Group #22589309 — subgroups

Subgroups — #22589309

Order 9088.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 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×#14556120	C₂	1	1	yes	maximal, characteristic
2	#17985869	C₂	1	1	yes	maximal, characteristic
3	#17985762	C₂	1	4	yes	maximal
4	C₂×#97×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#14556120	—	2	2	no	—
6	#14556120	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₇₁	—	2	2	no	—
8	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
10	C₂×Dic₇₁⋊C₄	—	2	2	no	—
11	C₂²×C₈×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
13	#97×C₇₁	—	2	2	no	—
14	C₂²×Dic₇₁	—	2	2	no	—
15	C₂²×Dic₇₁	—	2	2	no	—
16	C₄⋊Dic₇₁	—	2	2	no	—
17	C₄⋊Dic₇₁	—	2	4	no	—
18	C₄⋊Dic₇₁	—	4	4	no	—
19	C₄⋊Dic₇₁	C₂×C₄	1	4	yes	—
20	Dic₇₁⋊C₄	—	2	8	no	—
21	C₂²×C₄×C₇₁	D₄	1	2	yes	—
22	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₇₁	—	2	2	no	—
24	C₂×C₈×C₇₁	—	2	2	no	—
25	C₂×C₈×C₇₁	Q₈	1	4	yes	—
26	C₄²×C₇₁	C₂×C₄	1	2	yes	—
27	C₄²×C₇₁	D₄	1	2	yes	—
28	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₇₁	—	2	2	no	—
30	C₂×C₄×C₇₁	—	2	2	no	—
31	C₂×C₄×C₇₁	C₂²⋊C₄	1	4	yes	—
32	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₇₁	#41	1	2	yes	—
36	C₂×Dic₇₁	—	2	4	no	—
37	C₂×Dic₇₁	—	4	2	no	—
38	C₂×Dic₇₁	—	4	2	no	—
39	C₂×Dic₇₁	—	4	2	no	—
40	C₂×Dic₇₁	—	4	4	no	—
41	C₈×C₇₁	—	2	4	no	—
42	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₇₁	#124	1	1	yes	characteristic
44	C₂²×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₇₁	#137	1	2	yes	—
47	C₄×C₇₁	—	2	4	no	—
48	C₄×C₇₁	C₄×Q₈	1	2	yes	—
49	C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
50	Dic₇₁	—	4	4	no	—
51	Dic₇₁	—	8	2	no	—
52	C₂×C₇₁	#391	1	1	yes	characteristic
53	C₂×C₇₁	#417	1	2	yes	—
54	C₂×C₇₁	#490	1	1	yes	characteristic
55	C₂×C₇₁	#491	1	1	yes	characteristic
56	C₂×C₇₁	#501	1	2	yes	—
57	#1992	—	71	1	no	maximal, Sylow
58	C₇₁	#1992	1	1	yes	characteristic, complemented, Sylow
59	C₂×#125	—	71	1	no	—
60	C₂×#97	D₇₁	1	1	yes	characteristic
61	#384	—	71	1	no	—
62	#340	—	71	4	no	—
63	C₂²×C₈	C₂×D₇₁	1	2	yes	—
64	#125	—	71	4	no	—
65	#125	—	142	2	no	—
66	C₂×C₄⋊C₄	—	71	2	no	—
67	C₂×C₄⋊C₄	—	71	2	no	—
68	C₂×C₄⋊C₄	—	142	2	no	—
69	C₂×C₄⋊C₄	—	142	2	no	—
70	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
71	#97	—	2	2	no	—
72	C₂²×C₄	—	142	2	no	—
73	C₂²×C₄	—	142	2	no	—
74	C₂²×C₄	D₂₈₄	1	2	yes	—
75	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	Dic₁₄₂	1	4	yes	—
79	C₄⋊C₄	—	284	4	no	—
80	C₄⋊C₄	—	71	4	no	—
81	C₄⋊C₄	—	142	4	no	—
82	C₄⋊C₄	—	142	8	no	—
83	C₄⋊C₄	—	142	2	no	—
84	C₄²	C₄×D₇₁	1	2	yes	—
85	C₄²	C₂²⋊D₇₁	1	2	yes	—
86	C₂³	C₂×D₂₈₄	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	284	2	no	—
90	C₂×C₄	—	284	2	no	—
91	C₂×C₄	—	284	2	no	—
92	C₂×C₄	—	284	4	no	—
93	C₂×C₄	—	142	4	no	—
94	C₂×C₄	(C₂×D₇₁)⋊C₄	1	4	yes	—
95	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
97	C₂×C₄	C₂×Dic₁₄₂	1	2	yes	—
98	C₂×C₄	#11765865	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₂²	C₂×(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
101	C₂²	#14556116	1	1	yes	characteristic
102	C₂²	#14556128	1	1	yes	characteristic
103	C₂²	#14556156	1	2	yes	—
104	C₂²	#14556161	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	284	4	no	—
107	C₄	—	568	2	no	—
108	C₄	C₄×Dic₁₄₂	1	2	yes	—
109	C₄	#14556107	1	2	yes	—
110	C₂	#17985905	1	1	yes	characteristic
111	C₂	#17986001	1	2	yes	—
112	C₂	#17986364	1	1	yes	characteristic
113	C₂	#17986368	1	1	yes	characteristic
114	C₂	#17986408	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556120

