Group #22589740 — subgroups

Subgroups — #22589740

Order 9088.

196 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17986048	C₂	1	1	yes	maximal, characteristic
2	#17986047	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556046	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556046	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556046	C₂	1	1	yes	maximal, characteristic
6	C₇₁⋊#338	C₂	1	1	yes	maximal, characteristic
7	#430×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₁₄₂	C₂²	1	1	yes	characteristic
9	C₂×#11765849	C₂²	1	1	yes	characteristic
10	C₂×#11765849	C₂²	1	1	yes	characteristic
11	C₂×#11765849	C₂²	1	1	yes	characteristic
12	#14556064	—	2	2	no	—
13	#14556063	—	2	2	no	—
14	#14556046	—	2	2	no	—
15	#14556046	—	2	2	no	—
16	#14556046	—	2	2	no	—
17	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
20	#108×C₇₁	—	2	2	no	—
21	C₂×Dic₁₄₂	—	2	1	no	—
22	C₂×Dic₁₄₂	—	2	1	no	—
23	C₂×Dic₁₄₂	—	2	1	no	—
24	C₂×Dic₁₄₂	—	2	1	no	—
25	C₂×Dic₁₄₂	—	2	1	no	—
26	C₂×Dic₁₄₂	—	2	1	no	—
27	C₂²×Dic₇₁	Q₈	1	2	yes	—
28	#11765849	—	2	1	no	—
29	#11765849	—	2	1	no	—
30	#11765849	—	2	1	no	—
31	#11765849	—	2	1	no	—
32	#11765849	—	2	1	no	—
33	#11765849	D₄	1	2	yes	—
34	C₂×C₇₁⋊C₈	—	2	1	no	—
35	C₂×C₇₁⋊C₈	—	2	1	no	—
36	C₂×C₇₁⋊C₈	—	2	1	no	—
37	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
38	M₁₆×C₇₁	—	2	1	no	—
39	M₁₆×C₇₁	—	2	1	no	—
40	M₁₆×C₇₁	—	2	1	no	—
41	M₁₆×C₇₁	—	2	1	no	—
42	M₁₆×C₇₁	D₄	1	2	yes	—
43	M₁₆×C₇₁	D₄	1	2	yes	—
44	C₂×C₈×C₇₁	—	2	1	no	—
45	C₂×C₈×C₇₁	—	2	1	no	—
46	C₂×C₈×C₇₁	—	2	1	no	—
47	C₂³×C₇₁	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂×D₄	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×C₄×C₇₁	#41	1	1	yes	characteristic
52	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
53	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
54	C₂×Dic₇₁	—	2	2	no	—
55	C₂×Dic₇₁	—	2	2	no	—
56	C₂×Dic₇₁	—	2	2	no	—
57	Dic₁₄₂	—	4	1	no	—
58	Dic₁₄₂	—	4	1	no	—
59	Dic₁₄₂	—	4	1	no	—
60	Dic₁₄₂	—	4	1	no	—
61	C₈×C₇₁	—	2	2	no	—
62	C₈×C₇₁	—	2	2	no	—
63	C₈×C₇₁	—	4	1	no	—
64	C₇₁⋊C₈	—	2	2	no	—
65	C₇₁⋊C₈	—	4	1	no	—
66	C₇₁⋊C₈	—	4	1	no	—
67	C₂²×C₇₁	—	4	1	no	—
68	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂²×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
71	C₄×C₇₁	#121	1	1	yes	characteristic
72	C₄×C₇₁	#121	1	1	yes	characteristic
73	C₄×C₇₁	#121	1	1	yes	characteristic
74	C₄×C₇₁	#126	1	1	yes	characteristic
75	Dic₇₁	—	4	2	no	—
76	C₂×C₇₁	—	2	1	no	—
77	C₂×C₇₁	—	2	1	no	—
78	C₂×C₇₁	—	2	1	no	—
79	C₂×C₇₁	#396	1	1	yes	characteristic
80	#2070	—	71	1	no	maximal, Sylow
81	C₇₁	#2070	1	1	yes	characteristic, complemented, Sylow
82	#430	—	71	1	no	—
83	#430	—	71	1	no	—
84	#430	D₇₁	1	1	yes	characteristic
85	C₂×#101	—	71	1	no	—
86	C₂×#101	—	71	1	no	—
87	C₂×#101	—	71	1	no	—
88	#338	—	71	1	no	—
89	C₂²×Q₈	—	71	1	no	—
90	C₂×M₁₆	—	71	1	no	—
91	C₂×M₁₆	—	71	1	no	—
92	C₂×M₁₆	—	71	1	no	—
93	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
94	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
95	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
96	#108	—	2	2	no	—
97	#108	—	142	2	no	—
98	#108	—	142	2	no	—
99	#101	—	142	2	no	—
100	#101	—	142	2	no	—
101	#101	—	142	2	no	—
102	C₂×Q₈	—	142	1	no	—
103	C₂×Q₈	—	142	1	no	—
104	C₂×Q₈	—	142	1	no	—
105	C₂×Q₈	—	142	1	no	—
106	C₂×Q₈	—	142	1	no	—
107	C₂×Q₈	—	142	1	no	—
108	C₂²×C₄	—	71	2	no	—
109	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
110	M₁₆	—	2	1	no	—
111	M₁₆	—	2	1	no	—
112	M₁₆	—	2	1	no	—
113	M₁₆	—	2	1	no	—
114	M₁₆	—	71	2	no	—
115	M₁₆	—	142	1	no	—
116	M₁₆	—	142	1	no	—
117	M₁₆	—	142	1	no	—
118	M₁₆	—	142	1	no	—
119	M₁₆	—	142	1	no	—
120	M₁₆	D₂₈₄	1	2	yes	—
121	M₁₆	C₂²⋊D₇₁	1	2	yes	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	2	1	no	—
125	C₂×C₈	—	142	1	no	—
126	C₂×C₈	—	142	1	no	—
127	C₂×C₈	—	142	1	no	—
128	C₂³	Q₈×D₇₁	1	1	yes	characteristic
129	Q₈	—	284	1	no	—
130	Q₈	—	284	1	no	—
131	Q₈	—	284	1	no	—
132	Q₈	—	284	1	no	—
133	C₂×C₄	—	142	2	no	—
134	C₂×C₄	—	142	2	no	—
135	C₂×C₄	—	142	2	no	—
136	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
137	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
139	C₂×C₄	#11765864	1	1	yes	characteristic
140	C₂×C₄	#11765865	1	1	yes	characteristic
141	C₂×C₄	#11765866	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	4	1	no	—
145	C₈	—	284	1	no	—
146	C₈	—	284	1	no	—
147	C₈	—	142	2	no	—
148	C₂²	—	4	1	no	—
149	C₂²	#14556104	1	1	yes	characteristic
150	C₂²	#14556106	1	1	yes	characteristic
151	C₂²	#14556108	1	1	yes	characteristic
152	C₄	—	284	2	no	—
153	C₄	#14556100	1	1	yes	characteristic
154	C₄	#14556102	1	1	yes	characteristic
155	C₄	#14556103	1	1	yes	characteristic
156	C₄	#14556123	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#17985921	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986048

