Subgroups — #15682371

Order 2944.

380 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092954	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#13092496	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092490	C₂	1	2	yes	maximal, complemented
5	#13092482	C₂	1	1	yes	maximal, characteristic, complemented
6	#13092482	C₂	1	2	yes	maximal, complemented
7	#13092477	C₂	1	2	yes	maximal, complemented
8	#13092473	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092374	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₂₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
13	#10625118	C₄	1	1	yes	characteristic
14	#10625117	C₄	1	1	yes	characteristic
15	C₂×#67262	C₂²	1	2	yes	—
16	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₂₃	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₂₃	C₂²	1	2	yes	—
19	#10625069	C₄	1	2	yes	—
20	#10625066	C₄	1	1	yes	characteristic
21	#10625065	C₄	1	2	yes	—
22	#10625061	C₄	1	1	yes	characteristic
23	C₄×D₉₂	C₂²	1	2	yes	—
24	C₄×D₉₂	C₂²	1	1	yes	characteristic
25	#10625043	C₂²	1	2	yes	—
26	C₄²×D₂₃	C₂²	1	1	yes	characteristic
27	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
28	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
29	C₂₃×#97	C₂²	1	2	yes	—
30	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
31	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
32	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₂₃)⋊C₈	C₂²	1	2	yes	—
34	#10624997	C₂²	1	2	yes	—
35	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
36	C₈×Dic₂₃	C₂²	1	1	yes	characteristic
37	C₄⋊(C₂₃⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
39	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×C₂₃⋊C₈	C₂²	1	2	yes	—
41	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₉₂	C₂×C₄	1	2	yes	—
43	C₂×D₉₂	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×D₂₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×D₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×D₂₃	C₂³	1	2	yes	—
48	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
49	C₂×C₈×C₂₃	C₂³	1	2	yes	—
50	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
51	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₄×C₂₃	C₂×C₄	1	2	yes	—
53	C₄²×C₂₃	C₂³	1	1	yes	characteristic
54	C₄²×C₂₃	C₂³	1	2	yes	—
55	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
56	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
57	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
58	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₂₃)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
61	C₄⋊Dic₂₃	C₂×C₄	1	1	yes	characteristic
62	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
63	Dic₂₃⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₂₃	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₂₃	C₂³	1	1	yes	characteristic
