Group #30608563 — subgroups

Subgroups — #30608563

Order 15616.

184 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 76 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	#21462243	C₂	1	2	yes	maximal, complemented
2	#21460978	C₂	1	1	yes	maximal, characteristic, complemented
3	#21460937	C₂	1	2	yes	maximal, complemented
4	#21460926	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447655	C₂	1	1	yes	maximal, characteristic
6	#16840695	—	2	2	no	—
7	#16840694	C₂²	1	2	yes	—
8	D₆₁⋊C₃₂	—	2	2	no	—
9	C₂×#14519827	C₂²	1	1	yes	characteristic
10	C₂×#14519827	C₂²	1	1	yes	characteristic
11	#16840180	C₂²	1	2	yes	—
12	#16839880	C₄	1	1	yes	characteristic
13	#16839547	C₄	1	2	yes	—
14	C₄×#11758185	C₂²	1	1	yes	characteristic
15	C₆₁×#364	C₄	1	1	yes	characteristic
16	#14519827	—	2	2	no	—
17	#14519827	—	2	1	no	—
18	#14519827	D₄	1	2	yes	—
19	#14519799	C₈	1	1	yes	characteristic
20	#14519783	C₂×C₄	1	2	yes	—
21	#14519716	—	2	2	no	—
22	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
23	C₂×#11758185	C₂³	1	1	yes	characteristic
24	C₁₆×D₆₁	—	2	2	no	—
25	#14519713	C₈	1	1	yes	characteristic
26	#14519705	C₈	1	2	yes	—
27	M₃₂×C₆₁	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
29	#11758222	C₂×C₈	1	2	yes	—
30	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
33	#11758194	C₂×C₈	1	2	yes	—
34	Dic₂₄₄	C₁₆	1	1	yes	characteristic
35	D₄₈₈	C₁₆	1	1	yes	characteristic
36	#11758190	C₁₆	1	2	yes	—
37	#11758189	—	2	2	no	—
38	C₈×D₆₁	—	2	2	no	—
39	C₁₆×C₆₁	—	2	2	no	—
40	#11758185	—	2	1	no	—
41	#11758185	C₂×D₄	1	1	yes	characteristic
42	#11758185	#41	1	1	yes	characteristic
43	C₂×C₄×C₆₁	—	2	1	no	—
44	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
45	C₂²⋊D₆₁	—	2	2	no	—
46	D₂₄₄	C₂×C₁₆	1	2	yes	—
47	C₄×D₆₁	—	2	2	no	—
48	Dic₁₂₂	C₂×C₁₆	1	2	yes	—
49	C₈×C₆₁	C₂×C₁₆	1	1	yes	characteristic
50	C₈×C₆₁	C₂×C₁₆	1	1	yes	characteristic
51	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
52	C₈×C₆₁	#131	1	1	yes	characteristic
53	#92418	—	2	2	no	—
54	#13648	—	61	1	no	maximal, Sylow
55	C₂²×C₆₁	#504	1	1	yes	characteristic
56	C₂×D₆₁	—	2	2	no	—
57	C₄×C₆₁	—	4	1	no	—
58	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
59	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
60	Dic₆₁	—	2	2	no	—
61	#2258	—	61	2	no	—
62	#2170	—	61	1	no	—
63	#1422	—	61	1	no	—
64	#1401	—	61	2	no	—
65	#1397	—	61	1	no	—
66	C₂×C₆₁	—	2	1	no	—
67	C₂×C₆₁	D₄×C₁₆	1	1	yes	characteristic
68	D₆₁	—	4	2	no	—
69	#504	—	61	2	no	—
70	#443	—	61	1	no	—
71	M₆₄	—	61	2	no	—
72	M₆₄	—	122	2	no	—
73	C₂×C₃₂	—	61	1	no	—
74	C₂×C₃₂	—	61	1	no	—
75	C₂×C₃₂	—	122	2	no	—
76	#364	#6923	1	1	yes	characteristic
77	#350	—	61	2	no	—
78	C₄×C₁₆	—	61	1	no	—
79	C₆₁	#13648	1	1	yes	characteristic, complemented, Sylow
80	#135	—	61	1	no	—
81	#131	—	61	2	no	—
82	M₃₂	—	122	2	no	—
83	M₃₂	C₂×#6923	1	2	yes	—
84	C₂×C₁₆	—	61	1	no	—
85	C₂×C₁₆	—	61	1	no	—
86	C₂×C₁₆	—	122	2	no	—
87	#108	—	61	1	no	—
88	#104	—	61	2	no	—
89	C₄×C₈	C₂×#6923	1	1	yes	characteristic
90	C₃₂	—	61	2	no	—
91	C₃₂	—	122	1	no	—
92	C₃₂	—	122	2	no	—
93	#41	—	61	2	no	—
94	Q₁₆	—	61	1	no	—
95	SD₁₆	—	61	2	no	—
96	D₈	—	61	1	no	—
97	M₁₆	—	61	2	no	—
98	M₁₆	—	122	2	no	—
99	C₂×C₈	—	122	2	no	—
100	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
101	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
102	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	—	61	1	no	—
105	C₁₆	—	61	1	no	—
106	C₁₆	—	122	1	no	—
107	Q₈	—	61	2	no	—
108	D₄	—	61	2	no	—
109	D₄	—	122	2	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	122	2	no	—
112	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
113	C₈	—	122	2	no	—
114	C₈	D₆₁⋊C₁₆	1	1	yes	characteristic
115	C₈	D₆₁⋊C₁₆	1	1	yes	characteristic
116	C₈	D₄×#6923	1	1	yes	characteristic
117	C₈	#14519865	1	1	yes	characteristic
118	C₂²	—	122	2	no	—
119	C₂²	#16840821	1	1	yes	characteristic
120	C₄	—	4	1	no	—
121	C₄	—	122	2	no	—
122	C₄	#16840790	1	1	yes	characteristic
123	C₄	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	244	2	no	—
126	C₂	#21462198	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462243

