Group #21465553 — subgroups

Subgroups — C₃₁×#8794 #21465553

Order 7936.

270 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 126 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	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1835	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1286	C₂	1	4	yes	maximal
5	C₃₁×#431	C₂²	1	2	yes	—
6	C₂×C₃₁×#108	—	2	2	no	—
7	C₃₁×#423	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#382	C₂²	1	2	yes	—
10	C₃₁×#340	—	2	2	no	—
11	C₃₁×#322	C₄	1	4	yes	complemented
12	C₂×M₁₆×C₃₁	—	2	2	no	—
13	C₂²×C₈×C₃₁	D₄	1	2	yes	—
14	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
15	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
16	C₃₁×#108	—	4	4	no	—
17	C₃₁×C₄⋊C₈	—	2	2	no	—
18	C₃₁×C₄⋊C₈	—	2	4	no	—
19	C₃₁×C₂²⋊C₈	—	2	2	no	—
20	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
21	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₃₁	Q₈	1	2	yes	—
23	C₃₁×#95	—	2	4	no	—
24	C₂²×C₄×C₃₁	—	4	2	no	—
25	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
26	C₂²×C₄×C₃₁	#41	1	2	yes	—
27	M₁₆×C₃₁	—	4	4	no	—
28	C₂×C₈×C₃₁	—	2	2	no	—
29	C₂×C₈×C₃₁	—	2	2	no	—
30	C₂×C₈×C₃₁	—	2	4	no	—
31	C₂×C₈×C₃₁	—	4	2	no	—
32	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
33	C₄⋊C₄×C₃₁	—	4	2	no	—
34	C₄⋊C₄×C₃₁	—	4	2	no	—
35	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₃₁	#41	1	2	yes	—
38	#8794	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂³×C₃₁	#124	1	1	yes	characteristic
40	C₂×C₄×C₃₁	—	4	2	no	—
41	C₂×C₄×C₃₁	—	8	4	no	—
42	C₂×C₄×C₃₁	—	8	2	no	—
43	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₁	#117	1	2	yes	—
45	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
46	C₂×C₄×C₃₁	#125	1	2	yes	—
47	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
48	C₂×C₄×C₃₁	#137	1	2	yes	—
49	C₈×C₃₁	—	2	4	no	—
50	C₈×C₃₁	—	4	4	no	—
51	C₈×C₃₁	—	8	2	no	—
52	#1848	C₂×C₃₁	1	1	yes	characteristic
53	#1835	C₂×C₃₁	1	1	yes	characteristic
54	#1452	C₂×C₃₁	1	1	yes	characteristic
55	#1286	C₂×C₃₁	1	4	yes	—
56	C₂²×C₃₁	—	2	2	no	—
57	C₂²×C₃₁	#382	1	1	yes	characteristic
58	C₂²×C₃₁	#489	1	2	yes	—
59	C₄×C₃₁	—	2	2	no	—
60	C₄×C₃₁	—	2	2	no	—
61	C₄×C₃₁	—	16	2	no	—
62	C₄×C₃₁	#428	1	2	yes	—
63	C₄×C₃₁	#490	1	1	yes	characteristic
64	C₄×C₃₁	#491	1	1	yes	characteristic
65	#431	C₂²×C₃₁	1	2	yes	—
66	C₂×#108	—	2	2	no	—
67	#423	C₂²×C₃₁	1	2	yes	—
68	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
69	#382	C₂²×C₃₁	1	2	yes	—
70	#340	—	2	2	no	—
71	#322	C₄×C₃₁	1	4	yes	complemented
72	C₂×C₃₁	—	2	2	no	—
73	C₂×C₃₁	#1833	1	2	yes	—
74	C₂×C₃₁	#1836	1	1	yes	characteristic
75	C₂×M₁₆	—	2	2	no	—
76	C₂²×C₈	D₄×C₃₁	1	2	yes	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
79	#108	—	4	4	no	—
80	C₄⋊C₈	—	2	2	no	—
81	C₄⋊C₈	—	2	4	no	—
82	C₂²⋊C₈	—	2	2	no	—
83	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
84	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
85	C₄×C₈	Q₈×C₃₁	1	2	yes	—
86	#95	—	2	4	no	—
87	C₃₁	#8794	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
90	C₂²×C₄	#41×C₃₁	1	2	yes	—
91	M₁₆	—	4	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
100	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
101	C₄²	#41×C₃₁	1	2	yes	—
102	C₂³	C₃₁×#124	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
107	C₂×C₄	C₃₁×#117	1	2	yes	—
108	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
109	C₂×C₄	C₃₁×#125	1	2	yes	—
110	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
111	C₂×C₄	C₃₁×#137	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	4	4	no	—
114	C₈	—	8	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	C₃₁×#382	1	1	yes	characteristic
117	C₂²	C₃₁×#489	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	C₃₁×#428	1	2	yes	—
122	C₄	C₃₁×#490	1	1	yes	characteristic
123	C₄	C₃₁×#491	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	C₃₁×#1833	1	2	yes	—
126	C₂	C₃₁×#1836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#1848

