Subgroups — #26103613

Order 12032.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241524	C₂	1	1	yes	maximal, characteristic
2	#20241308	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1398	C₂	1	1	yes	maximal, characteristic
4	#15688957	C₄	1	2	yes	—
5	#15688899	C₄	1	1	yes	characteristic
6	#15688844	C₄	1	1	yes	characteristic
7	C₄₇×M₆₄	C₄	1	2	yes	—
8	C₄₇×#346	C₂²	1	1	yes	characteristic
9	C₂×C₄₇⋊C₁₆	—	2	1	no	—
10	C₄×C₄₇⋊C₈	—	2	1	no	—
11	C₄₇⋊C₃₂	—	2	2	no	—
12	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₃₂×C₄₇	C₈	1	4	yes	complemented
16	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
17	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
18	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
19	C₂×C₄₇⋊C₈	—	4	1	no	—
20	C₁₆×C₄₇	C₂×C₈	1	2	yes	—
21	C₁₆×C₄₇	M₁₆	1	2	yes	—
22	C₄₇⋊C₁₆	—	4	1	no	—
23	C₂×C₄×C₄₇	—	2	1	no	—
24	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
25	C₈×C₄₇	—	2	1	no	—
26	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
27	C₈×C₄₇	#97	1	1	yes	characteristic
28	C₄₇⋊C₈	—	8	1	no	—
29	#7333	—	47	1	no	maximal, Sylow
30	C₂²×C₄₇	#347	1	1	yes	characteristic
31	C₄×C₄₇	—	4	1	no	—
32	C₄×C₄₇	#322	1	1	yes	characteristic
33	C₄×C₄₇	#347	1	1	yes	characteristic
34	#1398	—	47	1	no	—
35	#1398	D₄₇	1	1	yes	characteristic
36	#1313	—	47	1	no	—
37	C₂×C₄₇	—	2	1	no	—
38	C₂×C₄₇	#1313	1	1	yes	characteristic
39	M₆₄	—	47	2	no	—
40	M₆₄	Dic₄₇	1	2	yes	—
41	#346	—	47	1	no	—
42	#346	C₂×D₄₇	1	1	yes	characteristic
43	#322	—	47	1	no	—
44	C₄₇	#7333	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₁₆	—	94	1	no	—
46	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
47	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
48	C₄×C₈	—	94	1	no	—
49	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
50	C₃₂	—	94	2	no	—
51	C₃₂	C₄₇⋊C₈	1	4	yes	complemented
52	C₂×C₈	—	188	1	no	—
53	C₂×C₈	#10625324	1	1	yes	characteristic
54	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
55	C₄²	#10625324	1	1	yes	characteristic
56	C₁₆	—	188	1	no	—
57	C₁₆	#10625324	1	2	yes	—
58	C₁₆	C₂×C₄₇⋊C₈	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	#13093934	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	376	1	no	—
63	C₈	C₄×C₄₇⋊C₈	1	1	yes	characteristic
64	C₈	#13093934	1	1	yes	characteristic
65	C₂²	#15688900	1	1	yes	characteristic
66	C₄	—	4	1	no	—
67	C₄	#15688844	1	1	yes	characteristic
68	C₄	#15688900	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	#20241308	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15688844
commutator	C₈×C₄₇	C₄×C₈
Frattini	#346	C₂×D₄₇
Fitting	C₄₇×#1398	C₂
socle	C₂×C₄₇	#1313
prime radical	C₂²×C₄₇	#347
cosocle	C₄₇×#346	C₂²
hypercenter	#1398	D₄₇
hypocenter	C₄₇	#7333
LCS	G ▷ C₈×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ #346 ◁ #1398

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₄₇	C₈	semidirect
44. C₄₇	#7333	semidirect
51. 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
#20241524	2
#20241308	2
C₄₇×#1398	2
#7333	47