Group #22587243 — subgroups

Subgroups — #22587243

Order 9088.

162 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 71 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	#17986130	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986077	C₂	1	1	yes	maximal, characteristic
3	#17985752	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
6	#17985720	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×SD₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#11765849	—	2	1	no	—
9	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
10	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
11	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
13	#14556039	—	2	1	no	—
14	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×SD₁₆×C₇₁	C₄	1	1	yes	characteristic
17	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
19	#106×C₇₁	C₄	1	1	yes	characteristic
20	#103×C₇₁	C₄	1	1	yes	characteristic
21	#102×C₇₁	C₄	1	1	yes	characteristic
22	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
23	#11765849	—	4	2	no	—
24	C₂×C₇₁⋊C₈	—	2	1	no	—
25	C₂×C₇₁⋊C₈	—	2	1	no	—
26	C₂×C₇₁⋊C₈	—	2	1	no	—
27	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
28	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
29	C₂×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₁	—	2	1	no	—
32	SD₁₆×C₇₁	—	2	2	no	—
33	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₇₁	—	2	1	no	—
36	C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₂²⋊C₄×C₇₁	—	2	1	no	—
39	C₄²×C₇₁	—	2	1	no	—
40	C₄²×C₇₁	C₂³	1	1	yes	characteristic
41	C₂³×C₇₁	—	2	1	no	—
42	Q₈×C₇₁	—	2	1	no	—
43	Q₈×C₇₁	—	2	1	no	—
44	D₄×C₇₁	—	2	1	no	—
45	D₄×C₇₁	—	2	1	no	—
46	C₂×C₄×C₇₁	—	2	1	no	—
47	C₂×C₄×C₇₁	—	2	1	no	—
48	C₂×C₄×C₇₁	—	2	1	no	—
49	C₂×C₄×C₇₁	—	4	1	no	—
50	C₂×C₄×C₇₁	—	4	1	no	—
51	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
54	C₈×C₇₁	—	2	1	no	—
55	C₈×C₇₁	M₁₆	1	2	yes	complemented
56	C₇₁⋊C₈	—	2	1	no	—
57	C₇₁⋊C₈	—	2	1	no	—
58	C₇₁⋊C₈	—	4	1	no	—
59	C₇₁⋊C₈	—	4	1	no	—
60	C₇₁⋊C₈	—	4	1	no	—
61	C₂²×C₇₁	—	4	1	no	—
62	C₂²×C₇₁	—	4	1	no	—
63	C₂²×C₇₁	—	4	1	no	—
64	C₂²×C₇₁	C₄×D₄	1	1	yes	characteristic
65	C₄×C₇₁	—	2	1	no	—
66	C₄×C₇₁	—	4	1	no	—
67	C₄×C₇₁	—	4	1	no	—
68	C₄×C₇₁	—	4	1	no	—
69	C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
70	C₄×C₇₁	#131	1	1	yes	characteristic
71	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₇₁	#137	1	1	yes	characteristic
73	C₂×C₇₁	—	8	1	no	—
74	C₂×C₇₁	#436	1	1	yes	characteristic
75	C₂×C₇₁	#440	1	1	yes	characteristic
76	C₂×C₇₁	#444	1	1	yes	characteristic
77	#1592	—	71	1	no	maximal, Sylow
78	C₇₁	#1592	1	1	yes	characteristic, complemented, Sylow
79	#446	—	71	1	no	—
80	C₄×SD₁₆	D₇₁	1	1	yes	characteristic
81	#436	—	71	1	no	—
82	#335	—	71	1	no	—
83	Q₈⋊C₈	—	71	1	no	—
84	D₄⋊C₈	—	71	1	no	—
85	#322	—	71	1	no	—
86	C₂×SD₁₆	Dic₇₁	1	1	yes	characteristic
87	C₂×M₁₆	—	142	1	no	—
88	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
89	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
90	#106	Dic₇₁	1	1	yes	characteristic
91	C₄⋊C₈	—	71	1	no	—
92	C₄⋊C₈	—	71	1	no	—
93	C₄⋊C₈	—	142	1	no	—
94	#103	Dic₇₁	1	1	yes	characteristic
95	#102	Dic₇₁	1	1	yes	characteristic
96	C₂²⋊C₈	—	142	1	no	—
97	#97	—	142	1	no	—
98	C₄×C₈	—	71	1	no	—
99	C₄×C₈	—	71	1	no	—
100	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
101	C₂×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
102	C₂×D₄	C₂×Dic₇₁	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	—	2	2	no	—
105	M₁₆	—	284	2	no	—
106	C₂×C₈	—	71	1	no	—
107	C₂×C₈	—	71	1	no	—
108	C₂×C₈	—	142	1	no	—
109	C₂×C₈	—	142	1	no	—
110	C₂×C₈	—	142	1	no	—
111	C₂×C₈	C₂×Dic₇₁	1	1	yes	characteristic
112	C₂×C₈	C₂×Dic₇₁	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
115	C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂²×D₇₁	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	2	1	no	—
122	D₄	—	2	1	no	—
123	D₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
130	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
131	C₂×C₄	#11765864	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	284	1	no	—
134	C₈	—	284	1	no	—
135	C₈	—	284	1	no	—
136	C₈	—	142	1	no	—
137	C₈	—	142	1	no	—
138	C₈	#11765849	1	2	yes	complemented
139	C₂²	—	4	1	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	D₄×Dic₇₁	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	4	1	no	—
147	C₄	C₂×#11765849	1	1	yes	characteristic
148	C₄	#14556135	1	1	yes	characteristic
149	C₄	#14556155	1	1	yes	characteristic
150	C₄	#14556157	1	1	yes	characteristic
151	C₂	—	8	1	no	—
152	C₂	#17986077	1	1	yes	characteristic
153	C₂	#17986100	1	1	yes	characteristic
154	C₂	#17986121	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986130

