Group #30608242 — subgroups

Subgroups — #30608242

Order 15616.

238 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 152 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×#16840694	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14519827	C₂	1	1	yes	maximal, characteristic
3	#21460927	C₂	1	2	yes	maximal, complemented
4	#21460924	C₂	1	2	yes	maximal, complemented
5	#21447418	C₂	1	1	yes	maximal, characteristic
6	#16840694	C₄	1	4	yes	complemented
7	C₂×#14519827	C₄	1	4	yes	complemented
8	C₂×#14519827	C₂²	1	1	yes	characteristic
9	C₂×#14519827	C₂²	1	2	yes	—
10	C₂×#14519827	C₂²	1	1	yes	characteristic
11	C₂²×#11758185	C₂²	1	1	yes	characteristic
12	#16839717	C₄	1	1	yes	characteristic
13	#16839537	C₄	1	2	yes	—
14	C₁₆×Dic₆₁	C₂²	1	2	yes	—
15	C₂×M₃₂×C₆₁	C₄	1	1	yes	characteristic
16	#14519827	C₂×C₄	1	2	yes	—
17	#14519827	C₂×C₄	1	2	yes	—
18	#14519827	C₂×C₄	1	4	yes	—
19	C₂×#11758194	C₈	1	1	yes	characteristic
20	#14519736	C₈	1	1	yes	characteristic
21	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
22	C₂×#11758185	C₂×C₄	1	2	yes	—
23	C₂×#11758185	C₂×C₄	1	2	yes	—
24	C₂×#11758185	C₂³	1	1	yes	characteristic
25	#14519687	C₈	1	2	yes	—
26	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₆₁	C₈	1	4	yes	—
29	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
31	C₂×C₈×C₆₁	C₄²	1	2	yes	—
32	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₂²⋊Dic₆₁	C₁₆	1	2	yes	—
36	C₄⋊Dic₆₁	C₁₆	1	2	yes	—
37	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
38	#11758194	C₁₆	1	4	yes	—
39	C₂×#92418	C₂×C₈	1	2	yes	—
40	C₁₆×C₆₁	C₂×C₈	1	4	yes	—
41	#11758185	C₄²	1	2	yes	—
42	#11758185	C₂²×C₄	1	1	yes	characteristic
43	#11758185	C₂²×C₄	1	1	yes	characteristic
44	C₂³×C₆₁	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₄×C₈	1	2	yes	—
46	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
48	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
49	C₂×Dic₆₁	C₂×C₁₆	1	4	yes	—
50	C₈×C₆₁	C₄×C₈	1	2	yes	—
51	C₈×C₆₁	C₂×C₄²	1	1	yes	characteristic
52	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
53	#92418	C₂×C₁₆	1	4	yes	—
54	#13549	—	61	1	no	maximal, Sylow
55	C₂²×C₆₁	—	2	2	no	—
56	C₂²×C₆₁	C₄×C₁₆	1	2	yes	—
57	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
58	C₄×C₆₁	C₄×C₁₆	1	2	yes	—
59	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
60	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
61	Dic₆₁	—	2	4	no	—
62	C₂×M₆₄	—	61	1	no	—
63	C₂²×C₃₂	—	61	1	no	—
64	#2108	—	61	1	no	—
65	#1397	—	61	2	no	—
66	C₄×C₃₂	—	61	2	no	—
67	C₂×C₆₁	—	2	2	no	—
68	C₂×C₆₁	C₂×C₄×C₁₆	1	1	yes	characteristic
69	C₂×C₆₁	#2258	1	2	yes	—
70	C₂×M₃₂	#6923	1	1	yes	characteristic
71	C₂²×C₁₆	—	61	1	no	—
72	#405	—	61	1	no	—
73	M₆₄	—	61	4	no	—
74	C₂×C₃₂	—	61	1	no	—
75	C₂×C₃₂	—	61	2	no	—
76	C₂×C₃₂	—	61	1	no	—
77	C₂×C₃₂	—	61	4	no	—
78	#346	—	61	2	no	—
79	C₄×C₁₆	—	61	2	no	—
80	C₆₁	#13549	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₁₆	—	61	1	no	—
82	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
83	#117	—	61	1	no	—
84	M₃₂	#92420	1	4	yes	—
85	C₂×C₁₆	—	61	2	no	—
86	C₂×C₁₆	—	61	2	no	—
87	C₂×C₁₆	—	61	1	no	—
88	C₂×C₁₆	—	61	1	no	—
89	C₂×C₁₆	C₂×#6923	1	2	yes	—
90	#97	—	61	2	no	—
91	C₄×C₈	—	61	2	no	—
92	C₃₂	—	61	2	no	—
93	C₃₂	—	61	2	no	—
94	C₃₂	—	61	4	no	—
95	C₂²×C₄	D₆₁⋊C₈	1	1	yes	characteristic
96	M₁₆	—	61	4	no	—
97	C₂×C₈	—	61	2	no	—
98	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
99	C₂×C₈	C₄×#6923	1	2	yes	—
100	C₂×C₈	C₂×#92420	1	2	yes	—
101	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
102	C₄⋊C₄	—	61	2	no	—
103	C₂²⋊C₄	—	61	2	no	—
104	C₄²	—	61	2	no	—
105	C₁₆	—	61	1	no	—
106	C₁₆	—	61	1	no	—
107	C₁₆	—	61	2	no	—
108	C₁₆	C₂×#92420	1	4	yes	—
109	C₂³	D₆₁⋊C₁₆	1	1	yes	characteristic
110	C₂×C₄	—	61	4	no	—
111	C₂×C₄	C₄×#92420	1	2	yes	—
112	C₂×C₄	D₆₁⋊C₁₆	1	2	yes	—
113	C₂×C₄	D₆₁⋊C₁₆	1	1	yes	characteristic
114	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
115	C₈	—	61	4	no	—
116	C₈	C₄×#92420	1	2	yes	—
117	C₈	C₂×C₄×#6923	1	1	yes	characteristic
118	C₈	C₂²×#92420	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	#16840648	1	2	yes	—
121	C₂²	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
122	C₄	—	122	4	no	—
123	C₄	#16840648	1	2	yes	—
124	C₄	C₂×C₄×#92420	1	1	yes	characteristic
125	C₄	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
126	C₂	—	2	2	no	—
127	C₂	C₂×#16840648	1	1	yes	characteristic
128	C₂	#21462243	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840694

