Group #30594523 — subgroups

Subgroups — #30594523

Order 15616.

190 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 66 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	#21461381	C₂	1	2	yes	maximal
2	C₂×#16839580	C₂	1	1	yes	maximal, characteristic
3	#16840783	—	2	2	no	—
4	#16840625	—	2	2	no	—
5	#16840624	C₄	1	4	yes	—
6	C₂²×#11758185	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
8	#16839580	—	2	2	no	—
9	C₂²×C₁₆×C₆₁	C₄	1	1	yes	characteristic
10	C₄⋊#92420	—	2	4	no	—
11	C₂²⋊#92420	—	4	2	no	—
12	#14519832	—	4	2	no	—
13	C₂²×#92418	Q₈	1	1	yes	characteristic
14	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
15	C₂×#11758185	—	2	1	no	—
16	C₂×#11758185	—	4	1	no	—
17	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
18	C₈×Dic₆₁	D₄	1	2	yes	—
19	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₆₁	—	2	1	no	—
21	C₂×C₁₆×C₆₁	—	4	1	no	—
22	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×Dic₆₁	D₈	1	1	yes	characteristic
24	C₂²×Dic₆₁	Q₁₆	1	1	yes	characteristic
25	C₂×#92420	—	4	4	no	—
26	C₂×C₈×C₆₁	—	2	1	no	—
27	C₂×C₈×C₆₁	—	2	1	no	—
28	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×Dic₆₁	C₂²⋊C₄	1	2	yes	—
31	C₄×Dic₆₁	SD₁₆	1	2	yes	—
32	C₂×#92418	—	4	1	no	—
33	C₂×#92418	C₄⋊C₄	1	2	yes	—
34	C₁₆×C₆₁	—	4	1	no	—
35	#11758185	—	4	1	no	—
36	C₂³×C₆₁	#107	1	1	yes	characteristic
37	C₂×C₄×C₆₁	—	2	1	no	—
38	C₂×C₄×C₆₁	—	2	1	no	—
39	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
40	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
41	C₂×Dic₆₁	—	2	1	no	—
42	C₂×Dic₆₁	—	2	1	no	—
43	C₂×Dic₆₁	—	4	1	no	—
44	C₂×Dic₆₁	#102	1	2	yes	—
45	C₂×Dic₆₁	#103	1	2	yes	—
46	#92420	—	8	4	no	—
47	C₈×C₆₁	—	2	1	no	—
48	C₈×C₆₁	#100	1	1	yes	characteristic
49	C₈×C₆₁	#101	1	1	yes	characteristic
50	#92418	—	2	2	no	—
51	#7242	—	61	1	no	maximal, Sylow
52	C₂²×C₆₁	—	2	1	no	—
53	C₂²×C₆₁	—	2	1	no	—
54	C₂²×C₆₁	#340	1	1	yes	characteristic
55	C₂²×C₆₁	#368	1	1	yes	characteristic
56	C₂²×C₆₁	#368	1	1	yes	characteristic
57	C₄×C₆₁	—	2	1	no	—
58	C₄×C₆₁	#338	1	1	yes	characteristic
59	C₄×C₆₁	#343	1	1	yes	characteristic
60	Dic₆₁	—	2	2	no	—
61	Dic₆₁	—	2	2	no	—
62	C₂×C₄⋊C₁₆	—	61	1	no	—
63	#1568	—	61	2	no	—
64	C₂×C₆₁	—	2	1	no	—
65	C₂×C₆₁	—	2	1	no	—
66	C₂×C₆₁	#1292	1	1	yes	characteristic
67	C₂×C₆₁	#1381	1	1	yes	characteristic
68	C₂×C₆₁	#1384	1	1	yes	characteristic
69	C₂²×C₁₆	—	61	1	no	—
70	C₂²×C₁₆	#6923	1	1	yes	characteristic
71	#432	—	122	2	no	—
72	C₂×C₄×C₈	—	61	1	no	—
73	C₄⋊C₁₆	—	122	2	no	—
74	#335	—	122	2	no	—
75	#334	—	61	4	no	—
76	C₆₁	#7242	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₈	—	61	1	no	—
78	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
79	C₂×C₄²	—	61	1	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	4	1	no	—
82	C₂×C₁₆	—	122	1	no	—
83	C₂×C₁₆	—	244	1	no	—
84	C₄⋊C₈	—	122	4	no	—
85	C₂²⋊C₈	—	244	2	no	—
86	#97	—	244	2	no	—
87	C₄×C₈	—	61	2	no	—
88	C₄×C₈	—	61	2	no	—
89	C₂²×C₄	—	61	1	no	—
90	C₂²×C₄	—	61	1	no	—
91	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	61	2	no	—
95	C₂×C₈	—	244	1	no	—
96	C₂×C₈	—	244	4	no	—
97	C₂×C₈	C₄×#6923	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
99	C₄²	—	61	2	no	—
100	C₄²	—	61	2	no	—
101	C₁₆	—	4	1	no	—
102	C₁₆	—	244	1	no	—
103	C₂³	#14519846	1	1	yes	characteristic
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	61	2	no	—
107	C₂×C₄	—	61	2	no	—
108	C₂×C₄	—	122	1	no	—
109	C₂×C₄	—	122	1	no	—
110	C₂×C₄	—	244	1	no	—
111	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
112	C₂×C₄	#14519845	1	1	yes	characteristic
113	C₈	—	2	1	no	—
114	C₈	—	122	2	no	—
115	C₈	—	488	4	no	—
116	C₈	#14519837	1	1	yes	characteristic
117	C₈	#14519838	1	1	yes	characteristic
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#16840636	1	1	yes	characteristic
121	C₂²	#16840690	1	1	yes	characteristic
122	C₂²	#16840691	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	122	2	no	—
125	C₄	—	122	2	no	—
126	C₄	#16840631	1	1	yes	characteristic
127	C₄	#16840642	1	1	yes	characteristic
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#21460689	1	1	yes	characteristic
131	C₂	#21460879	1	1	yes	characteristic
132	C₂	#21460887	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461381

