Group #22586222 — subgroups

Subgroups — #22586222

Order 9088.

180 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 73 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×#14556039	C₂	1	1	yes	maximal, characteristic
2	#17985865	C₂	1	1	yes	maximal, characteristic
3	C₂×#97×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₈	—	2	1	no	—
5	C₂×C₄⋊Dic₇₁	C₄	1	2	yes	—
6	#14556039	—	2	1	no	—
7	#14556039	C₄	1	2	yes	complemented
8	C₇₁⋊#95	—	2	1	no	—
9	C₇₁⋊#95	—	2	1	no	—
10	C₂²×C₈×C₇₁	—	2	1	no	—
11	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
12	#97×C₇₁	—	2	1	no	—
13	#97×C₇₁	C₄	1	2	yes	complemented
14	C₂²×Dic₇₁	—	2	2	no	—
15	C₂×C₇₁⋊C₈	—	2	4	no	—
16	C₂×C₇₁⋊C₈	—	2	2	no	—
17	C₄⋊Dic₇₁	—	4	2	no	—
18	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
21	C₂×C₈×C₇₁	—	2	4	no	—
22	C₂×C₈×C₇₁	—	2	2	no	—
23	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₇₁	D₄	1	1	yes	characteristic
26	C₄²×C₇₁	Q₈	1	1	yes	characteristic
27	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₇₁	—	2	1	no	—
29	C₂×C₄×C₇₁	—	2	1	no	—
30	C₂×C₄×C₇₁	—	2	2	no	—
31	C₂×C₄×C₇₁	—	2	2	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×Dic₇₁	—	4	2	no	—
39	C₂×Dic₇₁	—	4	2	no	—
40	C₂×Dic₇₁	—	4	2	no	—
41	C₈×C₇₁	—	2	4	no	—
42	C₇₁⋊C₈	—	2	4	no	—
43	C₂²×C₇₁	#95	1	1	yes	characteristic
44	C₂²×C₇₁	#100	1	1	yes	characteristic
45	C₂²×C₇₁	#101	1	1	yes	characteristic
46	C₂²×C₇₁	#104	1	2	yes	—
47	C₂²×C₇₁	#104	1	2	yes	—
48	C₄×C₇₁	—	2	1	no	—
49	C₄×C₇₁	—	2	1	no	—
50	C₄×C₇₁	—	2	1	no	—
51	C₄×C₇₁	—	2	1	no	—
52	C₄×C₇₁	—	2	1	no	—
53	C₄×C₇₁	—	2	1	no	—
54	Dic₇₁	—	8	2	no	—
55	C₂×C₇₁	#329	1	2	yes	—
56	C₂×C₇₁	#330	1	2	yes	—
57	C₂×C₇₁	#339	1	1	yes	characteristic
58	C₂×C₇₁	#339	1	1	yes	characteristic
59	C₂×C₇₁	#343	1	1	yes	characteristic
60	#1295	—	71	1	no	maximal, Sylow
61	C₇₁	#1295	1	1	yes	characteristic, complemented, Sylow
62	C₂×#97	—	71	1	no	—
63	C₂×#97	D₇₁	1	1	yes	characteristic
64	#382	—	71	1	no	—
65	C₂²×C₈	—	2	1	no	—
66	C₂²×C₈	—	142	1	no	—
67	C₂×C₄⋊C₄	—	71	2	no	—
68	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
69	#97	—	2	1	no	—
70	#97	—	71	2	no	—
71	#97	—	142	1	no	—
72	#97	Dic₇₁	1	2	yes	complemented
73	#95	—	142	1	no	—
74	#95	—	142	1	no	—
75	C₂²×C₄	—	142	2	no	—
76	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
77	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	142	4	no	—
82	C₂×C₈	—	142	2	no	—
83	C₄⋊C₄	—	284	2	no	—
84	C₄²	Dic₁₄₂	1	1	yes	characteristic
85	C₄²	C₄×D₇₁	1	1	yes	characteristic
86	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
87	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
88	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	284	2	no	—
94	C₂×C₄	—	284	2	no	—
95	C₂×C₄	—	284	2	no	—
96	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
97	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
98	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
99	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
100	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
101	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
102	C₈	—	2	4	no	—
103	C₈	—	142	4	no	—
104	C₂²	C₇₁⋊#95	1	1	yes	characteristic
105	C₂²	#14556044	1	1	yes	characteristic
106	C₂²	#14556046	1	1	yes	characteristic
107	C₂²	#14556054	1	2	yes	—
108	C₂²	#14556056	1	2	yes	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	568	2	no	—
116	C₂	#17985738	1	2	yes	—
117	C₂	#17985740	1	2	yes	—
118	C₂	#17985759	1	1	yes	characteristic
119	C₂	#17985760	1	1	yes	characteristic
120	C₂	#17985767	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556039

C₂

yes

maximal, characteristic

#17985865

C₂

yes

maximal, characteristic

C₂×#97×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

—

C₂×C₄⋊Dic₇₁

C₄

yes

—

#14556039

—

#14556039

C₄

yes

complemented

C₇₁⋊#95

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#97×C₇₁

—

#97×C₇₁

C₄

yes

complemented

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

—

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

—

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×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

#104

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

Dic₇₁

—

C₂×C₇₁

#329

yes

—

C₂×C₇₁

#330

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×#97

—

C₂×#97

D₇₁

yes

characteristic

#382

—

C₂²×C₈

—

C₂²×C₈

—

142

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇₁

yes

—

—

—

142

—

#97

Dic₇₁

yes

complemented

#95

—

142

—

#95

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄⋊C₄

—

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

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

C₄×Dic₇₁

yes

characteristic

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

100

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

101

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

102

C₈

—

103

C₈

—

142

—

104

C₂²

C₇₁⋊#95

yes

characteristic

105

C₂²

#14556044

yes

characteristic

106

C₂²

#14556046

yes

characteristic

107

C₂²

#14556054

yes

—

108

C₂²

#14556056

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

568

—

116

C₂

#17985738

yes

—

117

C₂

#17985740

yes

—

118

C₂

#17985759

yes

characteristic

119

C₂

#17985760

yes

characteristic

120

C₂

#17985767

yes

characteristic

	subgroup	G/H
center	C₂³	(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₄²
Frattini	C₂×C₄²	C₂×D₇₁
Fitting	C₂×#97×C₇₁	C₂
socle	C₂³×C₇₁	C₂²⋊C₄
hypercenter	C₂×#97	D₇₁
hypocenter	C₇₁	#1295
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

(C₂×D₇₁)⋊C₄

commutator

C₂×C₄×C₇₁

C₄²

Frattini

C₂×C₄²

C₂×D₇₁

Fitting

C₂×#97×C₇₁

C₂

socle

C₂³×C₇₁

C₂²⋊C₄

hypercenter

C₂×#97

D₇₁

hypocenter

C₇₁

#1295

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #14556039	C₄	semidirect
13. #97×C₇₁	C₄	semidirect
61. C₇₁	#1295	semidirect
72. #97	Dic₇₁	semidirect

H ≅ G/N

type

7. #14556039

C₄

semidirect

13. #97×C₇₁

C₄

semidirect

61. C₇₁

#1295

semidirect

72. #97

Dic₇₁

semidirect

Subgroups — #22586222

Conjugacy classes of subgroups

Complements