Subgroups — #26131022

Order 12032.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245670	C₂	1	4	yes	maximal
2	#20243006	C₂	1	2	yes	maximal
3	C₈×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
4	#15689528	—	2	2	no	—
5	#15689528	C₂²	1	4	yes	—
6	#15689503	—	2	2	no	—
7	#15688951	—	2	8	no	—
8	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
9	C₈²×C₄₇	C₂²	1	1	yes	characteristic
10	C₂×Dic₁₈₈	—	4	4	no	—
11	#13094024	—	2	4	no	—
12	#13093958	—	2	8	no	—
13	#13093958	—	4	4	no	—
14	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
15	C₄×C₈×C₄₇	D₄	1	2	yes	—
16	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
17	C₂×Dic₉₄	—	4	4	no	—
18	C₂×C₈×C₄₇	D₈	1	4	yes	—
19	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
20	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
21	C₄⋊Dic₄₇	—	4	4	no	—
22	C₄⋊Dic₄₇	—	4	4	no	—
23	Dic₁₈₈	—	8	8	no	—
24	C₁₆×C₄₇	Q₁₆	1	8	yes	complemented
25	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
26	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
27	C₂×Dic₄₇	—	8	4	no	—
28	Dic₉₄	—	8	8	no	—
29	C₈×C₄₇	SD₃₂	1	8	yes	—
30	C₈×C₄₇	C₂×Q₁₆	1	4	yes	—
31	#12523	—	47	1	no	maximal, Sylow
32	C₂²×C₄₇	#493	1	1	yes	characteristic
33	C₄×C₄₇	#494	1	2	yes	—
34	C₄×C₄₇	C₂×SD₃₂	1	4	yes	—
35	Dic₄₇	—	16	4	no	—
36	#2254	—	47	4	no	—
37	#1713	—	47	2	no	—
38	C₈×C₁₆	D₄₇	1	1	yes	characteristic
39	C₂×C₄₇	#1713	1	1	yes	characteristic
40	C₂×C₄₇	#2248	1	2	yes	—
41	#500	—	47	4	no	—
42	#500	—	94	2	no	—
43	#494	—	94	2	no	—
44	#367	—	94	8	no	—
45	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
46	C₈²	C₂×D₄₇	1	1	yes	characteristic
47	C₄₇	#12523	1	1	yes	characteristic, complemented, Sylow
48	C₂×Q₁₆	—	188	4	no	—
49	C₄⋊Q₈	—	94	4	no	—
50	C₂×C₁₆	D₁₈₈	1	4	yes	—
51	#107	—	94	8	no	—
52	#107	—	188	4	no	—
53	C₄×C₈	D₁₈₈	1	2	yes	—
54	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
55	C₂×Q₈	—	188	4	no	—
56	Q₁₆	—	376	8	no	—
57	C₂×C₈	D₃₇₆	1	4	yes	—
58	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
59	C₄⋊C₄	—	188	4	no	—
60	C₄⋊C₄	—	188	4	no	—
61	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
62	C₁₆	Dic₁₈₈	1	8	yes	complemented
63	Q₈	—	376	8	no	—
64	C₂×C₄	—	376	4	no	—
65	C₂×C₄	#13094021	1	1	yes	characteristic
66	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
67	C₈	#13093969	1	8	yes	—
68	C₈	C₂×Dic₁₈₈	1	4	yes	—
69	C₂²	#15689500	1	1	yes	characteristic
70	C₄	—	752	4	no	—
71	C₄	#15689503	1	2	yes	—
72	C₄	C₂×#13093969	1	4	yes	—
73	C₂	#20243006	1	1	yes	characteristic
74	C₂	#20245644	1	2	yes	—

	subgroup	G/H
center	C₂²	#15689500
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₈×C₁₆×C₄₇	C₂
socle, prime radical	C₂²×C₄₇	#493
hypercenter	C₈×C₁₆	D₄₇
hypocenter	C₄₇	#12523
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
24. C₁₆×C₄₇	Q₁₆	semidirect
47. C₄₇	#12523	semidirect
62. C₁₆	Dic₁₈₈	semidirect

Maximal subgroups

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

subgroup	index
#20245670	2
#20243006	2
C₈×C₁₆×C₄₇	2
#12523	47