Group #20290884 — subgroups

Subgroups — #20290884

Order 6784.

292 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 118 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×C₄⋊#88324	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#88324	C₂	1	1	yes	maximal, characteristic
3	#16811489	C₂	1	4	yes	maximal
4	C₂×Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88324	C₂²	1	2	yes	—
6	C₂²×#88324	C₂²	1	2	yes	—
7	C₄⋊#88324	—	2	2	no	—
8	C₄×#88324	—	2	2	no	—
9	Q₈×Dic₅₃	—	2	2	no	—
10	Q₈×Dic₅₃	—	2	2	no	—
11	C₂×C₄⋊Dic₅₃	C₄	1	1	yes	characteristic
12	C₂×C₄⋊Dic₅₃	C₄	1	2	yes	—
13	C₂×C₄×Dic₅₃	C₂²	1	2	yes	—
14	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₅₃	C₄	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	—	2	2	no	—
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂×Q₈×C₅₃	C₈	1	4	yes	complemented
19	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
22	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
24	C₂×#88324	—	2	2	no	—
25	C₂×#88324	—	2	2	no	—
26	C₂×#88324	—	2	2	no	—
27	C₂×#88324	—	2	4	no	—
28	C₂×#88324	Q₈	1	4	yes	complemented
29	C₄⋊Dic₅₃	—	2	2	no	—
30	C₄⋊Dic₅₃	—	2	4	no	—
31	C₄×Dic₅₃	—	2	2	no	—
32	C₄×Dic₅₃	—	2	2	no	—
33	C₄×Dic₅₃	D₄	1	2	yes	—
34	C₄×Dic₅₃	D₄	1	2	yes	—
35	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	Q₈×C₅₃	—	2	8	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	—
41	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
42	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
43	C₂×Dic₅₃	—	2	2	no	—
44	C₂×Dic₅₃	—	2	2	no	—
45	C₂×Dic₅₃	—	2	4	no	—
46	C₂×Dic₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₅₃	C₂×Q₈	1	2	yes	—
50	C₂×Dic₅₃	#41	1	2	yes	—
51	#88324	—	2	4	no	—
52	#88324	—	4	2	no	—
53	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₅₃	C₄×Q₈	1	2	yes	—
55	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₅₃	#131	1	2	yes	—
58	C₄×C₅₃	—	2	4	no	—
59	C₄×C₅₃	C₂²⋊C₈	1	4	yes	—
60	Dic₅₃	—	2	2	no	—
61	Dic₅₃	—	4	2	no	—
62	Dic₅₃	C₂²⋊Q₈	1	2	yes	—
63	Dic₅₃	#124	1	1	yes	characteristic
64	Dic₅₃	C₄⋊Q₈	1	1	yes	characteristic
65	#1984	—	53	1	no	maximal, Sylow
66	C₂×C₅₃	#391	1	1	yes	characteristic
67	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₅₃	#408	1	1	yes	characteristic
69	C₂×C₅₃	C₈×Q₈	1	2	yes	—
70	C₂×C₅₃	#446	1	2	yes	—
71	C₂×C₄×Q₈	—	53	1	no	—
72	C₂×C₄⋊C₈	—	53	1	no	—
73	C₂×C₄×C₈	—	53	1	no	—
74	#336	—	53	4	no	—
75	C₅₃	#1984	1	1	yes	characteristic, complemented, Sylow
76	C₂²×Q₈	#6157	1	1	yes	characteristic
77	C₂²×C₈	—	53	2	no	—
78	C₂²×C₈	—	53	2	no	—
79	C₄×Q₈	—	106	2	no	—
80	C₄×Q₈	—	106	2	no	—
81	C₂×C₄⋊C₄	—	53	2	no	—
82	C₂×C₄⋊C₄	—	53	1	no	—
83	C₂×C₄²	—	53	1	no	—
84	C₂×C₄²	—	53	2	no	—
85	C₄⋊C₈	—	106	2	no	—
86	C₄×C₈	—	106	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	#88324	1	4	yes	complemented
90	C₂²×C₄	—	53	1	no	—
91	C₂²×C₄	—	53	1	no	—
92	C₂²×C₄	—	53	2	no	—
93	C₂²×C₄	C₂×#6157	1	2	yes	—
94	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
95	C₂×C₈	—	53	4	no	—
96	C₂×C₈	—	106	2	no	—
97	C₂×C₈	—	106	2	no	—
98	C₂×C₈	—	106	2	no	—
99	C₂×C₈	—	106	4	no	—
100	C₄⋊C₄	—	106	4	no	—
101	C₄⋊C₄	—	106	2	no	—
102	C₄²	—	53	2	no	—
103	C₄²	—	53	2	no	—
104	C₄²	—	106	2	no	—
105	C₄²	—	106	2	no	—
106	C₂³	C₂²×#6157	1	1	yes	characteristic
107	Q₈	—	2	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	53	1	no	—
111	C₂×C₄	—	53	1	no	—
112	C₂×C₄	—	53	2	no	—
113	C₂×C₄	—	53	2	no	—
114	C₂×C₄	—	53	2	no	—
115	C₂×C₄	—	106	2	no	—
116	C₂×C₄	—	106	2	no	—
117	C₂×C₄	—	106	4	no	—
118	C₂×C₄	C₂×#88324	1	4	yes	—
119	C₂×C₄	C₂×#88324	1	2	yes	—
120	C₂×C₄	#11719940	1	2	yes	—
121	C₂×C₄	C₂²⋊#6157	1	2	yes	—
122	C₈	—	106	4	no	—
123	C₈	—	212	2	no	—
124	C₂²	C₂×C₂²⋊#6157	1	1	yes	characteristic
125	C₂²	Q₈×#6157	1	2	yes	—
126	C₂²	C₂²×#88324	1	1	yes	characteristic
127	C₂²	C₂×#11719940	1	1	yes	characteristic
128	C₂²	#13133305	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	53	2	no	—
131	C₄	—	53	1	no	—
132	C₄	—	53	1	no	—
133	C₄	—	106	2	no	—
134	C₄	—	212	2	no	—
135	C₄	C₂²⋊#88324	1	4	yes	—
136	C₂	(C₂×#6157)⋊Q₈	1	1	yes	characteristic
137	C₂	C₂×C₂²⋊#88324	1	1	yes	characteristic
138	C₂	#16811598	1	1	yes	characteristic
139	C₂	Q₈×#88324	1	2	yes	—
140	C₂	#16811674	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88324

