Group #22601209 — subgroups

Subgroups — #22601209

Order 9088.

352 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 125 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	#17986710	C₂	1	2	yes	maximal, complemented
2	#17986400	C₂	1	2	yes	maximal, complemented
3	#17986302	C₂	1	4	yes	maximal, complemented
4	#17986286	C₂	1	2	yes	maximal, complemented
5	C₈×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₅₆₈	C₂	1	1	yes	maximal, characteristic
7	#17986081	C₂	1	2	yes	maximal
8	#500×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
10	#14556119	C₂²	1	4	yes	—
11	#14556108	—	2	2	no	—
12	#14556104	—	2	4	no	—
13	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
14	C₄×D₂₈₄	C₂²	1	2	yes	—
15	#14556090	C₂²	1	4	yes	—
16	C₄⋊C₄×D₇₁	—	2	2	no	—
17	#14556062	C₂²	1	1	yes	characteristic
18	#14556059	C₂²	1	4	yes	—
19	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₇₁⋊C₈	C₂²	1	2	yes	—
21	#14556050	C₂²	1	2	yes	—
22	#14556048	C₂²	1	4	yes	—
23	C₈×Dic₇₁	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
25	#107×C₇₁	C₂²	1	2	yes	—
26	#107×C₇₁	C₂²	1	2	yes	—
27	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
28	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
29	C₂×D₂₈₄	C₂³	1	2	yes	—
30	C₂×C₄×D₇₁	—	2	2	no	—
31	C₂×C₄×D₇₁	—	2	4	no	—
32	D₅₆₈	Q₈	1	4	yes	complemented
33	C₂×C₇₁⋊C₈	C₂³	1	2	yes	—
34	C₄⋊Dic₇₁	—	2	4	no	—
35	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
36	C₄⋊Dic₇₁	C₂³	1	2	yes	—
37	Dic₇₁⋊C₄	—	2	4	no	—
38	Dic₇₁⋊C₄	D₄	1	2	yes	—
39	(C₂×D₇₁)⋊C₄	—	2	2	no	—
40	(C₂×D₇₁)⋊C₄	—	2	4	no	—
41	C₄×Dic₇₁	C₂³	1	2	yes	—
42	C₂×Q₈×C₇₁	—	2	2	no	—
43	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
44	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
45	C₄⋊C₄×C₇₁	—	2	2	no	—
46	C₄⋊C₄×C₇₁	C₂³	1	4	yes	—
47	C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	C₂²×D₇₁	—	2	2	no	—
49	Q₈×C₇₁	—	8	2	no	—
50	C₂×C₄×C₇₁	—	2	4	no	—
51	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
54	C₂×Dic₇₁	—	2	2	no	—
55	C₂×Dic₇₁	C₂×D₄	1	2	yes	—
56	C₂×Dic₇₁	C₂×D₄	1	2	yes	—
57	D₂₈₄	—	2	2	no	—
58	D₂₈₄	C₂×Q₈	1	4	yes	—
59	C₄×D₇₁	—	4	4	no	—
60	C₄×D₇₁	—	8	4	no	—
61	Dic₁₄₂	—	2	2	no	—
62	C₈×C₇₁	—	2	1	no	—
63	C₈×C₇₁	C₂×Q₈	1	2	yes	—
64	C₇₁⋊C₈	—	2	2	no	—
65	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
66	C₂×D₇₁	—	2	4	no	—
67	C₂×D₇₁	—	4	2	no	—
68	C₂×D₇₁	—	4	2	no	—
69	C₄×C₇₁	—	2	1	no	—
70	C₄×C₇₁	—	4	4	no	—
71	C₄×C₇₁	#135	1	2	yes	—
72	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
73	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
