Group #22589284 — subgroups

Subgroups — #22589284

Order 9088.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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×C₄⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#17985760	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#14556128	C₄	1	1	yes	characteristic, complemented
6	#14556128	C₄	1	1	yes	characteristic, complemented
7	#14556120	C₄	1	2	yes	complemented
8	#14556107	—	2	1	no	—
9	#14556107	—	2	1	no	—
10	C₄×Dic₁₄₂	—	2	1	no	—
11	C₄×Dic₁₄₂	—	2	1	no	—
12	#14556087	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₇₁	—	2	1	no	—
14	C₂×C₄×Dic₇₁	C₂²	1	2	yes	—
15	C₇₁⋊#95	—	2	1	no	—
16	C₂×M₁₆×C₇₁	C₂²	1	2	yes	—
17	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
18	#97×C₇₁	—	2	1	no	—
19	C₄×C₈×C₇₁	—	2	1	no	—
20	C₂×Dic₁₄₂	—	2	1	no	—
21	C₂×Dic₁₄₂	—	2	1	no	—
22	C₂²×Dic₇₁	—	2	2	no	—
23	C₄⋊Dic₇₁	—	2	4	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	C₄×Dic₇₁	—	2	4	no	—
33	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
36	M₁₆×C₇₁	—	2	2	no	—
37	M₁₆×C₇₁	Q₈	1	4	yes	complemented
38	C₂×C₈×C₇₁	—	2	2	no	—
39	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₇₁	D₄	1	1	yes	characteristic
42	C₄²×C₇₁	D₄	1	1	yes	characteristic
43	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	—	2	2	no	—
45	C₂×C₄×C₇₁	—	4	1	no	—
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	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
53	C₂×C₄×C₇₁	#41	1	2	yes	—
54	C₂×Dic₇₁	—	2	4	no	—
55	C₂×Dic₇₁	—	2	2	no	—
56	C₂×Dic₇₁	—	2	2	no	—
57	C₂×Dic₇₁	—	4	2	no	—
58	C₂×Dic₇₁	—	4	2	no	—
59	Dic₁₄₂	—	4	2	no	—
60	Dic₁₄₂	—	4	2	no	—
61	C₈×C₇₁	—	2	4	no	—
62	C₂²×C₇₁	—	2	1	no	—
63	C₂²×C₇₁	—	2	1	no	—
64	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₇₁	#124	1	1	yes	characteristic
66	C₂²×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
67	C₄×C₇₁	—	2	1	no	—
68	C₄×C₇₁	—	2	1	no	—
69	C₄×C₇₁	—	2	1	no	—
70	C₄×C₇₁	#104	1	2	yes	—
71	C₄×C₇₁	C₄×Q₈	1	2	yes	—
72	C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
73	Dic₇₁	—	4	4	no	—
74	Dic₇₁	—	4	2	no	—
75	Dic₇₁	—	4	2	no	—
76	C₂×C₇₁	—	2	1	no	—
77	C₂×C₇₁	—	2	1	no	—
78	C₂×C₇₁	#391	1	1	yes	characteristic
79	C₂×C₇₁	C₂×#104	1	1	yes	characteristic
80	C₂×C₇₁	#421	1	1	yes	characteristic
81	#1986	—	71	1	no	maximal, Sylow
82	C₇₁	#1986	1	1	yes	characteristic, complemented, Sylow
83	#533	—	71	1	no	—
84	C₄×M₁₆	D₇₁	1	1	yes	characteristic
85	C₄×C₄⋊C₄	—	71	1	no	—
86	#339	—	71	4	no	—
87	C₂×M₁₆	C₂×D₇₁	1	2	yes	—
88	C₄⋊Q₈	—	71	1	no	—
89	C₄⋊Q₈	—	71	1	no	—
90	#125	—	71	2	no	—
91	C₂²⋊Q₈	—	142	1	no	—
92	C₂²⋊Q₈	—	142	1	no	—
93	C₄×Q₈	—	142	1	no	—
94	C₄×Q₈	—	142	1	no	—
95	#117	—	71	2	no	—
96	C₂×C₄⋊C₄	—	142	1	no	—
97	C₂×C₄²	—	71	2	no	—
98	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
99	#97	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	#95	—	142	1	no	—
102	C₂×Q₈	—	142	1	no	—
103	C₂×Q₈	—	142	1	no	—
104	C₂²×C₄	—	142	2	no	—
105	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
106	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	Dic₁₄₂	1	4	yes	complemented
110	C₂×C₈	—	2	2	no	—
111	C₄⋊C₄	—	71	2	no	—
112	C₄⋊C₄	—	71	2	no	—
113	C₄⋊C₄	—	142	4	no	—
114	C₄⋊C₄	—	142	2	no	—
115	C₄⋊C₄	—	142	2	no	—
116	C₄⋊C₄	—	142	1	no	—
117	C₄⋊C₄	—	142	1	no	—
118	C₂²⋊C₄	—	142	2	no	—
119	C₄²	—	142	4	no	—
120	C₄²	—	142	2	no	—
121	C₄²	C₄×D₇₁	1	1	yes	characteristic
122	C₄²	C₄×D₇₁	1	1	yes	characteristic
123	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
124	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
125	C₂³	C₂×D₂₈₄	1	1	yes	characteristic
126	Q₈	—	284	2	no	—
127	Q₈	—	284	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	284	2	no	—
131	C₂×C₄	—	284	2	no	—
132	C₂×C₄	—	142	4	no	—
133	C₂×C₄	—	142	2	no	—
134	C₂×C₄	—	142	2	no	—
135	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
138	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
141	C₂×C₄	C₂×Dic₁₄₂	1	2	yes	—
142	C₂×C₄	#11765865	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
147	C₂²	#14556116	1	1	yes	characteristic
148	C₂²	#14556128	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	284	4	no	—
153	C₄	—	284	2	no	—
154	C₄	—	284	2	no	—
155	C₄	#14556054	1	2	yes	—
156	C₄	C₄×Dic₁₄₂	1	2	yes	—
157	C₄	#14556107	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#17985905	1	1	yes	characteristic
161	C₂	C₂×#14556054	1	1	yes	characteristic
162	C₂	#17986015	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986556

