	maximal, complemented
6	C₄×SD₁₆×C₂₃	C₂	1	2	yes	maximal, complemented
7	C₄×D₈×C₂₃	C₂	1	2	yes	maximal, complemented
8	C₂₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#416	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#102	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
14	C₂₃×AGL(1,ℤ/8)	C₄	1	8	yes	complemented
15	C₂×SD₁₆×C₂₃	C₂²	1	2	yes	—
16	C₂×D₈×C₂₃	C₂²	1	2	yes	—
17	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
20	C₄×D₄×C₂₃	—	2	1	no	—
21	C₄×D₄×C₂₃	—	2	1	no	—
22	C₄×D₄×C₂₃	—	2	2	no	—
23	C₄×D₄×C₂₃	C₂²	1	2	yes	—
24	C₄×D₄×C₂₃	C₂²	1	2	yes	—
25	C₄×D₄×C₂₃	C₂²	1	2	yes	complemented
26	C₂₃×#117	—	2	1	no	—
27	C₂₃×#117	C₂²	1	1	yes	characteristic
28	C₂₃×#117	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₂₃	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
32	C₂₃×#107	C₂²	1	2	yes	complemented
33	C₂₃×#106	C₂²	1	2	yes	complemented
34	C₂₃×#103	C₂²	1	1	yes	characteristic, complemented
35	C₂₃×#103	C₂²	1	1	yes	characteristic, complemented
36	C₂₃×#102	C₂²	1	2	yes	—
37	C₂₃×#102	C₂²	1	2	yes	—
38	C₂₃×#102	C₂²	1	1	yes	characteristic, complemented
39	C₂₃×#102	C₂²	1	1	yes	characteristic, complemented
40	C₂₃×#97	C₂²	1	2	yes	complemented
41	C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
42	C₂⁴×C₂₃	—	2	1	no	—
43	#41×C₂₃	—	2	2	no	—
44	#41×C₂₃	C₂×C₄	1	4	yes	complemented
45	C₂×Q₈×C₂₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	—	2	1	no	—
48	C₂×D₄×C₂₃	—	2	2	no	—
49	C₂×D₄×C₂₃	C₂×C₄	1	4	yes	complemented
50	C₂×D₄×C₂₃	C₂³	1	2	yes	—
51	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	—	2	1	no	—
53	C₂²×C₄×C₂₃	—	2	1	no	—
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	C₂²×C₄×C₂₃	—	2	2	no	—
56	C₂²×C₄×C₂₃	—	2	2	no	—
57	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
58	SD₁₆×C₂₃	C₂×C₄	1	8	yes	complemented
59	D₈×C₂₃	C₂×C₄	1	8	yes	complemented
60	M₁₆×C₂₃	C₂×C₄	1	4	yes	complemented
61	C₂×C₈×C₂₃	—	2	2	no	—
62	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
63	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
64	C₂×C₈×C₂₃	C₂³	1	2	yes	—
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	D₄	1	1	yes	characteristic
69	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
70	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₂₃	—	2	1	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	D₄	1	2	yes	—
77	C₄²×C₂₃	—	2	2	no	—
78	C₄²×C₂₃	—	2	1	no	—
79	C₄²×C₂₃	C₂³	1	2	yes	—
80	C₂³×C₂₃	—	2	1	no	—
81	C₂³×C₂₃	—	2	2	no	—
82	C₂³×C₂₃	—	2	4	no	—
83	C₂³×C₂₃	—	8	1	no	—
84	C₂³×C₂₃	—	8	1	no	—
85	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₂₃	—	2	1	no	—
87	Q₈×C₂₃	C₂²×C₄	1	2	yes	—
88	D₄×C₂₃	—	2	4	no	—
89	D₄×C₂₃	—	2	4	no	—
90	D₄×C₂₃	—	2	2	no	—
91	D₄×C₂₃	—	2	1	no	—
92	D₄×C₂₃	C₂²×C₄	1	2	yes	—
93	D₄×C₂₃	C₂²×C₄	1	4	yes	—
94	C₂×C₄×C₂₃	—	2	1	no	—
95	C₂×C₄×C₂₃	—	2	4	no	—
96	C₂×C₄×C₂₃	—	2	1	no	—
97	C₂×C₄×C₂₃	—	2	1	no	—
98	C₂×C₄×C₂₃	—	2	2	no	—
99	C₂×C₄×C₂₃	—	8	2	no	—
100	C₂×C₄×C₂₃	—	8	2	no	—
101	C₂×C₄×C₂₃	—	8	2	no	—
102	C₂×C₄×C₂₃	—	8	1	no	—
103	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₃	#41	1	2	yes	—
108	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
109	C₈×C₂₃	—	2	1	no	—
110	C₈×C₂₃	—	2	1	no	—
111	C₈×C₂₃	C₂²×C₄	1	4	yes	—
112	#2946	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₂₃	—	2	1	no	—
114	C₂²×C₂₃	—	2	1	no	—
