Group #22590374 — subgroups

Subgroups — #22590374

Order 9088.

186 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 49 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×D₁₁₃₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14556069	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986374	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986081	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985799	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
9	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
10	C₂×#11765850	—	2	1	no	—
11	#14556124	—	2	1	no	—
12	#14556090	—	2	1	no	—
13	D₁₁₃₆	—	2	2	no	—
14	#14556069	—	2	2	no	—
15	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
16	#14556048	—	2	1	no	—
17	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
18	C₂×D₈×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×D₂₈₄	—	2	1	no	—
22	C₂×D₂₈₄	—	2	1	no	—
23	C₂×C₂²⋊D₇₁	—	4	1	no	—
24	C₂×C₂²⋊D₇₁	—	4	1	no	—
25	C₂×C₄×D₇₁	—	4	1	no	—
26	D₅₆₈	—	2	1	no	—
27	D₅₆₈	—	2	1	no	—
28	D₅₆₈	—	2	1	no	—
29	D₅₆₈	D₄	1	2	yes	complemented
30	#11765850	—	4	2	no	—
31	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
32	(C₂×D₇₁)⋊C₄	—	4	1	no	—
33	C₄×Dic₇₁	D₄	1	1	yes	characteristic
34	C₇₁⋊C₁₆	—	2	1	no	—
35	C₂×D₄×C₇₁	—	2	1	no	—
36	D₈×C₇₁	—	2	1	no	—
37	D₈×C₇₁	—	2	1	no	—
38	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₄×C₇₁	—	2	1	no	—
40	C₁₆×C₇₁	—	2	1	no	—
41	C₂³×C₇₁	—	4	1	no	—
42	C₂²×D₇₁	—	4	1	no	—
43	C₂²×D₇₁	—	4	1	no	—
44	D₄×C₇₁	—	4	1	no	—
45	D₄×C₇₁	—	4	1	no	—
46	C₂×C₄×C₇₁	—	4	1	no	—
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₇₁	—	8	2	no	—
49	C₂²⋊D₇₁	—	8	2	no	—
50	C₂×Dic₇₁	D₈	1	1	yes	characteristic
51	C₂×Dic₇₁	D₈	1	1	yes	characteristic
52	D₂₈₄	—	2	2	no	—
53	D₂₈₄	—	4	1	no	—
54	D₂₈₄	—	4	1	no	—
55	D₂₈₄	—	4	1	no	—
56	C₄×D₇₁	—	8	2	no	—
57	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₈×C₇₁	#41	1	1	yes	characteristic
59	C₇₁⋊C₈	—	2	1	no	—
60	C₂²×C₇₁	—	8	1	no	—
61	C₂²×C₇₁	—	8	1	no	—
62	C₂²×C₇₁	—	8	1	no	—
63	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
64	C₂×D₇₁	—	4	2	no	—
65	C₂×D₇₁	—	8	1	no	—
66	C₂×D₇₁	—	8	1	no	—
67	C₂×D₇₁	—	8	1	no	—
68	C₂×D₇₁	—	8	1	no	—
69	C₂×D₇₁	—	8	1	no	—
70	C₄×C₇₁	—	8	1	no	—
71	C₄×C₇₁	#121	1	1	yes	characteristic
72	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
73	Dic₇₁	—	2	1	no	—
74	Dic₇₁	D₁₆	1	2	yes	complemented
