Subgroups — #26103609

Order 12032.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄₇⋊C₃₂	C₂	1	2	yes	maximal
2	C₈×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₇⋊C₃₂	C₄	1	4	yes	—
4	C₄×C₁₆×C₄₇	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
6	C₈²×C₄₇	C₄	1	1	yes	characteristic
7	C₄₇⋊C₃₂	—	2	4	no	—
8	C₂×C₁₆×C₄₇	C₈	1	2	yes	—
9	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
10	C₄×C₈×C₄₇	C₈	1	1	yes	characteristic
11	C₄×C₈×C₄₇	C₈	1	1	yes	characteristic
12	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₄₇	C₁₆	1	2	yes	—
14	C₂×C₈×C₄₇	C₁₆	1	2	yes	—
15	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
16	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
17	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
18	C₁₆×C₄₇	—	2	2	no	—
19	C₁₆×C₄₇	M₁₆	1	4	yes	—
20	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
21	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
22	C₂×C₄×C₄₇	C₂×C₁₆	1	1	yes	characteristic
23	C₈×C₄₇	—	2	2	no	—
24	C₈×C₄₇	C₃₂	1	4	yes	complemented
25	C₈×C₄₇	#97	1	2	yes	—
26	C₈×C₄₇	M₃₂	1	2	yes	—
27	#7331	—	47	1	no	maximal, Sylow
28	C₂²×C₄₇	C₄×C₁₆	1	1	yes	characteristic
29	C₄×C₄₇	#322	1	1	yes	characteristic
30	C₄×C₄₇	#346	1	1	yes	characteristic
31	C₄×C₄₇	C₂×C₃₂	1	2	yes	—
32	C₄×C₄₇	M₆₄	1	2	yes	—
33	C₄×C₃₂	—	47	2	no	—
34	C₈×C₁₆	D₄₇	1	1	yes	characteristic
35	C₂×C₄₇	#1312	1	1	yes	characteristic
36	C₂×C₄₇	C₄×C₃₂	1	1	yes	characteristic
37	C₂×C₄₇	#1397	1	1	yes	characteristic
38	C₂×C₃₂	—	47	4	no	—
39	C₄×C₁₆	Dic₄₇	1	1	yes	characteristic
40	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
41	C₈²	Dic₄₇	1	1	yes	characteristic
42	C₄₇	#7331	1	1	yes	characteristic, complemented, Sylow
43	C₂×C₁₆	C₄₇⋊C₈	1	2	yes	—
44	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
45	C₄×C₈	C₄₇⋊C₈	1	1	yes	characteristic
46	C₄×C₈	C₄₇⋊C₈	1	1	yes	characteristic
47	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
48	C₃₂	—	94	4	no	—
49	C₂×C₈	C₄₇⋊C₁₆	1	2	yes	—
50	C₂×C₈	C₄₇⋊C₁₆	1	2	yes	—
51	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
52	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
53	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
54	C₁₆	—	2	2	no	—
55	C₁₆	#10625324	1	4	yes	—
56	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
57	C₂×C₄	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
58	C₂×C₄	C₂×C₄₇⋊C₁₆	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	C₄₇⋊C₃₂	1	4	yes	complemented
61	C₈	#13093934	1	2	yes	—
62	C₈	#13093964	1	2	yes	—
63	C₂²	C₄×C₄₇⋊C₁₆	1	1	yes	characteristic
64	C₄	#15688844	1	1	yes	characteristic
65	C₄	#15688899	1	1	yes	characteristic
66	C₄	C₂×C₄₇⋊C₃₂	1	2	yes	—
67	C₄	#15688957	1	2	yes	—
68	C₂	#20241306	1	1	yes	characteristic
69	C₂	C₄×C₄₇⋊C₃₂	1	1	yes	characteristic
70	C₂	#20241522	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄×C₁₆	C₂×D₄₇
commutator	C₂×C₄₇	C₄×C₃₂
Fitting	C₈×C₁₆×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₄×C₁₆
cosocle	C₄×C₁₆×C₄₇	C₂²
hypercenter	C₈×C₁₆	D₄₇
hypocenter	C₄₇	#7331
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄×C₁₆ ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
24. C₈×C₄₇	C₃₂	semidirect
42. C₄₇	#7331	semidirect
60. C₈	C₄₇⋊C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄₇⋊C₃₂	2
C₈×C₁₆×C₄₇	2
#7331	47