Group #22586162 — subgroups

Subgroups — #22586162

Order 9088.

158 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17985897	C₂	1	1	yes	maximal, characteristic
2	#17985754	C₂	1	1	yes	maximal, characteristic
3	#336×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₈	—	2	1	no	—
5	C₂×C₄⋊Dic₇₁	C₄	1	1	yes	characteristic
6	C₂×C₄⋊Dic₇₁	C₄	1	1	yes	characteristic
7	C₂×Dic₇₁⋊C₄	—	2	1	no	—
8	C₇₁⋊#95	—	2	1	no	—
9	C₂²×C₈×C₇₁	—	2	1	no	—
10	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
11	C₂²×Dic₇₁	—	2	1	no	—
12	C₂²×Dic₇₁	—	2	1	no	—
13	C₂×C₇₁⋊C₈	—	2	2	no	—
14	C₂×C₇₁⋊C₈	—	4	1	no	—
15	C₂×C₇₁⋊C₈	—	4	1	no	—
16	C₄⋊Dic₇₁	—	2	1	no	—
17	C₄⋊Dic₇₁	—	4	1	no	—
18	C₄⋊Dic₇₁	C₈	1	2	yes	complemented
19	Dic₇₁⋊C₄	—	4	1	no	—
20	Dic₇₁⋊C₄	—	4	1	no	—
21	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₇₁	—	2	2	no	—
25	C₂×C₈×C₇₁	—	4	1	no	—
26	C₂×C₈×C₇₁	—	4	1	no	—
27	C₄²×C₇₁	—	4	1	no	—
28	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₇₁	—	2	1	no	—
30	C₂×C₄×C₇₁	—	2	1	no	—
31	C₂×C₄×C₇₁	—	2	1	no	—
32	C₂×C₄×C₇₁	—	2	1	no	—
33	C₂×C₄×C₇₁	—	2	1	no	—
34	C₂×C₄×C₇₁	—	2	1	no	—
35	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₇₁	M₁₆	1	1	yes	characteristic
37	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
38	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
41	C₂×Dic₇₁	—	2	2	no	—
42	C₂×Dic₇₁	—	4	1	no	—
43	C₂×Dic₇₁	—	4	1	no	—
44	C₂×Dic₇₁	—	4	1	no	—
45	C₂×Dic₇₁	—	4	1	no	—
46	C₂×Dic₇₁	—	4	1	no	—
47	C₈×C₇₁	—	4	2	no	—
48	C₇₁⋊C₈	—	4	2	no	—
49	C₂²×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
50	C₂²×C₇₁	#99	1	1	yes	characteristic
51	C₂²×C₇₁	#101	1	1	yes	characteristic
52	C₂²×C₇₁	#102	1	1	yes	characteristic
53	C₂²×C₇₁	#103	1	1	yes	characteristic
54	C₂²×C₇₁	#104	1	1	yes	characteristic
55	C₂²×C₇₁	#104	1	1	yes	characteristic
56	C₄×C₇₁	—	2	1	no	—
57	C₄×C₇₁	—	2	1	no	—
58	C₄×C₇₁	—	2	1	no	—
59	C₄×C₇₁	—	2	1	no	—
60	C₄×C₇₁	—	2	1	no	—
61	C₄×C₇₁	—	2	1	no	—
62	Dic₇₁	—	4	2	no	—
63	Dic₇₁	—	8	1	no	—
64	C₂×C₇₁	#324	1	1	yes	characteristic
65	C₂×C₇₁	D₄⋊C₈	1	1	yes	characteristic
66	C₂×C₇₁	Q₈⋊C₈	1	1	yes	characteristic
67	C₂×C₇₁	#327	1	1	yes	characteristic
68	C₂×C₇₁	#328	1	1	yes	characteristic
69	C₂×C₇₁	#330	1	1	yes	characteristic
70	C₂×C₇₁	#332	1	1	yes	characteristic
71	#1271	—	71	1	no	maximal, Sylow
72	C₇₁	#1271	1	1	yes	characteristic, complemented, Sylow
73	#389	—	71	1	no	—
74	#336	—	71	1	no	—
75	#336	D₇₁	1	1	yes	characteristic
76	C₂²×C₈	—	2	1	no	—
77	C₂²×C₈	—	142	1	no	—
78	C₂×C₄⋊C₄	—	71	1	no	—
79	C₂×C₄⋊C₄	—	71	1	no	—
80	C₂×C₄⋊C₄	—	142	1	no	—
81	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
82	#95	—	142	1	no	—
83	C₂²×C₄	—	142	1	no	—
84	C₂²×C₄	—	142	1	no	—
85	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
86	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	4	1	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	284	1	no	—
92	C₂×C₈	—	284	1	no	—
93	C₂×C₈	—	142	2	no	—
94	C₄⋊C₄	—	284	1	no	—
95	C₄⋊C₄	—	284	1	no	—
96	C₄⋊C₄	—	284	1	no	—
97	C₄⋊C₄	—	71	2	no	—
98	C₄⋊C₄	—	142	1	no	—
99	C₄²	—	4	1	no	—
100	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	284	1	no	—
108	C₂×C₄	—	284	1	no	—
109	C₂×C₄	—	284	1	no	—
110	C₂×C₄	—	284	1	no	—
111	C₂×C₄	—	284	1	no	—
112	C₂×C₄	—	142	2	no	—
113	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
114	C₂×C₄	#11765848	1	1	yes	characteristic
115	C₂×C₄	D₅₆₈	1	1	yes	characteristic
116	C₂×C₄	#11765853	1	1	yes	characteristic
117	C₂×C₄	#11765854	1	1	yes	characteristic
118	C₂×C₄	#11765855	1	1	yes	characteristic
119	C₈	—	4	2	no	—
120	C₈	—	284	2	no	—
121	C₂²	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
122	C₂²	#14556042	1	1	yes	characteristic
123	C₂²	#14556046	1	1	yes	characteristic
124	C₂²	#14556050	1	1	yes	characteristic
125	C₂²	#14556051	1	1	yes	characteristic
126	C₂²	#14556054	1	1	yes	characteristic
127	C₂²	#14556056	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	284	2	no	—
135	C₄	—	568	1	no	—
136	C₂	#17985724	1	1	yes	characteristic
137	C₂	#17985728	1	1	yes	characteristic
138	C₂	#17985729	1	1	yes	characteristic
139	C₂	#17985734	1	1	yes	characteristic
140	C₂	#17985735	1	1	yes	characteristic
141	C₂	#17985740	1	1	yes	characteristic
142	C₂	#17985744	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17985897

