Group #21465561 — subgroups

Subgroups — C₃₁×#8802 #21465561

Order 7936.

258 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 110 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×#1745	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1281	C₂	1	2	yes	maximal
3	C₃₁×#1276	C₂	1	2	yes	maximal
4	C₃₁×#1275	C₂	1	2	yes	maximal
5	C₃₁×#431	C₂²	1	4	yes	—
6	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
8	C₄³×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#336	C₄	1	4	yes	—
10	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
11	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
12	C₂×C₄²×C₃₁	—	2	2	no	—
13	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
15	C₃₁×C₄⋊C₈	—	2	4	no	—
16	C₃₁×C₂²⋊C₈	—	2	4	no	—
17	C₃₁×#97	—	2	2	no	—
18	C₃₁×#97	—	2	2	no	—
19	C₄×C₈×C₃₁	—	2	4	no	—
20	C₂²×C₄×C₃₁	—	2	2	no	—
21	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₁	#41	1	2	yes	—
23	C₂×C₈×C₃₁	—	2	2	no	—
24	C₂×C₈×C₃₁	—	2	2	no	—
25	C₂×C₈×C₃₁	—	2	4	no	—
26	C₂×C₈×C₃₁	—	2	4	no	—
27	C₂×C₈×C₃₁	—	4	4	no	—
28	C₄²×C₃₁	—	2	4	no	—
29	C₄²×C₃₁	—	4	4	no	—
30	C₄²×C₃₁	#41	1	2	yes	—
31	C₄²×C₃₁	#41	1	2	yes	—
32	#8802	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₃₁	#117	1	1	yes	characteristic
34	C₂×C₄×C₃₁	—	2	4	no	—
35	C₂×C₄×C₃₁	—	4	2	no	—
36	C₂×C₄×C₃₁	—	4	2	no	—
37	C₂×C₄×C₃₁	—	4	2	no	—
38	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
39	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
40	C₂×C₄×C₃₁	#126	1	4	yes	—
41	C₂×C₄×C₃₁	#131	1	4	yes	—
42	C₈×C₃₁	—	4	2	no	—
43	C₈×C₃₁	—	4	2	no	—
44	C₈×C₃₁	—	8	4	no	—
45	#1745	C₂×C₃₁	1	1	yes	characteristic
46	#1281	C₂×C₃₁	1	2	yes	—
47	#1276	C₂×C₃₁	1	2	yes	—
48	#1275	C₂×C₃₁	1	2	yes	—
49	C₂²×C₃₁	—	2	2	no	—
50	C₂²×C₃₁	#383	1	1	yes	characteristic
51	C₂²×C₃₁	#433	1	2	yes	—
52	C₄×C₃₁	—	2	4	no	—
53	C₄×C₃₁	—	4	4	no	—
54	C₄×C₃₁	#433	1	2	yes	—
55	C₄×C₃₁	#433	1	2	yes	—
56	#431	C₂²×C₃₁	1	4	yes	—
57	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
58	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
59	C₄³	C₂²×C₃₁	1	1	yes	characteristic
60	#336	C₄×C₃₁	1	4	yes	—
61	C₂×C₃₁	—	2	2	no	—
62	C₂×C₃₁	#1840	1	1	yes	characteristic
63	C₂×C₃₁	#1841	1	2	yes	—
64	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
65	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
66	C₂×C₄²	—	2	2	no	—
67	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
68	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
69	C₄⋊C₈	—	2	4	no	—
70	C₂²⋊C₈	—	2	4	no	—
71	#97	—	2	2	no	—
72	#97	—	2	2	no	—
73	C₄×C₈	—	2	4	no	—
74	C₃₁	#8802	1	1	yes	characteristic, direct product factor, Sylow
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
77	C₂²×C₄	#41×C₃₁	1	2	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	2	4	no	—
82	C₂×C₈	—	4	4	no	—
83	C₄²	—	2	4	no	—
84	C₄²	—	4	4	no	—
85	C₄²	#41×C₃₁	1	2	yes	—
86	C₄²	#41×C₃₁	1	2	yes	—
87	C₂³	C₃₁×#117	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
93	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
94	C₂×C₄	C₃₁×#126	1	4	yes	—
95	C₂×C₄	C₃₁×#131	1	4	yes	—
96	C₈	—	4	2	no	—
97	C₈	—	4	2	no	—
98	C₈	—	8	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₃₁×#383	1	1	yes	characteristic
101	C₂²	C₃₁×#433	1	2	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	4	4	no	—
104	C₄	C₃₁×#433	1	2	yes	—
105	C₄	C₃₁×#433	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	C₃₁×#1840	1	1	yes	characteristic
108	C₂	C₃₁×#1841	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×#1745

