Subgroups — C₂₃×#3366 #15670985

Order 2944.

396 conjugacy classes of subgroups.

The subgroup lattice has 260 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	C₂₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#490	C₂	1	2	yes	maximal, complemented
4	C₂₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×#478	C₂	1	2	yes	maximal, complemented
6	C₂₃×#476	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂₃×#459	C₂	1	2	yes	maximal, complemented
8	C₂₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂₃×#442	C₂	1	2	yes	maximal, complemented
11	C₄×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#41×C₂₃	—	2	1	no	—
13	C₂×D₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂×D₈×C₂₃	C₂²	1	2	yes	—
15	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂₃×#127	C₂²	1	1	yes	characteristic
17	C₂₃×#127	C₂²	1	2	yes	—
18	C₂₃×#125	C₂²	1	2	yes	—
19	C₂₃×C₂²⋊Q₈	—	2	1	no	—
20	C₂₃×C₂²⋊Q₈	—	2	2	no	—
21	C₂₃×#121	—	2	1	no	—
22	C₂₃×#121	—	2	2	no	—
23	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₂₃	C₂²	1	2	yes	—
26	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₂₃	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
29	C₂₃×#107	C₂²	1	1	yes	characteristic
30	C₂₃×#106	C₂²	1	2	yes	—
31	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
33	C₂₃×#102	C₂²	1	1	yes	characteristic
34	C₂₃×#102	C₂²	1	1	yes	characteristic
35	C₂₃×#102	C₂²	1	2	yes	—
36	C₂₃×#102	C₂²	1	2	yes	—
37	C₂₃×#102	C₂²	1	2	yes	—
38	C₂₃×#102	C₂²	1	2	yes	—
39	C₂₃×#97	C₂²	1	2	yes	—
40	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
41	#41×C₂₃	—	4	4	no	—
42	C₂×Q₈×C₂₃	—	2	1	no	—
43	C₂×Q₈×C₂₃	D₄	1	1	yes	characteristic
44	C₂×D₄×C₂₃	—	2	2	no	—
45	C₂×D₄×C₂₃	—	2	1	no	—
46	C₂×D₄×C₂₃	—	2	1	no	—
47	C₂×D₄×C₂₃	—	2	2	no	—
48	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₂₃	C₂³	1	2	yes	—
50	C₂×D₄×C₂₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	—	2	1	no	—
52	C₂²×C₄×C₂₃	—	2	2	no	—
53	C₂²×C₄×C₂₃	—	2	1	no	—
54	C₂²×C₄×C₂₃	—	2	2	no	—
55	D₈×C₂₃	—	2	2	no	—
56	D₈×C₂₃	—	2	4	no	—
57	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
58	C₂×C₈×C₂₃	C₂³	1	2	yes	—
59	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₂₃	—	2	2	no	—
61	C₄⋊C₄×C₂₃	—	2	1	no	—
62	C₄⋊C₄×C₂₃	—	2	2	no	—
63	C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₂₃	D₄	1	2	yes	—
65	C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
68	C₂²⋊C₄×C₂₃	—	2	1	no	—
69	C₂²⋊C₄×C₂₃	—	2	2	no	—
70	C₂²⋊C₄×C₂₃	—	2	1	no	—
71	C₂²⋊C₄×C₂₃	—	2	2	no	—
72	C₄²×C₂₃	C₂³	1	2	yes	—
73	C₄²×C₂₃	C₂³	1	1	yes	characteristic
74	C₂³×C₂₃	—	2	1	no	—
75	C₂³×C₂₃	—	2	2	no	—
76	C₂³×C₂₃	—	2	1	no	—
77	Q₈×C₂₃	—	2	2	no	—
78	Q₈×C₂₃	—	8	1	no	—
79	D₄×C₂₃	—	2	1	no	—
80	D₄×C₂₃	—	2	2	no	—
81	D₄×C₂₃	—	2	1	no	—
82	D₄×C₂₃	—	2	1	no	—
83	D₄×C₂₃	—	2	2	no	—
84	D₄×C₂₃	—	4	4	no	—
85	D₄×C₂₃	—	4	2	no	—
86	D₄×C₂₃	—	8	2	no	—
87	D₄×C₂₃	—	8	1	no	—
88	D₄×C₂₃	#41	1	2	yes	complemented
89	C₂×C₄×C₂₃	—	2	1	no	—
90	C₂×C₄×C₂₃	—	2	1	no	—
91	C₂×C₄×C₂₃	—	2	2	no	—
