Subgroups — #26103009

Order 12032.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241458	C₂	1	2	yes	maximal, complemented
2	C₄₇×#1712	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
4	C₄₇×#493	—	2	2	no	—
5	C₄₇×#493	C₄	1	2	yes	—
6	C₈²×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×C₄₇⋊C₁₆	—	4	2	no	—
8	C₂×D₈×C₄₇	—	4	2	no	—
9	C₂×D₈×C₄₇	—	4	2	no	—
10	C₂×D₈×C₄₇	—	4	2	no	—
11	#127×C₄₇	—	2	2	no	—
12	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₄×C₈×C₄₇	D₄	1	2	yes	—
14	C₂×D₄×C₄₇	—	4	2	no	—
15	C₂×D₄×C₄₇	—	4	4	no	—
16	D₈×C₄₇	—	8	4	no	—
17	D₈×C₄₇	—	8	4	no	—
18	D₈×C₄₇	—	8	2	no	—
19	D₈×C₄₇	—	8	2	no	—
20	C₂×C₈×C₄₇	—	2	1	no	—
21	C₂×C₈×C₄₇	D₈	1	2	yes	—
22	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
23	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
24	C₄₇⋊C₁₆	—	8	2	no	—
25	C₂³×C₄₇	—	8	2	no	—
26	D₄×C₄₇	—	8	4	no	—
27	D₄×C₄₇	—	8	4	no	—
28	D₄×C₄₇	—	8	4	no	—
29	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
30	C₂×C₄×C₄₇	#102	1	2	yes	—
31	C₈×C₄₇	—	2	2	no	—
32	C₈×C₄₇	—	2	2	no	—
33	C₈×C₄₇	D₁₆	1	2	yes	—
34	C₈×C₄₇	SD₃₂	1	2	yes	—
35	#7096	—	47	1	no	maximal, Sylow
36	C₂²×C₄₇	—	16	2	no	—
37	C₂²×C₄₇	—	16	4	no	—
38	C₂²×C₄₇	#331	1	1	yes	characteristic
39	C₄×C₄₇	—	2	1	no	—
40	C₄×C₄₇	#357	1	2	yes	—
41	C₄×C₄₇	#361	1	2	yes	—
42	#1712	D₄₇	1	1	yes	characteristic
43	#1370	—	47	2	no	—
44	C₂×C₄₇	—	32	2	no	—
45	C₂×C₄₇	#1357	1	1	yes	characteristic
46	C₂×C₄₇	#1361	1	2	yes	—
47	#493	—	2	2	no	—
48	#493	Dic₄₇	1	2	yes	—
49	C₄⋊C₁₆	—	94	2	no	—
50	C₈²	C₂×D₄₇	1	1	yes	characteristic
51	C₄₇	#7096	1	1	yes	characteristic, complemented, Sylow
52	C₂×D₈	—	4	2	no	—
53	C₂×D₈	—	4	2	no	—
54	C₂×D₈	—	4	2	no	—
55	#127	—	2	2	no	—
56	C₂×C₁₆	—	188	2	no	—
57	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
58	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
59	C₂×D₄	—	4	2	no	—
60	C₂×D₄	—	4	4	no	—
61	D₈	—	8	4	no	—
62	D₈	—	8	4	no	—
63	D₈	—	8	2	no	—
64	D₈	—	8	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	#10625334	1	2	yes	—
67	C₂×C₈	#10625335	1	2	yes	—
68	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
69	C₁₆	—	376	2	no	—
70	C₂³	—	8	2	no	—
71	D₄	—	8	4	no	—
72	D₄	—	8	4	no	—
73	D₄	—	8	4	no	—
74	C₂×C₄	#13093941	1	1	yes	characteristic
75	C₂×C₄	#13093945	1	2	yes	—
76	C₈	—	2	2	no	—
77	C₈	—	2	2	no	—
78	C₈	#13093965	1	2	yes	—
79	C₈	#13093968	1	2	yes	—
80	C₂²	—	16	2	no	—
81	C₂²	—	16	4	no	—
82	C₂²	#15688867	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	#15688924	1	2	yes	—
85	C₄	#15688936	1	2	yes	—
86	C₂	—	32	2	no	—
87	C₂	#20241425	1	1	yes	characteristic
88	C₂	#20241434	1	2	yes	—

	subgroup	G/H
center	C₂²	#15688867
commutator	C₄×C₈×C₄₇	C₂×C₄
Frattini	C₈²	C₂×D₄₇
Fitting, prime radical	C₄₇×#1712	C₂
socle	C₂²×C₄₇	#331
cosocle	C₈²×C₄₇	C₂²
hypercenter	#1712	D₄₇
hypocenter	C₄₇	#7096
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

Complements

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

N	H ≅ G/N	type
1. #20241458	C₂	semidirect
51. C₄₇	#7096	semidirect

Maximal subgroups

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

subgroup	index
#20241458	2
C₄₇×#1712	2
#7096	47