Group #22586165 — subgroups

Subgroups — #22586165

Order 9088.

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

normal subgroups only

This group has 166 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	#17985905	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×Dic₁₄₂	C₄	1	1	yes	characteristic
5	C₂²×C₇₁⋊C₈	—	2	1	no	—
6	C₂×C₄⋊Dic₇₁	C₄	1	1	yes	characteristic
7	C₇₁⋊#95	—	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	C₂×Dic₁₄₂	—	2	1	no	—
12	C₂×Dic₁₄₂	—	4	1	no	—
13	C₂×Dic₁₄₂	—	4	1	no	—
14	C₂×Dic₁₄₂	C₈	1	2	yes	complemented
15	C₂²×Dic₇₁	—	2	1	no	—
16	C₂²×Dic₇₁	—	2	1	no	—
17	C₂×C₇₁⋊C₈	—	2	2	no	—
18	C₂×C₇₁⋊C₈	—	4	1	no	—
19	C₂×C₇₁⋊C₈	—	4	1	no	—
20	C₄⋊Dic₇₁	—	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	SD₁₆	1	1	yes	characteristic
38	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₁	Q₁₆	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	Dic₁₄₂	—	4	2	no	—
48	Dic₁₄₂	—	4	2	no	—
49	C₈×C₇₁	—	4	2	no	—
50	C₇₁⋊C₈	—	4	2	no	—
51	C₂²×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
52	C₂²×C₇₁	#99	1	1	yes	characteristic
53	C₂²×C₇₁	#100	1	1	yes	characteristic
54	C₂²×C₇₁	#103	1	1	yes	characteristic
55	C₂²×C₇₁	#103	1	1	yes	characteristic
56	C₂²×C₇₁	#104	1	1	yes	characteristic
57	C₂²×C₇₁	#104	1	1	yes	characteristic
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	C₄×C₇₁	—	2	1	no	—
63	C₄×C₇₁	—	2	1	no	—
64	Dic₇₁	—	4	2	no	—
65	Dic₇₁	—	8	1	no	—
66	C₂×C₇₁	#323	1	1	yes	characteristic
67	C₂×C₇₁	Q₈⋊C₈	1	1	yes	characteristic
68	C₂×C₇₁	Q₈⋊C₈	1	1	yes	characteristic
69	C₂×C₇₁	#328	1	1	yes	characteristic
70	C₂×C₇₁	#328	1	1	yes	characteristic
71	C₂×C₇₁	#329	1	1	yes	characteristic
72	C₂×C₇₁	#333	1	1	yes	characteristic
73	#1272	—	71	1	no	maximal, Sylow
74	C₇₁	#1272	1	1	yes	characteristic, complemented, Sylow
75	#391	—	71	1	no	—
76	#336	—	71	1	no	—
77	#336	D₇₁	1	1	yes	characteristic
78	C₂²×Q₈	—	71	1	no	—
79	C₂²×C₈	—	2	1	no	—
80	C₂²×C₈	—	142	1	no	—
81	C₂×C₄⋊C₄	—	71	1	no	—
82	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
83	#95	—	142	1	no	—
84	#95	—	142	1	no	—
85	C₂×Q₈	—	284	1	no	—
86	C₂×Q₈	—	284	1	no	—
87	C₂×Q₈	—	71	2	no	—
88	C₂×Q₈	—	142	1	no	—
89	C₂²×C₄	—	142	1	no	—
90	C₂²×C₄	—	142	1	no	—
91	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
92	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	—	4	1	no	—
97	C₂×C₈	—	284	1	no	—
98	C₂×C₈	—	284	1	no	—
99	C₂×C₈	—	142	2	no	—
100	C₄⋊C₄	—	284	1	no	—
101	C₄²	—	4	1	no	—
102	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
103	Q₈	—	284	2	no	—
104	Q₈	—	284	2	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	284	1	no	—
112	C₂×C₄	—	284	1	no	—
113	C₂×C₄	—	284	1	no	—
114	C₂×C₄	—	284	1	no	—
115	C₂×C₄	—	284	1	no	—
116	C₂×C₄	—	142	2	no	—
117	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
118	C₂×C₄	#11765848	1	1	yes	characteristic
119	C₂×C₄	#11765853	1	1	yes	characteristic
120	C₂×C₄	#11765854	1	1	yes	characteristic
121	C₂×C₄	#11765855	1	1	yes	characteristic
122	C₂×C₄	Dic₂₈₄	1	1	yes	characteristic
123	C₈	—	4	2	no	—
124	C₈	—	284	2	no	—
125	C₂²	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
126	C₂²	#14556042	1	1	yes	characteristic
127	C₂²	#14556044	1	1	yes	characteristic
128	C₂²	#14556051	1	1	yes	characteristic
129	C₂²	#14556053	1	1	yes	characteristic
130	C₂²	#14556054	1	1	yes	characteristic
131	C₂²	#14556056	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	284	2	no	—
139	C₄	—	568	1	no	—
140	C₂	#17985722	1	1	yes	characteristic
141	C₂	#17985729	1	1	yes	characteristic
142	C₂	#17985731	1	1	yes	characteristic
143	C₂	#17985735	1	1	yes	characteristic
144	C₂	#17985737	1	1	yes	characteristic
145	C₂	#17985738	1	1	yes	characteristic
146	C₂	#17985747	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17985905

