Group #22586194 — subgroups

Subgroups — #22586194

Order 9088.

226 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17985862	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#407×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊Dic₇₁	—	2	1	no	—
5	C₂×C₂²⋊Dic₇₁	—	2	1	no	—
6	C₂×C₄×Dic₇₁	C₄	1	2	yes	complemented
7	C₇₁⋊#95	—	2	1	no	—
8	C₇₁⋊#95	—	2	1	no	—
9	C₇₁⋊#95	C₄	1	2	yes	complemented
10	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₇₁	—	2	1	no	—
12	C₂²⋊C₈×C₇₁	—	2	1	no	—
13	C₂²⋊C₈×C₇₁	C₄	1	2	yes	complemented
14	C₂²×Dic₇₁	—	2	2	no	—
15	C₂²×Dic₇₁	—	2	2	no	—
16	C₂²⋊Dic₇₁	—	2	4	no	—
17	C₂²⋊Dic₇₁	—	4	2	no	—
18	C₄×Dic₇₁	—	4	2	no	—
19	C₂⁴×C₇₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₇₁	—	4	2	no	—
21	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
27	M₁₆×C₇₁	—	4	2	no	—
28	C₂×C₈×C₇₁	—	2	2	no	—
29	C₂³×C₇₁	—	2	2	no	—
30	C₂³×C₇₁	—	2	2	no	—
31	C₂³×C₇₁	—	4	1	no	—
32	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂³×C₇₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	—	2	2	no	—
36	C₂×C₄×C₇₁	—	2	2	no	—
37	C₂×C₄×C₇₁	—	4	1	no	—
38	C₂×C₄×C₇₁	—	4	1	no	—
39	C₂×C₄×C₇₁	—	4	1	no	—
40	C₂×C₄×C₇₁	—	4	1	no	—
41	C₂×C₄×C₇₁	C₄²	1	1	yes	characteristic
42	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂×Dic₇₁	—	2	4	no	—
46	C₂×Dic₇₁	—	4	2	no	—
47	C₂×Dic₇₁	—	4	2	no	—
48	C₂×Dic₇₁	—	4	2	no	—
49	C₂×Dic₇₁	—	4	2	no	—
50	C₂×Dic₇₁	—	4	2	no	—
51	C₈×C₇₁	—	4	2	no	—
52	C₂²×C₇₁	—	2	1	no	—
53	C₂²×C₇₁	—	2	1	no	—
54	C₂²×C₇₁	—	2	1	no	—
55	C₂²×C₇₁	—	2	1	no	—
56	C₂²×C₇₁	—	2	1	no	—
57	C₂²×C₇₁	—	2	1	no	—
58	C₂²×C₇₁	—	2	1	no	—
59	C₂²×C₇₁	—	4	4	no	—
60	C₂²×C₇₁	#95	1	1	yes	characteristic
61	C₂²×C₇₁	#99	1	2	yes	—
62	C₂²×C₇₁	#104	1	2	yes	—
63	C₄×C₇₁	—	2	1	no	—
64	C₄×C₇₁	—	2	1	no	—
65	C₄×C₇₁	—	2	1	no	—
66	C₄×C₇₁	—	2	1	no	—
67	Dic₇₁	—	4	4	no	—
68	Dic₇₁	—	8	2	no	—
69	C₂×C₇₁	—	2	2	no	—
70	C₂×C₇₁	—	2	2	no	—
71	C₂×C₇₁	—	4	1	no	—
72	C₂×C₇₁	H(3,ℤ/4)	1	1	yes	characteristic
73	C₂×C₇₁	#339	1	1	yes	characteristic
74	C₂×C₇₁	#342	1	1	yes	characteristic
75	#1284	—	71	1	no	maximal, Sylow
76	C₇₁	#1284	1	1	yes	characteristic, complemented, Sylow
77	#407	D₇₁	1	1	yes	characteristic
78	#381	—	71	1	no	—
79	C₄×C₂²⋊C₄	—	71	1	no	—
80	C₂³×C₄	C₂×D₇₁	1	1	yes	characteristic
81	C₂×M₁₆	—	2	1	no	—
82	C₂×C₂²⋊C₄	—	142	1	no	—
83	C₂×C₂²⋊C₄	—	142	1	no	—
84	C₂×C₄²	—	71	2	no	—
85	C₂²⋊C₈	—	2	1	no	—
86	C₂²⋊C₈	Dic₇₁	1	2	yes	complemented
87	#95	—	71	2	no	—
88	#95	—	142	1	no	—
89	#95	—	142	1	no	—
90	C₂⁴	D₂₈₄	1	1	yes	characteristic
91	C₂²×C₄	—	4	2	no	—
92	C₂²×C₄	—	142	2	no	—
93	C₂²×C₄	—	142	2	no	—
94	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
96	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
98	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
100	M₁₆	—	4	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂²⋊C₄	—	284	2	no	—
103	C₂²⋊C₄	—	142	4	no	—
104	C₄²	—	284	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	—	2	2	no	—
107	C₂³	—	4	1	no	—
108	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
109	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
110	C₂³	C₄⋊Dic₇₁	1	1	yes	characteristic
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	284	2	no	—
118	C₂×C₄	—	284	2	no	—
119	C₂×C₄	—	284	2	no	—
120	C₂×C₄	—	284	2	no	—
121	C₂×C₄	—	284	2	no	—
122	C₂×C₄	—	142	4	no	—
123	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
124	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
125	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
126	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	4	4	no	—
136	C₂²	C₇₁⋊#95	1	1	yes	characteristic
137	C₂²	#14556042	1	2	yes	—
138	C₂²	#14556054	1	2	yes	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	284	4	no	—
144	C₄	—	568	2	no	—
145	C₂	—	2	2	no	—
146	C₂	—	2	2	no	—
147	C₂	—	4	1	no	—
148	C₂	#17985756	1	1	yes	characteristic
149	C₂	#17985760	1	1	yes	characteristic
150	C₂	#17985765	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17985862

