Group #20291260 — subgroups

Subgroups — #20291260

Order 6784.

424 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	(C₂×#6157)⋊Q₈	C₂	1	4	yes	maximal
2	#16811572	C₂	1	8	yes	maximal
3	#16811568	C₂	1	2	yes	maximal
4	C₄⋊Q₈×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#6157	C₂²	1	4	yes	—
6	C₂×C₄×#6157	C₂²	1	4	yes	—
7	(C₂×#6157)⋊C₄	C₂²	1	8	yes	—
8	(C₂×#6157)⋊C₄	C₂²	1	4	yes	—
9	(C₂×#6157)⋊C₄	C₂²	1	8	yes	—
10	C₂×Q₈×D₅₃	C₂²	1	2	yes	—
11	#13133217	C₄	1	1	yes	characteristic, complemented
12	#13133216	C₄	1	4	yes	complemented
13	#13133215	C₄	1	2	yes	complemented
14	C₄⋊C₄×D₅₃	C₂²	1	4	yes	—
15	C₄²×D₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic, complemented
17	C₂²×#6157	C₂³	1	8	yes	—
18	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
19	Q₈×D₅₃	—	4	2	no	—
20	Q₈×D₅₃	—	4	2	no	—
21	C₂×C₄×D₅₃	C₂³	1	2	yes	—
22	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
23	C₂×C₄×D₅₃	C₂³	1	4	yes	—
24	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
25	C₂×Dic₁₀₆	C₂×C₄	1	4	yes	—
26	C₄⋊#6157	—	4	4	no	—
27	C₄×#6157	—	4	4	no	—
28	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
29	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
31	#11719919	C₂×C₄	1	8	yes	—
32	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
34	C₂²×D₅₃	C₂⁴	1	1	yes	characteristic
35	C₂×#6157	—	2	8	no	—
36	C₂×#6157	—	2	16	no	—
37	Q₈×C₅₃	—	4	2	no	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
41	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
43	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
44	C₄×D₅₃	—	2	4	no	—
45	C₄×D₅₃	—	2	4	no	—
46	C₄×D₅₃	—	2	2	no	—
47	C₄×D₅₃	—	2	4	no	—
48	Dic₁₀₆	—	4	4	no	—
49	Dic₁₀₆	—	4	2	no	—
50	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
51	C₂×D₅₃	H(5,𝔽₂)	1	2	yes	—
52	C₂×D₅₃	2₋¹⁺⁴	1	4	yes	—
53	#6157	—	4	8	no	—
54	C₄×C₅₃	—	2	1	no	—
55	C₄×C₅₃	—	2	4	no	—
56	C₄×C₅₃	—	2	2	no	—
57	Dic₅₃	—	2	1	no	—
58	Dic₅₃	—	2	4	no	—
59	Dic₅₃	—	2	2	no	—
60	#2381	—	53	1	no	maximal, Sylow
61	C₂×C₅₃	#518	1	1	yes	characteristic
62	C₂×C₅₃	#519	1	2	yes	—
63	D₅₃	#563	1	4	yes	—
64	C₂×C₄⋊Q₈	—	53	1	no	—
65	#391	—	53	4	no	—
66	#387	—	53	8	no	—
67	#383	—	53	2	no	—
68	C₅₃	#2381	1	1	yes	characteristic, complemented, Sylow
69	C₂²×Q₈	—	53	2	no	—
70	C₄⋊Q₈	—	53	2	no	—
71	C₄⋊Q₈	—	53	4	no	—
72	C₄⋊Q₈	—	53	1	no	—
73	C₄⋊Q₈	#6157	1	1	yes	characteristic, complemented
74	C₂×C₄⋊C₄	—	53	4	no	—
75	C₂×C₄⋊C₄	—	53	4	no	—
76	C₂×C₄²	—	53	1	no	—
77	C₂×C₄²	—	53	4	no	—
78	#95	—	53	8	no	—
79	#95	—	53	4	no	—
80	#95	—	53	8	no	—
81	C₂×Q₈	—	53	4	no	—
82	C₂×Q₈	—	53	2	no	—
83	C₂×Q₈	—	212	2	no	—
84	C₂×Q₈	—	212	2	no	—
85	C₂×Q₈	C₂×#6157	1	2	yes	—
86	C₂²×C₄	—	53	2	no	—
87	C₂²×C₄	—	53	1	no	—
88	C₂²×C₄	—	53	4	no	—
89	C₂²×C₄	—	53	8	no	—
90	C₄⋊C₄	—	53	8	no	—
91	C₄⋊C₄	—	53	4	no	—
92	C₄⋊C₄	—	212	4	no	—
93	C₄⋊C₄	C₂×#6157	1	4	yes	—
94	C₄²	—	53	1	no	—
95	C₄²	—	53	2	no	—
96	C₄²	—	212	4	no	—
97	C₄²	C₂×#6157	1	1	yes	characteristic
98	C₂³	—	53	1	no	—
99	Q₈	—	4	2	no	—
100	Q₈	—	212	2	no	—
101	Q₈	—	212	4	no	—
102	C₂×C₄	—	53	2	no	—
103	C₂×C₄	—	53	1	no	—
104	C₂×C₄	—	53	4	no	—
105	C₂×C₄	—	106	4	no	—
106	C₂×C₄	—	106	4	no	—
107	C₂×C₄	—	106	4	no	—
108	C₂×C₄	—	106	2	no	—
109	C₂×C₄	—	106	8	no	—
110	C₂×C₄	—	106	16	no	—
111	C₂×C₄	C₂²×#6157	1	2	yes	—
112	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
113	C₂×C₄	C₂²×#6157	1	4	yes	—
114	C₂²	—	53	2	no	—
115	C₂²	—	53	4	no	—
116	C₂²	C₂³×#6157	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	2	4	no	—
119	C₄	—	2	2	no	—
120	C₄	—	106	4	no	—
121	C₄	—	106	1	no	—
122	C₄	—	106	2	no	—
123	C₄	—	212	8	no	—
124	C₂	—	53	4	no	—
125	C₂	#16811690	1	1	yes	characteristic
126	C₂	#16811691	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

