Group #22597437 — subgroups

Subgroups — #22597437

Order 9088.

322 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 161 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	#17986127	C₂	1	4	yes	maximal, complemented
3	#17986126	C₂	1	2	yes	maximal, complemented
4	#17986123	C₂	1	2	yes	maximal, complemented
5	#17986057	C₂	1	4	yes	maximal
6	#17986052	C₂	1	1	yes	maximal, characteristic
7	#432×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#14556128	C₄	1	1	yes	characteristic
9	#14556128	C₄	1	1	yes	characteristic
10	#14556120	C₄	1	2	yes	—
11	#14556107	C₄	1	2	yes	—
12	#14556107	C₄	1	2	yes	—
13	C₄×Dic₁₄₂	C₂²	1	2	yes	—
14	C₄×Dic₁₄₂	C₂²	1	2	yes	—
15	#14556087	C₂²	1	2	yes	—
16	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
17	Dic₇₁⋊C₈	C₂²	1	4	yes	—
18	Dic₇₁⋊C₈	C₂²	1	4	yes	—
19	C₂²⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
20	#14556038	C₂²	1	4	yes	—
21	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
22	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
23	C₈×Dic₇₁	C₂²	1	4	yes	—
24	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
25	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
26	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
27	#97×C₇₁	C₂²	1	2	yes	—
28	C₂×Dic₁₄₂	C₂×C₄	1	2	yes	—
29	C₂×Dic₁₄₂	C₂×C₄	1	2	yes	—
30	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
31	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
32	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
33	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
34	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
35	C₄⋊Dic₇₁	C₂×C₄	1	2	yes	—
36	Dic₇₁⋊C₄	C₂×C₄	1	4	yes	—
37	Dic₇₁⋊C₄	C₂×C₄	1	4	yes	—
38	C₂²⋊Dic₇₁	C₂×C₄	1	4	yes	—
39	C₄×Dic₇₁	C₂³	1	4	yes	—
40	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
43	C₂×C₈×C₇₁	C₂³	1	4	yes	—
44	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
46	C₄²×C₇₁	C₂³	1	1	yes	characteristic
47	C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	—	2	2	no	—
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	4	1	no	—
52	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
58	C₂×Dic₇₁	C₂²×C₄	1	4	yes	—
59	C₂×Dic₇₁	C₂²×C₄	1	4	yes	—
60	Dic₁₄₂	—	2	4	no	—
61	Dic₁₄₂	—	4	2	no	—
62	C₈×C₇₁	—	2	4	no	—
63	C₇₁⋊C₈	—	2	2	no	—
64	C₇₁⋊C₈	—	2	2	no	—
65	C₂²×C₇₁	—	2	1	no	—
66	C₂²×C₇₁	—	2	1	no	—
67	C₂²×C₇₁	—	2	1	no	—
68	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₇₁	—	2	1	no	—
70	C₄×C₇₁	—	2	1	no	—
71	C₄×C₇₁	—	2	1	no	—
72	C₄×C₇₁	—	2	1	no	—
73	C₄×C₇₁	#131	1	2	yes	—
74	C₄×C₇₁	2₋¹⁺⁴	1	2	yes	—
75	Dic₇₁	—	2	4	no	—
76	Dic₇₁	—	2	4	no	—
77	C₂×C₇₁	—	4	1	no	—
78	C₂×C₇₁	#519	1	1	yes	characteristic
79	C₂×C₇₁	C₂×#131	1	1	yes	characteristic
80	C₂×C₇₁	#568	1	1	yes	characteristic
81	#2985	—	71	1	no	maximal, Sylow
82	C₇₁	#2985	1	1	yes	characteristic, complemented, Sylow
83	#533	—	71	1	no	—
84	#446	—	71	2	no	—
85	#446	—	71	4	no	—
86	C₈×Q₈	—	71	2	no	—
87	#433	—	71	4	no	—
88	#432	D₇₁	1	1	yes	characteristic
89	#431	—	71	1	no	—
90	C₄⋊Q₈	—	71	1	no	—
91	C₄⋊Q₈	—	71	1	no	—
92	#125	—	71	2	no	—
93	C₂²⋊Q₈	—	71	2	no	—
94	C₂²⋊Q₈	—	71	2	no	—
95	C₄×Q₈	—	71	2	no	—
96	C₄×Q₈	—	71	2	no	—
97	#117	—	71	2	no	—
98	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
99	C₄⋊C₈	—	71	4	no	—
100	C₄⋊C₈	—	71	4	no	—
101	C₄⋊C₈	—	71	2	no	—
102	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
103	C₂²⋊C₈	—	71	2	no	—
104	C₂²⋊C₈	C₂×D₇₁	1	2	yes	—
105	#97	—	71	4	no	—
106	#97	C₂×D₇₁	1	2	yes	—
107	C₄×C₈	—	71	4	no	—
108	C₄×C₈	—	71	1	no	—
109	C₄×C₈	—	71	1	no	—
110	C₂×Q₈	—	71	2	no	—
111	C₂×Q₈	—	71	2	no	—
112	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
113	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
115	C₂×C₈	—	71	2	no	—
116	C₂×C₈	—	71	2	no	—
117	C₂×C₈	C₂²×D₇₁	1	4	yes	—
118	C₄⋊C₄	—	71	4	no	—
119	C₄⋊C₄	—	71	4	no	—
120	C₄⋊C₄	—	71	2	no	—
121	C₄⋊C₄	—	71	2	no	—
122	C₄⋊C₄	—	71	2	no	—
123	C₄⋊C₄	—	71	2	no	—
124	C₂²⋊C₄	—	71	4	no	—
125	C₄²	—	71	4	no	—
126	C₄²	C₄×D₇₁	1	1	yes	characteristic
127	C₄²	C₄×D₇₁	1	1	yes	characteristic
128	C₄²	C₂²×D₇₁	1	1	yes	characteristic
129	C₄²	C₂²×D₇₁	1	1	yes	characteristic
130	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
131	Q₈	—	284	2	no	—
132	Q₈	—	142	4	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	71	4	no	—
137	C₂×C₄	—	71	4	no	—
138	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
139	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₈	—	142	2	no	—
146	C₈	—	142	2	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂²×C₄×D₇₁	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	142	4	no	—
156	C₄	—	142	4	no	—
157	C₄	#14556136	1	2	yes	—
158	C₄	#14556175	1	2	yes	—
159	C₂	—	4	1	no	—
160	C₂	#17986476	1	1	yes	characteristic
161	C₂	C₂×#14556136	1	1	yes	characteristic
162	C₂	#17986765	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986556

