Group #21467119 — subgroups

Subgroups — C₃₁×#10360 #21467119

Order 7936.

476 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2845	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#1961	C₂	1	3	yes	maximal
3	C₃₁×#1295	C₂	1	3	yes	maximal, complemented
4	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
5	C₂×C₃₁×#102	—	2	6	no	—
6	C₂×C₃₁×#97	C₂²	1	3	yes	—
7	C₃₁×#398	—	2	6	no	—
8	C₃₁×#382	C₂²	1	3	yes	—
9	C₂²×D₄×C₃₁	—	2	3	no	—
10	C₂²×C₈×C₃₁	—	2	3	no	—
11	C₃₁×#127	—	2	3	no	—
12	C₃₁×#127	Q₈	1	2	yes	—
13	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
14	C₂×C₄⋊C₄×C₃₁	D₄	1	6	yes	—
15	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
16	C₃₁×#102	—	4	12	no	—
17	C₃₁×#97	—	2	6	no	—
18	C₃₁×#95	—	2	2	no	—
19	C₃₁×#95	—	2	6	no	—
20	C₂⁴×C₃₁	—	4	1	no	—
21	C₂×D₄×C₃₁	—	2	6	no	—
22	C₂×D₄×C₃₁	—	4	6	no	—
23	C₂×D₄×C₃₁	—	4	6	no	—
24	C₂×D₄×C₃₁	—	4	3	no	—
25	C₂²×C₄×C₃₁	—	2	6	no	—
26	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
27	C₂×C₈×C₃₁	—	2	6	no	—
28	C₂×C₈×C₃₁	—	4	3	no	—
29	C₂×C₈×C₃₁	—	4	3	no	—
30	C₄⋊C₄×C₃₁	—	4	6	no	—
31	C₄⋊C₄×C₃₁	—	8	6	no	—
32	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₃₁	#41	1	3	yes	—
34	#10360	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂³×C₃₁	—	4	2	no	—
36	C₂³×C₃₁	—	8	3	no	—
37	C₂³×C₃₁	—	8	3	no	—
38	C₂³×C₃₁	#120	1	1	yes	characteristic
39	D₄×C₃₁	—	4	12	no	—
40	D₄×C₃₁	—	8	6	no	—
41	C₂×C₄×C₃₁	—	2	3	no	—
42	C₂×C₄×C₃₁	—	2	3	no	—
43	C₂×C₄×C₃₁	—	4	6	no	—
44	C₂×C₄×C₃₁	—	4	6	no	—
45	C₂×C₄×C₃₁	—	4	6	no	—
46	C₂×C₄×C₃₁	C₂²⋊Q₈	1	3	yes	—
47	C₂×C₄×C₃₁	#124	1	3	yes	—
48	C₈×C₃₁	—	4	6	no	—
49	#2845	C₂×C₃₁	1	1	yes	characteristic, complemented
50	#1961	C₂×C₃₁	1	3	yes	—
51	#1295	C₂×C₃₁	1	3	yes	complemented
52	C₂²×C₃₁	—	8	6	no	—
53	C₂²×C₃₁	—	16	3	no	—
54	C₂²×C₃₁	—	16	1	no	—
55	C₂²×C₃₁	#393	1	1	yes	characteristic
56	C₂²×C₃₁	#453	1	3	yes	—
57	C₂²×C₃₁	#454	1	3	yes	—
58	C₄×C₃₁	—	2	3	no	—
59	C₄×C₃₁	—	2	3	no	—
60	C₄×C₃₁	—	8	6	no	—
61	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
62	C₂×#102	—	2	6	no	—
63	C₂×#97	C₂²×C₃₁	1	3	yes	—
64	#398	—	2	6	no	—
65	#382	C₂²×C₃₁	1	3	yes	—
66	C₂×C₃₁	—	16	2	no	—
67	C₂×C₃₁	#1656	1	3	yes	—
68	C₂×C₃₁	#1658	1	1	yes	characteristic
69	C₂×C₃₁	#2027	1	3	yes	—
70	C₂²×D₄	—	2	3	no	—
71	C₂²×C₈	—	2	3	no	—
72	#127	—	2	3	no	—
73	#127	Q₈×C₃₁	1	2	yes	—
74	C₂×C₄⋊C₄	—	2	6	no	—
75	C₂×C₄⋊C₄	D₄×C₃₁	1	6	yes	—
76	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
77	#102	—	4	12	no	—
78	#97	—	2	6	no	—
79	#95	—	2	2	no	—
80	#95	—	2	6	no	—
81	C₃₁	#10360	1	1	yes	characteristic, direct product factor, Sylow
82	C₂⁴	—	4	1	no	—
83	C₂×D₄	—	2	6	no	—
84	C₂×D₄	—	4	6	no	—
85	C₂×D₄	—	4	6	no	—
86	C₂×D₄	—	4	3	no	—
87	C₂²×C₄	—	2	6	no	—
88	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
89	C₂×C₈	—	2	6	no	—
90	C₂×C₈	—	4	3	no	—
91	C₂×C₈	—	4	3	no	—
92	C₄⋊C₄	—	4	6	no	—
93	C₄⋊C₄	—	8	6	no	—
94	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
95	C₄²	#41×C₃₁	1	3	yes	—
96	C₂³	—	4	2	no	—
97	C₂³	—	8	3	no	—
98	C₂³	—	8	3	no	—
99	C₂³	C₃₁×#120	1	1	yes	characteristic
100	D₄	—	4	12	no	—
101	D₄	—	8	6	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	C₃₁×C₂²⋊Q₈	1	3	yes	—
108	C₂×C₄	C₃₁×#124	1	3	yes	—
109	C₈	—	4	6	no	—
110	C₂²	—	8	6	no	—
111	C₂²	—	16	3	no	—
112	C₂²	—	16	1	no	—
113	C₂²	C₃₁×#393	1	1	yes	characteristic
114	C₂²	C₃₁×#453	1	3	yes	—
115	C₂²	C₃₁×#454	1	3	yes	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	8	6	no	—
119	C₂	—	16	2	no	—
120	C₂	C₃₁×#1656	1	3	yes	—
121	C₂	C₃₁×#1658	1	1	yes	characteristic
122	C₂	C₃₁×#2027	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#2845

