Group #22588195 — subgroups

Subgroups — #22588195

Order 9088.

292 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 153 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⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556038	C₂	1	2	yes	maximal
3	C₄²×Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	#17985753	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₇₁	C₄	1	2	yes	—
8	C₂×C₄×Dic₇₁	C₄	1	2	yes	—
9	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₄⋊(C₇₁⋊C₈)	C₄	1	4	yes	complemented
12	#14556038	—	2	4	no	—
13	C₂²×C₈×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₇₁	C₄	1	4	yes	complemented
16	C₂²×Dic₇₁	C₂×C₄	1	2	yes	—
17	C₂²×Dic₇₁	C₂×C₄	1	2	yes	—
18	C₂×C₇₁⋊C₈	—	2	2	no	—
19	C₂×C₇₁⋊C₈	—	2	2	no	—
20	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
21	C₄×Dic₇₁	—	2	4	no	—
22	C₄×Dic₇₁	—	2	1	no	—
23	C₄×Dic₇₁	—	2	1	no	—
24	C₄×Dic₇₁	—	2	4	no	—
25	C₄×Dic₇₁	D₄	1	2	yes	—
26	C₄×Dic₇₁	Q₈	1	2	yes	—
27	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₇₁	—	2	2	no	—
31	C₂×C₈×C₇₁	—	2	2	no	—
32	C₂×C₈×C₇₁	C₂×C₄	1	4	yes	—
33	C₄²×C₇₁	C₂×C₄	1	2	yes	—
34	C₄²×C₇₁	C₂×C₄	1	2	yes	—
35	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	—	2	2	no	—
37	C₂×C₄×C₇₁	C₄²	1	2	yes	—
38	C₂×C₄×C₇₁	C₄²	1	2	yes	—
39	C₂×C₄×C₇₁	M₁₆	1	4	yes	—
40	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
45	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
46	C₂×Dic₇₁	—	2	2	no	—
47	C₂×Dic₇₁	—	2	2	no	—
48	C₂×Dic₇₁	—	2	2	no	—
49	C₂×Dic₇₁	—	2	2	no	—
50	C₂×Dic₇₁	C₄⋊C₄	1	4	yes	—
51	C₂×Dic₇₁	M₁₆	1	4	yes	—
52	C₈×C₇₁	—	2	4	no	—
53	C₇₁⋊C₈	—	2	4	no	—
54	C₂²×C₇₁	C₂×C₄²	1	1	yes	characteristic
55	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₇₁	#117	1	1	yes	characteristic
57	C₂²×C₇₁	C₂×M₁₆	1	2	yes	—
58	C₂²×C₇₁	C₂×M₁₆	1	2	yes	—
59	C₄×C₇₁	—	2	2	no	—
60	C₄×C₇₁	#97	1	4	yes	complemented
61	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
62	C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
63	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
64	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
65	Dic₇₁	—	2	4	no	—
66	Dic₇₁	—	2	4	no	—
67	C₂×C₇₁	C₄×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₇₁	C₂×#97	1	1	yes	characteristic
69	C₂×C₇₁	C₄×M₁₆	1	1	yes	characteristic
70	C₂×C₇₁	#423	1	2	yes	—
71	C₂×C₇₁	#432	1	2	yes	—
72	#1767	—	71	1	no	maximal, Sylow
73	C₇₁	#1767	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊C₈	—	71	1	no	—
75	C₂×C₄⋊C₈	D₇₁	1	1	yes	characteristic
76	C₂×#97	—	71	2	no	—
77	C₄³	—	71	1	no	—
78	#336	—	71	2	no	—
79	C₂²×C₈	—	71	2	no	—
80	C₂²×C₈	C₂×D₇₁	1	2	yes	—
81	C₂×C₄²	—	71	1	no	—
82	C₂×C₄²	—	71	2	no	—
83	C₂×C₄²	—	71	2	no	—
84	C₂×C₄²	—	71	1	no	—
85	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
86	C₄⋊C₈	—	71	4	no	—
87	C₄⋊C₈	Dic₇₁	1	4	yes	complemented
88	#97	—	142	4	no	—
89	C₂²×C₄	—	71	2	no	—
90	C₂²×C₄	—	71	2	no	—
91	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
92	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	71	4	no	—
97	C₂×C₈	—	142	2	no	—
98	C₂×C₈	—	142	2	no	—
99	C₂×C₈	C₂×Dic₇₁	1	4	yes	—
100	C₄²	—	71	2	no	—
101	C₄²	—	71	2	no	—
102	C₄²	—	142	4	no	—
103	C₄²	—	142	4	no	—
104	C₄²	—	142	1	no	—
105	C₄²	—	142	1	no	—
106	C₄²	C₄×D₇₁	1	2	yes	—
107	C₄²	C₂×Dic₇₁	1	2	yes	—
108	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	71	4	no	—
111	C₂×C₄	—	71	4	no	—
112	C₂×C₄	—	142	2	no	—
113	C₂×C₄	—	142	2	no	—
114	C₂×C₄	—	142	2	no	—
115	C₂×C₄	—	142	2	no	—
116	C₂×C₄	C₄×Dic₇₁	1	2	yes	—
117	C₂×C₄	C₄×Dic₇₁	1	2	yes	—
118	C₂×C₄	#11765848	1	4	yes	—
119	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
120	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
121	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
122	C₂×C₄	Q₈×D₇₁	1	1	yes	characteristic
123	C₂×C₄	#11765864	1	1	yes	characteristic
124	C₂×C₄	#11765866	1	1	yes	characteristic
125	C₈	—	2	4	no	—
126	C₈	—	142	4	no	—
127	C₂²	C₂×C₄×Dic₇₁	1	1	yes	characteristic
128	C₂²	C₄⋊C₄×D₇₁	1	1	yes	characteristic
129	C₂²	#14556086	1	1	yes	characteristic
130	C₂²	M₁₆×D₇₁	1	2	yes	—
131	C₂²	C₂×#11765848	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	142	4	no	—
134	C₄	—	142	4	no	—
135	C₄	#14556038	1	4	yes	complemented
136	C₄	#14556090	1	1	yes	characteristic
137	C₄	D₄×Dic₇₁	1	1	yes	characteristic
138	C₄	#14556093	1	1	yes	characteristic
139	C₄	Q₈×Dic₇₁	1	1	yes	characteristic
140	C₂	C₄⋊C₄×Dic₇₁	1	1	yes	characteristic
141	C₂	C₂×#14556038	1	1	yes	characteristic
142	C₂	M₁₆×Dic₇₁	1	1	yes	characteristic
143	C₂	#17986021	1	2	yes	—
144	C₂	#17986055	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×#14556038

