Group #16856215 — subgroups

Subgroups — C₂×#14521451 #16856215

Order 3968.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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×#11758447	C₂	1	2	yes	maximal
2	#14521451	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11758406	C₂	1	2	yes	maximal
4	C₂×#11758405	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
10	C₂×Dic₃₁⋊C₄	—	2	4	no	—
11	#11758447	—	2	4	no	—
12	#11758447	C₂²	1	8	yes	—
13	C₄×C₈×C₃₁	C₂²	1	4	yes	—
14	#11758406	C₂²	1	8	yes	—
15	#11758405	C₂²	1	8	yes	—
16	C₂²×C₄×C₃₁	D₄	1	2	yes	—
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₁	—	2	4	no	—
19	C₂×C₈×C₃₁	Q₈	1	4	yes	—
20	C₂×C₈×C₃₁	Q₈	1	4	yes	—
21	C₂×C₈×C₃₁	C₂³	1	2	yes	—
22	C₂×C₈×C₃₁	C₂³	1	2	yes	—
23	C₄²×C₃₁	D₄	1	2	yes	—
24	C₄²×C₃₁	C₂³	1	2	yes	—
25	C₄⋊Dic₃₁	—	2	4	no	—
26	C₄⋊Dic₃₁	C₂³	1	8	yes	—
27	Dic₃₁⋊C₄	—	2	16	no	—
28	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₃₁	—	2	4	no	—
30	C₂×C₄×C₃₁	C₂×D₄	1	4	yes	—
31	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
33	C₂×C₄×C₃₁	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₃₁	—	2	8	no	—
36	C₂×Dic₃₁	—	4	8	no	—
37	C₈×C₃₁	C₂×Q₈	1	4	yes	—
38	C₈×C₃₁	C₂×Q₈	1	4	yes	—
39	C₂×#499	—	31	1	no	maximal, Sylow
40	C₂²×C₃₁	C₄⋊Q₈	1	2	yes	—
41	C₂²×C₃₁	#135	1	4	yes	—
42	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
43	C₄×C₃₁	—	2	4	no	—
44	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
45	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
46	C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
47	Dic₃₁	—	4	8	no	—
48	C₂×#125	—	31	2	no	—
49	#499	—	31	8	no	—
50	C₂×#107	—	31	2	no	—
51	C₂×#106	—	31	2	no	—
52	C₂×C₄×C₈	D₃₁	1	1	yes	characteristic
53	C₂×C₃₁	#499	1	4	yes	complemented
54	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₃₁	C₂×#135	1	2	yes	—
56	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
58	#125	—	31	8	no	—
59	#125	—	62	4	no	—
60	C₂×C₄⋊C₄	—	31	4	no	—
61	C₂×C₄⋊C₄	—	62	4	no	—
62	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
63	#107	—	31	8	no	—
64	#106	—	31	8	no	—
65	C₄×C₈	C₂×D₃₁	1	4	yes	—
66	C₃₁	C₂×#499	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	4	yes	—
71	C₂×C₈	Dic₆₂	1	4	yes	—
72	C₂×C₈	C₂²×D₃₁	1	2	yes	—
73	C₂×C₈	C₂²×D₃₁	1	2	yes	—
74	C₄⋊C₄	—	31	8	no	—
75	C₄⋊C₄	—	62	16	no	—
76	C₄⋊C₄	—	62	4	no	—
77	C₄²	D₁₂₄	1	2	yes	—
78	C₄²	C₂²×D₃₁	1	2	yes	—
79	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	62	8	no	—
82	C₂×C₄	—	124	8	no	—
83	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
84	C₂×C₄	C₂×Dic₆₂	1	2	yes	—
85	C₂×C₄	C₂×D₁₂₄	1	4	yes	—
86	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
87	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
88	C₈	C₂×Dic₆₂	1	4	yes	—
89	C₈	C₂×Dic₆₂	1	4	yes	—
90	C₂²	#11758446	1	2	yes	—
91	C₂²	#11758481	1	4	yes	—
92	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	124	8	no	—
95	C₄	#11758446	1	2	yes	—
96	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
97	C₄	C₂²×Dic₆₂	1	1	yes	characteristic
98	C₂	#14521451	1	4	yes	direct product factor
99	C₂	C₂×#11758446	1	1	yes	characteristic
100	C₂	C₂×#11758481	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758447

