Group #22589318 — subgroups

Subgroups — #22589318

Order 9088.

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

normal subgroups only

This group has 284 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	#17986023	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#17985758	C₂	1	4	yes	maximal
4	#533×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11765849	C₂²	1	2	yes	—
6	C₂×C₄×Dic₇₁	—	2	2	no	—
7	C₂×C₄×Dic₇₁	C₂²	1	2	yes	—
8	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
9	C₄⋊Q₈×C₇₁	C₄	1	2	yes	complemented
10	#125×C₇₁	C₄	1	2	yes	complemented
11	C₂²⋊Q₈×C₇₁	—	2	2	no	—
12	C₄×Q₈×C₇₁	—	2	2	no	—
13	#117×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₇₁	—	2	2	no	—
16	#11765849	—	2	4	no	—
17	C₂×C₇₁⋊C₈	—	2	2	no	—
18	C₄×Dic₇₁	—	2	8	no	—
19	C₄×Dic₇₁	—	2	2	no	—
20	C₄×Dic₇₁	Q₈	1	4	yes	complemented
21	C₂×Q₈×C₇₁	—	2	2	no	—
22	C₂²×C₄×C₇₁	D₄	1	2	yes	—
23	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₇₁	—	2	4	no	—
25	C₄⋊C₄×C₇₁	—	2	2	no	—
26	C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
27	C₂²⋊C₄×C₇₁	—	2	2	no	—
28	C₄²×C₇₁	—	2	2	no	—
29	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₇₁	D₄	1	2	yes	—
32	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₇₁	—	4	4	no	—
34	C₂×C₄×C₇₁	—	2	4	no	—
35	C₂×C₄×C₇₁	—	2	4	no	—
36	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
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	2	yes	—
41	C₂×C₄×C₇₁	#41	1	2	yes	—
42	C₂×Dic₇₁	—	2	8	no	—
43	C₂×Dic₇₁	—	2	4	no	—
44	C₇₁⋊C₈	—	4	2	no	—
45	C₂²×C₇₁	—	2	2	no	—
46	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₇₁	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₇₁	—	2	2	no	—
49	C₄×C₇₁	—	4	4	no	—
50	C₄×C₇₁	#104	1	4	yes	—
51	C₄×C₇₁	C₄×Q₈	1	2	yes	—
52	C₄×C₇₁	#124	1	1	yes	characteristic
53	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
54	Dic₇₁	—	2	8	no	—
55	C₂×C₇₁	—	2	2	no	—
56	C₂×C₇₁	#391	1	1	yes	characteristic
57	C₂×C₇₁	C₂×#104	1	2	yes	—
58	#1994	—	71	1	no	maximal, Sylow
59	C₇₁	#1994	1	1	yes	characteristic, complemented, Sylow
60	#533	D₇₁	1	1	yes	characteristic
61	#423	—	71	1	no	—
62	C₄³	—	71	1	no	—
63	#339	—	71	4	no	—
64	C₂×M₁₆	—	71	2	no	—
65	C₄⋊Q₈	Dic₇₁	1	2	yes	complemented
66	#125	Dic₇₁	1	2	yes	complemented
67	C₂²⋊Q₈	—	2	2	no	—
68	C₄×Q₈	—	2	2	no	—
69	#117	C₂×D₇₁	1	2	yes	—
70	C₂×C₄²	—	71	2	no	—
71	C₂×C₄²	—	142	2	no	—
72	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
73	C₄⋊C₈	—	142	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂²×C₄	—	142	2	no	—
76	C₂²×C₄	C₂²⋊D₇₁	1	2	yes	—
77	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
78	M₁₆	—	142	4	no	—
79	C₂×C₈	—	142	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	C₂×Dic₇₁	1	4	yes	—
83	C₂²⋊C₄	—	2	2	no	—
84	C₄²	—	2	2	no	—
85	C₄²	—	71	4	no	—
86	C₄²	—	142	8	no	—
87	C₄²	—	142	2	no	—
88	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
89	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
90	C₄²	C₂²⋊D₇₁	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
92	Q₈	—	4	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	142	8	no	—
96	C₂×C₄	—	142	4	no	—
97	C₂×C₄	C₂²⋊Dic₇₁	1	2	yes	—
98	C₂×C₄	C₂²⋊Dic₇₁	1	2	yes	—
99	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
101	C₂×C₄	Q₈×D₇₁	1	2	yes	—
102	C₂×C₄	#11765866	1	2	yes	—
103	C₈	—	284	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
106	C₂²	#14556104	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	4	4	no	—
109	C₄	—	142	8	no	—
110	C₄	#14556055	1	4	yes	—
111	C₄	Q₈×Dic₇₁	1	2	yes	—
112	C₄	#14556117	1	1	yes	characteristic
113	C₄	#14556126	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#17985906	1	1	yes	characteristic
116	C₂	C₂×#14556055	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17986023

