Group #22584255 — subgroups

Subgroups — C₇₁×#1693 #22584255

Order 9088.

166 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 68 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	#500×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#461×C₇₁	C₂	1	2	yes	maximal
3	#436×C₇₁	C₂	1	1	yes	maximal, characteristic
4	#334×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#332×C₇₁	C₂	1	2	yes	maximal, complemented
6	C₂×SD₁₆×C₇₁	—	2	2	no	—
7	C₂×M₁₆×C₇₁	—	2	1	no	—
8	C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
9	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
10	#107×C₇₁	—	2	1	no	—
11	#107×C₇₁	—	2	1	no	—
12	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
14	#103×C₇₁	—	2	2	no	—
15	C₂²⋊C₈×C₇₁	—	2	1	no	—
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₇₁	—	2	2	no	—
18	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₇₁	—	2	1	no	—
20	SD₁₆×C₇₁	—	4	4	no	—
21	M₁₆×C₇₁	—	4	2	no	—
22	C₂×C₈×C₇₁	—	2	1	no	—
23	C₂×C₈×C₇₁	—	2	2	no	—
24	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₇₁	—	2	2	no	—
27	C₄⋊C₄×C₇₁	—	2	2	no	—
28	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
29	C₂²⋊C₄×C₇₁	—	2	1	no	—
30	C₄²×C₇₁	C₂³	1	1	yes	characteristic
31	C₂³×C₇₁	—	2	1	no	—
32	Q₈×C₇₁	—	4	2	no	—
33	Q₈×C₇₁	—	4	2	no	—
34	D₄×C₇₁	—	2	2	no	—
35	C₂×C₄×C₇₁	—	2	1	no	—
36	C₂×C₄×C₇₁	—	4	1	no	—
37	C₂×C₄×C₇₁	—	4	1	no	—
38	C₂×C₄×C₇₁	—	4	2	no	—
39	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
42	C₈×C₇₁	—	2	1	no	—
43	C₈×C₇₁	—	4	1	no	—
44	C₈×C₇₁	—	4	2	no	—
45	C₈×C₇₁	SD₁₆	1	2	yes	complemented
46	C₂²×C₇₁	—	4	1	no	—
47	C₂²×C₇₁	—	4	1	no	—
48	C₂²×C₇₁	—	4	1	no	—
49	C₂²×C₇₁	#121	1	1	yes	characteristic
50	C₄×C₇₁	—	2	1	no	—
51	C₄×C₇₁	—	4	1	no	—
52	C₄×C₇₁	—	8	2	no	—
53	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₇₁	#137	1	1	yes	characteristic
56	C₂×C₇₁	—	8	1	no	—
57	C₂×C₇₁	#461	1	1	yes	characteristic
58	C₂×C₇₁	#469	1	1	yes	characteristic
59	C₂×C₇₁	#473	1	1	yes	characteristic
60	#1693	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₇₁	#1693	1	1	yes	characteristic, direct product factor, Sylow
62	#500	C₂×C₇₁	1	1	yes	characteristic, complemented
63	#461	C₂×C₇₁	1	2	yes	—
64	#436	C₂×C₇₁	1	1	yes	characteristic
65	#334	C₂×C₇₁	1	1	yes	characteristic, complemented
66	#332	C₂×C₇₁	1	2	yes	complemented
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×M₁₆	—	2	1	no	—
69	C₄⋊Q₈	C₂²×C₇₁	1	2	yes	—
70	C₄×D₄	C₂²×C₇₁	1	1	yes	characteristic
71	#107	—	2	1	no	—
72	#107	—	2	1	no	—
73	C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
74	C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
75	#103	—	2	2	no	—
76	C₂²⋊C₈	—	2	1	no	—
77	C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
78	C₂×Q₈	—	2	2	no	—
79	C₂×D₄	D₄×C₇₁	1	1	yes	characteristic
80	C₂²×C₄	—	2	1	no	—
81	SD₁₆	—	4	4	no	—
82	M₁₆	—	4	2	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	D₄×C₇₁	1	1	yes	characteristic
86	C₂×C₈	D₄×C₇₁	1	1	yes	characteristic
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	D₄×C₇₁	1	1	yes	characteristic
90	C₂²⋊C₄	—	2	1	no	—
91	C₄²	C₂³×C₇₁	1	1	yes	characteristic
92	C₂³	—	2	1	no	—
93	Q₈	—	4	2	no	—
94	Q₈	—	4	2	no	—
95	D₄	—	2	2	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
101	C₂×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
102	C₂×C₄	#41×C₇₁	1	1	yes	characteristic
103	C₈	—	2	1	no	—
104	C₈	—	4	1	no	—
105	C₈	—	4	2	no	—
106	C₈	SD₁₆×C₇₁	1	2	yes	complemented
107	C₂²	—	4	1	no	—
108	C₂²	—	4	1	no	—
109	C₂²	—	4	1	no	—
110	C₂²	#121×C₇₁	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	4	1	no	—
113	C₄	—	8	2	no	—
114	C₄	C₂×SD₁₆×C₇₁	1	1	yes	characteristic
115	C₄	AGL(1,ℤ/8)×C₇₁	1	2	yes	—
116	C₄	#137×C₇₁	1	1	yes	characteristic
117	C₂	—	8	1	no	—
118	C₂	#461×C₇₁	1	1	yes	characteristic
119	C₂	#469×C₇₁	1	1	yes	characteristic
120	C₂	#473×C₇₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#500×C₇₁

