Group #22587739 — subgroups

Subgroups — #22587739

Order 9088.

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

normal subgroups only

This group has 160 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	#17986404	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986289	C₂	1	2	yes	maximal
3	#17985752	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985747	C₂	1	2	yes	maximal, complemented
5	C₈×D₄×C₇₁	C₂	1	1	yes	maximal, characteristic
6	#14556128	C₂²	1	2	yes	—
7	#14556062	—	2	1	no	—
8	#14556062	—	2	1	no	—
9	#14556059	—	2	2	no	—
10	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
11	#14556049	—	2	2	no	—
12	C₂²×C₈×C₇₁	—	2	1	no	—
13	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₇₁	—	2	1	no	—
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₄₂	—	2	2	no	—
18	C₂×C₇₁⋊C₈	—	2	2	no	—
19	C₄⋊Dic₇₁	—	2	2	no	—
20	C₄⋊Dic₇₁	—	2	2	no	—
21	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	—	2	1	no	—
23	C₂×C₈×C₇₁	—	2	1	no	—
24	C₂×C₈×C₇₁	—	4	2	no	—
25	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
28	C₂²⋊C₄×C₇₁	—	2	1	no	—
29	C₄²×C₇₁	C₂³	1	1	yes	characteristic
30	C₂³×C₇₁	—	2	1	no	—
31	D₄×C₇₁	—	2	1	no	—
32	D₄×C₇₁	Q₁₆	1	2	yes	—
33	C₂×C₄×C₇₁	—	2	1	no	—
34	C₂×C₄×C₇₁	—	4	1	no	—
35	C₂×C₄×C₇₁	—	4	1	no	—
36	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
39	C₂×Dic₇₁	—	4	2	no	—
40	Dic₁₄₂	—	8	2	no	—
41	C₈×C₇₁	—	2	1	no	—
42	C₈×C₇₁	—	4	1	no	—
43	C₈×C₇₁	SD₁₆	1	2	yes	—
44	C₇₁⋊C₈	—	4	2	no	—
45	C₂²×C₇₁	—	2	2	no	—
46	C₂²×C₇₁	—	4	1	no	—
47	C₂²×C₇₁	—	4	1	no	—
48	C₂²×C₇₁	#121	1	1	yes	characteristic
49	C₄×C₇₁	—	2	1	no	—
50	C₄×C₇₁	—	4	1	no	—
51	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
52	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
53	C₄×C₇₁	#135	1	1	yes	characteristic
54	C₄×C₇₁	#137	1	1	yes	characteristic
55	Dic₇₁	—	8	2	no	—
56	C₂×C₇₁	—	4	2	no	—
57	C₂×C₇₁	#461	1	1	yes	characteristic
58	C₂×C₇₁	#467	1	1	yes	characteristic
59	C₂×C₇₁	#472	1	1	yes	characteristic
60	#1675	—	71	1	no	maximal, Sylow
61	C₇₁	#1675	1	1	yes	characteristic, complemented, Sylow
62	#500	—	71	1	no	—
63	#476	—	71	2	no	—
64	C₈×D₄	D₇₁	1	1	yes	characteristic
65	#335	—	71	1	no	—
66	#333	—	71	2	no	—
67	C₂²×C₈	—	2	1	no	—
68	C₄⋊Q₈	—	71	2	no	—
69	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
70	#107	—	142	1	no	—
71	#107	—	142	1	no	—
72	#106	—	142	2	no	—
73	C₄⋊C₈	—	71	2	no	—
74	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
75	#102	—	142	2	no	—
76	C₂²⋊C₈	—	2	1	no	—
77	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
78	C₂×Q₈	—	142	2	no	—
79	C₂×D₄	D₂₈₄	1	1	yes	characteristic
80	C₂²×C₄	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	4	2	no	—
83	C₂×C₈	—	142	2	no	—
84	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
85	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
86	C₄⋊C₄	—	142	2	no	—
87	C₄⋊C₄	—	142	2	no	—
88	C₄⋊C₄	D₂₈₄	1	1	yes	characteristic
89	C₂²⋊C₄	—	2	1	no	—
90	C₄²	C₂²×D₇₁	1	1	yes	characteristic
91	C₂³	—	2	1	no	—
92	Q₈	—	568	2	no	—
93	D₄	—	2	1	no	—
94	D₄	Dic₂₈₄	1	2	yes	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	284	2	no	—
99	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
100	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
101	C₂×C₄	#11765865	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	4	1	no	—
104	C₈	—	284	2	no	—
105	C₈	#11765853	1	2	yes	—
106	C₂²	—	2	2	no	—
107	C₂²	—	4	1	no	—
108	C₂²	—	4	1	no	—
109	C₂²	#14556102	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	4	1	no	—
112	C₄	—	568	2	no	—
113	C₄	C₂×#11765853	1	1	yes	characteristic
114	C₄	C₂×Dic₂₈₄	1	1	yes	characteristic
115	C₄	#14556150	1	1	yes	characteristic
116	C₄	#14556158	1	1	yes	characteristic
117	C₂	—	4	2	no	—
118	C₂	#17986204	1	1	yes	characteristic
119	C₂	#17986240	1	1	yes	characteristic
120	C₂	#17986269	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986404

C₂

yes

maximal, characteristic, complemented

#17986289

C₂

yes

maximal

#17985752

C₂

yes

maximal, characteristic, complemented

#17985747

C₂

yes

maximal, complemented

C₈×D₄×C₇₁

C₂

yes

maximal, characteristic

#14556128

C₂²

yes

—

#14556062

—

#14556062

—

#14556059

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556049

—

C₂²×C₈×C₇₁

—

C₄×D₄×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

Q₁₆

yes

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

SD₁₆

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#121

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

C₂×Q₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#500

—

#476

—

C₈×D₄

D₇₁

yes

characteristic

#335

—

#333

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄×D₄

C₂×D₇₁

yes

characteristic

#107

—

142

—

#107

—

142

—

#106

—

142

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₁

yes

characteristic

#102

—

142

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

142

—

C₂×D₄

D₂₈₄

yes

characteristic

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

D₂₈₄

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

—

Q₈

—

568

—

D₄

—

D₄

Dic₂₈₄

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

100

C₂×C₄

C₂×D₂₈₄

yes

characteristic

101

C₂×C₄

#11765865

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

284

—

105

C₈

#11765853

yes

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#14556102

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

568

—

113

C₄

C₂×#11765853

yes

characteristic

114

C₄

C₂×Dic₂₈₄

yes

characteristic

115

C₄

#14556150

yes

characteristic

116

C₄

#14556158

yes

characteristic

117

C₂

—

118

C₂

#17986204

yes

characteristic

119

C₂

#17986240

yes

characteristic

120

C₂

#17986269

yes

characteristic

	subgroup	G/H
center	C₂²	#14556102
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	C₈×D₄×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	C₈×D₄	D₇₁
hypocenter	C₇₁	#1675
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

subgroup

G/H

center

C₂²

#14556102

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

C₈×D₄×C₇₁

C₂

socle

C₂²×C₇₁

#121

hypercenter

C₈×D₄

D₇₁

hypocenter

C₇₁

#1675

LCS

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

N	H ≅ G/N	type
1. #17986404	C₂	semidirect
3. #17985752	C₂	semidirect
4. #17985747	C₂	semidirect
61. C₇₁	#1675	semidirect

H ≅ G/N

type

1. #17986404

C₂

semidirect

3. #17985752

C₂

semidirect

4. #17985747

C₂

semidirect

61. C₇₁

#1675

semidirect

Subgroups — #22587739

Conjugacy classes of subgroups

Complements