Group #22590029 — subgroups

Subgroups — #22590029

Order 9088.

230 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 73 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×#14556047	C₂	1	2	yes	maximal, complemented
2	#17985798	C₂	1	2	yes	maximal, complemented
3	#17985796	C₂	1	2	yes	maximal, complemented
4	#536×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11765849	—	2	2	no	—
6	#14556047	—	2	2	no	—
7	#14556047	C₂²	1	4	yes	—
8	C₂×#41×C₇₁	C₄	1	1	yes	characteristic
9	C₂²×Q₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
11	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
12	#124×C₇₁	C₄	1	1	yes	characteristic
13	#124×C₇₁	C₂²	1	1	yes	characteristic
14	#123×C₇₁	—	2	2	no	—
15	C₂²⋊Q₈×C₇₁	—	2	2	no	—
16	#117×C₇₁	C₄	1	1	yes	characteristic
17	#11765849	—	2	4	no	—
18	#11765849	—	4	2	no	—
19	C₂×C₇₁⋊C₈	—	4	2	no	—
20	#41×C₇₁	—	4	2	no	—
21	C₂×Q₈×C₇₁	—	4	2	no	—
22	C₂×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×Q₈×C₇₁	D₄	1	2	yes	—
25	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
26	C₂×D₄×C₇₁	—	2	1	no	—
27	C₂×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	—	2	1	no	—
29	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₁	D₄	1	2	yes	—
31	C₄⋊C₄×C₇₁	—	2	1	no	—
32	C₄⋊C₄×C₇₁	—	2	2	no	—
33	C₄⋊C₄×C₇₁	—	2	2	no	—
34	C₂²⋊C₄×C₇₁	—	2	1	no	—
35	C₂²⋊C₄×C₇₁	—	2	2	no	—
36	C₂²⋊C₄×C₇₁	—	2	2	no	—
37	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₇₁	—	2	1	no	—
40	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈×C₇₁	—	4	1	no	—
42	Q₈×C₇₁	—	4	2	no	—
43	Q₈×C₇₁	—	4	1	no	—
44	Q₈×C₇₁	—	4	1	no	—
45	D₄×C₇₁	—	4	1	no	—
46	D₄×C₇₁	—	4	2	no	—
47	C₂×C₄×C₇₁	—	2	2	no	—
48	C₂×C₄×C₇₁	—	2	2	no	—
49	C₂×C₄×C₇₁	—	2	2	no	—
50	C₂×C₄×C₇₁	—	2	1	no	—
51	C₂×C₄×C₇₁	—	2	1	no	—
52	C₂×C₄×C₇₁	—	2	1	no	—
53	C₂×C₄×C₇₁	—	4	2	no	—
54	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
58	C₇₁⋊C₈	—	4	4	no	—
59	C₂²×C₇₁	—	4	1	no	—
60	C₂²×C₇₁	—	4	2	no	—
61	C₂²×C₇₁	—	4	1	no	—
62	C₂²×C₇₁	#99	1	2	yes	—
63	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄×C₇₁	—	2	2	no	—
65	C₄×C₇₁	—	2	1	no	—
66	C₄×C₇₁	—	2	2	no	—
67	C₄×C₇₁	—	2	1	no	—
68	C₄×C₇₁	—	4	1	no	—
69	C₄×C₇₁	—	4	1	no	—
70	C₄×C₇₁	—	4	1	no	—
71	C₂×C₇₁	—	2	1	no	—
72	C₂×C₇₁	—	2	2	no	—
73	C₂×C₇₁	—	8	1	no	—
74	C₂×C₇₁	C₂×#99	1	1	yes	characteristic
75	#2133	—	71	1	no	maximal, Sylow
76	C₇₁	#2133	1	1	yes	characteristic, complemented, Sylow
77	#536	D₇₁	1	1	yes	characteristic
78	C₂×#101	—	71	2	no	—
79	#356	—	71	2	no	—
80	#355	—	71	2	no	—
81	C₂×#41	Dic₇₁	1	1	yes	characteristic
82	C₂²×Q₈	C₂×D₇₁	1	1	yes	characteristic
83	C₂×M₁₆	—	142	2	no	—
84	C₄⋊Q₈	Dic₇₁	1	1	yes	characteristic
85	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
86	#124	Dic₇₁	1	1	yes	characteristic
87	#124	C₂×D₇₁	1	1	yes	characteristic
88	#123	—	2	2	no	—
89	C₂²⋊Q₈	—	2	2	no	—
90	#117	Dic₇₁	1	1	yes	characteristic
91	#101	—	71	4	no	—
92	#101	—	142	2	no	—
93	#41	—	4	2	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
96	C₂×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
97	C₂×Q₈	C₂²⋊D₇₁	1	2	yes	—
98	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	C₂×Dic₇₁	1	1	yes	characteristic
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
103	C₂²×C₄	C₂²⋊D₇₁	1	2	yes	—
104	M₁₆	—	284	2	no	—
105	M₁₆	—	142	4	no	—
106	C₂×C₈	—	284	2	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	2	1	no	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
114	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
115	C₂³	—	2	1	no	—
116	C₂³	C₂²⋊Dic₇₁	1	1	yes	characteristic
117	Q₈	—	4	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	4	1	no	—
121	D₄	—	4	1	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	C₂²⋊Dic₇₁	1	2	yes	—
131	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
132	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₇₁	1	2	yes	—
134	C₈	—	284	4	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	—	4	1	no	—
138	C₂²	#14556043	1	2	yes	—
139	C₂²	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	2	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	4	1	no	—
147	C₂	—	2	1	no	—
148	C₂	—	2	2	no	—
149	C₂	—	8	1	no	—
150	C₂	C₂×#14556043	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556047