C₂

yes

maximal, characteristic, complemented

#461×C₇₁

C₂

yes

maximal

#436×C₇₁

C₂

yes

maximal, characteristic

#334×C₇₁

C₂

yes

maximal, characteristic, complemented

#332×C₇₁

C₂

yes

maximal, complemented

C₂×SD₁₆×C₇₁

—

C₂×M₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

C₄×D₄×C₇₁

C₂²

yes

characteristic

#107×C₇₁

—

#107×C₇₁

—

C₄⋊C₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

—

#103×C₇₁

—

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

—

SD₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

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

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

SD₁₆

yes

complemented

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#121

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₄×C₇₁

#137

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₇₁

#1693

yes

characteristic, direct product factor, Sylow

#500

C₂×C₇₁

yes

characteristic, complemented

#461

C₂×C₇₁

yes

—

#436

C₂×C₇₁

yes

characteristic

#334

C₂×C₇₁

yes

characteristic, complemented

#332

C₂×C₇₁

yes

complemented

C₂×SD₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

C₂²×C₇₁

yes

—

C₄×D₄

C₂²×C₇₁

yes

characteristic

#107

—

#107

—

C₄⋊C₈

C₂²×C₇₁

yes

characteristic

C₄⋊C₈

C₂²×C₇₁

yes

—

#103

—

C₂²⋊C₈

—

C₄×C₈

C₂²×C₇₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

D₄×C₇₁

yes

characteristic

C₂²×C₄

—

SD₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₇₁

yes

characteristic

C₂×C₈

D₄×C₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₇₁

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂³×C₇₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂×D₄×C₇₁

yes

characteristic

101

C₂×C₄

C₂×D₄×C₇₁

yes

characteristic

102

C₂×C₄

#41×C₇₁

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

SD₁₆×C₇₁

yes

complemented

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

#121×C₇₁

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

C₂×SD₁₆×C₇₁

yes

characteristic

115

C₄

AGL(1,ℤ/8)×C₇₁

yes

—

116

C₄

#137×C₇₁

yes

characteristic

117

C₂

—

118

C₂

#461×C₇₁

yes

characteristic

119

C₂

#469×C₇₁

yes

characteristic

120

C₂

#473×C₇₁

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₇₁
prime radical	C₂×D₄×C₇₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₄²×C₇₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₇₁

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₇₁

prime radical

C₂×D₄×C₇₁

D₄

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₇₁ ◁ C₄²×C₇₁ ◁ G

N	H ≅ G/N	type
1. #500×C₇₁	C₂	semidirect
4. #334×C₇₁	C₂	semidirect
5. #332×C₇₁	C₂	semidirect
45. C₈×C₇₁	SD₁₆	semidirect
60. #1693	C₇₁	direct (N × H)
61. C₇₁	#1693	direct (N × H)
62. #500	C₂×C₇₁	semidirect
65. #334	C₂×C₇₁	semidirect
66. #332	C₂×C₇₁	semidirect
106. C₈	SD₁₆×C₇₁	semidirect

H ≅ G/N

type

1. #500×C₇₁

C₂

semidirect

4. #334×C₇₁

C₂

semidirect

5. #332×C₇₁

C₂

semidirect

45. C₈×C₇₁

SD₁₆

semidirect

60. #1693

C₇₁

direct (N × H)

61. C₇₁

#1693

direct (N × H)

62. #500

C₂×C₇₁

semidirect

65. #334

C₂×C₇₁

semidirect

66. #332

C₂×C₇₁

semidirect

106. C₈

SD₁₆×C₇₁

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#500×C₇₁	2
#461×C₇₁	2
#436×C₇₁	2
#334×C₇₁	2
#332×C₇₁	2
#1693	71

subgroup

index

#500×C₇₁

#461×C₇₁

#436×C₇₁

#334×C₇₁

#332×C₇₁

#1693

Subgroups — C₇₁×#1693 #22584255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#1693 #22584255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#1693 #22584255