Group #22589950 — subgroups

Subgroups — #22589950

Order 9088.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂²×C₈×D₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556068	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₇₁)⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×D₇₁	C₄	1	1	yes	characteristic
6	C₂²×C₇₁⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₈×D₇₁	—	2	2	no	—
8	C₂×C₈×D₇₁	—	2	2	no	—
9	C₂×C₈×D₇₁	C₄	1	2	yes	—
10	C₂×C₈×D₇₁	C₂²	1	2	yes	—
11	#14556068	—	2	2	no	—
12	C₂×C₇₁⋊C₁₆	C₂²	1	2	yes	complemented
13	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	M₃₂×C₇₁	—	2	2	no	—
15	C₂×C₁₆×C₇₁	C₂²	1	2	yes	complemented
16	C₂³×D₇₁	C₈	1	1	yes	characteristic
17	C₂²×Dic₇₁	C₈	1	1	yes	characteristic
18	C₂×C₄×D₇₁	—	2	2	no	—
19	C₂×C₄×D₇₁	—	2	2	no	—
20	C₂×C₄×D₇₁	C₈	1	2	yes	—
21	C₂×C₄×D₇₁	C₂×C₄	1	2	yes	—
22	C₂×C₇₁⋊C₈	—	2	2	no	—
23	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
24	C₈×D₇₁	—	2	4	no	—
25	C₈×D₇₁	—	2	2	no	—
26	C₈×D₇₁	—	2	2	no	—
27	C₇₁⋊C₁₆	—	2	2	no	—
28	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₁	D₄	1	2	yes	—
31	C₂×C₈×C₇₁	D₄	1	2	yes	—
32	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
33	C₁₆×C₇₁	—	2	2	no	—
34	C₂³×C₇₁	C₂×C₈	1	1	yes	characteristic
35	C₂²×D₇₁	—	2	2	no	—
36	C₂²×D₇₁	—	2	4	no	—
37	C₂²×D₇₁	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×C₇₁	M₁₆	1	2	yes	—
41	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₇₁	—	2	2	no	—
43	C₂×Dic₇₁	C₂×C₈	1	2	yes	—
44	C₄×D₇₁	—	2	4	no	—
45	C₄×D₇₁	—	2	2	no	—
46	C₄×D₇₁	—	2	2	no	—
47	C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
48	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
49	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
50	C₇₁⋊C₈	—	2	2	no	—
51	C₂²×C₇₁	—	2	2	no	—
52	C₂²×C₇₁	C₂²⋊C₈	1	2	yes	—
53	C₂²×C₇₁	C₂²×C₈	1	1	yes	characteristic
54	C₂×D₇₁	—	2	8	no	—
55	C₂×D₇₁	—	2	2	no	—
56	C₂×D₇₁	—	2	2	no	—
57	C₂×D₇₁	M₃₂	1	4	yes	complemented
58	C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
59	C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
61	Dic₇₁	—	2	2	no	—
62	C₂×C₇₁	—	2	2	no	—
63	C₂×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
64	C₂×C₇₁	C₂×M₃₂	1	2	yes	—
65	D₇₁	—	2	4	no	—
66	#2112	—	71	1	no	maximal, Sylow
67	C₇₁	#2112	1	1	yes	characteristic, complemented, Sylow
68	C₂³×C₈	—	71	1	no	—
69	C₂×M₃₂	—	71	1	no	—
70	C₂×M₃₂	D₇₁	1	1	yes	characteristic, complemented
71	C₂²⋊C₁₆	—	71	4	no	—
72	C₂³×C₄	—	71	1	no	—
73	C₂²×C₈	—	71	1	no	—
74	C₂²×C₈	—	71	2	no	—
75	C₂²×C₈	—	71	2	no	—
76	C₂²×C₈	—	142	2	no	—
77	C₂²×C₈	—	142	2	no	—
78	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
79	M₃₂	—	2	2	no	—
80	M₃₂	—	142	2	no	—
81	C₂×C₁₆	—	71	2	no	—
82	C₂×C₁₆	C₂×D₇₁	1	2	yes	complemented
83	C₂⁴	—	71	1	no	—
84	C₂²×C₄	—	71	1	no	—
85	C₂²×C₄	—	71	2	no	—
86	C₂²×C₄	—	71	2	no	—
87	C₂²×C₄	—	142	2	no	—
88	C₂²×C₄	—	142	2	no	—
89	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
90	C₂×C₈	—	71	2	no	—
91	C₂×C₈	—	142	2	no	—
92	C₂×C₈	—	142	4	no	—
93	C₂×C₈	—	142	2	no	—
94	C₂×C₈	—	142	2	no	—
95	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
96	C₂×C₈	D₂₈₄	1	2	yes	—
97	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
98	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
99	C₁₆	—	2	2	no	—
100	C₁₆	—	142	2	no	—
101	C₂³	—	71	2	no	—
102	C₂³	—	142	4	no	—
103	C₂³	—	142	2	no	—
104	C₂³	C₈×D₇₁	1	1	yes	characteristic
105	C₂×C₄	—	71	2	no	—
106	C₂×C₄	—	142	2	no	—
107	C₂×C₄	—	142	4	no	—
108	C₂×C₄	—	142	2	no	—
109	C₂×C₄	—	142	2	no	—
110	C₂×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
111	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
112	C₂×C₄	#11765848	1	2	yes	—
113	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
114	C₈	—	142	2	no	—
115	C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
116	C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
117	C₈	C₂×D₂₈₄	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	—	71	4	no	—
120	C₂²	—	142	8	no	—
121	C₂²	—	142	2	no	—
122	C₂²	—	142	2	no	—
123	C₂²	(C₂×D₇₁)⋊C₈	1	2	yes	—
124	C₂²	C₂×C₈×D₇₁	1	1	yes	characteristic
125	C₄	—	142	2	no	—
126	C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
127	C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
128	C₄	C₂×#11765848	1	1	yes	characteristic
129	C₂	—	2	2	no	—
130	C₂	—	142	4	no	—
131	C₂	C₂×(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
132	C₂	M₃₂×D₇₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₇₁

