Group #30606356 — subgroups

Subgroups — #30606356

Order 15616.

338 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 148 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	C₄×#14519851	C₂	1	2	yes	maximal
2	#21460861	C₂	1	4	yes	maximal
3	#21445507	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519851	C₂²	1	4	yes	—
5	#16840650	C₄	1	4	yes	complemented
6	C₄×#11758187	C₄	1	4	yes	complemented
7	C₂×#14519687	C₂²	1	2	yes	—
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₈×Dic₆₁	C₄	1	2	yes	—
10	#16839608	C₄	1	1	yes	characteristic
11	C₂×#97×C₆₁	C₄	1	1	yes	characteristic
12	#14519851	—	2	8	no	—
13	C₂×#11758187	C₂×C₄	1	8	yes	—
14	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
15	C₂²×#92418	C₂³	1	1	yes	characteristic
16	C₂×C₄×Dic₆₁	C₂×C₄	1	2	yes	—
17	#14519687	—	2	4	no	—
18	C₄×#92418	C₂×C₄	1	2	yes	—
19	C₄×#92418	C₂×C₄	1	2	yes	—
20	C₈×Dic₆₁	C₈	1	8	yes	—
21	#14519684	—	2	2	no	—
22	C₂²×C₈×C₆₁	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
24	#97×C₆₁	—	2	2	no	—
25	C₂²×C₄×C₆₁	M₁₆	1	2	yes	—
26	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
27	C₂²×Dic₆₁	—	2	2	no	—
28	C₂×C₈×C₆₁	—	2	2	no	—
29	C₂×C₈×C₆₁	—	2	2	no	—
30	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
31	C₄²×C₆₁	C₄²	1	2	yes	—
32	C₄²×C₆₁	M₁₆	1	2	yes	—
33	C₄×Dic₆₁	—	2	2	no	—
34	C₄×Dic₆₁	C₂×C₈	1	4	yes	—
35	C₂×#92418	C₄²	1	1	yes	characteristic
36	C₂×#92418	C₄²	1	1	yes	characteristic
37	C₂×#92418	C₂×C₈	1	4	yes	—
38	C₂×#92418	C₂²×C₄	1	2	yes	—
39	C₂×#92418	#41	1	4	yes	—
40	#11758187	—	2	8	no	—
41	C₂³×C₆₁	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₄×C₆₁	—	4	2	no	—
43	C₂×C₄×C₆₁	#97	1	4	yes	—
44	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
45	C₂×C₄×C₆₁	#117	1	2	yes	—
46	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
47	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
48	C₂×Dic₆₁	—	2	4	no	—
49	C₂×Dic₆₁	—	4	4	no	—
50	C₈×C₆₁	—	2	4	no	—
51	#92418	C₄×C₈	1	2	yes	—
52	#92418	C₄×C₈	1	2	yes	—
53	#92418	#117	1	4	yes	—
54	#12593	—	61	1	no	maximal, Sylow
55	C₂²×C₆₁	—	2	2	no	—
56	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
