Group #16853980 — subgroups

Subgroups — #16853980

Order 3968.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#14520988	C₂	1	3	yes	maximal
2	#14520982	C₂	1	6	yes	maximal
3	#14520969	C₂	1	3	yes	maximal
4	#14520917	C₂	1	2	yes	maximal
5	C₃₁×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₆₂	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
8	C₂×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
9	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃₁	C₂²	1	3	yes	—
11	C₂×Dic₃₁⋊C₄	C₂²	1	6	yes	—
12	C₃₁⋊#95	C₂²	1	6	yes	—
13	C₃₁⋊#95	C₂²	1	6	yes	—
14	C₃₁⋊#95	C₂²	1	3	yes	—
15	C₃₁⋊#95	C₂²	1	3	yes	—
16	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
17	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
18	C₂²×C₄×C₃₁	C₂³	1	3	yes	—
19	C₂²×Dic₃₁	C₂³	1	6	yes	—
20	C₂²×Dic₃₁	C₂³	1	2	yes	—
21	C₂×Dic₆₂	—	4	3	no	—
22	C₄⋊C₄×C₃₁	—	4	3	no	—
23	C₄⋊C₄×C₃₁	—	4	3	no	—
24	C₄²×C₃₁	Q₈	1	4	yes	complemented
25	C₄⋊Dic₃₁	—	4	3	no	—
26	Dic₃₁⋊C₄	—	4	6	no	—
27	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₃₁	—	2	3	no	—
29	C₂×C₄×C₃₁	—	2	3	no	—
30	C₂×C₄×C₃₁	—	2	6	no	—
31	C₂×C₄×C₃₁	—	2	3	no	—
32	C₂×C₄×C₃₁	—	2	3	no	—
33	C₂×C₄×C₃₁	C₂×Q₈	1	6	yes	—
34	C₂×Dic₃₁	—	2	6	no	—
35	C₂×Dic₃₁	—	2	6	no	—
36	C₂×Dic₃₁	—	2	12	no	—
37	Dic₆₂	—	4	4	no	—
38	#2843	—	31	1	no	maximal, Sylow
39	C₂²×C₃₁	C₂²×Q₈	1	1	yes	characteristic
40	C₂²×C₃₁	H(5,𝔽₂)	1	3	yes	—
41	C₂²×C₃₁	2₋¹⁺⁴	1	3	yes	—
42	C₄×C₃₁	—	2	6	no	—
43	C₄×C₃₁	—	4	1	no	—
44	C₄×C₃₁	—	4	3	no	—
45	Dic₃₁	—	4	2	no	—
46	Dic₃₁	—	4	6	no	—
47	#400	—	31	3	no	—
48	#398	—	31	6	no	—
49	#395	—	31	3	no	—
50	#384	D₃₁	1	1	yes	characteristic
51	#383	—	31	2	no	—
52	C₂×C₃₁	#544	1	3	yes	—
53	C₂×C₃₁	#560	1	1	yes	characteristic
54	C₂×C₃₁	#563	1	3	yes	—
55	C₂²×Q₈	—	31	1	no	—
56	C₂×C₄⋊C₄	—	31	6	no	—
57	C₂×C₄⋊C₄	—	31	3	no	—
58	C₂×C₄⋊C₄	C₂×D₃₁	1	3	yes	—
59	C₂×C₄⋊C₄	C₂×D₃₁	1	3	yes	—
60	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
61	#95	—	31	6	no	—
62	#95	—	31	6	no	—
63	#95	—	31	3	no	—
64	#95	—	31	3	no	—
65	C₃₁	#2843	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₈	—	124	3	no	—
67	C₂²×C₄	—	31	6	no	—
68	C₂²×C₄	—	31	2	no	—
69	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
71	C₂²×C₄	C₂²×D₃₁	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	4	3	no	—
74	C₄⋊C₄	—	124	6	no	—
75	C₄⋊C₄	—	124	3	no	—
76	C₄²	Dic₆₂	1	4	yes	complemented
77	C₂³	C₂³×D₃₁	1	1	yes	characteristic
78	Q₈	—	124	4	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	62	6	no	—
85	C₂×C₄	—	62	6	no	—
86	C₂×C₄	—	62	12	no	—
87	C₂×C₄	C₂×Dic₆₂	1	6	yes	—
88	C₂²	C₂²×Dic₆₂	1	1	yes	characteristic
89	C₂²	#11758561	1	3	yes	—
90	C₂²	#11758564	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	1	no	—
93	C₄	—	4	3	no	—
94	C₄	—	124	2	no	—
95	C₄	—	124	6	no	—
96	C₂	#14521680	1	3	yes	—
97	C₂	#14521786	1	1	yes	characteristic
98	C₂	#14521798	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14520988

C₂

yes

maximal

#14520982

C₂

yes

maximal

#14520969

C₂

yes

maximal

#14520917

C₂

yes

maximal

C₃₁×#384

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×Dic₃₁⋊C₄

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₃₁⋊#95

C₂²

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂×Dic₆₂

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

Q₈

yes

complemented

C₄⋊Dic₃₁

—

Dic₃₁⋊C₄

—

C₂³×C₃₁

C₂⁴

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

#2843

—

maximal, Sylow

C₂²×C₃₁

C₂²×Q₈

yes

characteristic

C₂²×C₃₁

H(5,𝔽₂)

yes

—

C₂²×C₃₁

2₋¹⁺⁴

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

Dic₃₁

—

Dic₃₁

—

—

—

—

D₃₁

yes

characteristic

#383

—

C₂×C₃₁

#544

yes

—

C₂×C₃₁

#560

yes

characteristic

C₂×C₃₁

#563

yes

—

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

—

—

—

—

C₃₁

#2843

yes

characteristic, complemented, Sylow

C₂×Q₈

—

124

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

124

—

C₄⋊C₄

—

124

—

C₄²

Dic₆₂

yes

complemented

C₂³

C₂³×D₃₁

yes

characteristic

Q₈

—

124

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₆₂

yes

—

C₂²

C₂²×Dic₆₂

yes

characteristic

C₂²

#11758561

yes

—

C₂²

#11758564

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

124

—

C₄

—

124

—

C₂

#14521680

yes

—

C₂

#14521786

yes

characteristic

C₂

#14521798

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator, socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
Fitting	C₃₁×#384	C₂
hypercenter	#384	D₃₁
hypocenter	C₃₁	#2843
LCS	G ▷ C₂³×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator, socle, prime radical, cosocle

C₂³×C₃₁

C₂⁴

Fitting

C₃₁×#384

C₂

D₃₁

C₃₁

G ▷ C₂³×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
24. C₄²×C₃₁	Q₈	semidirect
65. C₃₁	#2843	semidirect
76. C₄²	Dic₆₂	semidirect

H ≅ G/N

type

24. C₄²×C₃₁

Q₈

semidirect

65. C₃₁

#2843

semidirect

76. C₄²

Dic₆₂

semidirect

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

subgroup	index
#14520988	2
#14520982	2
#14520969	2
#14520917	2
C₃₁×#384	2
#2843	31

subgroup

index

C₃₁×#384

Subgroups — #16853980

Conjugacy classes of subgroups

Complements

Maximal subgroups