C₂

yes

maximal

C₂×#16839580

C₂

yes

maximal, characteristic

#16840783

—

#16840625

—

#16840624

C₄

yes

—

C₂²×#11758185

C₄

yes

characteristic

C₂×C₈×Dic₆₁

C₂²

yes

characteristic

#16839580

—

C₂²×C₁₆×C₆₁

C₄

yes

—

—

—

C₂²×#92418

Q₈

yes

characteristic

C₂×C₄×Dic₆₁

D₄

yes

characteristic

C₂×#11758185

—

C₂×#11758185

—

C₈×Dic₆₁

C₂×C₄

yes

—

C₈×Dic₆₁

D₄

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂²×Dic₆₁

D₈

yes

characteristic

C₂²×Dic₆₁

Q₁₆

yes

characteristic

C₂×#92420

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₄×Dic₆₁

C₂²⋊C₄

yes

—

C₄×Dic₆₁

SD₁₆

yes

—

C₂×#92418

—

C₂×#92418

C₄⋊C₄

yes

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

#107

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#102

yes

—

C₂×Dic₆₁

#103

yes

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

C₂×C₄⋊C₁₆

—

#1568

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂²×C₁₆

—

C₂²×C₁₆

#6923

yes

characteristic

#432

—

122

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

122

—

#335

—

122

—

#334

—

C₆₁

#7242

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×#6923

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

—

244

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

244

—

#97

—

244

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄⋊#6923

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

—

244

—

C₂×C₈

C₄×#6923

yes

characteristic

C₂×C₈

C₂²⋊#6923

yes

characteristic

C₄²

—

100

C₄²

—

101

C₁₆

—

102

C₁₆

—

244

—

103

C₂³

#14519846

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

122

—

109

C₂×C₄

—

122

—

110

C₂×C₄

—

244

—

111

C₂×C₄

(C₂×#6923)⋊C₄

yes

characteristic

112

C₂×C₄

#14519845

yes

characteristic

113

C₈

—

114

C₈

—

122

—

115

C₈

—

488

—

116

C₈

#14519837

yes

characteristic

117

C₈

#14519838

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

#16840636

yes

characteristic

121

C₂²

#16840690

yes

characteristic

122

C₂²

#16840691

yes

characteristic

123

C₄

—

124

C₄

—

122

—

125

C₄

—

122

—

126

C₄

#16840631

yes

characteristic

127

C₄

#16840642

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

#21460689

yes

characteristic

131

C₂

#21460879

yes

characteristic

132

C₂

#21460887

yes

characteristic

	subgroup	G/H
center	C₂²	#16840636
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂²×C₈	C₂×#6923
Fitting	C₂²×C₁₆×C₆₁	C₄
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#107
cosocle	C₂×C₈×Dic₆₁	C₂²
hypercenter	C₂²×C₁₆	#6923
hypocenter	C₆₁	#7242
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²

#16840636

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₂²×C₈

C₂×#6923

Fitting

C₂²×C₁₆×C₆₁

C₄

C₂²×C₆₁

C₂³×C₆₁

C₂×C₈×Dic₆₁

C₂²

C₂²×C₁₆

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
76. C₆₁	#7242	semidirect

H ≅ G/N

type

76. C₆₁

#7242

semidirect

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

subgroup	index
#21461381	2
C₂×#16839580	2
#7242	61

subgroup

index

#21461381

C₂×#16839580

#7242

Subgroups — #30594523

Conjugacy classes of subgroups

Complements

Maximal subgroups