Group #22598880 — subgroups

Subgroups — #22598880

Order 9088.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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	#17986627	C₂	1	2	yes	maximal
2	#17986406	C₂	1	4	yes	maximal, complemented
3	#17986403	C₂	1	2	yes	maximal, complemented
4	#17986400	C₂	1	2	yes	maximal, complemented
5	#17986042	C₂	1	2	yes	maximal
6	C₂×#14556062	C₂	1	1	yes	maximal, characteristic
7	#17985954	C₂	1	1	yes	maximal, characteristic
8	#427×C₇₁	C₂	1	1	yes	maximal, characteristic
9	C₂×#11765849	C₂²	1	1	yes	characteristic
10	#14556128	—	2	2	no	—
11	#14556127	C₂²	1	4	yes	—
12	#14556120	—	2	2	no	—
13	#14556119	C₂²	1	4	yes	—
14	#14556105	—	2	4	no	—
15	#14556087	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
17	#14556062	C₂²	1	2	yes	—
18	#14556062	C₂²	1	2	yes	—
19	#14556061	C₂²	1	4	yes	—
20	#14556059	C₂²	1	4	yes	—
21	#14556038	C₂²	1	2	yes	—
22	C₈×Dic₇₁	C₂²	1	2	yes	—
23	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
24	#117×C₇₁	C₂²	1	2	yes	—
25	#107×C₇₁	C₂²	1	1	yes	characteristic
26	#107×C₇₁	C₂²	1	1	yes	characteristic
27	#106×C₇₁	C₂²	1	2	yes	—
28	C₂×Dic₁₄₂	—	2	4	no	—
29	C₂²×Dic₇₁	—	2	2	no	—
30	#11765849	Q₈	1	4	yes	complemented
31	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
32	C₄⋊Dic₇₁	—	2	2	no	—
33	C₄⋊Dic₇₁	—	2	4	no	—
34	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
35	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
36	C₄⋊Dic₇₁	C₂³	1	4	yes	—
37	Dic₇₁⋊C₄	—	2	4	no	—
38	Dic₇₁⋊C₄	—	2	4	no	—
39	C₂²⋊Dic₇₁	D₄	1	2	yes	—
40	C₄×Dic₇₁	C₂³	1	2	yes	—
41	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
42	C₂×C₈×C₇₁	Q₈	1	4	yes	complemented
43	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
46	C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
47	C₂²⋊C₄×C₇₁	—	2	2	no	—
48	C₄²×C₇₁	—	2	2	no	—
49	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	—	2	2	no	—
51	C₂×C₄×C₇₁	—	2	2	no	—
52	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
54	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
55	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
56	C₂×Dic₇₁	—	2	4	no	—
57	C₂×Dic₇₁	—	8	2	no	—
58	C₂×Dic₇₁	C₂×D₄	1	2	yes	—
59	C₂×Dic₇₁	C₂×D₄	1	2	yes	—
60	Dic₁₄₂	—	8	4	no	—
61	C₈×C₇₁	C₂×Q₈	1	2	yes	—
62	C₈×C₇₁	C₂×Q₈	1	2	yes	—
63	C₇₁⋊C₈	C₂×Q₈	1	4	yes	—
64	C₂²×C₇₁	—	2	1	no	—
65	C₂²×C₇₁	—	2	1	no	—
66	C₂²×C₇₁	C₄⋊Q₈	1	2	yes	—
67	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
68	C₄×C₇₁	—	4	2	no	—
69	C₄×C₇₁	—	4	2	no	—
70	C₄×C₇₁	C₄⋊Q₈	1	2	yes	—
71	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
72	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
