Group #31828704 — subgroups

Subgroups — C₇₁×#7832 #31828704

Order 18176.

322 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₇₁×#2810	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2100	C₂	1	3	yes	maximal
3	C₇₁×#1271	C₂	1	3	yes	maximal
4	#400×C₇₁	—	2	3	no	—
5	#389×C₇₁	C₂²	1	3	yes	—
6	#387×C₇₁	—	2	3	no	—
7	#384×C₇₁	C₂²	1	1	yes	characteristic
8	#340×C₇₁	—	2	3	no	—
9	#340×C₇₁	—	2	3	no	—
10	#336×C₇₁	C₂²	1	3	yes	—
11	C₂²×C₈×C₇₁	—	2	3	no	—
12	C₂×C₄⋊C₄×C₇₁	—	2	3	no	—
13	C₂×C₄⋊C₄×C₇₁	—	2	3	no	—
14	C₂×C₄⋊C₄×C₇₁	D₄	1	3	yes	—
15	C₂×C₄⋊C₄×C₇₁	D₄	1	3	yes	—
16	C₂×C₄²×C₇₁	—	2	1	no	—
17	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
18	#95×C₇₁	—	2	3	no	—
19	#95×C₇₁	—	2	3	no	—
20	#95×C₇₁	—	2	1	no	—
21	#95×C₇₁	—	2	3	no	—
22	C₂²×C₄×C₇₁	—	2	3	no	—
23	C₂²×C₄×C₇₁	—	2	3	no	—
24	C₂²×C₄×C₇₁	—	4	1	no	—
25	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
26	C₂×C₈×C₇₁	—	4	3	no	—
27	C₂×C₈×C₇₁	—	4	3	no	—
28	C₂×C₈×C₇₁	—	4	3	no	—
29	C₄⋊C₄×C₇₁	—	2	3	no	—
30	C₄⋊C₄×C₇₁	—	2	3	no	—
31	C₄⋊C₄×C₇₁	—	4	3	no	—
32	C₄⋊C₄×C₇₁	—	4	6	no	—
33	C₄⋊C₄×C₇₁	—	4	3	no	—
34	C₄⋊C₄×C₇₁	—	4	3	no	—
35	C₄²×C₇₁	—	4	1	no	—
36	C₄²×C₇₁	—	8	1	no	—
37	C₂³×C₇₁	#120	1	1	yes	characteristic
38	C₂×C₄×C₇₁	—	2	3	no	—
39	C₂×C₄×C₇₁	—	2	3	no	—
40	C₂×C₄×C₇₁	—	4	3	no	—
41	C₂×C₄×C₇₁	—	4	3	no	—
42	C₂×C₄×C₇₁	—	4	3	no	—
43	C₂×C₄×C₇₁	—	4	3	no	—
44	C₂×C₄×C₇₁	—	4	3	no	—
45	C₂×C₄×C₇₁	—	4	3	no	—
46	C₂×C₄×C₇₁	—	8	3	no	—
47	C₂×C₄×C₇₁	#135	1	3	yes	—
48	C₂×C₄×C₇₁	#137	1	3	yes	—
49	C₈×C₇₁	—	8	3	no	—
50	C₂²×C₇₁	#452	1	3	yes	—
51	C₂²×C₇₁	#454	1	3	yes	—
52	C₂²×C₇₁	#458	1	1	yes	characteristic
53	C₄×C₇₁	—	2	3	no	—
54	C₄×C₇₁	—	2	3	no	—
55	C₄×C₇₁	—	8	3	no	—
56	C₄×C₇₁	—	8	3	no	—
57	C₄×C₇₁	—	16	1	no	—
58	#7832	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×C₇₁	#1615	1	3	yes	—
60	C₂×C₇₁	#1638	1	3	yes	—
61	C₂×C₇₁	#1658	1	1	yes	characteristic
62	#2810	C₂×C₇₁	1	1	yes	characteristic
63	#2100	C₂×C₇₁	1	3	yes	—
64	#1271	C₂×C₇₁	1	3	yes	—
65	C₇₁	#7832	1	1	yes	characteristic, direct product factor, Sylow
66	#400	—	2	3	no	—
67	#389	C₂²×C₇₁	1	3	yes	—
68	#387	—	2	3	no	—
69	#384	C₂²×C₇₁	1	1	yes	characteristic
70	#340	—	2	3	no	—
71	#340	—	2	3	no	—
72	#336	C₂²×C₇₁	1	3	yes	—
73	C₂²×C₈	—	2	3	no	—
74	C₂×C₄⋊C₄	—	2	3	no	—
75	C₂×C₄⋊C₄	—	2	3	no	—
76	C₂×C₄⋊C₄	D₄×C₇₁	1	3	yes	—
77	C₂×C₄⋊C₄	D₄×C₇₁	1	3	yes	—
78	C₂×C₄²	—	2	1	no	—
79	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
80	#95	—	2	3	no	—
81	#95	—	2	3	no	—
82	#95	—	2	1	no	—
83	#95	—	2	3	no	—
84	C₂²×C₄	—	2	3	no	—
85	C₂²×C₄	—	2	3	no	—
86	C₂²×C₄	—	4	1	no	—
87	C₂²×C₄	C₂×D₄×C₇₁	1	3	yes	—
88	C₂×C₈	—	4	3	no	—
89	C₂×C₈	—	4	3	no	—
90	C₂×C₈	—	4	3	no	—
91	C₄⋊C₄	—	2	3	no	—
92	C₄⋊C₄	—	2	3	no	—
93	C₄⋊C₄	—	4	3	no	—
94	C₄⋊C₄	—	4	6	no	—
95	C₄⋊C₄	—	4	3	no	—
96	C₄⋊C₄	—	4	3	no	—
97	C₄²	—	4	1	no	—
98	C₄²	—	8	1	no	—
99	C₂³	#120×C₇₁	1	1	yes	characteristic
100	C₂×C₄	—	2	3	no	—
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	4	3	no	—
103	C₂×C₄	—	4	3	no	—
104	C₂×C₄	—	4	3	no	—
105	C₂×C₄	—	4	3	no	—
106	C₂×C₄	—	4	3	no	—
107	C₂×C₄	—	4	3	no	—
108	C₂×C₄	—	8	3	no	—
109	C₂×C₄	#135×C₇₁	1	3	yes	—
110	C₂×C₄	#137×C₇₁	1	3	yes	—
111	C₈	—	8	3	no	—
112	C₂²	#452×C₇₁	1	3	yes	—
113	C₂²	#454×C₇₁	1	3	yes	—
114	C₂²	#458×C₇₁	1	1	yes	characteristic
115	C₄	—	2	3	no	—
116	C₄	—	2	3	no	—
117	C₄	—	8	3	no	—
118	C₄	—	8	3	no	—
119	C₄	—	16	1	no	—
120	C₂	C₇₁×#1615	1	3	yes	—
121	C₂	C₇₁×#1638	1	3	yes	—
122	C₂	C₇₁×#1658	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₁×#2810