115	C₂²×C₂₃	—	2	2	no	—
116	C₂²×C₂₃	—	2	4	no	—
117	C₂²×C₂₃	—	4	1	no	—
118	C₂²×C₂₃	—	4	2	no	—
119	C₂²×C₂₃	—	4	2	no	—
120	C₂²×C₂₃	—	4	1	no	—
121	C₂²×C₂₃	—	8	4	no	—
122	C₂²×C₂₃	C₄×D₄	1	2	yes	—
123	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
124	C₄×C₂₃	—	2	2	no	—
125	C₄×C₂₃	—	2	2	no	—
126	C₄×C₂₃	—	2	2	no	—
127	C₄×C₂₃	—	4	1	no	—
128	C₄×C₂₃	—	4	2	no	—
129	C₄×C₂₃	—	4	1	no	—
130	C₄×C₂₃	C₄×D₄	1	2	yes	—
131	C₄×C₂₃	C₂³×C₄	1	1	yes	characteristic
132	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
133	C₂×AGL(1,ℤ/8)	C₂×C₂₃	1	1	yes	characteristic
134	#520	C₂×C₂₃	1	1	yes	characteristic, complemented
135	#518	C₂×C₂₃	1	1	yes	characteristic, complemented
136	#442	C₂×C₂₃	1	2	yes	complemented
137	#440	C₂×C₂₃	1	2	yes	complemented
138	C₄×SD₁₆	C₂×C₂₃	1	2	yes	complemented
139	C₄×D₈	C₂×C₂₃	1	2	yes	complemented
140	#428	C₂×C₂₃	1	1	yes	characteristic, complemented
141	#416	C₂×C₂₃	1	1	yes	characteristic, complemented
142	C₂×#102	C₂×C₂₃	1	1	yes	characteristic, complemented
143	#405	C₂×C₂₃	1	1	yes	characteristic, complemented
144	C₂×C₂₃	—	2	2	no	—
145	C₂×C₂₃	—	4	2	no	—
146	C₂×C₂₃	—	4	4	no	—
147	C₂×C₂₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂×C₂₃	#576	1	1	yes	characteristic
149	C₂×C₂₃	#577	1	1	yes	characteristic
150	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
151	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
152	AGL(1,ℤ/8)	C₄×C₂₃	1	8	yes	complemented
153	C₂×SD₁₆	C₂²×C₂₃	1	2	yes	—
154	C₂×D₈	C₂²×C₂₃	1	2	yes	—
155	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
156	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
157	C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	2	no	—
161	C₄×D₄	C₂²×C₂₃	1	2	yes	—
162	C₄×D₄	C₂²×C₂₃	1	2	yes	—
163	C₄×D₄	C₂²×C₂₃	1	2	yes	complemented
164	#117	—	2	1	no	—
165	#117	C₂²×C₂₃	1	1	yes	characteristic
166	#117	C₂²×C₂₃	1	1	yes	characteristic
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	C₂²×C₂₃	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	—	2	2	no	—
170	#107	C₂²×C₂₃	1	2	yes	complemented
171	#106	C₂²×C₂₃	1	2	yes	complemented
172	#103	C₂²×C₂₃	1	1	yes	characteristic, complemented
173	#103	C₂²×C₂₃	1	1	yes	characteristic, complemented
174	#102	C₂²×C₂₃	1	2	yes	—
175	#102	C₂²×C₂₃	1	2	yes	—
176	#102	C₂²×C₂₃	1	1	yes	characteristic, complemented
177	#102	C₂²×C₂₃	1	1	yes	characteristic, complemented
178	#97	C₂²×C₂₃	1	2	yes	complemented
179	C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
180	C₂₃	#2946	1	1	yes	characteristic, direct product factor, Sylow
181	C₂⁴	—	2	1	no	—
182	#41	—	2	2	no	—
183	#41	C₂×C₄×C₂₃	1	4	yes	complemented
184	C₂×Q₈	C₂³×C₂₃	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	C₂×C₄×C₂₃	1	4	yes	complemented
189	C₂×D₄	C₂³×C₂₃	1	2	yes	—
190	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
197	SD₁₆	C₂×C₄×C₂₃	1	8	yes	complemented
198	D₈	C₂×C₄×C₂₃	1	8	yes	complemented
199	M₁₆	C₂×C₄×C₂₃	1	4	yes	complemented
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
202	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
203	C₂×C₈	C₂³×C₂₃	1	2	yes	—
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
207	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
208	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
209	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