C₂

yes

maximal, characteristic

C₃₁×#1281

C₂

yes

maximal

C₃₁×#1276

C₂

yes

maximal

C₃₁×#1275

C₂

yes

maximal

C₃₁×#431

C₂²

yes

—

C₂×C₃₁×#97

C₂²

yes

characteristic

C₂×C₃₁×#97

C₂²

yes

characteristic

C₄³×C₃₁

C₂²

yes

characteristic

C₃₁×#336

C₄

yes

—

C₂²×C₈×C₃₁

C₂×C₄

yes

—

C₂²×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₄²×C₃₁

—

C₂×C₄²×C₃₁

C₂×C₄

yes

—

C₂×C₄²×C₃₁

C₂³

yes

characteristic

C₃₁×C₄⋊C₈

—

C₃₁×C₂²⋊C₈

—

C₃₁×#97

—

C₃₁×#97

—

C₄×C₈×C₃₁

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂²×C₄×C₃₁

#41

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

—

C₄²×C₃₁

#41

yes

—

C₄²×C₃₁

#41

yes

—

#8802

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

#117

yes

characteristic

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

#131

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

—

#1745

C₂×C₃₁

yes

characteristic

#1281

C₂×C₃₁

yes

—

#1276

C₂×C₃₁

yes

—

#1275

C₂×C₃₁

yes

—

C₂²×C₃₁

—

C₂²×C₃₁

#383

yes

characteristic

C₂²×C₃₁

#433

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#433

yes

—

C₄×C₃₁

#433

yes

—

#431

C₂²×C₃₁

yes

—

C₂×#97

C₂²×C₃₁

yes

characteristic

C₂×#97

C₂²×C₃₁

yes

characteristic

C₄³

C₂²×C₃₁

yes

characteristic

#336

C₄×C₃₁

yes

—

C₂×C₃₁

—

C₂×C₃₁

#1840

yes

characteristic

C₂×C₃₁

#1841

yes

—

C₂²×C₈

C₂×C₄×C₃₁

yes

—

C₂²×C₈

C₂×C₄×C₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄×C₃₁

yes

—

C₂×C₄²

C₂³×C₃₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₃₁

#8802

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

C₂²×C₄×C₃₁

yes

characteristic

C₂²×C₄

#41×C₃₁

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

#41×C₃₁

yes

—

C₄²

#41×C₃₁

yes

—

C₂³

C₃₁×#117

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₃₁×#117

yes

characteristic

C₂×C₄

C₃₁×#117

yes

characteristic

C₂×C₄

C₃₁×#126

yes

—

C₂×C₄

C₃₁×#131

yes

—

C₈

—

C₈

—

C₈

—

C₂²

—

100

C₂²

C₃₁×#383

yes

characteristic

101

C₂²

C₃₁×#433

yes

—

102

C₄

—

103

C₄

—

104

C₄

C₃₁×#433

yes

—

105

C₄

C₃₁×#433

yes

—

106

C₂

—

107

C₂

C₃₁×#1840

yes

characteristic

108

C₂

C₃₁×#1841

yes

—

	subgroup	G/H
center	C₂×C₄×C₃₁	#117
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#383
prime radical	C₂³×C₃₁	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

subgroup

G/H

center

C₂×C₄×C₃₁

#117

commutator

C₂²×C₄

C₂²×C₄×C₃₁

Frattini, cosocle

C₂×C₄²

C₂³×C₃₁

C₂²×C₃₁

C₂³×C₃₁

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

N	H ≅ G/N	type
32. #8802	C₃₁	direct (N × H)
74. C₃₁	#8802	direct (N × H)

H ≅ G/N

type

32. #8802

C₃₁

direct (N × H)

74. C₃₁

#8802

direct (N × H)

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

subgroup	index
C₃₁×#1745	2
C₃₁×#1281	2
C₃₁×#1276	2
C₃₁×#1275	2
#8802	31

subgroup

index

C₃₁×#1745

C₃₁×#1281

C₃₁×#1276

C₃₁×#1275

#8802

Subgroups — C₃₁×#8802 #21465561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#8802 #21465561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8802 #21465561