Group #30664749 — subgroups

Subgroups — #30664749

Order 16768.

158 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 77 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	#22553365	C₂	1	1	yes	maximal, characteristic
2	#22553336	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	C₂×#14523497	—	2	1	no	—
7	C₄⋊(C₁₃₁⋊C₈)	—	2	1	no	—
8	C₄⋊(C₁₃₁⋊C₈)	C₄	1	2	yes	—
9	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
10	C₂²⋊(C₁₃₁⋊C₈)	C₄	1	2	yes	—
11	C₄×C₁₃₁⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₁₃₁	—	2	1	no	—
16	C₄×C₈×C₁₃₁	—	2	1	no	—
17	#14523497	—	4	2	no	—
18	C₂×C₁₃₁⋊C₈	—	2	2	no	—
19	C₂×C₁₃₁⋊C₈	C₂×C₄	1	4	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	1	no	—
24	C₂×C₈×C₁₃₁	—	2	1	no	—
25	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₃₁	D₄	1	1	yes	characteristic
27	C₄²×C₁₃₁	Q₈	1	1	yes	characteristic
28	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
29	C₁₃₁⋊C₈	—	2	4	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	—	2	1	no	—
35	C₂×C₄×C₁₃₁	—	2	1	no	—
36	C₂×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
42	C₈×C₁₃₁	—	4	1	no	—
43	C₈×C₁₃₁	—	4	1	no	—
44	C₂²×C₁₃₁	—	2	1	no	—
45	C₂²×C₁₃₁	—	2	1	no	—
46	C₂²×C₁₃₁	—	2	1	no	—
47	C₂²×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₄×C₁₃₁	—	2	1	no	—
49	C₄×C₁₃₁	—	2	1	no	—
50	C₄×C₁₃₁	—	2	1	no	—
51	C₄×C₁₃₁	#108	1	2	yes	—
52	C₄×C₁₃₁	#131	1	2	yes	—
53	C₄×C₁₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
54	C₄×C₁₃₁	#137	1	1	yes	characteristic
55	C₂×C₁₃₁	—	4	1	no	—
56	C₂×C₁₃₁	#424	1	1	yes	characteristic
57	C₂×C₁₃₁	#428	1	1	yes	characteristic
58	C₂×C₁₃₁	C₂×#108	1	1	yes	characteristic
59	C₁₃₁	#1574	1	1	yes	characteristic, complemented, Sylow
60	#1574	—	131	1	no	maximal, Sylow
61	#431	—	131	1	no	—
62	#431	D₁₃₁	1	1	yes	characteristic
63	#423	—	131	1	no	—
64	#335	—	131	2	no	—
65	#334	—	131	2	no	—
66	C₂×M₁₆	—	262	1	no	—
67	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
68	C₄⋊C₈	—	262	1	no	—
69	C₄⋊C₈	—	131	2	no	—
70	C₄⋊C₈	—	131	2	no	—
71	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
72	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
73	C₂²⋊C₈	—	2	1	no	—
74	C₂²⋊C₈	—	131	2	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	131	2	no	—
77	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
79	C₂²×C₄	Dic₂₆₂	1	1	yes	characteristic
80	M₁₆	—	524	2	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	262	2	no	—
84	C₂×C₈	—	131	4	no	—
85	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
86	C₄²	C₂²⋊D₁₃₁	1	1	yes	characteristic
87	C₄²	Dic₂₆₂	1	1	yes	characteristic
88	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
89	C₂³	Dic₁₃₁⋊C₄	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	Dic₁₃₁⋊C₄	1	1	yes	characteristic
95	C₂×C₄	Dic₁₃₁⋊C₄	1	1	yes	characteristic
96	C₂×C₄	Dic₁₃₁⋊C₄	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₂₆₂	1	1	yes	characteristic
100	C₈	—	4	1	no	—
101	C₈	—	4	1	no	—
102	C₈	—	262	4	no	—
103	C₈	—	524	2	no	—
104	C₂²	—	2	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₂×Dic₁₃₁⋊C₄	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	#16862181	1	2	yes	—
112	C₄	#16862252	1	2	yes	—
113	C₄	#16862271	1	1	yes	characteristic
114	C₄	#16862276	1	1	yes	characteristic
115	C₂	—	4	1	no	—
116	C₂	#22553338	1	1	yes	characteristic
117	C₂	#22553355	1	1	yes	characteristic
118	C₂	C₂×#16862181	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553365

C₂

yes

maximal, characteristic

#22553336

C₂

yes

maximal, characteristic

#22553064

C₂

yes

maximal, complemented

#22553062

C₂

yes

maximal, complemented

#431×C₁₃₁

C₂

yes

maximal, characteristic

C₂×#14523497

—

C₄⋊(C₁₃₁⋊C₈)

—

C₄⋊(C₁₃₁⋊C₈)

C₄

yes

—

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₁₃₁⋊C₈)

C₄

yes

—

C₄×C₁₃₁⋊C₈

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

C₂²

yes

characteristic

C₂²⋊C₈×C₁₃₁

—

C₄×C₈×C₁₃₁

—

#14523497

—

C₂×C₁₃₁⋊C₈

—

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

—

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

—

C₂³×C₁₃₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₁

—

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

—

C₈×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#108

yes

—

C₄×C₁₃₁

#131

yes

—

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

C₂×C₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#1574

—

131

maximal, Sylow

#431

—

131

—

#431

D₁₃₁

yes

characteristic

#423

—

131

—

#335

—

131

—

#334

—

131

—

C₂×M₁₆

—

262

—

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

C₄⋊C₈

—

262

—

C₄⋊C₈

—

131

—

C₄⋊C₈

—

131

—

C₄⋊C₈

C₂×D₁₃₁

yes

characteristic

C₄⋊C₈

C₂×D₁₃₁

yes

characteristic

C₂²⋊C₈

—

C₂²⋊C₈

—

131

—

C₄×C₈

—

C₄×C₈

—

131

—

C₂²×C₄

C₄×D₁₃₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₃₁

yes

characteristic

C₂²×C₄

Dic₂₆₂

yes

characteristic

M₁₆

—

524

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

262

—

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

—

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

100

C₈

—

101

C₈

—

102

C₈

—

262

—

103

C₈

—

524

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×Dic₁₃₁⋊C₄

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#16862181

yes

—

112

C₄

#16862252

yes

—

113

C₄

#16862271

yes

characteristic

114

C₄

#16862276

yes

characteristic

115

C₂

—

116

C₂

#22553338

yes

characteristic

117

C₂

#22553355

yes

characteristic

118

C₂

C₂×#16862181

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	#1574
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
59. C₁₃₁	#1574	semidirect

H ≅ G/N

type

3. #22553064

C₂

semidirect

4. #22553062

C₂

semidirect

59. C₁₃₁

#1574

semidirect

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

subgroup	index
#22553365	2
#22553336	2
#22553064	2
#22553062	2
#431×C₁₃₁	2
#1574	131

subgroup

index

#431×C₁₃₁

131

Subgroups — #30664749

Conjugacy classes of subgroups

Complements

Maximal subgroups