92	C₂×C₄×C₂₃	—	4	4	no	—
93	C₂×C₄×C₂₃	—	4	2	no	—
94	C₂×C₄×C₂₃	—	4	2	no	—
95	C₂×C₄×C₂₃	—	8	2	no	—
96	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
101	C₈×C₂₃	—	2	2	no	—
102	C₈×C₂₃	—	2	1	no	—
103	C₈×C₂₃	—	2	1	no	—
104	#3366	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₂²×C₂₃	—	2	2	no	—
106	C₂²×C₂₃	—	4	1	no	—
107	C₂²×C₂₃	—	4	1	no	—
108	C₂²×C₂₃	—	4	2	no	—
109	C₂²×C₂₃	—	4	1	no	—
110	C₂²×C₂₃	—	4	2	no	—
111	C₂²×C₂₃	—	4	1	no	—
112	C₂²×C₂₃	—	4	1	no	—
113	C₂²×C₂₃	—	4	2	no	—
114	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
115	C₄×C₂₃	—	2	2	no	—
116	C₄×C₂₃	—	2	2	no	—
117	C₄×C₂₃	—	2	1	no	—
118	C₄×C₂₃	—	4	1	no	—
119	C₄×C₂₃	—	4	1	no	—
120	C₄×C₂₃	—	4	2	no	—
121	C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
122	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
123	C₄×C₂₃	2₋¹⁺⁴	1	1	yes	characteristic
124	#554	C₂×C₂₃	1	1	yes	characteristic, complemented
125	#550	C₂×C₂₃	1	1	yes	characteristic, complemented
126	#490	C₂×C₂₃	1	2	yes	complemented
127	#486	C₂×C₂₃	1	1	yes	characteristic, complemented
128	#478	C₂×C₂₃	1	2	yes	complemented
129	#476	C₂×C₂₃	1	1	yes	characteristic, complemented
130	#459	C₂×C₂₃	1	2	yes	complemented
131	#459	C₂×C₂₃	1	1	yes	characteristic, complemented
132	#446	C₂×C₂₃	1	1	yes	characteristic, complemented
133	#442	C₂×C₂₃	1	2	yes	complemented
134	C₄×D₈	C₂×C₂₃	1	1	yes	characteristic, complemented
135	C₂×C₂₃	—	4	2	no	—
136	C₂×C₂₃	—	8	2	no	—
137	C₂×C₂₃	—	8	1	no	—
138	C₂×C₂₃	#548	1	1	yes	characteristic
139	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×C₂₃	#576	1	1	yes	characteristic
141	C₂×#41	—	2	1	no	—
142	C₂×D₈	C₂²×C₂₃	1	1	yes	characteristic
143	C₂×D₈	C₂²×C₂₃	1	2	yes	—
144	C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
145	#127	C₂²×C₂₃	1	1	yes	characteristic
146	#127	C₂²×C₂₃	1	2	yes	—
147	#125	C₂²×C₂₃	1	2	yes	—
148	C₂²⋊Q₈	—	2	1	no	—
149	C₂²⋊Q₈	—	2	2	no	—
150	#121	—	2	1	no	—
151	#121	—	2	2	no	—
152	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
153	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
154	C₄×D₄	C₂²×C₂₃	1	2	yes	—
155	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
156	C₄×D₄	C₂²×C₂₃	1	2	yes	—
157	C₂×C₄⋊C₄	—	2	1	no	—
158	#107	C₂²×C₂₃	1	1	yes	characteristic
159	#106	C₂²×C₂₃	1	2	yes	—
160	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
161	C₄⋊C₈	C₂²×C₂₃	1	2	yes	—
162	#102	C₂²×C₂₃	1	1	yes	characteristic
163	#102	C₂²×C₂₃	1	1	yes	characteristic
164	#102	C₂²×C₂₃	1	2	yes	—
165	#102	C₂²×C₂₃	1	2	yes	—
166	#102	C₂²×C₂₃	1	2	yes	—
167	#102	C₂²×C₂₃	1	2	yes	—
168	#97	C₂²×C₂₃	1	2	yes	—
169	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
170	C₂₃	#3366	1	1	yes	characteristic, direct product factor, Sylow
171	#41	—	4	4	no	—
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	D₄×C₂₃	1	1	yes	characteristic
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	1	no	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
179	C₂×D₄	C₂³×C₂₃	1	2	yes	—
