Group #22586795 — subgroups

Subgroups — #22586795

Order 9088.

232 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 101 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	#17986023	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊(C₇₁⋊C₈)	C₂	1	4	yes	maximal, complemented
4	#533×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11765849	—	2	1	no	—
6	C₂²×C₇₁⋊C₈	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
9	C₄×C₇₁⋊C₈	—	2	1	no	—
10	C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
12	C₄⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
13	#125×C₇₁	—	2	1	no	—
14	C₂²⋊Q₈×C₇₁	—	2	1	no	—
15	C₂²⋊Q₈×C₇₁	C₄	1	2	yes	—
16	C₄×Q₈×C₇₁	—	2	1	no	—
17	C₄×Q₈×C₇₁	C₂²	1	2	yes	—
18	#117×C₇₁	—	2	1	no	—
19	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
20	#11765849	—	4	2	no	—
21	C₂×C₇₁⋊C₈	—	2	4	no	—
22	C₂×C₇₁⋊C₈	—	2	2	no	—
23	C₂×C₇₁⋊C₈	—	2	2	no	—
24	C₂×Q₈×C₇₁	—	2	1	no	—
25	C₂×Q₈×C₇₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₇₁	—	2	1	no	—
30	C₄⋊C₄×C₇₁	—	2	1	no	—
31	C₄⋊C₄×C₇₁	—	2	1	no	—
32	C₄⋊C₄×C₇₁	—	2	2	no	—
33	C₄⋊C₄×C₇₁	—	2	2	no	—
34	C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
35	C₂²⋊C₄×C₇₁	—	2	2	no	—
36	C₄²×C₇₁	—	2	2	no	—
37	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₇₁	D₄	1	1	yes	characteristic
39	C₄²×C₇₁	D₄	1	1	yes	characteristic
40	C₄²×C₇₁	C₂³	1	1	yes	characteristic
41	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₇₁	—	2	4	no	—
43	Q₈×C₇₁	—	4	2	no	—
44	C₂×C₄×C₇₁	—	2	2	no	—
45	C₂×C₄×C₇₁	—	2	2	no	—
46	C₂×C₄×C₇₁	—	2	2	no	—
47	C₂×C₄×C₇₁	—	2	2	no	—
48	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
52	C₂×C₄×C₇₁	Q₁₆	1	2	yes	—
53	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
56	C₇₁⋊C₈	—	2	4	no	—
57	C₇₁⋊C₈	—	4	2	no	—
58	C₂²×C₇₁	—	2	1	no	—
59	C₂²×C₇₁	—	2	1	no	—
60	C₂²×C₇₁	#103	1	2	yes	—
61	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₇₁	—	2	1	no	—
63	C₄×C₇₁	—	2	1	no	—
64	C₄×C₇₁	—	4	2	no	—
65	C₄×C₇₁	—	4	2	no	—
66	C₄×C₇₁	#103	1	2	yes	—
67	C₄×C₇₁	#104	1	2	yes	—
68	C₄×C₇₁	#131	1	2	yes	—
69	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₇₁	—	2	2	no	—
72	C₂×C₇₁	#408	1	1	yes	characteristic
73	C₂×C₇₁	C₂×#103	1	1	yes	characteristic
74	C₂×C₇₁	C₂×#104	1	1	yes	characteristic
75	#1493	—	71	1	no	maximal, Sylow
76	C₇₁	#1493	1	1	yes	characteristic, complemented, Sylow
77	#533	D₇₁	1	1	yes	characteristic
78	#423	—	71	1	no	—
79	C₂×C₄×C₈	—	71	1	no	—
80	Q₈⋊C₈	—	71	4	no	—
81	C₂×M₁₆	—	142	1	no	—
82	C₂²×C₈	—	142	1	no	—
83	C₄⋊Q₈	Dic₇₁	1	1	yes	characteristic
84	C₄⋊Q₈	Dic₇₁	1	1	yes	characteristic
85	#125	—	2	1	no	—
86	C₂²⋊Q₈	—	2	1	no	—
87	C₂²⋊Q₈	Dic₇₁	1	2	yes	—
88	C₄×Q₈	—	2	1	no	—
89	C₄×Q₈	C₂×D₇₁	1	2	yes	—
90	#117	—	2	1	no	—
91	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
92	C₄⋊C₈	—	71	2	no	—
93	C₄⋊C₈	—	142	1	no	—
94	C₄×C₈	—	71	2	no	—
95	C₄×C₈	—	142	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	C₂×Dic₇₁	1	2	yes	—
98	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
100	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
101	M₁₆	—	284	2	no	—
102	C₂×C₈	—	142	2	no	—
103	C₂×C₈	—	142	4	no	—
104	C₂×C₈	—	142	2	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	C₂×Dic₇₁	1	2	yes	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
114	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
115	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
116	C₄²	C₂²×D₇₁	1	1	yes	characteristic
117	C₂³	C₂²⋊Dic₇₁	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	4	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
125	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
126	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
127	C₂×C₄	#11765852	1	2	yes	—
128	C₂×C₄	#11765855	1	2	yes	—
129	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
132	C₈	—	284	2	no	—
133	C₈	—	142	4	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	#14556052	1	2	yes	—
137	C₂²	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	2	no	—
141	C₄	—	4	2	no	—
142	C₄	#14556052	1	2	yes	—
143	C₄	#14556055	1	2	yes	—
144	C₄	#14556135	1	2	yes	—
145	C₄	C₂×#11765852	1	1	yes	characteristic
146	C₄	C₂×#11765855	1	1	yes	characteristic
147	C₂	—	2	2	no	—
148	C₂	#17985966	1	1	yes	characteristic
149	C₂	C₂×#14556052	1	1	yes	characteristic
150	C₂	C₂×#14556055	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986023