C₂

yes

maximal, characteristic, complemented

#17986077

C₂

yes

maximal, characteristic

#17985752

C₂

yes

maximal, characteristic, complemented

Q₈⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

#17985720

C₂

yes

maximal, characteristic, complemented

C₄×SD₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₇₁⋊C₈)

—

#14556039

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₇₁

C₄

yes

characteristic

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄×D₄×C₇₁

C₂²

yes

characteristic

#106×C₇₁

C₄

yes

characteristic

#103×C₇₁

C₄

yes

characteristic

#102×C₇₁

C₄

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₂×Q₈×C₇₁

C₂×C₄

yes

characteristic

C₂×D₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

—

SD₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

M₁₆

yes

complemented

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×M₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#446

—

C₄×SD₁₆

D₇₁

yes

characteristic

#436

—

#335

—

Q₈⋊C₈

—

D₄⋊C₈

—

#322

—

C₂×SD₁₆

Dic₇₁

yes

characteristic

C₂×M₁₆

—

142

—

C₄×Q₈

C₂×D₇₁

yes

characteristic

C₄×D₄

C₂×D₇₁

yes

characteristic

#106

Dic₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

#103

Dic₇₁

yes

characteristic

#102

Dic₇₁

yes

characteristic

C₂²⋊C₈

—

142

—

#97

—

142

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂×D₇₁

yes

characteristic

101

C₂×Q₈

C₂×Dic₇₁

yes

characteristic

102

C₂×D₄

C₂×Dic₇₁

yes

characteristic

103

C₂²×C₄

—

104

SD₁₆

—

105

M₁₆

—

284

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

142

—

109

C₂×C₈

—

142

—

110

C₂×C₈

—

142

—

111

C₂×C₈

C₂×Dic₇₁

yes

characteristic

112

C₂×C₈

C₂×Dic₇₁

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂×Dic₇₁

yes

characteristic

115

C₄⋊C₄

C₂×Dic₇₁

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂²×D₇₁

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

130

C₂×C₄

D₄×D₇₁

yes

characteristic

131

C₂×C₄

#11765864

yes

characteristic

132

C₈

—

133

C₈

—

284

—

134

C₈

—

284

—

135

C₈

—

284

—

136

C₈

—

142

—

137

C₈

—

142

—

138

C₈

#11765849

yes

complemented

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

D₄×Dic₇₁

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

C₂×#11765849

yes

characteristic

148

C₄

#14556135

yes

characteristic

149

C₄

#14556155

yes

characteristic

150

C₄

#14556157

yes

characteristic

151

C₂

—

152

C₂

#17986077

yes

characteristic

153

C₂

#17986100

yes

characteristic

154

C₂

#17986121

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	C₄×SD₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₄×D₄
hypercenter	C₄×SD₁₆	D₇₁
hypocenter	C₇₁	#1592
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

subgroup

G/H

center

C₂×C₄

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

C₄×SD₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₄×D₄

hypercenter

C₄×SD₁₆

D₇₁

hypocenter

C₇₁

#1592

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

N	H ≅ G/N	type
1. #17986130	C₂	semidirect
3. #17985752	C₂	semidirect
4. Q₈⋊(C₇₁⋊C₈)	C₂	semidirect
6. #17985720	C₂	semidirect
55. C₈×C₇₁	M₁₆	semidirect
78. C₇₁	#1592	semidirect
138. C₈	#11765849	semidirect

H ≅ G/N

type

1. #17986130

C₂

semidirect

3. #17985752

C₂

semidirect

4. Q₈⋊(C₇₁⋊C₈)

C₂

semidirect

6. #17985720

C₂

semidirect

55. C₈×C₇₁

M₁₆

semidirect

78. C₇₁

#1592

semidirect

138. C₈

#11765849

semidirect

Subgroups — #22587243

Conjugacy classes of subgroups

Complements