C₂

yes

maximal, characteristic

#17986127

C₂

yes

maximal, complemented

#17986126

C₂

yes

maximal, complemented

#17986123

C₂

yes

maximal, complemented

#17986057

C₂

yes

maximal

#17986052

C₂

yes

maximal, characteristic

#432×C₇₁

C₂

yes

maximal, characteristic

#14556128

C₄

yes

characteristic

#14556128

C₄

yes

characteristic

#14556120

C₄

yes

—

#14556107

C₄

yes

—

#14556107

C₄

yes

—

C₄×Dic₁₄₂

C₂²

yes

—

C₄×Dic₁₄₂

C₂²

yes

—

#14556087

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

Dic₇₁⋊C₈

C₂²

yes

—

Dic₇₁⋊C₈

C₂²

yes

—

C₂²⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556038

C₂²

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₄×C₇₁⋊C₈

C₂²

yes

characteristic

C₈×Dic₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

—

C₂²⋊C₈×C₇₁

C₂²

yes

—

#97×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

C₂×C₄

yes

—

C₂×Dic₁₄₂

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

C₂³

yes

—

C₂×C₇₁⋊C₈

C₂³

yes

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

Dic₇₁⋊C₄

C₂×C₄

yes

—

Dic₇₁⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₇₁

C₂×C₄

yes

—

C₄×Dic₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂³

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

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

C₂²×C₄

yes

—

C₂×Dic₇₁

C₂²×C₄

yes

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂³×C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#131

yes

—

C₄×C₇₁

2₋¹⁺⁴

yes

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#533

—

#446

—

#446

—

C₈×Q₈

—

#433

—

#432

D₇₁

yes

characteristic

#431

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

C₄⋊C₈

C₂×D₇₁

yes

—

103

C₂²⋊C₈

—

104

C₂²⋊C₈

C₂×D₇₁

yes

—

105

#97

—

106

#97

C₂×D₇₁

yes

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂²×C₄

C₄×D₇₁

yes

characteristic

113

C₂²×C₄

C₄×D₇₁

yes

characteristic

114

C₂²×C₄

C₂²×D₇₁

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₇₁

yes

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

C₄×D₇₁

yes

characteristic

127

C₄²

C₄×D₇₁

yes

characteristic

128

C₄²

C₂²×D₇₁

yes

characteristic

129

C₄²

C₂²×D₇₁

yes

characteristic

130

C₂³

C₂×C₄×D₇₁

yes

characteristic

131

Q₈

—

284

—

132

Q₈

—

142

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

139

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

140

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

141

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

142

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

143

C₂×C₄

C₂³×D₇₁

yes

characteristic

144

C₈

—

145

C₈

—

142

—

146

C₈

—

142

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂²×C₄×D₇₁

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

142

—

156

C₄

—

142

—

157

C₄

#14556136

yes

—

158

C₄

#14556175

yes

—

159

C₂

—

160

C₂

#17986476

yes

characteristic

161

C₂

C₂×#14556136

yes

characteristic

162

C₂

#17986765

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₁
commutator, socle	C₂²×C₇₁	C₂³×C₄
Fitting	#432×C₇₁	C₂
hypercenter	#432	D₇₁
hypocenter	C₇₁	#2985
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #432

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₇₁

commutator, socle

C₂²×C₇₁

C₂³×C₄

Fitting

#432×C₇₁

C₂

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₄ ◁ #432

N	H ≅ G/N	type
2. #17986127	C₂	semidirect
3. #17986126	C₂	semidirect
4. #17986123	C₂	semidirect
82. C₇₁	#2985	semidirect

H ≅ G/N

type

2. #17986127

C₂

semidirect

3. #17986126

C₂

semidirect

4. #17986123

C₂

semidirect

82. C₇₁

#2985

semidirect

Subgroups — #22597437

Conjugacy classes of subgroups

Complements