Group #22586234 — subgroups

Subgroups — #22586234

Order 9088.

172 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17986056	C₂	1	1	yes	maximal, characteristic
2	#17985845	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Dic₇₁	—	2	1	no	—
5	C₄⋊(C₇₁⋊C₈)	C₄	1	2	yes	complemented
6	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
7	#14556039	—	2	1	no	—
8	C₇₁⋊#95	—	2	1	no	—
9	C₇₁⋊#95	C₄	1	2	yes	—
10	C₂×M₁₆×C₇₁	—	2	1	no	—
11	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
12	#97×C₇₁	C₄	1	2	yes	complemented
13	C₄×C₈×C₇₁	—	2	1	no	—
14	C₂²×Dic₇₁	—	2	2	no	—
15	C₂×C₇₁⋊C₈	—	2	2	no	—
16	C₄×Dic₇₁	—	2	4	no	—
17	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
20	M₁₆×C₇₁	—	2	4	no	—
21	C₂×C₈×C₇₁	—	2	2	no	—
22	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₇₁	D₄	1	1	yes	characteristic
25	C₄²×C₇₁	Q₈	1	1	yes	characteristic
26	C₂³×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₇₁	—	2	2	no	—
28	C₂×C₄×C₇₁	—	2	2	no	—
29	C₂×C₄×C₇₁	—	4	1	no	—
30	C₂×C₄×C₇₁	C₄²	1	1	yes	characteristic
31	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×Dic₇₁	—	2	4	no	—
37	C₂×Dic₇₁	—	4	2	no	—
38	C₂×Dic₇₁	—	4	2	no	—
39	C₈×C₇₁	—	2	4	no	—
40	C₇₁⋊C₈	—	4	2	no	—
41	C₂²×C₇₁	—	2	1	no	—
42	C₂²×C₇₁	—	2	1	no	—
43	C₂²×C₇₁	#95	1	1	yes	characteristic
44	C₂²×C₇₁	#104	1	2	yes	—
45	C₄×C₇₁	—	2	1	no	—
46	C₄×C₇₁	—	2	1	no	—
47	C₄×C₇₁	—	2	1	no	—
48	C₄×C₇₁	—	2	1	no	—
49	C₄×C₇₁	—	2	1	no	—
50	C₄×C₇₁	#104	1	2	yes	—
51	Dic₇₁	—	4	4	no	—
52	C₂×C₇₁	—	2	2	no	—
53	C₂×C₇₁	#339	1	1	yes	characteristic
54	C₂×C₇₁	#339	1	1	yes	characteristic
55	C₂×C₇₁	#344	1	1	yes	characteristic
56	#1300	—	71	1	no	maximal, Sylow
57	C₇₁	#1300	1	1	yes	characteristic, complemented, Sylow
58	#432	—	71	1	no	—
59	C₄×M₁₆	D₇₁	1	1	yes	characteristic
60	#376	—	71	1	no	—
61	C₂×M₁₆	—	2	1	no	—
62	C₂×C₄²	—	142	1	no	—
63	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
64	C₄⋊C₈	—	71	2	no	—
65	C₂²⋊C₈	—	142	1	no	—
66	#97	—	142	1	no	—
67	#97	Dic₇₁	1	2	yes	complemented
68	C₄×C₈	—	2	1	no	—
69	#95	—	71	2	no	—
70	#95	—	142	1	no	—
71	C₂²×C₄	—	142	2	no	—
72	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
73	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
75	M₁₆	—	2	4	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	142	2	no	—
78	C₄²	—	142	4	no	—
79	C₄²	Dic₁₄₂	1	1	yes	characteristic
80	C₄²	C₄×D₇₁	1	1	yes	characteristic
81	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
82	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
83	C₂³	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	284	2	no	—
88	C₂×C₄	—	284	2	no	—
89	C₂×C₄	—	142	4	no	—
90	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
91	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
93	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
94	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
95	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
96	C₈	—	2	4	no	—
97	C₈	—	284	2	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	C₇₁⋊#95	1	1	yes	characteristic
101	C₂²	#14556056	1	2	yes	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	—	284	4	no	—
108	C₄	#14556054	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#17985759	1	1	yes	characteristic
111	C₂	#17985760	1	1	yes	characteristic
112	C₂	#17985771	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986056

C₂

yes

maximal, characteristic

#17985845

C₂

yes

maximal, characteristic

C₄×M₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×C₄×Dic₇₁

—

C₄⋊(C₇₁⋊C₈)

C₄

yes

complemented

C₂²⋊(C₇₁⋊C₈)

—

#14556039

—

C₇₁⋊#95

—

C₇₁⋊#95

C₄

yes

—

C₂×M₁₆×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#97×C₇₁

C₄

yes

complemented

C₄×C₈×C₇₁

—

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₄×Dic₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

M₁₆×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

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

—

C₂²×C₇₁

—

C₂²×C₇₁

#95

yes

characteristic

C₂²×C₇₁

#104

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#104

yes

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#432

—

C₄×M₁₆

D₇₁

yes

characteristic

#376

—

C₂×M₁₆

—

C₂×C₄²

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

142

—

#97

—

142

—

#97

Dic₇₁

yes

complemented

C₄×C₈

—

#95

—

#95

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

142

—

C₄²

—

142

—

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

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

142

—

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

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₈

—

C₈

—

284

—

C₂²

—

C₂²

—

100

C₂²

C₇₁⋊#95

yes

characteristic

101

C₂²

#14556056

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

284

—

108

C₄

#14556054

yes

—

109

C₂

—

110

C₂

#17985759

yes

characteristic

111

C₂

#17985760

yes

characteristic

112

C₂

#17985771

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₇₁)⋊C₄
commutator	C₂×C₄×C₇₁	C₄²
Frattini	C₂×C₄²	C₂×D₇₁
Fitting	C₄×M₁₆×C₇₁	C₂
socle	C₂²×C₇₁	#95
hypercenter	C₄×M₁₆	D₇₁
hypocenter	C₇₁	#1300
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

(C₂×D₇₁)⋊C₄

commutator

C₂×C₄×C₇₁

C₄²

Frattini

C₂×C₄²

C₂×D₇₁

Fitting

C₄×M₁₆×C₇₁

C₂

socle

C₂²×C₇₁

#95

hypercenter

C₄×M₁₆

D₇₁

hypocenter

C₇₁

#1300

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
5. C₄⋊(C₇₁⋊C₈)	C₄	semidirect
12. #97×C₇₁	C₄	semidirect
57. C₇₁	#1300	semidirect
67. #97	Dic₇₁	semidirect

H ≅ G/N

type

5. C₄⋊(C₇₁⋊C₈)

C₄

semidirect

12. #97×C₇₁

C₄

semidirect

57. C₇₁

#1300

semidirect

67. #97

Dic₇₁

semidirect

Subgroups — #22586234

Conjugacy classes of subgroups

Complements