Group #30665102 — subgroups

Subgroups — #30665102

Order 16768.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22554042	C₂	1	1	yes	maximal, characteristic
2	#22553691	C₂	1	1	yes	maximal, characteristic
3	#22553608	C₂	1	2	yes	maximal
4	#22553042	C₂	1	2	yes	maximal
5	#333×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14523503	—	2	2	no	—
7	#16862246	C₂²	1	1	yes	characteristic
8	#16862245	—	2	2	no	—
9	#16862244	—	2	2	no	—
10	C₄×Dic₂₆₂	C₂²	1	2	yes	—
11	#16862211	—	2	2	no	—
12	#16862180	—	2	2	no	—
13	#16862171	—	2	2	no	—
14	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
17	C₂×Dic₂₆₂	—	2	2	no	—
18	C₂×Dic₂₆₂	—	2	1	no	—
19	C₂×Dic₂₆₂	D₄	1	2	yes	—
20	#14523503	—	4	4	no	—
21	C₂×C₁₃₁⋊C₈	—	2	1	no	—
22	C₄⋊Dic₁₃₁	—	2	2	no	—
23	C₄⋊Dic₁₃₁	D₄	1	2	yes	—
24	Dic₁₃₁⋊C₄	—	2	4	no	—
25	Dic₁₃₁⋊C₄	—	2	2	no	—
26	C₄×Dic₁₃₁	—	2	2	no	—
27	C₄×Dic₁₃₁	—	2	2	no	—
28	C₂×Q₈×C₁₃₁	D₄	1	2	yes	—
29	C₂×C₈×C₁₃₁	—	2	2	no	—
30	C₄⋊C₄×C₁₃₁	—	2	2	no	—
31	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
32	Dic₂₆₂	—	2	2	no	—
33	Dic₂₆₂	—	4	2	no	—
34	Dic₂₆₂	—	8	2	no	—
35	Dic₂₆₂	Q₁₆	1	4	yes	complemented
36	C₂×Dic₁₃₁	—	2	2	no	—
37	C₂×Dic₁₃₁	—	2	2	no	—
38	C₂×Dic₁₃₁	—	4	1	no	—
39	C₁₃₁⋊C₈	—	2	2	no	—
40	Q₈×C₁₃₁	—	4	2	no	—
41	C₂×C₄×C₁₃₁	—	2	2	no	—
42	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
44	C₈×C₁₃₁	—	4	2	no	—
45	C₂²×C₁₃₁	#120	1	1	yes	characteristic
46	C₄×C₁₃₁	—	2	1	no	—
47	C₄×C₁₃₁	—	4	2	no	—
48	C₄×C₁₃₁	C₂×Q₁₆	1	2	yes	—
49	C₄×C₁₃₁	#137	1	2	yes	—
50	Dic₁₃₁	—	2	4	no	—
51	Dic₁₃₁	—	4	2	no	—
52	Dic₁₃₁	—	8	1	no	—
53	C₂×C₁₃₁	#451	1	2	yes	—
54	C₂×C₁₃₁	#453	1	1	yes	characteristic
55	C₁₃₁	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	131	1	no	maximal, Sylow
57	Q₈²	—	131	1	no	—
58	#494	—	131	1	no	—
59	#477	—	131	2	no	—
60	#333	D₁₃₁	1	1	yes	characteristic
61	Q₈⋊C₈	—	131	2	no	—
62	C₂×Q₁₆	—	262	2	no	—
63	C₄⋊Q₈	—	262	2	no	—
64	C₄⋊Q₈	—	262	2	no	—
65	C₄⋊Q₈	—	131	1	no	—
66	C₄⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
67	C₄×Q₈	—	262	2	no	—
68	C₄×Q₈	—	131	2	no	—
69	#107	—	262	2	no	—
70	C₄⋊C₈	C₂×D₁₃₁	1	2	yes	—
71	#103	—	262	2	no	—
72	C₄×C₈	—	131	1	no	—
73	C₂×Q₈	—	262	2	no	—
74	C₂×Q₈	—	262	1	no	—
75	C₂×Q₈	—	131	2	no	—
76	C₂×Q₈	D₅₂₄	1	2	yes	—
77	Q₁₆	—	524	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	262	1	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	262	4	no	—
82	C₄⋊C₄	—	262	2	no	—
83	C₄⋊C₄	—	262	2	no	—
84	C₄⋊C₄	—	131	2	no	—
85	C₄²	—	262	2	no	—
86	C₄²	—	262	2	no	—
87	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	262	2	no	—
90	Q₈	—	524	2	no	—
91	Q₈	—	1048	2	no	—
92	Q₈	—	131	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	262	2	no	—
95	C₂×C₄	—	262	2	no	—
96	C₂×C₄	—	524	1	no	—
97	C₂×C₄	D₄×D₁₃₁	1	2	yes	—
98	C₂×C₄	C₂×D₅₂₄	1	1	yes	characteristic
99	C₈	—	4	2	no	—
100	C₈	—	262	2	no	—
101	C₂²	#16862216	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	262	4	no	—
105	C₄	—	524	2	no	—
106	C₄	—	1048	1	no	—
107	C₄	Q₁₆×D₁₃₁	1	2	yes	—
108	C₄	#16862275	1	2	yes	—
109	C₂	#22553469	1	2	yes	—
110	C₂	#22553482	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22554042

