Group #22586934 — subgroups

Subgroups — #22586934

Order 9088.

226 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 77 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	#17986556	C₂	1	1	yes	maximal, characteristic
2	C₄×#11765849	C₂	1	1	yes	maximal, characteristic
3	#17985740	C₂	1	4	yes	maximal, complemented
4	C₄×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11765849	—	2	1	no	—
6	#14556128	C₄	1	1	yes	characteristic
7	#14556128	C₄	1	1	yes	characteristic
8	#14556120	C₂²	1	2	yes	—
9	#14556107	—	2	1	no	—
10	#14556107	—	2	1	no	—
11	C₄×Dic₁₄₂	—	2	1	no	—
12	C₄×Dic₁₄₂	—	2	1	no	—
13	#14556087	C₄	1	2	yes	—
14	#14556039	C₂²	1	2	yes	—
15	C₄×C₇₁⋊C₈	—	2	1	no	—
16	C₂×M₁₆×C₇₁	—	2	1	no	—
17	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
18	#97×C₇₁	C₂²	1	2	yes	—
19	C₄×C₈×C₇₁	—	2	1	no	—
20	C₂×Dic₁₄₂	—	2	1	no	—
21	C₂×Dic₁₄₂	—	2	1	no	—
22	#11765849	—	2	4	no	—
23	C₂×C₇₁⋊C₈	—	2	2	no	—
24	C₄⋊Dic₇₁	—	2	1	no	—
25	C₄⋊Dic₇₁	—	2	1	no	—
26	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
27	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
28	Dic₇₁⋊C₄	—	2	2	no	—
29	Dic₇₁⋊C₄	—	2	2	no	—
30	C₂²⋊Dic₇₁	—	2	2	no	—
31	C₄×Dic₇₁	—	2	2	no	—
32	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
35	M₁₆×C₇₁	—	2	4	no	—
36	C₂×C₈×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	C₂×C₄×C₇₁	—	2	2	no	—
43	C₂×C₄×C₇₁	—	2	2	no	—
44	C₂×C₄×C₇₁	—	2	2	no	—
45	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₇₁	—	2	2	no	—
52	C₂×Dic₇₁	—	2	2	no	—
53	Dic₁₄₂	—	4	2	no	—
54	Dic₁₄₂	—	4	2	no	—
55	C₈×C₇₁	—	2	4	no	—
56	C₇₁⋊C₈	—	2	4	no	—
57	C₂²×C₇₁	—	2	1	no	—
58	C₂²×C₇₁	—	2	1	no	—
59	C₂²×C₇₁	—	2	1	no	—
60	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₄×C₇₁	—	2	1	no	—
62	C₄×C₇₁	—	2	1	no	—
63	C₄×C₇₁	—	2	1	no	—
64	C₄×C₇₁	#101	1	2	yes	—
65	C₄×C₇₁	#104	1	2	yes	—
66	C₄×C₇₁	#104	1	2	yes	—
67	Dic₇₁	—	4	2	no	—
68	Dic₇₁	—	4	2	no	—
69	C₂×C₇₁	—	4	1	no	—
70	C₂×C₇₁	C₂×#101	1	1	yes	characteristic
71	C₂×C₇₁	C₂×#104	1	1	yes	characteristic
72	C₂×C₇₁	C₂×#104	1	1	yes	characteristic
73	#1526	—	71	1	no	maximal, Sylow
74	C₇₁	#1526	1	1	yes	characteristic, complemented, Sylow
75	#533	—	71	1	no	—
76	C₄×M₁₆	—	71	1	no	—
77	C₄×M₁₆	D₇₁	1	1	yes	characteristic
78	#330	—	71	4	no	—
79	C₂×M₁₆	—	2	1	no	—
80	C₂×M₁₆	—	142	1	no	—
81	C₄⋊Q₈	—	71	1	no	—
82	C₄⋊Q₈	—	71	1	no	—
83	#125	—	71	2	no	—
84	C₂²⋊Q₈	—	142	1	no	—
85	C₂²⋊Q₈	—	142	1	no	—
86	C₄×Q₈	—	142	1	no	—
87	C₄×Q₈	—	142	1	no	—
88	#117	—	71	2	no	—
89	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
90	#97	—	71	2	no	—
91	#97	C₂×D₇₁	1	2	yes	—
92	C₄×C₈	—	2	1	no	—
93	C₄×C₈	—	142	1	no	—
94	C₂×Q₈	—	142	1	no	—
95	C₂×Q₈	—	142	1	no	—
96	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
97	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	M₁₆	—	142	4	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	142	2	no	—
103	C₄⋊C₄	—	71	2	no	—
104	C₄⋊C₄	—	71	2	no	—
105	C₄⋊C₄	—	142	1	no	—
106	C₄⋊C₄	—	142	1	no	—
107	C₄⋊C₄	—	142	2	no	—
108	C₄⋊C₄	—	142	2	no	—
109	C₂²⋊C₄	—	142	2	no	—
110	C₄²	—	142	2	no	—
111	C₄²	C₄×D₇₁	1	1	yes	characteristic
112	C₄²	D₂₈₄	1	1	yes	characteristic
113	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
114	C₄²	C₂²×D₇₁	1	1	yes	characteristic
115	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
116	Q₈	—	284	2	no	—
117	Q₈	—	284	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	142	2	no	—
122	C₂×C₄	—	142	2	no	—
123	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
124	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
125	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
126	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
127	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
128	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
129	C₈	—	2	4	no	—
130	C₈	—	142	4	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	284	2	no	—
139	C₄	—	284	2	no	—
140	C₄	#14556046	1	2	yes	—
141	C₄	#14556054	1	2	yes	—
142	C₄	#14556056	1	2	yes	—
143	C₂	—	4	1	no	—
144	C₂	C₂×#14556046	1	1	yes	characteristic
145	C₂	C₂×#14556054	1	1	yes	characteristic
146	C₂	C₂×#14556056	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986556

