Group #22586242 — subgroups

Subgroups — #22586242

Order 9088.

176 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 77 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	#17986023	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#431×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11765849	—	2	1	no	—
5	C₂×C₄⋊Dic₇₁	—	2	1	no	—
6	C₂×C₄×Dic₇₁	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₈)	C₄	1	2	yes	complemented
9	C₇₁⋊#95	C₄	1	2	yes	—
10	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₇₁	C₄	1	2	yes	complemented
12	C₂²⋊C₈×C₇₁	—	2	1	no	—
13	C₄×C₈×C₇₁	—	2	1	no	—
14	C₂²×Dic₇₁	—	2	2	no	—
15	#11765849	—	4	1	no	—
16	#11765849	—	4	1	no	—
17	C₂×C₇₁⋊C₈	—	2	2	no	—
18	C₄⋊Dic₇₁	—	2	4	no	—
19	C₄×Dic₇₁	—	4	1	no	—
20	C₄×Dic₇₁	—	4	1	no	—
21	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₇₁	—	2	2	no	—
25	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₇₁	D₄	1	1	yes	characteristic
28	C₄²×C₇₁	Q₈	1	1	yes	characteristic
29	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₁	—	2	2	no	—
31	C₂×C₄×C₇₁	—	4	1	no	—
32	C₂×C₄×C₇₁	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
39	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
41	C₂×C₄×C₇₁	Q₁₆	1	1	yes	characteristic
42	C₂×Dic₇₁	—	2	4	no	—
43	C₂×Dic₇₁	—	4	2	no	—
44	C₂×Dic₇₁	—	4	2	no	—
45	C₈×C₇₁	—	4	2	no	—
46	C₇₁⋊C₈	—	4	2	no	—
47	C₂²×C₇₁	—	2	1	no	—
48	C₂²×C₇₁	—	2	1	no	—
49	C₂²×C₇₁	#95	1	1	yes	characteristic
50	C₂²×C₇₁	#102	1	1	yes	characteristic
51	C₂²×C₇₁	#103	1	1	yes	characteristic
52	C₄×C₇₁	—	2	1	no	—
53	C₄×C₇₁	—	2	1	no	—
54	C₄×C₇₁	—	2	1	no	—
55	C₄×C₇₁	#102	1	1	yes	characteristic
56	C₄×C₇₁	#103	1	1	yes	characteristic
57	C₄×C₇₁	#104	1	2	yes	—
58	C₄×C₇₁	#106	1	1	yes	characteristic
59	C₄×C₇₁	#107	1	1	yes	characteristic
60	Dic₇₁	—	4	4	no	—
61	C₂×C₇₁	—	2	1	no	—
62	C₂×C₇₁	—	2	1	no	—
63	C₂×C₇₁	#339	1	1	yes	characteristic
64	C₂×C₇₁	#340	1	1	yes	characteristic
65	C₂×C₇₁	#344	1	1	yes	characteristic
66	#1303	—	71	1	no	maximal, Sylow
67	C₇₁	#1303	1	1	yes	characteristic, complemented, Sylow
68	#431	D₇₁	1	1	yes	characteristic
69	#423	—	71	1	no	—
70	C₄×C₄⋊C₄	—	71	1	no	—
71	C₂×M₁₆	—	142	1	no	—
72	C₂×C₄⋊C₄	—	142	1	no	—
73	C₂×C₄²	—	142	1	no	—
74	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
75	C₄⋊C₈	—	71	2	no	—
76	C₄⋊C₈	—	142	1	no	—
77	C₄⋊C₈	Dic₇₁	1	2	yes	complemented
78	C₂²⋊C₈	—	2	1	no	—
79	C₄×C₈	—	2	1	no	—
80	#95	—	71	2	no	—
81	C₂²×C₄	—	142	2	no	—
82	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
83	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
84	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
85	M₁₆	—	284	1	no	—
86	M₁₆	—	284	1	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	142	2	no	—
89	C₄⋊C₄	—	142	4	no	—
90	C₄²	—	284	1	no	—
91	C₄²	—	284	1	no	—
92	C₄²	Dic₁₄₂	1	1	yes	characteristic
93	C₄²	C₄×D₇₁	1	1	yes	characteristic
94	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
95	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
96	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	284	2	no	—
100	C₂×C₄	—	284	2	no	—
101	C₂×C₄	—	142	4	no	—
102	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
103	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
104	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
105	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
106	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
107	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
108	C₂×C₄	D₅₆₈	1	1	yes	characteristic
109	C₂×C₄	#11765854	1	1	yes	characteristic
110	C₂×C₄	#11765854	1	1	yes	characteristic
111	C₂×C₄	Dic₂₈₄	1	1	yes	characteristic
112	C₈	—	4	2	no	—
113	C₈	—	284	2	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	C₇₁⋊#95	1	1	yes	characteristic
117	C₂²	#14556050	1	1	yes	characteristic
118	C₂²	#14556053	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	284	4	no	—
123	C₄	#14556050	1	1	yes	characteristic
124	C₄	#14556053	1	1	yes	characteristic
125	C₄	#14556055	1	2	yes	—
126	C₄	#14556060	1	1	yes	characteristic
127	C₄	#14556062	1	1	yes	characteristic
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#17985758	1	1	yes	characteristic
131	C₂	#17985762	1	1	yes	characteristic
132	C₂	#17985771	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986023

