Group #30667554 — subgroups

Subgroups — #30667554

Order 16768.

196 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 113 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×#16862185	C₂	1	2	yes	maximal
2	C₄×C₈×D₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#22553088	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₆×Dic₁₃₁	C₂	1	2	yes	maximal, complemented
5	#346×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₁₃₁	C₄	1	1	yes	characteristic
7	C₂×C₈×D₁₃₁	C₂²	1	1	yes	characteristic
8	C₄²×D₁₃₁	C₄	1	1	yes	characteristic
9	#16862185	C₄	1	8	yes	complemented
10	C₂×C₁₃₁⋊C₁₆	C₂²	1	2	yes	—
11	C₄×C₁₃₁⋊C₈	C₄	1	1	yes	characteristic
12	C₈×Dic₁₃₁	C₄	1	1	yes	characteristic
13	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₁₃₁	C₂²	1	2	yes	—
15	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₃₁	C₈	1	2	yes	—
17	C₂×C₄×D₁₃₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃₁⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₈×D₁₃₁	C₈	1	4	yes	—
21	C₈×D₁₃₁	C₂×C₄	1	4	yes	—
22	C₄×Dic₁₃₁	C₈	1	2	yes	—
23	C₄×Dic₁₃₁	C₂×C₄	1	1	yes	characteristic
24	C₁₃₁⋊C₁₆	C₂×C₄	1	4	yes	complemented
25	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
27	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₁₃₁	C₂×C₄	1	4	yes	complemented
29	C₂²×D₁₃₁	C₂×C₈	1	1	yes	characteristic
30	C₂×Dic₁₃₁	C₂×C₈	1	1	yes	characteristic
31	C₂×Dic₁₃₁	C₂×C₈	1	2	yes	—
32	C₄×D₁₃₁	—	2	4	no	—
33	C₄×D₁₃₁	C₄²	1	2	yes	—
34	C₄×D₁₃₁	C₂×C₈	1	2	yes	—
35	C₁₃₁⋊C₈	C₄²	1	2	yes	—
36	C₁₃₁⋊C₈	C₂×C₈	1	2	yes	—
37	C₂×C₄×C₁₃₁	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₈×C₁₃₁	C₂×C₈	1	2	yes	—
40	C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
41	C₂²×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
42	C₂×D₁₃₁	—	2	2	no	—
43	C₂×D₁₃₁	C₄×C₈	1	2	yes	—
44	C₄×C₁₃₁	—	2	2	no	—
45	C₄×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
46	C₄×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
47	Dic₁₃₁	C₄×C₈	1	2	yes	—
48	Dic₁₃₁	M₃₂	1	4	yes	complemented
49	C₂×C₁₃₁	C₂×C₄×C₈	1	1	yes	characteristic
50	C₂×C₁₃₁	C₂×M₃₂	1	2	yes	—
51	D₁₃₁	—	2	2	no	—
52	C₁₃₁	C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
53	C₄×M₃₂	—	131	1	no	maximal, Sylow
54	C₂×M₃₂	—	131	2	no	—
55	C₂×C₄×C₈	—	131	1	no	—
56	#346	—	131	1	no	—
57	#346	D₁₃₁	1	1	yes	characteristic, complemented
58	C₄×C₁₆	—	131	2	no	—
59	C₂²×C₈	—	131	1	no	—
60	C₂²×C₈	—	131	1	no	—
61	C₂×C₄²	—	131	1	no	—
62	M₃₂	—	131	8	no	—
63	C₂×C₁₆	—	131	2	no	—
64	C₂×C₁₆	C₂×D₁₃₁	1	2	yes	—
65	C₄×C₈	—	131	1	no	—
66	C₄×C₈	—	131	1	no	—
67	C₄×C₈	—	131	1	no	—
68	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
69	C₂²×C₄	—	131	2	no	—
70	C₂²×C₄	—	131	1	no	—
71	C₂×C₈	—	131	1	no	—
72	C₂×C₈	—	131	1	no	—
73	C₂×C₈	—	131	4	no	—
74	C₂×C₈	—	131	4	no	—
75	C₂×C₈	C₄×D₁₃₁	1	1	yes	characteristic
76	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
77	C₄²	—	131	2	no	—
78	C₄²	—	131	1	no	—
79	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
80	C₁₆	—	131	4	no	—
81	C₁₆	C₄×D₁₃₁	1	4	yes	complemented
82	C₂³	—	131	1	no	—
83	C₂×C₄	—	262	4	no	—
84	C₂×C₄	—	131	2	no	—
85	C₂×C₄	—	131	1	no	—
86	C₂×C₄	—	131	2	no	—
87	C₂×C₄	—	131	2	no	—
88	C₂×C₄	C₈×D₁₃₁	1	2	yes	—
89	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
90	C₈	—	131	2	no	—
91	C₈	—	131	2	no	—
92	C₈	C₈×D₁₃₁	1	2	yes	—
93	C₈	C₂×C₄×D₁₃₁	1	2	yes	—
94	C₂²	—	262	2	no	—
95	C₂²	—	131	2	no	—
96	C₂²	C₂×C₈×D₁₃₁	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	131	4	no	—
99	C₄	—	131	2	no	—
100	C₄	C₄²×D₁₃₁	1	1	yes	characteristic
101	C₄	C₂×C₈×D₁₃₁	1	1	yes	characteristic
102	C₂	—	262	2	no	—
103	C₂	C₄×C₈×D₁₃₁	1	1	yes	characteristic
104	C₂	M₃₂×D₁₃₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16862185