C₂

yes

maximal, characteristic, complemented

C₃₁×#1961

C₂

yes

maximal

C₃₁×#1295

C₂

yes

maximal, complemented

C₂×C₃₁×#127

C₂²

yes

characteristic

C₂×C₃₁×#102

—

C₂×C₃₁×#97

C₂²

yes

—

C₃₁×#398

—

C₃₁×#382

C₂²

yes

—

C₂²×D₄×C₃₁

—

C₂²×C₈×C₃₁

—

C₃₁×#127

—

C₃₁×#127

Q₈

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

D₄

yes

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×#102

—

C₃₁×#97

—

C₃₁×#95

—

C₃₁×#95

—

C₂⁴×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×Q₈

yes

characteristic

C₄²×C₃₁

#41

yes

—

#10360

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

#120

yes

characteristic

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊Q₈

yes

—

C₂×C₄×C₃₁

#124

yes

—

C₈×C₃₁

—

#2845

C₂×C₃₁

yes

characteristic, complemented

#1961

C₂×C₃₁

yes

—

#1295

C₂×C₃₁

yes

complemented

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#393

yes

characteristic

C₂²×C₃₁

#453

yes

—

C₂²×C₃₁

#454

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₂×#127

C₂²×C₃₁

yes

characteristic

C₂×#102

—

C₂×#97

C₂²×C₃₁

yes

—

#398

—

#382

C₂²×C₃₁

yes

—

C₂×C₃₁

—

C₂×C₃₁

#1656

yes

—

C₂×C₃₁

#1658

yes

characteristic

C₂×C₃₁

#2027

yes

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

Q₈×C₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₃₁

yes

—

C₂×C₄²

C₂³×C₃₁

yes

—

—

—

—

C₃₁

#10360

yes

characteristic, direct product factor, Sylow

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₃₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Q₈×C₃₁

yes

characteristic

C₄²

#41×C₃₁

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₃₁×#120

yes

characteristic

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₃₁×C₂²⋊Q₈

yes

—

108

C₂×C₄

C₃₁×#124

yes

—

109

C₈

—

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₃₁×#393

yes

characteristic

114

C₂²

C₃₁×#453

yes

—

115

C₂²

C₃₁×#454

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

120

C₂

C₃₁×#1656

yes

—

121

C₂

C₃₁×#1658

yes

characteristic

122

C₂

C₃₁×#2027

yes

—

	subgroup	G/H
center, socle	C₂³×C₃₁	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂³×C₃₁

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

prime radical

C₂×C₃₁×#127

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

N	H ≅ G/N	type
1. C₃₁×#2845	C₂	semidirect
3. C₃₁×#1295	C₂	semidirect
34. #10360	C₃₁	direct (N × H)
49. #2845	C₂×C₃₁	semidirect
51. #1295	C₂×C₃₁	semidirect
81. C₃₁	#10360	direct (N × H)

H ≅ G/N

type

1. C₃₁×#2845

C₂

semidirect

3. C₃₁×#1295

C₂

semidirect

34. #10360

C₃₁

direct (N × H)

49. #2845

C₂×C₃₁

semidirect

51. #1295

C₂×C₃₁

semidirect

81. C₃₁

#10360

direct (N × H)

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

subgroup	index
C₃₁×#2845	2
C₃₁×#1961	2
C₃₁×#1295	2
#10360	31

subgroup

index

C₃₁×#2845

C₃₁×#1961

C₃₁×#1295

#10360

Subgroups — C₃₁×#10360 #21467119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#10360 #21467119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10360 #21467119