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₇₁

C₂

yes

maximal, characteristic

#431×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

—

C₂×C₄⋊Dic₇₁

—

C₂×C₄×Dic₇₁

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₄

yes

complemented

C₇₁⋊#95

C₄

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₄

yes

complemented

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

C₂²×Dic₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

Q₈

yes

characteristic

C₂³×C₇₁

C₂²⋊C₄

yes

characteristic

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×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

Q₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#431

D₇₁

yes

characteristic

#423

—

C₄×C₄⋊C₄

—

C₂×M₁₆

—

142

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄²

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

Dic₇₁

yes

complemented

C₂²⋊C₈

—

C₄×C₈

—

#95

—

C₂²×C₄

—

142

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

M₁₆

—

284

—

M₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

142

—

C₄⋊C₄

—

142

—

C₄²

—

284

—

C₄²

—

284

—

C₄²

Dic₁₄₂

yes

characteristic

C₄²

C₄×D₇₁

yes

characteristic

C₄²

C₂×Dic₇₁

yes

characteristic

C₄²

C₂²⋊D₇₁

yes

characteristic

C₂³

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

100

C₂×C₄

—

284

—

101

C₂×C₄

—

142

—

102

C₂×C₄

C₄×Dic₇₁

yes

characteristic

103

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

104

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

105

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

106

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

107

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

108

C₂×C₄

D₅₆₈

yes

characteristic

109

C₂×C₄

#11765854

yes

characteristic

110

C₂×C₄

#11765854

yes

characteristic

111

C₂×C₄

Dic₂₈₄

yes

characteristic

112

C₈

—

113

C₈

—

284

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₇₁⋊#95

yes

characteristic

117

C₂²

#14556050

yes

characteristic

118

C₂²

#14556053

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

284

—

123

C₄

#14556050

yes

characteristic

124

C₄

#14556053

yes

characteristic

125

C₄

#14556055

yes

—

126

C₄

#14556060

yes

characteristic

127

C₄

#14556062

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

#17985758

yes

characteristic

131

C₂

#17985762

yes

characteristic

132

C₂

#17985771

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₄²
Frattini	C₂×C₄²	C₂×D₇₁
Fitting	#431×C₇₁	C₂
socle	C₂²×C₇₁	#95
hypercenter	#431	D₇₁
hypocenter	C₇₁	#1303
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₇₁)⋊C₄

commutator

C₂×C₄×C₇₁

C₄²

Frattini

C₂×C₄²

C₂×D₇₁

Fitting

#431×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
8. C₄⋊(C₇₁⋊C₈)	C₄	semidirect
11. C₄⋊C₈×C₇₁	C₄	semidirect
67. C₇₁	#1303	semidirect
77. C₄⋊C₈	Dic₇₁	semidirect

H ≅ G/N

type

8. C₄⋊(C₇₁⋊C₈)

C₄

semidirect

11. C₄⋊C₈×C₇₁

C₄

semidirect

67. C₇₁

#1303

semidirect

77. C₄⋊C₈

Dic₇₁

semidirect

Subgroups — #22586242

Conjugacy classes of subgroups

Complements