57	C₂²×C₆₁	#431	1	2	yes	—
58	C₄×C₆₁	—	2	4	no	—
59	C₄×C₆₁	#376	1	1	yes	characteristic
60	C₄×C₆₁	C₂×C₄×C₈	1	1	yes	characteristic
61	C₄×C₆₁	#431	1	2	yes	—
62	Dic₆₁	—	4	4	no	—
63	C₄×M₃₂	—	61	2	no	—
64	#1726	—	61	1	no	—
65	#1375	—	61	4	no	—
66	C₂×C₆₁	—	2	2	no	—
67	C₂×C₆₁	#1744	1	1	yes	characteristic
68	C₂×C₆₁	#2110	1	2	yes	—
69	C₂×M₃₂	—	61	4	no	—
70	C₂×#97	—	61	2	no	—
71	C₂×#97	#6923	1	1	yes	characteristic
72	C₂×C₄×C₈	—	61	2	no	—
73	C₂×C₄×C₈	—	61	1	no	—
74	#376	—	61	1	no	—
75	#346	—	61	4	no	—
76	C₄×C₁₆	—	61	4	no	—
77	C₆₁	#12593	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₈	—	61	1	no	—
79	C₂²×C₈	—	61	1	no	—
80	C₂²×C₈	C₂×#6923	1	2	yes	—
81	C₂×C₄²	—	61	2	no	—
82	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
83	M₃₂	—	122	8	no	—
84	C₂×C₁₆	—	61	8	no	—
85	#97	—	2	2	no	—
86	#97	—	122	4	no	—
87	C₄×C₈	—	61	8	no	—
88	C₄×C₈	—	61	2	no	—
89	C₄×C₈	—	61	2	no	—
90	#95	—	122	2	no	—
91	C₂²×C₄	—	122	2	no	—
92	C₂²×C₄	#11758213	1	2	yes	—
93	C₂²×C₄	C₂²×#6923	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	61	2	no	—
97	C₂×C₈	—	61	1	no	—
98	C₂×C₈	—	61	1	no	—
99	C₂×C₈	—	61	4	no	—
100	C₂×C₈	—	61	4	no	—
101	C₂×C₈	D₆₁⋊C₈	1	4	yes	—
102	C₄²	—	61	4	no	—
103	C₄²	—	122	2	no	—
104	C₄²	C₄×#6923	1	2	yes	—
105	C₄²	#11758217	1	2	yes	—
106	C₁₆	—	122	8	no	—
107	C₂³	C₂×#11758213	1	1	yes	characteristic
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	122	4	no	—
110	C₂×C₄	—	244	4	no	—
111	C₂×C₄	#14519832	1	4	yes	—
112	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
113	C₂×C₄	#14519856	1	2	yes	—
114	C₂×C₄	C₂×D₆₁⋊C₈	1	2	yes	—
115	C₂×C₄	C₂×#11758217	1	1	yes	characteristic
116	C₈	—	2	4	no	—
117	C₈	—	61	2	no	—
118	C₈	—	61	2	no	—
119	C₈	—	61	4	no	—
120	C₂²	—	2	2	no	—
121	C₂²	C₂×#14519832	1	1	yes	characteristic
122	C₂²	#16840781	1	2	yes	—
123	C₄	—	2	4	no	—
124	C₄	—	244	4	no	—
125	C₄	#16840698	1	1	yes	characteristic
126	C₄	C₄×D₆₁⋊C₈	1	1	yes	characteristic
127	C₄	#16840779	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	#21461534	1	1	yes	characteristic
130	C₂	#21462042	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14519851

