Group #20290078 — subgroups

Subgroups — #20290078

Order 6784.

202 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 60 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	#16811645	C₂	1	1	yes	maximal, characteristic
2	#16811641	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#88324	C₂	1	4	yes	maximal, complemented
4	#16811178	C₂	1	1	yes	maximal, characteristic
5	#13133284	—	2	1	no	—
6	C₄⋊#88324	C₂²	1	2	yes	—
7	#13133274	—	2	1	no	—
8	#13133274	—	2	1	no	—
9	#13133272	—	2	1	no	—
10	C₄×#88324	C₂²	1	2	yes	—
11	#13133217	C₄	1	1	yes	characteristic
12	#13133208	—	2	1	no	—
13	#13133197	—	2	1	no	—
14	#13133193	C₄	1	2	yes	—
15	Q₈×Dic₅₃	C₂²	1	2	yes	—
16	#13133182	—	2	1	no	—
17	#13133175	—	2	1	no	—
18	C₄²×D₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
23	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
24	C₂×Dic₁₀₆	—	2	1	no	—
25	C₂×#88324	—	2	2	no	—
26	C₂×#88324	—	2	2	no	—
27	C₄⋊C₄×C₅₃	—	2	1	no	—
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	#11719921	—	2	2	no	—
31	C₄⋊Dic₅₃	—	2	1	no	—
32	C₄⋊Dic₅₃	—	2	2	no	—
33	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
34	#11719919	—	2	2	no	—
35	C₄×Dic₅₃	—	2	2	no	—
36	C₄×Dic₅₃	D₄	1	1	yes	characteristic
37	C₄×Dic₅₃	D₄	1	1	yes	characteristic
38	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
39	C₂²×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	Q₈×C₅₃	—	2	4	no	—
41	C₂×C₄×C₅₃	—	2	2	no	—
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₅₃	—	2	2	no	—
46	C₂×Dic₅₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
49	C₄×D₅₃	—	2	2	no	—
50	C₄×D₅₃	—	2	2	no	—
51	C₄×D₅₃	—	4	1	no	—
52	Dic₁₀₆	—	4	2	no	—
53	#88324	—	4	2	no	—
54	#88324	—	4	2	no	—
55	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×D₅₃	—	2	1	no	—
57	C₂×D₅₃	—	2	1	no	—
58	C₂×D₅₃	—	2	1	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	—	4	2	no	—
62	C₄×C₅₃	#131	1	2	yes	—
63	Dic₅₃	—	2	1	no	—
64	Dic₅₃	—	2	1	no	—
65	Dic₅₃	—	4	2	no	—
66	Dic₅₃	#137	1	2	yes	—
67	#1497	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	#408	1	1	yes	characteristic
69	C₂×C₅₃	#419	1	1	yes	characteristic
70	C₂×C₅₃	#421	1	1	yes	characteristic
71	D₅₃	—	4	1	no	—
72	#533	—	53	1	no	—
73	#432	—	53	1	no	—
74	#431	—	53	1	no	—
75	Q₈⋊C₈	—	53	4	no	—
76	C₅₃	#1497	1	1	yes	characteristic, complemented, Sylow
77	C₄⋊Q₈	—	53	1	no	—
78	C₄⋊Q₈	#6157	1	1	yes	characteristic
79	#125	—	106	1	no	—
80	C₂²⋊Q₈	—	53	2	no	—
81	C₂²⋊Q₈	—	106	1	no	—
82	C₄×Q₈	—	53	2	no	—
83	C₄×Q₈	—	106	1	no	—
84	#117	—	106	1	no	—
85	C₂×C₄²	—	53	1	no	—
86	C₄⋊C₈	—	53	2	no	—
87	C₄⋊C₈	—	106	1	no	—
88	C₂²⋊C₈	—	106	1	no	—
89	C₂²⋊C₈	—	106	1	no	—
90	#97	—	106	1	no	—
91	C₄×C₈	—	53	2	no	—
92	C₂×Q₈	—	106	1	no	—
93	C₂×Q₈	C₂×#6157	1	2	yes	—
94	C₂²×C₄	—	53	1	no	—
95	C₂²×C₄	—	53	1	no	—
96	C₂²×C₄	—	53	1	no	—
97	C₂×C₈	—	106	2	no	—
98	C₂×C₈	—	106	2	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	53	2	no	—
102	C₄⋊C₄	—	106	2	no	—
103	C₄⋊C₄	—	106	2	no	—
104	C₄⋊C₄	—	106	1	no	—
105	C₂²⋊C₄	—	106	2	no	—
106	C₄²	—	53	1	no	—
107	C₄²	—	53	1	no	—
108	C₄²	—	53	1	no	—
109	C₄²	—	106	2	no	—
110	C₄²	C₂×#6157	1	1	yes	characteristic
111	C₂³	—	53	1	no	—
112	Q₈	—	2	4	no	—
113	Q₈	—	212	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	53	1	no	—
116	C₂×C₄	—	53	1	no	—
117	C₂×C₄	—	53	1	no	—
118	C₂×C₄	—	106	2	no	—
119	C₂×C₄	—	106	2	no	—
120	C₂×C₄	—	106	2	no	—
121	C₂×C₄	—	212	1	no	—
122	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
124	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
125	C₈	—	212	2	no	—
126	C₈	—	212	2	no	—
127	C₂²	—	106	1	no	—
128	C₂²	—	106	1	no	—
129	C₂²	—	106	1	no	—
130	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	4	2	no	—
134	C₄	—	53	2	no	—
135	C₄	—	106	1	no	—
136	C₄	—	106	1	no	—
137	C₄	—	212	2	no	—
138	C₄	#13133304	1	2	yes	—
139	C₂	—	212	1	no	—
140	C₂	#16811597	1	1	yes	characteristic
141	C₂	#16811616	1	1	yes	characteristic
142	C₂	#16811622	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811645