C₂

yes

maximal, characteristic

C₇₁×#2100

C₂

yes

maximal

C₇₁×#1271

C₂

yes

maximal

#400×C₇₁

—

#389×C₇₁

C₂²

yes

—

#387×C₇₁

—

#384×C₇₁

C₂²

yes

characteristic

#340×C₇₁

—

#340×C₇₁

—

#336×C₇₁

C₂²

yes

—

C₂²×C₈×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

D₄

yes

—

C₂×C₄⋊C₄×C₇₁

D₄

yes

—

C₂×C₄²×C₇₁

—

C₂×C₄²×C₇₁

C₂³

yes

characteristic

#95×C₇₁

—

#95×C₇₁

—

#95×C₇₁

—

#95×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

#120

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

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#135

yes

—

C₂×C₄×C₇₁

#137

yes

—

C₈×C₇₁

—

C₂²×C₇₁

#452

yes

—

C₂²×C₇₁

#454

yes

—

C₂²×C₇₁

#458

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

#7832

C₇₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇₁

#1615

yes

—

C₂×C₇₁

#1638

yes

—

C₂×C₇₁

#1658

yes

characteristic

#2810

C₂×C₇₁

yes

characteristic

#2100

C₂×C₇₁

yes

—

#1271

C₂×C₇₁

yes

—

C₇₁

#7832

yes

characteristic, direct product factor, Sylow

#400

—

#389

C₂²×C₇₁

yes

—

#387

—

#384

C₂²×C₇₁

yes

—

—

C₂²×C₇₁

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₇₁

yes

—

C₂×C₄⋊C₄

D₄×C₇₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³×C₇₁

yes

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₇₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂³

#120×C₇₁

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#135×C₇₁

yes

—

110

C₂×C₄

#137×C₇₁

yes

—

111

C₈

—

112

C₂²

#452×C₇₁

yes

—

113

C₂²

#454×C₇₁

yes

—

114

C₂²

#458×C₇₁

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₂

C₇₁×#1615

yes

—

121

C₂

C₇₁×#1638

yes

—

122

C₂

C₇₁×#1658

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇₁	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₇₁

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₇₁

LCS

G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

N	H ≅ G/N	type
58. #7832	C₇₁	direct (N × H)
65. C₇₁	#7832	direct (N × H)

H ≅ G/N

type

58. #7832

C₇₁

direct (N × H)

65. C₇₁

#7832

direct (N × H)

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

subgroup	index
C₇₁×#2810	2
C₇₁×#2100	2
C₇₁×#1271	2
#7832	71

subgroup

index

C₇₁×#2810

C₇₁×#2100

C₇₁×#1271

#7832

Subgroups — C₇₁×#7832 #31828704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C71×#7832 #31828704

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#7832 #31828704