Group #22590055 — subgroups

Subgroups — #22590055

Order 9088.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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×D₅₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556068	C₂	1	1	yes	maximal, characteristic, complemented
3	#17985799	C₂	1	4	yes	maximal, complemented
4	#427×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₂₈₄	—	2	1	no	—
6	C₂×D₅₆₈	—	2	2	no	—
7	C₂×D₅₆₈	—	2	1	no	—
8	C₂×D₅₆₈	C₄	1	4	yes	complemented
9	C₂×D₅₆₈	C₂²	1	2	yes	—
10	#14556068	—	2	2	no	—
11	C₂×C₇₁⋊C₁₆	C₂²	1	2	yes	complemented
12	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	#117×C₇₁	—	2	1	no	—
14	#107×C₇₁	C₂²	1	2	yes	complemented
15	#106×C₇₁	—	2	1	no	—
16	C₂³×D₇₁	—	4	1	no	—
17	C₂×D₂₈₄	—	2	4	no	—
18	C₂×D₂₈₄	—	2	2	no	—
19	C₂×D₂₈₄	—	4	1	no	—
20	C₂×D₂₈₄	—	4	2	no	—
21	D₅₆₈	—	2	4	no	—
22	D₅₆₈	—	2	2	no	—
23	D₅₆₈	C₂×C₄	1	4	yes	—
24	C₇₁⋊C₁₆	—	2	2	no	—
25	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₇₁	D₄	1	2	yes	—
28	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
29	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₇₁	—	2	2	no	—
31	C₂²⋊C₄×C₇₁	—	4	1	no	—
32	C₄²×C₇₁	—	4	1	no	—
33	C₂³×C₇₁	D₈	1	1	yes	characteristic
34	C₂²×D₇₁	—	4	4	no	—
35	C₂²×D₇₁	—	4	2	no	—
36	C₂²×D₇₁	—	8	4	no	—
37	C₂×C₄×C₇₁	—	4	2	no	—
38	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
40	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
41	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
42	D₂₈₄	—	2	4	no	—
43	D₂₈₄	—	4	4	no	—
44	D₂₈₄	—	4	2	no	—
45	C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
46	C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₇₁	—	2	2	no	—
49	C₂²×C₇₁	#102	1	2	yes	—
50	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
51	C₂×D₇₁	—	4	4	no	—
52	C₂×D₇₁	—	8	8	no	—
53	C₂×D₇₁	—	8	4	no	—
54	C₄×C₇₁	—	8	2	no	—
55	C₄×C₇₁	#102	1	2	yes	—
56	C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₇₁	—	2	2	no	—
59	C₂×C₇₁	C₂×#102	1	1	yes	characteristic
60	C₂×C₇₁	#509	1	2	yes	—
61	D₇₁	—	8	4	no	—
62	#2140	—	71	1	no	maximal, Sylow
63	C₇₁	#2140	1	1	yes	characteristic, complemented, Sylow
64	C₂²×D₈	—	71	1	no	—
65	C₂×M₃₂	—	71	1	no	—
66	#427	D₇₁	1	1	yes	characteristic, complemented
67	#357	—	71	4	no	—
68	C₂²×D₄	—	142	1	no	—
69	C₂×D₈	—	71	4	no	—
70	C₂×D₈	—	71	2	no	—
71	C₂×D₈	—	142	2	no	—
72	C₂×D₈	—	142	1	no	—
73	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
74	#117	—	2	1	no	—
75	M₃₂	—	142	2	no	—
76	C₂×C₁₆	—	71	2	no	—
77	#107	C₂×D₇₁	1	2	yes	complemented
78	#106	—	2	1	no	—
79	C₂⁴	—	284	1	no	—
80	C₂×D₄	—	284	1	no	—
81	C₂×D₄	—	284	2	no	—
82	C₂×D₄	—	142	4	no	—
83	C₂×D₄	—	142	2	no	—
84	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
85	D₈	—	71	4	no	—
86	D₈	—	142	4	no	—
87	D₈	—	142	2	no	—
88	C₂×C₈	C₄×D₇₁	1	2	yes	—
89	C₂×C₈	D₂₈₄	1	2	yes	—
90	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
91	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	4	1	no	—
94	C₄²	—	4	1	no	—
95	C₁₆	—	142	2	no	—
96	C₂³	—	284	4	no	—
97	C₂³	—	284	2	no	—
98	C₂³	—	568	4	no	—
99	C₂³	#11765850	1	1	yes	characteristic
100	D₄	—	284	4	no	—
101	D₄	—	284	2	no	—
102	D₄	—	142	4	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	(C₂×D₇₁)⋊C₄	1	2	yes	—
105	C₂×C₄	#11765850	1	1	yes	characteristic
106	C₂×C₄	#11765852	1	2	yes	—
107	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
108	C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
109	C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
110	C₈	C₂×D₂₈₄	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	284	4	no	—
113	C₂²	—	568	8	no	—
114	C₂²	—	568	4	no	—
115	C₂²	#14556048	1	2	yes	—
116	C₂²	C₂×#11765850	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	#14556048	1	2	yes	—
119	C₄	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
120	C₄	C₂×#11765852	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	568	4	no	—
123	C₂	C₂×#14556048	1	1	yes	characteristic
124	C₂	#17986434	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₆₈