C₂

yes

maximal, characteristic

#22553691

C₂

yes

maximal, characteristic

#22553608

C₂

yes

maximal

#22553042

C₂

yes

maximal

#333×C₁₃₁

C₂

yes

maximal, characteristic

C₂×#14523503

—

#16862246

C₂²

yes

characteristic

#16862245

—

#16862244

—

C₄×Dic₂₆₂

C₂²

yes

—

#16862211

—

#16862180

—

#16862171

—

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

C₂²

yes

—

C₂×Dic₂₆₂

—

C₂×Dic₂₆₂

—

C₂×Dic₂₆₂

D₄

yes

—

#14523503

—

C₂×C₁₃₁⋊C₈

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

D₄

yes

—

Dic₁₃₁⋊C₄

—

Dic₁₃₁⋊C₄

—

C₄×Dic₁₃₁

—

C₄×Dic₁₃₁

—

C₂×Q₈×C₁₃₁

D₄

yes

—

C₂×C₈×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄²×C₁₃₁

C₂³

yes

characteristic

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

—

Dic₂₆₂

Q₁₆

yes

complemented

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₁₃₁⋊C₈

—

Q₈×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₈×C₁₃₁

—

C₂²×C₁₃₁

#120

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×Q₁₆

yes

—

C₄×C₁₃₁

#137

yes

—

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

#451

yes

—

C₂×C₁₃₁

#453

yes

characteristic

C₁₃₁

#1633

yes

characteristic, complemented, Sylow

#1633

—

131

maximal, Sylow

Q₈²

—

131

—

#494

—

131

—

#477

—

131

—

#333

D₁₃₁

yes

characteristic

Q₈⋊C₈

—

131

—

C₂×Q₁₆

—

262

—

C₄⋊Q₈

—

262

—

C₄⋊Q₈

—

262

—

C₄⋊Q₈

—

131

—

C₄⋊Q₈

C₂×D₁₃₁

yes

characteristic

C₄×Q₈

—

262

—

C₄×Q₈

—

131

—

#107

—

262

—

C₄⋊C₈

C₂×D₁₃₁

yes

—

#103

—

262

—

C₄×C₈

—

131

—

C₂×Q₈

—

262

—

C₂×Q₈

—

262

—

C₂×Q₈

—

131

—

C₂×Q₈

D₅₂₄

yes

—

Q₁₆

—

524

—

C₂×C₈

—

C₂×C₈

—

262

—

C₄⋊C₄

—

C₄⋊C₄

—

262

—

C₄⋊C₄

—

262

—

C₄⋊C₄

—

262

—

C₄⋊C₄

—

131

—

C₄²

—

262

—

C₄²

—

262

—

C₄²

C₂²×D₁₃₁

yes

characteristic

Q₈

—

Q₈

—

262

—

Q₈

—

524

—

Q₈

—

1048

—

Q₈

—

131

—

C₂×C₄

—

C₂×C₄

—

262

—

C₂×C₄

—

262

—

C₂×C₄

—

524

—

C₂×C₄

D₄×D₁₃₁

yes

—

C₂×C₄

C₂×D₅₂₄

yes

characteristic

C₈

—

100

C₈

—

262

—

101

C₂²

#16862216

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

262

—

105

C₄

—

524

—

106

C₄

—

1048

—

107

C₄

Q₁₆×D₁₃₁

yes

—

108

C₄

#16862275

yes

—

109

C₂

#22553469

yes

—

110

C₂

#22553482

yes

characteristic

	subgroup	G/H
center	C₂²	#16862216
commutator, cosocle	C₄²×C₁₃₁	C₂³
Frattini	C₄²	C₂²×D₁₃₁
Fitting	#333×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	#120
hypercenter	#333	D₁₃₁
hypocenter	C₁₃₁	#1633
LCS	G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#16862216

commutator, cosocle

C₄²×C₁₃₁

C₂³

Frattini

C₄²

C₂²×D₁₃₁

Fitting

#333×C₁₃₁

C₂

C₂²×C₁₃₁

D₁₃₁

C₁₃₁

G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
35. Dic₂₆₂	Q₁₆	semidirect
55. C₁₃₁	#1633	semidirect

H ≅ G/N

type

35. Dic₂₆₂

Q₁₆

semidirect

55. C₁₃₁

#1633

semidirect

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

subgroup	index
#22554042	2
#22553691	2
#22553608	2
#22553042	2
#333×C₁₃₁	2
#1633	131

subgroup

index

#333×C₁₃₁

131

Subgroups — #30665102

Conjugacy classes of subgroups

Complements

Maximal subgroups