C₂

yes

maximal, characteristic

#17985869

C₂

yes

maximal, characteristic

#17985762

C₂

yes

maximal

C₂×#97×C₇₁

C₂

yes

maximal, characteristic

#14556120

—

#14556120

C₄

yes

complemented

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×Dic₇₁⋊C₄

—

C₂²×C₈×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#97×C₇₁

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂×C₄

yes

—

Dic₇₁⋊C₄

—

C₂²×C₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

Q₈

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

D₄

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

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

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₁

#124

yes

characteristic

C₂²×C₇₁

C₄⋊Q₈

yes

characteristic

C₂²×C₇₁

AGL(1,ℤ/8)

yes

—

C₂²×C₇₁

#137

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₄×Q₈

yes

—

C₄×C₇₁

C₂²⋊Q₈

yes

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#391

yes

characteristic

C₂×C₇₁

#417

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

#1992

—

maximal, Sylow

C₇₁

#1992

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×#97

D₇₁

yes

characteristic

#384

—

#340

—

C₂²×C₈

C₂×D₇₁

yes

—

#125

—

#125

—

142

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#97

—

C₂²×C₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

D₂₈₄

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₄₂

yes

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

C₄×D₇₁

yes

—

C₄²

C₂²⋊D₇₁

yes

—

C₂³

C₂×D₂₈₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

142

—

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₂×C₄

C₂×Dic₁₄₂

yes

—

C₂×C₄

#11765865

yes

—

C₈

—

100

C₂²

C₂×(C₂×D₇₁)⋊C₄

yes

characteristic

101

C₂²

#14556116

yes

characteristic

102

C₂²

#14556128

yes

characteristic

103

C₂²

#14556156

yes

—

104

C₂²

#14556161

yes

—

105

C₄

—

106

C₄

—

284

—

107

C₄

—

568

—

108

C₄

C₄×Dic₁₄₂

yes

—

109

C₄

#14556107

yes

—

110

C₂

#17985905

yes

characteristic

111

C₂

#17986001

yes

—

112

C₂

#17986364

yes

characteristic

113

C₂

#17986368

yes

characteristic

114

C₂

#17986408

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#97×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×#97	D₇₁
hypocenter	C₇₁	#1992
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#97×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

hypercenter

C₂×#97

D₇₁

hypocenter

C₇₁

#1992

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
6. #14556120	C₄	semidirect
58. C₇₁	#1992	semidirect

H ≅ G/N

type

6. #14556120

C₄

semidirect

58. C₇₁

#1992

semidirect

Subgroups — #22589309

Conjugacy classes of subgroups

Complements