Group #22590447 — subgroups

Subgroups — #22590447

Order 9088.

150 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 47 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, complemented
2	C₂×#14556067	C₂	1	1	yes	maximal, characteristic
3	#17986241	C₂	1	1	yes	maximal, characteristic
4	#17986237	C₂	1	1	yes	maximal, characteristic
5	#17985826	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985804	C₂	1	1	yes	maximal, characteristic, complemented
7	#358×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
10	#14556108	—	2	1	no	—
11	#14556104	—	2	1	no	—
12	#14556074	—	2	2	no	—
13	#14556067	—	2	2	no	—
14	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
15	#14556062	C₂²	1	1	yes	characteristic
16	#14556052	—	2	1	no	—
17	#14556051	—	2	1	no	—
18	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
20	#107×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₄₂	—	2	1	no	—
22	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
23	Dic₂₈₄	—	2	1	no	—
24	Dic₂₈₄	—	2	1	no	—
25	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
26	C₈×D₇₁	—	2	1	no	—
27	C₈×D₇₁	—	2	1	no	—
28	C₄⋊Dic₇₁	—	2	1	no	—
29	C₄⋊Dic₇₁	—	4	1	no	—
30	Dic₇₁⋊C₄	—	4	1	no	—
31	(C₂×D₇₁)⋊C₄	—	4	1	no	—
32	(C₂×D₇₁)⋊C₄	—	4	1	no	—
33	C₇₁⋊C₁₆	D₄	1	2	yes	—
34	C₂×Q₈×C₇₁	—	2	1	no	—
35	Q₁₆×C₇₁	—	2	1	no	—
36	Q₁₆×C₇₁	—	2	1	no	—
37	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₇₁	—	2	1	no	—
39	C₁₆×C₇₁	—	2	1	no	—
40	C₂²×D₇₁	D₈	1	1	yes	characteristic
41	Q₈×C₇₁	—	4	1	no	—
42	Q₈×C₇₁	—	4	1	no	—
43	C₂×C₄×C₇₁	—	4	1	no	—
44	C₂×C₄×C₇₁	—	4	1	no	—
45	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₇₁	—	4	1	no	—
47	C₂×Dic₇₁	—	4	1	no	—
48	C₂×Dic₇₁	D₈	1	1	yes	characteristic
49	C₄×D₇₁	—	2	1	no	—
50	C₄×D₇₁	—	2	1	no	—
51	Dic₁₄₂	—	4	1	no	—
52	Dic₁₄₂	—	4	1	no	—
53	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
54	C₈×C₇₁	#41	1	1	yes	characteristic
55	C₇₁⋊C₈	—	2	1	no	—
56	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
57	C₂×D₇₁	—	2	1	no	—
58	C₂×D₇₁	—	2	1	no	—
59	C₂×D₇₁	—	2	1	no	—
60	C₄×C₇₁	—	8	1	no	—
61	C₄×C₇₁	—	8	1	no	—
62	C₄×C₇₁	#121	1	1	yes	characteristic
63	C₄×C₇₁	#135	1	1	yes	characteristic
64	Dic₇₁	—	2	1	no	—
65	Dic₇₁	—	8	1	no	—
66	Dic₇₁	—	8	1	no	—
67	C₂×C₇₁	#466	1	1	yes	characteristic
68	C₂×C₇₁	#510	1	1	yes	characteristic
69	C₂×C₇₁	#510	1	1	yes	characteristic
70	D₇₁	—	4	1	no	—
71	#2219	—	71	1	no	maximal, Sylow
72	C₇₁	#2219	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₃₂	—	71	1	no	—
74	C₂×M₃₂	—	71	1	no	—
75	#467	—	71	1	no	—
76	#467	—	71	1	no	—
77	#367	—	71	1	no	—
78	#358	—	71	1	no	—
79	#358	D₇₁	1	1	yes	characteristic, complemented
80	C₂×Q₁₆	—	71	1	no	—
81	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
82	C₂²×C₈	—	71	1	no	—
83	C₂²⋊Q₈	—	142	1	no	—
84	C₂²⋊Q₈	—	142	1	no	—
85	Q₃₂	—	142	2	no	—
86	M₃₂	—	142	2	no	—
87	C₂×C₁₆	—	71	1	no	—
88	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
89	#107	—	71	1	no	—
90	#107	C₂×D₇₁	1	1	yes	characteristic
91	#103	—	142	1	no	—
92	#103	—	142	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	142	1	no	—
95	C₂²×C₄	—	71	1	no	—
96	Q₁₆	—	2	1	no	—
97	Q₁₆	—	2	1	no	—
98	Q₁₆	—	142	1	no	—
99	Q₁₆	—	142	1	no	—
100	C₂×C₈	—	71	1	no	—
101	C₂×C₈	—	142	1	no	—
102	C₂×C₈	—	142	1	no	—
103	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	284	1	no	—
106	C₄⋊C₄	—	284	1	no	—
107	C₄⋊C₄	—	142	1	no	—
108	C₂²⋊C₄	—	284	1	no	—
109	C₂²⋊C₄	—	284	1	no	—
110	C₁₆	—	2	1	no	—
111	C₁₆	—	71	2	no	—
112	C₂³	—	71	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	—	142	1	no	—
123	C₂×C₄	—	142	1	no	—
124	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
125	C₈	—	142	1	no	—
126	C₈	D₄×D₇₁	1	1	yes	characteristic
127	C₈	#11765865	1	1	yes	characteristic
128	C₂²	—	142	1	no	—
129	C₂²	—	142	1	no	—
130	C₂²	—	142	1	no	—
131	C₂²	D₈×D₇₁	1	1	yes	characteristic
132	C₄	—	8	1	no	—
133	C₄	—	8	1	no	—
134	C₄	—	568	1	no	—
135	C₄	—	568	1	no	—
136	C₄	—	142	1	no	—
137	C₄	#14556101	1	1	yes	characteristic
138	C₄	#14556149	1	1	yes	characteristic
139	C₂	—	284	1	no	—
140	C₂	#17986234	1	1	yes	characteristic
141	C₂	#17986438	1	1	yes	characteristic
142	C₂	#17986441	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556074

