Group #16845875 — subgroups

Subgroups — #16845875

Order 3968.

248 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 141 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×#11758406	C₂	1	2	yes	maximal
2	C₂×#11758405	C₂	1	2	yes	maximal
3	#14520915	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
9	C₄×C₈×C₃₁	C₄	1	4	yes	complemented
10	C₃₁⋊#95	—	2	4	no	—
11	#11758406	—	2	4	no	—
12	#11758405	—	2	4	no	—
13	C₂²×C₄×C₃₁	D₄	1	2	yes	—
14	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
15	C₂²×Dic₃₁	—	2	4	no	—
16	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₃₁	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₃₁	D₄	1	2	yes	—
19	C₂×C₈×C₃₁	D₄	1	2	yes	—
20	C₂×C₈×C₃₁	Q₈	1	2	yes	—
21	C₂×C₈×C₃₁	Q₈	1	2	yes	—
22	C₄²×C₃₁	C₂×C₄	1	2	yes	—
23	C₄²×C₃₁	Q₈	1	2	yes	—
24	C₄⋊Dic₃₁	—	2	4	no	—
25	C₄⋊Dic₃₁	—	2	4	no	—
26	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₃₁	—	2	4	no	—
28	C₂×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
29	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
35	C₂×Dic₃₁	—	4	4	no	—
36	C₂×Dic₃₁	—	4	8	no	—
37	C₈×C₃₁	C₄⋊C₄	1	4	yes	—
38	C₈×C₃₁	C₄⋊C₄	1	4	yes	—
39	#1846	—	31	1	no	maximal, Sylow
40	C₂²×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₃₁	#127	1	1	yes	characteristic
42	C₂²×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
43	C₂²×C₃₁	#135	1	4	yes	—
44	C₄×C₃₁	—	2	4	no	—
45	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
49	Dic₃₁	—	8	4	no	—
50	C₂×#107	—	31	2	no	—
51	C₂×#106	—	31	2	no	—
52	C₂×C₄×C₈	D₃₁	1	1	yes	characteristic
53	#382	—	31	2	no	—
54	C₂×C₃₁	#384	1	1	yes	characteristic
55	C₂×C₃₁	#427	1	2	yes	—
56	C₂×C₃₁	#495	1	2	yes	—
57	C₂×C₃₁	#499	1	2	yes	—
58	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
59	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	31	4	no	—
61	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
62	#107	—	62	4	no	—
63	#106	—	62	4	no	—
64	C₄×C₈	Dic₃₁	1	4	yes	complemented
65	#95	—	62	4	no	—
66	C₃₁	#1846	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₄	—	62	4	no	—
68	C₂²×C₄	D₁₂₄	1	2	yes	—
69	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
70	C₂×C₈	Dic₆₂	1	2	yes	—
71	C₂×C₈	Dic₆₂	1	2	yes	—
72	C₂×C₈	D₁₂₄	1	2	yes	—
73	C₂×C₈	D₁₂₄	1	2	yes	—
74	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
75	C₂×C₈	C₂×Dic₃₁	1	2	yes	—
76	C₄⋊C₄	—	62	4	no	—
77	C₄⋊C₄	—	62	4	no	—
78	C₄²	Dic₆₂	1	2	yes	—
79	C₄²	C₂×Dic₃₁	1	2	yes	—
80	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	124	4	no	—
83	C₂×C₄	—	124	8	no	—
84	C₂×C₄	C₄⋊Dic₃₁	1	4	yes	—
85	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
86	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
87	C₂×C₄	C₂×Dic₆₂	1	1	yes	characteristic
88	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
89	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
90	C₂×C₄	C₂²×Dic₃₁	1	1	yes	characteristic
91	C₈	C₄⋊Dic₃₁	1	4	yes	—
92	C₈	C₄⋊Dic₃₁	1	4	yes	—
93	C₂²	#11758446	1	1	yes	characteristic
94	C₂²	#11758451	1	1	yes	characteristic
95	C₂²	#11758481	1	4	yes	—
96	C₂²	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
97	C₄	—	2	4	no	—
98	C₄	—	248	4	no	—
99	C₄	#11758446	1	1	yes	characteristic
100	C₄	#11758446	1	1	yes	characteristic
101	C₄	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
102	C₄	C₂×C₄⋊Dic₃₁	1	1	yes	characteristic
103	C₂	#14520919	1	1	yes	characteristic
104	C₂	#14521087	1	2	yes	—
105	C₂	#14521433	1	2	yes	—
106	C₂	#14521451	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758406

