Group #30664739 — subgroups

Subgroups — #30664739

Order 16768.

150 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 73 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	#22553369	C₂	1	1	yes	maximal, characteristic
2	#22553365	C₂	1	1	yes	maximal, characteristic
3	#22553064	C₂	1	2	yes	maximal, complemented
4	#22553062	C₂	1	2	yes	maximal, complemented
5	#431×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₄⋊(C₁₃₁⋊C₈)	C₄	1	2	yes	—
7	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
8	C₂²⋊(C₁₃₁⋊C₈)	—	2	1	no	—
9	C₂²⋊(C₁₃₁⋊C₈)	C₄	1	2	yes	—
10	#16862157	—	2	1	no	—
11	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₁₃₁	—	2	1	no	—
16	C₄×C₈×C₁₃₁	—	2	1	no	—
17	C₂×C₁₃₁⋊C₈	—	2	2	no	—
18	C₂×C₁₃₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₁₃₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₁	Q₈	1	1	yes	characteristic
23	C₂×C₈×C₁₃₁	—	2	2	no	—
24	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
26	C₄²×C₁₃₁	Q₈	1	1	yes	characteristic
27	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
28	C₁₃₁⋊C₈	—	2	2	no	—
29	C₁₃₁⋊C₈	—	2	2	no	—
30	C₁₃₁⋊C₈	—	4	2	no	—
31	C₂³×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₃₁	—	2	2	no	—
33	C₂×C₄×C₁₃₁	—	2	2	no	—
34	C₂×C₄×C₁₃₁	—	4	1	no	—
35	C₂×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₁₃₁	—	4	2	no	—
42	C₂²×C₁₃₁	—	2	1	no	—
43	C₂²×C₁₃₁	—	2	1	no	—
44	C₂²×C₁₃₁	—	2	1	no	—
45	C₂²×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₁₃₁	—	2	1	no	—
47	C₄×C₁₃₁	—	2	1	no	—
48	C₄×C₁₃₁	—	2	1	no	—
49	C₄×C₁₃₁	—	2	1	no	—
50	C₄×C₁₃₁	#131	1	2	yes	—
51	C₄×C₁₃₁	#135	1	2	yes	—
52	C₂×C₁₃₁	—	4	1	no	—
53	C₂×C₁₃₁	#424	1	1	yes	characteristic
54	C₂×C₁₃₁	#427	1	1	yes	characteristic
55	C₂×C₁₃₁	#430	1	1	yes	characteristic
56	C₁₃₁	#1572	1	1	yes	characteristic, complemented, Sylow
57	#1572	—	131	1	no	maximal, Sylow
58	#432	—	131	1	no	—
59	#431	—	131	1	no	—
60	#431	D₁₃₁	1	1	yes	characteristic
61	#335	—	131	2	no	—
62	#334	—	131	2	no	—
63	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
64	C₄⋊C₈	—	131	2	no	—
65	C₄⋊C₈	—	131	2	no	—
66	C₄⋊C₈	C₂×D₁₃₁	1	2	yes	—
67	C₂²⋊C₈	—	2	1	no	—
68	C₂²⋊C₈	—	262	1	no	—
69	C₂²⋊C₈	—	131	2	no	—
70	#97	—	262	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	131	1	no	—
73	C₄×C₈	—	131	1	no	—
74	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
75	C₂²×C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
76	C₂²×C₄	Dic₂₆₂	1	1	yes	characteristic
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	262	2	no	—
79	C₂×C₈	—	131	2	no	—
80	C₂×C₈	—	131	2	no	—
81	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
82	C₄²	C₂²⋊D₁₃₁	1	1	yes	characteristic
83	C₄²	Dic₂₆₂	1	1	yes	characteristic
84	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
85	C₂³	Dic₁₃₁⋊C₄	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	4	1	no	—
89	C₂×C₄	Dic₁₃₁⋊C₄	1	1	yes	characteristic
90	C₂×C₄	Dic₁₃₁⋊C₄	1	1	yes	characteristic
91	C₂×C₄	Dic₁₃₁⋊C₄	1	1	yes	characteristic
92	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
93	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₂₆₂	1	1	yes	characteristic
95	C₈	—	4	2	no	—
96	C₈	—	262	2	no	—
97	C₈	—	262	2	no	—
98	C₈	—	524	2	no	—
99	C₂²	—	2	1	no	—
100	C₂²	—	2	1	no	—
101	C₂²	—	2	1	no	—
102	C₂²	C₂×Dic₁₃₁⋊C₄	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	#16862252	1	2	yes	—
108	C₄	#16862266	1	2	yes	—
109	C₂	—	4	1	no	—
110	C₂	#22553338	1	1	yes	characteristic
111	C₂	#22553348	1	1	yes	characteristic
112	C₂	#22553360	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553369

