Subgroups — #26110057

Order 12032.

390 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20243516	C₂	1	6	yes	maximal
2	C₂×C₈²×C₄₇	C₂	1	1	yes	maximal, characteristic
3	#15688993	C₂²	1	4	yes	—
4	#15688886	—	2	12	no	—
5	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
6	C₈²×C₄₇	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₄₇	—	2	12	no	—
8	C₂²×C₈×C₄₇	D₄	1	6	yes	—
9	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
10	C₄×C₈×C₄₇	C₂×C₄	1	6	yes	—
11	C₄×C₈×C₄₇	Q₈	1	6	yes	—
12	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
13	C₂²×Dic₄₇	—	4	4	no	—
14	C₂×C₈×C₄₇	—	2	12	no	—
15	C₂×C₈×C₄₇	C₄⋊C₄	1	12	yes	—
16	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
17	C₄²×C₄₇	C₂×Q₈	1	3	yes	—
18	C₄⋊Dic₄₇	—	4	24	no	—
19	C₂³×C₄₇	#127	1	1	yes	characteristic
20	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	3	yes	—
21	C₂×C₄×C₄₇	C₄⋊Q₈	1	3	yes	—
22	C₂×C₄×C₄₇	#135	1	12	yes	—
23	C₂×Dic₄₇	—	8	12	no	—
24	C₈×C₄₇	—	2	12	no	—
25	#8817	—	47	1	no	maximal, Sylow
26	C₂²×C₄₇	#384	1	1	yes	characteristic
27	C₂²×C₄₇	#495	1	6	yes	—
28	C₄×C₄₇	#427	1	6	yes	—
29	C₄×C₄₇	#499	1	6	yes	—
30	Dic₄₇	—	16	4	no	—
31	#1834	—	47	6	no	—
32	C₂×C₈²	D₄₇	1	1	yes	characteristic
33	C₂×C₄₇	#1711	1	4	yes	—
34	C₂×C₄₇	#1846	1	3	yes	—
35	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
36	#384	—	47	4	no	—
37	#340	—	94	12	no	—
38	C₈²	Dic₄₇	1	4	yes	complemented
39	C₄₇	#8817	1	1	yes	characteristic, complemented, Sylow
40	C₂²×C₈	D₁₈₈	1	6	yes	—
41	C₂×C₄⋊C₄	—	94	12	no	—
42	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
43	C₄×C₈	Dic₉₄	1	6	yes	—
44	C₄×C₈	C₂×Dic₄₇	1	6	yes	—
45	C₂²×C₄	—	188	4	no	—
46	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
47	C₂×C₈	—	2	12	no	—
48	C₂×C₈	C₄⋊Dic₄₇	1	12	yes	—
49	C₄⋊C₄	—	188	24	no	—
50	C₄²	C₂×Dic₉₄	1	3	yes	—
51	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
52	C₂³	#13094021	1	1	yes	characteristic
53	C₂×C₄	—	376	12	no	—
54	C₂×C₄	C₂×C₄⋊Dic₄₇	1	3	yes	—
55	C₂×C₄	#13094024	1	3	yes	—
56	C₂×C₄	#13094046	1	12	yes	—
57	C₈	—	2	12	no	—
58	C₂²	#15688993	1	1	yes	characteristic
59	C₂²	#15689507	1	6	yes	—
60	C₄	—	752	4	no	—
61	C₄	#15689161	1	6	yes	—
62	C₄	#15689525	1	6	yes	—
63	C₂	#20243000	1	4	yes	—
64	C₂	#20243552	1	3	yes	—

	subgroup	G/H
center	C₂³	#13094021
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₈²×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×C₈²	D₄₇
hypocenter	C₄₇	#8817
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
6. C₈²×C₄₇	C₄	semidirect
38. C₈²	Dic₄₇	semidirect
39. C₄₇	#8817	semidirect

Maximal subgroups

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

subgroup	index
#20243516	2
C₂×C₈²×C₄₇	2
#8817	47