Group #22586247 — subgroups

Subgroups — #22586247

Order 9088.

200 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17985963	C₂	1	1	yes	maximal, characteristic
2	#17985859	C₂	1	1	yes	maximal, characteristic
3	#407×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11765849	C₄	1	2	yes	complemented
5	C₂×C₄⋊Dic₇₁	—	2	1	no	—
6	C₂×C₂²⋊Dic₇₁	C₄	1	2	yes	—
7	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
9	C₇₁⋊#95	—	2	1	no	—
10	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₇₁	—	2	1	no	—
12	C₂²⋊C₈×C₇₁	—	2	1	no	—
13	C₂²⋊C₈×C₇₁	C₄	1	2	yes	complemented
14	C₂²×Dic₇₁	—	2	2	no	—
15	#11765849	—	4	2	no	—
16	C₂×C₇₁⋊C₈	—	2	2	no	—
17	C₄⋊Dic₇₁	—	4	2	no	—
18	C₂²⋊Dic₇₁	—	4	2	no	—
19	C₂⁴×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₇₁	—	4	2	no	—
21	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
27	M₁₆×C₇₁	—	4	2	no	—
28	C₂×C₈×C₇₁	—	2	2	no	—
29	C₂³×C₇₁	—	2	1	no	—
30	C₂³×C₇₁	—	2	1	no	—
31	C₂³×C₇₁	—	4	1	no	—
32	C₂³×C₇₁	—	4	1	no	—
33	C₂³×C₇₁	C₄²	1	1	yes	characteristic
34	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂³×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	—	2	2	no	—
37	C₂×C₄×C₇₁	—	2	2	no	—
38	C₂×C₄×C₇₁	—	2	2	no	—
39	C₂×C₄×C₇₁	—	4	1	no	—
40	C₂×C₄×C₇₁	—	4	1	no	—
41	C₂×C₄×C₇₁	—	4	1	no	—
42	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₇₁	—	4	2	no	—
47	C₂×Dic₇₁	—	4	2	no	—
48	C₂×Dic₇₁	—	4	2	no	—
49	C₈×C₇₁	—	4	2	no	—
50	C₇₁⋊C₈	—	4	2	no	—
51	C₂²×C₇₁	—	2	1	no	—
52	C₂²×C₇₁	—	2	1	no	—
53	C₂²×C₇₁	—	2	1	no	—
54	C₂²×C₇₁	—	2	1	no	—
55	C₂²×C₇₁	—	2	1	no	—
56	C₂²×C₇₁	—	2	1	no	—
57	C₂²×C₇₁	—	2	1	no	—
58	C₂²×C₇₁	—	2	1	no	—
59	C₂²×C₇₁	—	4	2	no	—
60	C₂²×C₇₁	—	4	2	no	—
61	C₂²×C₇₁	#95	1	1	yes	characteristic
62	C₂²×C₇₁	#100	1	1	yes	characteristic
63	C₂²×C₇₁	#101	1	1	yes	characteristic
64	C₄×C₇₁	—	2	1	no	—
65	C₄×C₇₁	—	2	1	no	—
66	C₄×C₇₁	—	2	1	no	—
67	C₄×C₇₁	#99	1	2	yes	—
68	Dic₇₁	—	8	2	no	—
69	C₂×C₇₁	—	2	1	no	—
70	C₂×C₇₁	—	2	1	no	—
71	C₂×C₇₁	—	4	1	no	—
72	C₂×C₇₁	—	4	1	no	—
73	C₂×C₇₁	#342	1	1	yes	characteristic
74	C₂×C₇₁	#343	1	1	yes	characteristic
75	C₂×C₇₁	#344	1	1	yes	characteristic
76	#1305	—	71	1	no	maximal, Sylow
77	C₇₁	#1305	1	1	yes	characteristic, complemented, Sylow
78	#407	—	71	1	no	—
79	#407	D₇₁	1	1	yes	characteristic
80	#380	—	71	1	no	—
81	C₂³×C₄	C₂×D₇₁	1	1	yes	characteristic
82	C₂×M₁₆	—	2	1	no	—
83	C₂×M₁₆	—	71	2	no	—
84	C₂×C₄⋊C₄	—	142	1	no	—
85	C₂×C₂²⋊C₄	—	71	2	no	—
86	C₂²⋊C₈	—	2	1	no	—
87	C₂²⋊C₈	—	142	1	no	—
88	C₂²⋊C₈	—	142	1	no	—
89	C₂²⋊C₈	Dic₇₁	1	2	yes	complemented
90	#95	—	142	1	no	—
91	C₂⁴	C₄×D₇₁	1	1	yes	characteristic
92	C₂²×C₄	—	4	2	no	—
93	C₂²×C₄	—	142	2	no	—
94	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
96	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
100	M₁₆	—	4	2	no	—
101	M₁₆	—	284	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	142	2	no	—
104	C₄⋊C₄	—	284	2	no	—
105	C₂²⋊C₄	—	284	2	no	—
106	C₂³	—	2	1	no	—
107	C₂³	—	2	1	no	—
108	C₂³	—	4	1	no	—
109	C₂³	—	4	1	no	—
110	C₂³	C₄×Dic₇₁	1	1	yes	characteristic
111	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
112	C₂³	Dic₇₁⋊C₄	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	284	2	no	—
120	C₂×C₄	—	284	2	no	—
121	C₂×C₄	—	284	2	no	—
122	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
124	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
125	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
126	C₈	—	4	2	no	—
127	C₈	—	284	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	—	4	2	no	—
138	C₂²	C₇₁⋊#95	1	1	yes	characteristic
139	C₂²	#14556044	1	1	yes	characteristic
140	C₂²	#14556046	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	568	2	no	—
145	C₄	#14556042	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	—	4	1	no	—
149	C₂	—	4	1	no	—
150	C₂	#17985765	1	1	yes	characteristic
151	C₂	#17985767	1	1	yes	characteristic
152	C₂	#17985770	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17985963

