Group #22600982 — subgroups

Subgroups — #22600982

Order 9088.

370 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 141 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	#17986728	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986351	C₂	1	3	yes	maximal, complemented
3	#17986281	C₂	1	3	yes	maximal, complemented
4	#17986204	C₂	1	3	yes	maximal
5	Q₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11765853	C₂	1	3	yes	maximal
7	D₄×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765853	C₂²	1	3	yes	—
9	#14556128	C₂²	1	3	yes	—
10	#14556119	—	2	3	no	—
11	Q₈×Dic₇₁	—	2	1	no	—
12	C₄×Dic₁₄₂	C₂²	1	3	yes	—
13	#14556059	C₂²	1	3	yes	—
14	C₄⋊(C₇₁⋊C₈)	C₂²	1	3	yes	—
15	#14556051	C₂²	1	3	yes	—
16	#14556051	C₂²	1	6	yes	—
17	#14556049	C₂²	1	1	yes	characteristic
18	C₄×C₇₁⋊C₈	C₂²	1	3	yes	—
19	C₂²×Q₈×C₇₁	—	2	1	no	—
20	C₄⋊Q₈×C₇₁	C₂²	1	3	yes	—
21	C₂²⋊Q₈×C₇₁	—	2	3	no	—
22	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₇₁	C₂²	1	3	yes	—
24	C₂×Dic₁₄₂	C₂³	1	3	yes	—
25	#11765853	—	2	6	no	—
26	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
27	C₂×C₇₁⋊C₈	C₂³	1	3	yes	—
28	C₄⋊Dic₇₁	—	2	3	no	—
29	C₄⋊Dic₇₁	—	2	3	no	—
30	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
31	Dic₇₁⋊C₄	—	2	3	no	—
32	C₄×Dic₇₁	—	2	3	no	—
33	C₂×Q₈×C₇₁	—	2	3	no	—
34	C₂×Q₈×C₇₁	—	4	4	no	—
35	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
36	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₇₁	—	2	3	no	—
38	C₄⋊C₄×C₇₁	—	2	3	no	—
39	C₄⋊C₄×C₇₁	D₄	1	3	yes	—
40	C₄⋊C₄×C₇₁	C₂³	1	3	yes	—
41	C₂²⋊C₄×C₇₁	—	2	3	no	—
42	C₄²×C₇₁	C₂³	1	3	yes	—
43	C₂³×C₇₁	—	2	1	no	—
44	Q₈×C₇₁	—	4	6	no	—
45	Q₈×C₇₁	SD₁₆	1	4	yes	complemented
46	D₄×C₇₁	—	2	1	no	—
47	D₄×C₇₁	#41	1	2	yes	—
48	C₂×C₄×C₇₁	—	2	3	no	—
49	C₂×C₄×C₇₁	—	4	6	no	—
50	C₂×C₄×C₇₁	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₇₁	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₇₁	—	2	1	no	—
54	C₂×Dic₇₁	—	2	3	no	—
55	Dic₁₄₂	—	2	3	no	—
56	Dic₁₄₂	—	2	3	no	—
57	C₇₁⋊C₈	—	2	3	no	—
58	C₇₁⋊C₈	—	2	1	no	—
59	C₂²×C₇₁	—	2	2	no	—
60	C₂²×C₇₁	—	4	1	no	—
61	C₂²×C₇₁	—	4	1	no	—
62	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₇₁	—	2	3	no	—
64	C₄×C₇₁	—	4	3	no	—
65	C₄×C₇₁	C₂×SD₁₆	1	6	yes	—
66	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
67	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₇₁	—	4	3	no	—
69	Dic₇₁	—	4	1	no	—
70	C₂×C₇₁	—	4	2	no	—
71	C₂×C₇₁	#548	1	1	yes	characteristic
72	C₂×C₇₁	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₇₁	#578	1	1	yes	characteristic
74	#3359	—	71	1	no	maximal, Sylow
75	C₇₁	#3359	1	1	yes	characteristic, complemented, Sylow
76	#557	—	71	1	no	—
77	D₄×Q₈	D₇₁	1	1	yes	characteristic
78	#487	—	71	3	no	—
79	#475	—	71	3	no	—
80	#461	—	71	3	no	—
81	C₈×Q₈	—	71	1	no	—
82	C₄×SD₁₆	—	71	3	no	—
83	C₂²×Q₈	—	2	1	no	—
84	C₂×SD₁₆	—	71	3	no	—
85	C₄⋊Q₈	—	71	3	no	—
86	C₄⋊Q₈	C₂×D₇₁	1	3	yes	—
87	#125	—	142	3	no	—
88	C₂²⋊Q₈	—	2	3	no	—
89	C₄×Q₈	—	71	3	no	—
90	C₄×Q₈	—	142	1	no	—
91	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
92	C₄×D₄	C₂×D₇₁	1	3	yes	—
93	#106	—	71	3	no	—
94	C₄⋊C₈	—	71	3	no	—
95	#103	—	71	3	no	—
96	#103	—	71	6	no	—
97	#102	—	71	1	no	—
98	C₄×C₈	—	71	3	no	—
99	C₂×Q₈	—	2	3	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	—	71	3	no	—
102	C₂×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
103	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
104	C₂²×C₄	—	2	3	no	—
105	SD₁₆	—	142	6	no	—
106	C₂×C₈	—	71	1	no	—
107	C₂×C₈	—	71	3	no	—
108	C₄⋊C₄	—	2	3	no	—
109	C₄⋊C₄	—	71	1	no	—
110	C₄⋊C₄	—	142	3	no	—
111	C₄⋊C₄	—	142	3	no	—
112	C₄⋊C₄	—	142	3	no	—
113	C₄⋊C₄	C₂²⋊D₇₁	1	3	yes	—
114	C₄⋊C₄	C₂²×D₇₁	1	3	yes	—
115	C₂²⋊C₄	—	2	3	no	—
116	C₄²	—	142	3	no	—
117	C₄²	C₂²×D₇₁	1	3	yes	—
118	C₂³	—	2	1	no	—
119	Q₈	—	4	6	no	—
120	Q₈	—	142	3	no	—
121	Q₈	—	142	3	no	—
122	Q₈	#11765853	1	4	yes	complemented
123	D₄	—	2	1	no	—
124	D₄	#11765866	1	2	yes	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	4	6	no	—
127	C₂×C₄	—	142	1	no	—
128	C₂×C₄	—	142	3	no	—
129	C₂×C₄	C₂×C₂²⋊D₇₁	1	3	yes	—
130	C₂×C₄	C₂×C₂²⋊D₇₁	1	3	yes	—
131	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
132	C₈	—	142	3	no	—
133	C₈	—	142	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	1	no	—
137	C₂²	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	4	3	no	—
140	C₄	—	284	3	no	—
141	C₄	—	284	1	no	—
142	C₄	C₂×#11765853	1	6	yes	—
143	C₄	C₂×#11765866	1	1	yes	characteristic
144	C₄	#14556176	1	1	yes	characteristic
145	C₂	—	4	2	no	—
146	C₂	#17986660	1	1	yes	characteristic
147	C₂	C₂²×#11765853	1	1	yes	characteristic
148	C₂	#17986814	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986728