C₂

yes

maximal, characteristic

C₂×#14556068

C₂

yes

maximal, characteristic, complemented

#17985799

C₂

yes

maximal, complemented

#427×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₈₄

—

C₂×D₅₆₈

—

C₂×D₅₆₈

—

C₂×D₅₆₈

C₄

yes

complemented

C₂×D₅₆₈

C₂²

yes

—

#14556068

—

C₂×C₇₁⋊C₁₆

C₂²

yes

complemented

C₂²×C₈×C₇₁

C₂²

yes

characteristic

#117×C₇₁

—

#107×C₇₁

C₂²

yes

complemented

#106×C₇₁

—

C₂³×D₇₁

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

D₅₆₈

—

D₅₆₈

—

D₅₆₈

C₂×C₄

yes

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

D₈

yes

characteristic

C₂²×D₇₁

—

C₂²×D₇₁

—

C₂²×D₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

C₈×C₇₁

C₂²⋊C₄

yes

—

C₈×C₇₁

C₂²×C₄

yes

characteristic

C₈×C₇₁

C₂×D₄

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

#102

yes

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#102

yes

—

C₄×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

C₂×#102

yes

characteristic

C₂×C₇₁

#509

yes

—

D₇₁

—

#2140

—

maximal, Sylow

C₇₁

#2140

yes

characteristic, complemented, Sylow

C₂²×D₈

—

C₂×M₃₂

—

#427

D₇₁

yes

characteristic, complemented

#357

—

C₂²×D₄

—

142

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

142

—

C₂×D₈

—

142

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

#117

—

M₃₂

—

142

—

C₂×C₁₆

—

#107

C₂×D₇₁

yes

complemented

#106

—

C₂⁴

—

284

—

C₂×D₄

—

284

—

C₂×D₄

—

284

—

C₂×D₄

—

142

—

C₂×D₄

—

142

—

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

D₈

—

D₈

—

142

—

D₈

—

142

—

C₂×C₈

C₄×D₇₁

yes

—

C₂×C₈

D₂₈₄

yes

—

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

142

—

C₂³

—

284

—

C₂³

—

284

—

C₂³

—

568

—

C₂³

#11765850

yes

characteristic

100

D₄

—

284

—

101

D₄

—

284

—

102

D₄

—

142

—

103

C₂×C₄

—

104

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

—

105

C₂×C₄

#11765850

yes

characteristic

106

C₂×C₄

#11765852

yes

—

107

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

108

C₈

(C₂×D₇₁)⋊C₄

yes

—

109

C₈

C₂×C₄×D₇₁

yes

characteristic

110

C₈

C₂×D₂₈₄

yes

characteristic

111

C₂²

—

112

C₂²

—

284

—

113

C₂²

—

568

—

114

C₂²

—

568

—

115

C₂²

#14556048

yes

—

116

C₂²

C₂×#11765850

yes

characteristic

117

C₄

—

118

C₄

#14556048

yes

—

119

C₄

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

120

C₄

C₂×#11765852

yes

characteristic

121

C₂

—

122

C₂

—

568

—

123

C₂

C₂×#14556048

yes

characteristic

124

C₂

#17986434

yes

—

	subgroup	G/H
center	C₂²	C₂×#11765850
commutator	C₈×C₇₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#427×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#427	D₇₁
hypocenter	C₇₁	#2140
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₂²

C₂×#11765850

commutator

C₈×C₇₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#427×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

D₇₁

C₇₁

G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
2. C₂×#14556068	C₂	semidirect
3. #17985799	C₂	semidirect
4. #427×C₇₁	C₂	semidirect
8. C₂×D₅₆₈	C₄	semidirect
11. C₂×C₇₁⋊C₁₆	C₂²	semidirect
14. #107×C₇₁	C₂²	semidirect
63. C₇₁	#2140	semidirect
66. #427	D₇₁	semidirect
77. #107	C₂×D₇₁	semidirect

H ≅ G/N

type

2. C₂×#14556068

C₂

semidirect

3. #17985799

C₂

semidirect

4. #427×C₇₁

C₂

semidirect

8. C₂×D₅₆₈

C₄

semidirect

11. C₂×C₇₁⋊C₁₆

C₂²

semidirect

14. #107×C₇₁

C₂²

semidirect

63. C₇₁

#2140

semidirect

66. #427

D₇₁

semidirect

77. #107

C₂×D₇₁

semidirect

Subgroups — #22590055

Conjugacy classes of subgroups

Complements