Group #31828028 — subgroups

Subgroups — C₇₁×#7156 #31828028

Order 18176.

234 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 142 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×#346×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#334×C₇₁	C₂	1	2	yes	maximal
3	C₂²×C₁₆×C₇₁	C₄	1	2	yes	—
4	C₂×C₄⋊C₈×C₇₁	—	2	2	no	—
5	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
6	#346×C₇₁	—	2	2	no	—
7	#334×C₇₁	—	2	2	no	—
8	#334×C₇₁	C₄	1	4	yes	—
9	C₂²×C₈×C₇₁	—	4	2	no	—
10	C₂²×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₈×C₇₁	Q₈	1	1	yes	characteristic
12	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₇₁	—	2	2	no	—
14	C₂×C₁₆×C₇₁	—	2	2	no	—
15	C₂×C₁₆×C₇₁	C₈	1	4	yes	complemented
16	C₄⋊C₈×C₇₁	—	2	4	no	—
17	C₄⋊C₈×C₇₁	—	4	2	no	—
18	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₇₁	D₄	1	2	yes	—
20	C₂²×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₇₁	SD₁₆	1	2	yes	—
22	C₂×C₈×C₇₁	—	2	2	no	—
23	C₂×C₈×C₇₁	—	4	4	no	—
24	C₂×C₈×C₇₁	—	8	4	no	—
25	C₂×C₈×C₇₁	C₄²	1	1	yes	characteristic
26	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₇₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₈×C₇₁	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₇₁	M₁₆	1	2	yes	—
30	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
31	C₄²×C₇₁	D₈	1	1	yes	characteristic
32	C₄²×C₇₁	Q₁₆	1	1	yes	characteristic
33	C₁₆×C₇₁	—	2	4	no	—
34	C₂³×C₇₁	#106	1	1	yes	characteristic
35	C₂×C₄×C₇₁	—	2	4	no	—
36	C₂×C₄×C₇₁	#95	1	1	yes	characteristic
37	C₂×C₄×C₇₁	#102	1	2	yes	—
38	C₂×C₄×C₇₁	#103	1	2	yes	—
39	C₂×C₄×C₇₁	C₄⋊C₈	1	2	yes	—
40	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
41	C₂×C₄×C₇₁	#108	1	2	yes	—
42	C₈×C₇₁	—	2	2	no	—
43	C₈×C₇₁	—	8	4	no	—
44	C₈×C₇₁	C₄×C₈	1	1	yes	characteristic
45	C₈×C₇₁	#97	1	1	yes	characteristic
46	C₈×C₇₁	C₂²⋊C₈	1	2	yes	—
47	C₂²×C₇₁	#334	1	2	yes	—
48	C₂²×C₇₁	#340	1	1	yes	characteristic
49	C₂²×C₇₁	#365	1	4	yes	—
50	C₄×C₇₁	—	2	2	no	—
51	C₄×C₇₁	—	2	2	no	—
52	C₄×C₇₁	#335	1	2	yes	—
53	C₄×C₇₁	#336	1	1	yes	characteristic
54	C₄×C₇₁	#341	1	1	yes	characteristic
55	#7156	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂×C₇₁	#1277	1	1	yes	characteristic
