Group #21469657 — subgroups

Subgroups — C₃₁×#12898 #21469657

Order 7936.

414 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 286 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×C₄⋊C₁₆	C₂	1	6	yes	maximal
2	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆×C₃₁	C₂²	1	4	yes	—
4	C₂×C₄×C₈×C₃₁	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
6	C₄³×C₃₁	C₄	1	1	yes	characteristic
7	C₃₁×C₄⋊C₁₆	—	2	12	no	—
8	C₂²×C₈×C₃₁	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₃₁	C₈	1	3	yes	—
11	C₂×C₄²×C₃₁	C₈	1	1	yes	characteristic
12	C₂×C₄²×C₃₁	C₂×C₄	1	3	yes	—
13	C₂×C₁₆×C₃₁	—	2	12	no	—
14	C₄×C₈×C₃₁	—	2	6	no	—
15	C₄×C₈×C₃₁	D₄	1	6	yes	—
16	C₄×C₈×C₃₁	Q₈	1	6	yes	—
17	C₂²×C₄×C₃₁	C₂×C₈	1	3	yes	—
18	C₂²×C₄×C₃₁	C₂×C₈	1	3	yes	—
19	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
20	C₂×C₈×C₃₁	—	2	6	no	—
21	C₂×C₈×C₃₁	C₄⋊C₄	1	6	yes	—
22	C₂×C₈×C₃₁	C₂×D₄	1	3	yes	—
23	C₂×C₈×C₃₁	C₂×Q₈	1	3	yes	—
24	C₄²×C₃₁	—	2	6	no	—
25	C₄²×C₃₁	C₁₆	1	4	yes	complemented
26	C₄²×C₃₁	C₄⋊C₄	1	6	yes	—
27	C₄²×C₃₁	M₁₆	1	6	yes	—
28	C₁₆×C₃₁	—	4	4	no	—
29	#12898	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
31	C₂×C₄×C₃₁	—	2	6	no	—
32	C₂×C₄×C₃₁	C₄⋊C₈	1	6	yes	—
33	C₂×C₄×C₃₁	C₄⋊C₈	1	6	yes	—
34	C₂×C₄×C₃₁	C₂×C₁₆	1	6	yes	—
35	C₂×C₄×C₃₁	M₃₂	1	6	yes	—
36	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	3	yes	—
37	C₂×C₄×C₃₁	C₂×M₁₆	1	3	yes	—
38	C₈×C₃₁	—	2	6	no	—
39	C₈×C₃₁	#127	1	1	yes	characteristic
40	C₈×C₃₁	C₄⋊Q₈	1	3	yes	—
41	C₂×C₄⋊C₁₆	C₂×C₃₁	1	6	yes	—
42	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
43	C₂²×C₃₁	C₂×C₄⋊C₈	1	3	yes	—
44	C₂²×C₃₁	C₂²×C₁₆	1	1	yes	characteristic
45	C₂²×C₃₁	C₂×M₃₂	1	3	yes	—
46	C₄×C₃₁	—	2	6	no	—
47	C₄×C₃₁	C₄⋊C₁₆	1	12	yes	complemented
48	C₄×C₃₁	#384	1	1	yes	characteristic
49	C₄×C₃₁	#423	1	3	yes	—
50	C₂²×C₁₆	C₂²×C₃₁	1	4	yes	—
51	C₂×C₄×C₈	C₄×C₃₁	1	3	yes	—
52	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
53	C₄³	C₄×C₃₁	1	1	yes	characteristic
54	C₄⋊C₁₆	—	2	12	no	—
55	C₂×C₃₁	#1842	1	1	yes	characteristic
56	C₂×C₃₁	C₂×C₄⋊C₁₆	1	3	yes	—
57	C₂×C₃₁	#2150	1	3	yes	—
58	C₂²×C₈	C₂×C₄×C₃₁	1	3	yes	—
59	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
60	C₂×C₄²	C₈×C₃₁	1	3	yes	—
61	C₂×C₄²	C₈×C₃₁	1	1	yes	characteristic
62	C₂×C₄²	C₂×C₄×C₃₁	1	3	yes	—
63	C₂×C₁₆	—	2	12	no	—
64	C₄×C₈	—	2	6	no	—
65	C₄×C₈	D₄×C₃₁	1	6	yes	—
66	C₄×C₈	Q₈×C₃₁	1	6	yes	—
67	C₃₁	#12898	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×C₄	C₂×C₈×C₃₁	1	3	yes	—
69	C₂²×C₄	C₂×C₈×C₃₁	1	3	yes	—
70	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	C₄⋊C₄×C₃₁	1	6	yes	—
73	C₂×C₈	C₂×D₄×C₃₁	1	3	yes	—
74	C₂×C₈	C₂×Q₈×C₃₁	1	3	yes	—
75	C₄²	—	2	6	no	—
76	C₄²	C₁₆×C₃₁	1	4	yes	complemented
77	C₄²	C₄⋊C₄×C₃₁	1	6	yes	—
78	C₄²	M₁₆×C₃₁	1	6	yes	—
79	C₁₆	—	4	4	no	—
80	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	C₃₁×C₄⋊C₈	1	6	yes	—
83	C₂×C₄	C₃₁×C₄⋊C₈	1	6	yes	—
84	C₂×C₄	C₂×C₁₆×C₃₁	1	6	yes	—
85	C₂×C₄	C₃₁×M₃₂	1	6	yes	—
86	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	3	yes	—
87	C₂×C₄	C₂×M₁₆×C₃₁	1	3	yes	—
88	C₈	—	2	6	no	—
89	C₈	C₃₁×#127	1	1	yes	characteristic
90	C₈	C₃₁×C₄⋊Q₈	1	3	yes	—
91	C₂²	C₂×C₃₁×C₄⋊C₈	1	3	yes	—
92	C₂²	C₂²×C₁₆×C₃₁	1	1	yes	characteristic
93	C₂²	C₂×C₃₁×M₃₂	1	3	yes	—
94	C₄	—	2	6	no	—
95	C₄	C₃₁×C₄⋊C₁₆	1	12	yes	complemented
96	C₄	C₃₁×#384	1	1	yes	characteristic
97	C₄	C₃₁×#423	1	3	yes	—
98	C₂	C₃₁×#1842	1	1	yes	characteristic
99	C₂	C₂×C₃₁×C₄⋊C₁₆	1	3	yes	—
100	C₂	C₃₁×#2150	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₁×C₄⋊C₁₆