C₂

yes

maximal, characteristic

#17985859

C₂

yes

maximal, characteristic

#407×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₄

yes

complemented

C₂×C₄⋊Dic₇₁

—

C₂×C₂²⋊Dic₇₁

C₄

yes

—

C₂²⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₇₁⋊#95

—

C₂³×C₄×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

C₄

yes

complemented

C₂²×Dic₇₁

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₂⁴×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

Q₈

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₄²

yes

characteristic

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

C₂³×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#99

yes

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#407

—

#407

D₇₁

yes

characteristic

#380

—

C₂³×C₄

C₂×D₇₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×C₄⋊C₄

—

142

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

C₂²⋊C₈

—

142

—

C₂²⋊C₈

—

142

—

C₂²⋊C₈

Dic₇₁

yes

complemented

#95

—

142

—

C₂⁴

C₄×D₇₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

100

M₁₆

—

101

M₁₆

—

284

—

102

C₂×C₈

—

103

C₂×C₈

—

142

—

104

C₄⋊C₄

—

284

—

105

C₂²⋊C₄

—

284

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₄×Dic₇₁

yes

characteristic

111

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

112

C₂³

Dic₇₁⋊C₄

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

284

—

120

C₂×C₄

—

284

—

121

C₂×C₄

—

284

—

122

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

123

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

124

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

125

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

126

C₈

—

127

C₈

—

284

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₇₁⋊#95

yes

characteristic

139

C₂²

#14556044

yes

characteristic

140

C₂²

#14556046

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

568

—

145

C₄

#14556042

yes

—

146

C₂

—

147

C₂

—

148

C₂

—

149

C₂

—

150

C₂

#17985765

yes

characteristic

151

C₂

#17985767

yes

characteristic

152

C₂

#17985770

yes

characteristic

	subgroup	G/H
center	C₂²	C₇₁⋊#95
commutator	C₂³×C₇₁	C₄²
Frattini	C₂³×C₄	C₂×D₇₁
Fitting	#407×C₇₁	C₂
socle	C₂²×C₇₁	#95
hypercenter	#407	D₇₁
hypocenter	C₇₁	#1305
LCS	G ▷ C₂³×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₇₁⋊#95

commutator

C₂³×C₇₁

C₄²

Frattini

C₂³×C₄

C₂×D₇₁

Fitting

#407×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
4. C₂×#11765849	C₄	semidirect
13. C₂²⋊C₈×C₇₁	C₄	semidirect
77. C₇₁	#1305	semidirect
89. C₂²⋊C₈	Dic₇₁	semidirect

H ≅ G/N

type

4. C₂×#11765849

C₄

semidirect

13. C₂²⋊C₈×C₇₁

C₄

semidirect

77. C₇₁

#1305

semidirect

89. C₂²⋊C₈

Dic₇₁

semidirect

Subgroups — #22586247

Conjugacy classes of subgroups

Complements