Group #30667558 — subgroups

Subgroups — #30667558

Order 16768.

184 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×#16862186	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#22553088	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₃₁⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	#405×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	#16862186	C₄	1	4	yes	complemented
7	C₂×C₁₃₁⋊C₁₆	C₄	1	4	yes	complemented
8	C₂×C₁₃₁⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₁₃₁⋊C₁₆	C₂²	1	2	yes	—
10	C₂×C₁₃₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×C₁₃₁	C₄	1	1	yes	characteristic
12	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
13	#117×C₁₃₁	C₄	1	1	yes	characteristic
14	#97×C₁₃₁	C₄	1	2	yes	—
15	C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
16	C₁₃₁⋊C₁₆	C₂×C₄	1	2	yes	—
17	C₁₃₁⋊C₁₆	C₂×C₄	1	2	yes	—
18	C₁₃₁⋊C₁₆	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
20	M₁₆×C₁₃₁	C₈	1	4	yes	—
21	C₂×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₁₃₁	C₈	1	2	yes	—
27	C₂²⋊C₄×C₁₃₁	C₈	1	2	yes	—
28	C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
29	C₂³×C₁₃₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₃₁	C₄²	1	2	yes	—
31	C₂×C₄×C₁₃₁	C₂×C₈	1	4	yes	—
32	C₂×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₁₃₁	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₁₃₁	C₄²	1	2	yes	—
36	C₈×C₁₃₁	C₂×C₈	1	4	yes	—
37	C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₁₃₁	—	2	2	no	—
40	C₂²×C₁₃₁	C₄×C₈	1	2	yes	—
41	C₂²×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₃₁	—	2	4	no	—
43	C₄×C₁₃₁	C₄×C₈	1	2	yes	—
44	C₄×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
45	C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
46	C₂×C₁₃₁	—	2	2	no	—
47	C₂×C₁₃₁	C₂×C₄×C₈	1	1	yes	characteristic
48	C₂×C₁₃₁	#504	1	2	yes	—
49	C₁₃₁	#2108	1	1	yes	characteristic, complemented, Sylow
50	#2108	—	131	1	no	maximal, Sylow
51	C₂×M₃₂	—	131	1	no	—
52	C₂²×C₁₆	—	131	1	no	—
53	#405	D₁₃₁	1	1	yes	characteristic
54	#346	—	131	2	no	—
55	C₄×C₁₆	—	131	2	no	—
56	C₂×M₁₆	Dic₁₃₁	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₁₃₁	1	1	yes	characteristic
58	#117	Dic₁₃₁	1	1	yes	characteristic
59	M₃₂	—	131	4	no	—
60	C₂×C₁₆	—	131	4	no	—
61	C₂×C₁₆	—	131	1	no	—
62	C₂×C₁₆	—	131	2	no	—
63	C₂×C₁₆	—	131	1	no	—
64	#97	Dic₁₃₁	1	2	yes	—
65	C₄×C₈	C₂×D₁₃₁	1	2	yes	—
66	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
67	M₁₆	C₁₃₁⋊C₈	1	4	yes	—
68	C₂×C₈	C₄×D₁₃₁	1	2	yes	—
69	C₂×C₈	C₄×D₁₃₁	1	2	yes	—
70	C₂×C₈	C₂×Dic₁₃₁	1	1	yes	characteristic
71	C₂×C₈	C₂×Dic₁₃₁	1	2	yes	—
72	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
73	C₄⋊C₄	C₁₃₁⋊C₈	1	2	yes	—
74	C₂²⋊C₄	C₁₃₁⋊C₈	1	2	yes	—
75	C₄²	C₂×Dic₁₃₁	1	2	yes	—
76	C₁₆	—	131	2	no	—
77	C₁₆	—	131	2	no	—
78	C₁₆	—	131	4	no	—
79	C₂³	C₂×C₁₃₁⋊C₈	1	1	yes	characteristic
80	C₂×C₄	C₄×Dic₁₃₁	1	2	yes	—
81	C₂×C₄	C₂×C₁₃₁⋊C₈	1	4	yes	—
82	C₂×C₄	C₂×C₁₃₁⋊C₈	1	2	yes	—
83	C₂×C₄	C₂×C₁₃₁⋊C₈	1	1	yes	characteristic
84	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
85	C₈	C₄×Dic₁₃₁	1	2	yes	—
86	C₈	C₂×C₁₃₁⋊C₈	1	4	yes	—
87	C₈	C₂×C₄×D₁₃₁	1	1	yes	characteristic
88	C₈	C₂×C₄×D₁₃₁	1	1	yes	characteristic
89	C₂²	—	2	2	no	—
90	C₂²	C₄×C₁₃₁⋊C₈	1	2	yes	—
91	C₂²	C₂²×C₁₃₁⋊C₈	1	1	yes	characteristic
92	C₄	—	2	4	no	—
93	C₄	C₄×C₁₃₁⋊C₈	1	2	yes	—
94	C₄	C₂×C₄×Dic₁₃₁	1	1	yes	characteristic
95	C₄	C₂²×C₁₃₁⋊C₈	1	1	yes	characteristic
96	C₂	—	2	2	no	—
97	C₂	C₂×C₄×C₁₃₁⋊C₈	1	1	yes	characteristic
98	C₂	#22553729	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16862186