C₂

yes

maximal, complemented

#21460978

C₂

yes

maximal, characteristic, complemented

#21460937

C₂

yes

maximal, complemented

#21460926

C₂

yes

maximal, characteristic, complemented

#21447655

C₂

yes

maximal, characteristic

#16840695

—

#16840694

C₂²

yes

—

D₆₁⋊C₃₂

—

C₂×#14519827

C₂²

yes

characteristic

C₂×#14519827

C₂²

yes

characteristic

#16840180

C₂²

yes

—

#16839880

C₄

yes

characteristic

#16839547

C₄

yes

—

C₄×#11758185

C₂²

yes

characteristic

C₆₁×#364

C₄

yes

—

—

D₄

yes

—

#14519799

C₈

yes

characteristic

#14519783

C₂×C₄

yes

—

#14519716

—

C₂×#11758185

C₂×C₄

yes

characteristic

C₂×#11758185

C₂³

yes

characteristic

C₁₆×D₆₁

—

#14519713

C₈

yes

characteristic

#14519705

C₈

yes

—

M₃₂×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

#11758222

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

characteristic

#11758194

C₂×C₈

yes

—

Dic₂₄₄

C₁₆

yes

characteristic

D₄₈₈

C₁₆

yes

characteristic

#11758190

C₁₆

yes

—

#11758189

—

C₈×D₆₁

—

C₁₆×C₆₁

—

#11758185

—

#11758185

C₂×D₄

yes

yes

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂²⋊D₆₁

—

D₂₄₄

C₂×C₁₆

yes

—

C₄×D₆₁

—

Dic₁₂₂

C₂×C₁₆

yes

—

C₈×C₆₁

C₂×C₁₆

yes

characteristic

C₈×C₆₁

C₂×C₁₆

yes

characteristic

C₈×C₆₁

C₄×D₄

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₈×D₄

yes

characteristic

C₄×C₆₁

C₂²×C₁₆

yes

characteristic

Dic₆₁

—

—

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

D₄×C₁₆

yes

characteristic

D₆₁

—

#504

—

#443

—

M₆₄

—

M₆₄

—

122

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

122

—

yes

—

C₄×C₁₆

—

C₆₁

#13648

yes

characteristic, complemented, Sylow

#135

—

#131

—

M₃₂

—

122

—

M₃₂

C₂×#6923

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

#108

—

#104

—

C₄×C₈

C₂×#6923

yes

characteristic

C₃₂

—

C₃₂

—

122

—

C₃₂

—

122

—

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

122

—

C₂×C₈

—

122

—

100

C₂×C₈

D₆₁⋊C₈

yes

characteristic

101

C₂×C₈

C₂²×#6923

yes

characteristic

102

C₄²

D₆₁⋊C₈

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

—

106

C₁₆

—

122

—

107

Q₈

—

108

D₄

—

109

D₄

—

122

—

110

C₂×C₄

—

111

C₂×C₄

—

122

—

112

C₂×C₄

C₂×D₆₁⋊C₈

yes

characteristic

113

C₈

—

122

—

114

C₈

D₆₁⋊C₁₆

yes

characteristic

115

C₈

D₆₁⋊C₁₆

yes

characteristic

116

C₈

D₄×#6923

yes

characteristic

117

C₈

#14519865

yes

characteristic

118

C₂²

—

122

—

119

C₂²

#16840821

yes

characteristic

120

C₄

—

121

C₄

—

122

—

122

C₄

#16840790

yes

characteristic

123

C₄

C₂×D₆₁⋊C₁₆

yes

characteristic

124

C₂

—

125

C₂

—

244

—

126

C₂

#21462198

yes

characteristic

	subgroup	G/H
center	C₈	D₄×#6923
commutator	C₄×C₆₁	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#364	C₄
socle	C₂×C₆₁	D₄×C₁₆
prime radical	#14519799	C₈
cosocle	C₂×#11758185	C₂³
hypercenter	#364	#6923
hypocenter	C₆₁	#13648
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

subgroup

G/H

center

C₈

D₄×#6923

commutator

C₄×C₆₁

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₆₁×#364

C₄

socle

C₂×C₆₁

D₄×C₁₆

C₈

C₂³

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #364

N	H ≅ G/N	type
1. #21462243	C₂	semidirect
2. #21460978	C₂	semidirect
3. #21460937	C₂	semidirect
4. #21460926	C₂	semidirect
79. C₆₁	#13648	semidirect

H ≅ G/N

type

1. #21462243

C₂

semidirect

2. #21460978

C₂

semidirect

3. #21460937

C₂

semidirect

4. #21460926

C₂

semidirect

79. C₆₁

#13648

semidirect

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

subgroup	index
#21462243	2
#21460978	2
#21460937	2
#21460926	2
#21447655	2
#13648	61

subgroup

index

Subgroups — #30608563

Conjugacy classes of subgroups

Complements

Maximal subgroups