Group #21464027 — subgroups

Subgroups — C₃₁×#7268 #21464027

Order 7936.

238 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 118 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×C₃₁×#364	C₂	1	2	yes	maximal
2	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×M₃₂	—	2	2	no	—
4	C₂×C₃₁×#106	—	2	2	no	—
5	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₃₁×#384	C₄	1	2	yes	—
7	C₃₁×#364	—	2	2	no	—
8	C₃₁×#364	C₄	1	4	yes	complemented
9	C₂²×C₈×C₃₁	D₄	1	2	yes	—
10	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
11	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
12	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
13	C₃₁×M₃₂	—	2	4	no	—
14	C₃₁×M₃₂	—	4	2	no	—
15	C₂×C₁₆×C₃₁	—	4	2	no	—
16	C₃₁×#106	—	4	2	no	—
17	C₃₁×#106	—	4	2	no	—
18	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
20	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₃₁	—	2	1	no	—
22	C₂²×C₄×C₃₁	—	4	2	no	—
23	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₃₁	D₈	1	2	yes	—
27	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
28	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
29	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
30	C₄⋊C₄×C₃₁	—	4	4	no	—
31	C₄⋊C₄×C₃₁	—	8	4	no	—
32	C₄²×C₃₁	—	2	1	no	—
33	C₄²×C₃₁	C₄²	1	1	yes	characteristic
34	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
35	C₁₆×C₃₁	—	4	4	no	—
36	#7268	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂³×C₃₁	#100	1	1	yes	characteristic
38	C₂×C₄×C₃₁	—	2	4	no	—
39	C₂×C₄×C₃₁	—	2	1	no	—
40	C₂×C₄×C₃₁	—	2	1	no	—
41	C₂×C₄×C₃₁	—	8	4	no	—
42	C₂×C₄×C₃₁	—	8	2	no	—
43	C₂×C₄×C₃₁	#95	1	1	yes	characteristic
44	C₂×C₄×C₃₁	#101	1	1	yes	characteristic
45	C₂×C₄×C₃₁	#102	1	2	yes	—
46	C₂×C₄×C₃₁	#103	1	2	yes	—
47	C₈×C₃₁	#102	1	2	yes	—
48	C₈×C₃₁	#103	1	2	yes	—
49	C₈×C₃₁	#106	1	2	yes	—
50	C₈×C₃₁	#107	1	2	yes	—
51	C₂×#364	C₂×C₃₁	1	2	yes	—
52	#1844	C₂×C₃₁	1	1	yes	characteristic
53	C₂²×C₃₁	—	2	2	no	—
54	C₂²×C₃₁	#331	1	1	yes	characteristic
55	C₂²×C₃₁	#333	1	1	yes	characteristic
56	C₂²×C₃₁	#343	1	1	yes	characteristic
57	C₄×C₃₁	—	2	2	no	—
58	C₄×C₃₁	—	2	2	no	—
59	C₄×C₃₁	—	16	2	no	—
60	C₄×C₃₁	#332	1	2	yes	—
61	C₄×C₃₁	#340	1	2	yes	—
62	C₂×M₃₂	—	2	2	no	—
63	C₂×#106	—	2	2	no	—
64	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
65	#384	C₄×C₃₁	1	2	yes	—
66	#364	—	2	2	no	—
67	#364	C₄×C₃₁	1	4	yes	complemented
68	C₂×C₃₁	—	2	1	no	—
69	C₂×C₃₁	—	2	1	no	—
70	C₂×C₃₁	#1296	1	1	yes	characteristic
71	C₂×C₃₁	#1359	1	2	yes	—
72	C₂²×C₈	D₄×C₃₁	1	2	yes	—
73	C₂×C₄⋊C₄	—	2	2	no	—
74	C₂×C₄⋊C₄	—	4	2	no	—
75	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
76	M₃₂	—	2	4	no	—
77	M₃₂	—	4	2	no	—
78	C₂×C₁₆	—	4	2	no	—
79	#106	—	4	2	no	—
80	#106	—	4	2	no	—
81	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
82	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
83	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
84	C₃₁	#7268	1	1	yes	characteristic, direct product factor, Sylow
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
88	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
89	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
90	C₂×C₈	D₈×C₃₁	1	2	yes	—
91	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
92	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
93	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	—	8	4	no	—
96	C₄²	—	2	1	no	—
97	C₄²	C₄²×C₃₁	1	1	yes	characteristic
98	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
99	C₁₆	—	4	4	no	—
100	C₂³	C₃₁×#100	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	8	4	no	—
105	C₂×C₄	—	8	2	no	—
106	C₂×C₄	C₃₁×#95	1	1	yes	characteristic
107	C₂×C₄	C₃₁×#101	1	1	yes	characteristic
108	C₂×C₄	C₃₁×#102	1	2	yes	—
109	C₂×C₄	C₃₁×#103	1	2	yes	—
110	C₈	C₃₁×#102	1	2	yes	—
111	C₈	C₃₁×#103	1	2	yes	—
112	C₈	C₃₁×#106	1	2	yes	—
113	C₈	C₃₁×#107	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	C₃₁×#331	1	1	yes	characteristic
116	C₂²	C₃₁×#333	1	1	yes	characteristic
117	C₂²	C₃₁×#343	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	16	2	no	—
121	C₄	C₃₁×#332	1	2	yes	—
122	C₄	C₃₁×#340	1	2	yes	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	C₃₁×#1296	1	1	yes	characteristic
126	C₂	C₃₁×#1359	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×#364