C₂

yes

maximal, characteristic

C₂²×C₁₃₁⋊C₁₆

C₂

yes

maximal, characteristic

#22553088

C₂

yes

maximal, complemented

C₄×C₁₃₁⋊C₁₆

C₂

yes

maximal, complemented

#405×C₁₃₁

C₂

yes

maximal, characteristic

#16862186

C₄

yes

complemented

C₂×C₁₃₁⋊C₁₆

C₄

yes

complemented

C₂×C₁₃₁⋊C₁₆

C₂²

yes

characteristic

C₂×C₁₃₁⋊C₁₆

C₂²

yes

—

C₂×C₁₃₁⋊C₁₆

C₂²

yes

characteristic

C₂×M₁₆×C₁₃₁

C₄

yes

characteristic

C₂²×C₈×C₁₃₁

C₂²

yes

characteristic

#117×C₁₃₁

C₄

yes

characteristic

#97×C₁₃₁

C₄

yes

—

C₄×C₈×C₁₃₁

C₂²

yes

—

C₁₃₁⋊C₁₆

C₂×C₄

yes

—

C₁₃₁⋊C₁₆

C₂×C₄

yes

—

C₁₃₁⋊C₁₆

C₂×C₄

yes

—

C₂²×C₄×C₁₃₁

C₂×C₄

yes

characteristic

M₁₆×C₁₃₁

C₈

yes

—

C₂×C₈×C₁₃₁

C₂×C₄

yes

—

C₂×C₈×C₁₃₁

C₂×C₄

yes

—

C₂×C₈×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₁

C₂×C₄

yes

—

C₂×C₈×C₁₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₁

C₈

yes

—

C₂²⋊C₄×C₁₃₁

C₈

yes

—

C₄²×C₁₃₁

C₂×C₄

yes

—

C₂³×C₁₃₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₁

C₄²

yes

—

C₂×C₄×C₁₃₁

C₂×C₈

yes

—

C₂×C₄×C₁₃₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₁

C₂×C₈

yes

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₈×C₁₃₁

C₄²

yes

—

C₈×C₁₃₁

C₂×C₈

yes

—

C₈×C₁₃₁

C₂²×C₄

yes

characteristic

C₈×C₁₃₁

C₂²×C₄

yes

characteristic

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₄×C₈

yes

—

C₂²×C₁₃₁

C₂²×C₈

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

C₄×C₈

yes

—

C₄×C₁₃₁

C₂×C₄²

yes

characteristic

C₄×C₁₃₁

C₂²×C₈

yes

characteristic

C₂×C₁₃₁

—

C₂×C₁₃₁

C₂×C₄×C₈

yes

characteristic

C₂×C₁₃₁

#504

yes

—

C₁₃₁

#2108

yes

characteristic, complemented, Sylow

#2108

—

131

maximal, Sylow

C₂×M₃₂

—

131

—

C₂²×C₁₆

—

131

—

#405

D₁₃₁

yes

characteristic

#346

—

131

—

C₄×C₁₆

—

131

—

C₂×M₁₆

Dic₁₃₁

yes

characteristic

C₂²×C₈

C₂×D₁₃₁

yes

characteristic

#117

Dic₁₃₁

yes

characteristic

M₃₂

—

131

—

C₂×C₁₆

—

131

—

C₂×C₁₆

—

131

—

C₂×C₁₆

—

131

—

C₂×C₁₆

—

131

—

#97

Dic₁₃₁

yes

—

C₄×C₈

C₂×D₁₃₁

yes

—

C₂²×C₄

C₂×Dic₁₃₁

yes

characteristic

M₁₆

