Group #22588390 — subgroups

Subgroups — #22588390

Order 9088.

252 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 105 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	2	yes	maximal, complemented
2	#17985983	C₂	1	2	yes	maximal
3	C₇₁⋊#338	C₂	1	2	yes	maximal, complemented
4	C₂×#131×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11765849	C₂²	1	4	yes	—
6	#14556064	—	2	4	no	—
7	#14556047	—	2	2	no	—
8	#14556045	—	2	2	no	—
9	C₂×#41×C₇₁	C₂²	1	1	yes	characteristic
10	#131×C₇₁	—	2	4	no	—
11	C₂×M₁₆×C₇₁	C₄	1	2	yes	—
12	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₇₁	C₄	1	2	yes	—
14	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	#11765849	—	2	4	no	—
16	#11765849	—	2	4	no	—
17	C₂×C₇₁⋊C₈	—	2	4	no	—
18	#41×C₇₁	—	2	2	no	—
19	#41×C₇₁	—	2	2	no	—
20	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
21	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
22	C₂×D₄×C₇₁	Q₈	1	2	yes	—
23	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
25	M₁₆×C₇₁	—	2	2	no	—
26	M₁₆×C₇₁	—	2	4	no	—
27	C₂×C₈×C₇₁	—	2	4	no	—
28	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₇₁	D₄	1	4	yes	—
32	C₂³×C₇₁	C₄⋊C₄	1	2	yes	—
33	C₂³×C₇₁	C₂×Q₈	1	1	yes	characteristic
34	Q₈×C₇₁	—	2	2	no	—
35	D₄×C₇₁	—	2	4	no	—
36	D₄×C₇₁	—	2	2	no	—
37	C₂×C₄×C₇₁	—	2	2	no	—
38	C₂×C₄×C₇₁	—	2	2	no	—
39	C₂×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇₁	#41	1	2	yes	—
44	C₈×C₇₁	—	2	2	no	—
45	C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
46	C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
47	C₇₁⋊C₈	—	4	4	no	—
48	C₂²×C₇₁	—	2	2	no	—
49	C₂²×C₇₁	—	4	1	no	—
50	C₂²×C₇₁	—	4	2	no	—
51	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₇₁	C₂²⋊Q₈	1	2	yes	—
53	C₄×C₇₁	—	2	2	no	—
54	C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₇₁	#117	1	1	yes	characteristic
56	C₄×C₇₁	#121	1	2	yes	—
57	C₂×C₇₁	—	2	1	no	—
58	C₂×C₇₁	—	2	2	no	—
59	C₂×C₇₁	—	4	2	no	—
60	C₂×C₇₁	#380	1	1	yes	characteristic
61	#1813	—	71	1	no	maximal, Sylow
62	C₇₁	#1813	1	1	yes	characteristic, complemented, Sylow
63	C₂×#131	D₇₁	1	1	yes	characteristic
64	C₂×#108	—	71	2	no	—
65	#413	—	71	2	no	—
66	#338	—	71	2	no	—
67	C₂×#41	C₂×D₇₁	1	1	yes	characteristic
68	#131	—	2	4	no	—
69	C₂×M₁₆	—	71	4	no	—
70	C₂×M₁₆	Dic₇₁	1	2	yes	—
71	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
72	C₂²×C₈	Dic₇₁	1	2	yes	—
73	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
74	#108	—	142	4	no	—
75	#101	—	142	2	no	—
76	#100	—	142	2	no	—
77	#41	—	2	2	no	—
78	#41	—	2	2	no	—
79	C₂×Q₈	D₂₈₄	1	1	yes	characteristic
80	C₂×D₄	Dic₁₄₂	1	2	yes	—
81	C₂×D₄	D₂₈₄	1	1	yes	characteristic
82	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
83	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	2	4	no	—
86	M₁₆	—	142	4	no	—
87	M₁₆	—	142	4	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	142	4	no	—
90	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
91	C₂×C₈	C₂×Dic₇₁	1	1	yes	characteristic
92	C₂×C₈	C₂×Dic₇₁	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₇₁	1	4	yes	—
94	C₂³	C₄⋊Dic₇₁	1	2	yes	—
95	C₂³	C₂×Dic₁₄₂	1	1	yes	characteristic
96	Q₈	—	2	2	no	—
97	D₄	—	2	4	no	—
98	D₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₄⋊Dic₇₁	1	2	yes	—
102	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
103	C₂×C₄	C₂×C₂²⋊D₇₁	1	2	yes	—
104	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
105	C₂×C₄	#11765865	1	2	yes	—
106	C₈	—	2	2	no	—
107	C₈	—	284	4	no	—
108	C₈	C₂²⋊Dic₇₁	1	2	yes	—
109	C₈	C₂²⋊Dic₇₁	1	2	yes	—
110	C₂²	—	2	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	4	2	no	—
113	C₂²	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
114	C₂²	#14556107	1	2	yes	—
115	C₄	—	2	2	no	—
116	C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
117	C₄	#14556087	1	1	yes	characteristic
118	C₄	#14556102	1	2	yes	—
119	C₂	—	2	1	no	—
120	C₂	—	2	2	no	—
121	C₂	—	4	2	no	—
122	C₂	#17985859	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556064

