Group #22588463 — subgroups

Subgroups — #22588463

Order 9088.

172 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×#14556064	C₂	1	1	yes	maximal, characteristic
2	#17985954	C₂	1	1	yes	maximal, characteristic
3	#17985769	C₂	1	2	yes	maximal
4	#17985755	C₂	1	2	yes	maximal
5	#427×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765849	C₂²	1	2	yes	—
7	C₂×#11765849	C₂²	1	1	yes	characteristic
8	#14556087	C₂²	1	1	yes	characteristic
9	#14556064	—	2	1	no	—
10	#14556064	—	2	1	no	—
11	#14556038	C₄	1	2	yes	complemented
12	C₈×Dic₇₁	C₄	1	2	yes	complemented
13	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	#117×C₇₁	C₂²	1	2	yes	—
15	#107×C₇₁	—	2	1	no	—
16	#106×C₇₁	—	2	1	no	—
17	#11765849	—	2	1	no	—
18	#11765849	—	2	1	no	—
19	#11765849	—	2	2	no	—
20	#11765849	—	2	2	no	—
21	C₂×C₇₁⋊C₈	—	2	2	no	—
22	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
23	C₄⋊Dic₇₁	—	2	1	no	—
24	C₂²⋊Dic₇₁	—	2	1	no	—
25	C₄×Dic₇₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₂×C₈×C₇₁	Q₈	1	1	yes	characteristic
32	C₂×C₈×C₇₁	Q₈	1	1	yes	characteristic
33	C₄⋊C₄×C₇₁	—	2	2	no	—
34	C₂²⋊C₄×C₇₁	—	2	2	no	—
35	C₄²×C₇₁	—	2	2	no	—
36	C₂³×C₇₁	#41	1	1	yes	characteristic
37	C₂×C₄×C₇₁	—	4	2	no	—
38	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
42	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
43	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
44	C₂×Dic₇₁	—	2	2	no	—
45	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
46	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
49	C₇₁⋊C₈	—	2	2	no	—
50	C₇₁⋊C₈	—	4	2	no	—
51	C₂²×C₇₁	—	4	1	no	—
52	C₂²×C₇₁	#117	1	1	yes	characteristic
53	C₂²×C₇₁	#124	1	1	yes	characteristic
54	C₂²×C₇₁	#125	1	1	yes	characteristic
55	C₄×C₇₁	—	8	2	no	—
56	C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₄×C₇₁	#117	1	1	yes	characteristic
58	C₄×C₇₁	#124	1	1	yes	characteristic
59	C₄×C₇₁	#125	1	1	yes	characteristic
60	Dic₇₁	—	4	2	no	—
61	C₂×C₇₁	—	2	1	no	—
62	C₂×C₇₁	—	2	1	no	—
63	C₂×C₇₁	—	2	1	no	—
64	C₂×C₇₁	#382	1	1	yes	characteristic
65	#1833	—	71	1	no	maximal, Sylow
66	C₇₁	#1833	1	1	yes	characteristic, complemented, Sylow
67	C₂×#108	—	71	1	no	—
68	#427	D₇₁	1	1	yes	characteristic
69	#405	—	71	1	no	—
70	#344	—	71	2	no	—
71	H(3,ℤ/4)	—	71	2	no	—
72	C₂×M₁₆	—	71	1	no	—
73	C₂×M₁₆	—	71	2	no	—
74	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
75	#117	—	71	1	no	—
76	#117	C₂×D₇₁	1	2	yes	—
77	#108	—	142	1	no	—
78	#108	—	142	1	no	—
79	#107	—	2	1	no	—
80	#106	—	2	1	no	—
81	#97	—	71	2	no	—
82	C₄×C₈	—	71	2	no	—
83	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
84	M₁₆	—	142	2	no	—
85	M₁₆	—	142	2	no	—
86	M₁₆	—	142	1	no	—
87	M₁₆	—	142	1	no	—
88	C₂×C₈	—	71	2	no	—
89	C₂×C₈	—	142	2	no	—
90	C₂×C₈	Dic₁₄₂	1	1	yes	characteristic
91	C₂×C₈	Dic₁₄₂	1	1	yes	characteristic
92	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
93	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
94	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
95	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	142	1	no	—
98	C₂²⋊C₄	—	2	2	no	—
99	C₂²⋊C₄	—	142	1	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	71	2	no	—
102	C₂³	#11765864	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	142	2	no	—
105	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
108	C₂×C₄	#11765864	1	1	yes	characteristic
109	C₂×C₄	#11765866	1	1	yes	characteristic
110	C₂×C₄	#11765866	1	1	yes	characteristic
111	C₈	—	284	2	no	—
112	C₈	—	142	2	no	—
113	C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
114	C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
115	C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
116	C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
117	C₂²	—	4	1	no	—
118	C₂²	#14556086	1	1	yes	characteristic
119	C₂²	#14556115	1	1	yes	characteristic
120	C₂²	#14556119	1	1	yes	characteristic
121	C₄	—	8	2	no	—
122	C₄	—	284	2	no	—
123	C₄	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
124	C₄	#14556086	1	1	yes	characteristic
125	C₄	#14556117	1	1	yes	characteristic
126	C₄	#14556119	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#17985864	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556064