C₂

yes

maximal, characteristic

#17985754

C₂

yes

maximal, characteristic

#336×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₄

yes

characteristic

C₂²×C₇₁⋊C₈

—

C₂×C₄⋊Dic₇₁

C₄

yes

characteristic

C₇₁⋊#95

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

C₈

yes

complemented

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

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

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

Q₁₆

yes

characteristic

C₂×C₄×C₇₁

Q₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

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

#323

yes

characteristic

C₂×C₇₁

Q₈⋊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

#391

—

#336

—

#336

D₇₁

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

142

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#95

—

142

—

#95

—

142

—

C₂×Q₈

—

284

—

C₂×Q₈

—

284

—

C₂×Q₈

—

C₂×Q₈

—

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

—

100

C₄⋊C₄

—

284

—

101

C₄²

—

102

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

103

Q₈

—

284

—

104

Q₈

—

284

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

284

—

112

C₂×C₄

—

284

—

113

C₂×C₄

—

284

—

114

C₂×C₄

—

284

—

115

C₂×C₄

—

284

—

116

C₂×C₄

—

142

—

117

C₂×C₄

C₈×D₇₁

yes

characteristic

118

C₂×C₄

#11765848

yes

characteristic

119

C₂×C₄

#11765853

yes

characteristic

120

C₂×C₄

#11765854

yes

characteristic

121

C₂×C₄

#11765855

yes

characteristic

122

C₂×C₄

Dic₂₈₄

yes

characteristic

123

C₈

—

124

C₈

—

284

—

125

C₂²

(C₂×D₇₁)⋊C₈

yes

characteristic

126

C₂²

#14556042

yes

characteristic

127

C₂²

#14556044

yes

characteristic

128

C₂²

#14556051

yes

characteristic

129

C₂²

#14556053

yes

characteristic

130

C₂²

#14556054

yes

characteristic

131

C₂²

#14556056

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

284

—

139

C₄

—

568

—

140

C₂

#17985722

yes

characteristic

141

C₂

#17985729

yes

characteristic

142

C₂

#17985731

yes

characteristic

143

C₂

#17985735

yes

characteristic

144

C₂

#17985737

yes

characteristic

145

C₂

#17985738

yes

characteristic

146

C₂

#17985747

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	#1272
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
14. C₂×Dic₁₄₂	C₈	semidirect
74. C₇₁	#1272	semidirect

H ≅ G/N

type

14. C₂×Dic₁₄₂

C₈

semidirect

74. C₇₁

#1272

semidirect

Subgroups — #22586165

Conjugacy classes of subgroups

Complements