C₂

yes

maximal, characteristic

C₂²×#14519827

C₂

yes

maximal, characteristic

#21460927

C₂

yes

maximal, complemented

#21460924

C₂

yes

maximal, complemented

#21447418

C₂

yes

maximal, characteristic

#16840694

C₄

yes

complemented

C₂×#14519827

C₄

yes

complemented

C₂×#14519827

C₂²

yes

characteristic

C₂×#14519827

C₂²

yes

—

C₂×#14519827

C₂²

yes

characteristic

C₂²×#11758185

C₂²

yes

characteristic

#16839717

C₄

yes

characteristic

#16839537

C₄

yes

—

C₁₆×Dic₆₁

C₂²

yes

—

C₂×M₃₂×C₆₁

C₄

yes

characteristic

#14519827

C₂×C₄

yes

—

#14519827

C₂×C₄

yes

—

#14519827

C₂×C₄

yes

—

C₂×#11758194

C₈

yes

characteristic

#14519736

C₈

yes

characteristic

C₂×#11758185

C₂×C₄

yes

characteristic

C₂×#11758185

C₂×C₄

yes

—

C₂×#11758185

C₂×C₄

yes

—

C₂×#11758185

C₂³

yes

characteristic

#14519687

C₈

yes

—

C₈×Dic₆₁

C₂×C₄

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

M₃₂×C₆₁

C₈

yes

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₄²

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂²⋊Dic₆₁

C₁₆

yes

—

C₄⋊Dic₆₁

C₁₆

yes

—

C₄×Dic₆₁

C₂×C₈

yes

—

#11758194

C₁₆

yes

—

C₂×#92418

C₂×C₈

yes

—

C₁₆×C₆₁

C₂×C₈

yes

—

#11758185

C₄²

yes

—

#11758185

C₂²×C₄

yes

characteristic

#11758185

C₂²×C₄

yes

characteristic

C₂³×C₆₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₁

C₄×C₈

yes

—

C₂×C₄×C₆₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×C₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×Dic₆₁