C₂

yes

maximal, characteristic, complemented

C₂×#14556067

C₂

yes

maximal, characteristic

#17986241

C₂

yes

maximal, characteristic

#17986237

C₂

yes

maximal, characteristic

#17985826

C₂

yes

maximal, characteristic, complemented

#17985804

C₂

yes

maximal, characteristic, complemented

#358×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₈₄

C₂²

yes

characteristic

C₂×C₈×D₇₁

C₂²

yes

—

—

—

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

#14556062

C₂²

yes

characteristic

#14556052

—

#14556051

—

C₂×Q₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

#107×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×C₄×D₇₁

D₄

yes

characteristic

Dic₂₈₄

—

Dic₂₈₄

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

(C₂×D₇₁)⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₇₁⋊C₁₆

D₄

yes

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

—

C₂²×D₇₁

D₈

yes

characteristic

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

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#2219

—

maximal, Sylow

C₇₁

#2219

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×M₃₂

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₈

—

C₂²⋊Q₈

—

142

—

C₂²⋊Q₈

—

142

—

Q₃₂

—

142

—

M₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

#107

—

#107

C₂×D₇₁

yes

characteristic

#103

—

142

—

#103

—

142

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

142

—

Q₁₆

—

142

—

100

C₂×C₈

—

101

C₂×C₈

—

142

—

102

C₂×C₈

—

142

—

103

C₂×C₈

C₂²×D₇₁

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

284

—

106

C₄⋊C₄

—

284

—

107

C₄⋊C₄

—

142

—

108

C₂²⋊C₄

—

284

—

109

C₂²⋊C₄

—

284

—

110

C₁₆

—

111

C₁₆

—

112

C₂³

—

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

—

142

—

123

C₂×C₄

—

142

—

124

C₂×C₄

D₄×D₇₁

yes

characteristic

125

C₈

—

142

—

126

C₈

D₄×D₇₁

yes

characteristic

127

C₈

#11765865

yes

characteristic

128

C₂²

—

142

—

129

C₂²

—

142

—

130

C₂²

—

142

—

131

C₂²

D₈×D₇₁

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

568

—

135

C₄

—

568

—

136

C₄

—

142

—

137

C₄

#14556101

yes

characteristic

138

C₄

#14556149

yes

characteristic

139

C₂

—

284

—

140

C₂

#17986234

yes

characteristic

141

C₂

#17986438

yes

characteristic

142

C₂

#17986441

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#358×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#358	D₇₁
hypocenter	C₇₁	#2219
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#358×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
1. C₂×#14556074	C₂	semidirect
5. #17985826	C₂	semidirect
6. #17985804	C₂	semidirect
7. #358×C₇₁	C₂	semidirect
72. C₇₁	#2219	semidirect
79. #358	D₇₁	semidirect

H ≅ G/N

type

1. C₂×#14556074

C₂

semidirect

5. #17985826

C₂

semidirect

6. #17985804

C₂

semidirect

7. #358×C₇₁

C₂

semidirect

72. C₇₁

#2219

semidirect

79. #358

D₇₁

semidirect

Subgroups — #22590447

Conjugacy classes of subgroups

Complements