C₂

yes

maximal, characteristic

C₂×C₄×#88324

C₂

yes

maximal, characteristic

#16811489

C₂

yes

maximal

C₂×Q₈×Dic₅₃

C₂

yes

maximal, characteristic

C₂²×#88324

C₂²

yes

—

C₂²×#88324

C₂²

yes

—

C₄⋊#88324

—

C₄×#88324

—

Q₈×Dic₅₃

—

Q₈×Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₄

yes

characteristic

C₂×C₄⋊Dic₅₃

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂²×Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₈

yes

complemented

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

—

—

—

—

Q₈

yes

complemented

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

—

C₄×Dic₅₃

D₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²⋊C₄

yes

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×Q₈

yes

—

C₂×Dic₅₃

#41

yes

—

#88324

—

#88324

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×Q₈

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

characteristic

C₂²×C₅₃

#131

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊C₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂²⋊Q₈

yes

—

Dic₅₃

#124

yes

characteristic

Dic₅₃

C₄⋊Q₈

yes

—

C₂×C₅₃

yes

C₂×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₅₃

#408

yes

characteristic

C₂×C₅₃

C₈×Q₈

yes

—

C₂×C₅₃

#446

yes

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

#336

—

C₅₃

#1984

yes

characteristic, complemented, Sylow

C₂²×Q₈

#6157

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

106

—

C₄×Q₈

—

106

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

106

—

C₄×C₈

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#88324

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6157

yes

—

C₂²×C₄

C₂×#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

100

C₄⋊C₄

—

106

—

101

C₄⋊C₄

—

106

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

106

—

105

C₄²

—

106

—

106

C₂³

C₂²×#6157

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

106

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

106

—

118

C₂×C₄

C₂×#88324

yes

—

119

C₂×C₄

C₂×#88324

yes

—

120

C₂×C₄

#11719940

yes

—

121

C₂×C₄

C₂²⋊#6157

yes

—

122

C₈

—

106

—

123

C₈

—

212

—

124

C₂²

C₂×C₂²⋊#6157

yes

characteristic

125

C₂²

Q₈×#6157

yes

—

126

C₂²

C₂²×#88324

yes

characteristic

127

C₂²

C₂×#11719940

yes

characteristic

128

C₂²

#13133305

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

106

—

134

C₄

—

212

—

135

C₄

C₂²⋊#88324

yes

—

136

C₂

(C₂×#6157)⋊Q₈

yes

characteristic

137

C₂

C₂×C₂²⋊#88324

yes

characteristic

138

C₂

#16811598

yes

characteristic

139

C₂

Q₈×#88324

yes

—

140

C₂

#16811674

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#6157
commutator	C₂²×C₅₃	C₂²×C₈
Fitting	C₂²×Q₈×C₅₃	C₄
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₂²×Q₈	#6157
hypocenter	C₅₃	#1984
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂²×#6157

commutator

C₂²×C₅₃

C₂²×C₈

Fitting

C₂²×Q₈×C₅₃

C₄

socle

C₂³×C₅₃

C₂²×C₄

C₂²×Q₈

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂³ ◁ C₂²×Q₈

N	H ≅ G/N	type
18. C₂×Q₈×C₅₃	C₈	semidirect
28. C₂×#88324	Q₈	semidirect
75. C₅₃	#1984	semidirect
89. C₂×Q₈	#88324	semidirect

H ≅ G/N

type

18. C₂×Q₈×C₅₃

C₈

semidirect

28. C₂×#88324

Q₈

semidirect

75. C₅₃

#1984

semidirect

89. C₂×Q₈

#88324

semidirect

Subgroups — #20290884

Conjugacy classes of subgroups

Complements