C₂

yes

maximal

C₄×C₈×D₁₃₁

C₂

yes

maximal, characteristic

#22553088

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₁₃₁

C₂

yes

maximal, complemented

#346×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₃₁

C₄

yes

characteristic

C₂×C₈×D₁₃₁

C₂²

yes

characteristic

C₄²×D₁₃₁

C₄

yes

characteristic

#16862185

C₄

yes

complemented

C₂×C₁₃₁⋊C₁₆

C₂²

yes

—

C₄×C₁₃₁⋊C₈

C₄

yes

characteristic

C₈×Dic₁₃₁

C₄

yes

characteristic

C₈×Dic₁₃₁

C₂²

yes

characteristic

C₂×C₁₆×C₁₃₁

C₂²

yes

—

C₄×C₈×C₁₃₁

C₂²

yes

characteristic

C₂×C₄×D₁₃₁

C₈

yes

—

C₂×C₄×D₁₃₁

C₂×C₄

yes

characteristic

C₂×C₁₃₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₃₁⋊C₈

C₂×C₄

yes

characteristic

C₈×D₁₃₁

C₈

yes

—

C₈×D₁₃₁

C₂×C₄

yes

—

C₄×Dic₁₃₁

C₈

yes

—

C₄×Dic₁₃₁

C₂×C₄

yes

characteristic

C₁₃₁⋊C₁₆

C₂×C₄

yes

complemented

C₂×C₈×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₁

C₂³

yes

characteristic

C₄²×C₁₃₁

C₂×C₄

yes

characteristic

C₁₆×C₁₃₁

C₂×C₄

yes

complemented

C₂²×D₁₃₁

C₂×C₈

yes

characteristic

C₂×Dic₁₃₁

C₂×C₈

yes

characteristic

C₂×Dic₁₃₁

C₂×C₈

yes

—

C₄×D₁₃₁

—

C₄×D₁₃₁

C₄²

yes

—

C₄×D₁₃₁

C₂×C₈

yes

—

C₁₃₁⋊C₈

C₄²

yes

—

C₁₃₁⋊C₈

C₂×C₈

yes

—

C₂×C₄×C₁₃₁

C₂×C₈

yes

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₈×C₁₃₁

C₂×C₈

yes

—

C₈×C₁₃₁

C₂²×C₄

yes

—

C₂²×C₁₃₁

C₂²×C₈

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

C₄×C₈

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×C₄²

yes

characteristic

C₄×C₁₃₁

C₂²×C₈

yes

characteristic

Dic₁₃₁

C₄×C₈

yes

—

Dic₁₃₁

M₃₂

yes

complemented

C₂×C₁₃₁

C₂×C₄×C₈

yes

characteristic

C₂×C₁₃₁

C₂×M₃₂

yes

—

D₁₃₁

—

C₁₃₁

C₄×M₃₂

yes

characteristic, complemented, Sylow

C₄×M₃₂

—

131

maximal, Sylow

C₂×M₃₂

—

131

—

C₂×C₄×C₈

—

131

—

#346

—

131

—

#346

D₁₃₁

yes

characteristic, complemented

C₄×C₁₆

—

131

—

C₂²×C₈

—

131

—

C₂²×C₈

—

131

—

C₂×C₄²

—

131

—

M₃₂

—

131

—

C₂×C₁₆

—

131

—

C₂×C₁₆

C₂×D₁₃₁

yes

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

C₄×C₈

C₂×D₁₃₁

yes

characteristic

C₂²×C₄

—

131

—

C₂²×C₄

—

