Group #16847728 — subgroups

Subgroups — #16847728

Order 3968.

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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	#14521221	C₂	1	3	yes	maximal, complemented
2	#14520846	C₂	1	3	yes	maximal, complemented
3	C₃₁×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758564	C₂²	1	1	yes	characteristic, complemented
5	C₃₁×#124	—	2	3	no	—
6	C₃₁×#124	C₂²	1	3	yes	—
7	C₃₁×#120	—	2	3	no	—
8	#11758509	—	2	3	no	—
9	#11758424	C₂²	1	3	yes	complemented
10	#11758393	—	2	3	no	—
11	C₂⁴×C₃₁	—	2	1	no	—
12	C₂×Q₈×C₃₁	—	2	1	no	—
13	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
14	C₂×D₄×C₃₁	—	2	3	no	—
15	C₂×D₄×C₃₁	D₄	1	3	yes	—
16	#92499	—	4	1	no	—
17	Q₈×D₃₁	—	4	1	no	—
18	#92495	—	2	3	no	—
19	C₂²⋊C₄×C₃₁	—	2	3	no	—
20	C₂²⋊C₄×C₃₁	—	2	6	no	—
21	C₄²×C₃₁	D₄	1	3	yes	—
22	#92480	—	4	3	no	—
23	#92479	—	4	3	no	—
24	#92474	—	2	3	no	—
25	C₂³×C₃₁	—	2	3	no	—
26	C₂³×C₃₁	—	8	2	no	—
27	C₂³×C₃₁	—	8	1	no	—
28	Q₈×C₃₁	—	4	1	no	—
29	Q₈×C₃₁	—	8	1	no	—
30	D₄×C₃₁	—	4	3	no	—
31	D₄×C₃₁	—	8	3	no	—
32	C₂×C₄×C₃₁	—	2	3	no	—
33	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
34	C₂²⋊D₃₁	—	4	1	no	—
35	D₁₂₄	—	2	3	no	—
36	C₄×D₃₁	—	4	3	no	—
37	Dic₆₂	—	2	3	no	—
38	C₃₁⋊C₈	—	4	3	no	—
39	#2202	—	31	1	no	maximal, Sylow
40	C₂²×C₃₁	—	4	6	no	—
41	C₂²×C₃₁	—	4	3	no	—
42	C₂²×C₃₁	—	8	4	no	—
43	C₂²×C₃₁	#120	1	1	yes	characteristic
44	C₂×D₃₁	—	4	1	no	—
45	C₄×C₃₁	—	2	3	no	—
46	C₄×C₃₁	—	4	3	no	—
47	Dic₃₁	—	4	1	no	—
48	H(3,𝔽₄)	D₃₁	1	1	yes	characteristic, complemented
49	#454	—	31	3	no	—
50	#355	—	31	3	no	—
51	C₂×C₃₁	—	2	1	no	—
52	C₂×C₃₁	—	8	3	no	—
53	C₂×C₃₁	B(4,𝔽₂)	1	1	yes	characteristic
54	D₃₁	—	8	1	no	—
55	2₋¹⁺⁴	—	31	1	no	—
56	AGL(1,ℤ/8)	—	62	3	no	—
57	#124	—	2	3	no	—
58	#124	C₂×D₃₁	1	3	yes	—
59	#120	—	2	3	no	—
60	#104	—	62	3	no	—
61	#101	—	31	3	no	—
62	C₃₁	#2202	1	1	yes	characteristic, complemented, Sylow
63	C₂⁴	—	2	1	no	—
64	#41	—	62	3	no	—
65	#41	—	124	1	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	124	1	no	—
68	C₂×Q₈	C₂²×D₃₁	1	1	yes	characteristic
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂²⋊D₃₁	1	3	yes	—
71	SD₁₆	—	124	3	no	—
72	D₈	—	124	3	no	—
73	M₁₆	—	62	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₄²	C₂²⋊D₃₁	1	3	yes	—
77	C₂³	—	2	3	no	—
78	C₂³	—	8	2	no	—
79	C₂³	—	8	1	no	—
80	Q₈	—	4	1	no	—
81	Q₈	—	8	1	no	—
82	Q₈	—	62	3	no	—
83	D₄	—	4	3	no	—
84	D₄	—	8	3	no	—
85	D₄	—	62	3	no	—
86	D₄	—	124	1	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	124	3	no	—
89	C₂×C₄	C₂×C₂²⋊D₃₁	1	3	yes	—
90	C₈	—	124	3	no	—
91	C₂²	—	4	6	no	—
92	C₂²	—	4	3	no	—
93	C₂²	—	8	4	no	—
94	C₂²	—	124	1	no	—
95	C₂²	#11758530	1	1	yes	characteristic
96	C₄	—	2	3	no	—
97	C₄	—	4	3	no	—
98	C₄	—	124	1	no	—
99	C₂	—	2	1	no	—
100	C₂	—	8	3	no	—
101	C₂	—	248	1	no	—
102	C₂	#14521236	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521221

