Subgroups — #15671370

Order 2944.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13092238	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#443	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#135	C₄	1	1	yes	characteristic
5	C₂₃×#131	—	2	1	no	—
6	C₂₃×#108	C₄	1	1	yes	characteristic
7	C₂₃×#104	—	2	1	no	—
8	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	#10624994	—	2	1	no	—
10	C₄×C₂₃⋊C₈	—	2	1	no	—
11	#41×C₂₃	—	2	1	no	—
12	Q₁₆×C₂₃	C₈	1	1	yes	characteristic, complemented
13	SD₁₆×C₂₃	—	2	1	no	—
14	D₈×C₂₃	C₈	1	1	yes	characteristic, complemented
15	M₁₆×C₂₃	—	2	1	no	—
16	M₁₆×C₂₃	—	4	1	no	—
17	C₂×C₈×C₂₃	—	4	1	no	—
18	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
20	C₄²×C₂₃	D₄	1	1	yes	characteristic
21	C₂×C₂₃⋊C₈	—	2	2	no	—
22	C₂₃⋊C₁₆	—	4	1	no	—
23	Q₈×C₂₃	—	2	1	no	—
24	D₄×C₂₃	—	2	1	no	—
25	D₄×C₂₃	—	4	1	no	—
26	C₂×C₄×C₂₃	—	2	1	no	—
27	C₂×C₄×C₂₃	—	4	1	no	—
28	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₈×C₂₃	—	4	1	no	—
30	C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
31	C₈×C₂₃	M₁₆	1	1	yes	characteristic
32	C₈×C₂₃	D₈	1	1	yes	characteristic
33	C₈×C₂₃	SD₁₆	1	1	yes	characteristic
34	C₂₃⋊C₈	—	2	4	no	—
35	#1335	—	23	1	no	maximal, Sylow
36	C₂²×C₂₃	—	4	1	no	—
37	C₂²×C₂₃	#104	1	1	yes	characteristic
38	C₄×C₂₃	—	2	2	no	—
39	C₄×C₂₃	—	4	1	no	—
40	C₄×C₂₃	C₂²⋊C₈	1	1	yes	characteristic
41	C₄×C₂₃	#102	1	1	yes	characteristic
42	#443	D₂₃	1	1	yes	characteristic
43	#364	—	23	1	no	—
44	C₈²	—	23	1	no	—
45	C₂×C₂₃	—	2	1	no	—
46	C₂×C₂₃	—	8	1	no	—
47	C₂×C₂₃	D₄⋊C₈	1	1	yes	characteristic
48	#135	Dic₂₃	1	1	yes	characteristic
49	#131	—	2	1	no	—
50	M₃₂	—	46	1	no	—
51	#108	Dic₂₃	1	1	yes	characteristic
52	#104	—	2	1	no	—
53	C₄×C₈	—	46	1	no	—
54	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
55	C₂₃	#1335	1	1	yes	characteristic, complemented, Sylow
56	#41	—	2	1	no	—
57	Q₁₆	C₂₃⋊C₈	1	1	yes	characteristic, complemented
58	SD₁₆	—	2	1	no	—
59	D₈	C₂₃⋊C₈	1	1	yes	characteristic, complemented
60	M₁₆	—	2	1	no	—
61	M₁₆	—	4	1	no	—
62	C₂×C₈	—	4	1	no	—
63	C₂×C₈	—	46	2	no	—
64	C₂×C₈	C₂×Dic₂₃	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
66	C₄²	C₂²⋊D₂₃	1	1	yes	characteristic
67	C₁₆	—	92	1	no	—
68	Q₈	—	2	1	no	—
69	D₄	—	2	1	no	—
70	D₄	—	4	1	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	4	1	no	—
73	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
74	C₈	—	4	1	no	—
75	C₈	—	46	4	no	—
76	C₈	C₂×C₂₃⋊C₈	1	1	yes	characteristic
77	C₈	#67267	1	1	yes	characteristic
78	C₈	#67272	1	1	yes	characteristic
79	C₈	#67273	1	1	yes	characteristic
80	C₂²	—	4	1	no	—
81	C₂²	#10625019	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	4	1	no	—
84	C₄	C₂²⋊(C₂₃⋊C₈)	1	1	yes	characteristic
85	C₄	#10625014	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	8	1	no	—
88	C₂	D₄⋊(C₂₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#67272
commutator	C₈×C₂₃	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₃
Fitting	C₂₃×#443	C₂
socle	C₂×C₂₃	D₄⋊C₈
prime radical	C₂₃×#135	C₄
cosocle	C₄×C₈×C₂₃	C₂²
hypercenter	#443	D₂₃
hypocenter	C₂₃	#1335
LCS	G ▷ C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #13092238	C₂	semidirect
2. C₈×C₂₃⋊C₈	C₂	semidirect
12. Q₁₆×C₂₃	C₈	semidirect
14. D₈×C₂₃	C₈	semidirect
55. C₂₃	#1335	semidirect
57. Q₁₆	C₂₃⋊C₈	semidirect
59. D₈	C₂₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#13092238	2
C₈×C₂₃⋊C₈	2
C₂₃×#443	2
#1335	23