Subgroups — #26103938

Order 12032.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20241586	C₂	1	1	yes	maximal, characteristic
2	#20241464	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#2254	C₂	1	1	yes	maximal, characteristic
4	#15688957	—	2	1	no	—
5	#15688944	—	2	1	no	—
6	C₄₇×#500	—	2	1	no	—
7	C₄₇×#367	—	2	1	no	—
8	C₄₇×#367	C₄	1	2	yes	—
9	C₄×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
10	#13093964	—	4	1	no	—
11	C₄₇⋊C₃₂	—	4	1	no	—
12	C₄⋊Q₈×C₄₇	—	4	1	no	—
13	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₄₇	D₄	1	1	yes	characteristic
15	#107×C₄₇	—	2	2	no	—
16	#107×C₄₇	—	4	1	no	—
17	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
18	C₂×Q₈×C₄₇	—	8	1	no	—
19	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
21	C₄⋊C₄×C₄₇	—	4	2	no	—
22	C₄⋊C₄×C₄₇	—	8	1	no	—
23	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
24	C₁₆×C₄₇	—	2	1	no	—
25	C₁₆×C₄₇	SD₁₆	1	1	yes	characteristic
26	C₁₆×C₄₇	Q₁₆	1	1	yes	characteristic
27	C₄₇⋊C₁₆	—	8	1	no	—
28	Q₈×C₄₇	—	16	2	no	—
29	C₂×C₄×C₄₇	—	2	1	no	—
30	C₂×C₄×C₄₇	—	8	2	no	—
31	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
32	C₈×C₄₇	#101	1	1	yes	characteristic
33	C₈×C₄₇	#103	1	1	yes	characteristic
34	C₈×C₄₇	SD₃₂	1	1	yes	characteristic
35	C₈×C₄₇	Q₃₂	1	1	yes	characteristic
36	#7453	—	47	1	no	maximal, Sylow
37	C₂²×C₄₇	#361	1	1	yes	characteristic
38	C₄×C₄₇	—	2	2	no	—
39	C₄×C₄₇	—	16	2	no	—
40	C₄×C₄₇	#332	1	1	yes	characteristic
41	C₄×C₄₇	#358	1	1	yes	characteristic
42	#2254	D₄₇	1	1	yes	characteristic
43	#1422	—	47	1	no	—
44	#1373	—	47	1	no	—
45	C₂×C₄₇	—	2	1	no	—
46	C₂×C₄₇	#1365	1	1	yes	characteristic
47	#500	—	2	1	no	—
48	M₆₄	—	94	1	no	—
49	#367	—	2	1	no	—
50	#367	Dic₄₇	1	2	yes	—
51	#364	—	94	1	no	—
52	C₄×C₁₆	C₂×D₄₇	1	1	yes	characteristic
53	C₄₇	#7453	1	1	yes	characteristic, complemented, Sylow
54	C₄⋊Q₈	—	4	1	no	—
55	M₃₂	—	188	1	no	—
56	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
57	C₂×C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
58	#107	—	2	2	no	—
59	#107	—	4	1	no	—
60	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
61	C₃₂	—	188	1	no	—
62	C₂×Q₈	—	8	1	no	—
63	C₂×C₈	#10625334	1	1	yes	characteristic
64	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
65	C₄⋊C₄	—	4	2	no	—
66	C₄⋊C₄	—	8	1	no	—
67	C₄²	#10625335	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	—	376	1	no	—
70	C₁₆	#10625336	1	1	yes	characteristic
71	C₁₆	#10625337	1	1	yes	characteristic
72	Q₈	—	16	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	#13093945	1	1	yes	characteristic
76	C₈	#13093943	1	1	yes	characteristic
77	C₈	#13093948	1	1	yes	characteristic
78	C₈	#13093967	1	1	yes	characteristic
79	C₈	#13093970	1	1	yes	characteristic
80	C₂²	#15688936	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	16	2	no	—
83	C₄	#15688869	1	1	yes	characteristic
84	C₄	#15688927	1	1	yes	characteristic
85	C₂	—	2	1	no	—
86	C₂	#20241446	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20241446
commutator	C₂×C₁₆×C₄₇	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₄₇
Fitting	C₄₇×#2254	C₂
socle	C₂×C₄₇	#1365
prime radical	C₂²×C₄₇	#361
cosocle	C₄×C₁₆×C₄₇	C₂²
hypercenter	#2254	D₄₇
hypocenter	C₄₇	#7453
LCS	G ▷ C₂×C₁₆×C₄₇ ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2254

Complements

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

N	H ≅ G/N	type
53. C₄₇	#7453	semidirect

Maximal subgroups

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

subgroup	index
#20241586	2
#20241464	2
C₄₇×#2254	2
#7453	47