C₂

yes

maximal, characteristic

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

Q₈⋊(C₇₁⋊C₈)

C₂

yes

maximal, complemented

#533×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

—

C₂²×C₇₁⋊C₈

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₇₁

C₄

yes

characteristic

C₄⋊Q₈×C₇₁

C₄

yes

characteristic

#125×C₇₁

—

C₂²⋊Q₈×C₇₁

—

C₂²⋊Q₈×C₇₁

C₄

yes

—

C₄×Q₈×C₇₁

—

C₄×Q₈×C₇₁

C₂²

yes

—

#117×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×Q₈×C₇₁

—

C₂×Q₈×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

Q₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#103

yes

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#103

yes

—

C₄×C₇₁

#104

yes

—

C₄×C₇₁

#131

yes

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

C₂×Q₁₆

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#533

D₇₁

yes

characteristic

#423

—

C₂×C₄×C₈

—

Q₈⋊C₈

—

C₂×M₁₆

—

142

—

C₂²×C₈

—

142

—

C₄⋊Q₈

Dic₇₁

yes

characteristic

C₄⋊Q₈

Dic₇₁

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Dic₇₁

yes

—

C₄×Q₈

—

C₄×Q₈

C₂×D₇₁

yes

—

#117

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄×C₈

—

C₄×C₈

—

142

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

100

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

101

M₁₆

—

284

—

102

C₂×C₈

—

142

—

103

C₂×C₈

—

142

—

104

C₂×C₈

—

142

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

C₂×Dic₇₁

yes

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

C₂×Dic₇₁

yes

characteristic

114

C₄²

C₂²⋊D₇₁

yes

characteristic

115

C₄²

C₂²⋊D₇₁

yes

characteristic

116

C₄²

C₂²×D₇₁

yes

characteristic

117

C₂³

C₂²⋊Dic₇₁

yes

characteristic

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

125

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

126

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

127

C₂×C₄

#11765852

yes

—

128

C₂×C₄

#11765855

yes

—

129

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

132

C₈

—

284

—

133

C₈

—

142

—

134

C₂²

—

135

C₂²

—

136

C₂²

#14556052

yes

—

137

C₂²

C₂×C₂²⋊Dic₇₁

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#14556052

yes

—

143

C₄

#14556055

yes

—

144

C₄

#14556135

yes

—

145

C₄

C₂×#11765852

yes

characteristic

146

C₄

C₂×#11765855

yes

characteristic

147

C₂

—

148

C₂

#17985966

yes

characteristic

149

C₂

C₂×#14556052

yes

characteristic

150

C₂

C₂×#14556055

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	#533×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	#533	D₇₁
hypocenter	C₇₁	#1493
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

#533×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. Q₈⋊(C₇₁⋊C₈)	C₂	semidirect
76. C₇₁	#1493	semidirect

H ≅ G/N

type

3. Q₈⋊(C₇₁⋊C₈)

C₂

semidirect

76. C₇₁

#1493

semidirect

Subgroups — #22586795

Conjugacy classes of subgroups

Complements