210	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
216	C₄²	—	2	2	no	—
217	C₄²	—	2	1	no	—
218	C₄²	C₂³×C₂₃	1	2	yes	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	8	1	no	—
224	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂²×C₄×C₂₃	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	2	no	—
230	D₄	—	2	1	no	—
231	D₄	C₂²×C₄×C₂₃	1	2	yes	—
232	D₄	C₂²×C₄×C₂₃	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	C₂²×C₄×C₂₃	1	2	yes	—
243	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
244	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
245	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
246	C₂×C₄	#41×C₂₃	1	2	yes	—
247	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	C₂²×C₄×C₂₃	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	8	4	no	—
260	C₂²	C₄×D₄×C₂₃	1	2	yes	—
261	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	—	4	1	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	1	no	—
268	C₄	C₄×D₄×C₂₃	1	2	yes	—
269	C₄	C₂³×C₄×C₂₃	1	1	yes	characteristic
270	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	4	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂×C₄×D₄×C₂₃	1	1	yes	characteristic
275	C₂	C₂₃×#576	1	1	yes	characteristic
276	C₂	C₂₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator	C₄	C₂³×C₄×C₂₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
prime radical	C₂×C₂₃×AGL(1,ℤ/8)	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂₃×#520	C₂	semidirect
3. C₂₃×#518	C₂	semidirect
4. C₂₃×#442	C₂	semidirect
5. C₂₃×#440	C₂	semidirect
6. C₄×SD₁₆×C₂₃	C₂	semidirect
7. C₄×D₈×C₂₃	C₂	semidirect
8. C₂₃×#428	C₂	semidirect
9. C₂₃×#416	C₂	semidirect
10. C₂×C₂₃×#102	C₂	semidirect
11. C₂₃×#405	C₂	semidirect
14. C₂₃×AGL(1,ℤ/8)	C₄	semidirect
19. C₄×Q₈×C₂₃	C₂²	semidirect
25. C₄×D₄×C₂₃	C₂²	semidirect
32. C₂₃×#107	C₂²	semidirect
33. C₂₃×#106	C₂²	semidirect
34. C₂₃×#103	C₂²	semidirect
35. C₂₃×#103	C₂²	semidirect
38. C₂₃×#102	C₂²	semidirect
39. C₂₃×#102	C₂²	semidirect
40. C₂₃×#97	C₂²	semidirect
41. C₄×C₈×C₂₃	C₂²	semidirect
44. #41×C₂₃	C₂×C₄	semidirect
49. C₂×D₄×C₂₃	C₂×C₄	semidirect
58. SD₁₆×C₂₃	C₂×C₄	semidirect
59. D₈×C₂₃	C₂×C₄	semidirect
60. M₁₆×C₂₃	C₂×C₄	semidirect
112. #2946	C₂₃	direct (N × H)
134. #520	C₂×C₂₃	semidirect
135. #518	C₂×C₂₃	semidirect
136. #442	C₂×C₂₃	semidirect
137. #440	C₂×C₂₃	semidirect
138. C₄×SD₁₆	C₂×C₂₃	semidirect
139. C₄×D₈	C₂×C₂₃	semidirect
140. #428	C₂×C₂₃	semidirect
141. #416	C₂×C₂₃	semidirect
142. C₂×#102	C₂×C₂₃	semidirect
143. #405	C₂×C₂₃	semidirect
152. AGL(1,ℤ/8)	C₄×C₂₃	semidirect
157. C₄×Q₈	C₂²×C₂₃	semidirect
163. C₄×D₄	C₂²×C₂₃	semidirect
170. #107	C₂²×C₂₃	semidirect
171. #106	C₂²×C₂₃	semidirect
172. #103	C₂²×C₂₃	semidirect
173. #103	C₂²×C₂₃	semidirect
176. #102	C₂²×C₂₃	semidirect
177. #102	C₂²×C₂₃	semidirect
178. #97	C₂²×C₂₃	semidirect
179. C₄×C₈	C₂²×C₂₃	semidirect
180. C₂₃	#2946	direct (N × H)
183. #41	C₂×C₄×C₂₃	semidirect
188. C₂×D₄	C₂×C₄×C₂₃	semidirect
197. SD₁₆	C₂×C₄×C₂₃	semidirect
198. D₈	C₂×C₄×C₂₃	semidirect
199. M₁₆	C₂×C₄×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×AGL(1,ℤ/8)	2
C₂₃×#520	2
C₂₃×#518	2
C₂₃×#442	2
C₂₃×#440	2
C₄×SD₁₆×C₂₃	2
C₄×D₈×C₂₃	2
C₂₃×#428	2
C₂₃×#416	2
C₂×C₂₃×#102	2
C₂₃×#405	2
#2946	23

Subgroups — C23×#2946 #15670565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2946 #15670565