73	Dic₇₁	—	2	2	no	—
74	Dic₇₁	—	2	2	no	—
75	Dic₇₁	—	4	4	no	—
76	C₂×C₇₁	—	2	2	no	—
77	C₂×C₇₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₇₁	#576	1	1	yes	characteristic
79	C₂×C₇₁	#578	1	1	yes	characteristic
80	#3164	—	71	1	no	maximal, Sylow
81	C₇₁	#3164	1	1	yes	characteristic, complemented, Sylow
82	#544	—	71	2	no	—
83	#501	—	71	4	no	—
84	#500	—	71	2	no	—
85	#499	—	71	2	no	—
86	#428	—	71	2	no	—
87	#427	D₇₁	1	1	yes	characteristic
88	C₂×#107	—	71	1	no	—
89	#405	—	71	1	no	—
90	C₂×M₁₆	—	71	1	no	—
91	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
92	C₄⋊Q₈	—	71	4	no	—
93	C₄⋊Q₈	—	142	2	no	—
94	#125	—	71	4	no	—
95	#125	—	142	2	no	—
96	C₂²⋊Q₈	—	142	4	no	—
97	#117	—	71	1	no	—
98	#117	C₂×D₇₁	1	2	yes	—
99	C₂×C₄⋊C₄	—	71	2	no	—
100	#107	—	71	2	no	—
101	#107	—	71	2	no	—
102	#107	—	71	4	no	—
103	#107	C₂×D₇₁	1	1	yes	characteristic
104	#107	C₂×D₇₁	1	1	yes	characteristic
105	#106	—	71	4	no	—
106	#106	C₂×D₇₁	1	2	yes	—
107	#97	—	71	2	no	—
108	C₄×C₈	—	71	2	no	—
109	C₂×Q₈	—	142	4	no	—
110	C₂²×C₄	—	142	2	no	—
111	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
112	M₁₆	—	71	4	no	—
113	C₂×C₈	—	71	2	no	—
114	C₂×C₈	Dic₁₄₂	1	4	yes	complemented
115	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
116	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
117	C₄⋊C₄	—	71	1	no	—
118	C₄⋊C₄	—	71	1	no	—
119	C₄⋊C₄	—	71	4	no	—
120	C₄⋊C₄	—	142	4	no	—
121	C₄⋊C₄	—	142	4	no	—
122	C₄⋊C₄	—	142	4	no	—
123	C₄⋊C₄	—	142	2	no	—
124	C₄⋊C₄	C₂²×D₇₁	1	2	yes	—
125	C₄⋊C₄	C₂²×D₇₁	1	2	yes	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₂²⋊C₄	—	71	2	no	—
128	C₄²	—	2	2	no	—
129	C₄²	—	71	2	no	—
130	C₂³	D₄×D₇₁	1	1	yes	characteristic
131	Q₈	—	568	4	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	568	2	no	—
135	C₂×C₄	—	71	2	no	—
136	C₂×C₄	—	71	2	no	—
137	C₂×C₄	—	142	4	no	—
138	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
139	C₂×C₄	Q₈×D₇₁	1	2	yes	—
140	C₂×C₄	C₂×Dic₁₄₂	1	2	yes	—
141	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
142	C₈	—	71	4	no	—
143	C₈	C₂×Dic₁₄₂	1	2	yes	—
144	C₈	C₂×Dic₁₄₂	1	2	yes	—
145	C₂²	—	2	1	no	—
146	C₂²	—	2	1	no	—
147	C₂²	#14556127	1	2	yes	—
148	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
149	C₄	—	4	2	no	—
150	C₄	—	4	2	no	—
151	C₄	—	284	4	no	—
152	C₄	—	142	2	no	—
153	C₄	—	142	2	no	—
154	C₄	#14556127	1	2	yes	—
155	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
156	C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	C₂×#14556127	1	1	yes	characteristic
159	C₂	#17986801	1	1	yes	characteristic
160	C₂	#17986813	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986627