C₂

yes

maximal, complemented

#17985983

C₂

yes

maximal

C₇₁⋊#338

C₂

yes

maximal, complemented

C₂×#131×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

—

#14556064

—

#14556047

—

#14556045

—

C₂×#41×C₇₁

C₂²

yes

characteristic

#131×C₇₁

—

C₂×M₁₆×C₇₁

C₄

yes

—

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

C₄

yes

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

D₄

yes

characteristic

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂×D₄×C₇₁

Q₈

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

—

C₂³×C₇₁

C₄⋊C₄

yes

—

C₂³×C₇₁

C₂×Q₈

yes

characteristic

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊C₄

yes

—

C₈×C₇₁

C₂²⋊C₄

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₂²⋊C₄

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×#131

D₇₁

yes

—

—

—

C₂×#41

C₂×D₇₁

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

Dic₇₁

yes

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

C₂²×C₈

Dic₇₁

yes

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

#108

—

142

—

#101

—

142

—

#100

—

142

—

#41

—

#41

—

C₂×Q₈

D₂₈₄

yes

characteristic

C₂×D₄

Dic₁₄₂

yes

—

C₂×D₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

142

—

M₁₆

—

142

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₂×C₈

C₂×Dic₇₁

yes

characteristic

C₂×C₈

C₂×Dic₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

—

C₂³

C₄⋊Dic₇₁

yes

—

C₂³

C₂×Dic₁₄₂

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₄⋊Dic₇₁

yes

—

102

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

103

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

104

C₂×C₄

C₂×D₂₈₄

yes

characteristic

105

C₂×C₄

#11765865

yes

—

106

C₈

—

107

C₈

—

284

—

108

C₈

C₂²⋊Dic₇₁

yes

—

109

C₈

C₂²⋊Dic₇₁

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×C₄⋊Dic₇₁

yes

characteristic

114

C₂²

#14556107

yes

—

115

C₄

—

116

C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

117

C₄

#14556087

yes

characteristic

118

C₄

#14556102

yes

—

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

#17985859

yes

characteristic

	subgroup	G/H
center	C₄	#14556087
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#131×C₇₁	C₂
socle	C₂×C₇₁	#380
hypercenter	C₂×#131	D₇₁
hypocenter	C₇₁	#1813
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#14556087

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#131×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. C₂×#14556064	C₂	semidirect
3. C₇₁⋊#338	C₂	semidirect
62. C₇₁	#1813	semidirect

H ≅ G/N

type

1. C₂×#14556064

C₂

semidirect

3. C₇₁⋊#338

C₂

semidirect

62. C₇₁

#1813

semidirect

Subgroups — #22588390

Conjugacy classes of subgroups

Complements