C₂

yes

maximal, characteristic

C₂×#14556068

C₂

yes

maximal, characteristic, complemented

(C₂×D₇₁)⋊C₁₆

C₂

yes

maximal, complemented

C₂×M₃₂×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₇₁

C₄

yes

characteristic

C₂²×C₇₁⋊C₈

C₄

yes

characteristic

C₂×C₈×D₇₁

—

C₂×C₈×D₇₁

—

C₂×C₈×D₇₁

C₄

yes

—

C₂×C₈×D₇₁

C₂²

yes

—

#14556068

—

C₂×C₇₁⋊C₁₆

C₂²

yes

complemented

C₂²×C₈×C₇₁

C₂²

yes

characteristic

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂²

yes

complemented

C₂³×D₇₁

C₈

yes

characteristic

C₂²×Dic₇₁

C₈

yes

characteristic

C₂×C₄×D₇₁

—

C₂×C₄×D₇₁

—

C₂×C₄×D₇₁

C₈

yes

—

C₂×C₄×D₇₁

C₂×C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₈×D₇₁

—

C₈×D₇₁

—

C₈×D₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×C₈

yes

characteristic

C₂²×D₇₁

—

C₂²×D₇₁

—

C₂²×D₇₁

C₂×C₈

yes

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

M₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

C₂×C₈

yes

—

C₄×D₇₁

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

C₂²⋊C₄

yes

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

C₂×D₄

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²⋊C₈

yes

—

C₂²×C₇₁

C₂²×C₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

M₃₂

yes

complemented

C₄×C₇₁

C₂²⋊C₈

yes

—

C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

C₂×M₁₆

yes

characteristic

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₇₁

C₂×M₃₂

yes

—

D₇₁

—

#2112

—

maximal, Sylow

C₇₁

#2112

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂×M₃₂

—

C₂×M₃₂

D₇₁

yes

characteristic, complemented

C₂²⋊C₁₆

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

142

—

C₂²×C₈

—

142

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

M₃₂

—

M₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

complemented

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

D₂₈₄

yes

—

C₂×C₈

C₂²⋊D₇₁

yes

—

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₁₆

—

100

C₁₆

—

142

—

101

C₂³

—

102

C₂³

—

142

—

103

C₂³

—

142

—

104

C₂³

C₈×D₇₁

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

142

—

107

C₂×C₄

—

142

—

108

C₂×C₄

—

142

—

109

C₂×C₄

—

142

—

110

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

—

111

C₂×C₄

C₈×D₇₁

yes

characteristic

112

C₂×C₄

#11765848

yes

—

113

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

114

C₈

—

142

—

115

C₈

(C₂×D₇₁)⋊C₄

yes

—

116

C₈

C₂×C₂²⋊D₇₁

yes

characteristic

117

C₈

C₂×D₂₈₄

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

142

—

121

C₂²

—

142

—

122

C₂²

—

142

—

123

C₂²

(C₂×D₇₁)⋊C₈

yes

—

124

C₂²

C₂×C₈×D₇₁

yes

characteristic

125

C₄

—

142

—

126

C₄

(C₂×D₇₁)⋊C₈

yes

—

127

C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

128

C₄

C₂×#11765848

yes

characteristic

129

C₂

—

130

C₂

—

142

—

131

C₂

C₂×(C₂×D₇₁)⋊C₈

yes

characteristic

132

C₂

M₃₂×D₇₁

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂²×C₈
Fitting	C₂×M₃₂×C₇₁	C₂
hypercenter	C₂×M₃₂	D₇₁
hypocenter	C₇₁	#2112
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₇₁

commutator, socle

C₂²×C₇₁

C₂²×C₈

Fitting

C₂×M₃₂×C₇₁

C₂

hypercenter

C₂×M₃₂

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. C₂×#14556068	C₂	semidirect
3. (C₂×D₇₁)⋊C₁₆	C₂	semidirect
4. C₂×M₃₂×C₇₁	C₂	semidirect
12. C₂×C₇₁⋊C₁₆	C₂²	semidirect
15. C₂×C₁₆×C₇₁	C₂²	semidirect
57. C₂×D₇₁	M₃₂	semidirect
67. C₇₁	#2112	semidirect
70. C₂×M₃₂	D₇₁	semidirect
82. C₂×C₁₆	C₂×D₇₁	semidirect

H ≅ G/N

type

2. C₂×#14556068

C₂

semidirect

3. (C₂×D₇₁)⋊C₁₆

C₂

semidirect

4. C₂×M₃₂×C₇₁

C₂

semidirect

12. C₂×C₇₁⋊C₁₆

C₂²

semidirect

15. C₂×C₁₆×C₇₁

C₂²

semidirect

57. C₂×D₇₁

M₃₂

semidirect

67. C₇₁

#2112

semidirect

70. C₂×M₃₂

D₇₁

semidirect

82. C₂×C₁₆

C₂×D₇₁

semidirect

Subgroups — #22589950

Conjugacy classes of subgroups

Complements