Subgroups — #26103960

Order 12032.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241522	C₂	1	1	yes	maximal, characteristic
2	#20241462	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1372	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇⋊C₃₂	C₄	1	1	yes	characteristic
5	C₂×C₄₇⋊C₃₂	C₄	1	1	yes	characteristic
6	#15688876	—	2	1	no	—
7	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×#335	—	2	1	no	—
9	C₄⋊(C₄₇⋊C₈)	—	4	1	no	—
10	C₄₇⋊C₃₂	—	2	1	no	—
11	C₄₇⋊C₃₂	C₈	1	2	yes	complemented
12	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₄₇	Q₈	1	1	yes	characteristic
14	C₄⋊C₈×C₄₇	—	4	1	no	—
15	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₄₇	—	8	1	no	—
17	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
19	C₄²×C₄₇	D₈	1	1	yes	characteristic
20	C₂×C₄₇⋊C₈	—	8	1	no	—
21	C₁₆×C₄₇	—	2	1	no	—
22	C₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₄₇	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
25	C₂×C₄×C₄₇	SD₃₂	1	1	yes	characteristic
26	C₂×C₄×C₄₇	SD₃₂	1	1	yes	characteristic
27	C₈×C₄₇	—	2	1	no	—
28	C₈×C₄₇	—	16	1	no	—
29	C₈×C₄₇	C₄⋊C₈	1	1	yes	characteristic
30	C₈×C₄₇	#108	1	1	yes	characteristic
31	C₄₇⋊C₈	—	16	1	no	—
32	#7463	—	47	1	no	maximal, Sylow
33	C₂²×C₄₇	#367	1	1	yes	characteristic
34	C₄×C₄₇	—	2	1	no	—
35	C₄×C₄₇	—	2	1	no	—
36	C₄×C₄₇	#335	1	1	yes	characteristic
37	C₄×C₄₇	#368	1	1	yes	characteristic
38	#1397	—	47	1	no	—
39	#1372	—	47	1	no	—
40	#1372	D₄₇	1	1	yes	characteristic
41	C₂×C₄₇	#1372	1	1	yes	characteristic
42	C₂×C₄₇	#1426	1	1	yes	characteristic
43	C₂×C₄₇	#1426	1	1	yes	characteristic
44	C₂×C₃₂	—	47	1	no	—
45	C₂×C₃₂	—	47	1	no	—
46	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
47	#335	—	2	1	no	—
48	#335	—	94	1	no	—
49	C₄₇	#7463	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₁₆	Dic₉₄	1	1	yes	characteristic
51	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
52	C₄⋊C₈	—	4	1	no	—
53	C₄⋊C₈	—	188	1	no	—
54	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
55	C₃₂	—	47	2	no	—
56	C₃₂	—	94	1	no	—
57	C₂×C₈	—	8	1	no	—
58	C₂×C₈	—	376	1	no	—
59	C₂×C₈	Dic₄₇⋊C₄	1	1	yes	characteristic
60	C₂×C₈	#10625337	1	1	yes	characteristic
61	C₄²	#10625334	1	1	yes	characteristic
62	C₁₆	—	2	1	no	—
63	C₁₆	C₈×D₄₇	1	1	yes	characteristic
64	C₁₆	#10625324	1	1	yes	characteristic
65	C₂×C₄	#13093957	1	1	yes	characteristic
66	C₂×C₄	#13093967	1	1	yes	characteristic
67	C₂×C₄	#13093968	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	16	1	no	—
70	C₈	—	752	1	no	—
71	C₈	Dic₄₇⋊C₈	1	1	yes	characteristic
72	C₈	#13093959	1	1	yes	characteristic
73	C₂²	#15688950	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	#15688875	1	1	yes	characteristic
77	C₄	#15688952	1	1	yes	characteristic
78	C₂	#20241461	1	1	yes	characteristic
79	C₂	#20241593	1	1	yes	characteristic
80	C₂	#20241595	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#15688950
commutator	C₁₆×C₄₇	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₄₇
Fitting	C₄₇×#1372	C₂
socle, prime radical	C₂²×C₄₇	#367
cosocle	C₄×C₁₆×C₄₇	C₂²
hypercenter	#1372	D₄₇
hypocenter	C₄₇	#7463
LCS	G ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1372

Complements

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

N	H ≅ G/N	type
11. C₄₇⋊C₃₂	C₈	semidirect
49. C₄₇	#7463	semidirect

Maximal subgroups

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

subgroup	index
#20241522	2
#20241462	2
C₄₇×#1372	2
#7463	47