C₂

yes

maximal, characteristic

C₄×C₂²⋊Dic₇₁

C₂

yes

maximal, characteristic

#407×C₇₁

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₇₁

—

C₂×C₂²⋊Dic₇₁

—

C₂×C₄×Dic₇₁

C₄

yes

complemented

C₇₁⋊#95

—

C₇₁⋊#95

—

C₇₁⋊#95

C₄

yes

complemented

C₂³×C₄×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₂²⋊C₈×C₇₁

C₄

yes

complemented

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₂²⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₂⁴×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

Q₈

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

C₂³×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄²

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#95

yes

characteristic

C₂²×C₇₁

#99

yes

—

C₂²×C₇₁

#104

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#407

D₇₁

yes

characteristic

#381

—

C₄×C₂²⋊C₄

—

C₂³×C₄

C₂×D₇₁

yes

characteristic

C₂×M₁₆

—

C₂×C₂²⋊C₄

—

142

—

C₂×C₂²⋊C₄

—

142

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

Dic₇₁

yes

complemented

#95

—

#95

—

142

—

#95

—

142

—

C₂⁴

D₂₈₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

100

M₁₆

—

101

C₂×C₈

—

102

C₂²⋊C₄

—

284

—

103

C₂²⋊C₄

—

142

—

104

C₄²

—

284

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

109

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

110

C₂³

C₄⋊Dic₇₁

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

284

—

118

C₂×C₄

—

284

—

119

C₂×C₄

—

284

—

120

C₂×C₄

—

284

—

121

C₂×C₄

—

284

—

122

C₂×C₄

—

142

—

123

C₂×C₄

C₄×Dic₇₁

yes

characteristic

124

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

125

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

126

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₇₁⋊#95

yes

characteristic

137

C₂²

#14556042

yes

—

138

C₂²

#14556054

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

284

—

144

C₄

—

568

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

#17985756

yes

characteristic

149

C₂

#17985760

yes

characteristic

150

C₂

#17985765

yes

characteristic

	subgroup	G/H
center	C₂²	C₇₁⋊#95
commutator	C₂×C₄×C₇₁	C₄²
Frattini	C₂³×C₄	C₂×D₇₁
Fitting	#407×C₇₁	C₂
socle	C₂²×C₇₁	#95
hypercenter	#407	D₇₁
hypocenter	C₇₁	#1284
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₇₁⋊#95

commutator

C₂×C₄×C₇₁

C₄²

Frattini

C₂³×C₄

C₂×D₇₁

Fitting

#407×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
6. C₂×C₄×Dic₇₁	C₄	semidirect
9. C₇₁⋊#95	C₄	semidirect
13. C₂²⋊C₈×C₇₁	C₄	semidirect
76. C₇₁	#1284	semidirect
86. C₂²⋊C₈	Dic₇₁	semidirect

H ≅ G/N

type

6. C₂×C₄×Dic₇₁

C₄

semidirect

9. C₇₁⋊#95

C₄

semidirect

13. C₂²⋊C₈×C₇₁

C₄

semidirect

76. C₇₁

#1284

semidirect

86. C₂²⋊C₈

Dic₇₁

semidirect

Subgroups — #22586194

Conjugacy classes of subgroups

Complements