Group #30668136 — subgroups

Subgroups — #30668136

Order 16768.

120 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 43 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	#22553712	C₂	1	1	yes	maximal, characteristic, complemented
2	#22553428	C₂	1	1	yes	maximal, characteristic
3	#22553136	C₂	1	1	yes	maximal, characteristic, complemented
4	#22553132	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553123	C₂	1	1	yes	maximal, characteristic
6	#22553121	C₂	1	1	yes	maximal, characteristic
7	#365×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#16862268	C₂²	1	1	yes	characteristic
9	#16862254	—	2	1	no	—
10	#16862237	—	2	1	no	—
11	#16862186	C₂²	1	1	yes	characteristic
12	#16862179	—	2	1	no	—
13	#16862178	C₂²	1	1	yes	characteristic
14	#16862172	—	2	1	no	—
15	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
16	M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
17	#108×C₁₃₁	C₂²	1	1	yes	characteristic
18	#106×C₁₃₁	C₂²	1	1	yes	characteristic
19	#14523513	—	2	1	no	—
20	Dic₅₂₄	Q₈	1	1	yes	characteristic
21	#14523502	—	2	1	no	—
22	D₁₀₄₈	Q₈	1	1	yes	characteristic
23	#14523496	—	4	1	no	—
24	C₂×C₁₃₁⋊C₈	D₄	1	1	yes	characteristic
25	C₈×D₁₃₁	—	4	1	no	—
26	C₄⋊Dic₁₃₁	—	2	1	no	—
27	C₄⋊Dic₁₃₁	—	4	1	no	—
28	Dic₁₃₁⋊C₄	—	4	1	no	—
29	C₄×Dic₁₃₁	D₄	1	1	yes	characteristic
30	C₁₃₁⋊C₁₆	—	2	1	no	—
31	M₁₆×C₁₃₁	—	2	1	no	—
32	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₃₁	—	2	1	no	—
34	C₁₆×C₁₃₁	—	2	1	no	—
35	Dic₂₆₂	—	2	1	no	—
36	D₅₂₄	—	2	1	no	—
37	C₂²⋊D₁₃₁	—	4	1	no	—
38	C₂×Dic₁₃₁	—	2	1	no	—
39	C₂×Dic₁₃₁	—	4	1	no	—
40	C₄×D₁₃₁	—	4	1	no	—
41	C₁₃₁⋊C₈	SD₁₆	1	1	yes	characteristic
42	C₁₃₁⋊C₈	SD₁₆	1	1	yes	characteristic
43	C₂×C₄×C₁₃₁	—	4	1	no	—
44	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
45	C₈×C₁₃₁	—	4	1	no	—
46	C₈×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
47	C₈×C₁₃₁	#41	1	1	yes	characteristic
48	C₂²×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
