Subgroups — #26103957

Order 12032.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241460	C₂	1	2	yes	maximal
2	C₄×C₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic
3	#15688876	—	2	2	no	—
4	C₂×C₃₂×C₄₇	C₄	1	1	yes	characteristic
5	C₂×C₃₂×C₄₇	C₄	1	1	yes	characteristic
6	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
7	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
8	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₄₇	Q₈	1	1	yes	characteristic
10	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
11	C₃₂×C₄₇	—	2	1	no	—
12	C₃₂×C₄₇	C₈	1	2	yes	complemented
13	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
14	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
15	C₄²×C₄₇	D₈	1	1	yes	characteristic
16	C₂×C₄₇⋊C₈	—	8	2	no	—
17	C₁₆×C₄₇	—	2	1	no	—
18	C₁₆×C₄₇	C₂×C₈	1	1	yes	characteristic
19	C₁₆×C₄₇	M₁₆	1	1	yes	characteristic
20	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
21	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
22	C₂×C₄×C₄₇	Q₃₂	1	1	yes	characteristic
23	C₈×C₄₇	—	2	1	no	—
24	C₈×C₄₇	C₄⋊C₈	1	1	yes	characteristic
25	C₈×C₄₇	#108	1	1	yes	characteristic
26	C₄₇⋊C₈	—	16	2	no	—
27	#7461	—	47	1	no	maximal, Sylow
28	C₂²×C₄₇	#366	1	1	yes	characteristic
29	C₄×C₄₇	—	2	1	no	—
30	C₄×C₄₇	#335	1	1	yes	characteristic
31	C₄×C₄₇	#368	1	1	yes	characteristic
32	C₄×C₄₇	SD₆₄	1	2	yes	—
33	C₄×C₃₂	D₄₇	1	1	yes	characteristic
34	#1371	—	47	2	no	—
35	C₂×C₄₇	#1371	1	1	yes	characteristic
36	C₂×C₄₇	#1424	1	1	yes	characteristic
37	C₂×C₄₇	#1425	1	1	yes	characteristic
38	C₂×C₃₂	Dic₄₇	1	1	yes	characteristic
39	C₂×C₃₂	Dic₄₇	1	1	yes	characteristic
40	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
41	#335	—	94	2	no	—
42	C₄₇	#7461	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₁₆	Dic₉₄	1	1	yes	characteristic
44	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
45	C₄⋊C₈	—	188	2	no	—
46	C₄×C₈	D₁₈₈	1	1	yes	characteristic
47	C₃₂	—	2	1	no	—
48	C₃₂	C₄₇⋊C₈	1	2	yes	complemented
49	C₂×C₈	—	376	2	no	—
50	C₂×C₈	Dic₁₈₈	1	1	yes	characteristic
51	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
52	C₄²	D₃₇₆	1	1	yes	characteristic
53	C₁₆	—	2	1	no	—
54	C₁₆	C₂×C₄₇⋊C₈	1	1	yes	characteristic
55	C₁₆	#10625329	1	1	yes	characteristic
56	C₂×C₄	#13093958	1	1	yes	characteristic
57	C₂×C₄	D₇₅₂	1	1	yes	characteristic
58	C₂×C₄	Dic₃₇₆	1	1	yes	characteristic
59	C₈	—	2	1	no	—
60	C₈	—	752	2	no	—
61	C₈	C₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
62	C₈	#13093960	1	1	yes	characteristic
63	C₂²	#15688949	1	1	yes	characteristic
64	C₄	—	2	1	no	—
65	C₄	#15688876	1	1	yes	characteristic
66	C₄	#15688953	1	1	yes	characteristic
67	C₄	#15688962	1	2	yes	—
68	C₂	#20241460	1	1	yes	characteristic
69	C₂	#20241590	1	1	yes	characteristic
70	C₂	#20241592	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₇₅₂
commutator	C₁₆×C₄₇	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₄₇
Fitting	C₄×C₃₂×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	#366
cosocle	C₄×C₁₆×C₄₇	C₂²
hypercenter	C₄×C₃₂	D₄₇
hypocenter	C₄₇	#7461
LCS	G ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
12. C₃₂×C₄₇	C₈	semidirect
42. C₄₇	#7461	semidirect
48. C₃₂	C₄₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20241460	2
C₄×C₃₂×C₄₇	2
#7461	47