Group #22587828 — subgroups

Subgroups — #22587828

Order 9088.

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

normal subgroups only

This group has 166 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	#17986378	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986280	C₂	1	1	yes	maximal, characteristic
3	#17986276	C₂	1	1	yes	maximal, characteristic
4	#17986076	C₂	1	1	yes	maximal, characteristic
5	#17985749	C₂	1	1	yes	maximal, characteristic, complemented
6	#17985746	C₂	1	1	yes	maximal, characteristic, complemented
7	#332×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11765855	—	2	1	no	—
9	C₂×#11765855	—	2	1	no	—
10	C₂×#11765848	—	2	1	no	—
11	#14556128	C₂²	1	1	yes	characteristic
12	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
13	#14556062	—	2	1	no	—
14	#14556061	—	2	1	no	—
15	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
16	#14556050	—	2	1	no	—
17	#14556048	—	2	1	no	—
18	(C₂×D₇₁)⋊C₈	—	2	1	no	—
19	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₂	—	2	1	no	—
24	C₂×D₂₈₄	D₄	1	1	yes	characteristic
25	C₂×C₄×D₇₁	—	2	1	no	—
26	#11765855	—	4	2	no	—
27	#11765855	—	4	2	no	—
28	#11765848	—	4	2	no	—
29	C₂×C₇₁⋊C₈	—	2	1	no	—
30	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
31	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
32	C₄⋊Dic₇₁	—	2	1	no	—
33	C₄⋊Dic₇₁	—	2	1	no	—
34	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
35	(C₂×D₇₁)⋊C₄	—	2	1	no	—
36	C₂×Q₈×C₇₁	—	2	1	no	—
37	C₂×C₈×C₇₁	—	2	1	no	—
38	C₂×C₈×C₇₁	—	2	1	no	—
39	C₄⋊C₄×C₇₁	—	2	1	no	—
40	C₄⋊C₄×C₇₁	—	2	1	no	—
41	C₄²×C₇₁	C₂³	1	1	yes	characteristic
42	C₂²×D₇₁	—	2	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	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
49	C₂×Dic₇₁	—	2	1	no	—
50	C₂×Dic₇₁	—	4	1	no	—
51	D₂₈₄	—	2	1	no	—
52	D₂₈₄	—	2	1	no	—
53	C₄×D₇₁	—	4	1	no	—
54	C₄×D₇₁	—	4	1	no	—
55	Dic₁₄₂	—	4	1	no	—
56	Dic₁₄₂	—	4	1	no	—
57	C₈×C₇₁	—	4	1	no	—
58	C₈×C₇₁	—	4	1	no	—
59	C₇₁⋊C₈	—	2	1	no	—
60	C₇₁⋊C₈	—	4	1	no	—
61	C₇₁⋊C₈	D₈	1	2	yes	—
62	C₂²×C₇₁	#121	1	1	yes	characteristic
63	C₂×D₇₁	—	4	1	no	—
64	C₂×D₇₁	—	4	1	no	—
65	C₂×D₇₁	—	4	1	no	—
66	C₄×C₇₁	—	2	1	no	—
67	C₄×C₇₁	—	8	1	no	—
68	C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
69	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
70	C₄×C₇₁	#137	1	1	yes	characteristic
71	C₄×C₇₁	#137	1	1	yes	characteristic
72	Dic₇₁	—	4	1	no	—
73	Dic₇₁	—	8	1	no	—
74	C₂×C₇₁	#459	1	1	yes	characteristic
75	C₂×C₇₁	#470	1	1	yes	characteristic
76	C₂×C₇₁	#473	1	1	yes	characteristic
77	D₇₁	—	8	1	no	—
78	#1689	—	71	1	no	maximal, Sylow
79	C₇₁	#1689	1	1	yes	characteristic, complemented, Sylow
80	#494	—	71	1	no	—
81	#474	—	71	1	no	—
82	#474	—	71	1	no	—
83	#436	—	71	1	no	—
84	#334	—	71	1	no	—
85	#332	—	71	1	no	—
86	#332	D₇₁	1	1	yes	characteristic, complemented
87	C₂×Q₁₆	—	142	1	no	—
88	C₂×Q₁₆	—	142	1	no	—
89	C₂×M₁₆	—	142	1	no	—
90	C₄⋊Q₈	—	71	1	no	—
91	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
92	C₄×D₄	—	71	1	no	—
93	#107	—	142	1	no	—
94	#107	—	142	1	no	—
95	C₄⋊C₈	—	71	1	no	—
96	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
97	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
98	#102	—	142	1	no	—
99	#102	—	142	1	no	—
100	C₂²⋊C₈	—	142	1	no	—
101	C₄×C₈	—	71	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	142	1	no	—
104	C₂×D₄	—	71	1	no	—
105	C₂²×C₄	—	142	1	no	—
106	Q₁₆	—	284	2	no	—
107	Q₁₆	—	284	2	no	—
108	M₁₆	—	284	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	71	1	no	—
112	C₂×C₈	—	71	1	no	—
113	C₂×C₈	—	142	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	71	1	no	—
117	C₄⋊C₄	—	142	1	no	—
118	C₄⋊C₄	—	142	1	no	—
119	C₂²⋊C₄	—	142	1	no	—
120	C₄²	C₂²×D₇₁	1	1	yes	characteristic
121	C₂³	—	142	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	284	1	no	—
125	Q₈	—	284	1	no	—
126	D₄	—	142	1	no	—
127	D₄	—	142	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	284	1	no	—
130	C₂×C₄	—	284	1	no	—
131	C₂×C₄	—	284	1	no	—
132	C₂×C₄	—	142	1	no	—
133	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
134	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
135	C₂×C₄	#11765865	1	1	yes	characteristic
136	C₈	—	4	1	no	—
137	C₈	—	4	1	no	—
138	C₈	—	284	1	no	—
139	C₈	—	71	2	no	—
140	C₈	—	142	1	no	—
141	C₂²	—	284	1	no	—
142	C₂²	—	284	1	no	—
143	C₂²	—	284	1	no	—
144	C₂²	#14556101	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	8	1	no	—
147	C₄	—	284	1	no	—
148	C₄	—	568	1	no	—
149	C₄	D₈×D₇₁	1	1	yes	characteristic
150	C₄	#14556155	1	1	yes	characteristic
151	C₄	#14556157	1	1	yes	characteristic
152	C₄	#14556160	1	1	yes	characteristic
153	C₂	—	568	1	no	—
154	C₂	#17986193	1	1	yes	characteristic
155	C₂	#17986256	1	1	yes	characteristic
156	C₂	#17986273	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986378

