Group #16856219 — subgroups

Subgroups — C₂×#14521454 #16856219

Order 3968.

424 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 213 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×#11758446	C₂	1	2	yes	maximal
2	#14521454	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11758406	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₆₂	—	2	2	no	—
6	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₃₁	—	2	2	no	—
9	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
10	#11758446	—	2	4	no	—
11	#11758446	C₂²	1	8	yes	—
12	C₄×C₈×C₃₁	C₂²	1	4	yes	—
13	#11758406	C₂²	1	16	yes	—
14	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
17	C₂²×Dic₃₁	—	2	4	no	—
18	C₂×Dic₆₂	—	2	8	no	—
19	C₂×Dic₆₂	—	4	8	no	—
20	C₂×C₈×C₃₁	Q₈	1	8	yes	complemented
21	C₂×C₈×C₃₁	C₂³	1	4	yes	—
22	C₄²×C₃₁	D₄	1	2	yes	—
23	C₄²×C₃₁	C₂³	1	2	yes	—
24	C₄⋊Dic₃₁	—	2	4	no	—
25	C₄⋊Dic₃₁	—	2	8	no	—
26	C₄⋊Dic₃₁	C₂³	1	8	yes	—
27	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₃₁	D₈	1	4	yes	—
29	C₂×C₄×C₃₁	Q₁₆	1	4	yes	—
30	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₃₁	—	2	8	no	—
36	C₂×Dic₃₁	—	4	8	no	—
37	Dic₆₂	—	4	16	no	—
38	C₈×C₃₁	C₂×Q₈	1	8	yes	complemented
39	C₂×#500	—	31	1	no	maximal, Sylow
40	C₂²×C₃₁	C₄⋊Q₈	1	2	yes	—
41	C₂²×C₃₁	C₂×D₈	1	2	yes	—
42	C₂²×C₃₁	C₂×Q₁₆	1	2	yes	—
43	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
44	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
45	C₄×C₃₁	C₂×D₈	1	4	yes	—
46	C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
47	C₄×C₃₁	C₂²×Q₈	1	2	yes	—
48	Dic₃₁	—	4	8	no	—
49	C₂×C₄⋊Q₈	—	31	2	no	—
50	#500	—	31	8	no	—
51	C₂×#107	—	31	4	no	—
52	C₂×C₄×C₈	D₃₁	1	1	yes	characteristic
53	C₂×C₃₁	#500	1	4	yes	complemented
54	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₃₁	C₂²×D₈	1	1	yes	characteristic
56	C₂×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
57	C₂²×Q₈	—	62	2	no	—
58	C₂²×C₈	C₂×D₃₁	1	2	yes	—
59	C₄⋊Q₈	—	31	8	no	—
60	C₄⋊Q₈	—	62	4	no	—
61	C₂×C₄⋊C₄	—	31	4	no	—
62	C₂×C₄⋊C₄	—	62	2	no	—
63	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
64	#107	—	31	16	no	—
65	C₄×C₈	C₂×D₃₁	1	4	yes	—
66	C₃₁	C₂×#500	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₈	—	62	8	no	—
68	C₂×Q₈	—	124	8	no	—
69	C₂²×C₄	—	62	4	no	—
70	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
71	C₂²×C₄	D₁₂₄	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
73	C₂×C₈	Dic₆₂	1	8	yes	complemented
74	C₂×C₈	C₂²×D₃₁	1	4	yes	—
75	C₄⋊C₄	—	31	8	no	—
76	C₄⋊C₄	—	62	4	no	—
77	C₄⋊C₄	—	62	8	no	—
78	C₄²	D₁₂₄	1	2	yes	—
79	C₄²	C₂²×D₃₁	1	2	yes	—
80	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
81	Q₈	—	124	16	no	—
82	C₂×C₄	—	62	8	no	—
83	C₂×C₄	—	124	8	no	—
84	C₂×C₄	D₂₄₈	1	4	yes	—
85	C₂×C₄	Dic₁₂₄	1	4	yes	—
86	C₂×C₄	C₂×Dic₆₂	1	4	yes	—
87	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
88	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
89	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
90	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
91	C₈	C₂×Dic₆₂	1	8	yes	complemented
92	C₂²	#11758446	1	2	yes	—
93	C₂²	C₂×D₂₄₈	1	2	yes	—
94	C₂²	C₂×Dic₁₂₄	1	2	yes	—
95	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
96	C₄	—	124	8	no	—
97	C₄	#11758446	1	2	yes	—
98	C₄	C₂×D₂₄₈	1	4	yes	—
99	C₄	C₂×Dic₁₂₄	1	4	yes	—
100	C₄	C₂²×Dic₆₂	1	2	yes	—
101	C₂	#14521454	1	4	yes	direct product factor
102	C₂	C₂×#11758446	1	1	yes	characteristic
103	C₂	C₂²×D₂₄₈	1	1	yes	characteristic
104	C₂	C₂²×Dic₁₂₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758446