C₂

yes

maximal

C₃₁×#1844

C₂

yes

maximal, characteristic

C₂×C₃₁×M₃₂

—

C₂×C₃₁×#106

—

C₂×C₄×C₈×C₃₁

C₂²

yes

characteristic

C₃₁×#384

C₄

yes

—

C₃₁×#364

—

C₃₁×#364

C₄

yes

complemented

C₂²×C₈×C₃₁

D₄

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄²×C₃₁

C₂×C₄

yes

characteristic

C₃₁×M₃₂

—

C₃₁×M₃₂

—

C₂×C₁₆×C₃₁

—

C₃₁×#106

—

C₃₁×#106

—

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₄×C₈×C₃₁

D₄

yes

characteristic

C₄×C₈×C₃₁

Q₈

yes

characteristic

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₃₁

C₂²⋊C₄

yes

—

C₂×C₈×C₃₁

C₄⋊C₄

yes

—

C₂×C₈×C₃₁

D₈

yes

—

C₂×C₈×C₃₁

SD₁₆

yes

—

C₂×C₈×C₃₁

SD₁₆

yes

—

C₂×C₈×C₃₁

Q₁₆

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

C₄²

yes

characteristic

C₄²×C₃₁

C₄⋊C₄

yes

characteristic

C₁₆×C₃₁

—

#7268

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#100

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

C₂×C₄×C₃₁

yes

—

C₂×C₄×C₃₁

#103

yes

—

C₈×C₃₁

#102

yes

—

C₈×C₃₁

#103

yes

—

C₈×C₃₁

#106

yes

—

C₈×C₃₁

#107

yes

—

C₂×#364

C₂×C₃₁

yes

—

#1844

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

—

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#332

yes

—

C₄×C₃₁

#340

yes

—

C₂×M₃₂

—

C₂×#106

—

C₂×C₄×C₈

C₂²×C₃₁

yes

characteristic

#384

C₄×C₃₁

yes

—

#364

—

#364

C₄×C₃₁

yes

complemented

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

#1296

yes

characteristic

C₂×C₃₁

#1359

yes

—

C₂²×C₈

D₄×C₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×C₄×C₃₁

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#106

—

#106

—

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₄×C₈

D₄×C₃₁

yes

characteristic

C₄×C₈

Q₈×C₃₁

yes

characteristic

C₃₁

#7268

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄×C₃₁

yes

characteristic

C₂×C₈

C₂²⋊C₄×C₃₁

yes

—

C₂×C₈

C₄⋊C₄×C₃₁

yes

—

C₂×C₈

D₈×C₃₁

yes

—

C₂×C₈

SD₁₆×C₃₁

yes

—

C₂×C₈

SD₁₆×C₃₁

yes

—

C₂×C₈

Q₁₆×C₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₄²×C₃₁

yes

characteristic

C₄²

C₄⋊C₄×C₃₁

yes

characteristic

C₁₆

—

100

C₂³

C₃₁×#100

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₃₁×#95

yes

characteristic

107

C₂×C₄

C₃₁×#101

yes

characteristic

108

C₂×C₄

C₃₁×#102

yes

—

109

C₂×C₄

C₃₁×#103

yes

—

110

C₈

C₃₁×#102

yes

—

111

C₈

C₃₁×#103

yes

—

112

C₈

C₃₁×#106

yes

—

113

C₈

C₃₁×#107

yes

—

114

C₂²

—

115

C₂²

C₃₁×#331

yes

characteristic

116

C₂²

C₃₁×#333

yes

characteristic

117

C₂²

C₃₁×#343

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₃₁×#332

yes

—

122

C₄

C₃₁×#340

yes

—

123

C₂

—

124

C₂

—

125

C₂

C₃₁×#1296

yes

characteristic

126

C₂

C₃₁×#1359

yes

—

	subgroup	G/H
center, socle	C₂²×C₃₁	#331
commutator	C₄²	C₄²×C₃₁
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃₁
prime radical	C₂³×C₃₁	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₁

#331

commutator

C₄²

C₄²×C₃₁

Frattini, cosocle

C₂×C₄×C₈

C₂²×C₃₁

prime radical

C₂³×C₃₁

#100

LCS

G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

N	H ≅ G/N	type
8. C₃₁×#364	C₄	semidirect
36. #7268	C₃₁	direct (N × H)
67. #364	C₄×C₃₁	semidirect
84. C₃₁	#7268	direct (N × H)

H ≅ G/N

type

8. C₃₁×#364

C₄

semidirect

36. #7268

C₃₁

direct (N × H)

67. #364

C₄×C₃₁

semidirect

84. C₃₁

#7268

direct (N × H)

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

subgroup	index
C₂×C₃₁×#364	2
C₃₁×#1844	2
#7268	31

subgroup

index

C₂×C₃₁×#364

C₃₁×#1844

#7268

Subgroups — C₃₁×#7268 #21464027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#7268 #21464027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7268 #21464027