Subgroups — #26103989

Order 12032.

170 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245172	C₂	1	2	yes	maximal
2	C₂×C₄₇×M₆₄	C₂	1	1	yes	maximal, characteristic
3	#15689161	—	2	2	no	—
4	#15688951	C₄	1	4	yes	complemented
5	#15688949	—	2	2	no	—
6	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
7	C₄₇×M₆₄	—	2	2	no	—
8	C₂×C₃₂×C₄₇	C₄	1	2	yes	complemented
9	#13093983	—	4	2	no	—
10	#13093958	—	2	4	no	—
11	#13093956	—	4	2	no	—
12	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
15	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
16	C₂×C₁₆×C₄₇	Q₈	1	1	yes	characteristic
17	C₃₂×C₄₇	—	2	2	no	—
18	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
19	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
20	C₂×C₈×C₄₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
22	C₂×C₈×C₄₇	Q₁₆	1	1	yes	characteristic
23	C₂²⋊Dic₄₇	—	8	2	no	—
24	C₄⋊Dic₄₇	—	4	4	no	—
25	C₄×Dic₄₇	—	8	2	no	—
26	C₁₆×C₄₇	C₄²	1	1	yes	characteristic
27	C₁₆×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₁₆×C₄₇	C₄⋊C₄	1	2	yes	—
29	C₂³×C₄₇	D₁₆	1	1	yes	characteristic
30	C₂×C₄×C₄₇	#102	1	2	yes	—
31	C₂×C₄×C₄₇	#107	1	1	yes	characteristic
32	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
33	C₂×C₄×C₄₇	Q₃₂	1	1	yes	characteristic
34	C₂×Dic₄₇	—	8	4	no	—
35	C₈×C₄₇	#95	1	1	yes	characteristic
36	C₈×C₄₇	#103	1	2	yes	—
37	C₈×C₄₇	#106	1	1	yes	characteristic
38	#7475	—	47	1	no	maximal, Sylow
39	C₂²×C₄₇	—	4	1	no	—
40	C₂²×C₄₇	#357	1	2	yes	—
41	C₂²×C₄₇	#366	1	1	yes	characteristic
42	C₄×C₄₇	#340	1	1	yes	characteristic
43	C₄×C₄₇	#358	1	2	yes	—
44	C₄×C₄₇	#367	1	1	yes	characteristic
45	Dic₄₇	—	16	4	no	—
46	C₂×M₆₄	D₄₇	1	1	yes	characteristic
47	#2158	—	47	2	no	—
48	C₂×C₄₇	—	2	2	no	—
49	C₂×C₄₇	—	2	1	no	—
50	C₂×C₄₇	#1380	1	1	yes	characteristic
51	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
52	#427	—	94	2	no	—
53	M₆₄	—	2	2	no	—
54	C₂×C₃₂	Dic₄₇	1	2	yes	complemented
55	#367	—	47	4	no	—
56	#366	—	94	2	no	—
57	C₄₇	#7475	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
59	#117	—	188	2	no	—
60	C₂×C₁₆	Dic₉₄	1	1	yes	characteristic
61	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
62	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
63	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
64	#107	—	94	4	no	—
65	#106	—	188	2	no	—
66	C₃₂	—	2	2	no	—
67	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
68	C₂×C₈	#10625325	1	2	yes	—
69	C₂×C₈	Dic₁₈₈	1	1	yes	characteristic
70	C₂×C₈	C₄⋊Dic₄₇	1	1	yes	characteristic
71	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
72	C₄⋊C₄	—	188	4	no	—
73	C₂²⋊C₄	—	376	2	no	—
74	C₄²	—	376	2	no	—
75	C₁₆	C₄×Dic₄₇	1	1	yes	characteristic
76	C₁₆	Dic₄₇⋊C₄	1	2	yes	—
77	C₁₆	C₂²⋊Dic₄₇	1	1	yes	characteristic
78	C₂³	D₇₅₂	1	1	yes	characteristic
79	C₂×C₄	—	376	4	no	—
80	C₂×C₄	#13093946	1	2	yes	—
81	C₂×C₄	#13093958	1	1	yes	characteristic
82	C₂×C₄	#13093969	1	2	yes	—
83	C₂×C₄	Dic₃₇₆	1	1	yes	characteristic
84	C₈	C₄₇⋊#95	1	1	yes	characteristic
85	C₈	#13093949	1	2	yes	—
86	C₈	#13093956	1	1	yes	characteristic
87	C₂²	—	4	1	no	—
88	C₂²	#15688925	1	2	yes	—
89	C₂²	#15688949	1	1	yes	characteristic
90	C₄	—	752	4	no	—
91	C₄	#15688886	1	1	yes	characteristic
92	C₄	#15688928	1	2	yes	—
93	C₄	#15688951	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	—	2	1	no	—
96	C₂	#20241480	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#15688951
commutator	C₁₆×C₄₇	C₄²
Frattini	C₂²×C₁₆	C₂×D₄₇
Fitting	C₂×C₄₇×M₆₄	C₂
socle	C₂×C₄₇	#1380
prime radical	C₂³×C₄₇	D₁₆
cosocle	C₂²×C₁₆×C₄₇	C₂²
hypercenter	C₂×M₆₄	D₄₇
hypocenter	C₄₇	#7475
LCS	G ▷ C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
4. #15688951	C₄	semidirect
8. C₂×C₃₂×C₄₇	C₄	semidirect
54. C₂×C₃₂	Dic₄₇	semidirect
57. C₄₇	#7475	semidirect

Maximal subgroups

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

subgroup	index
#20245172	2
C₂×C₄₇×M₆₄	2
#7475	47