C₂

yes

maximal, characteristic

C₄²×Dic₇₁

C₂

yes

maximal, characteristic

#17985758

C₂

yes

maximal

#533×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

—

C₂×C₄×Dic₇₁

—

C₂×C₄×Dic₇₁

C₂²

yes

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊Q₈×C₇₁

C₄

yes

complemented

#125×C₇₁

C₄

yes

complemented

C₂²⋊Q₈×C₇₁

—

C₄×Q₈×C₇₁

—

#117×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

Q₈

yes

complemented

C₂×Q₈×C₇₁

—

C₂²×C₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#104

yes

—

C₄×C₇₁

C₄×Q₈

yes

—

C₄×C₇₁

#124

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#391

yes

characteristic

C₂×C₇₁

C₂×#104

yes

—

#1994

—

maximal, Sylow

C₇₁

#1994

yes

characteristic, complemented, Sylow

#533

D₇₁

yes

characteristic

#423

—

C₄³

—

#339

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₇₁

yes

complemented

#125

Dic₇₁

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

142

—

C₂×Q₈

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²⋊D₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

142

—

C₂×C₈

—

142

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₇₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

142

—

C₄²

—

142

—

C₄²

C₂×Dic₇₁

yes

characteristic

C₄²

C₂×Dic₇₁

yes

characteristic

C₄²

C₂²⋊D₇₁

yes

—

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

C₂²⋊Dic₇₁

yes

—

C₂×C₄

C₂²⋊Dic₇₁

yes

—

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

101

C₂×C₄

Q₈×D₇₁

yes

—

102

C₂×C₄

#11765866

yes

—

103

C₈

—

284

—

104

C₂²

—

105

C₂²

C₂×C₂²⋊Dic₇₁

yes

characteristic

106

C₂²

#14556104

yes

—

107

C₄

—

108

C₄

—

109

C₄

—

142

—

110

C₄

#14556055

yes

—

111

C₄

Q₈×Dic₇₁

yes

—

112

C₄

#14556117

yes

characteristic

113

C₄

#14556126

yes

characteristic

114

C₂

—

115

C₂

#17985906

yes

characteristic

116

C₂

C₂×#14556055

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#533×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₄
hypercenter	#533	D₇₁
hypocenter	C₇₁	#1994
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#533×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₂²⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
9. C₄⋊Q₈×C₇₁	C₄	semidirect
10. #125×C₇₁	C₄	semidirect
20. C₄×Dic₇₁	Q₈	semidirect
59. C₇₁	#1994	semidirect
65. C₄⋊Q₈	Dic₇₁	semidirect
66. #125	Dic₇₁	semidirect

H ≅ G/N

type

9. C₄⋊Q₈×C₇₁

C₄

semidirect

10. #125×C₇₁

C₄

semidirect

20. C₄×Dic₇₁

Q₈

semidirect

59. C₇₁

#1994

semidirect

65. C₄⋊Q₈

Dic₇₁

semidirect

66. #125

Dic₇₁

semidirect

Subgroups — #22589318

Conjugacy classes of subgroups

Complements