75	C₂×C₇₁	—	16	1	no	—
76	C₂×C₇₁	#459	1	1	yes	characteristic
77	C₂×C₇₁	C₂×D₁₆	1	1	yes	characteristic
78	C₂×C₇₁	#509	1	1	yes	characteristic
79	D₇₁	—	8	2	no	—
80	D₇₁	—	16	1	no	—
81	#2206	—	71	1	no	maximal, Sylow
82	C₇₁	#2206	1	1	yes	characteristic, complemented, Sylow
83	C₂×D₁₆	—	71	1	no	—
84	C₂×D₁₆	—	71	1	no	—
85	#493	—	71	1	no	—
86	C₄×D₈	—	71	1	no	—
87	C₄⋊C₁₆	—	71	1	no	—
88	#357	—	71	1	no	—
89	#357	D₇₁	1	1	yes	characteristic, complemented
90	C₂×D₈	—	71	1	no	—
91	C₂×D₈	—	71	1	no	—
92	C₂×D₈	—	142	1	no	—
93	C₂×D₈	C₂×D₇₁	1	1	yes	characteristic
94	#127	—	142	1	no	—
95	C₄×D₄	—	142	1	no	—
96	D₁₆	—	142	2	no	—
97	D₁₆	—	142	2	no	—
98	C₂×C₁₆	—	71	1	no	—
99	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
100	#107	C₂×D₇₁	1	1	yes	characteristic
101	#102	—	142	1	no	—
102	C₄×C₈	—	71	1	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	—	284	1	no	—
105	C₂×D₄	—	284	1	no	—
106	C₂×D₄	—	142	1	no	—
107	C₂×D₄	—	142	1	no	—
108	C₂²×C₄	—	284	1	no	—
109	D₈	—	2	1	no	—
110	D₈	—	2	1	no	—
111	D₈	—	284	2	no	—
112	D₈	—	71	2	no	—
113	D₈	—	142	1	no	—
114	D₈	—	142	1	no	—
115	D₈	—	142	1	no	—
116	C₂×C₈	—	71	1	no	—
117	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
118	C₄⋊C₄	—	2	1	no	—
119	C₂²⋊C₄	—	284	1	no	—
120	C₄²	—	71	1	no	—
121	C₁₆	—	2	1	no	—
122	C₁₆	—	142	1	no	—
123	C₂³	—	4	1	no	—
124	C₂³	—	284	1	no	—
125	C₂³	—	284	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	284	1	no	—
129	D₄	—	284	1	no	—
130	D₄	—	284	1	no	—
131	D₄	—	568	2	no	—
132	D₄	—	568	2	no	—
133	D₄	—	142	2	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	568	2	no	—
136	C₂×C₄	—	71	1	no	—
137	C₂×C₄	—	71	1	no	—
138	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
139	C₈	—	142	1	no	—
140	C₈	D₄×D₇₁	1	1	yes	characteristic
141	C₈	#11765865	1	1	yes	characteristic
142	C₂²	—	8	1	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	284	2	no	—
146	C₂²	—	568	1	no	—
147	C₂²	—	568	1	no	—
148	C₂²	—	568	1	no	—
149	C₂²	—	568	1	no	—
150	C₂²	—	568	1	no	—
151	C₂²	D₈×D₇₁	1	1	yes	characteristic
152	C₄	—	8	1	no	—
153	C₄	—	71	2	no	—
154	C₄	—	142	1	no	—
155	C₄	#14556101	1	1	yes	characteristic
156	C₄	#14556155	1	1	yes	characteristic
157	C₂	—	16	1	no	—
158	C₂	—	568	2	no	—
159	C₂	—	1136	1	no	—
160	C₂	#17986193	1	1	yes	characteristic
161	C₂	D₁₆×D₇₁	1	1	yes	characteristic
162	C₂	#17986436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₁₃₆