C₂

yes

maximal

C₂×#11758405

C₂

yes

maximal

#14520915

C₂

yes

maximal

C₂×C₄×C₈×C₃₁

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₄×C₈×C₃₁

C₄

yes

complemented

C₃₁⋊#95

—

#11758406

—

#11758405

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

C₂×C₄

yes

—

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

Q₈

yes

—

C₂×C₈×C₃₁

Q₈

yes

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

Q₈

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₄⋊C₄

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

C₄⋊C₄

yes

—

C₈×C₃₁

C₄⋊C₄

yes

—

#1846

—

maximal, Sylow

C₂²×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₁

#127

yes

characteristic

C₂²×C₃₁

C₄⋊Q₈

yes

characteristic

C₂²×C₃₁

#135

yes

—

C₄×C₃₁

—

C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

characteristic

Dic₃₁

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

D₃₁

yes

characteristic

#382

—

C₂×C₃₁

#384

yes

characteristic

C₂×C₃₁

#427

yes

—

C₂×C₃₁

#495

yes

—

C₂×C₃₁

#499

yes

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#107

—

#106

—

C₄×C₈

Dic₃₁

yes

complemented

#95

—

C₃₁

#1846

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

Dic₆₂

yes

—

C₂×C₈

Dic₆₂

yes

—

C₂×C₈

D₁₂₄

yes

—

C₂×C₈

D₁₂₄

yes

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₂×C₈

C₂×Dic₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₆₂

yes

—

C₄²

C₂×Dic₃₁

yes

—

C₂³

C₂×D₁₂₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

—

124

—

C₂×C₄

C₄⋊Dic₃₁

yes

—

C₂×C₄

C₂×Dic₆₂

yes

characteristic

C₂×C₄

C₂×Dic₆₂

yes

characteristic

C₂×C₄

C₂×Dic₆₂

yes

characteristic

C₂×C₄

C₂×D₁₂₄

yes

characteristic

C₂×C₄

C₂×D₁₂₄

yes

characteristic

C₂×C₄

C₂²×Dic₃₁

yes

characteristic

C₈

C₄⋊Dic₃₁

yes

—

C₈

C₄⋊Dic₃₁

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×C₄⋊Dic₃₁

yes

characteristic

C₄

—

C₄

—

248

—

C₄

#11758446

yes

characteristic

100

C₄

#11758446

yes

characteristic

101

C₄

C₂×C₄⋊Dic₃₁

yes

characteristic

102

C₄

C₂×C₄⋊Dic₃₁

yes

characteristic

103

C₂

#14520919

yes

characteristic

104

C₂

#14521087

yes

—

105

C₂

#14521433

yes

—

106

C₂

#14521451

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₂×C₄×C₈×C₃₁	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	C₂×C₄×C₈	D₃₁
hypocenter	C₃₁	#1846
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₂₄

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₂×C₄×C₈×C₃₁

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

hypercenter

C₂×C₄×C₈

D₃₁

hypocenter

C₃₁

#1846

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. C₄×C₈×C₃₁	C₄	semidirect
64. C₄×C₈	Dic₃₁	semidirect
66. C₃₁	#1846	semidirect

H ≅ G/N

type

9. C₄×C₈×C₃₁

C₄

semidirect

64. C₄×C₈

Dic₃₁

semidirect

66. C₃₁

#1846

semidirect

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

subgroup	index
C₂×#11758406	2
C₂×#11758405	2
#14520915	2
C₂×C₄×C₈×C₃₁	2
#1846	31

subgroup

index

C₂×C₄×C₈×C₃₁

Subgroups — #16845875

Conjugacy classes of subgroups

Complements

Maximal subgroups