Subgroups — #26103016

Order 12032.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241452	C₂	1	2	yes	maximal
2	C₄₇×#1311	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
4	C₄×C₁₆×C₄₇	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₄₇	C₄	1	1	yes	characteristic
6	C₈²×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×C₄₇⋊C₁₆	—	4	2	no	—
8	C₂×C₁₆×C₄₇	—	2	1	no	—
9	C₂×C₁₆×C₄₇	C₈	1	1	yes	characteristic
10	C₂×C₁₆×C₄₇	C₈	1	1	yes	characteristic
11	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
12	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
13	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
14	C₂×C₈×C₄₇	—	2	1	no	—
15	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
16	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
17	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
18	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
19	C₁₆×C₄₇	—	2	2	no	—
20	C₁₆×C₄₇	—	2	1	no	—
21	C₁₆×C₄₇	C₁₆	1	2	yes	complemented
22	C₄₇⋊C₁₆	—	8	2	no	—
23	C₂×C₄×C₄₇	C₄⋊C₈	1	1	yes	characteristic
24	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
25	C₂×C₄×C₄₇	#108	1	1	yes	characteristic
26	C₈×C₄₇	—	2	1	no	—
27	C₈×C₄₇	—	2	2	no	—
28	C₈×C₄₇	—	4	1	no	—
29	C₈×C₄₇	C₂×C₁₆	1	1	yes	characteristic
30	C₈×C₄₇	M₃₂	1	1	yes	characteristic
31	#7099	—	47	1	no	maximal, Sylow
32	C₂²×C₄₇	#334	1	1	yes	characteristic
33	C₄×C₄₇	—	2	1	no	—
34	C₄×C₄₇	C₄⋊C₁₆	1	1	yes	characteristic
35	C₄×C₄₇	#364	1	1	yes	characteristic
36	C₄×C₄₇	#365	1	2	yes	—
37	#1367	—	47	2	no	—
38	#1311	D₄₇	1	1	yes	characteristic
39	C₂×C₄₇	#1367	1	1	yes	characteristic
40	C₂×C₄₇	#1368	1	1	yes	characteristic
41	C₂×C₄₇	#1369	1	1	yes	characteristic
42	C₄⋊C₁₆	—	94	2	no	—
43	C₄×C₁₆	Dic₄₇	1	1	yes	characteristic
44	C₄×C₁₆	Dic₄₇	1	1	yes	characteristic
45	C₈²	C₂×D₄₇	1	1	yes	characteristic
46	C₄₇	#7099	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	—	188	2	no	—
49	C₂×C₁₆	C₄₇⋊C₈	1	1	yes	characteristic
50	C₂×C₁₆	C₄₇⋊C₈	1	1	yes	characteristic
51	C₄×C₈	Dic₉₄	1	1	yes	characteristic
52	C₄×C₈	D₁₈₈	1	1	yes	characteristic
53	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
54	C₂×C₈	—	2	1	no	—
55	C₂×C₈	#10625325	1	2	yes	—
56	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
57	C₂×C₈	#10625329	1	1	yes	characteristic
58	C₄²	C₄⋊Dic₄₇	1	1	yes	characteristic
59	C₁₆	—	2	2	no	—
60	C₁₆	—	2	1	no	—
61	C₁₆	—	376	2	no	—
62	C₁₆	C₄₇⋊C₁₆	1	2	yes	complemented
63	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
64	C₂×C₄	#13093956	1	1	yes	characteristic
65	C₂×C₄	#13093960	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	2	2	no	—
68	C₈	—	4	1	no	—
69	C₈	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
70	C₈	#13093964	1	1	yes	characteristic
71	C₂²	#15688874	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	C₄⋊(C₄₇⋊C₁₆)	1	1	yes	characteristic
74	C₄	#15688944	1	1	yes	characteristic
75	C₄	#15688946	1	2	yes	—
76	C₂	#20241452	1	1	yes	characteristic
77	C₂	#20241454	1	1	yes	characteristic
78	C₂	#20241456	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13093956
commutator	C₈×C₄₇	C₂×C₁₆
Frattini	C₈²	C₂×D₄₇
Fitting	C₄₇×#1311	C₂
socle, prime radical	C₂²×C₄₇	#334
cosocle	C₈²×C₄₇	C₂²
hypercenter	#1311	D₄₇
hypocenter	C₄₇	#7099
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

Complements

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

N	H ≅ G/N	type
21. C₁₆×C₄₇	C₁₆	semidirect
46. C₄₇	#7099	semidirect
62. C₁₆	C₄₇⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#20241452	2
C₄₇×#1311	2
#7099	47