C₂

yes

maximal, characteristic

C₄×#11765849

C₂

yes

maximal, characteristic

#17985740

C₂

yes

maximal, complemented

C₄×M₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

—

#14556128

C₄

yes

characteristic

#14556128

C₄

yes

characteristic

#14556120

C₂²

yes

—

#14556107

—

#14556107

—

C₄×Dic₁₄₂

—

C₄×Dic₁₄₂

—

#14556087

C₄

yes

—

#14556039

C₂²

yes

—

C₄×C₇₁⋊C₈

—

C₂×M₁₆×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#97×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×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

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#101

yes

—

C₄×C₇₁

#104

yes

—

C₄×C₇₁

#104

yes

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#533

—

C₄×M₁₆

—

C₄×M₁₆

D₇₁

yes

characteristic

#330

—

C₂×M₁₆

—

C₂×M₁₆

—

142

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

142

—

C₂²⋊Q₈

—

142

—

C₄×Q₈

—

142

—

C₄×Q₈

—

142

—

#117

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#97

—

#97

C₂×D₇₁

yes

—

C₄×C₈

—

C₄×C₈

—

142

—

C₂×Q₈

—

142

—

C₂×Q₈

—

142

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

M₁₆

—

100

M₁₆

—

142

—

101

C₂×C₈

—

102

C₂×C₈

—

142

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

142

—

106

C₄⋊C₄

—

142

—

107

C₄⋊C₄

—

142

—

108

C₄⋊C₄

—

142

—

109

C₂²⋊C₄

—

142

—

110

C₄²

—

142

—

111

C₄²

C₄×D₇₁

yes

characteristic

112

C₄²

D₂₈₄

yes

characteristic

113

C₄²

C₂²⋊D₇₁

yes

characteristic

114

C₄²

C₂²×D₇₁

yes

characteristic

115

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

116

Q₈

—

284

—

117

Q₈

—

284

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

142

—

122

C₂×C₄

—

142

—

123

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

124

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

125

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

126

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

127

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

128

C₂×C₄

C₂×D₂₈₄

yes

characteristic

129

C₈

—

130

C₈

—

142

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

284

—

139

C₄

—

284

—

140

C₄

#14556046

yes

—

141

C₄

#14556054

yes

—

142

C₄

#14556056

yes

—

143

C₂

—

144

C₂

C₂×#14556046

yes

characteristic

145

C₂

C₂×#14556054

yes

characteristic

146

C₂

C₂×#14556056

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	C₄×M₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	C₄×M₁₆	D₇₁
hypocenter	C₇₁	#1526
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

(C₂×D₇₁)⋊C₄

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

C₄×M₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

hypercenter

C₄×M₁₆

D₇₁

hypocenter

C₇₁

#1526

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
3. #17985740	C₂	semidirect
74. C₇₁	#1526	semidirect

H ≅ G/N

type

3. #17985740

C₂

semidirect

74. C₇₁

#1526

semidirect

Subgroups — #22586934

Conjugacy classes of subgroups

Complements