Group #30664869 — subgroups

Subgroups — #30664869

Order 16768.

148 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 63 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	#22553439	C₂	1	2	yes	maximal
2	C₄×Dic₅₂₄	C₂	1	1	yes	maximal, characteristic
3	#22553042	C₂	1	2	yes	maximal
4	#22553033	C₂	1	1	yes	maximal, characteristic
5	#334×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₂₄	C₄	1	1	yes	characteristic
7	C₄×Dic₂₆₂	C₂²	1	2	yes	—
8	#16862180	C₄	1	1	yes	characteristic
9	Dic₁₃₁⋊C₈	—	2	2	no	—
10	#16862171	C₄	1	2	yes	—
11	#16862156	—	2	2	no	—
12	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
15	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₆₂	C₂×C₄	1	2	yes	—
17	Dic₅₂₄	—	2	2	no	—
18	C₂×C₁₃₁⋊C₈	—	2	1	no	—
19	C₂×C₁₃₁⋊C₈	D₄	1	2	yes	—
20	C₄⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
21	Dic₁₃₁⋊C₄	—	2	2	no	—
22	C₄×Dic₁₃₁	—	2	2	no	—
23	C₂×C₈×C₁₃₁	—	2	2	no	—
24	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
27	Dic₂₆₂	—	2	2	no	—
28	Dic₂₆₂	—	2	2	no	—
29	C₂×Dic₁₃₁	—	2	2	no	—
30	C₂×Dic₁₃₁	—	2	2	no	—
31	C₁₃₁⋊C₈	—	2	2	no	—
32	C₁₃₁⋊C₈	—	4	1	no	—
33	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₁	#41	1	1	yes	characteristic
36	C₈×C₁₃₁	—	2	1	no	—
37	C₈×C₁₃₁	—	4	2	no	—
38	C₈×C₁₃₁	M₁₆	1	2	yes	—
39	C₂²×C₁₃₁	C₄×D₄	1	1	yes	characteristic
40	C₄×C₁₃₁	—	2	1	no	—
41	C₄×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
42	C₄×C₁₃₁	#131	1	1	yes	characteristic
43	C₄×C₁₃₁	#137	1	2	yes	—
44	Dic₁₃₁	—	4	2	no	—
45	Dic₁₃₁	—	4	2	no	—
46	C₂×C₁₃₁	#436	1	1	yes	characteristic
47	C₂×C₁₃₁	#441	1	1	yes	characteristic
48	C₂×C₁₃₁	#444	1	1	yes	characteristic
49	C₁₃₁	#1593	1	1	yes	characteristic, complemented, Sylow
50	#1593	—	131	1	no	maximal, Sylow
51	#446	—	131	2	no	—
52	C₄×Q₁₆	—	131	1	no	—
53	#334	D₁₃₁	1	1	yes	characteristic
54	Q₈⋊C₈	—	131	2	no	—
55	#322	—	131	1	no	—
56	C₂×Q₁₆	—	131	1	no	—
57	C₄×Q₈	—	131	2	no	—
58	#107	—	131	1	no	—
59	C₄⋊C₈	—	262	2	no	—
60	C₄⋊C₈	C₂×D₁₃₁	1	2	yes	—
61	#103	—	131	2	no	—
62	#97	—	262	2	no	—
63	C₄×C₈	—	131	1	no	—
64	C₄×C₈	—	131	1	no	—
65	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
66	C₂×Q₈	—	131	2	no	—
67	Q₁₆	—	262	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	262	1	no	—
70	C₂×C₈	—	131	2	no	—
71	C₂×C₈	C₄×D₁₃₁	1	1	yes	characteristic
72	C₂×C₈	C₄×D₁₃₁	1	1	yes	characteristic
73	C₄⋊C₄	—	262	2	no	—
74	C₄⋊C₄	—	131	2	no	—
75	C₄²	—	262	2	no	—
76	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
77	Q₈	—	262	2	no	—
78	Q₈	—	262	2	no	—
79	C₂×C₄	—	262	2	no	—
80	C₂×C₄	—	262	2	no	—
81	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
82	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
83	C₂×C₄	#14523514	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	4	2	no	—
86	C₈	—	262	2	no	—
87	C₈	—	524	1	no	—
88	C₈	#14523496	1	2	yes	—
89	C₂²	#16862208	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	524	2	no	—
92	C₄	—	524	2	no	—
93	C₄	C₂×#14523496	1	1	yes	characteristic
94	C₄	#16862254	1	1	yes	characteristic
95	C₄	#16862275	1	2	yes	—
96	C₂	#22553389	1	1	yes	characteristic
97	C₂	#22553418	1	1	yes	characteristic
98	C₂	#22553433	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553439

