Group #20285264 — subgroups

Subgroups — #20285264

Order 6784.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#16811180	C₂	1	1	yes	maximal, characteristic
2	#16811033	C₂	1	4	yes	maximal, complemented
3	#16811022	C₂	1	4	yes	maximal, complemented
4	#16810663	C₂	1	4	yes	maximal
5	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#11719917	C₂²	1	2	yes	—
8	#13133217	C₂²	1	2	yes	—
9	#13133209	C₂²	1	2	yes	—
10	#13133196	—	2	2	no	—
11	C₄×Dic₁₀₆	—	2	2	no	—
12	#13133176	C₂²	1	2	yes	—
13	#13133150	C₂²	1	8	yes	—
14	#13133148	C₂²	1	8	yes	—
15	#13133128	C₂²	1	2	yes	—
16	C₄×#88322	C₂²	1	2	yes	—
17	#125×C₅₃	—	2	2	no	—
18	#125×C₅₃	C₂²	1	2	yes	—
19	#125×C₅₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
21	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
22	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	—	2	2	no	—
24	C₂²×C₄×C₅₃	D₄	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₀₆	—	2	2	no	—
27	C₄⋊C₄×C₅₃	—	2	4	no	—
28	C₄⋊C₄×C₅₃	—	2	4	no	—
29	C₄⋊C₄×C₅₃	—	2	2	no	—
30	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
31	C₄²×C₅₃	D₄	1	2	yes	—
32	C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₅₃	—	2	2	no	—
35	C₄⋊Dic₅₃	—	2	2	no	—
36	C₄⋊Dic₅₃	C₂³	1	4	yes	—
37	#11719919	—	2	4	no	—
38	C₄×Dic₅₃	—	2	2	no	—
39	#11719917	Q₈	1	8	yes	complemented
40	C₂×#88322	C₂³	1	4	yes	—
41	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	—	2	4	no	—
43	C₂×C₄×C₅₃	—	4	2	no	—
44	C₂×C₄×C₅₃	—	4	4	no	—
45	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	4	no	—
51	Dic₁₀₆	—	4	4	no	—
52	#88322	C₂×Q₈	1	8	yes	—
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	—	4	4	no	—
59	C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
60	C₄×C₅₃	C₂²×Q₈	1	2	yes	—
61	Dic₅₃	—	4	4	no	—
62	#3166	—	53	1	no	maximal, Sylow
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₅₃	#575	1	2	yes	—
66	#533	—	53	1	no	—
67	C₂×#125	D₅₃	1	1	yes	characteristic
68	#501	—	53	4	no	—
69	#499	—	53	4	no	—
70	#428	—	53	4	no	—
71	C₄×M₁₆	—	53	1	no	—
72	C₅₃	#3166	1	1	yes	characteristic, complemented, Sylow
73	C₂×M₁₆	—	53	2	no	—
74	C₄⋊Q₈	—	53	2	no	—
75	#125	—	2	2	no	—
76	#125	—	53	2	no	—
77	#125	C₂×D₅₃	1	2	yes	—
78	#125	C₂×D₅₃	1	2	yes	—
79	C₂²⋊Q₈	—	106	2	no	—
80	C₄×Q₈	—	106	2	no	—
81	#117	—	53	2	no	—
82	C₂×C₄⋊C₄	—	2	2	no	—
83	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
84	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
85	#107	—	53	8	no	—
86	#106	—	53	8	no	—
87	#97	—	53	2	no	—
88	C₄×C₈	—	53	2	no	—
89	C₂×Q₈	—	106	2	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
92	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
93	M₁₆	—	53	8	no	—
94	C₂×C₈	—	53	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	53	4	no	—
99	C₄⋊C₄	—	106	4	no	—
100	C₄⋊C₄	—	106	2	no	—
101	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
102	C₂²⋊C₄	—	106	2	no	—
103	C₄²	—	106	2	no	—
104	C₄²	C₂²⋊D₅₃	1	2	yes	—
105	C₄²	C₂²×D₅₃	1	1	yes	characteristic
106	C₄²	C₂²×D₅₃	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
108	Q₈	—	212	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	106	4	no	—
113	C₂×C₄	Q₈×D₅₃	1	4	yes	—
114	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
117	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
118	C₈	—	53	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#13133215	1	2	yes	—
121	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	212	4	no	—
126	C₄	#13133215	1	2	yes	—
127	C₄	C₂×Q₈×D₅₃	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#13133215	1	1	yes	characteristic
130	C₂	#16811421	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811180

C₂

yes

maximal, characteristic

#16811033

C₂

yes

maximal, complemented

#16811022

C₂

yes

maximal, complemented

#16810663

C₂

yes

maximal

C₄×#11719917

C₂

yes

maximal, characteristic

C₂×#125×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

#13133217

C₂²

yes

—

#13133209

C₂²

yes

—

#13133196

—

C₄×Dic₁₀₆

—

#13133176

C₂²

yes

—

#13133150

C₂²

yes

—

#13133148

C₂²

yes

—

#13133128

C₂²

yes

—

C₄×#88322

C₂²

yes

—

#125×C₅₃

—

#125×C₅₃

C₂²

yes

—

#125×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

#11719919

—

C₄×Dic₅₃

—

#11719917

Q₈

yes

complemented

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

#88322

C₂×Q₈

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊Q₈

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄⋊Q₈

yes

—

C₄×C₅₃

C₂²×Q₈

yes

—

Dic₅₃

—

#3166

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₅₃

#575

yes

—

#533

—

C₂×#125

D₅₃

yes

—

—

—

C₄×M₁₆

—

C₅₃

#3166

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

#125

—

#125

C₂×D₅₃

yes

—

#125

C₂×D₅₃

yes

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

106

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

—

—

C₄×C₈

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

100

C₄⋊C₄

—

106

—

101

C₄⋊C₄

C₂²×D₅₃

yes

—

102

C₂²⋊C₄

—

106

—

103

C₄²

—

106

—

104

C₄²

C₂²⋊D₅₃

yes

—

105

C₄²

C₂²×D₅₃

yes

characteristic

106

C₄²

C₂²×D₅₃

yes

characteristic

107

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

108

Q₈

—

212

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

106

—

113

C₂×C₄

Q₈×D₅₃

yes

—

114

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

115

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

116

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

117

C₂×C₄

C₂³×D₅₃

yes

characteristic

118

C₈

—

119

C₂²

—

120

C₂²

#13133215

yes

—

121

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

212

—

126

C₄

#13133215

yes

—

127

C₄

C₂×Q₈×D₅₃

yes

—

128

C₂

—

129

C₂

C₂×#13133215

yes

characteristic

130

C₂

#16811421

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#125×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	C₂×#125	D₅₃
hypocenter	C₅₃	#3166
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×#125×C₅₃

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #16811033	C₂	semidirect
3. #16811022	C₂	semidirect
39. #11719917	Q₈	semidirect
72. C₅₃	#3166	semidirect

H ≅ G/N

type

2. #16811033

C₂

semidirect

3. #16811022

C₂

semidirect

39. #11719917

Q₈

semidirect

72. C₅₃

#3166

semidirect

Subgroups — #20285264

Conjugacy classes of subgroups

Complements