C₂

yes

maximal, characteristic

#17985954

C₂

yes

maximal, characteristic

#17985769

C₂

yes

maximal

#17985755

C₂

yes

maximal

#427×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

—

C₂×#11765849

C₂²

yes

characteristic

#14556087

C₂²

yes

—

—

C₄

yes

complemented

C₈×Dic₇₁

C₄

yes

complemented

C₂²×C₈×C₇₁

C₂²

yes

characteristic

#117×C₇₁

C₂²

yes

—

#107×C₇₁

—

#106×C₇₁

—

—

—

—

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

Q₈

yes

characteristic

C₂×C₈×C₇₁

Q₈

yes

characteristic

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

#41

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₄

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×#108

—

#427

D₇₁

yes

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

#117

—

#117

C₂×D₇₁

yes

—

#108

—

142

—

#108

—

142

—

#107

—

#106

—

#97

—

C₄×C₈

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

142

—

M₁₆

—

142

—

M₁₆

—

142

—

M₁₆

—

142

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

Dic₁₄₂

yes

characteristic

C₂×C₈

Dic₁₄₂

yes

characteristic

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

C₂²⋊C₄

—

142

—

100

C₄²

—

101

C₄²

—

102

C₂³

#11765864

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

142

—

105

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

107

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

108

C₂×C₄

#11765864

yes

characteristic

109

C₂×C₄

#11765866

yes

characteristic

110

C₂×C₄

#11765866

yes

characteristic

111

C₈

—

284

—

112

C₈

—

142

—

113

C₈

Dic₇₁⋊C₄

yes

characteristic

114

C₈

Dic₇₁⋊C₄

yes

characteristic

115

C₈

Dic₇₁⋊C₄

yes

characteristic

116

C₈

Dic₇₁⋊C₄

yes

characteristic

117

C₂²

—

118

C₂²

#14556086

yes

characteristic

119

C₂²

#14556115

yes

characteristic

120

C₂²

#14556119

yes

characteristic

121

C₄

—

122

C₄

—

284

—

123

C₄

C₂×Dic₇₁⋊C₄

yes

characteristic

124

C₄

#14556086

yes

characteristic

125

C₄

#14556117

yes

characteristic

126

C₄

#14556119

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#17985864

yes

characteristic

	subgroup	G/H
center	C₄	#14556086
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#427×C₇₁	C₂
socle	C₂×C₇₁	#382
hypercenter	#427	D₇₁
hypocenter	C₇₁	#1833
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₄

#14556086

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#427×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
11. #14556038	C₄	semidirect
12. C₈×Dic₇₁	C₄	semidirect
66. C₇₁	#1833	semidirect

H ≅ G/N

type

11. #14556038

C₄

semidirect

12. C₈×Dic₇₁

C₄

semidirect

66. C₇₁

#1833

semidirect

Subgroups — #22588463

Conjugacy classes of subgroups

Complements