Group #22586428 — subgroups

Subgroups — #22586428

Order 9088.

136 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₂×#14556068	C₂	1	1	yes	maximal, characteristic
2	#17985954	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11765849	C₄	1	1	yes	characteristic
5	#14556087	C₄	1	1	yes	characteristic
6	#14556068	—	2	1	no	—
7	#14556068	C₄	1	2	yes	complemented
8	C₂×C₇₁⋊C₁₆	—	2	1	no	—
9	#14556038	—	2	1	no	—
10	C₈×Dic₇₁	—	2	1	no	—
11	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₇₁	—	2	1	no	—
13	C₂×C₁₆×C₇₁	—	2	1	no	—
14	C₂×C₁₆×C₇₁	C₄	1	2	yes	complemented
15	#11765849	—	2	1	no	—
16	#11765849	C₈	1	2	yes	—
17	C₂×C₇₁⋊C₈	—	2	1	no	—
18	C₄⋊Dic₇₁	C₈	1	2	yes	—
19	C₂²⋊Dic₇₁	—	2	1	no	—
20	C₄×Dic₇₁	—	2	1	no	—
21	C₇₁⋊C₁₆	—	2	2	no	—
22	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₂×C₈×C₇₁	Q₈	1	1	yes	characteristic
29	C₁₆×C₇₁	—	2	2	no	—
30	C₂³×C₇₁	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₇₁	C₄²	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	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₇₁	M₁₆	1	1	yes	characteristic
37	C₂×Dic₇₁	—	2	2	no	—
38	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
41	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
42	C₇₁⋊C₈	—	2	2	no	—
43	C₂²×C₇₁	—	2	2	no	—
44	C₂²×C₇₁	#97	1	1	yes	characteristic
45	C₂²×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
46	C₂²×C₇₁	C₄⋊C₈	1	1	yes	characteristic
47	C₄×C₇₁	#95	1	1	yes	characteristic
48	C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
49	C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
50	C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
51	Dic₇₁	—	4	2	no	—
52	C₂×C₇₁	—	2	1	no	—
53	C₂×C₇₁	—	2	1	no	—
54	C₂×C₇₁	#336	1	1	yes	characteristic
55	C₂×C₇₁	#350	1	2	yes	—
56	#1379	—	71	1	no	maximal, Sylow
57	C₇₁	#1379	1	1	yes	characteristic, complemented, Sylow
58	C₂×M₃₂	—	71	1	no	—
59	C₂²×C₁₆	D₇₁	1	1	yes	characteristic
60	#405	—	71	1	no	—
61	C₂×M₁₆	—	71	1	no	—
62	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
63	#117	—	71	1	no	—
64	M₃₂	—	71	2	no	—
65	M₃₂	—	142	1	no	—
66	C₂×C₁₆	—	2	1	no	—
67	C₂×C₁₆	—	2	1	no	—
68	C₂×C₁₆	—	142	1	no	—
69	C₂×C₁₆	Dic₇₁	1	2	yes	complemented
70	#97	—	142	1	no	—
71	C₄×C₈	—	142	1	no	—
72	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
73	M₁₆	—	71	2	no	—
74	M₁₆	—	142	1	no	—
75	C₂×C₈	—	142	1	no	—
76	C₂×C₈	Dic₁₄₂	1	1	yes	characteristic
77	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
78	C₂×C₈	D₂₈₄	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₇₁	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
82	C₄⋊C₄	—	71	2	no	—
83	C₂²⋊C₄	—	142	1	no	—
84	C₄²	—	142	1	no	—
85	C₁₆	—	2	2	no	—
86	C₁₆	—	142	2	no	—
87	C₂³	#11765848	1	1	yes	characteristic
88	C₂×C₄	—	142	2	no	—
89	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
90	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
91	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
93	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
94	C₂×C₄	#11765848	1	1	yes	characteristic
95	C₈	—	142	2	no	—
96	C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
97	C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
98	C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
99	C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
100	C₂²	—	2	2	no	—
101	C₂²	#14556038	1	1	yes	characteristic
102	C₂²	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
103	C₂²	Dic₇₁⋊C₈	1	1	yes	characteristic
104	C₄	—	284	2	no	—
105	C₄	C₇₁⋊#95	1	1	yes	characteristic
106	C₄	C₈×Dic₇₁	1	1	yes	characteristic
107	C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
108	C₄	Dic₇₁⋊C₈	1	1	yes	characteristic
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	#17985753	1	1	yes	characteristic
112	C₂	#17985782	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556068