74	Dic₇₁	—	2	2	no	—
75	Dic₇₁	—	2	2	no	—
76	Dic₇₁	—	4	2	no	—
77	C₂×C₇₁	D₄×Q₈	1	1	yes	characteristic
78	C₂×C₇₁	C₂×#135	1	1	yes	characteristic
79	C₂×C₇₁	#576	1	1	yes	characteristic
80	D₇₁	—	4	4	no	—
81	#3380	—	71	1	no	maximal, Sylow
82	C₇₁	#3380	1	1	yes	characteristic, complemented, Sylow
83	#554	—	71	2	no	—
84	#500	D₇₁	1	1	yes	characteristic, complemented
85	#499	—	71	2	no	—
86	#478	—	71	4	no	—
87	#476	—	71	2	no	—
88	C₈×Q₈	—	71	1	no	—
89	C₄×D₈	—	71	2	no	—
90	C₄×D₈	—	71	1	no	—
91	C₂×D₈	—	71	1	no	—
92	C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
93	#125	—	71	4	no	—
94	C₂²⋊Q₈	—	142	2	no	—
95	C₂²⋊Q₈	—	142	4	no	—
96	C₄×Q₈	—	71	1	no	—
97	C₄×D₄	—	71	4	no	—
98	C₄×D₄	—	71	2	no	—
99	C₂×C₄⋊C₄	—	142	2	no	—
100	#107	—	71	1	no	—
101	#107	C₂×D₇₁	1	2	yes	—
102	#107	C₂×D₇₁	1	2	yes	—
103	#106	—	71	4	no	—
104	C₄⋊C₈	—	71	2	no	—
105	C₄⋊C₈	—	71	1	no	—
106	#102	—	71	2	no	—
107	#102	—	71	4	no	—
108	C₄×C₈	—	71	2	no	—
109	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	71	1	no	—
112	C₂×D₄	—	71	2	no	—
113	C₂²×C₄	—	142	4	no	—
114	C₂²×C₄	—	142	2	no	—
115	D₈	—	71	4	no	—
116	C₂×C₈	—	71	2	no	—
117	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
118	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	71	2	no	—
121	C₄⋊C₄	—	71	1	no	—
122	C₄⋊C₄	—	71	2	no	—
123	C₄⋊C₄	—	142	4	no	—
124	C₄⋊C₄	—	142	4	no	—
125	C₄⋊C₄	C₂²×D₇₁	1	4	yes	—
126	C₂²⋊C₄	—	142	4	no	—
127	C₂²⋊C₄	—	142	2	no	—
128	C₄²	—	71	2	no	—
129	C₄²	C₂²×D₇₁	1	1	yes	characteristic
130	C₂³	—	142	2	no	—
131	Q₈	—	8	2	no	—
132	Q₈	—	142	2	no	—
133	D₄	—	71	4	no	—
134	D₄	—	142	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	284	4	no	—
137	C₂×C₄	—	568	4	no	—
138	C₂×C₄	—	71	2	no	—
139	C₂×C₄	—	71	2	no	—
140	C₂×C₄	—	142	2	no	—
141	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
142	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
143	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	142	2	no	—
146	C₈	Q₈×D₇₁	1	2	yes	—
147	C₂²	—	284	2	no	—
148	C₂²	—	284	2	no	—
149	C₂²	—	142	4	no	—
150	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
151	C₄	—	2	1	no	—
152	C₄	—	4	4	no	—
153	C₄	—	284	2	no	—
154	C₄	—	142	2	no	—
155	C₄	—	142	2	no	—
156	C₄	#14556151	1	2	yes	—
157	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
158	C₄	#14556176	1	1	yes	characteristic
159	C₂	—	284	4	no	—
160	C₂	#17986664	1	1	yes	characteristic
161	C₂	C₂×#14556151	1	1	yes	characteristic
162	C₂	#17986800	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986710

