Group #22586470 — subgroups

Subgroups — #22586470

Order 9088.

162 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 53 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	#17985973	C₂	1	1	yes	maximal, characteristic
2	#17985966	C₂	1	1	yes	maximal, characteristic
3	#413×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11765849	C₄	1	1	yes	characteristic, complemented
5	C₂²×C₇₁⋊C₈	C₄	1	1	yes	characteristic, complemented
6	#14556087	—	2	1	no	—
7	#14556043	—	2	1	no	—
8	#14556043	C₄	1	2	yes	—
9	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
10	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
11	C₂×#41×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×M₁₆×C₇₁	—	2	1	no	—
13	#101×C₇₁	C₄	1	2	yes	complemented
14	#100×C₇₁	—	2	1	no	—
15	#11765849	—	4	1	no	—
16	C₂×C₇₁⋊C₈	—	2	1	no	—
17	C₂×C₇₁⋊C₈	—	2	1	no	—
18	C₂×C₇₁⋊C₈	—	4	1	no	—
19	C₄⋊Dic₇₁	—	4	1	no	—
20	C₂²⋊Dic₇₁	—	2	2	no	—
21	C₄×Dic₇₁	—	4	1	no	—
22	#41×C₇₁	—	4	1	no	—
23	#41×C₇₁	—	4	1	no	—
24	C₂×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
27	C₂×D₄×C₇₁	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
31	M₁₆×C₇₁	—	2	2	no	—
32	M₁₆×C₇₁	—	4	1	no	—
33	C₂×C₈×C₇₁	—	4	1	no	—
34	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂³×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂³×C₇₁	C₄⋊C₄	1	1	yes	characteristic
37	Q₈×C₇₁	—	4	1	no	—
38	D₄×C₇₁	—	4	1	no	—
39	D₄×C₇₁	—	4	1	no	—
40	D₄×C₇₁	—	4	1	no	—
41	C₂×C₄×C₇₁	—	2	1	no	—
42	C₂×C₄×C₇₁	—	2	1	no	—
43	C₂×C₄×C₇₁	—	2	1	no	—
44	C₂×C₄×C₇₁	—	2	1	no	—
45	C₂×C₄×C₇₁	—	2	1	no	—
46	C₂×C₄×C₇₁	—	2	1	no	—
47	C₂×C₄×C₇₁	C₄²	1	1	yes	characteristic
48	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂×Dic₇₁	—	4	2	no	—
52	C₈×C₇₁	—	4	2	no	—
53	C₇₁⋊C₈	—	4	1	no	—
54	C₇₁⋊C₈	—	4	1	no	—
55	C₂²×C₇₁	—	2	1	no	—
56	C₂²×C₇₁	—	2	1	no	—
57	C₂²×C₇₁	—	2	1	no	—
58	C₂²×C₇₁	—	4	1	no	—
59	C₂²×C₇₁	—	4	1	no	—
60	C₂²×C₇₁	—	4	1	no	—
61	C₂²×C₇₁	#95	1	1	yes	characteristic
62	C₄×C₇₁	—	2	1	no	—
63	C₄×C₇₁	—	2	1	no	—
64	C₄×C₇₁	—	2	1	no	—
65	C₄×C₇₁	#99	1	1	yes	characteristic
66	C₄×C₇₁	#99	1	1	yes	characteristic
67	Dic₇₁	—	8	2	no	—
68	C₂×C₇₁	—	2	1	no	—
69	C₂×C₇₁	—	4	1	no	—
70	C₂×C₇₁	—	4	1	no	—
71	C₂×C₇₁	—	4	1	no	—
72	C₂×C₇₁	#342	1	1	yes	characteristic
73	#1395	—	71	1	no	maximal, Sylow
74	C₇₁	#1395	1	1	yes	characteristic, complemented, Sylow
75	#413	D₇₁	1	1	yes	characteristic
76	#410	—	71	1	no	—
77	#408	—	71	1	no	—
78	C₂×#41	C₂×D₇₁	1	1	yes	characteristic
79	C₂×M₁₆	—	2	1	no	—
80	C₂×M₁₆	—	71	1	no	—
81	C₂²×C₈	—	71	1	no	—
82	#117	—	142	1	no	—
83	#101	Dic₇₁	1	2	yes	complemented
84	#100	—	2	1	no	—
85	#99	—	71	2	no	—
86	#99	—	142	1	no	—
87	C₂²⋊C₈	—	142	1	no	—
88	C₂²⋊C₈	—	142	1	no	—
89	#41	—	4	1	no	—
90	#41	—	4	1	no	—
91	C₂×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
92	C₂×D₄	Dic₁₄₂	1	1	yes	characteristic
93	C₂×D₄	C₄×D₇₁	1	1	yes	characteristic
94	C₂×D₄	C₂²⋊D₇₁	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
96	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
98	M₁₆	—	2	2	no	—
99	M₁₆	—	4	1	no	—
100	M₁₆	—	284	1	no	—
101	C₂×C₈	—	4	1	no	—
102	C₂×C₈	—	284	1	no	—
103	C₂×C₈	—	142	1	no	—
104	C₂×C₈	—	142	1	no	—
105	C₄⋊C₄	—	284	1	no	—
106	C₂²⋊C₄	—	142	2	no	—
107	C₄²	—	284	1	no	—
108	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
109	C₂³	Dic₇₁⋊C₄	1	1	yes	characteristic
110	C₂³	Dic₇₁⋊C₄	1	1	yes	characteristic
111	Q₈	—	4	1	no	—
112	D₄	—	4	1	no	—
113	D₄	—	4	1	no	—
114	D₄	—	4	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	284	2	no	—
122	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
123	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
124	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
125	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
126	C₈	—	4	2	no	—
127	C₈	—	284	1	no	—
128	C₈	—	284	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	4	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	C₇₁⋊#95	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	568	2	no	—
140	C₄	#14556042	1	1	yes	characteristic
141	C₄	#14556042	1	1	yes	characteristic
142	C₂	—	2	1	no	—
143	C₂	—	4	1	no	—
144	C₂	—	4	1	no	—
145	C₂	—	4	1	no	—
146	C₂	#17985765	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17985973