67	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₂₃⋊C₈	C₂³	1	2	yes	—
69	C₂×C₂₃⋊C₈	C₂³	1	1	yes	characteristic
70	#67262	—	2	4	no	—
71	C₈×D₂₃	—	2	2	no	—
72	C₈×D₂₃	—	2	2	no	—
73	C₈×D₂₃	—	2	4	no	—
74	C₂²×D₂₃	C₂²×C₄	1	2	yes	—
75	C₂²×D₂₃	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₂₃	—	2	2	no	—
77	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₂₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
82	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×Dic₂₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₂₃	C₂²×C₄	1	2	yes	—
86	D₉₂	—	2	4	no	—
87	D₉₂	—	2	2	no	—
88	C₄×D₂₃	—	2	2	no	—
89	C₄×D₂₃	—	2	2	no	—
90	C₄×D₂₃	—	2	1	no	—
91	C₄×D₂₃	—	2	1	no	—
92	C₄×D₂₃	—	2	2	no	—
93	C₄×D₂₃	—	2	2	no	—
94	C₈×C₂₃	—	2	1	no	—
95	C₈×C₂₃	—	2	2	no	—
96	C₈×C₂₃	#41	1	2	yes	—
97	C₂₃⋊C₈	—	2	1	no	—
98	C₂₃⋊C₈	—	2	2	no	—
99	C₂₃⋊C₈	#41	1	2	yes	—
100	#2966	—	23	1	no	maximal, Sylow
101	C₂²×C₂₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×D₂₃	—	2	1	no	—
103	C₂×D₂₃	—	2	1	no	—
104	C₂×D₂₃	—	2	2	no	—
105	C₂×D₂₃	—	2	2	no	—
106	C₂×D₂₃	—	2	2	no	—
107	C₂×D₂₃	#131	1	2	yes	—
108	C₄×C₂₃	—	2	1	no	—
109	C₄×C₂₃	—	2	2	no	—
110	C₄×C₂₃	—	2	2	no	—
111	C₄×C₂₃	#131	1	2	yes	—
112	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
113	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
114	Dic₂₃	—	2	1	no	—
115	Dic₂₃	—	2	2	no	—
116	Dic₂₃	—	2	1	no	—
117	Dic₂₃	—	2	1	no	—
118	#529	—	23	1	no	—
119	#446	D₂₃	1	1	yes	characteristic, complemented
120	C₈×Q₈	—	23	1	no	—
121	#436	—	23	1	no	—
122	#435	—	23	2	no	—
123	C₈×D₄	—	23	1	no	—
124	C₈×D₄	—	23	2	no	—
125	#433	—	23	2	no	—
126	#432	—	23	1	no	—
127	#405	—	23	2	no	—
128	C₂×C₄×C₈	—	23	1	no	—
129	C₂×C₂₃	C₄×#41	1	1	yes	characteristic
130	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
131	C₂×C₂₃	C₂×#131	1	1	yes	characteristic
132	D₂₃	—	2	2	no	—
133	D₂₃	—	4	2	no	—
134	C₂×M₁₆	—	23	2	no	—
135	C₂²×C₈	—	23	1	no	—
136	C₂²×C₈	—	23	1	no	—
137	C₂²×C₈	—	23	2	no	—
138	#126	—	23	2	no	—
139	#125	—	23	1	no	—
140	#124	—	23	1	no	—
141	#123	—	23	2	no	—
142	C₂²⋊Q₈	—	23	1	no	—
143	#121	—	23	1	no	—
144	C₄×Q₈	C₂×D₂₃	1	1	yes	characteristic
145	C₄×D₄	—	23	2	no	—
146	C₄×D₄	—	23	1	no	—
147	#117	—	23	2	no	—
148	C₂×C₄²	—	23	1	no	—
149	C₄⋊C₈	—	23	1	no	—
150	C₄⋊C₈	—	23	2	no	—
151	C₄⋊C₈	C₂×D₂₃	1	1	yes	characteristic
152	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
153	C₂²⋊C₈	—	23	2	no	—
154	C₂²⋊C₈	—	23	2	no	—
155	C₂²⋊C₈	—	23	2	no	—
156	#97	—	23	2	no	—
157	#97	C₂×D₂₃	1	2	yes	—
158	C₄×C₈	—	23	1	no	—
159	C₄×C₈	—	23	1	no	—
160	C₄×C₈	—	23	1	no	—
161	C₄×C₈	—	23	2	no	—
162	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
163	C₂₃	#2966	1	1	yes	characteristic, complemented, Sylow
164	C₂×Q₈	C₄×D₂₃	1	1	yes	characteristic
165	C₂×D₄	—	23	2	no	—
166	C₂×D₄	—	23	1	no	—
167	C₂²×C₄	—	23	1	no	—
168	C₂²×C₄	—	23	1	no	—