C₂

yes

maximal, complemented

#17986400

C₂

yes

maximal, complemented

#17986302

C₂

yes

maximal, complemented

#17986286

C₂

yes

maximal, complemented

C₈×Dic₁₄₂

C₂

yes

maximal, characteristic, complemented

C₄×D₅₆₈

C₂

yes

maximal, characteristic

#17986081

C₂

yes

maximal

#500×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×D₅₆₈

C₂²

yes

characteristic

#14556119

C₂²

yes

—

#14556108

—

#14556104

—

C₄×Dic₁₄₂

C₂²

yes

characteristic

C₄×D₂₈₄

C₂²

yes

—

#14556090

C₂²

yes

—

C₄⋊C₄×D₇₁

—

#14556062

C₂²

yes

characteristic

#14556059

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

Dic₇₁⋊C₈

C₂²

yes

—

#14556050

C₂²

yes

—

#14556048

C₂²

yes

—

C₈×Dic₇₁

C₂²

yes

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

#107×C₇₁

C₂²

yes

—

#107×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

D₄

yes

characteristic

C₂×D₂₈₄

C₂³

yes

—

C₂×C₄×D₇₁

—

C₂×C₄×D₇₁

—

D₅₆₈

Q₈

yes

complemented

C₂×C₇₁⋊C₈

C₂³

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

D₄

yes

—

(C₂×D₇₁)⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₄×Dic₇₁

C₂³

yes

—

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂³

yes

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂×D₄

yes

—

C₂×Dic₇₁

C₂×D₄

yes

—

D₂₈₄

—

D₂₈₄

C₂×Q₈

yes

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×Q₈

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#135

yes

—

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₄×C₇₁

2₋¹⁺⁴

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

D₄×Q₈

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#3380

—

maximal, Sylow

C₇₁

#3380

yes

characteristic, complemented, Sylow

#554

—

#500

D₇₁

yes

characteristic, complemented

#499

—

#478

—

#476

—

C₈×Q₈

—

C₄×D₈

—

C₄×D₈

—

C₂×D₈

—

C₄⋊Q₈

C₂×D₇₁

yes

—

#125

—

C₂²⋊Q₈

—

142

—

C₂²⋊Q₈

—

142

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

142

—

100

#107

—

101

#107

C₂×D₇₁

yes

—

102

#107

C₂×D₇₁

yes

—

103

#106

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

#102

—

107

#102

—

108

C₄×C₈

—

109

C₄×C₈

C₂×D₇₁

yes

characteristic

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×D₄

—

113

C₂²×C₄

—

142

—

114

C₂²×C₄

—

142

—

115

D₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂²×D₇₁

yes

characteristic

118

C₂×C₈

C₂²×D₇₁

yes

characteristic

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

142

—

124

C₄⋊C₄

—

142

—

125

C₄⋊C₄

C₂²×D₇₁

yes

—

126

C₂²⋊C₄

—

142

—

127

C₂²⋊C₄

—

142

—

128

C₄²

—

129

C₄²

C₂²×D₇₁

yes

characteristic

130

C₂³

—

142

—

131

Q₈

—

132

Q₈

—

142

—

133

D₄

—

134

D₄

—

142

—

135

C₂×C₄

—

136

C₂×C₄

—

284

—

137

C₂×C₄

—

568

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

142

—

141

C₂×C₄

D₄×D₇₁

yes

characteristic

142

C₂×C₄

D₄×D₇₁

yes

characteristic

143

C₂×C₄

C₂³×D₇₁

yes

characteristic

144

C₈

—

145

C₈

—

142

—

146

C₈

Q₈×D₇₁

yes

—

147

C₂²

—

284

—

148

C₂²

—

284

—

149

C₂²

—

142

—

150

C₂²

C₂×D₄×D₇₁

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

284

—

154

C₄

—

142

—

155

C₄

—

142

—

156

C₄

#14556151

yes

—

157

C₄

C₂×Q₈×D₇₁

yes

characteristic

158

C₄

#14556176

yes

characteristic

159

C₂

—

284

—

160

C₂

#17986664

yes

characteristic

161

C₂

C₂×#14556151

yes

characteristic

162

C₂

#17986800

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#500×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#500	D₇₁
hypocenter	C₇₁	#3380
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

#500×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #17986710	C₂	semidirect
2. #17986400	C₂	semidirect
3. #17986302	C₂	semidirect
4. #17986286	C₂	semidirect
5. C₈×Dic₁₄₂	C₂	semidirect
8. #500×C₇₁	C₂	semidirect
32. D₅₆₈	Q₈	semidirect
82. C₇₁	#3380	semidirect
84. #500	D₇₁	semidirect

H ≅ G/N

type

1. #17986710

C₂

semidirect

2. #17986400

C₂

semidirect

3. #17986302

C₂

semidirect

4. #17986286

C₂

semidirect

5. C₈×Dic₁₄₂

C₂

semidirect

8. #500×C₇₁

C₂

semidirect

32. D₅₆₈

Q₈

semidirect

82. C₇₁

#3380

semidirect

84. #500

D₇₁

semidirect

Subgroups — #22601209

Conjugacy classes of subgroups

Complements