C₂

yes

maximal

C₄×Dic₅₂₄

C₂

yes

maximal, characteristic

#22553042

C₂

yes

maximal

#22553033

C₂

yes

maximal, characteristic

#334×C₁₃₁

C₂

yes

maximal, characteristic

C₂×Dic₅₂₄

C₄

yes

characteristic

C₄×Dic₂₆₂

C₂²

yes

—

#16862180

C₄

yes

characteristic

Dic₁₃₁⋊C₈

—

#16862171

C₄

yes

—

#16862156

—

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

C₂²

yes

—

C₄×C₈×C₁₃₁

C₂²

yes

characteristic

C₂×Dic₂₆₂

C₂×C₄

yes

—

Dic₅₂₄

—

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

D₄

yes

—

C₄⋊Dic₁₃₁

C₂×C₄

yes

—

Dic₁₃₁⋊C₄

—

C₄×Dic₁₃₁

—

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₁

C₂×C₄

yes

characteristic

C₄²×C₁₃₁

C₂³

yes

characteristic

Dic₂₆₂

—

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

#41

yes

characteristic

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₈×C₁₃₁

M₁₆

yes

—

C₂²×C₁₃₁

C₄×D₄

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×M₁₆

yes

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#1593

—

131

maximal, Sylow

#446

—

131

—

C₄×Q₁₆

—

131

—

#334

D₁₃₁

yes

characteristic

Q₈⋊C₈

—

131

—

#322

—

131

—

C₂×Q₁₆

—

131

—

C₄×Q₈

—

131

—

#107

—

131

—

C₄⋊C₈

—

262

—

C₄⋊C₈

C₂×D₁₃₁

yes

—

#103

—

131

—

#97

—

262

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

C₄×C₈

C₂×D₁₃₁

yes

characteristic

C₂×Q₈

—

131

—

Q₁₆

—

262

—

C₂×C₈

—

C₂×C₈

—

262

—

C₂×C₈

—

131

—

C₂×C₈

C₄×D₁₃₁

yes

characteristic

C₂×C₈

C₄×D₁₃₁

yes

characteristic

C₄⋊C₄

—

262

—

C₄⋊C₄

—

131

—

C₄²

—

262

—

C₄²

C₂²×D₁₃₁

yes

characteristic

Q₈

—

262

—

Q₈

—

262

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

C₂×C₄

D₄×D₁₃₁

yes

characteristic

C₂×C₄

#14523514

yes

characteristic

C₈

—

C₈

—

C₈

—

262

—

C₈

—

524

—

C₈

#14523496

yes

—

C₂²

#16862208

yes

characteristic

C₄

—

C₄

—

524

—

C₄

—

524

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	D₄×D₁₃₁
commutator	C₂×C₄×C₁₃₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃₁
Fitting	#334×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	C₄×D₄
cosocle	C₄²×C₁₃₁	C₂³
hypercenter	#334	D₁₃₁
hypocenter	C₁₃₁	#1593
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₁₃₁

commutator

C₂×C₄×C₁₃₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃₁

Fitting

#334×C₁₃₁

C₂

socle, prime radical

C₂²×C₁₃₁

C₄×D₄

cosocle

C₄²×C₁₃₁

C₂³

D₁₃₁

C₁₃₁

G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
49. C₁₃₁	#1593	semidirect

H ≅ G/N

type

49. C₁₃₁

#1593

semidirect

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

subgroup	index
#22553439	2
C₄×Dic₅₂₄	2
#22553042	2
#22553033	2
#334×C₁₃₁	2
#1593	131

subgroup

index

C₄×Dic₅₂₄

#334×C₁₃₁

131

Subgroups — #30664869

Conjugacy classes of subgroups

Complements

Maximal subgroups