Group #16844529 — subgroups

Subgroups — #16844529

Order 3968.

160 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 71 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	#14521173	C₂	1	2	yes	maximal
2	C₄×Dic₁₂₄	C₂	1	1	yes	maximal, characteristic
3	#14520779	C₂	1	2	yes	maximal
4	#14520770	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₂₄	C₄	1	1	yes	characteristic
7	C₄×Dic₆₂	C₂²	1	2	yes	—
8	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	#11758406	C₄	1	1	yes	characteristic
11	#11758404	C₄	1	2	yes	—
12	Dic₃₁⋊C₈	—	2	2	no	—
13	C₈×Dic₃₁	—	2	2	no	—
14	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×Dic₆₂	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₃₁	—	2	2	no	—
18	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₃₁	C₂³	1	1	yes	characteristic
21	C₄⋊Dic₃₁	C₂×C₄	1	2	yes	—
22	Dic₃₁⋊C₄	—	2	2	no	—
23	C₄×Dic₃₁	—	2	2	no	—
24	C₂×C₃₁⋊C₈	—	2	1	no	—
25	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
26	Dic₁₂₄	C₈	1	4	yes	complemented
27	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₃₁	#41	1	1	yes	characteristic
30	C₂×Dic₃₁	—	2	2	no	—
31	C₂×Dic₃₁	—	2	2	no	—
32	Dic₆₂	—	2	2	no	—
33	Dic₆₂	C₂×C₈	1	4	yes	—
34	C₈×C₃₁	—	2	1	no	—
35	C₈×C₃₁	—	4	2	no	—
36	C₈×C₃₁	C₂×C₈	1	2	yes	—
37	C₃₁⋊C₈	—	2	2	no	—
38	C₃₁⋊C₈	—	4	1	no	—
39	#1579	—	31	1	no	maximal, Sylow
40	C₂²×C₃₁	C₄×D₄	1	1	yes	characteristic
41	C₄×C₃₁	—	2	1	no	—
42	C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
43	C₄×C₃₁	#131	1	1	yes	characteristic
44	C₄×C₃₁	#137	1	2	yes	—
45	Dic₃₁	—	2	4	no	—
46	Dic₃₁	—	4	2	no	—
47	C₈×Q₈	—	31	2	no	—
48	C₄×Q₁₆	—	31	1	no	—
49	#334	D₃₁	1	1	yes	characteristic
50	Q₈⋊C₈	—	31	2	no	—
51	#322	—	31	1	no	—
52	C₂×C₃₁	C₈×D₄	1	1	yes	characteristic
53	C₂×C₃₁	#441	1	1	yes	characteristic
54	C₂×C₃₁	#444	1	1	yes	characteristic
55	C₂×Q₁₆	—	31	1	no	—
56	C₄×Q₈	—	31	2	no	—
57	#107	—	31	1	no	—
58	C₄⋊C₈	—	62	2	no	—
59	C₄⋊C₈	C₂×D₃₁	1	2	yes	—
60	#103	—	31	2	no	—
61	C₄×C₈	—	31	1	no	—
62	C₄×C₈	—	31	1	no	—
63	C₄×C₈	—	62	2	no	—
64	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
65	C₃₁	#1579	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₈	—	31	2	no	—
67	Q₁₆	—	31	4	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	31	2	no	—
70	C₂×C₈	—	62	1	no	—
71	C₂×C₈	C₄×D₃₁	1	1	yes	characteristic
72	C₂×C₈	C₄×D₃₁	1	1	yes	characteristic
73	C₄⋊C₄	—	31	2	no	—
74	C₄⋊C₄	—	62	2	no	—
75	C₄²	—	62	2	no	—
76	C₄²	C₂²×D₃₁	1	1	yes	characteristic
77	Q₈	—	31	4	no	—
78	Q₈	—	62	2	no	—
79	C₂×C₄	—	62	2	no	—
80	C₂×C₄	—	62	2	no	—
81	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
82	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
83	C₂×C₄	#92499	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	4	2	no	—
86	C₈	—	62	2	no	—
87	C₈	—	124	1	no	—
88	C₈	C₈×D₃₁	1	2	yes	—
89	C₂²	#11758470	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	62	4	no	—
92	C₄	—	124	2	no	—
93	C₄	C₂×C₈×D₃₁	1	1	yes	characteristic
94	C₄	#11758484	1	1	yes	characteristic
95	C₄	#11758492	1	2	yes	—
96	C₂	#14521114	1	1	yes	characteristic
97	C₂	#14521155	1	1	yes	characteristic
98	C₂	#14521170	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521173

C₂

yes

maximal

C₄×Dic₁₂₄

C₂

yes

maximal, characteristic

#14520779

C₂

yes

maximal

#14520770

C₂

yes

maximal, characteristic

C₃₁×#334

C₂

yes

maximal, characteristic

C₂×Dic₁₂₄

C₄

yes

characteristic

C₄×Dic₆₂

C₂²

yes

—

C₃₁×C₄⋊C₈

C₂²

yes

—

C₄×C₈×C₃₁

C₂²

yes

characteristic

#11758406

C₄

yes

characteristic

#11758404

C₄

yes

—

Dic₃₁⋊C₈

—

C₈×Dic₃₁

—

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×Dic₆₂

C₂×C₄

yes

—

C₂×C₈×C₃₁

—

C₂×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₃₁

C₂×C₄

yes

characteristic

C₄²×C₃₁

C₂³

yes

characteristic

C₄⋊Dic₃₁

C₂×C₄

yes

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

D₄

yes

—

Dic₁₂₄

C₈

yes

complemented

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

#41

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

C₂×C₈

yes

—

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

C₂×C₈

yes

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#1579

—

maximal, Sylow

C₂²×C₃₁

C₄×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₂²×C₈

yes

C₄×C₃₁

yes

C₄×C₃₁

yes

—

Dic₃₁

—

Dic₃₁

—

C₈×Q₈

—

C₄×Q₁₆

—

#334

D₃₁

yes

characteristic

Q₈⋊C₈

—

#322

—

C₂×C₃₁

C₈×D₄

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₁

yes

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1579

yes

characteristic, complemented, Sylow

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₁

yes

characteristic

C₂×C₈

C₄×D₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²×D₃₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

C₂×C₄

D₄×D₃₁

yes

characteristic

C₂×C₄

#92499

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

124

—

C₈

C₈×D₃₁

yes

—

C₂²

#11758470

yes

characteristic

C₄

—

C₄

—

C₄

—

124

—

C₄

C₂×C₈×D₃₁

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	D₄×D₃₁
commutator	C₄×C₃₁	C₂²×C₈
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#334	C₂
socle, prime radical	C₂²×C₃₁	C₄×D₄
cosocle	C₄²×C₃₁	C₂³
hypercenter	#334	D₃₁
hypocenter	C₃₁	#1579
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₃₁

commutator

C₄×C₃₁

C₂²×C₈

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#334

C₂

socle, prime radical

C₂²×C₃₁

C₄×D₄

cosocle

C₄²×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
26. Dic₁₂₄	C₈	semidirect
65. C₃₁	#1579	semidirect

H ≅ G/N

type

26. Dic₁₂₄

C₈

semidirect

65. C₃₁

#1579

semidirect

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

subgroup	index
#14521173	2
C₄×Dic₁₂₄	2
#14520779	2
#14520770	2
C₃₁×#334	2
#1579	31

subgroup

index

C₄×Dic₁₂₄

C₃₁×#334

Subgroups — #16844529

Conjugacy classes of subgroups

Complements

Maximal subgroups