Group #22586176 — subgroups

Subgroups — #22586176

Order 9088.

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

normal subgroups only

This group has 172 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	C₄×#11765849	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#431×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11765849	C₄	1	2	yes	—
5	C₂²×C₇₁⋊C₈	—	2	1	no	—
6	#14556039	—	2	1	no	—
7	C₄×C₇₁⋊C₈	—	2	1	no	—
8	C₄×C₇₁⋊C₈	—	2	1	no	—
9	C₄×C₇₁⋊C₈	C₄	1	2	yes	—
10	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₇₁	C₄	1	2	yes	—
12	C₂²⋊C₈×C₇₁	—	2	1	no	—
13	C₄×C₈×C₇₁	—	2	1	no	—
14	#11765849	—	2	2	no	—
15	#11765849	C₈	1	4	yes	complemented
16	C₂×C₇₁⋊C₈	—	2	4	no	—
17	C₂×C₇₁⋊C₈	—	2	2	no	—
18	C₂×C₇₁⋊C₈	—	2	2	no	—
19	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
27	C₂³×C₇₁	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₇₁	—	2	2	no	—
29	C₂×C₄×C₇₁	—	2	2	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	C₂²⋊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×C₄×C₇₁	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₇₁	M₁₆	1	2	yes	—
38	C₈×C₇₁	—	4	2	no	—
39	C₇₁⋊C₈	—	2	4	no	—
40	C₇₁⋊C₈	—	2	4	no	—
41	C₂²×C₇₁	—	2	1	no	—
42	C₂²×C₇₁	—	2	1	no	—
43	C₂²×C₇₁	#95	1	1	yes	characteristic
44	C₂²×C₇₁	C₄⋊C₈	1	2	yes	—
45	C₄×C₇₁	—	2	1	no	—
46	C₄×C₇₁	—	2	1	no	—
47	C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
48	C₄×C₇₁	#97	1	1	yes	characteristic
49	C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
50	C₄×C₇₁	#104	1	2	yes	—
51	C₄×C₇₁	#108	1	2	yes	—
52	C₂×C₇₁	—	2	2	no	—
53	C₂×C₇₁	#336	1	1	yes	characteristic
54	C₂×C₇₁	#339	1	1	yes	characteristic
55	C₂×C₇₁	#341	1	1	yes	characteristic
56	#1278	—	71	1	no	maximal, Sylow
57	C₇₁	#1278	1	1	yes	characteristic, complemented, Sylow
58	#431	D₇₁	1	1	yes	characteristic
59	C₄×M₁₆	—	71	1	no	—
60	C₂×C₄×C₈	—	71	1	no	—
61	C₂×M₁₆	—	71	2	no	—
62	C₂²×C₈	—	142	1	no	—
63	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
64	C₄⋊C₈	Dic₇₁	1	2	yes	—
65	C₂²⋊C₈	—	2	1	no	—
66	#97	—	142	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	71	2	no	—
69	C₄×C₈	—	142	1	no	—
70	C₄×C₈	—	142	1	no	—
71	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
72	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
73	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
74	M₁₆	—	71	4	no	—
75	M₁₆	—	142	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	142	4	no	—
78	C₂×C₈	—	142	2	no	—
79	C₂×C₈	—	142	2	no	—
80	C₄²	C₄×D₇₁	1	1	yes	characteristic
81	C₄²	D₂₈₄	1	1	yes	characteristic
82	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
83	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
84	C₂³	Dic₇₁⋊C₄	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
88	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
89	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
91	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
93	C₂×C₄	C₈×D₇₁	1	2	yes	—
94	C₂×C₄	#11765848	1	2	yes	—
95	C₈	—	4	2	no	—
96	C₈	—	142	4	no	—
97	C₈	—	142	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	C₇₁⋊#95	1	1	yes	characteristic
101	C₂²	Dic₇₁⋊C₈	1	2	yes	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₇₁	1	1	yes	characteristic
105	C₄	#14556038	1	1	yes	characteristic
106	C₄	(C₂×D₇₁)⋊C₈	1	2	yes	—
107	C₄	#14556055	1	2	yes	—
108	C₄	#14556063	1	2	yes	—
109	C₂	—	2	2	no	—
110	C₂	#17985753	1	1	yes	characteristic
111	C₂	#17985758	1	1	yes	characteristic
112	C₂	#17985763	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#11765849

C₂

yes

maximal, characteristic

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#431×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₄

yes

—

C₂²×C₇₁⋊C₈

—

#14556039

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

—

C₄×C₇₁⋊C₈

C₄

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₄

yes

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

—

#11765849

—

#11765849

C₈

yes

complemented

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

Q₈

yes

characteristic

C₂×C₈×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₂³×C₇₁

C₄⋊C₄

yes

characteristic

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

—

C₂×C₄×C₇₁

M₁₆

yes

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#95

yes

characteristic

C₂²×C₇₁

C₄⋊C₈

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×C₈

yes

characteristic

C₄×C₇₁

#97

yes

characteristic

C₄×C₇₁

C₂²⋊C₈

yes

—

C₄×C₇₁

#104

yes

—

C₄×C₇₁

#108

yes

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#431

D₇₁

yes

characteristic

C₄×M₁₆

—

C₂×C₄×C₈

—

C₂×M₁₆

—

C₂²×C₈

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

Dic₇₁

yes

—

C₂²⋊C₈

—

#97

—

142

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

142

—

C₄×C₈

—

142

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

142

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄²

C₄×D₇₁

yes

characteristic

C₄²

D₂₈₄

yes

characteristic

C₄²

C₂×Dic₇₁

yes

characteristic

C₄²

C₂²⋊D₇₁

yes

characteristic

C₂³

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₇₁

yes

characteristic

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

C₈×D₇₁

yes

—

C₂×C₄

#11765848

yes

—

C₈

—

C₈

—

142

—

C₈

—

142

—

C₂²

—

C₂²

—

100

C₂²

C₇₁⋊#95

yes

characteristic

101

C₂²

Dic₇₁⋊C₈

yes

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₇₁

yes

characteristic

105

C₄

#14556038

yes

characteristic

106

C₄

(C₂×D₇₁)⋊C₈

yes

—

107

C₄

#14556055

yes

—

108

C₄

#14556063

yes

—

109

C₂

—

110

C₂

#17985753

yes

characteristic

111

C₂

#17985758

yes

characteristic

112

C₂

#17985763

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊D₇₁
commutator	C₄×C₇₁	C₄×C₈
Frattini	C₂×C₄²	C₂×D₇₁
Fitting	#431×C₇₁	C₂
socle	C₂²×C₇₁	#95
hypercenter	#431	D₇₁
hypocenter	C₇₁	#1278
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊D₇₁

commutator

C₄×C₇₁

C₄×C₈

Frattini

C₂×C₄²

C₂×D₇₁

Fitting

#431×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
15. #11765849	C₈	semidirect
57. C₇₁	#1278	semidirect

H ≅ G/N

type

15. #11765849

C₈

semidirect

57. C₇₁

#1278

semidirect

Subgroups — #22586176

Conjugacy classes of subgroups

Complements