C₂

yes

maximal, characteristic

#22553365

C₂

yes

maximal, characteristic

#22553064

C₂

yes

maximal, complemented

#22553062

C₂

yes

maximal, complemented

#431×C₁₃₁

C₂

yes

maximal, characteristic

C₄⋊(C₁₃₁⋊C₈)

C₄

yes

—

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₃₁⋊C₈)

—

C₂²⋊(C₁₃₁⋊C₈)

C₄

yes

—

#16862157

—

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

C₂²

yes

—

C₂²⋊C₈×C₁₃₁

—

C₄×C₈×C₁₃₁

—

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₃₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₁₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₁

D₄

yes

characteristic

C₂²×C₄×C₁₃₁

Q₈

yes

characteristic

C₂×C₈×C₁₃₁

—

C₄²×C₁₃₁

C₂×C₄

yes

characteristic

C₄²×C₁₃₁

D₄

yes

characteristic

C₄²×C₁₃₁

Q₈

yes

characteristic

C₄²×C₁₃₁

C₂³

yes

characteristic

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

C₂³×C₁₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×Q₈

yes

characteristic

C₈×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#131

yes

—

C₄×C₁₃₁

#135

yes

—

C₂×C₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#1572

—

131

maximal, Sylow

#432

—

131

—

#431

—

131

—

#431

D₁₃₁

yes

characteristic

#335

—

131

—

#334

—

131

—

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

C₄⋊C₈

—

131

—

C₄⋊C₈

—

131

—

C₄⋊C₈

C₂×D₁₃₁

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

262

—

C₂²⋊C₈

—

131

—

#97

—

262

—

C₄×C₈

—

C₄×C₈

—

131

—

C₄×C₈

—

131

—

C₂²×C₄

C₄×D₁₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃₁

yes

characteristic

C₂²×C₄

Dic₂₆₂

yes

characteristic

C₂×C₈

—

C₂×C₈

—

262

—

C₂×C₈

—

131

—

C₂×C₈

—

131

—

C₄²

C₄×D₁₃₁

yes

characteristic

C₄²

C₂²⋊D₁₃₁

yes

characteristic

C₄²

Dic₂₆₂

yes

characteristic

C₄²

C₂²×D₁₃₁

yes

characteristic

C₂³

Dic₁₃₁⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₁₃₁⋊C₄

yes

characteristic

C₂×C₄

Dic₁₃₁⋊C₄

yes

characteristic

C₂×C₄

Dic₁₃₁⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

characteristic

C₂×C₄

C₂×Dic₂₆₂

yes

characteristic

C₈

—

C₈

—

262

—

C₈

—

262

—

C₈

—

524

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₂×Dic₁₃₁⋊C₄

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

#16862252

yes

—

108

C₄

#16862266

yes

—

109

C₂

—

110

C₂

#22553338

yes

characteristic

111

C₂

#22553348

yes

characteristic

112

C₂

#22553360

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₁
commutator	C₂×C₄×C₁₃₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃₁
Fitting	#431×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₁	C₄⋊C₄
cosocle	C₄²×C₁₃₁	C₂³
hypercenter	#431	D₁₃₁
hypocenter	C₁₃₁	#1572
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₃₁

commutator

C₂×C₄×C₁₃₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃₁

Fitting

#431×C₁₃₁

C₂

socle

C₂²×C₁₃₁

C₂×C₄⋊C₄

prime radical

C₂³×C₁₃₁

C₄⋊C₄

cosocle

C₄²×C₁₃₁

C₂³

D₁₃₁

C₁₃₁

G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #22553064	C₂	semidirect
4. #22553062	C₂	semidirect
56. C₁₃₁	#1572	semidirect

H ≅ G/N

type

3. #22553064

C₂

semidirect

4. #22553062

C₂

semidirect

56. C₁₃₁

#1572

semidirect

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

subgroup	index
#22553369	2
#22553365	2
#22553064	2
#22553062	2
#431×C₁₃₁	2
#1572	131

subgroup

index

#431×C₁₃₁

131

Subgroups — #30664739

Conjugacy classes of subgroups

Complements

Maximal subgroups