C₂

yes

maximal, complemented

#17985798

C₂

yes

maximal, complemented

#17985796

C₂

yes

maximal, complemented

#536×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

—

#14556047

—

#14556047

C₂²

yes

—

C₂×#41×C₇₁

C₄

yes

characteristic

C₂²×Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊Q₈×C₇₁

C₄

yes

characteristic

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

#124×C₇₁

C₄

yes

characteristic

#124×C₇₁

C₂²

yes

characteristic

#123×C₇₁

—

C₂²⋊Q₈×C₇₁

—

#117×C₇₁

C₄

yes

characteristic

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

#41×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

C₂×C₄

yes

characteristic

C₂×Q₈×C₇₁

C₂×C₄

yes

characteristic

C₂×Q₈×C₇₁

D₄

yes

—

C₂×Q₈×C₇₁

C₂³

yes

characteristic

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₂³×C₇₁

—

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#99

yes

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₂×#99

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#536

D₇₁

yes

—

—

—

C₂×#41

Dic₇₁

yes

characteristic

C₂²×Q₈

C₂×D₇₁

yes

characteristic

C₂×M₁₆

—

142

—

C₄⋊Q₈

Dic₇₁

yes

characteristic

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

#124

Dic₇₁

yes

characteristic

#124

C₂×D₇₁

yes

characteristic

#123

—

C₂²⋊Q₈

—

#117

Dic₇₁

yes

characteristic

#101

—

#101

—

142

—

#41

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₇₁

yes

characteristic

C₂×Q₈

C₂×Dic₇₁

yes

characteristic

C₂×Q₈

C₂²⋊D₇₁

yes

—

C₂×Q₈

C₂²×D₇₁

yes

characteristic

C₂×D₄

—

100

C₂×D₄

C₂×Dic₇₁

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

103

C₂²×C₄

C₂²⋊D₇₁

yes

—

104

M₁₆

—

284

—

105

M₁₆

—

142

—

106

C₂×C₈

—

284

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

C₂×Dic₇₁

yes

characteristic

114

C₄²

C₂×Dic₇₁

yes

characteristic

115

C₂³

—

116

C₂³

C₂²⋊Dic₇₁

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂²⋊Dic₇₁

yes

—

131

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

132

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

134

C₈

—

284

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#14556043

yes

—

139

C₂²

C₂×C₂²⋊Dic₇₁

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₂

—

148

C₂

—

149

C₂

—

150

C₂

C₂×#14556043

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#14556043
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂×Q₈	C₂²×D₇₁
Fitting	#536×C₇₁	C₂
socle	C₂×C₇₁	C₂×#99
hypercenter	#536	D₇₁
hypocenter	C₇₁	#2133
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×Q₈ ◁ #536

subgroup

G/H

center

C₂

C₂×#14556043

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂×Q₈

C₂²×D₇₁

Fitting

#536×C₇₁

C₂

socle

C₂×C₇₁

C₂×#99

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×Q₈ ◁ #536

N	H ≅ G/N	type
1. C₂×#14556047	C₂	semidirect
2. #17985798	C₂	semidirect
3. #17985796	C₂	semidirect
76. C₇₁	#2133	semidirect

H ≅ G/N

type

1. C₂×#14556047

C₂

semidirect

2. #17985798

C₂

semidirect

3. #17985796

C₂

semidirect

76. C₇₁

#2133

semidirect

Subgroups — #22590029

Conjugacy classes of subgroups

Complements