169	C₂²×C₄	—	23	1	no	—
170	C₂²×C₄	—	23	2	no	—
171	M₁₆	—	46	4	no	—
172	C₂×C₈	—	23	1	no	—
173	C₂×C₈	—	23	1	no	—
174	C₂×C₈	—	23	2	no	—
175	C₂×C₈	—	46	2	no	—
176	C₂×C₈	—	46	4	no	—
177	C₂×C₈	—	46	2	no	—
178	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
179	C₂×C₈	C₂²×D₂₃	1	2	yes	—
180	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
181	C₄⋊C₄	—	23	2	no	—
182	C₄⋊C₄	—	23	2	no	—
183	C₄⋊C₄	—	23	1	no	—
184	C₄⋊C₄	—	23	2	no	—
185	C₄⋊C₄	C₄×D₂₃	1	1	yes	characteristic
186	C₄⋊C₄	C₄×D₂₃	1	2	yes	—
187	C₂²⋊C₄	—	23	2	no	—
188	C₂²⋊C₄	—	23	2	no	—
189	C₂²⋊C₄	—	23	2	no	—
190	C₂²⋊C₄	—	23	2	no	—
191	C₂²⋊C₄	—	23	2	no	—
192	C₄²	—	23	1	no	—
193	C₄²	—	23	1	no	—
194	C₄²	—	23	1	no	—
195	C₄²	C₂²×D₂₃	1	1	yes	characteristic
196	C₄²	C₂²×D₂₃	1	2	yes	—
197	C₂³	—	23	1	no	—
198	C₂³	—	23	2	no	—
199	Q₈	—	2	2	no	—
200	D₄	—	46	4	no	—
201	D₄	—	46	2	no	—
202	C₂×C₄	—	23	1	no	—
203	C₂×C₄	—	23	1	no	—
204	C₂×C₄	—	23	1	no	—
205	C₂×C₄	—	23	2	no	—
206	C₂×C₄	—	46	2	no	—
207	C₂×C₄	—	46	2	no	—
208	C₂×C₄	—	46	1	no	—
209	C₂×C₄	—	46	2	no	—
210	C₂×C₄	—	46	1	no	—
211	C₂×C₄	—	46	2	no	—
212	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
214	C₂×C₄	C₂×C₄×D₂₃	1	2	yes	—
215	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
216	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	2	no	—
219	C₈	—	23	2	no	—
220	C₈	—	46	1	no	—
221	C₈	—	46	2	no	—
222	C₈	#67292	1	2	yes	—
223	C₂²	—	23	2	no	—
224	C₂²	—	46	1	no	—
225	C₂²	—	46	1	no	—
226	C₂²	—	46	2	no	—
227	C₂²	—	46	2	no	—
228	C₂²	—	46	2	no	—
229	C₂²	C₂²×C₄×D₂₃	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	2	no	—
233	C₄	—	46	1	no	—
234	C₄	—	46	2	no	—
235	C₄	—	46	1	no	—
236	C₄	—	46	1	no	—
237	C₄	#10625078	1	2	yes	—
238	C₄	C₂×#67292	1	1	yes	characteristic
239	C₄	#41×D₂₃	1	1	yes	characteristic
240	C₂	—	46	2	no	—
241	C₂	—	92	2	no	—
242	C₂	C₄×#67292	1	1	yes	characteristic
243	C₂	#131×D₂₃	1	1	yes	characteristic
244	C₂	C₂×#10625078	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₃
commutator, socle	C₂²×C₂₃	C₂³×C₄
Fitting	C₂₃×#446	C₂
prime radical	#10625066	C₄
cosocle	C₂×C₄×C₂₃	C₂⁴
hypercenter	#446	D₂₃
hypocenter	C₂₃	#2966
LCS	G ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ #446

Complements

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

N	H ≅ G/N	type
2. Q₈×C₂₃⋊C₈	C₂	semidirect
3. #13092496	C₂	semidirect
4. #13092490	C₂	semidirect
5. #13092482	C₂	semidirect
6. #13092482	C₂	semidirect
7. #13092477	C₂	semidirect
8. #13092473	C₂	semidirect
9. #13092374	C₂	semidirect
10. C₄×C₈×D₂₃	C₂	semidirect
11. C₂₃×#446	C₂	semidirect
119. #446	D₂₃	semidirect
163. C₂₃	#2966	semidirect

Maximal subgroups

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

subgroup	index
#13092954	2
Q₈×C₂₃⋊C₈	2
#13092496	2
#13092490	2
#13092482	2
#13092477	2
#13092473	2
#13092374	2
C₄×C₈×D₂₃	2
C₂₃×#446	2
#2966	23