(C₂×#6157)⋊Q₈

C₂

yes

maximal

#16811572

C₂

yes

maximal

#16811568

C₂

yes

maximal

C₄⋊Q₈×D₅₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#6157

C₂²

yes

—

C₂×C₄×#6157

C₂²

yes

—

(C₂×#6157)⋊C₄

C₂²

yes

—

(C₂×#6157)⋊C₄

C₂²

yes

—

(C₂×#6157)⋊C₄

C₂²

yes

—

C₂×Q₈×D₅₃

C₂²

yes

—

#13133217

C₄

yes

characteristic, complemented

#13133216

C₄

yes

complemented

#13133215

C₄

yes

complemented

C₄⋊C₄×D₅₃

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

characteristic, complemented

C₂²×#6157

C₂³

yes

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

Q₈×D₅₃

—

Q₈×D₅₃

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₄⋊#6157

—

C₄×#6157

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂×C₄

yes

—

C₂²×D₅₃

C₂⁴

yes

characteristic

C₂×#6157

—

C₂×#6157

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

H(5,𝔽₂)

yes

—

C₂×D₅₃

2₋¹⁺⁴

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

D₅₃

#563

yes

—

C₂×C₄⋊Q₈

—

#391

—

#387

—

#383

—

C₅₃

#2381

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#6157

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

#95

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

212

—

C₂×Q₈

—

212

—

C₂×Q₈

C₂×#6157

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

212

—

C₄⋊C₄

C₂×#6157

yes

—

C₄²

—

C₄²

—

C₄²

—

212

—

C₄²

C₂×#6157

yes

characteristic

C₂³

—

Q₈

—

100

Q₈

—

212

—

101

Q₈

—

212

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

106

—

108

C₂×C₄

—

106

—

109

C₂×C₄

—

106

—

110

C₂×C₄

—

106

—

111

C₂×C₄

C₂²×#6157

yes

—

112

C₂×C₄

C₂²×#6157

yes

characteristic

113

C₂×C₄

C₂²×#6157

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₂³×#6157

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

106

—

121

C₄

—

106

—

122

C₄

—

106

—

123

C₄

—

212

—

124

C₂

—

125

C₂

#16811690

yes

characteristic

126

C₂

#16811691

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₄⋊Q₈×C₅₃	C₄
hypercenter	C₄⋊Q₈	#6157
hypocenter	C₅₃	#2381
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#6157

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₄⋊Q₈×C₅₃

C₄

C₄⋊Q₈

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #13133217	C₄	semidirect
12. #13133216	C₄	semidirect
13. #13133215	C₄	semidirect
16. C₄⋊Q₈×C₅₃	C₄	semidirect
68. C₅₃	#2381	semidirect
73. C₄⋊Q₈	#6157	semidirect

H ≅ G/N

type

11. #13133217

C₄

semidirect

12. #13133216

C₄

semidirect

13. #13133215

C₄

semidirect

16. C₄⋊Q₈×C₅₃

C₄

semidirect

68. C₅₃

#2381

semidirect

73. C₄⋊Q₈

#6157

semidirect

Subgroups — #20291260

Conjugacy classes of subgroups

Complements