Subgroups — #26103607

Order 12032.

110 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	#20241304	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1397	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇⋊C₃₂	C₄	1	2	yes	—
5	C₄×C₄₇⋊C₁₆	C₄	1	1	yes	characteristic
6	C₈×C₄₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₄₇	C₄	1	2	yes	—
8	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
9	C₂×C₄₇⋊C₁₆	C₈	1	2	yes	—
10	C₄×C₄₇⋊C₈	C₈	1	2	yes	—
11	C₄₇⋊C₃₂	C₈	1	4	yes	complemented
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	C₂×C₈	1	1	yes	characteristic
18	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
19	C₂×C₄₇⋊C₈	—	2	2	no	—
20	C₁₆×C₄₇	C₂×C₈	1	2	yes	—
21	C₁₆×C₄₇	C₂×C₈	1	2	yes	—
22	C₄₇⋊C₁₆	—	2	2	no	—
23	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
24	C₂×C₄×C₄₇	M₃₂	1	2	yes	—
25	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
26	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
27	C₈×C₄₇	M₃₂	1	2	yes	—
28	C₄₇⋊C₈	—	4	2	no	—
29	#7330	—	47	1	no	maximal, Sylow
30	C₂²×C₄₇	#346	1	1	yes	characteristic
31	C₄×C₄₇	—	2	2	no	—
32	C₄×C₄₇	C₈²	1	1	yes	characteristic
33	C₄×C₄₇	#346	1	1	yes	characteristic
34	#1397	—	47	1	no	—
35	#1397	D₄₇	1	1	yes	characteristic
36	#1311	—	47	1	no	—
37	C₂×C₄₇	#1311	1	1	yes	characteristic
38	C₂×C₄₇	#1398	1	2	yes	—
39	C₂×C₃₂	—	47	2	no	—
40	C₂×C₃₂	Dic₄₇	1	2	yes	—
41	C₄×C₁₆	—	47	1	no	—
42	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
43	C₈²	—	47	1	no	—
44	C₄₇	#7330	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₁₆	—	47	2	no	—
46	C₂×C₁₆	C₄×D₄₇	1	1	yes	characteristic
47	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
48	C₄×C₈	—	47	2	no	—
49	C₄×C₈	C₄×D₄₇	1	1	yes	characteristic
50	C₃₂	—	47	4	no	—
51	C₃₂	C₄₇⋊C₈	1	4	yes	complemented
52	C₂×C₈	—	94	2	no	—
53	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
54	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
55	C₄²	C₈×D₄₇	1	1	yes	characteristic
56	C₁₆	—	94	2	no	—
57	C₁₆	C₈×D₄₇	1	2	yes	—
58	C₁₆	C₂×C₄₇⋊C₈	1	2	yes	—
59	C₂×C₄	C₈×Dic₄₇	1	1	yes	characteristic
60	C₂×C₄	#13093963	1	2	yes	—
61	C₈	—	188	2	no	—
62	C₈	C₈×Dic₄₇	1	1	yes	characteristic
63	C₈	C₄×C₄₇⋊C₈	1	1	yes	characteristic
64	C₈	#13093963	1	2	yes	—
65	C₂²	#15688898	1	1	yes	characteristic
66	C₄	—	2	2	no	—
67	C₄	C₈×C₄₇⋊C₈	1	1	yes	characteristic
68	C₄	#15688898	1	1	yes	characteristic
69	C₂	#20241304	1	1	yes	characteristic
70	C₂	#20241523	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₄×Dic₄₇
commutator	C₄×C₄₇	C₈²
Frattini	C₄×C₁₆	C₂×D₄₇
Fitting	C₄₇×#1397	C₂
socle, prime radical	C₂²×C₄₇	#346
cosocle	C₄×C₁₆×C₄₇	C₂²
hypercenter	#1397	D₄₇
hypocenter	C₄₇	#7330
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ #1397

Complements

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

N	H ≅ G/N	type
11. C₄₇⋊C₃₂	C₈	semidirect
15. C₃₂×C₄₇	C₈	semidirect
44. C₄₇	#7330	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
#20241522	2
#20241304	2
C₄₇×#1397	2
#7330	47