C₂

yes

maximal, characteristic, complemented

C₂×#14556069

C₂

yes

maximal, characteristic, complemented

#17986374

C₂

yes

maximal, characteristic, complemented

#17986081

C₂

yes

maximal, characteristic, complemented

Dic₇₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#17985799

C₂

yes

maximal, characteristic, complemented

#357×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×D₅₆₈

C₂²

yes

characteristic

C₂×D₅₆₈

C₂²

yes

—

—

—

D₁₁₃₆

—

#14556069

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

#14556048

—

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×D₈×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

#107×C₇₁

C₂²

yes

characteristic

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×C₂²⋊D₇₁

—

C₂×C₂²⋊D₇₁

—

C₂×C₄×D₇₁

—

D₅₆₈

—

D₅₆₈

—

D₅₆₈

—

D₅₆₈

D₄

yes

complemented

#11765850

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

(C₂×D₇₁)⋊C₄

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

—

C₂²×D₇₁

—

C₂²×D₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²⋊D₇₁

—

C₂²⋊D₇₁

—

C₂×Dic₇₁

D₈

yes

characteristic

C₂×Dic₇₁

D₈

yes

characteristic

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

D₁₆

yes

complemented

C₂×C₇₁

—

C₂×C₇₁

#459

yes

characteristic

C₂×C₇₁

C₂×D₁₆

yes

characteristic

C₂×C₇₁

#509

yes

characteristic

D₇₁

—

D₇₁

—

#2206

—

maximal, Sylow

C₇₁

#2206

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

C₂×D₁₆

—

#493

—

C₄×D₈

—

C₄⋊C₁₆

—

#357

—

#357

D₇₁

yes

characteristic, complemented

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

142

—

C₂×D₈

C₂×D₇₁

yes

characteristic

#127

—

142

—

C₄×D₄

—

142

—

D₁₆

—

142

—

D₁₆

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

100

#107

C₂×D₇₁

yes

characteristic

101

#102

—

142

—

102

C₄×C₈

—

103

C₂×D₄

—

104

C₂×D₄

—

284

—

105

C₂×D₄

—

284

—

106

C₂×D₄

—

142

—

107

C₂×D₄

—

142

—

108

C₂²×C₄

—

284

—

109

D₈

—

110

D₈

—

111

D₈

—

284

—

112

D₈

—

113

D₈

—

142

—

114

D₈

—

142

—

115

D₈

—

142

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₇₁

yes

characteristic

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

284

—

120

C₄²

—

121

C₁₆

—

122

C₁₆

—

142

—

123

C₂³

—

124

C₂³

—

284

—

125

C₂³

—

284

—

126

D₄

—

127

D₄

—

128

D₄

—

284

—

129

D₄

—

284

—

130

D₄

—

284

—

131

D₄

—

568

—

132

D₄

—

568

—

133

D₄

—

142

—

134

C₂×C₄

—

135

C₂×C₄

—

568

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

D₄×D₇₁

yes

characteristic

139

C₈

—

142

—

140

C₈

D₄×D₇₁

yes

characteristic

141

C₈

#11765865

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

284

—

146

C₂²

—

568

—

147

C₂²

—

568

—

148

C₂²

—

568

—

149

C₂²

—

568

—

150

C₂²

—

568

—

151

C₂²

D₈×D₇₁

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

142

—

155

C₄

#14556101

yes

characteristic

156

C₄

#14556155

yes

characteristic

157

C₂

—

158

C₂

—

568

—

159

C₂

—

1136

—

160

C₂

#17986193

yes

characteristic

161

C₂

D₁₆×D₇₁

yes

characteristic

162

C₂

#17986436

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#357×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#357	D₇₁
hypocenter	C₇₁	#2206
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#357×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 ◁ #357

N	H ≅ G/N	type
1. C₂×D₁₁₃₆	C₂	semidirect
2. C₂×#14556069	C₂	semidirect
3. #17986374	C₂	semidirect
4. #17986081	C₂	semidirect
5. Dic₇₁⋊C₁₆	C₂	semidirect
6. #17985799	C₂	semidirect
7. #357×C₇₁	C₂	semidirect
29. D₅₆₈	D₄	semidirect
74. Dic₇₁	D₁₆	semidirect
82. C₇₁	#2206	semidirect
89. #357	D₇₁	semidirect

H ≅ G/N

type

1. C₂×D₁₁₃₆

C₂

semidirect

2. C₂×#14556069

C₂

semidirect

3. #17986374

C₂

semidirect

4. #17986081

C₂

semidirect

5. Dic₇₁⋊C₁₆

C₂

semidirect

6. #17985799

C₂

semidirect

7. #357×C₇₁

C₂

semidirect

29. D₅₆₈

D₄

semidirect

74. Dic₇₁

D₁₆

semidirect

82. C₇₁

#2206

semidirect

89. #357

D₇₁

semidirect

Subgroups — #22590374

Conjugacy classes of subgroups

Complements