Group #22587730 — subgroups

Subgroups — #22587730

Order 9088.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17986404	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986205	C₂	1	2	yes	maximal
3	C₈×D₂₈₄	C₂	1	1	yes	maximal, characteristic
4	#17985746	C₂	1	2	yes	maximal, complemented
5	#335×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11765854	—	2	2	no	—
7	C₂×C₈×D₇₁	—	2	1	no	—
8	#14556128	C₂²	1	2	yes	—
9	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
10	#14556062	—	2	1	no	—
11	#14556062	—	2	1	no	—
12	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
13	#14556053	—	2	2	no	—
14	(C₂×D₇₁)⋊C₈	—	2	1	no	—
15	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₄₂	—	2	2	no	—
18	C₂×D₂₈₄	D₄	1	1	yes	characteristic
19	C₂×C₄×D₇₁	—	2	1	no	—
20	#11765854	—	4	4	no	—
21	C₂×C₇₁⋊C₈	—	2	1	no	—
22	C₈×D₇₁	—	4	2	no	—
23	C₄⋊Dic₇₁	—	2	2	no	—
24	C₄⋊Dic₇₁	—	2	2	no	—
25	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
26	(C₂×D₇₁)⋊C₄	—	2	1	no	—
27	C₂×C₈×C₇₁	—	2	2	no	—
28	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
30	C₄²×C₇₁	C₂³	1	1	yes	characteristic
31	C₂²×D₇₁	—	2	1	no	—
32	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
35	C₂×Dic₇₁	—	2	1	no	—
36	C₂×Dic₇₁	—	4	2	no	—
37	D₂₈₄	—	2	1	no	—
38	D₂₈₄	D₈	1	2	yes	—
39	C₄×D₇₁	—	4	1	no	—
40	C₄×D₇₁	—	4	1	no	—
41	Dic₁₄₂	—	4	2	no	—
42	Dic₁₄₂	—	4	2	no	—
43	C₈×C₇₁	—	2	1	no	—
44	C₈×C₇₁	—	4	2	no	—
45	C₈×C₇₁	SD₁₆	1	2	yes	complemented
46	C₇₁⋊C₈	—	4	1	no	—
47	C₂²×C₇₁	#121	1	1	yes	characteristic
48	C₂×D₇₁	—	2	2	no	—
49	C₂×D₇₁	—	4	1	no	—
50	C₂×D₇₁	—	4	1	no	—
51	C₄×C₇₁	—	2	1	no	—
52	C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
53	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₄×C₇₁	#135	1	1	yes	characteristic
55	C₄×C₇₁	#137	1	1	yes	characteristic
56	Dic₇₁	—	4	1	no	—
57	Dic₇₁	—	8	2	no	—
58	C₂×C₇₁	#461	1	1	yes	characteristic
59	C₂×C₇₁	#466	1	1	yes	characteristic
60	C₂×C₇₁	#473	1	1	yes	characteristic
61	D₇₁	—	4	2	no	—
62	#1673	—	71	1	no	maximal, Sylow
63	C₇₁	#1673	1	1	yes	characteristic, complemented, Sylow
64	#500	—	71	1	no	—
65	#461	—	71	2	no	—
66	C₈×D₄	—	71	1	no	—
67	#335	D₇₁	1	1	yes	characteristic, complemented
68	#332	—	71	2	no	—
69	C₂×SD₁₆	—	142	2	no	—
70	C₂²×C₈	—	142	1	no	—
71	C₄⋊Q₈	—	71	2	no	—
72	C₄×D₄	—	71	1	no	—
73	#107	—	142	1	no	—
74	#107	—	142	1	no	—
75	C₄⋊C₈	—	71	1	no	—
76	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
77	#103	—	142	2	no	—
78	C₂²⋊C₈	—	142	1	no	—
79	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
80	C₂×Q₈	—	142	2	no	—
81	C₂×D₄	—	71	1	no	—
82	C₂²×C₄	—	142	1	no	—
83	SD₁₆	—	284	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	284	2	no	—
86	C₂×C₈	—	142	1	no	—
87	C₂×C₈	D₂₈₄	1	1	yes	characteristic
88	C₂×C₈	D₂₈₄	1	1	yes	characteristic
89	C₄⋊C₄	—	71	1	no	—
90	C₄⋊C₄	—	142	2	no	—
91	C₄⋊C₄	—	142	2	no	—
92	C₂²⋊C₄	—	142	1	no	—
93	C₄²	C₂²×D₇₁	1	1	yes	characteristic
94	C₂³	—	142	1	no	—
95	Q₈	—	284	2	no	—
96	Q₈	—	284	2	no	—
97	D₄	—	71	2	no	—
98	D₄	—	142	1	no	—
99	C₂×C₄	—	284	1	no	—
100	C₂×C₄	—	284	1	no	—
101	C₂×C₄	—	284	2	no	—
102	C₂×C₄	—	142	1	no	—
103	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
104	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
105	C₂×C₄	#11765866	1	1	yes	characteristic
106	C₈	—	2	1	no	—
107	C₈	—	4	2	no	—
108	C₈	—	284	1	no	—
109	C₈	#11765854	1	2	yes	complemented
110	C₂²	—	284	1	no	—
111	C₂²	—	284	1	no	—
112	C₂²	—	142	2	no	—
113	C₂²	#14556103	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	284	1	no	—
116	C₄	—	568	2	no	—
117	C₄	D₈×D₇₁	1	1	yes	characteristic
118	C₄	C₂×#11765854	1	1	yes	characteristic
119	C₄	#14556151	1	1	yes	characteristic
120	C₄	#14556161	1	1	yes	characteristic
121	C₂	—	284	2	no	—
122	C₂	#17986205	1	1	yes	characteristic
123	C₂	#17986236	1	1	yes	characteristic
124	C₂	#17986275	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986404

