Subgroups — #15673557

Order 2944.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625000	C₂	1	4	yes	maximal
2	#13092287	C₂	1	2	yes	maximal
3	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
5	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₄⋊Dic₂₃	—	2	4	no	—
7	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
8	C₄×C₈×C₂₃	C₄	1	4	yes	complemented
9	#10625000	—	2	8	no	—
10	C₂²×C₄×C₂₃	D₄	1	2	yes	—
11	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
12	C₂²×Dic₂₃	—	2	4	no	—
13	C₂×C₈×C₂₃	C₂×C₄	1	4	yes	—
14	C₂×C₈×C₂₃	D₄	1	4	yes	—
15	C₂×C₈×C₂₃	Q₈	1	4	yes	—
16	C₄²×C₂₃	C₂×C₄	1	2	yes	—
17	C₄²×C₂₃	Q₈	1	2	yes	—
18	C₄⋊Dic₂₃	—	2	8	no	—
19	C₄⋊Dic₂₃	—	4	8	no	—
20	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
22	C₂×C₄×C₂₃	D₈	1	4	yes	—
23	C₂×C₄×C₂₃	Q₁₆	1	4	yes	—
24	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
26	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
27	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
28	C₂×Dic₂₃	—	4	4	no	—
29	C₂×Dic₂₃	—	4	8	no	—
30	C₈×C₂₃	C₄⋊C₄	1	8	yes	complemented
31	#1845	—	23	1	no	maximal, Sylow
32	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
33	C₂²×C₂₃	#127	1	1	yes	characteristic
34	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
35	C₂²×C₂₃	C₂×D₈	1	2	yes	—
36	C₂²×C₂₃	C₂×Q₁₆	1	2	yes	—
37	C₄×C₂₃	#107	1	8	yes	complemented
38	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
39	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
40	Dic₂₃	—	8	4	no	—
41	C₂×#107	—	23	4	no	—
42	C₂×C₄×C₈	D₂₃	1	1	yes	characteristic
43	#384	—	23	2	no	—
44	C₂×C₂₃	#384	1	1	yes	characteristic
45	C₂×C₂₃	C₂×#107	1	2	yes	—
46	C₂×C₂₃	#493	1	1	yes	characteristic
47	C₂×C₂₃	#494	1	1	yes	characteristic
48	C₂×C₂₃	#500	1	2	yes	—
49	C₂²×C₈	C₂×D₂₃	1	2	yes	—
50	C₂×C₄⋊C₄	—	23	4	no	—
51	C₂×C₄⋊C₄	—	46	4	no	—
52	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
53	#107	—	46	8	no	—
54	C₄×C₈	Dic₂₃	1	4	yes	complemented
55	C₂₃	#1845	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₄	—	46	4	no	—
57	C₂²×C₄	D₉₂	1	2	yes	—
58	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
59	C₂×C₈	Dic₄₆	1	4	yes	—
60	C₂×C₈	D₉₂	1	4	yes	—
61	C₂×C₈	C₂×Dic₂₃	1	4	yes	—
62	C₄⋊C₄	—	46	8	no	—
63	C₄⋊C₄	—	92	8	no	—
64	C₄²	Dic₄₆	1	2	yes	—
65	C₄²	C₂×Dic₂₃	1	2	yes	—
66	C₂³	C₂×D₉₂	1	1	yes	characteristic
67	C₂×C₄	—	92	4	no	—
68	C₂×C₄	—	92	8	no	—
69	C₂×C₄	D₁₈₄	1	4	yes	—
70	C₂×C₄	Dic₉₂	1	4	yes	—
71	C₂×C₄	C₄⋊Dic₂₃	1	4	yes	—
72	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
73	C₂×C₄	C₂×Dic₄₆	1	2	yes	—
74	C₂×C₄	C₂×D₉₂	1	2	yes	—
75	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
76	C₈	C₄⋊Dic₂₃	1	8	yes	complemented
77	C₂²	#10625040	1	1	yes	characteristic
78	C₂²	#10625045	1	1	yes	characteristic
79	C₂²	C₂×D₁₈₄	1	2	yes	—
80	C₂²	C₂×Dic₉₂	1	2	yes	—
81	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
82	C₄	—	184	4	no	—
83	C₄	#10625000	1	8	yes	complemented
84	C₄	#10625040	1	2	yes	—
85	C₄	C₂×C₄⋊Dic₂₃	1	2	yes	—
86	C₂	#13092287	1	1	yes	characteristic
87	C₂	C₂×#10625000	1	2	yes	—
88	C₂	#13092794	1	1	yes	characteristic
89	C₂	#13092797	1	1	yes	characteristic
90	C₂	#13092822	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₉₂
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₄×C₈×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄×C₈	D₂₃
hypocenter	C₂₃	#1845
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₄×C₈×C₂₃	C₄	semidirect
30. C₈×C₂₃	C₄⋊C₄	semidirect
37. C₄×C₂₃	#107	semidirect
54. C₄×C₈	Dic₂₃	semidirect
55. C₂₃	#1845	semidirect
76. C₈	C₄⋊Dic₂₃	semidirect
83. C₄	#10625000	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625000	2
#13092287	2
C₂×C₄×C₈×C₂₃	2
#1845	23