C₂

yes

maximal, characteristic

#17986047

C₂

yes

maximal, characteristic

C₂×#14556046

C₂

yes

maximal, characteristic

C₂×#14556046

C₂

yes

maximal, characteristic

C₂×#14556046

C₂

yes

maximal, characteristic

C₇₁⋊#338

C₂

yes

maximal, characteristic

#430×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₂

C₂²

yes

characteristic

C₂×#11765849

C₂²

yes

characteristic

C₂×#11765849

C₂²

yes

characteristic

C₂×#11765849

C₂²

yes

—

—

—

—

—

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

#108×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

Q₈

yes

—

—

—

—

—

—

D₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

D₄

yes

—

M₁₆×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂³×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#430

—

#430

—

#430

D₇₁

yes

—

—

—

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂×M₁₆

C₂×D₇₁

yes

characteristic

#108

—

#108

—

142

—

#108

—

142

—

#101

—

142

—

100

#101

—

142

—

101

#101

—

142

—

102

C₂×Q₈

—

142

—

103

C₂×Q₈

—

142

—

104

C₂×Q₈

—

142

—

105

C₂×Q₈

—

142

—

106

C₂×Q₈

—

142

—

107

C₂×Q₈

—

142

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂²×D₇₁

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

142

—

116

M₁₆

—

142

—

117

M₁₆

—

142

—

118

M₁₆

—

142

—

119

M₁₆

—

142

—

120

M₁₆

D₂₈₄

yes

—

121

M₁₆

C₂²⋊D₇₁

yes

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

142

—

126

C₂×C₈

—

142

—

127

C₂×C₈

—

142

—

128

C₂³

Q₈×D₇₁

yes

characteristic

129

Q₈

—

284

—

130

Q₈

—

284

—

131

Q₈

—

284

—

132

Q₈

—

284

—

133

C₂×C₄

—

142

—

134

C₂×C₄

—

142

—

135

C₂×C₄

—

142

—

136

C₂×C₄

D₄×D₇₁

yes

characteristic

137

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

138

C₂×C₄

C₂×D₂₈₄

yes

characteristic

139

C₂×C₄

#11765864

yes

characteristic

140

C₂×C₄

#11765865

yes

characteristic

141

C₂×C₄

#11765866

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

284

—

146

C₈

—

284

—

147

C₈

—

142

—

148

C₂²

—

149

C₂²

#14556104

yes

characteristic

150

C₂²

#14556106

yes

characteristic

151

C₂²

#14556108

yes

characteristic

152

C₄

—

284

—

153

C₄

#14556100

yes

characteristic

154

C₄

#14556102

yes

characteristic

155

C₄

#14556103

yes

characteristic

156

C₄

#14556123

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#17985921

yes

characteristic

	subgroup	G/H
center	C₂	#17985921
commutator	C₂²×C₄×C₇₁	C₂³
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#430×C₇₁	C₂
socle	C₂×C₇₁	#396
hypercenter	#430	D₇₁
hypocenter	C₇₁	#2070
LCS	G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #430

subgroup

G/H

center

C₂

#17985921

commutator

C₂²×C₄×C₇₁

C₂³

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#430×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂²×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #430

N	H ≅ G/N	type
81. C₇₁	#2070	semidirect

H ≅ G/N

type

81. C₇₁

#2070

semidirect

Subgroups — #22589740

Conjugacy classes of subgroups

Complements