Group #22586429 — subgroups

Subgroups — #22586429

Order 9088.

136 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂²×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#405×C₇₁	C₂	1	1	yes	maximal, characteristic
4	#14556068	—	2	1	no	—
5	#14556068	C₄	1	2	yes	complemented
6	C₂×C₇₁⋊C₁₆	—	2	1	no	—
7	C₂×C₇₁⋊C₁₆	—	2	1	no	—
8	C₂×C₇₁⋊C₁₆	—	2	1	no	—
9	C₂×C₇₁⋊C₁₆	C₄	1	2	yes	complemented
10	C₂×M₁₆×C₇₁	C₄	1	1	yes	characteristic
11	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	#117×C₇₁	C₄	1	1	yes	characteristic
13	#97×C₇₁	—	2	1	no	—
14	C₄×C₈×C₇₁	—	2	1	no	—
15	C₇₁⋊C₁₆	—	2	2	no	—
16	C₇₁⋊C₁₆	—	2	2	no	—
17	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₇₁	—	2	1	no	—
19	M₁₆×C₇₁	C₈	1	2	yes	—
20	C₂×C₈×C₇₁	—	2	1	no	—
21	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₁	Q₈	1	1	yes	characteristic
27	C₄⋊C₄×C₇₁	C₈	1	2	yes	—
28	C₂²⋊C₄×C₇₁	—	2	1	no	—
29	C₄²×C₇₁	—	2	1	no	—
30	C₂³×C₇₁	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₇₁	—	2	2	no	—
32	C₂×C₄×C₇₁	C₄²	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	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₇₁	M₁₆	1	1	yes	characteristic
38	C₈×C₇₁	—	2	2	no	—
39	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
42	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
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	—	4	2	no	—
48	C₄×C₇₁	#95	1	1	yes	characteristic
49	C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
50	C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
51	C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
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	—	71	1	no	—
60	#405	D₇₁	1	1	yes	characteristic
61	C₂×M₁₆	Dic₇₁	1	1	yes	characteristic
62	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
63	#117	Dic₇₁	1	1	yes	characteristic
64	M₃₂	—	71	2	no	—
65	M₃₂	—	142	1	no	—
66	C₂×C₁₆	—	71	2	no	—
67	C₂×C₁₆	—	142	1	no	—
68	C₂×C₁₆	—	142	1	no	—
69	C₂×C₁₆	—	142	1	no	—
70	#97	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
73	M₁₆	—	2	1	no	—
74	M₁₆	C₇₁⋊C₈	1	2	yes	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	Dic₁₄₂	1	1	yes	characteristic
77	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
78	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
79	C₂×C₈	D₂₈₄	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	C₇₁⋊C₈	1	2	yes	—
83	C₂²⋊C₄	—	2	1	no	—
84	C₄²	—	2	1	no	—
85	C₁₆	—	142	2	no	—
86	C₁₆	—	142	2	no	—
87	C₂³	#11765849	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
90	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
91	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
92	C₂×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
93	C₂×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
94	C₂×C₄	#11765849	1	1	yes	characteristic
95	C₈	—	2	2	no	—
96	C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
97	C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
98	C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
99	C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
100	C₂²	—	2	2	no	—
101	C₂²	#14556039	1	1	yes	characteristic
102	C₂²	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
103	C₂²	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
104	C₄	—	4	2	no	—
105	C₄	C₇₁⋊#95	1	1	yes	characteristic
106	C₄	C₄×C₇₁⋊C₈	1	1	yes	characteristic
107	C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
108	C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	#17985754	1	1	yes	characteristic
112	C₂	#17985783	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556068

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₁₆

C₂

yes

maximal, characteristic

#405×C₇₁

C₂

yes

maximal, characteristic

#14556068

—

#14556068

C₄

yes

complemented

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₄

yes

complemented

C₂×M₁₆×C₇₁

C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

characteristic

#117×C₇₁

C₄

yes

characteristic

#97×C₇₁

—

C₄×C₈×C₇₁

—

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

C₈

yes

—

C₂×C₈×C₇₁

—

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

C₄⋊C₄×C₇₁

C₈

yes

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

M₁₆

yes

characteristic

C₂×C₄×C₇₁

—

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

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

#97

yes

characteristic

C₂²×C₇₁

C₂²⋊C₈

yes

characteristic

C₂²×C₇₁

C₄⋊C₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

#95

yes

characteristic

C₄×C₇₁

C₄×C₈

yes

characteristic

C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₄×C₇₁

C₄⋊C₈

yes

characteristic

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

—

#405

D₇₁

yes

characteristic

C₂×M₁₆

Dic₇₁

yes

characteristic

C₂²×C₈

C₂×D₇₁

yes

characteristic

#117

Dic₇₁

yes

characteristic

M₃₂

—

M₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₂×C₁₆

—

142

—

C₂×C₁₆

—

142

—

#97

—

C₄×C₈

—

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

M₁₆

—

M₁₆

C₇₁⋊C₈

yes

—

C₂×C₈

—

C₂×C₈

Dic₁₄₂

yes

characteristic

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

D₂₈₄

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

C₇₁⋊C₈

yes

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

142

—

C₁₆

—

142

—

C₂³

#11765849

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×Dic₇₁

yes

characteristic

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

C₂×C₇₁⋊C₈

yes

characteristic

C₂×C₄

C₂×C₇₁⋊C₈

yes

characteristic

C₂×C₄

#11765849

yes

characteristic

C₈

—

C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

C₈

Dic₇₁⋊C₄

yes

characteristic

C₈

Dic₇₁⋊C₄

yes

characteristic

100

C₂²

—

101

C₂²

#14556039

yes

characteristic

102

C₂²

C₂²⋊(C₇₁⋊C₈)

yes

characteristic

103

C₂²

C₄⋊(C₇₁⋊C₈)

yes

characteristic

104

C₄

—

105

C₄

C₇₁⋊#95

yes

characteristic

106

C₄

C₄×C₇₁⋊C₈

yes

characteristic

107

C₄

C₂²⋊(C₇₁⋊C₈)

yes

characteristic

108

C₄

C₄⋊(C₇₁⋊C₈)

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

#17985754

yes

characteristic

112

C₂

#17985783

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₇₁
commutator	C₄×C₇₁	C₄×C₈
Frattini	C₂²×C₈	C₂×D₇₁
Fitting	#405×C₇₁	C₂
socle	C₂²×C₇₁	C₂²⋊C₈
hypercenter	#405	D₇₁
hypocenter	C₇₁	#1379
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #405

subgroup

G/H

center

C₂×C₈

C₂²⋊D₇₁

commutator

C₄×C₇₁

C₄×C₈

Frattini

C₂²×C₈

C₂×D₇₁

Fitting

#405×C₇₁

C₂

socle

C₂²×C₇₁

C₂²⋊C₈

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #405

N	H ≅ G/N	type
5. #14556068	C₄	semidirect
9. C₂×C₇₁⋊C₁₆	C₄	semidirect
57. C₇₁	#1379	semidirect

H ≅ G/N

type

5. #14556068

C₄

semidirect

9. C₂×C₇₁⋊C₁₆

C₄

semidirect

57. C₇₁

#1379

semidirect

Subgroups — #22586429

Conjugacy classes of subgroups

Complements