C₂

yes

maximal

#14521451

C₂

yes

maximal, direct product factor

C₂×#11758406

C₂

yes

maximal

C₂×#11758405

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

—

C₂×Dic₃₁⋊C₄

—

#11758447

—

#11758447

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

—

#11758406

C₂²

yes

—

#11758405

C₂²

yes

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×C₈×C₃₁

Q₈

yes

—

C₂×C₈×C₃₁

Q₈

yes

—

C₂×C₈×C₃₁

C₂³

yes

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₈×C₃₁

C₂×Q₈

yes

—

C₈×C₃₁

C₂×Q₈

yes

—

C₂×#499

—

maximal, Sylow

C₂²×C₃₁

C₄⋊Q₈

yes

—

C₂²×C₃₁

#135

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₄⋊Q₈

yes

—

C₄×C₃₁

C₂²×Q₈

yes

characteristic

C₄×C₃₁

C₂²×Q₈

yes

characteristic

Dic₃₁

—

—

—

—

—

C₂×C₄×C₈

D₃₁

yes

characteristic

C₂×C₃₁

#499

yes

complemented

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

C₂×#135

yes

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

C₂²×C₈

C₂×D₃₁

yes

characteristic

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂×D₃₁

yes

—

C₃₁

C₂×#499

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

C₂²×D₃₁

yes

—

C₂×C₈

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₁₂₄

yes

—

C₄²

C₂²×D₃₁

yes

—

C₂³

C₂×D₁₂₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

C₂×Dic₆₂

yes

—

C₂×C₄

C₂×Dic₆₂

yes

—

C₂×C₄

C₂×D₁₂₄

yes

—

C₂×C₄

C₂×D₁₂₄

yes

characteristic

C₂×C₄

C₂³×D₃₁

yes

characteristic

C₈

C₂×Dic₆₂

yes

—

C₈

C₂×Dic₆₂

yes

—

C₂²

#11758446

yes

—

C₂²

#11758481

yes

—

C₂²

C₂²×D₁₂₄

yes

characteristic

C₄

—

C₄

—

124

—

C₄

#11758446

yes

—

C₄

C₂²×Dic₆₂

yes

characteristic

C₄

C₂²×Dic₆₂

yes

characteristic

C₂

#14521451

yes

direct product factor

C₂

C₂×#11758446

yes

characteristic

100

C₂

C₂×#11758481

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₂₄
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₄×C₈×C₃₁	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	C₂×C₄×C₈	D₃₁
hypocenter	C₃₁	C₂×#499
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₂₄

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂×C₄×C₈×C₃₁

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

hypercenter

C₂×C₄×C₈

D₃₁

hypocenter

C₃₁

C₂×#499

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #14521451	C₂	direct (N × H)
53. C₂×C₃₁	#499	semidirect
66. C₃₁	C₂×#499	semidirect
98. C₂	#14521451	direct (N × H)

H ≅ G/N

type

2. #14521451

C₂

direct (N × H)

53. C₂×C₃₁

#499

semidirect

66. C₃₁

C₂×#499

semidirect

98. C₂

#14521451

direct (N × H)

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

subgroup	index
C₂×#11758447	2
#14521451	2
C₂×#11758406	2
C₂×#11758405	2
C₂×C₄×C₈×C₃₁	2
C₂×#499	31

subgroup

index

C₂×C₄×C₈×C₃₁

Subgroups — C₂×#14521451 #16856215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14521451 #16856215

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521451 #16856215