49	C₂×D₁₃₁	—	4	1	no	—
50	C₄×C₁₃₁	—	8	1	no	—
51	C₄×C₁₃₁	C₂²⋊Q₈	1	1	yes	characteristic
52	C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
53	Dic₁₃₁	—	4	1	no	—
54	Dic₁₃₁	—	4	1	no	—
55	Dic₁₃₁	—	8	1	no	—
56	C₂×C₁₃₁	—	2	1	no	—
57	C₂×C₁₃₁	#476	1	1	yes	characteristic
58	D₁₃₁	—	8	1	no	—
59	C₁₃₁	#2231	1	1	yes	characteristic, complemented, Sylow
60	#2231	—	131	1	no	maximal, Sylow
61	#499	—	131	1	no	—
62	#443	—	131	1	no	—
63	#365	—	131	1	no	—
64	#365	D₁₃₁	1	1	yes	characteristic, complemented
65	#364	—	131	1	no	—
66	#360	—	131	1	no	—
67	#360	—	131	1	no	—
68	#135	—	131	1	no	—
69	#131	—	262	1	no	—
70	#125	—	262	1	no	—
71	M₃₂	—	131	1	no	—
72	M₃₂	C₂×D₁₃₁	1	1	yes	characteristic
73	#108	C₂×D₁₃₁	1	1	yes	characteristic
74	#107	—	262	1	no	—
75	#106	—	131	1	no	—
76	#106	C₂×D₁₃₁	1	1	yes	characteristic
77	#104	—	262	1	no	—
78	C₄×C₈	—	131	1	no	—
79	#41	—	262	1	no	—
80	Q₁₆	—	131	1	no	—
81	SD₁₆	—	262	1	no	—
82	D₈	—	131	1	no	—
83	M₁₆	—	2	1	no	—
84	M₁₆	—	524	1	no	—
85	C₂×C₈	—	524	1	no	—
86	C₂×C₈	—	131	1	no	—
87	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	262	1	no	—
90	C₄⋊C₄	—	524	1	no	—
91	C₄⋊C₄	—	524	1	no	—
92	C₄²	—	131	1	no	—
93	C₁₆	—	2	1	no	—
94	C₁₆	—	262	1	no	—
95	Q₈	—	262	1	no	—
96	D₄	—	262	1	no	—
97	D₄	—	524	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	262	1	no	—
100	C₂×C₄	—	524	1	no	—
101	C₂×C₄	—	524	1	no	—
102	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
103	C₈	—	4	1	no	—
104	C₈	—	131	1	no	—
105	C₈	—	131	1	no	—
106	C₈	Q₈×D₁₃₁	1	1	yes	characteristic
107	C₈	#14523513	1	1	yes	characteristic
108	C₂²	—	524	1	no	—
109	C₂²	#16862273	1	1	yes	characteristic
110	C₄	—	8	1	no	—
111	C₄	—	524	1	no	—
112	C₄	—	524	1	no	—
113	C₄	—	1048	1	no	—
114	C₄	#16862224	1	1	yes	characteristic
115	C₄	SD₁₆×D₁₃₁	1	1	yes	characteristic
116	C₂	—	2	1	no	—
117	C₂	—	1048	1	no	—
118	C₂	#22553601	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553712