C₂

yes

maximal, characteristic, complemented

#17986205

C₂

yes

maximal

C₈×D₂₈₄

C₂

yes

maximal, characteristic

#17985746

C₂

yes

maximal, complemented

#335×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765854

—

C₂×C₈×D₇₁

—

#14556128

C₂²

yes

—

C₄×D₂₈₄

C₂²

yes

characteristic

#14556062

—

#14556062

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

#14556053

—

(C₂×D₇₁)⋊C₈

—

C₄⋊C₈×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×D₂₈₄

D₄

yes

characteristic

C₂×C₄×D₇₁

—

#11765854

—

C₂×C₇₁⋊C₈

—

C₈×D₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

D₄

yes

characteristic

(C₂×D₇₁)⋊C₄

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂²×D₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

D₈

yes

—

C₄×D₇₁

—

C₄×D₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

SD₁₆

yes

complemented

C₇₁⋊C₈

—

C₂²×C₇₁

#121

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1673

—

maximal, Sylow

C₇₁

#1673

yes

characteristic, complemented, Sylow

#500

—

#461

—

C₈×D₄

—

#335

D₇₁

yes

characteristic, complemented

#332

—

C₂×SD₁₆

—

142

—

C₂²×C₈

—

142

—

C₄⋊Q₈

—

C₄×D₄

—

#107

—

142

—

#107

—

142

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₁

yes

—

#103

—

142

—

C₂²⋊C₈

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

142

—

C₂×D₄

—

C₂²×C₄

—

142

—

SD₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

284

—

C₂×C₈

—

142

—

C₂×C₈

D₂₈₄

yes

characteristic

C₂×C₈

D₂₈₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

142

—

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

—

142

—

Q₈

—

284

—

Q₈

—

284

—

D₄

—

D₄

—

142

—

C₂×C₄

—

284

—

100

C₂×C₄

—

284

—

101

C₂×C₄

—

284

—

102

C₂×C₄

—

142

—

103

C₂×C₄

D₄×D₇₁

yes

characteristic

104

C₂×C₄

C₂×D₂₈₄

yes

characteristic

105

C₂×C₄

#11765866

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

—

284

—

109

C₈

#11765854

yes

complemented

110

C₂²

—

284

—

111

C₂²

—

284

—

112

C₂²

—

142

—

113

C₂²

#14556103

yes

characteristic

114

C₄

—

115

C₄

—

284

—

116

C₄

—

568

—

117

C₄

D₈×D₇₁

yes

characteristic

118

C₄

C₂×#11765854

yes

characteristic

119

C₄

#14556151

yes

characteristic

120

C₄

#14556161

yes

characteristic

121

C₂

—

284

—

122

C₂

#17986205

yes

characteristic

123

C₂

#17986236

yes

characteristic

124

C₂

#17986275

yes

characteristic

	subgroup	G/H
center	C₂²	#14556103
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#335×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#335	D₇₁
hypocenter	C₇₁	#1673
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#14556103

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#335×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #17986404	C₂	semidirect
4. #17985746	C₂	semidirect
5. #335×C₇₁	C₂	semidirect
45. C₈×C₇₁	SD₁₆	semidirect
63. C₇₁	#1673	semidirect
67. #335	D₇₁	semidirect
109. C₈	#11765854	semidirect

H ≅ G/N

type

1. #17986404

C₂

semidirect

4. #17985746

C₂

semidirect

5. #335×C₇₁

C₂

semidirect

45. C₈×C₇₁

SD₁₆

semidirect

63. C₇₁

#1673

semidirect

67. #335

D₇₁

semidirect

109. C₈

#11765854

semidirect

Subgroups — #22587730

Conjugacy classes of subgroups

Complements