Group #20283650 — subgroups

Subgroups — #20283650

Order 6784.

364 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 229 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	#16810754	C₂	1	4	yes	maximal, complemented
2	Q₈×#88322	C₂	1	4	yes	maximal, complemented
3	#16810650	C₂	1	2	yes	maximal
4	#16810647	C₂	1	1	yes	maximal, characteristic
5	#16810579	C₂	1	2	yes	maximal
6	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#533	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719917	C₂²	1	2	yes	—
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	C₄⋊#88322	C₂²	1	4	yes	—
12	C₄⋊#88322	C₂²	1	8	yes	—
13	#13133128	C₂²	1	4	yes	—
14	C₄×#88322	C₂²	1	2	yes	—
15	C₄×#88322	C₂²	1	2	yes	—
16	C₄×#88322	C₂²	1	4	yes	—
17	C₄⋊Q₈×C₅₃	C₄	1	2	yes	—
18	#125×C₅₃	C₄	1	2	yes	—
19	C₂²⋊Q₈×C₅₃	C₄	1	4	yes	—
20	C₄×Q₈×C₅₃	C₂²	1	4	yes	—
21	#117×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
27	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
28	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
29	C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
30	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄²×C₅₃	C₂³	1	2	yes	—
33	C₄²×C₅₃	C₂³	1	4	yes	—
34	#11719917	—	2	4	no	—
35	C₂×#88322	—	2	2	no	—
36	C₂×#88322	Q₈	1	4	yes	complemented
37	C₂×#88322	C₂³	1	2	yes	—
38	C₂×#88322	C₂³	1	4	yes	—
39	C₂×#88322	C₂³	1	1	yes	characteristic
40	C₂×#88322	C₂³	1	1	yes	characteristic
41	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	Q₈×C₅₃	—	2	8	no	—
43	C₂×C₄×C₅₃	—	2	4	no	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₅₃	#41	1	2	yes	—
50	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
51	#88322	—	2	2	no	—
52	#88322	—	2	4	no	—
53	#88322	C₂×Q₈	1	2	yes	—
54	#88322	C₂×Q₈	1	2	yes	—
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
57	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₅₃	—	2	2	no	—
59	C₄×C₅₃	—	2	8	no	—
60	C₄×C₅₃	C₄×Q₈	1	2	yes	—
61	C₄×C₅₃	#131	1	4	yes	—
62	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
64	#2960	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	2	no	—
66	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
67	C₂×C₅₃	C₂×#131	1	2	yes	—
68	#533	D₅₃	1	1	yes	characteristic
69	#446	—	53	4	no	—
70	C₈×Q₈	—	53	4	no	—
71	#424	—	53	2	no	—
72	#423	—	53	1	no	—
73	#405	—	53	2	no	—
74	C₂×C₄×C₈	—	53	1	no	—
75	C₅₃	#2960	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	53	2	no	—
77	C₂²×C₈	—	53	1	no	—
78	C₂²×C₈	—	53	1	no	—
79	C₄⋊Q₈	Dic₅₃	1	2	yes	—
80	#125	Dic₅₃	1	2	yes	—
81	C₂²⋊Q₈	Dic₅₃	1	4	yes	—
82	C₄×Q₈	C₂×D₅₃	1	4	yes	—
83	#117	C₂×D₅₃	1	2	yes	—
84	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
85	C₄⋊C₈	—	53	4	no	—
86	C₄⋊C₈	—	53	8	no	—
87	#97	—	53	4	no	—
88	C₄×C₈	—	53	2	no	—
89	C₄×C₈	—	53	2	no	—
90	C₄×C₈	—	53	4	no	—
91	C₂×Q₈	C₂×Dic₅₃	1	4	yes	—
92	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
93	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
94	M₁₆	—	106	4	no	—
95	C₂×C₈	—	53	4	no	—
96	C₂×C₈	—	53	1	no	—
97	C₂×C₈	—	53	1	no	—
98	C₂×C₈	—	53	2	no	—
99	C₂×C₈	—	53	4	no	—
100	C₂×C₈	—	106	2	no	—
101	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
102	C₄⋊C₄	C₂×Dic₅₃	1	8	yes	—
103	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
104	C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
105	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
106	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
107	C₄²	C₂²×D₅₃	1	2	yes	—
108	C₄²	C₂²×D₅₃	1	4	yes	—
109	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
110	Q₈	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	Q₈×D₅₃	1	2	yes	—
113	C₂×C₄	#11719949	1	2	yes	—
114	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
115	C₂×C₄	C₂²×Dic₅₃	1	8	yes	—
116	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
118	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
119	C₈	—	53	2	no	—
120	C₈	—	53	2	no	—
121	C₈	—	106	2	no	—
122	C₈	—	106	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	Q₈×Dic₅₃	1	2	yes	—
125	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	2	8	no	—
128	C₄	Q₈×Dic₅₃	1	2	yes	—
129	C₄	#13133224	1	4	yes	—
130	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
131	C₄	C₂×#11719949	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
134	C₂	C₂×#13133224	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810754

