Group #30667725 — subgroups

Subgroups — #30667725

Order 16768.

152 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 75 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	M₃₂×D₁₃₁	C₂	1	2	yes	maximal
2	#22553266	C₂	1	1	yes	maximal, characteristic
3	#22553133	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553132	C₂	1	2	yes	maximal, complemented
5	#364×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523496	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₃₁	C₂²	1	1	yes	characteristic
8	#16862203	C₄	1	1	yes	characteristic
9	#16862186	C₂²	1	2	yes	—
10	#16862185	—	2	2	no	—
11	C₁₆×D₁₃₁	—	2	2	no	—
12	#16862157	C₄	1	1	yes	characteristic
13	#16862156	C₄	1	1	yes	characteristic
14	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
15	M₃₂×C₁₃₁	C₂²	1	2	yes	—
16	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
18	#14523496	C₈	1	4	yes	—
19	C₂×C₁₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₈×D₁₃₁	D₄	1	2	yes	—
22	C₈×D₁₃₁	Q₈	1	2	yes	—
23	Dic₁₃₁⋊C₄	—	2	1	no	—
24	(C₂×D₁₃₁)⋊C₄	—	2	1	no	—
25	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
26	C₁₃₁⋊C₁₆	—	2	2	no	—
27	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
29	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
30	C₁₆×C₁₃₁	—	2	2	no	—
31	C₂²×D₁₃₁	M₁₆	1	1	yes	characteristic
32	C₂×Dic₁₃₁	—	2	1	no	—
33	C₂×Dic₁₃₁	M₁₆	1	1	yes	characteristic
34	C₄×D₁₃₁	C₄⋊C₄	1	2	yes	—
35	C₄×D₁₃₁	C₂×C₈	1	2	yes	—
36	C₁₃₁⋊C₈	C₄⋊C₄	1	2	yes	—
37	C₁₃₁⋊C₈	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₁₃₁	—	2	1	no	—
39	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₁₃₁	C₂×C₈	1	2	yes	—
41	C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
42	C₈×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
44	C₂×D₁₃₁	—	4	1	no	—
45	C₂×D₁₃₁	C₄⋊C₈	1	2	yes	—
46	C₄×C₁₃₁	—	4	1	no	—
47	C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
49	Dic₁₃₁	—	4	1	no	—
50	Dic₁₃₁	C₄⋊C₈	1	2	yes	—
51	C₂×C₁₃₁	—	2	1	no	—
52	C₂×C₁₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
53	D₁₃₁	—	2	2	no	—
54	C₁₃₁	#2153	1	1	yes	characteristic, complemented, Sylow
55	#2153	—	131	1	no	maximal, Sylow
56	C₂×M₃₂	—	131	2	no	—
57	#405	—	131	1	no	—
58	#364	—	131	2	no	—
59	#364	—	131	1	no	—
60	#364	D₁₃₁	1	1	yes	characteristic, complemented
61	C₂×M₁₆	—	131	1	no	—
62	C₂²×C₈	—	131	1	no	—
63	#117	—	131	1	no	—
64	M₃₂	—	262	2	no	—
65	M₃₂	—	131	2	no	—
66	M₃₂	C₂×D₁₃₁	1	2	yes	—
67	C₂×C₁₆	—	262	2	no	—
68	#97	—	131	1	no	—
69	#97	—	131	1	no	—
70	C₄×C₈	—	131	1	no	—
71	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
72	C₂²×C₄	—	131	1	no	—
73	M₁₆	—	131	4	no	—
74	C₂×C₈	—	131	1	no	—
75	C₂×C₈	—	131	1	no	—
76	C₂×C₈	—	131	2	no	—
77	C₂×C₈	—	131	2	no	—
78	C₂×C₈	C₄×D₁₃₁	1	1	yes	characteristic
79	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
80	C₄⋊C₄	—	262	1	no	—
81	C₂²⋊C₄	—	262	1	no	—
82	C₄²	—	131	1	no	—
83	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
84	C₁₆	—	2	2	no	—
85	C₁₆	—	262	2	no	—
86	C₂³	—	131	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	262	1	no	—
89	C₂×C₄	—	131	1	no	—
90	C₂×C₄	—	131	2	no	—
91	C₂×C₄	—	131	2	no	—
92	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
93	C₈	—	131	2	no	—
94	C₈	—	131	2	no	—
95	C₈	C₈×D₁₃₁	1	2	yes	—
96	C₈	D₄×D₁₃₁	1	1	yes	characteristic
97	C₈	Q₈×D₁₃₁	1	1	yes	characteristic
98	C₂²	—	524	1	no	—
99	C₂²	—	131	2	no	—
100	C₂²	M₁₆×D₁₃₁	1	1	yes	characteristic
101	C₄	—	4	1	no	—
102	C₄	—	524	1	no	—
103	C₄	—	131	2	no	—
104	C₄	C₄⋊C₄×D₁₃₁	1	1	yes	characteristic
105	C₄	C₂×C₈×D₁₃₁	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	262	2	no	—
108	C₂	C₄⋊C₈×D₁₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₃₂×D₁₃₁