C₂

yes

maximal, characteristic, complemented

#22553428

C₂

yes

maximal, characteristic

#22553136

C₂

yes

maximal, characteristic, complemented

#22553132

C₂

yes

maximal, characteristic, complemented

#22553123

C₂

yes

maximal, characteristic

#22553121

C₂

yes

maximal, characteristic

#365×C₁₃₁

C₂

yes

maximal, characteristic, complemented

#16862268

C₂²

yes

—

—

C₂²

yes

characteristic

#16862179

—

#16862178

C₂²

yes

characteristic

#16862172

—

C₈×Dic₁₃₁

C₂²

yes

characteristic

M₃₂×C₁₃₁

C₂²

yes

characteristic

#108×C₁₃₁

C₂²

yes

characteristic

#106×C₁₃₁

C₂²

yes

characteristic

#14523513

—

Dic₅₂₄

Q₈

yes

characteristic

#14523502

—

D₁₀₄₈

Q₈

yes

characteristic

#14523496

—

C₂×C₁₃₁⋊C₈

D₄

yes

characteristic

C₈×D₁₃₁

—

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

Dic₁₃₁⋊C₄

—

C₄×Dic₁₃₁

D₄

yes

characteristic

C₁₃₁⋊C₁₆

—

M₁₆×C₁₃₁

—

C₂×C₈×C₁₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₁

—

C₁₆×C₁₃₁

—

Dic₂₆₂

—

D₅₂₄

—

C₂²⋊D₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₄×D₁₃₁

—

C₁₃₁⋊C₈

SD₁₆

yes

characteristic

C₁₃₁⋊C₈

SD₁₆

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₈×C₁₃₁

—

C₈×C₁₃₁

C₂×Q₈

yes

C₈×C₁₃₁

yes

C₂²×C₁₃₁

yes

C₂×D₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂²⋊Q₈

yes

characteristic

C₄×C₁₃₁

C₂×SD₁₆

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

#476

yes

characteristic

D₁₃₁

—

C₁₃₁

#2231

yes

characteristic, complemented, Sylow

#2231

—

131

maximal, Sylow

#499

—

131

—

#443

—

131

—

#365

—

131

—

#365

D₁₃₁

yes

characteristic, complemented

#364

—

131

—

#360

—

131

—

#360

—

131

—

#135

—

131

—

#131

—

262

—

#125

—

262

—

M₃₂

—

131

—

M₃₂

C₂×D₁₃₁

yes

characteristic

#108

C₂×D₁₃₁

yes

characteristic

#107

—

262

—

#106

—

131

—

#106

C₂×D₁₃₁

yes

characteristic

#104

—

262

—

C₄×C₈

—

131

—

#41

—

262

—

Q₁₆

—

131

—

SD₁₆

—

262

—

D₈

—

131

—

M₁₆

—

M₁₆

—

524

—

C₂×C₈

—

524

—

C₂×C₈

—

131

—

C₂×C₈

C₂²×D₁₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

262

—

C₄⋊C₄

—

524

—

C₄⋊C₄

—

524

—

C₄²

—

131

—

C₁₆

—

C₁₆

—

262

—

Q₈

—

262

—

D₄

—

262

—

D₄

—

524

—

C₂×C₄

—

C₂×C₄

—

262

—

100

C₂×C₄

—

524

—

101

C₂×C₄

—

524

—

102

C₂×C₄

D₄×D₁₃₁

yes

characteristic

103

C₈

—

104

C₈

—

131

—

105

C₈

—

131

—

106

C₈

Q₈×D₁₃₁

yes

characteristic

107

C₈

#14523513

yes

characteristic

108

C₂²

—

524

—

109

C₂²

#16862273

yes

characteristic

110

C₄

—

111

C₄

—

524

—

112

C₄

—

524

—

113

C₄

—

1048

—

114

C₄

#16862224

yes

characteristic

115

C₄

SD₁₆×D₁₃₁

yes

characteristic

116

C₂

—

117

C₂

—

1048

—

118

C₂

#22553601

yes

characteristic

	subgroup	G/H
center	C₂	#22553601
commutator, cosocle	C₂×C₈×C₁₃₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₁
Fitting	#365×C₁₃₁	C₂
socle	C₂×C₁₃₁	#476
prime radical	#16862268	C₂²
hypercenter	#365	D₁₃₁
hypocenter	C₁₃₁	#2231
LCS	G ▷ C₂×C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

subgroup

G/H

center

C₂

#22553601

commutator, cosocle

C₂×C₈×C₁₃₁

C₂³

Frattini

C₂×C₈

C₂²×D₁₃₁

Fitting

#365×C₁₃₁

C₂

C₂×C₁₃₁

C₂²

D₁₃₁

C₁₃₁

G ▷ C₂×C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #365

N	H ≅ G/N	type
1. #22553712	C₂	semidirect
3. #22553136	C₂	semidirect
4. #22553132	C₂	semidirect
7. #365×C₁₃₁	C₂	semidirect
59. C₁₃₁	#2231	semidirect
64. #365	D₁₃₁	semidirect

H ≅ G/N

type

1. #22553712

C₂

semidirect

3. #22553136

C₂

semidirect

4. #22553132

C₂

semidirect

7. #365×C₁₃₁

C₂

semidirect

59. C₁₃₁

#2231

semidirect

64. #365

D₁₃₁

semidirect

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

subgroup	index
#22553712	2
#22553428	2
#22553136	2
#22553132	2
#22553123	2
#22553121	2
#365×C₁₃₁	2
#2231	131

subgroup

index

#365×C₁₃₁

131

Subgroups — #30668136

Conjugacy classes of subgroups

Complements

Maximal subgroups