Group #22590388 — subgroups

Subgroups — #22590388

Order 9088.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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×#14556074	C₂	1	1	yes	maximal, characteristic
2	#17986377	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
4	Dic₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	#17985804	C₂	1	1	yes	maximal, characteristic
6	#17985803	C₂	1	1	yes	maximal, characteristic
7	C₂×Q₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
9	C₂×#11765855	—	2	1	no	—
10	#14556126	—	2	1	no	—
11	Q₈×Dic₇₁	—	2	1	no	—
12	#14556074	—	2	2	no	—
13	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
14	#14556062	C₂²	1	1	yes	characteristic
15	#14556052	—	2	1	no	—
16	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	Q₃₂×C₇₁	—	2	2	no	—
20	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₄₂	—	2	1	no	—
22	Dic₂₈₄	—	2	1	no	—
23	Dic₂₈₄	—	2	1	no	—
24	#11765855	—	4	2	no	—
25	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
26	C₄⋊Dic₇₁	—	2	1	no	—
27	C₄⋊Dic₇₁	—	4	1	no	—
28	Dic₇₁⋊C₄	—	4	1	no	—
29	Dic₇₁⋊C₄	—	4	1	no	—
30	C₄×Dic₇₁	—	4	1	no	—
31	C₄×Dic₇₁	D₄	1	1	yes	characteristic
32	C₇₁⋊C₁₆	—	2	1	no	—
33	C₂×Q₈×C₇₁	—	2	1	no	—
34	C₂×Q₈×C₇₁	—	2	1	no	—
35	Q₁₆×C₇₁	—	2	1	no	—
36	Q₁₆×C₇₁	—	2	1	no	—
37	Q₁₆×C₇₁	—	2	1	no	—
38	Q₁₆×C₇₁	D₄	1	2	yes	—
39	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
40	C₁₆×C₇₁	—	2	1	no	—
41	Q₈×C₇₁	—	2	2	no	—
42	Q₈×C₇₁	—	4	1	no	—
43	Q₈×C₇₁	—	4	1	no	—
44	Q₈×C₇₁	—	4	1	no	—
45	C₂×C₄×C₇₁	—	4	1	no	—
46	C₂×C₄×C₇₁	—	4	1	no	—
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₇₁	—	4	1	no	—
49	C₂×Dic₇₁	—	4	1	no	—
50	C₂×Dic₇₁	D₈	1	1	yes	characteristic
51	C₂×Dic₇₁	D₈	1	1	yes	characteristic
52	Dic₁₄₂	—	4	1	no	—
53	Dic₁₄₂	—	4	1	no	—
54	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₈×C₇₁	#41	1	1	yes	characteristic
56	C₇₁⋊C₈	—	2	1	no	—
57	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
58	C₄×C₇₁	—	4	2	no	—
59	C₄×C₇₁	—	8	1	no	—
60	C₄×C₇₁	#121	1	1	yes	characteristic
61	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₇₁	—	2	1	no	—
63	Dic₇₁	—	8	1	no	—
64	Dic₇₁	—	8	1	no	—
65	Dic₇₁	Q₃₂	1	2	yes	complemented
66	C₂×C₇₁	#459	1	1	yes	characteristic
67	C₂×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
68	C₂×C₇₁	#510	1	1	yes	characteristic
69	#2209	—	71	1	no	maximal, Sylow
70	C₇₁	#2209	1	1	yes	characteristic, complemented, Sylow
71	C₂×Q₃₂	—	71	1	no	—
72	C₂×Q₃₂	D₇₁	1	1	yes	characteristic
73	#494	—	71	1	no	—
74	C₄×Q₁₆	—	71	1	no	—
75	C₄⋊C₁₆	—	71	1	no	—
76	#358	—	71	1	no	—
77	#358	—	71	1	no	—
78	C₂×Q₁₆	—	71	1	no	—
79	C₂×Q₁₆	—	142	1	no	—
80	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
81	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
82	C₄⋊Q₈	—	142	1	no	—
83	C₄×Q₈	—	142	1	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	142	2	no	—
86	C₂×C₁₆	—	71	1	no	—
87	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
88	#107	—	71	1	no	—
89	#103	—	142	1	no	—
90	C₄×C₈	—	71	1	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	142	1	no	—
94	Q₁₆	—	2	1	no	—
95	Q₁₆	—	2	1	no	—
96	Q₁₆	—	2	1	no	—
97	Q₁₆	—	284	2	no	—
98	Q₁₆	—	142	1	no	—
99	Q₁₆	—	142	1	no	—
100	Q₁₆	C₂²⋊D₇₁	1	2	yes	—
101	C₂×C₈	—	71	1	no	—
102	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
103	C₄⋊C₄	—	284	1	no	—
104	C₄⋊C₄	—	284	1	no	—
105	C₄⋊C₄	—	284	1	no	—
106	C₄⋊C₄	—	142	1	no	—
107	C₄²	—	284	1	no	—
108	C₄²	—	71	1	no	—
109	C₁₆	—	2	1	no	—
110	C₁₆	—	142	1	no	—
111	Q₈	—	2	2	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	4	1	no	—
115	Q₈	—	284	1	no	—
116	Q₈	—	284	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	284	1	no	—
120	C₂×C₄	—	284	1	no	—
121	C₂×C₄	—	71	1	no	—
122	C₂×C₄	—	71	1	no	—
123	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
124	C₈	—	142	1	no	—
125	C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
126	C₈	#11765864	1	1	yes	characteristic
127	C₂²	D₈×D₇₁	1	1	yes	characteristic
128	C₄	—	4	2	no	—
129	C₄	—	8	1	no	—
130	C₄	—	568	1	no	—
131	C₄	—	568	1	no	—
132	C₄	—	71	2	no	—
133	C₄	—	142	1	no	—
134	C₄	#14556100	1	1	yes	characteristic
135	C₄	#14556152	1	1	yes	characteristic
136	C₂	#17986192	1	1	yes	characteristic
137	C₂	Q₃₂×D₇₁	1	1	yes	characteristic
138	C₂	#17986441	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556074