C₂

yes

maximal, characteristic

#17985966

C₂

yes

maximal, characteristic

#413×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₄

yes

characteristic, complemented

C₂²×C₇₁⋊C₈

C₄

yes

characteristic, complemented

#14556087

—

#14556043

—

#14556043

C₄

yes

—

C₂²⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

C₂×#41×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

—

#101×C₇₁

C₄

yes

complemented

#100×C₇₁

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

C₂×C₄

yes

characteristic

C₂×D₄×C₇₁

C₂×C₄

yes

characteristic

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂×D₄×C₇₁

Q₈

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

C₂³×C₇₁

C₄⋊C₄

yes

characteristic

C₂³×C₇₁

C₄⋊C₄

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#95

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#413

D₇₁

yes

characteristic

#410

—

#408

—

C₂×#41

C₂×D₇₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

142

—

#101

Dic₇₁

yes

—

—

—

142

—

C₂²⋊C₈

—

142

—

C₂²⋊C₈

—

142

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₇₁

yes

characteristic

C₂×D₄

Dic₁₄₂

yes

characteristic

C₂×D₄

C₄×D₇₁

yes

characteristic

C₂×D₄

C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

100

M₁₆

—

284

—

101

C₂×C₈

—

102

C₂×C₈

—

284

—

103

C₂×C₈

—

142

—

104

C₂×C₈

—

142

—

105

C₄⋊C₄

—

284

—

106

C₂²⋊C₄

—

142

—

107

C₄²

—

284

—

108

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

109

C₂³

Dic₇₁⋊C₄

yes

characteristic

110

C₂³

Dic₇₁⋊C₄

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

284

—

122

C₂×C₄

C₄×Dic₇₁

yes

characteristic

123

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

124

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

125

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

126

C₈

—

127

C₈

—

284

—

128

C₈

—

284

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

C₇₁⋊#95

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

568

—

140

C₄

#14556042

yes

characteristic

141

C₄

#14556042

yes

characteristic

142

C₂

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#17985765

yes

characteristic

	subgroup	G/H
center	C₄	#14556042
commutator	C₂×C₄×C₇₁	C₄²
Frattini	C₂×#41	C₂×D₇₁
Fitting	#413×C₇₁	C₂
socle	C₂×C₇₁	#342
hypercenter	#413	D₇₁
hypocenter	C₇₁	#1395
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #413

subgroup

G/H

center

C₄

#14556042

commutator

C₂×C₄×C₇₁

C₄²

Frattini

C₂×#41

C₂×D₇₁

Fitting

#413×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #413

N	H ≅ G/N	type
4. C₂×#11765849	C₄	semidirect
5. C₂²×C₇₁⋊C₈	C₄	semidirect
13. #101×C₇₁	C₄	semidirect
74. C₇₁	#1395	semidirect
83. #101	Dic₇₁	semidirect

H ≅ G/N

type

4. C₂×#11765849

C₄

semidirect

5. C₂²×C₇₁⋊C₈

C₄

semidirect

13. #101×C₇₁

C₄

semidirect

74. C₇₁

#1395

semidirect

83. #101

Dic₇₁

semidirect

Subgroups — #22586470

Conjugacy classes of subgroups

Complements