57	C₂×C₇₁	#1368	1	4	yes	—
58	C₂×C₇₁	#1383	1	2	yes	—
59	C₂×#346	C₂×C₇₁	1	1	yes	characteristic
60	C₂×#334	C₂×C₇₁	1	2	yes	—
61	C₇₁	#7156	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²×C₁₆	C₄×C₇₁	1	2	yes	—
63	C₂×C₄⋊C₈	—	2	2	no	—
64	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
65	#346	—	2	2	no	—
66	#334	—	2	2	no	—
67	#334	C₄×C₇₁	1	4	yes	—
68	C₂²×C₈	—	4	2	no	—
69	C₂²×C₈	C₂×C₄×C₇₁	1	1	yes	characteristic
70	C₂²×C₈	Q₈×C₇₁	1	1	yes	characteristic
71	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
72	C₂×C₁₆	—	2	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	C₈×C₇₁	1	4	yes	complemented
75	C₄⋊C₈	—	2	4	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₄×C₈	C₂×C₄×C₇₁	1	2	yes	—
78	C₄×C₈	D₄×C₇₁	1	2	yes	—
79	C₂²×C₄	C₄⋊C₄×C₇₁	1	1	yes	characteristic
80	C₂²×C₄	SD₁₆×C₇₁	1	2	yes	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	4	4	no	—
83	C₂×C₈	—	8	4	no	—
84	C₂×C₈	C₄²×C₇₁	1	1	yes	characteristic
85	C₂×C₈	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
86	C₂×C₈	C₄⋊C₄×C₇₁	1	2	yes	—
87	C₂×C₈	C₂×C₈×C₇₁	1	2	yes	—
88	C₂×C₈	M₁₆×C₇₁	1	2	yes	—
89	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
90	C₄²	D₈×C₇₁	1	1	yes	characteristic
91	C₄²	Q₁₆×C₇₁	1	1	yes	characteristic
92	C₁₆	—	2	4	no	—
93	C₂³	#106×C₇₁	1	1	yes	characteristic
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	#95×C₇₁	1	1	yes	characteristic
96	C₂×C₄	#102×C₇₁	1	2	yes	—
97	C₂×C₄	#103×C₇₁	1	2	yes	—
98	C₂×C₄	C₄⋊C₈×C₇₁	1	2	yes	—
99	C₂×C₄	#107×C₇₁	1	1	yes	characteristic
100	C₂×C₄	#108×C₇₁	1	2	yes	—
101	C₈	—	2	2	no	—
102	C₈	—	8	4	no	—
103	C₈	C₄×C₈×C₇₁	1	1	yes	characteristic
104	C₈	#97×C₇₁	1	1	yes	characteristic
105	C₈	C₂²⋊C₈×C₇₁	1	2	yes	—
106	C₂²	#334×C₇₁	1	2	yes	—
107	C₂²	#340×C₇₁	1	1	yes	characteristic
108	C₂²	#365×C₇₁	1	4	yes	—
109	C₄	—	2	2	no	—
110	C₄	—	2	2	no	—
111	C₄	#335×C₇₁	1	2	yes	—
112	C₄	#336×C₇₁	1	1	yes	characteristic
113	C₄	#341×C₇₁	1	1	yes	characteristic
114	C₂	C₇₁×#1277	1	1	yes	characteristic
115	C₂	C₇₁×#1368	1	4	yes	—
116	C₂	C₇₁×#1383	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#346×C₇₁