C₂

yes

maximal, characteristic

#17986377

C₂

yes

maximal, characteristic

Q₁₆×Dic₇₁

C₂

yes

maximal, characteristic

Dic₇₁⋊C₁₆

C₂

yes

maximal, characteristic

#17985804

C₂

yes

maximal, characteristic

#17985803

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

C₂²

yes

characteristic

C₂×#11765855

—

#14556126

—

Q₈×Dic₇₁

—

#14556074

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

#14556062

C₂²

yes

characteristic

#14556052

—

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×Q₁₆×C₇₁

C₂²

yes

characteristic

C₂×Q₁₆×C₇₁

C₂²

yes

characteristic

Q₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

#11765855

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₁₆×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

D₈

yes

characteristic

C₂×Dic₇₁

D₈

yes

characteristic

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

Q₃₂

yes

complemented

C₂×C₇₁

#459

yes

characteristic

C₂×C₇₁

C₂×Q₃₂

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

D₇₁

yes

characteristic

#494

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

#358

—

#358

—

C₂×Q₁₆

—

C₂×Q₁₆

—

142

—

C₂×Q₁₆

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

C₂×D₇₁

yes

characteristic

C₄⋊Q₈

—

142

—

C₄×Q₈

—

142

—

Q₃₂

—

Q₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

#107

—

#103

—

142

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

142

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

284

—

Q₁₆

—

142

—

Q₁₆

—

142

—

100

Q₁₆

C₂²⋊D₇₁

yes

—

101

C₂×C₈

—

102

C₂×C₈

C₂²×D₇₁

yes

characteristic

103

C₄⋊C₄

—

284

—

104

C₄⋊C₄

—

284

—

105

C₄⋊C₄

—

284

—

106

C₄⋊C₄

—

142

—

107

C₄²

—

284

—

108

C₄²

—

109

C₁₆

—

110

C₁₆

—

142

—

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

284

—

116

Q₈

—

284

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

284

—

120

C₂×C₄

—

284

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

D₄×D₇₁

yes

characteristic

124

C₈

—

142

—

125

C₈

C₂×C₂²⋊D₇₁

yes

characteristic

126

C₈

#11765864

yes

characteristic

127

C₂²

D₈×D₇₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

568

—

131

C₄

—

568

—

132

C₄

—

133

C₄

—

142

—

134

C₄

#14556100

yes

characteristic

135

C₄

#14556152

yes

characteristic

136

C₂

#17986192

yes

characteristic

137

C₂

Q₃₂×D₇₁

yes

characteristic

138

C₂

#17986441

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	C₂×Q₃₂×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	C₂×Q₃₂	D₇₁
hypocenter	C₇₁	#2209
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

C₂×Q₃₂×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

hypercenter

C₂×Q₃₂

D₇₁

hypocenter

C₇₁

#2209

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
65. Dic₇₁	Q₃₂	semidirect
70. C₇₁	#2209	semidirect

H ≅ G/N

type

65. Dic₇₁

Q₃₂

semidirect

70. C₇₁

#2209

semidirect

Subgroups — #22590388

Conjugacy classes of subgroups

Complements