Group #22586196 — subgroups

Subgroups — #22586196

Order 9088.

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

normal subgroups only

This group has 164 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	#17986056	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
5	C₄⋊(C₇₁⋊C₈)	C₄	1	2	yes	complemented
6	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
8	#14556039	—	2	1	no	—
9	#14556039	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
11	C₂×C₄²×C₇₁	C₄	1	2	yes	—
12	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
13	#95×C₇₁	—	2	1	no	—
14	C₂×C₇₁⋊C₈	—	2	2	no	—
15	C₂×C₇₁⋊C₈	—	2	2	no	—
16	C₂²×C₄×C₇₁	—	2	2	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	C₄⋊C₄×C₇₁	—	2	4	no	—
21	C₄²×C₇₁	—	4	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	1	no	—
28	C₂×C₄×C₇₁	—	2	1	no	—
29	C₂×C₄×C₇₁	—	2	2	no	—
30	C₂×C₄×C₇₁	—	2	4	no	—
31	C₂×C₄×C₇₁	—	4	1	no	—
32	C₂×C₄×C₇₁	—	4	2	no	—
33	C₂×C₄×C₇₁	—	4	2	no	—
34	C₂×C₄×C₇₁	C₄²	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
40	C₇₁⋊C₈	—	4	2	no	—
41	C₇₁⋊C₈	—	4	2	no	—
42	C₂²×C₇₁	—	2	1	no	—
43	C₂²×C₇₁	—	2	1	no	—
44	C₂²×C₇₁	#95	1	1	yes	characteristic
45	C₂²×C₇₁	#100	1	1	yes	characteristic
46	C₂²×C₇₁	#101	1	1	yes	characteristic
47	C₄×C₇₁	—	2	1	no	—
48	C₄×C₇₁	—	2	1	no	—
49	C₄×C₇₁	—	2	1	no	—
50	C₄×C₇₁	—	2	1	no	—
51	C₄×C₇₁	—	2	1	no	—
52	C₄×C₇₁	—	4	4	no	—
53	C₄×C₇₁	#104	1	2	yes	—
54	C₂×C₇₁	—	2	1	no	—
55	C₂×C₇₁	—	2	1	no	—
56	C₂×C₇₁	H(3,ℤ/4)	1	1	yes	characteristic
57	C₂×C₇₁	#339	1	1	yes	characteristic
58	C₂×C₇₁	#343	1	1	yes	characteristic
59	#1285	—	71	1	no	maximal, Sylow
60	C₇₁	#1285	1	1	yes	characteristic, complemented, Sylow
61	#432	—	71	1	no	—
62	#432	—	71	1	no	—
63	C₄×C₄⋊C₄	D₇₁	1	1	yes	characteristic
64	C₂×C₄⋊C₄	—	2	1	no	—
65	C₂×C₄²	Dic₇₁	1	2	yes	—
66	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
67	C₄⋊C₈	—	71	2	no	—
68	C₄⋊C₈	—	142	1	no	—
69	C₂²⋊C₈	—	142	1	no	—
70	C₂²⋊C₈	—	142	1	no	—
71	#97	—	71	2	no	—
72	#97	—	142	1	no	—
73	#95	—	2	1	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
76	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
77	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
78	C₂×C₈	—	142	2	no	—
79	C₂×C₈	—	142	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄²	—	4	2	no	—
82	C₄²	Dic₁₄₂	1	1	yes	characteristic
83	C₄²	C₄×D₇₁	1	1	yes	characteristic
84	C₄²	C₄×D₇₁	1	1	yes	characteristic
85	C₄²	D₂₈₄	1	1	yes	characteristic
86	C₂³	C₂²⋊Dic₇₁	1	1	yes	characteristic
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
95	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
96	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
97	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
98	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
99	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
100	C₈	—	284	2	no	—
101	C₈	—	284	2	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	C₇₁⋊#95	1	1	yes	characteristic
105	C₂²	#14556045	1	1	yes	characteristic
106	C₂²	#14556047	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	4	4	no	—
113	C₄	#14556056	1	2	yes	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#17985755	1	1	yes	characteristic
117	C₂	#17985759	1	1	yes	characteristic
118	C₂	#17985768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986056

C₂

yes

maximal, characteristic

#17986056

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄×C₇₁

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₄

yes

complemented

C₂²⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

—

#14556039

—

#14556039

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄²×C₇₁

C₄

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#95×C₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄²×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

—

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

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#104

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#432

—

#432

—

C₄×C₄⋊C₄

D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

Dic₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₂²⋊C₈

—

142

—

C₂²⋊C₈

—

142

—

#97

—

#97

—

142

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄⋊C₄

—

C₄²

—

C₄²

Dic₁₄₂

yes

characteristic

C₄²

C₄×D₇₁

yes

characteristic

C₄²

C₄×D₇₁

yes

characteristic

C₄²

D₂₈₄

yes

characteristic

C₂³

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

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

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

100

C₈

—

284

—

101

C₈

—

284

—

102

C₂²

—

103

C₂²

—

104

C₂²

C₇₁⋊#95

yes

characteristic

105

C₂²

#14556045

yes

characteristic

106

C₂²

#14556047

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#14556056

yes

—

114

C₂

—

115

C₂

—

116

C₂

#17985755

yes

characteristic

117

C₂

#17985759

yes

characteristic

118

C₂

#17985768

yes

characteristic

	subgroup	G/H
center	C₂²	C₇₁⋊#95
commutator	C₂×C₄×C₇₁	C₄²
Frattini	C₂×C₄²	C₂×D₇₁
Fitting	C₄×C₄⋊C₄×C₇₁	C₂
socle	C₂²×C₇₁	#95
hypercenter	C₄×C₄⋊C₄	D₇₁
hypocenter	C₇₁	#1285
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂²

C₇₁⋊#95

commutator

C₂×C₄×C₇₁

C₄²

Frattini

C₂×C₄²

C₂×D₇₁

Fitting

C₄×C₄⋊C₄×C₇₁

C₂

socle

C₂²×C₇₁

#95

hypercenter

C₄×C₄⋊C₄

D₇₁

hypocenter

C₇₁

#1285

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
5. C₄⋊(C₇₁⋊C₈)	C₄	semidirect
9. #14556039	C₄	semidirect
60. C₇₁	#1285	semidirect

H ≅ G/N

type

5. C₄⋊(C₇₁⋊C₈)

C₄

semidirect

9. #14556039

C₄

semidirect

60. C₇₁

#1285

semidirect

Subgroups — #22586196

Conjugacy classes of subgroups

Complements