C₂

yes

maximal

#22553266

C₂

yes

maximal, characteristic

#22553133

C₂

yes

maximal, characteristic, complemented

#22553132

C₂

yes

maximal, complemented

#364×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×#14523496

C₄

yes

characteristic

C₂×C₈×D₁₃₁

C₂²

yes

characteristic

#16862203

C₄

yes

characteristic

#16862186

C₂²

yes

—

#16862185

—

C₁₆×D₁₃₁

—

#16862157

C₄

yes

characteristic

#16862156

C₄

yes

characteristic

C₈×Dic₁₃₁

C₂²

yes

characteristic

M₃₂×C₁₃₁

C₂²

yes

—

C₄×C₈×C₁₃₁

C₂²

yes

characteristic

C₂×C₄×D₁₃₁

C₂×C₄

yes

characteristic

#14523496

C₈

yes

—

C₂×C₁₃₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₁⋊C₈

C₂×C₄

yes

characteristic

C₈×D₁₃₁

D₄

yes

—

C₈×D₁₃₁

Q₈

yes

—

Dic₁₃₁⋊C₄

—

(C₂×D₁₃₁)⋊C₄

—

C₄×Dic₁₃₁

C₂×C₄

yes

characteristic

C₁₃₁⋊C₁₆

—

C₂×C₈×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₁

C₂³

yes

characteristic

C₄²×C₁₃₁

C₂×C₄

yes

characteristic

C₁₆×C₁₃₁

—

C₂²×D₁₃₁

M₁₆

yes

characteristic

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

M₁₆

yes

characteristic

C₄×D₁₃₁

C₄⋊C₄

yes

—

C₄×D₁₃₁

C₂×C₈

yes

—

C₁₃₁⋊C₈

C₄⋊C₄

yes

—

C₁₃₁⋊C₈

C₂×C₈

yes

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₈×C₁₃₁

C₂×C₈

yes

—

C₈×C₁₃₁

C₂×D₄

yes

characteristic

C₈×C₁₃₁

C₂×Q₈

yes

characteristic

C₂²×C₁₃₁

C₂×M₁₆

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

C₄⋊C₈

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₁

C₂²×C₈

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

C₄⋊C₈

yes

—

C₂×C₁₃₁

—

C₂×C₁₃₁

C₂×C₄⋊C₈

yes

characteristic

D₁₃₁

—

C₁₃₁

#2153

yes

characteristic, complemented, Sylow

#2153

—

131

maximal, Sylow

C₂×M₃₂

—

131

—

#405

—

131

—

#364

—

131

—

#364

—

131

—

#364

D₁₃₁

yes

characteristic, complemented

C₂×M₁₆

—

131

—

C₂²×C₈

—

131

—

#117

—

131

—

M₃₂

—

262

—

M₃₂

—

131

—

M₃₂

C₂×D₁₃₁

yes

—

C₂×C₁₆

—

262

—

#97

—

131

—

#97

—

131

—

C₄×C₈

—

131

—

C₄×C₈

C₂×D₁₃₁

yes

characteristic

C₂²×C₄

—

131

—

M₁₆

—

131

—

C₂×C₈

—

131

—

C₂×C₈

—

131

—

C₂×C₈

—

131

—

C₂×C₈

—

131

—

C₂×C₈

C₄×D₁₃₁

yes

characteristic

C₂×C₈

C₂²×D₁₃₁

yes

characteristic

C₄⋊C₄

—

262

—

C₂²⋊C₄

—

262

—

C₄²

—

131

—

C₄²

C₄×D₁₃₁

yes

characteristic

C₁₆

—

C₁₆

—

262

—

C₂³

—

131

—

C₂×C₄

—

C₂×C₄

—

262

—

C₂×C₄

—

131

—

C₂×C₄

—

131

—

C₂×C₄

—

131

—

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

C₈

—

131

—

C₈

—

131

—

C₈

C₈×D₁₃₁

yes

—

C₈

D₄×D₁₃₁

yes

characteristic

C₈

Q₈×D₁₃₁

yes

characteristic

C₂²

—

524

—

C₂²

—

131

—

100

C₂²

M₁₆×D₁₃₁

yes

characteristic

101

C₄

—

102

C₄

—

524

—

103

C₄

—

131

—

104

C₄

C₄⋊C₄×D₁₃₁

yes

characteristic

105

C₄

C₂×C₈×D₁₃₁

yes

characteristic

106

C₂

—

107

C₂

—

262

—

108

C₂

C₄⋊C₈×D₁₃₁

yes

characteristic

	subgroup	G/H
center	C₈	D₄×D₁₃₁
commutator	C₄×C₁₃₁	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₁₃₁
Fitting	#364×C₁₃₁	C₂
socle	C₂×C₁₃₁	C₂×C₄⋊C₈
prime radical	C₂²×D₁₃₁	M₁₆
cosocle	C₂×C₈×C₁₃₁	C₂³
hypercenter	#364	D₁₃₁
hypocenter	C₁₃₁	#2153
LCS	G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₈

D₄×D₁₃₁

commutator

C₄×C₁₃₁

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₁₃₁

Fitting

#364×C₁₃₁

C₂

socle

C₂×C₁₃₁

C₂×C₄⋊C₈

prime radical

C₂²×D₁₃₁

M₁₆

cosocle

C₂×C₈×C₁₃₁

C₂³

D₁₃₁

C₁₃₁

G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
3. #22553133	C₂	semidirect
4. #22553132	C₂	semidirect
5. #364×C₁₃₁	C₂	semidirect
54. C₁₃₁	#2153	semidirect
60. #364	D₁₃₁	semidirect

H ≅ G/N

type

3. #22553133

C₂

semidirect

4. #22553132

C₂

semidirect

5. #364×C₁₃₁

C₂

semidirect

54. C₁₃₁

#2153

semidirect

60. #364

D₁₃₁

semidirect

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

subgroup	index
M₃₂×D₁₃₁	2
#22553266	2
#22553133	2
#22553132	2
#364×C₁₃₁	2
#2153	131

subgroup

index

M₃₂×D₁₃₁

#364×C₁₃₁

131

Subgroups — #30667725

Conjugacy classes of subgroups

Complements

Maximal subgroups