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

C₄²

—

131

—

C₄²

—

131

—

C₄²

C₄×D₁₃₁

yes

characteristic

C₁₆

—

131

—

C₁₆

C₄×D₁₃₁

yes

complemented

C₂³

—

131

—

C₂×C₄

—

262

—

C₂×C₄

—

131

—

C₂×C₄

—

131

—

C₂×C₄

—

131

—

C₂×C₄

—

131

—

C₂×C₄

C₈×D₁₃₁

yes

—

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

C₈

—

131

—

C₈

—

131

—

C₈

C₈×D₁₃₁

yes

—

C₈

C₂×C₄×D₁₃₁

yes

—

C₂²

—

262

—

C₂²

—

131

—

C₂²

C₂×C₈×D₁₃₁

yes

characteristic

C₄

—

C₄

—

131

—

C₄

—

131

—

100

C₄

C₄²×D₁₃₁

yes

characteristic

101

C₄

C₂×C₈×D₁₃₁

yes

characteristic

102

C₂

—

262

—

103

C₂

C₄×C₈×D₁₃₁

yes

characteristic

104

C₂

M₃₂×D₁₃₁

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₃₁
commutator	C₂×C₁₃₁	C₂×C₄×C₈
Fitting	#346×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂²×C₈
prime radical	C₂²×D₁₃₁	C₂×C₈
cosocle	C₂×C₈×C₁₃₁	C₂³
hypercenter	#346	D₁₃₁
hypocenter	C₁₃₁	C₄×M₃₂
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₃₁

commutator

C₂×C₁₃₁

C₂×C₄×C₈

Fitting

#346×C₁₃₁

C₂

socle

C₂²×C₁₃₁

C₂²×C₈

prime radical

C₂²×D₁₃₁

C₂×C₈

cosocle

C₂×C₈×C₁₃₁

C₂³

hypercenter

#346

D₁₃₁

hypocenter

C₁₃₁

C₄×M₃₂

LCS

G ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
3. #22553088	C₂	semidirect
4. C₁₆×Dic₁₃₁	C₂	semidirect
5. #346×C₁₃₁	C₂	semidirect
9. #16862185	C₄	semidirect
24. C₁₃₁⋊C₁₆	C₂×C₄	semidirect
28. C₁₆×C₁₃₁	C₂×C₄	semidirect
48. Dic₁₃₁	M₃₂	semidirect
52. C₁₃₁	C₄×M₃₂	semidirect
57. #346	D₁₃₁	semidirect
81. C₁₆	C₄×D₁₃₁	semidirect

H ≅ G/N

type

3. #22553088

C₂

semidirect

4. C₁₆×Dic₁₃₁

C₂

semidirect

5. #346×C₁₃₁

C₂

semidirect

9. #16862185

C₄

semidirect

24. C₁₃₁⋊C₁₆

C₂×C₄

semidirect

28. C₁₆×C₁₃₁

C₂×C₄

semidirect

48. Dic₁₃₁

M₃₂

semidirect

52. C₁₃₁

C₄×M₃₂

semidirect

57. #346

D₁₃₁

semidirect

81. C₁₆

C₄×D₁₃₁

semidirect

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

subgroup	index
C₂×#16862185	2
C₄×C₈×D₁₃₁	2
#22553088	2
C₁₆×Dic₁₃₁	2
#346×C₁₃₁	2
C₄×M₃₂	131

subgroup

index

C₂×#16862185

C₄×C₈×D₁₃₁

#22553088

C₁₆×Dic₁₃₁

#346×C₁₃₁

C₄×M₃₂

131

Subgroups — #30667554

Conjugacy classes of subgroups

Complements

Maximal subgroups