Group #22587125 — subgroups

Subgroups — #22587125

Order 9088.

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

normal subgroups only

This group has 160 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	#17986056	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#17985752	C₂	1	2	yes	maximal, complemented
4	#17985750	C₂	1	2	yes	maximal, complemented
5	#431×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇₁⋊C₈	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
10	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
11	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
12	#14556039	—	2	1	no	—
13	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₇₁	C₄	1	2	yes	—
15	C₂²⋊C₈×C₇₁	C₄	1	2	yes	—
16	C₄×C₈×C₇₁	C₂²	1	2	yes	—
17	C₂×C₇₁⋊C₈	—	2	1	no	—
18	C₂×C₇₁⋊C₈	—	2	1	no	—
19	C₂×C₇₁⋊C₈	—	2	2	no	—
20	C₂×C₇₁⋊C₈	—	4	2	no	—
21	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
24	C₂×C₈×C₇₁	C₂×C₄	1	4	yes	—
25	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₇₁	D₄	1	1	yes	characteristic
27	C₄²×C₇₁	Q₈	1	1	yes	characteristic
28	C₄²×C₇₁	C₂³	1	1	yes	characteristic
29	C₂³×C₇₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₁	—	2	2	no	—
31	C₂×C₄×C₇₁	—	2	2	no	—
32	C₂×C₄×C₇₁	—	2	1	no	—
33	C₂×C₄×C₇₁	—	2	1	no	—
34	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₇₁	—	2	4	no	—
41	C₇₁⋊C₈	—	4	1	no	—
42	C₇₁⋊C₈	—	4	2	no	—
43	C₇₁⋊C₈	—	4	1	no	—
44	C₂²×C₇₁	—	2	1	no	—
45	C₂²×C₇₁	—	2	1	no	—
46	C₂²×C₇₁	#108	1	2	yes	—
47	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
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	#131	1	2	yes	—
53	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₄×C₇₁	#137	1	1	yes	characteristic
55	C₂×C₇₁	—	2	2	no	—
56	C₂×C₇₁	#424	1	1	yes	characteristic
57	C₂×C₇₁	#428	1	1	yes	characteristic
58	C₂×C₇₁	C₂×#108	1	1	yes	characteristic
59	#1573	—	71	1	no	maximal, Sylow
60	C₇₁	#1573	1	1	yes	characteristic, complemented, Sylow
61	#432	—	71	1	no	—
62	#431	D₇₁	1	1	yes	characteristic
63	C₂×C₄⋊C₈	—	71	1	no	—
64	#335	—	71	2	no	—
65	#334	—	71	2	no	—
66	C₂²×C₈	—	142	1	no	—
67	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
68	C₄⋊C₈	—	71	1	no	—
69	C₄⋊C₈	—	71	2	no	—
70	C₄⋊C₈	—	71	1	no	—
71	C₄⋊C₈	—	142	1	no	—
72	C₄⋊C₈	Dic₇₁	1	2	yes	—
73	C₂²⋊C₈	—	142	1	no	—
74	C₂²⋊C₈	Dic₇₁	1	2	yes	—
75	#97	—	142	1	no	—
76	C₄×C₈	C₂×D₇₁	1	2	yes	—
77	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
78	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
79	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
80	C₂×C₈	—	284	2	no	—
81	C₂×C₈	—	142	1	no	—
82	C₂×C₈	—	142	1	no	—
83	C₂×C₈	—	142	2	no	—
84	C₂×C₈	C₂×Dic₇₁	1	4	yes	—
85	C₄²	Dic₁₄₂	1	1	yes	characteristic
86	C₄²	D₂₈₄	1	1	yes	characteristic
87	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
88	C₄²	C₂²×D₇₁	1	1	yes	characteristic
89	C₂³	C₄⋊Dic₇₁	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
95	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
97	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
98	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
100	C₈	—	2	4	no	—
101	C₈	—	284	1	no	—
102	C₈	—	284	2	no	—
103	C₈	—	284	1	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	#14556064	1	2	yes	—
107	C₂²	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	#14556135	1	2	yes	—
113	C₄	#14556156	1	1	yes	characteristic
114	C₄	#14556161	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	#17986026	1	1	yes	characteristic
117	C₂	#17986041	1	1	yes	characteristic
118	C₂	C₂×#14556064	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986056

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₇₁⋊C₈)

C₂

yes

maximal, characteristic

#17985752

C₂

yes

maximal, complemented

#17985750

C₂

yes

maximal, complemented

#431×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₇₁⋊C₈)

—

#14556039

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₄

yes

—

C₂²⋊C₈×C₇₁

C₄

yes

—

C₄×C₈×C₇₁

C₂²

yes

—

C₂×C₇₁⋊C₈

—

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

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

Q₈

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#108

yes

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#131

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#432

—

#431

D₇₁

yes

characteristic

C₂×C₄⋊C₈

—

#335

—

#334

—

C₂²×C₈

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

Dic₇₁

yes

—

C₂²⋊C₈

—

142

—

C₂²⋊C₈

Dic₇₁

yes

—

#97

—

142

—

C₄×C₈

C₂×D₇₁

yes

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂×C₈

—

284

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₄²

Dic₁₄₂

yes

characteristic

C₄²

D₂₈₄

yes

characteristic

C₄²

C₂×Dic₇₁

yes

characteristic

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

C₂×C₄

C₂×D₂₈₄

yes

characteristic

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

100

C₈

—

101

C₈

—

284

—

102

C₈

—

284

—

103

C₈

—

284

—

104

C₂²

—

105

C₂²

—

106

C₂²

#14556064

yes

—

107

C₂²

C₂×C₄⋊Dic₇₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

#14556135

yes

—

113

C₄

#14556156

yes

characteristic

114

C₄

#14556161

yes

characteristic

115

C₂

—

116

C₂

#17986026

yes

characteristic

117

C₂

#17986041

yes

characteristic

118

C₂

C₂×#14556064

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	#431×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₄
hypercenter	#431	D₇₁
hypocenter	C₇₁	#1573
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

#431×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #17985752	C₂	semidirect
4. #17985750	C₂	semidirect
60. C₇₁	#1573	semidirect

H ≅ G/N

type

3. #17985752

C₂

semidirect

4. #17985750

C₂

semidirect

60. C₇₁

#1573

semidirect

Subgroups — #22587125

Conjugacy classes of subgroups

Complements