C₂

yes

maximal

C₄²×C₈×C₃₁

C₂

yes

maximal, characteristic

C₂²×C₁₆×C₃₁

C₂²

yes

—

C₂×C₄×C₈×C₃₁

C₄

yes

—

C₂×C₄×C₈×C₃₁

C₂²

yes

—

C₄³×C₃₁

C₄

yes

characteristic

C₃₁×C₄⋊C₁₆

—

C₂²×C₈×C₃₁

C₂×C₄

yes

—

C₂²×C₈×C₃₁

C₂³

yes

characteristic

C₂×C₄²×C₃₁

C₈

yes

—

C₂×C₄²×C₃₁

C₈

yes

characteristic

C₂×C₄²×C₃₁

C₂×C₄

yes

—

C₂×C₁₆×C₃₁

—

C₄×C₈×C₃₁

—

C₄×C₈×C₃₁

D₄

yes

—

C₄×C₈×C₃₁

Q₈

yes

—

C₂²×C₄×C₃₁

C₂×C₈

yes

—

C₂²×C₄×C₃₁

C₂×C₈

yes

—

C₂²×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₄⋊C₄

yes

—

C₂×C₈×C₃₁

C₂×D₄

yes

—

C₂×C₈×C₃₁

C₂×Q₈

yes

—

C₄²×C₃₁

—

C₄²×C₃₁

C₁₆

yes

complemented

C₄²×C₃₁

C₄⋊C₄

yes

—

C₄²×C₃₁

M₁₆

yes

—

C₁₆×C₃₁

—

#12898

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₂²×C₈

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₈

yes

—

C₂×C₄×C₃₁

C₄⋊C₈

yes

—

C₂×C₄×C₃₁

C₂×C₁₆

yes

—

C₂×C₄×C₃₁

M₃₂

yes

—

C₂×C₄×C₃₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂×M₁₆

yes

—

C₈×C₃₁

—

C₈×C₃₁

#127

yes

characteristic

C₈×C₃₁

C₄⋊Q₈

yes

—

C₂×C₄⋊C₁₆

C₂×C₃₁

yes

—

C₄²×C₈

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

C₂×C₄⋊C₈

yes

—

C₂²×C₃₁

C₂²×C₁₆

yes

characteristic

C₂²×C₃₁

C₂×M₃₂

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₄⋊C₁₆

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

C₂²×C₁₆

C₂²×C₃₁

yes

—

C₂×C₄×C₈

C₄×C₃₁

yes

—

C₂×C₄×C₈

C₂²×C₃₁

yes

—

C₄³

C₄×C₃₁

yes

characteristic

C₄⋊C₁₆

—

C₂×C₃₁

#1842

yes

characteristic

C₂×C₃₁

C₂×C₄⋊C₁₆

yes

—

C₂×C₃₁

#2150

yes

—

C₂²×C₈

C₂×C₄×C₃₁

yes

—

C₂²×C₈

C₂³×C₃₁

yes

characteristic

C₂×C₄²

C₈×C₃₁

yes

—

C₂×C₄²

C₈×C₃₁

yes

characteristic

C₂×C₄²

C₂×C₄×C₃₁

yes

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

D₄×C₃₁

yes

—

C₄×C₈

Q₈×C₃₁

yes

—

C₃₁

#12898

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₂×C₈×C₃₁

yes

—

C₂²×C₄