C₂

yes

maximal, characteristic, complemented

#17986351

C₂

yes

maximal, complemented

#17986281

C₂

yes

maximal, complemented

#17986204

C₂

yes

maximal

Q₈×C₇₁⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#11765853

C₂

yes

maximal

D₄×Q₈×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765853

C₂²

yes

—

#14556128

C₂²

yes

—

#14556119

—

Q₈×Dic₇₁

—

C₄×Dic₁₄₂

C₂²

yes

—

#14556059

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556051

C₂²

yes

—

#14556051

C₂²

yes

—

#14556049

C₂²

yes

characteristic

C₄×C₇₁⋊C₈

C₂²

yes

—

C₂²×Q₈×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

C₂²⋊Q₈×C₇₁

—

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄×D₄×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

C₂³

yes

—

#11765853

—

C₂×C₇₁⋊C₈

C₂³

yes

characteristic

C₂×C₇₁⋊C₈

C₂³

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂³

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

D₄

yes

characteristic

C₂×D₄×C₇₁

C₂³

yes

characteristic

C₂²×C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

—

C₂³×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

SD₁₆

yes

complemented

D₄×C₇₁

—

D₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

—

C₄×C₇₁

C₂×#41

yes

characteristic

C₄×C₇₁

2₋¹⁺⁴

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#548

yes

characteristic

C₂×C₇₁

C₂²×SD₁₆

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#557

—

D₄×Q₈

D₇₁

yes

—

—

—

C₈×Q₈

—

C₄×SD₁₆

—

C₂²×Q₈

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

—

#125

—

142

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

142

—

C₄×Q₈

C₂×D₇₁

yes

characteristic

C₄×D₄

C₂×D₇₁

yes

—

#106

—

C₄⋊C₈

—

#103

—

#103

—

#102

—

C₄×C₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

C₂²⋊D₇₁

yes

characteristic

103

C₂×D₄

C₂²×D₇₁

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

142

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

142

—

111

C₄⋊C₄

—

142

—

112

C₄⋊C₄

—

142

—

113

C₄⋊C₄

C₂²⋊D₇₁

yes

—

114

C₄⋊C₄

C₂²×D₇₁

yes

—

115

C₂²⋊C₄

—

116

C₄²

—

142

—

117

C₄²

C₂²×D₇₁

yes

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

142

—

121

Q₈

—

142

—

122

Q₈

#11765853

yes

complemented

123

D₄

—

124

D₄

#11765866

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

142

—

128

C₂×C₄

—

142

—

129

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

130

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

131

C₂×C₄

C₂³×D₇₁

yes

characteristic

132

C₈

—

142

—

133

C₈

—

142

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×C₂²⋊D₇₁

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

284

—

141

C₄

—

284

—

142

C₄

C₂×#11765853

yes

—

143

C₄

C₂×#11765866

yes

characteristic

144

C₄

#14556176

yes

characteristic

145

C₂

—

146

C₂

#17986660

yes

characteristic

147

C₂

C₂²×#11765853

yes

characteristic

148

C₂

#17986814

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	D₄×Q₈×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	D₄×Q₈	D₇₁
hypocenter	C₇₁	#3359
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

D₄×Q₈×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

hypercenter

D₄×Q₈

D₇₁

hypocenter

C₇₁

#3359

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. #17986728	C₂	semidirect
2. #17986351	C₂	semidirect
3. #17986281	C₂	semidirect
5. Q₈×C₇₁⋊C₈	C₂	semidirect
45. Q₈×C₇₁	SD₁₆	semidirect
75. C₇₁	#3359	semidirect
122. Q₈	#11765853	semidirect

H ≅ G/N

type

1. #17986728

C₂

semidirect

2. #17986351

C₂

semidirect

3. #17986281

C₂

semidirect

5. Q₈×C₇₁⋊C₈

C₂

semidirect

45. Q₈×C₇₁

SD₁₆

semidirect

75. C₇₁

#3359

semidirect

122. Q₈

#11765853

semidirect

Subgroups — #22600982

Conjugacy classes of subgroups

Complements