C₂

yes

maximal, complemented

Q₈×#88322

C₂

yes

maximal, complemented

#16810650

C₂

yes

maximal

#16810647

C₂

yes

maximal, characteristic

#16810579

C₂

yes

maximal

C₂×C₄×#88322

C₂

yes

maximal, characteristic

C₅₃×#533

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

characteristic

C₂²×#88322

C₂²

yes

characteristic

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

#13133128

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₄

yes

—

#125×C₅₃

C₄

yes

—

C₂²⋊Q₈×C₅₃

C₄

yes

—

C₄×Q₈×C₅₃

C₂²

yes

—

#117×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719917

—

C₂×#88322

—

C₂×#88322

Q₈

yes

complemented

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

—

—

C₂×Q₈

yes

—

#88322

C₂×Q₈

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄×Q₈

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

C₂²×Q₈

yes

characteristic

C₄×C₅₃

C₂×#41

yes

characteristic

#2960

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₅₃

C₂×#131

yes

—

#533

D₅₃

yes

characteristic

#446

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

—

C₅₃

#2960

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

Dic₅₃

yes

—

#125

Dic₅₃

yes

—

C₂²⋊Q₈

Dic₅₃

yes

—

C₄×Q₈

C₂×D₅₃

yes

—

#117

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

106

—

101

C₄⋊C₄

C₂×Dic₅₃

yes

—

102

C₄⋊C₄

C₂×Dic₅₃

yes

—

103

C₄⋊C₄

C₂×Dic₅₃

yes

—

104

C₂²⋊C₄

C₂×Dic₅₃

yes

—

105

C₄²

C₂×Dic₅₃

yes

characteristic

106

C₄²

C₂×Dic₅₃

yes

characteristic

107

C₄²

C₂²×D₅₃

yes

—

108

C₄²

C₂²×D₅₃

yes

—

109

C₂³

C₂²×Dic₅₃

yes

characteristic

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

Q₈×D₅₃

yes

—

113

C₂×C₄

#11719949

yes

—

114

C₂×C₄

C₂²×Dic₅₃

yes

—

115

C₂×C₄

C₂²×Dic₅₃

yes

—

116

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

117

C₂×C₄

C₂²×Dic₅₃

yes

—

118

C₂×C₄

C₂³×D₅₃

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

106

—

122

C₈

—

106

—

123

C₂²

—

124

C₂²

Q₈×Dic₅₃

yes

—

125

C₂²

C₂³×Dic₅₃

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

Q₈×Dic₅₃

yes

—

129

C₄

#13133224

yes

—

130

C₄

C₂×Q₈×D₅₃

yes

characteristic

131

C₄

C₂×#11719949

yes

characteristic

132

C₂

—

133

C₂

C₂×Q₈×Dic₅₃

yes

characteristic

134

C₂

C₂×#13133224

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#533	C₂
hypercenter	#533	D₅₃
hypocenter	C₅₃	#2960
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₅₃×#533

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #16810754	C₂	semidirect
2. Q₈×#88322	C₂	semidirect
36. C₂×#88322	Q₈	semidirect
75. C₅₃	#2960	semidirect

H ≅ G/N

type

1. #16810754

C₂

semidirect

2. Q₈×#88322

C₂

semidirect

36. C₂×#88322

Q₈

semidirect

75. C₅₃

#2960

semidirect

Subgroups — #20283650

Conjugacy classes of subgroups

Complements