Subgroups — #26103606

Order 12032.

126 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	C₈	1	8	yes	complemented
8	C₂×C₁₆×C₄₇	C₈	1	2	yes	—
9	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
10	C₄×C₈×C₄₇	C₈	1	2	yes	—
11	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₄₇	C₁₆	1	4	yes	—
13	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
14	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
16	C₁₆×C₄₇	C₁₆	1	4	yes	—
17	C₁₆×C₄₇	C₂×C₈	1	4	yes	—
18	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
19	C₂×C₄×C₄₇	C₂×C₁₆	1	2	yes	—
20	C₈×C₄₇	—	2	4	no	—
21	C₈×C₄₇	C₄×C₈	1	2	yes	—
22	C₈×C₄₇	C₂×C₁₆	1	2	yes	—
23	#7329	—	47	1	no	maximal, Sylow
24	C₂²×C₄₇	C₄×C₁₆	1	1	yes	characteristic
25	C₄×C₄₇	C₈²	1	1	yes	characteristic
26	C₄×C₄₇	C₄×C₁₆	1	1	yes	characteristic
27	C₄×C₄₇	M₆₄	1	4	yes	—
28	C₄×C₃₂	—	47	2	no	—
29	C₈×C₁₆	D₄₇	1	1	yes	characteristic
30	C₂×C₄₇	C₈×C₁₆	1	1	yes	characteristic
31	C₂×C₄₇	#1397	1	2	yes	—
32	C₂×C₃₂	—	47	4	no	—
33	C₄×C₁₆	Dic₄₇	1	1	yes	characteristic
34	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
35	C₈²	Dic₄₇	1	1	yes	characteristic
36	C₄₇	#7329	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₁₆	C₄₇⋊C₈	1	2	yes	—
38	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
39	C₄×C₈	C₄₇⋊C₈	1	2	yes	—
40	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
41	C₃₂	—	47	8	no	—
42	C₂×C₈	C₄₇⋊C₁₆	1	4	yes	—
43	C₂×C₈	C₂×C₄₇⋊C₈	1	1	yes	characteristic
44	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
45	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
46	C₁₆	C₄₇⋊C₁₆	1	4	yes	—
47	C₁₆	C₈×D₄₇	1	4	yes	—
48	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
49	C₂×C₄	C₂×C₄₇⋊C₁₆	1	2	yes	—
50	C₈	—	2	4	no	—
51	C₈	C₈×Dic₄₇	1	2	yes	—
52	C₈	C₂×C₄₇⋊C₁₆	1	2	yes	—
53	C₂²	C₄×C₄₇⋊C₁₆	1	1	yes	characteristic
54	C₄	C₈×C₄₇⋊C₈	1	1	yes	characteristic
55	C₄	C₄×C₄₇⋊C₁₆	1	1	yes	characteristic
56	C₄	#15688957	1	4	yes	—
57	C₂	C₈×C₄₇⋊C₁₆	1	1	yes	characteristic
58	C₂	#20241522	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₁₆	C₂×D₄₇
commutator	C₂×C₄₇	C₈×C₁₆
Fitting	C₈×C₁₆×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	C₄×C₁₆
cosocle	C₄×C₁₆×C₄₇	C₂²
hypercenter	C₈×C₁₆	D₄₇
hypocenter	C₄₇	#7329
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
7. C₄₇⋊C₃₂	C₈	semidirect
36. C₄₇	#7329	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
#7329	47