C₂

yes

maximal

#14521454

C₂

yes

maximal, direct product factor

C₂×#11758406

C₂

yes

maximal

C₂×C₄×C₈×C₃₁

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

—

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

#11758446

—

#11758446

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

—

#11758406

C₂²

yes

—

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

D₄

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

Q₈

yes

complemented

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

D₈

yes

—

C₂×C₄×C₃₁

Q₁₆

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

C₈×C₃₁

C₂×Q₈

yes

complemented

C₂×#500

—

maximal, Sylow

C₂²×C₃₁

C₄⋊Q₈

yes

—

C₂²×C₃₁

C₂×D₈

yes

—

C₂²×C₃₁

C₂×Q₁₆

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

C₄⋊Q₈

yes

—

C₄×C₃₁

C₂×D₈

yes

—

C₄×C₃₁

C₂×Q₁₆

yes

—

C₄×C₃₁

C₂²×Q₈

yes

—

Dic₃₁

—

C₂×C₄⋊Q₈

—

#500

—

C₂×#107

—

C₂×C₄×C₈

D₃₁

yes

characteristic

C₂×C₃₁

#500

yes

complemented

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

C₂²×D₈

yes

characteristic

C₂×C₃₁

C₂²×Q₁₆

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

C₂×D₃₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#107

—

C₄×C₈

C₂×D₃₁

yes

—

C₃₁

C₂×#500

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

124

—

C₂²×C₄

—

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

D₁₂₄

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

Dic₆₂

yes

complemented

C₂×C₈

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₁₂₄

yes

—

C₄²

C₂²×D₃₁

yes

—

C₂³

C₂×D₁₂₄

yes

characteristic

Q₈

—

124

—

C₂×C₄

—

C₂×C₄

—

124

—

C₂×C₄

D₂₄₈

yes

—

C₂×C₄

Dic₁₂₄

yes

—

C₂×C₄

C₂×Dic₆₂

yes

—

C₂×C₄

C₂×D₁₂₄

yes

characteristic

C₂×C₄

C₂×D₁₂₄

yes

—

C₂×C₄

C₂×D₁₂₄

yes

—

C₂×C₄

C₂³×D₃₁

yes

characteristic

C₈

C₂×Dic₆₂

yes

complemented

C₂²

#11758446

yes

—

C₂²

C₂×D₂₄₈

yes

—

C₂²

C₂×Dic₁₂₄

yes

—

C₂²

C₂²×D₁₂₄

yes

characteristic

C₄

—

124

—

C₄

#11758446

yes

—

C₄

C₂×D₂₄₈

yes

—

C₄

C₂×Dic₁₂₄

yes

—

100

C₄

C₂²×Dic₆₂

yes

—

101

C₂

#14521454

yes

direct product factor

102

C₂

C₂×#11758446

yes

characteristic

103

C₂

C₂²×D₂₄₈

yes

characteristic

104

C₂

C₂²×Dic₁₂₄

yes

characteristic

	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×#500
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×#500

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #14521454	C₂	direct (N × H)
20. C₂×C₈×C₃₁	Q₈	semidirect
38. C₈×C₃₁	C₂×Q₈	semidirect
53. C₂×C₃₁	#500	semidirect
66. C₃₁	C₂×#500	semidirect
73. C₂×C₈	Dic₆₂	semidirect
91. C₈	C₂×Dic₆₂	semidirect
101. C₂	#14521454	direct (N × H)

H ≅ G/N

type

2. #14521454

C₂

direct (N × H)

20. C₂×C₈×C₃₁

Q₈

semidirect

38. C₈×C₃₁

C₂×Q₈

semidirect

53. C₂×C₃₁

#500

semidirect

66. C₃₁

C₂×#500

semidirect

73. C₂×C₈

Dic₆₂

semidirect

91. C₈

C₂×Dic₆₂

semidirect

101. C₂

#14521454

direct (N × H)

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

subgroup	index
C₂×#11758446	2
#14521454	2
C₂×#11758406	2
C₂×C₄×C₈×C₃₁	2
C₂×#500	31

subgroup

index

C₂×C₄×C₈×C₃₁

Subgroups — C₂×#14521454 #16856219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#14521454 #16856219

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#14521454 #16856219