C₂

yes

maximal, characteristic

#17985754

C₂

yes

maximal, characteristic

#336×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

—

C₂×C₄⋊Dic₇₁

C₄

yes

characteristic

C₂×C₄⋊Dic₇₁

C₄

yes

characteristic

C₂×Dic₇₁⋊C₄

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₈

yes

complemented

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

M₁₆

yes

characteristic

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

Q₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂²⋊C₈

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#324

yes

characteristic

C₂×C₇₁

D₄⋊C₈

yes

characteristic

C₂×C₇₁

Q₈⋊C₈

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#389

—

#336

—

#336

D₇₁

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

142

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#95

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

—

284

—

C₂×C₈

—

142

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄²

—

100

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

284

—

108

C₂×C₄

—

284

—

109

C₂×C₄

—

284

—

110

C₂×C₄

—

284

—

111

C₂×C₄

—

284

—

112

C₂×C₄

—

142

—

113

C₂×C₄

C₈×D₇₁

yes

characteristic

114

C₂×C₄

#11765848

yes

characteristic

115

C₂×C₄

D₅₆₈

yes

characteristic

116

C₂×C₄

#11765853

yes

characteristic

117

C₂×C₄

#11765854

yes

characteristic

118

C₂×C₄

#11765855

yes

characteristic

119

C₈

—

120

C₈

—

284

—

121

C₂²

(C₂×D₇₁)⋊C₈

yes

characteristic

122

C₂²

#14556042

yes

characteristic

123

C₂²

#14556046

yes

characteristic

124

C₂²

#14556050

yes

characteristic

125

C₂²

#14556051

yes

characteristic

126

C₂²

#14556054

yes

characteristic

127

C₂²

#14556056

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

284

—

135

C₄

—

568

—

136

C₂

#17985724

yes

characteristic

137

C₂

#17985728

yes

characteristic

138

C₂

#17985729

yes

characteristic

139

C₂

#17985734

yes

characteristic

140

C₂

#17985735

yes

characteristic

141

C₂

#17985740

yes

characteristic

142

C₂

#17985744

yes

characteristic

	subgroup	G/H
center	C₂³	(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₂×C₈
Frattini	C₂×C₄²	C₂×D₇₁
Fitting	#336×C₇₁	C₂
socle	C₂³×C₇₁	C₂²⋊C₄
hypercenter	#336	D₇₁
hypocenter	C₇₁	#1271
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

(C₂×D₇₁)⋊C₄

commutator

C₂×C₄×C₇₁

C₂×C₈

Frattini

C₂×C₄²

C₂×D₇₁

Fitting

#336×C₇₁

C₂

socle

C₂³×C₇₁

C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
18. C₄⋊Dic₇₁	C₈	semidirect
72. C₇₁	#1271	semidirect

H ≅ G/N

type

18. C₄⋊Dic₇₁

C₈

semidirect

72. C₇₁

#1271

semidirect

Subgroups — #22586162

Conjugacy classes of subgroups

Complements