C₁₃₁⋊C₈

yes

—

C₂×C₈

C₄×D₁₃₁

yes

—

C₂×C₈

C₄×D₁₃₁

yes

—

C₂×C₈

C₂×Dic₁₃₁

yes

characteristic

C₂×C₈

C₂×Dic₁₃₁

yes

—

C₂×C₈

C₂²×D₁₃₁

yes

characteristic

C₄⋊C₄

C₁₃₁⋊C₈

yes

—

C₂²⋊C₄

C₁₃₁⋊C₈

yes

—

C₄²

C₂×Dic₁₃₁

yes

—

C₁₆

—

131

—

C₁₆

—

131

—

C₁₆

—

131

—

C₂³

C₂×C₁₃₁⋊C₈

yes

characteristic

C₂×C₄

C₄×Dic₁₃₁

yes

—

C₂×C₄

C₂×C₁₃₁⋊C₈

yes

—

C₂×C₄

C₂×C₁₃₁⋊C₈

yes

—

C₂×C₄

C₂×C₁₃₁⋊C₈

yes

characteristic

C₂×C₄

C₂²×Dic₁₃₁

yes

characteristic

C₈

C₄×Dic₁₃₁

yes

—

C₈

C₂×C₁₃₁⋊C₈

yes

—

C₈

C₂×C₄×D₁₃₁

yes

characteristic

C₈

C₂×C₄×D₁₃₁

yes

characteristic

C₂²

—

C₂²

C₄×C₁₃₁⋊C₈

yes

—

C₂²

C₂²×C₁₃₁⋊C₈

yes

characteristic

C₄

—

C₄

C₄×C₁₃₁⋊C₈

yes

—

C₄

C₂×C₄×Dic₁₃₁

yes

characteristic

C₄

C₂²×C₁₃₁⋊C₈

yes

characteristic

C₂

—

C₂

C₂×C₄×C₁₃₁⋊C₈

yes

characteristic

C₂

#22553729

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₃₁
commutator	C₂×C₁₃₁	C₂×C₄×C₈
Fitting	#405×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×C₈
prime radical	C₂³×C₁₃₁	C₂×C₈
cosocle	C₂×C₈×C₁₃₁	C₂³
hypercenter	#405	D₁₃₁
hypocenter	C₁₃₁	#2108
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₈ ◁ #405

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₃₁

commutator

C₂×C₁₃₁

C₂×C₄×C₈

Fitting

#405×C₁₃₁

C₂

socle

C₂²×C₁₃₁

C₂²×C₈

prime radical

C₂³×C₁₃₁

C₂×C₈

cosocle

C₂×C₈×C₁₃₁

C₂³

D₁₃₁

C₁₃₁

G ▷ C₂×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂×C₈ ◁ #405

N	H ≅ G/N	type
3. #22553088	C₂	semidirect
4. C₄×C₁₃₁⋊C₁₆	C₂	semidirect
6. #16862186	C₄	semidirect
7. C₂×C₁₃₁⋊C₁₆	C₄	semidirect
49. C₁₃₁	#2108	semidirect

H ≅ G/N

type

3. #22553088

C₂

semidirect

4. C₄×C₁₃₁⋊C₁₆

C₂

semidirect

6. #16862186

C₄

semidirect

7. C₂×C₁₃₁⋊C₁₆

C₄

semidirect

49. C₁₃₁

#2108

semidirect

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

subgroup	index
C₂×#16862186	2
C₂²×C₁₃₁⋊C₁₆	2
#22553088	2
C₄×C₁₃₁⋊C₁₆	2
#405×C₁₃₁	2
#2108	131

subgroup

index

C₂×#16862186

C₂²×C₁₃₁⋊C₁₆

#22553088

C₄×C₁₃₁⋊C₁₆

#405×C₁₃₁

#2108

131

Subgroups — #30667558

Conjugacy classes of subgroups

Complements

Maximal subgroups