C₂

yes

maximal, characteristic

C₂×#334×C₇₁

C₂

yes

maximal

C₂²×C₁₆×C₇₁

C₄

yes

—

C₂×C₄⋊C₈×C₇₁

—

C₂×C₄×C₈×C₇₁

C₂²

yes

characteristic

#346×C₇₁

—

#334×C₇₁

—

#334×C₇₁

C₄

yes

—

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₁

Q₈

yes

characteristic

C₂×C₄²×C₇₁

D₄

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₈

yes

complemented

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂×C₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₇₁

SD₁₆

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₄²

yes

characteristic

C₂×C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₁

C₄⋊C₄

yes

—

C₂×C₈×C₇₁

C₂×C₈

yes

—

C₂×C₈×C₇₁

M₁₆

yes

—

C₄²×C₇₁

C₂²⋊C₄

yes

—

C₄²×C₇₁

D₈

yes

characteristic

C₄²×C₇₁

Q₁₆

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

#106

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#95

yes

characteristic

C₂×C₄×C₇₁

#102

yes

—

C₂×C₄×C₇₁

#103

yes

—

C₂×C₄×C₇₁

C₄⋊C₈

yes

—

C₂×C₄×C₇₁

#107

yes

characteristic

C₂×C₄×C₇₁

#108

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

C₄×C₈

yes

characteristic

C₈×C₇₁

#97

yes

characteristic

C₈×C₇₁

C₂²⋊C₈

yes

—

C₂²×C₇₁

#334

yes

—

C₂²×C₇₁

#340

yes

characteristic

C₂²×C₇₁

#365

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#335

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

#1277

yes

characteristic

C₂×C₇₁

#1368

yes

—

C₂×C₇₁

#1383

yes

—

C₂×#346

C₂×C₇₁

yes

characteristic

C₂×#334

C₂×C₇₁

yes

—

C₇₁

#7156

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

C₄×C₇₁

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂²×C₇₁

yes

—

—

C₄×C₇₁

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₇₁

yes

characteristic

C₂²×C₈

Q₈×C₇₁

yes

characteristic

C₂×C₄²

D₄×C₇₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₈×C₇₁

yes

complemented

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₇₁

yes

—

C₄×C₈

D₄×C₇₁

yes

—

C₂²×C₄

C₄⋊C₄×C₇₁

yes

characteristic

C₂²×C₄

SD₁₆×C₇₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₇₁

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₇₁

yes

characteristic

C₂×C₈

C₄⋊C₄×C₇₁

yes

—

C₂×C₈

C₂×C₈×C₇₁

yes

—

C₂×C₈

M₁₆×C₇₁

yes

—

C₄²

C₂²⋊C₄×C₇₁

yes

—

C₄²

D₈×C₇₁

yes

characteristic

C₄²

Q₁₆×C₇₁

yes

characteristic

C₁₆

—

C₂³

#106×C₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

#95×C₇₁

yes

characteristic

C₂×C₄

#102×C₇₁

yes

—

C₂×C₄

#103×C₇₁

yes

—

C₂×C₄

C₄⋊C₈×C₇₁

yes

—

C₂×C₄

#107×C₇₁

yes

characteristic

100

C₂×C₄

#108×C₇₁

yes

—

101

C₈

—

102

C₈

—

103

C₈

C₄×C₈×C₇₁

yes

characteristic

104

C₈

#97×C₇₁

yes

characteristic

105

C₈

C₂²⋊C₈×C₇₁

yes

—

106

C₂²

#334×C₇₁

yes

—

107

C₂²

#340×C₇₁

yes

characteristic

108

C₂²

#365×C₇₁

yes

—

109

C₄

—

110

C₄

—

111

C₄

#335×C₇₁

yes

—

112

C₄

#336×C₇₁

yes

characteristic

113

C₄

#341×C₇₁

yes

characteristic

114

C₂

C₇₁×#1277

yes

characteristic

115

C₂

C₇₁×#1368

yes

—

116

C₂

C₇₁×#1383

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₁	#106
commutator	C₈	C₄×C₈×C₇₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₇₁
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₇₁

#106

commutator

C₈

C₄×C₈×C₇₁

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₇₁

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ C₂×C₄×C₈×C₇₁ ◁ G

N	H ≅ G/N	type
15. C₂×C₁₆×C₇₁	C₈	semidirect
55. #7156	C₇₁	direct (N × H)
61. C₇₁	#7156	direct (N × H)
74. C₂×C₁₆	C₈×C₇₁	semidirect

H ≅ G/N

type

15. C₂×C₁₆×C₇₁

C₈

semidirect

55. #7156

C₇₁

direct (N × H)

61. C₇₁

#7156

direct (N × H)

74. C₂×C₁₆

C₈×C₇₁

semidirect

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

subgroup	index
C₂×#346×C₇₁	2
C₂×#334×C₇₁	2
#7156	71

subgroup

index

C₂×#346×C₇₁

C₂×#334×C₇₁

#7156

Subgroups — C₇₁×#7156 #31828028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#7156 #31828028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#7156 #31828028