Group #20285027 — subgroups

Subgroups — C₂×#16810975 #20285027

Order 6784.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×#13133217	C₂	1	1	yes	maximal, characteristic
2	#16810975	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133140	C₂	1	4	yes	maximal
4	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
7	C₂²×#88322	C₂²	1	2	yes	—
8	#13133217	—	2	2	no	—
9	#13133217	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₅₃	—	2	1	no	—
11	C₂×C₄⋊Dic₅₃	—	2	1	no	—
12	#13133140	C₂²	1	16	yes	—
13	C₄×#88322	C₂²	1	4	yes	—
14	C₂²×Q₈×C₅₃	—	2	1	no	—
15	C₄⋊Q₈×C₅₃	—	2	2	no	—
16	C₄⋊Q₈×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
18	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	—	2	4	no	—
21	C₂×Q₈×C₅₃	—	4	4	no	—
22	C₂²×C₄×C₅₃	—	2	2	no	—
23	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₂²×Dic₅₃	—	2	2	no	—
27	C₂×Dic₁₀₆	—	2	8	no	—
28	C₂×Dic₁₀₆	—	2	2	no	—
29	C₂×Dic₁₀₆	C₂³	1	4	yes	—
30	C₄⋊C₄×C₅₃	—	2	4	no	—
31	C₄⋊C₄×C₅₃	—	2	2	no	—
32	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
33	C₄²×C₅₃	D₄	1	2	yes	—
34	C₄²×C₅₃	C₂³	1	2	yes	—
35	C₄⋊Dic₅₃	—	2	4	no	—
36	C₄⋊Dic₅₃	—	2	4	no	—
37	C₂×#88322	—	2	4	no	—
38	C₂×#88322	C₂³	1	4	yes	—
39	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₅₃	—	4	8	no	—
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	—	4	4	no	—
43	C₂×C₄×C₅₃	SD₁₆	1	4	yes	—
44	C₂×C₄×C₅₃	Q₁₆	1	4	yes	—
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₅₃	#41	1	4	yes	—
49	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	4	no	—
51	C₂×Dic₅₃	—	4	4	no	—
52	Dic₁₀₆	—	2	8	no	—
53	Dic₁₀₆	—	4	4	no	—
54	#88322	—	2	4	no	—
55	C₂²×C₅₃	#124	1	2	yes	—
56	C₂²×C₅₃	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
58	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₅₃	—	4	4	no	—
60	C₄×C₅₃	#124	1	2	yes	—
61	C₄×C₅₃	C₂×SD₁₆	1	4	yes	—
62	C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
63	C₄×C₅₃	C₂×#41	1	2	yes	—
64	Dic₅₃	—	4	4	no	—
65	C₂×#487	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	#487	1	4	yes	complemented
67	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
68	C₂×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
69	C₂×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
70	C₂×C₄⋊Q₈	—	53	1	no	—
71	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
72	#487	—	53	8	no	—
73	C₂×#103	—	53	4	no	—
74	C₂×C₄×C₈	—	53	1	no	—
75	C₅₃	C₂×#487	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	—	2	1	no	—
77	C₂²×Q₈	—	53	2	no	—
78	C₂²×C₈	—	53	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	C₄⋊Q₈	—	53	4	no	—
81	C₄⋊Q₈	—	106	2	no	—
82	C₄⋊Q₈	C₂×D₅₃	1	4	yes	—
83	C₂×C₄⋊C₄	—	2	1	no	—
84	C₂×C₄⋊C₄	—	106	1	no	—
85	C₂×C₄⋊C₄	—	106	1	no	—
86	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
87	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
88	#103	—	53	16	no	—
89	C₄×C₈	—	53	4	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂×Q₈	—	4	4	no	—
92	C₂×Q₈	—	53	4	no	—
93	C₂×Q₈	—	106	2	no	—
94	C₂×Q₈	—	106	8	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	106	2	no	—
97	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
100	C₂×C₈	—	53	4	no	—
101	C₂×C₈	—	106	4	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	106	4	no	—
105	C₄⋊C₄	—	106	4	no	—
106	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
107	C₄²	C₂²⋊D₅₃	1	2	yes	—
108	C₄²	C₂²×D₅₃	1	2	yes	—
109	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
110	Q₈	—	4	8	no	—
111	Q₈	—	106	8	no	—
112	Q₈	—	212	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	106	4	no	—
116	C₂×C₄	—	212	4	no	—
117	C₂×C₄	#11719923	1	4	yes	—
118	C₂×C₄	#11719925	1	4	yes	—
119	C₂×C₄	#11719949	1	4	yes	—
120	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
122	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
123	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
124	C₈	—	106	4	no	—
125	C₂²	#13133206	1	2	yes	—
126	C₂²	C₂×#11719923	1	2	yes	—
127	C₂²	C₂×#11719925	1	2	yes	—
128	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
129	C₄	—	4	4	no	—
130	C₄	—	212	4	no	—
131	C₄	#13133206	1	2	yes	—
132	C₄	C₂×#11719923	1	4	yes	—
133	C₄	C₂×#11719925	1	4	yes	—
134	C₄	C₂×#11719949	1	2	yes	—
135	C₂	#16810975	1	4	yes	direct product factor
136	C₂	C₂×#13133206	1	1	yes	characteristic
137	C₂	C₂²×#11719923	1	1	yes	characteristic
138	C₂	C₂²×#11719925	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic

#16810975

C₂

yes

maximal, direct product factor

C₂×#13133140

C₂

yes

maximal

C₂×C₄×#88322

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

#13133217

—

#13133217

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

#13133140

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₂²×Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

#124

yes

—

C₂²×C₅₃

C₂×SD₁₆

yes

—

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

C₂×SD₁₆

yes

—

C₄×C₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

C₂²×SD₁₆

yes

characteristic

C₂×C₅₃

C₂²×Q₁₆

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

#487

—

C₂×#103

—

C₂×C₄×C₈

—

C₅₃

C₂×#487

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂²×C₄

C₂²⋊D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

106

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

—

106

—

106

C₄⋊C₄

C₂²×D₅₃

yes

—

107

C₄²

C₂²⋊D₅₃

yes

—

108

C₄²

C₂²×D₅₃

yes

—

109

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

110

Q₈

—

111

Q₈

—

106

—

112

Q₈

—

212

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

106

—

116

C₂×C₄

—

212

—

117

C₂×C₄

#11719923

yes

—

118

C₂×C₄

#11719925

yes

—

119

C₂×C₄

#11719949

yes

—

120

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

122

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

123

C₂×C₄

C₂³×D₅₃

yes

characteristic

124

C₈

—

106

—

125

C₂²

#13133206

yes

—

126

C₂²

C₂×#11719923

yes

—

127

C₂²

C₂×#11719925

yes

—

128

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

129

C₄

—

130

C₄

—

212

—

131

C₄

#13133206

yes

—

132

C₄

C₂×#11719923

yes

—

133

C₄

C₂×#11719925

yes

—

134

C₄

C₂×#11719949

yes

—

135

C₂

#16810975

yes

direct product factor

136

C₂

C₂×#13133206

yes

characteristic

137

C₂

C₂²×#11719923

yes

characteristic

138

C₂

C₂²×#11719925

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	C₂×#487
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

C₂×#487

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #16810975	C₂	direct (N × H)
66. C₂×C₅₃	#487	semidirect
75. C₅₃	C₂×#487	semidirect
135. C₂	#16810975	direct (N × H)

H ≅ G/N

type

2. #16810975

C₂

direct (N × H)

66. C₂×C₅₃

#487

semidirect

75. C₅₃

C₂×#487

semidirect

135. C₂

#16810975

direct (N × H)

Subgroups — C₂×#16810975 #20285027

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810975 #20285027

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810975 #20285027