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₇₁

C₂

yes

maximal, characteristic

#17985760

C₂

yes

maximal

C₄×M₁₆×C₇₁

C₂

yes

maximal, characteristic

#14556128

C₄

yes

characteristic, complemented

#14556128

C₄

yes

characteristic, complemented

#14556120

C₄

yes

complemented

#14556107

—

#14556107

—

C₄×Dic₁₄₂

—

C₄×Dic₁₄₂

—

#14556087

C₂²

yes

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄×Dic₇₁

C₂²

yes

—

C₇₁⋊#95

—

C₂×M₁₆×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#97×C₇₁

—

C₄×C₈×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

Q₈

yes

complemented

C₂×C₈×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₂³×C₇₁

C₂×D₄

yes

characteristic

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₁

#124

yes

characteristic

C₂²×C₇₁

C₄⋊Q₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#104

yes

—

C₄×C₇₁

C₄×Q₈

yes

—

C₄×C₇₁

C₂²⋊Q₈

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#533

—

C₄×M₁₆

D₇₁

yes

characteristic

C₄×C₄⋊C₄

—

#339

—

C₂×M₁₆

C₂×D₇₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

142

—

C₂²⋊Q₈

—

142

—

C₄×Q₈

—

142

—

C₄×Q₈

—

142

—

#117

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#97

—

100

C₄×C₈

—

101

#95

—

142

—

102

C₂×Q₈

—

142

—

103

C₂×Q₈

—

142

—

104

C₂²×C₄

—

142

—

105

C₂²×C₄

D₂₈₄

yes

characteristic

106

C₂²×C₄

D₂₈₄

yes

characteristic

107

C₂²×C₄

C₂²×D₇₁

yes

characteristic

108

M₁₆

—

109

M₁₆

Dic₁₄₂

yes

complemented

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

142

—

114

C₄⋊C₄

—

142

—

115

C₄⋊C₄

—

142

—

116

C₄⋊C₄

—

142

—

117

C₄⋊C₄

—

142

—

118

C₂²⋊C₄

—

142

—

119

C₄²

—

142

—

120

C₄²

—

142

—

121

C₄²

C₄×D₇₁

yes

characteristic

122

C₄²

C₄×D₇₁

yes

characteristic

123

C₄²

C₂²⋊D₇₁

yes

characteristic

124

C₄²

C₂²⋊D₇₁

yes

characteristic

125

C₂³

C₂×D₂₈₄

yes

characteristic

126

Q₈

—

284

—

127

Q₈

—

284

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

284

—

131

C₂×C₄

—

284

—

132

C₂×C₄

—

142

—

133

C₂×C₄

—

142

—

134

C₂×C₄

—

142

—

135

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

136

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

137

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

138

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

139

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

140

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

141

C₂×C₄

C₂×Dic₁₄₂

yes

—

142

C₂×C₄

#11765865

yes

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

147

C₂²

#14556116

yes

characteristic

148

C₂²

#14556128

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

284

—

153

C₄

—

284

—

154

C₄

—

284

—

155

C₄

#14556054

yes

—

156

C₄

C₄×Dic₁₄₂

yes

—

157

C₄

#14556107

yes

—

158

C₂

—

159

C₂

—

160

C₂

#17985905

yes

characteristic

161

C₂

C₂×#14556054

yes

characteristic

162

C₂

#17986015

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₄×M₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	C₄×M₁₆	D₇₁
hypocenter	C₇₁	#1986
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₄×M₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

hypercenter

C₄×M₁₆

D₇₁

hypocenter

C₇₁

#1986

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
5. #14556128	C₄	semidirect
6. #14556128	C₄	semidirect
7. #14556120	C₄	semidirect
37. M₁₆×C₇₁	Q₈	semidirect
82. C₇₁	#1986	semidirect
109. M₁₆	Dic₁₄₂	semidirect

H ≅ G/N

type

5. #14556128

C₄

semidirect

6. #14556128

C₄

semidirect

7. #14556120

C₄

semidirect

37. M₁₆×C₇₁

Q₈

semidirect

82. C₇₁

#1986

semidirect

109. M₁₆

Dic₁₄₂

semidirect

Subgroups — #22589284

Conjugacy classes of subgroups

Complements