C₂

yes

maximal, characteristic

#16811641

C₂

yes

maximal, characteristic

Q₈⋊#88324

C₂

yes

maximal, complemented

#16811178

C₂

yes

maximal, characteristic

#13133284

—

C₄⋊#88324

C₂²

yes

—

—

—

—

C₂²

yes

—

#13133217

C₄

yes

—

—

C₄

yes

—

Q₈×Dic₅₃

C₂²

yes

—

#13133182

—

#13133175

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×#88324

—

C₂×#88324

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂²×D₅₃

C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

#88324

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#131

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#137

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#533

—

#432

—

#431

—

Q₈⋊C₈

—

C₅₃

#1497

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

#6157

yes

characteristic

#125

—

106

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

C₄×Q₈

—

106

—

#117

—

106

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

#97

—

106

—

C₄×C₈

—

C₂×Q₈

—

106

—

C₂×Q₈

C₂×#6157

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

106

—

103

C₄⋊C₄

—

106

—

104

C₄⋊C₄

—

106

—

105

C₂²⋊C₄

—

106

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

106

—

110

C₄²

C₂×#6157

yes

characteristic

111

C₂³

—

112

Q₈

—

113

Q₈

—

212

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

106

—

119

C₂×C₄

—

106

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

212

—

122

C₂×C₄

C₂²⋊#6157

yes

characteristic

123

C₂×C₄

C₂²⋊#6157

yes

characteristic

124

C₂×C₄

C₂²×#6157

yes

characteristic

125

C₈

—

212

—

126

C₈

—

212

—

127

C₂²

—

106

—

128

C₂²

—

106

—

129

C₂²

—

106

—

130

C₂²

C₂×C₂²⋊#6157

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

106

—

136

C₄

—

106

—

137

C₄

—

212

—

138

C₄

#13133304

yes

—

139

C₂

—

212

—

140

C₂

#16811597

yes

characteristic

141

C₂

#16811616

yes

characteristic

142

C₂

#16811622

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6157
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₄⋊Q₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6157
hypocenter	C₅₃	#1497
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6157

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₄⋊Q₈×C₅₃

C₄

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

C₄⋊Q₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. Q₈⋊#88324	C₂	semidirect
76. C₅₃	#1497	semidirect

H ≅ G/N

type

3. Q₈⋊#88324

C₂

semidirect

76. C₅₃

#1497

semidirect

Subgroups — #20290078

Conjugacy classes of subgroups

Complements