C₂

yes

maximal

#17986406

C₂

yes

maximal, complemented

#17986403

C₂

yes

maximal, complemented

#17986400

C₂

yes

maximal, complemented

#17986042

C₂

yes

maximal

C₂×#14556062

C₂

yes

maximal, characteristic

#17985954

C₂

yes

maximal, characteristic

#427×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

characteristic

#14556128

—

#14556127

C₂²

yes

—

#14556120

—

#14556119

C₂²

yes

—

#14556105

—

#14556087

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

—

#14556062

C₂²

yes

—

#14556062

C₂²

yes

—

#14556061

C₂²

yes

—

#14556059

C₂²

yes

—

#14556038

C₂²

yes

—

C₈×Dic₇₁

C₂²

yes

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

#117×C₇₁

C₂²

yes

—

#107×C₇₁

C₂²

yes

characteristic

#107×C₇₁

C₂²

yes

characteristic

#106×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

#11765849

Q₈

yes

complemented

C₂×C₇₁⋊C₈

C₂³

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

C₄⋊Dic₇₁

D₄

yes

characteristic

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

D₄

yes

—

C₄×Dic₇₁

C₂³

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

Q₈

yes

complemented

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

C₂³

yes

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂×D₄

yes

—

C₂×Dic₇₁

C₂×D₄

yes

—

Dic₁₄₂

—

C₈×C₇₁

C₂×Q₈

yes

—

C₈×C₇₁

C₂×Q₈

yes

—

C₇₁⋊C₈

C₂×Q₈

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₄⋊Q₈

yes

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄⋊Q₈

yes

—

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₄×C₇₁

C₂²×Q₈

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₂×C₄⋊Q₈

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₁

yes

characteristic

C₂×#107

—

#405

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

#125

—

#125

—

142

—

C₂²⋊Q₈

—

142

—

#117

—

#117

C₂×D₇₁

yes

—

C₂×C₄⋊C₄

—

100

#107

—

101

#107

—

102

#107

—

103

#107

C₂×D₇₁

yes

characteristic

104

#107

C₂×D₇₁

yes

characteristic

105

#106

—

106

#106

C₂×D₇₁

yes

—

107

#97

—

108

C₄×C₈

—

109

C₂×Q₈

—

142

—

110

C₂²×C₄

—

142

—

111

C₂²×C₄

C₂²×D₇₁

yes

characteristic

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

Dic₁₄₂

yes

complemented

115

C₂×C₈

C₂²×D₇₁

yes

characteristic

116

C₂×C₈

C₂²×D₇₁

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

142

—

121

C₄⋊C₄

—

142

—

122

C₄⋊C₄

—

142

—

123

C₄⋊C₄

—

142

—

124

C₄⋊C₄

C₂²×D₇₁

yes

—

125

C₄⋊C₄

C₂²×D₇₁

yes

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

—

130

C₂³

D₄×D₇₁

yes

characteristic

131

Q₈

—

568

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

568

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

142

—

138

C₂×C₄

D₄×D₇₁

yes

characteristic

139

C₂×C₄

Q₈×D₇₁

yes

—

140

C₂×C₄

C₂×Dic₁₄₂

yes

—

141

C₂×C₄

C₂³×D₇₁

yes

characteristic

142

C₈

—

143

C₈

C₂×Dic₁₄₂

yes

—

144

C₈

C₂×Dic₁₄₂

yes

—

145

C₂²

—

146

C₂²

—

147

C₂²

#14556127

yes

—

148

C₂²

C₂×D₄×D₇₁

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

284

—

152

C₄

—

142

—

153

C₄

—

142

—

154

C₄

#14556127

yes

—

155

C₄

C₂×Q₈×D₇₁

yes

characteristic

156

C₄

C₂²×Dic₁₄₂

yes

characteristic

157

C₂

—

158

C₂

C₂×#14556127

yes

characteristic

159

C₂

#17986801

yes

characteristic

160

C₂

#17986813

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#427×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#427	D₇₁
hypocenter	C₇₁	#3164
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂²

C₂×D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

#427×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #17986406	C₂	semidirect
3. #17986403	C₂	semidirect
4. #17986400	C₂	semidirect
30. #11765849	Q₈	semidirect
42. C₂×C₈×C₇₁	Q₈	semidirect
81. C₇₁	#3164	semidirect
114. C₂×C₈	Dic₁₄₂	semidirect

H ≅ G/N

type

2. #17986406

C₂

semidirect

3. #17986403

C₂

semidirect

4. #17986400

C₂

semidirect

30. #11765849

Q₈

semidirect

42. C₂×C₈×C₇₁

Q₈

semidirect

81. C₇₁

#3164

semidirect

114. C₂×C₈

Dic₁₄₂

semidirect

Subgroups — #22598880

Conjugacy classes of subgroups

Complements