C₂×C₈×C₃₁

yes

—

C₂²×C₄

C₂²×C₄×C₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₃₁

yes

—

C₂×C₈

C₂×D₄×C₃₁

yes

—

C₂×C₈

C₂×Q₈×C₃₁

yes

—

C₄²

—

C₄²

C₁₆×C₃₁

yes

complemented

C₄²

C₄⋊C₄×C₃₁

yes

—

C₄²

M₁₆×C₃₁

yes

—

C₁₆

—

C₂³

C₂²×C₈×C₃₁

yes

characteristic

C₂×C₄

—

C₂×C₄

C₃₁×C₄⋊C₈

yes

—

C₂×C₄

C₃₁×C₄⋊C₈

yes

—

C₂×C₄

C₂×C₁₆×C₃₁

yes

—

C₂×C₄

C₃₁×M₃₂

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₃₁

yes

—

C₂×C₄

C₂×M₁₆×C₃₁

yes

—

C₈

—

C₈

C₃₁×#127

yes

characteristic

C₈

C₃₁×C₄⋊Q₈

yes

—

C₂²

C₂×C₃₁×C₄⋊C₈

yes

—

C₂²

C₂²×C₁₆×C₃₁

yes

characteristic

C₂²

C₂×C₃₁×M₃₂

yes

—

C₄

—

C₄

C₃₁×C₄⋊C₁₆

yes

complemented

C₄

C₃₁×#384

yes

characteristic

C₄

C₃₁×#423

yes

—

C₂

C₃₁×#1842

yes

characteristic

C₂

C₂×C₃₁×C₄⋊C₁₆

yes

—

100

C₂

C₃₁×#2150

yes

—

	subgroup	G/H
center	C₂²×C₈×C₃₁	C₂³
commutator	C₂²	C₂²×C₁₆×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₃₁ ◁ G

subgroup

G/H

center

C₂²×C₈×C₃₁

C₂³

commutator

C₂²

C₂²×C₁₆×C₃₁

Frattini, cosocle

C₂²×C₈

C₂³×C₃₁

socle, prime radical

C₂³×C₃₁

C₂²×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₈×C₃₁ ◁ G

N	H ≅ G/N	type
25. C₄²×C₃₁	C₁₆	semidirect
29. #12898	C₃₁	direct (N × H)
47. C₄×C₃₁	C₄⋊C₁₆	semidirect
67. C₃₁	#12898	direct (N × H)
76. C₄²	C₁₆×C₃₁	semidirect
95. C₄	C₃₁×C₄⋊C₁₆	semidirect

H ≅ G/N

type

25. C₄²×C₃₁

C₁₆

semidirect

29. #12898

C₃₁

direct (N × H)

47. C₄×C₃₁

C₄⋊C₁₆

semidirect

67. C₃₁

#12898

direct (N × H)

76. C₄²

C₁₆×C₃₁

semidirect

95. C₄

C₃₁×C₄⋊C₁₆

semidirect

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

subgroup	index
C₂×C₃₁×C₄⋊C₁₆	2
C₄²×C₈×C₃₁	2
#12898	31

subgroup

index

C₂×C₃₁×C₄⋊C₁₆

C₄²×C₈×C₃₁

#12898

Subgroups — C₃₁×#12898 #21469657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#12898 #21469657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12898 #21469657