C₂

yes

maximal, characteristic

C₃₁×#1835

C₂

yes

maximal, characteristic

C₃₁×#1452

C₂

yes

maximal, characteristic

C₃₁×#1286

C₂

yes

maximal

C₃₁×#431

C₂²

yes

—

C₂×C₃₁×#108

—

C₃₁×#423

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#382

C₂²

yes

—

C₃₁×#340

—

C₃₁×#322

C₄

yes

complemented

C₂×M₁₆×C₃₁

—

C₂²×C₈×C₃₁

D₄

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×#108

—

C₃₁×C₄⋊C₈

—

C₃₁×C₄⋊C₈

—

C₃₁×C₂²⋊C₈

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

Q₈

yes

—

C₃₁×#95

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₃₁

#41

yes

—

M₁₆×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₄⋊C₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂²×C₄

yes

characteristic

C₄²×C₃₁

C₂×Q₈

yes

characteristic

C₄²×C₃₁

#41

yes

—

#8794

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#124

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₃₁

#117

yes

—

C₂×C₄×C₃₁

#124

yes

characteristic

C₂×C₄×C₃₁

#125

yes

—

C₂×C₄×C₃₁

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₃₁

#137

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

#1848

C₂×C₃₁

yes

characteristic

#1835

C₂×C₃₁

yes

characteristic

#1452

C₂×C₃₁

yes

characteristic

#1286

C₂×C₃₁

yes

—

C₂²×C₃₁

—

C₂²×C₃₁

#382

yes

characteristic

C₂²×C₃₁

#489

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#428

yes

—

C₄×C₃₁

yes

C₄×C₃₁

yes

C₂²×C₃₁

yes

—

C₂×#108

—

#423

C₂²×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

#382

C₂²×C₃₁

yes

—

#340

—

#322

C₄×C₃₁

yes

complemented

C₂×C₃₁

—

C₂×C₃₁

#1833

yes

—

C₂×C₃₁

#1836

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

D₄×C₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³×C₃₁

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₄×C₈

Q₈×C₃₁

yes

—

#95

—

C₃₁

#8794

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₃₁

yes

characteristic

C₂²×C₄

#41×C₃₁

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×C₄×C₃₁

yes

characteristic

100

C₄²

C₂×Q₈×C₃₁

yes

characteristic

101

C₄²

#41×C₃₁

yes

—

102

C₂³

C₃₁×#124

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂×C₄⋊C₄×C₃₁

yes

characteristic

107

C₂×C₄

C₃₁×#117

yes

—

108

C₂×C₄

C₃₁×#124

yes

characteristic

109

C₂×C₄

C₃₁×#125

yes

—

110

C₂×C₄

C₃₁×AGL(1,ℤ/8)

yes

—

111

C₂×C₄

C₃₁×#137

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₂²

—

116

C₂²

C₃₁×#382

yes

characteristic

117

C₂²

C₃₁×#489

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₃₁×#428

yes

—

122

C₄

C₃₁×#490

yes

characteristic

123

C₄

C₃₁×#491

yes

characteristic

124

C₂

—

125

C₂

C₃₁×#1833

yes

—

126

C₂

C₃₁×#1836

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#382
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂³×C₃₁	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₁

#382

commutator

C₄²

C₂²×C₄×C₃₁

Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

C₂³×C₃₁

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
11. C₃₁×#322	C₄	semidirect
38. #8794	C₃₁	direct (N × H)
71. #322	C₄×C₃₁	semidirect
87. C₃₁	#8794	direct (N × H)

H ≅ G/N

type

11. C₃₁×#322

C₄

semidirect

38. #8794

C₃₁

direct (N × H)

71. #322

C₄×C₃₁

semidirect

87. C₃₁

#8794

direct (N × H)

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

subgroup	index
C₃₁×#1848	2
C₃₁×#1835	2
C₃₁×#1452	2
C₃₁×#1286	2
#8794	31

subgroup

index

C₃₁×#1848

C₃₁×#1835

C₃₁×#1452

C₃₁×#1286

#8794

Subgroups — C₃₁×#8794 #21465553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#8794 #21465553

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8794 #21465553