C₂

yes

maximal

C₄²×Dic₇₁

C₂

yes

maximal, characteristic

#17985753

C₂

yes

maximal

C₂×C₄⋊C₈×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₇₁

C₄

yes

—

C₂×C₄×Dic₇₁

C₄

yes

—

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₄

yes

complemented

#14556038

—

C₂²×C₈×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₄

yes

complemented

C₂²×Dic₇₁

C₂×C₄

yes

—

C₂²×Dic₇₁

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

D₄

yes

—

C₄×Dic₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₂³×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄²

yes

—

C₂×C₄×C₇₁

C₄²

yes

—

C₂×C₄×C₇₁

M₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₂×Dic₇₁

M₁₆

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₄²

yes

characteristic

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

#117

yes

characteristic

C₂²×C₇₁

C₂×M₁₆

yes

—

C₂²×C₇₁

C₂×M₁₆

yes

—

C₄×C₇₁

—

C₄×C₇₁

#97

yes

complemented

C₄×C₇₁

C₄×D₄

yes

characteristic

C₄×C₇₁

C₄×D₄

yes

characteristic

C₄×C₇₁

C₄×Q₈

yes

characteristic

C₄×C₇₁

C₄×Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₇₁

C₂×#97

yes

characteristic

C₂×C₇₁

C₄×M₁₆

yes

characteristic

C₂×C₇₁

#423

yes

—

C₂×C₇₁

#432

yes

—

#1767

—

maximal, Sylow

C₇₁

#1767

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₇₁

yes

characteristic

C₂×#97

—

C₄³

—

#336

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

Dic₇₁

yes

complemented

#97

—

142

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₂×Dic₇₁

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

142

—

103

C₄²

—

142

—

104

C₄²

—

142

—

105

C₄²

—

142

—

106

C₄²

C₄×D₇₁

yes

—

107

C₄²

C₂×Dic₇₁

yes

—

108

C₂³

C₂×C₄×D₇₁

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

142

—

113

C₂×C₄

—

142

—

114

C₂×C₄

—

142

—

115

C₂×C₄

—

142

—

116

C₂×C₄

C₄×Dic₇₁

yes

—

117

C₂×C₄

C₄×Dic₇₁

yes

—

118

C₂×C₄

#11765848

yes

—

119

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

120

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

121

C₂×C₄

D₄×D₇₁

yes

characteristic

122

C₂×C₄

Q₈×D₇₁

yes

characteristic

123

C₂×C₄

#11765864

yes

characteristic

124

C₂×C₄

#11765866

yes

characteristic

125

C₈

—

126

C₈

—

142

—

127

C₂²

C₂×C₄×Dic₇₁

yes

characteristic

128

C₂²

C₄⋊C₄×D₇₁

yes

characteristic

129

C₂²

#14556086

yes

characteristic

130

C₂²

M₁₆×D₇₁

yes

—

131

C₂²

C₂×#11765848

yes

—

132

C₄

—

133

C₄

—

142

—

134

C₄

—

142

—

135

C₄

#14556038

yes

complemented

136

C₄

#14556090

yes

characteristic

137

C₄

D₄×Dic₇₁

yes

characteristic

138

C₄

#14556093

yes

characteristic

139

C₄

Q₈×Dic₇₁

yes

characteristic

140

C₂

C₄⋊C₄×Dic₇₁

yes

characteristic

141

C₂

C₂×#14556038

yes

characteristic

142

C₂

M₁₆×Dic₇₁

yes

characteristic

143

C₂

#17986021

yes

—

144

C₂

#17986055

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₁
commutator	C₂²×C₇₁	C₂×C₄²
Fitting	C₂×C₄⋊C₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂²×C₄
hypercenter	C₂×C₄⋊C₈	D₇₁
hypocenter	C₇₁	#1767
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₁

commutator

C₂²×C₇₁

C₂×C₄²

Fitting

C₂×C₄⋊C₈×C₇₁

C₂

socle

C₂³×C₇₁

C₂²×C₄

hypercenter

C₂×C₄⋊C₈

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
11. C₄⋊(C₇₁⋊C₈)	C₄	semidirect
15. C₄⋊C₈×C₇₁	C₄	semidirect
60. C₄×C₇₁	#97	semidirect
73. C₇₁	#1767	semidirect
87. C₄⋊C₈	Dic₇₁	semidirect
135. C₄	#14556038	semidirect

H ≅ G/N

type

11. C₄⋊(C₇₁⋊C₈)

C₄

semidirect

15. C₄⋊C₈×C₇₁

C₄

semidirect

60. C₄×C₇₁

#97

semidirect

73. C₇₁

#1767

semidirect

87. C₄⋊C₈

Dic₇₁

semidirect

135. C₄

#14556038

semidirect

Subgroups — #22588195

Conjugacy classes of subgroups

Complements