C₂

yes

maximal, complemented

#14520846

C₂

yes

maximal, complemented

C₃₁×H(3,𝔽₄)

C₂

yes

maximal, characteristic, complemented

#11758564

C₂²

yes

characteristic, complemented

C₃₁×#124

—

C₃₁×#124

C₂²

yes

—

C₃₁×#120

—

#11758509

—

#11758424

C₂²

yes

complemented

#11758393

—

C₂⁴×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

C₂³

yes

characteristic

C₂×D₄×C₃₁

—

C₂×D₄×C₃₁

D₄

yes

—

#92499

—

Q₈×D₃₁

—

#92495

—

C₂²⋊C₄×C₃₁

—

C₂²⋊C₄×C₃₁

—

C₄²×C₃₁

D₄

yes

—

#92480

—

#92479

—

#92474

—

C₂³×C₃₁

—

C₂³×C₃₁

—

C₂³×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

D₄×C₃₁

—

D₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂²⋊D₃₁

—

D₁₂₄

—

C₄×D₃₁

—

Dic₆₂

—

C₃₁⋊C₈

—

#2202

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

#120

yes

characteristic

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

Dic₃₁

—

H(3,𝔽₄)

D₃₁

yes

characteristic, complemented

#454

—

#355

—

C₂×C₃₁

—

C₂×C₃₁

—

C₂×C₃₁

B(4,𝔽₂)

yes

characteristic

D₃₁

—

2₋¹⁺⁴

—

AGL(1,ℤ/8)

—

#124

—

#124

C₂×D₃₁

yes

—

#120

—

#104

—

#101

—

C₃₁

#2202

yes

characteristic, complemented, Sylow

C₂⁴

—

#41

—

#41

—

124

—

C₂×Q₈

—

C₂×Q₈

—

124

—

C₂×Q₈

C₂²×D₃₁

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂²⋊D₃₁

yes

—

SD₁₆

—

124

—

D₈

—

124

—

M₁₆

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₃₁

yes

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

124

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

C₈

—

124

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

124

—

C₂²

#11758530

yes

characteristic

C₄

—

C₄

—

C₄

—

124

—

C₂

—

100

C₂

—

101

C₂

—

248

—

102

C₂

#14521236

yes

characteristic

	subgroup	G/H
center	C₂	#14521236
commutator, cosocle	C₂×Q₈×C₃₁	C₂³
Frattini	C₂×Q₈	C₂²×D₃₁
Fitting	C₃₁×H(3,𝔽₄)	C₂
socle	C₂×C₃₁	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₃₁
hypocenter	C₃₁	#2202
LCS	G ▷ C₂×Q₈×C₃₁ ▷ C₂²×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

subgroup

G/H

center

C₂

#14521236

commutator, cosocle

C₂×Q₈×C₃₁

C₂³

Frattini

C₂×Q₈

C₂²×D₃₁

Fitting

C₃₁×H(3,𝔽₄)

C₂

C₂×C₃₁

D₃₁

C₃₁

G ▷ C₂×Q₈×C₃₁ ▷ C₂²×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

N	H ≅ G/N	type
1. #14521221	C₂	semidirect
2. #14520846	C₂	semidirect
3. C₃₁×H(3,𝔽₄)	C₂	semidirect
4. #11758564	C₂²	semidirect
9. #11758424	C₂²	semidirect
48. H(3,𝔽₄)	D₃₁	semidirect
62. C₃₁	#2202	semidirect

H ≅ G/N

type

1. #14521221

C₂

semidirect

2. #14520846

C₂

semidirect

3. C₃₁×H(3,𝔽₄)

C₂

semidirect

4. #11758564

C₂²

semidirect

9. #11758424

C₂²

semidirect

48. H(3,𝔽₄)

D₃₁

semidirect

62. C₃₁

#2202

semidirect

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

subgroup	index
#14521221	2
#14520846	2
C₃₁×H(3,𝔽₄)	2
#2202	31

subgroup

index

#14521221

#14520846

C₃₁×H(3,𝔽₄)

#2202

Subgroups — #16847728

Conjugacy classes of subgroups

Complements

Maximal subgroups