C₂

yes

maximal, characteristic

#17985954

C₂

yes

maximal, characteristic

C₂²×C₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₄

yes

characteristic

#14556087

C₄

yes

characteristic

#14556068

—

#14556068

C₄

yes

complemented

C₂×C₇₁⋊C₁₆

—

#14556038

—

C₈×Dic₇₁

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₄

yes

complemented

#11765849

—

#11765849

C₈

yes

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

C₈

yes

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

Q₈

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

M₁₆

yes

characteristic

C₂×C₄×C₇₁

C₄²

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

M₁₆

yes

characteristic

C₂×Dic₇₁

—

C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#97

yes

characteristic

C₂²×C₇₁

C₂²⋊C₈

yes

characteristic

C₂²×C₇₁

C₄⋊C₈

yes

characteristic

C₄×C₇₁

#95

yes

characteristic

C₄×C₇₁

C₄×C₈

yes

characteristic

C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₄×C₇₁

C₄⋊C₈

yes

characteristic

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#336

yes

characteristic

C₂×C₇₁

#350

yes

—

#1379

—

maximal, Sylow

C₇₁

#1379

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

D₇₁

yes

characteristic

#405

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

#117

—

M₃₂

—

M₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₂×C₁₆

Dic₇₁

yes

complemented

#97

—

142

—

C₄×C₈

—

142

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

142

—

C₂×C₈

—

142

—

C₂×C₈

Dic₁₄₂

yes

characteristic

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

D₂₈₄

yes

characteristic

C₂×C₈

C₂×Dic₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

142

—

C₄²

—

142

—

C₁₆

—

C₁₆

—

142

—

C₂³

#11765848

yes

characteristic

C₂×C₄

—

142

—

C₂×C₄

C₄×Dic₇₁

yes

characteristic

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

C₈×D₇₁

yes

characteristic

C₂×C₄

C₈×D₇₁

yes

characteristic

C₂×C₄

#11765848

yes

characteristic

C₈

—

142

—

C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

C₈

C₂²⋊Dic₇₁

yes

characteristic

C₈

Dic₇₁⋊C₄

yes

characteristic

C₈

C₄⋊Dic₇₁

yes

characteristic

100

C₂²

—

101

C₂²

#14556038

yes

characteristic

102

C₂²

(C₂×D₇₁)⋊C₈

yes

characteristic

103

C₂²

Dic₇₁⋊C₈

yes

characteristic

104

C₄

—

284

—

105

C₄

C₇₁⋊#95

yes

characteristic

106

C₄

C₈×Dic₇₁

yes

characteristic

107

C₄

(C₂×D₇₁)⋊C₈

yes

characteristic

108

C₄

Dic₇₁⋊C₈

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

#17985753

yes

characteristic

112

C₂

#17985782

yes

—

	subgroup	G/H
center	C₂×C₈	D₂₈₄
commutator	C₄×C₇₁	C₄×C₈
Frattini	C₂²×C₈	C₂×D₇₁
Fitting	C₂²×C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	C₂²⋊C₈
hypercenter	C₂²×C₁₆	D₇₁
hypocenter	C₇₁	#1379
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₈

D₂₈₄

commutator

C₄×C₇₁

C₄×C₈

Frattini

C₂²×C₈

C₂×D₇₁

Fitting

C₂²×C₁₆×C₇₁

C₂

socle

C₂²×C₇₁

C₂²⋊C₈

hypercenter

C₂²×C₁₆

D₇₁

hypocenter

C₇₁

#1379

LCS

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
7. #14556068	C₄	semidirect
14. C₂×C₁₆×C₇₁	C₄	semidirect
57. C₇₁	#1379	semidirect
69. C₂×C₁₆	Dic₇₁	semidirect

H ≅ G/N

type

7. #14556068

C₄

semidirect

14. C₂×C₁₆×C₇₁

C₄

semidirect

57. C₇₁

#1379

semidirect

69. C₂×C₁₆

Dic₇₁

semidirect

Subgroups — #22586428

Conjugacy classes of subgroups

Complements