C₂

yes

maximal, characteristic, complemented

#17986280

C₂

yes

maximal, characteristic

#17986276

C₂

yes

maximal, characteristic

#17986076

C₂

yes

maximal, characteristic

#17985749

C₂

yes

maximal, characteristic, complemented

#17985746

C₂

yes

maximal, characteristic, complemented

#332×C₇₁

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

C₄×D₂₈₄

C₂²

yes

characteristic

#14556062

—

#14556061

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

#14556050

—

#14556048

—

(C₂×D₇₁)⋊C₈

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×D₂₈₄

D₄

yes

characteristic

C₂×C₄×D₇₁

—

#11765855

—

#11765855

—

#11765848

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

(C₂×D₇₁)⋊C₄

—

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

D₈

yes

—

C₂²×C₇₁

#121

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×D₈

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1689

—

maximal, Sylow

C₇₁

#1689

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₂×Q₁₆

—

142

—

C₂×Q₁₆

—

142

—

C₂×M₁₆

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₁

yes

characteristic

C₄×D₄

—

#107

—

142

—

#107

—

142

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₁

yes

characteristic

C₄⋊C₈

C₂×D₇₁

yes

characteristic

#102

—

142

—

#102

—

142

—

100

C₂²⋊C₈

—

142

—

101

C₄×C₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

142

—

104

C₂×D₄

—

105

C₂²×C₄

—

142

—

106

Q₁₆

—

284

—

107

Q₁₆

—

284

—

108

M₁₆

—

284

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

142

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

142

—

118

C₄⋊C₄

—

142

—

119

C₂²⋊C₄

—

142

—

120

C₄²

C₂²×D₇₁

yes

characteristic

121

C₂³

—

142

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

284

—

125

Q₈

—

284

—

126

D₄

—

142

—

127

D₄

—

142

—

128

C₂×C₄

—

129

C₂×C₄

—

284

—

130

C₂×C₄

—

284

—

131

C₂×C₄

—

284

—

132

C₂×C₄

—

142

—

133

C₂×C₄

D₄×D₇₁

yes

characteristic

134

C₂×C₄

D₄×D₇₁

yes

characteristic

135

C₂×C₄

#11765865

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

284

—

139

C₈

—

140

C₈

—

142

—

141

C₂²

—

284

—

142

C₂²

—

284

—

143

C₂²

—

284

—

144

C₂²

#14556101

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

284

—

148

C₄

—

568

—

149

C₄

D₈×D₇₁

yes

characteristic

150

C₄

#14556155

yes

characteristic

151

C₄

#14556157

yes

characteristic

152

C₄

#14556160

yes

characteristic

153

C₂

—

568

—

154

C₂

#17986193

yes

characteristic

155

C₂

#17986256

yes

characteristic

156

C₂

#17986273

yes

characteristic

	subgroup	G/H
center	C₂²	#14556101
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#332×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#332	D₇₁
hypocenter	C₇₁	#1689
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#14556101

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#332×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
1. #17986378	C₂	semidirect
5. #17985749	C₂	semidirect
6. #17985746	C₂	semidirect
7. #332×C₇₁	C₂	semidirect
79. C₇₁	#1689	semidirect
86. #332	D₇₁	semidirect

H ≅ G/N

type

1. #17986378

C₂

semidirect

5. #17985749

C₂

semidirect

6. #17985746

C₂

semidirect

7. #332×C₇₁

C₂

semidirect

79. C₇₁

#1689

semidirect

86. #332

D₇₁

semidirect

Subgroups — #22587828

Conjugacy classes of subgroups

Complements