Group #16856225 — subgroups

Subgroups — #16856225

Order 3968.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 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	#14521606	C₂	1	2	yes	maximal
2	#14521454	C₂	1	2	yes	maximal, complemented
3	#14521451	C₂	1	4	yes	maximal, complemented
4	#14521447	C₂	1	2	yes	maximal, complemented
5	#14521087	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	#11758503	C₂²	1	4	yes	—
10	#11758501	—	2	4	no	—
11	#11758447	C₂²	1	4	yes	—
12	#11758446	C₂²	1	4	yes	—
13	C₄×Dic₆₂	—	2	4	no	—
14	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₃₁	C₂²	1	2	yes	—
16	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	#11758406	C₂²	1	8	yes	—
18	#11758405	C₂²	1	8	yes	—
19	C₂²×C₄×C₃₁	D₄	1	2	yes	—
20	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×Dic₆₂	—	2	4	no	—
22	C₂×C₈×C₃₁	Q₈	1	8	yes	complemented
23	C₂×C₈×C₃₁	C₂³	1	2	yes	—
24	C₂×C₈×C₃₁	C₂³	1	2	yes	—
25	C₄²×C₃₁	D₄	1	2	yes	—
26	C₄²×C₃₁	C₂³	1	1	yes	characteristic
27	C₄²×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²⋊Dic₃₁	—	2	4	no	—
29	C₄⋊Dic₃₁	—	2	4	no	—
30	C₄⋊Dic₃₁	C₂³	1	8	yes	—
31	Dic₃₁⋊C₄	—	2	8	no	—
32	C₄×Dic₃₁	—	2	4	no	—
33	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₃₁	—	2	4	no	—
35	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
39	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
40	C₂×Dic₃₁	—	2	8	no	—
41	Dic₆₂	—	4	8	no	—
42	C₈×C₃₁	C₂×Q₈	1	4	yes	—
43	C₈×C₃₁	C₂×Q₈	1	4	yes	—
44	#3160	—	31	1	no	maximal, Sylow
45	C₂²×C₃₁	—	2	2	no	—
46	C₂²×C₃₁	C₄⋊Q₈	1	2	yes	—
47	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
48	C₄×C₃₁	—	2	2	no	—
49	C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
50	C₄×C₃₁	#135	1	4	yes	—
51	C₄×C₃₁	C₂²×Q₈	1	2	yes	—
52	Dic₃₁	—	4	8	no	—
53	#533	—	31	2	no	—
54	#500	—	31	2	no	—
55	#499	—	31	4	no	—
56	#498	—	31	2	no	—
57	#427	—	31	4	no	—
58	C₂×C₄×C₈	D₃₁	1	1	yes	characteristic
59	C₂×C₃₁	—	2	2	no	—
60	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₃₁	C₂×#135	1	2	yes	—
62	C₂²×C₈	C₂×D₃₁	1	2	yes	—
63	C₄⋊Q₈	—	31	4	no	—
64	#125	—	31	4	no	—
65	C₂²⋊Q₈	—	62	4	no	—
66	C₄×Q₈	—	62	4	no	—
67	#117	—	31	4	no	—
68	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
69	#107	—	31	8	no	—
70	#106	—	31	8	no	—
71	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
72	C₄×C₈	C₂×D₃₁	1	2	yes	—
73	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
74	C₃₁	#3160	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₈	—	62	4	no	—
76	C₂²×C₄	D₁₂₄	1	2	yes	—
77	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
78	C₂×C₈	Dic₆₂	1	8	yes	complemented
79	C₂×C₈	C₂²×D₃₁	1	2	yes	—
80	C₂×C₈	C₂²×D₃₁	1	2	yes	—
81	C₄⋊C₄	—	31	8	no	—
82	C₄⋊C₄	—	62	8	no	—
83	C₄⋊C₄	—	62	4	no	—
84	C₂²⋊C₄	—	62	4	no	—
85	C₄²	—	62	4	no	—
86	C₄²	D₁₂₄	1	2	yes	—
87	C₄²	C₂²×D₃₁	1	1	yes	characteristic
88	C₄²	C₂²×D₃₁	1	1	yes	characteristic
89	C₂³	C₂×D₁₂₄	1	1	yes	characteristic
90	Q₈	—	124	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	62	8	no	—
93	C₂×C₄	C₂×Dic₆₂	1	4	yes	—
94	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
95	C₂×C₄	C₂×D₁₂₄	1	2	yes	—
96	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
97	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
98	C₈	C₂×Dic₆₂	1	4	yes	—
99	C₈	C₂×Dic₆₂	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#11758446	1	2	yes	—
102	C₂²	C₂²×D₁₂₄	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	124	8	no	—
105	C₄	#11758446	1	2	yes	—
106	C₄	#11758481	1	4	yes	—
107	C₄	C₂²×Dic₆₂	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#11758446	1	1	yes	characteristic
110	C₂	C₂×#11758481	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14521606