C₂

yes

maximal

#21460861

C₂

yes

maximal

#21445507

C₂

yes

maximal, characteristic

C₂×#14519851

C₂²

yes

—

#16840650

C₄

yes

complemented

C₄×#11758187

C₄

yes

complemented

C₂×#14519687

C₂²

yes

—

C₂×C₄×#92418

C₂²

yes

characteristic

C₂×C₈×Dic₆₁

C₄

yes

—

#16839608

C₄

yes

characteristic

C₂×#97×C₆₁

C₄

yes

characteristic

#14519851

—

C₂×#11758187

C₂×C₄

yes

—

C₂²×#92418

C₂×C₄

yes

characteristic

C₂²×#92418

C₂³

yes

characteristic

C₂×C₄×Dic₆₁

C₂×C₄

yes

—

#14519687

—

C₄×#92418

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

—

C₈×Dic₆₁

C₈

yes

—

#14519684

—

C₂²×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

#97×C₆₁

—

C₂²×C₄×C₆₁

M₁₆

yes

—

C₂²×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₄²×C₆₁

C₄²

yes

—

C₄²×C₆₁

M₁₆

yes

—

C₄×Dic₆₁

—

C₄×Dic₆₁

C₂×C₈

yes

—

C₂×#92418

C₄²

yes

characteristic

C₂×#92418

C₄²

yes

characteristic

C₂×#92418

C₂×C₈

yes

—

C₂×#92418

C₂²×C₄

yes

—

C₂×#92418

#41

yes

—

#11758187

—

C₂³×C₆₁

C₂×M₁₆

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#97

yes

—

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

—

C₂×C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

—

#92418

C₄×C₈

yes

—

#92418

C₄×C₈

yes

—

#92418

#117

yes

—

#12593

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₂×#97

yes

characteristic

C₂²×C₆₁

#431

yes

—

C₄×C₆₁

—

C₄×C₆₁

#376

yes

characteristic

C₄×C₆₁

C₂×C₄×C₈

yes

characteristic

C₄×C₆₁

#431

yes

—

Dic₆₁

—

C₄×M₃₂

—

#1726

—

#1375

—

C₂×C₆₁

—

C₂×C₆₁

#1744

yes

characteristic

C₂×C₆₁

#2110

yes

—

C₂×M₃₂

—

C₂×#97

—

C₂×#97

#6923

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#376

—

#346

—

C₄×C₁₆

—

C₆₁

#12593

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#6923

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×#6923

yes

characteristic

M₃₂

—

122

—

C₂×C₁₆

—

#97

—

#97

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

#95

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

#11758213

yes

—

C₂²×C₄

C₂²×#6923

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

D₆₁⋊C₈

yes

—

102

C₄²

—

103

C₄²

—

122

—

104

C₄²

C₄×#6923

yes

—

105

C₄²

#11758217

yes

—

106

C₁₆

—

122

—

107

C₂³

C₂×#11758213

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

122

—

110

C₂×C₄

—

244

—

111

C₂×C₄

#14519832

yes

—

112

C₂×C₄

C₂×C₄×#6923

yes

characteristic

113

C₂×C₄

#14519856

yes

—

114

C₂×C₄

C₂×D₆₁⋊C₈

yes

—

115

C₂×C₄

C₂×#11758217

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₂²

—

121

C₂²

C₂×#14519832

yes

characteristic

122

C₂²

#16840781

yes

—

123

C₄

—

124

C₄

—

244

—

125

C₄

#16840698

yes

characteristic

126

C₄

C₄×D₆₁⋊C₈

yes

characteristic

127

C₄

#16840779

yes

—

128

C₂

—

129

C₂

#21461534

yes

characteristic

130

C₂

#21462042

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6923
commutator	C₄×C₆₁	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#6923
Fitting	C₂×#97×C₆₁	C₄
socle	C₂²×C₆₁	C₂×#97
prime radical	C₂³×C₆₁	C₂×M₁₆
cosocle	C₂²×#92418	C₂³
hypercenter	C₂×#97	#6923
hypocenter	C₆₁	#12593
LCS	G ▷ C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

subgroup

G/H

center

C₂×C₄

C₂×C₄×#6923

commutator

C₄×C₆₁

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×#6923

Fitting

C₂×#97×C₆₁

C₄

socle

C₂²×C₆₁

C₂×#97

prime radical

C₂³×C₆₁

C₂×M₁₆

cosocle

C₂²×#92418

C₂³

C₂×#97

C₆₁

G ▷ C₄×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ C₂×#97

N	H ≅ G/N	type
5. #16840650	C₄	semidirect
6. C₄×#11758187	C₄	semidirect
77. C₆₁	#12593	semidirect

H ≅ G/N

type

5. #16840650

C₄

semidirect

6. C₄×#11758187

C₄

semidirect

77. C₆₁

#12593

semidirect

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

subgroup	index
C₄×#14519851	2
#21460861	2
#21445507	2
#12593	61

subgroup

index

Subgroups — #30606356

Conjugacy classes of subgroups

Complements

Maximal subgroups