180	C₂×D₄	C₂³×C₂₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	D₈	—	2	2	no	—
186	D₈	—	2	4	no	—
187	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
188	C₂×C₈	C₂³×C₂₃	1	1	yes	characteristic
189	C₂×C₈	C₂³×C₂₃	1	2	yes	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
194	C₄⋊C₄	D₄×C₂₃	1	2	yes	—
195	C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
196	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
197	C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
198	C₂²⋊C₄	—	2	1	no	—
199	C₂²⋊C₄	—	2	2	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₄²	C₂³×C₂₃	1	1	yes	characteristic
203	C₄²	C₂³×C₂₃	1	2	yes	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	1	no	—
207	Q₈	—	2	2	no	—
208	Q₈	—	8	1	no	—
209	D₄	—	2	1	no	—
210	D₄	—	2	2	no	—
211	D₄	—	2	1	no	—
212	D₄	—	2	1	no	—
213	D₄	—	2	2	no	—
214	D₄	—	4	4	no	—
215	D₄	—	4	2	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	1	no	—
218	D₄	#41×C₂₃	1	2	yes	complemented
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
227	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
228	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
229	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
230	C₂×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₈	—	2	1	no	—
234	C₂²	—	2	2	no	—
235	C₂²	—	4	1	no	—
236	C₂²	—	4	1	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	4	1	no	—
241	C₂²	—	4	1	no	—
242	C₂²	—	4	2	no	—
243	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	1	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	2	no	—
250	C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
251	C₄	C₂×#41×C₂₃	1	1	yes	characteristic
252	C₄	C₂₃×2₋¹⁺⁴	1	1	yes	characteristic
253	C₂	—	4	2	no	—
254	C₂	—	8	2	no	—
255	C₂	—	8	1	no	—
256	C₂	C₂₃×#548	1	1	yes	characteristic
257	C₂	C₂×C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₂	C₂₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄×Q₈×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂₃×#554	C₂	semidirect
2. C₂₃×#550	C₂	semidirect
3. C₂₃×#490	C₂	semidirect
4. C₂₃×#486	C₂	semidirect
5. C₂₃×#478	C₂	semidirect
6. C₂₃×#476	C₂	semidirect
7. C₂₃×#459	C₂	semidirect
8. C₂₃×#459	C₂	semidirect
9. C₂₃×#446	C₂	semidirect
10. C₂₃×#442	C₂	semidirect
11. C₄×D₈×C₂₃	C₂	semidirect
88. D₄×C₂₃	#41	semidirect
104. #3366	C₂₃	direct (N × H)
124. #554	C₂×C₂₃	semidirect
125. #550	C₂×C₂₃	semidirect
126. #490	C₂×C₂₃	semidirect
127. #486	C₂×C₂₃	semidirect
128. #478	C₂×C₂₃	semidirect
129. #476	C₂×C₂₃	semidirect
130. #459	C₂×C₂₃	semidirect
131. #459	C₂×C₂₃	semidirect
132. #446	C₂×C₂₃	semidirect
133. #442	C₂×C₂₃	semidirect
134. C₄×D₈	C₂×C₂₃	semidirect
170. C₂₃	#3366	direct (N × H)
218. D₄	#41×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
C₂₃×#550	2
C₂₃×#490	2
C₂₃×#486	2
C₂₃×#478	2
C₂₃×#476	2
C₂₃×#459	2
C₂₃×#446	2
C₂₃×#442	2
C₄×D₈×C₂₃	2
#3366	23

Subgroups — C23×#3366 #15670985

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#3366 #15670985