C₂

yes

maximal

#14521454

C₂

yes

maximal, complemented

#14521451

C₂

yes

maximal, complemented

#14521447

C₂

yes

maximal, complemented

#14521087

C₂

yes

maximal

C₂×C₄×C₈×C₃₁

C₂

yes

maximal, characteristic

C₂²×C₈×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

#11758503

C₂²

yes

—

#11758501

—

#11758447

C₂²

yes

—

#11758446

C₂²

yes

—

C₄×Dic₆₂

—

C₄×C₈×C₃₁

C₂²

yes

characteristic

C₄×C₈×C₃₁

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

characteristic

#11758406

C₂²

yes

—

#11758405

C₂²

yes

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₆₂

—

C₂×C₈×C₃₁

Q₈

yes

complemented

C₂×C₈×C₃₁

C₂³

yes

—

C₂×C₈×C₃₁

C₂³

yes

—

C₄²×C₃₁

D₄

yes

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₂²⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

Dic₆₂

—

C₈×C₃₁

C₂×Q₈

yes

—

C₈×C₃₁

C₂×Q₈

yes

—

#3160

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₄⋊Q₈

yes

—

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₄⋊Q₈

yes

—

C₄×C₃₁

#135

yes

—

C₄×C₃₁

C₂²×Q₈

yes

—

Dic₃₁

—

—

—

—

—

—

C₂×C₄×C₈

D₃₁

yes

characteristic

C₂×C₃₁

—

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

C₂×#135

yes

—

C₂²×C₈

C₂×D₃₁

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₄×C₈

C₂×D₃₁

yes

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#3160

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

D₁₂₄

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

Dic₆₂

yes

complemented

C₂×C₈

C₂²×D₃₁

yes

—

C₂×C₈

C₂²×D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₁₂₄

yes

—

C₄²

C₂²×D₃₁

yes

characteristic

C₄²

C₂²×D₃₁

yes

characteristic

C₂³

C₂×D₁₂₄

yes

characteristic

Q₈

—

124

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₆₂

yes

—

C₂×C₄

C₂×D₁₂₄

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

—

100

C₂²

—

101

C₂²

#11758446

yes

—

102

C₂²

C₂²×D₁₂₄

yes

characteristic

103

C₄

—

104

C₄

—

124

—

105

C₄

#11758446

yes

—

106

C₄

#11758481

yes

—

107

C₄

C₂²×Dic₆₂

yes

—

108

C₂

—

109

C₂

C₂×#11758446

yes

characteristic

110

C₂

C₂×#11758481

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂₄
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₂×C₄×C₈×C₃₁	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	C₂³×C₃₁	C₂×D₄
hypercenter	C₂×C₄×C₈	D₃₁
hypocenter	C₃₁	#3160
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₁₂₄

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₂×C₄×C₈×C₃₁

C₂

socle

C₂²×C₃₁

C₂²×D₄

prime radical

C₂³×C₃₁

C₂×D₄

hypercenter

C₂×C₄×C₈

D₃₁

hypocenter

C₃₁

#3160

LCS

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #14521454	C₂	semidirect
3. #14521451	C₂	semidirect
4. #14521447	C₂	semidirect
22. C₂×C₈×C₃₁	Q₈	semidirect
74. C₃₁	#3160	semidirect
78. C₂×C₈	Dic₆₂	semidirect

H ≅ G/N

type

2. #14521454

C₂

semidirect

3. #14521451

C₂

semidirect

4. #14521447

C₂

semidirect

22. C₂×C₈×C₃₁

Q₈

semidirect

74. C₃₁

#3160

semidirect

78. C₂×C₈

Dic₆₂

semidirect

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

subgroup	index
#14521606	2
#14521454	2
#14521451	2
#14521447	2
#14521087	2
C₂×C₄×C₈×C₃₁	2
#3160	31

subgroup

index

C₂×C₄×C₈×C₃₁

Subgroups — #16856225

Conjugacy classes of subgroups

Complements

Maximal subgroups