C₂×C₁₆

yes

—

C₈×C₆₁

C₄×C₈

yes

—

C₈×C₆₁

C₂×C₄²

yes

characteristic

C₈×C₆₁

C₂²×C₈

yes

characteristic

#92418

C₂×C₁₆

yes

—

#13549

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₄×C₁₆

yes

—

C₂²×C₆₁

C₂²×C₁₆

yes

characteristic

C₄×C₆₁

C₄×C₁₆

yes

—

C₄×C₆₁

C₂×C₄×C₈

yes

characteristic

C₄×C₆₁

C₂²×C₁₆

yes

characteristic

Dic₆₁

—

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

—

#1397

—

C₄×C₃₂

—

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄×C₁₆

yes

characteristic

C₂×C₆₁

#2258

yes

—

C₂×M₃₂

#6923

yes

characteristic

C₂²×C₁₆

—

#405

—

M₆₄

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

#346

—

C₄×C₁₆

—

C₆₁

#13549

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×#6923

yes

characteristic

#117

—

M₃₂

#92420

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#6923

yes

—

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

C₃₂

—

C₂²×C₄

D₆₁⋊C₈

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

D₆₁⋊C₈

yes

characteristic

C₂×C₈

C₄×#6923

yes

—

100

C₂×C₈

C₂×#92420

yes

—

101

C₂×C₈

C₂²×#6923

yes

characteristic

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₁₆

—

106

C₁₆

—

107

C₁₆

—

108

C₁₆

C₂×#92420

yes

—

109

C₂³

D₆₁⋊C₁₆

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

C₄×#92420

yes

—

112

C₂×C₄

D₆₁⋊C₁₆

yes

—

113

C₂×C₄

D₆₁⋊C₁₆

yes

characteristic

114

C₂×C₄

C₂×D₆₁⋊C₈

yes

characteristic

115

C₈

—

116

C₈

C₄×#92420

yes

—

117

C₈

C₂×C₄×#6923

yes

characteristic

118

C₈

C₂²×#92420

yes

characteristic

119

C₂²

—

120

C₂²

#16840648

yes

—

121

C₂²

C₂×D₆₁⋊C₁₆

yes

characteristic

122

C₄

—

122

—

123

C₄

#16840648

yes

—

124

C₄

C₂×C₄×#92420

yes

characteristic

125

C₄

C₂×D₆₁⋊C₁₆

yes

characteristic

126

C₂

—

127

C₂

C₂×#16840648

yes

characteristic

128

C₂

#21462243

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6923
commutator	C₂×C₆₁	C₂×C₄×C₁₆
Fitting	C₂×M₃₂×C₆₁	C₄
socle	C₂²×C₆₁	C₂²×C₁₆
prime radical	C₂³×C₆₁	C₂×C₁₆
cosocle	C₂×#11758185	C₂³
hypercenter	C₂×M₃₂	#6923
hypocenter	C₆₁	#13549
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#6923

commutator

C₂×C₆₁

C₂×C₄×C₁₆

Fitting

C₂×M₃₂×C₆₁

C₄

socle

C₂²×C₆₁

C₂²×C₁₆

prime radical

C₂³×C₆₁

C₂×C₁₆

C₂³

C₂×M₃₂

C₆₁

G ▷ C₂×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #21460927	C₂	semidirect
4. #21460924	C₂	semidirect
6. #16840694	C₄	semidirect
7. C₂×#14519827	C₄	semidirect
80. C₆₁	#13549	semidirect

H ≅ G/N

type

3. #21460927

C₂

semidirect

4. #21460924

C₂

semidirect

6. #16840694

C₄

semidirect

7. C₂×#14519827

C₄

semidirect

80. C₆₁

#13549

semidirect

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

subgroup	index
C₂×#16840694	2
C₂²×#14519827	2
#21460927	2
#21460924	2
#21447418	2
#13549	61

subgroup

index

Subgroups — #30608242

Conjugacy classes of subgroups

Complements

Maximal subgroups