Group #30594602 — subgroups

Subgroups — #30594602

Order 15616.

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

normal subgroups only

This group has 218 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	C₂×#16840654	C₂	1	2	yes	maximal
2	#21445952	C₂	1	1	yes	maximal, characteristic
3	C₂×#14519851	—	2	2	no	—
4	#16840654	—	2	2	no	—
5	#16840654	C₄	1	4	yes	complemented
6	C₂×#14519688	C₂²	1	1	yes	characteristic
7	#16839632	C₄	1	1	yes	characteristic
8	#16839524	—	2	2	no	—
9	C₆₁×#382	C₄	1	1	yes	characteristic
10	#14519851	—	2	4	no	—
11	#14519851	—	2	4	no	—
12	C₂×#11758187	—	2	4	no	—
13	C₂²×#92418	D₄	1	2	yes	—
14	C₂×C₄⋊Dic₆₁	—	2	1	no	—
15	C₂×C₄⋊Dic₆₁	—	4	1	no	—
16	#14519688	C₂×C₄	1	2	yes	—
17	#14519688	D₄	1	1	yes	characteristic
18	#14519688	Q₈	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
20	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
21	#95×C₆₁	—	4	1	no	—
22	C₂²×C₄×C₆₁	—	2	1	no	—
23	C₂²×C₄×C₆₁	—	4	1	no	—
24	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×Dic₆₁	—	4	1	no	—
26	C₄⋊C₄×C₆₁	—	4	2	no	—
27	C₄²×C₆₁	—	2	1	no	—
28	C₄²×C₆₁	C₄²	1	1	yes	characteristic
29	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
30	C₄⋊Dic₆₁	—	4	2	no	—
31	C₄⋊Dic₆₁	—	8	2	no	—
32	C₂×#92418	—	2	2	no	—
33	C₂×#92418	—	2	2	no	—
34	C₂×#92418	C₂²⋊C₄	1	2	yes	—
35	C₂×#92418	C₄⋊C₄	1	2	yes	—
36	#11758187	—	2	8	no	—
37	C₂³×C₆₁	#100	1	1	yes	characteristic
38	C₂×C₄×C₆₁	—	2	1	no	—
39	C₂×C₄×C₆₁	—	2	1	no	—
40	C₂×C₄×C₆₁	—	4	1	no	—
41	C₂×C₄×C₆₁	—	4	1	no	—
42	C₂×C₄×C₆₁	—	8	1	no	—
43	C₂×C₄×C₆₁	—	8	1	no	—
44	C₂×C₄×C₆₁	—	8	1	no	—
45	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
46	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
47	C₂×C₄×C₆₁	#104	1	2	yes	—
48	C₂×C₄×C₆₁	#104	1	2	yes	—
49	C₂×Dic₆₁	—	8	2	no	—
50	C₂×Dic₆₁	—	8	1	no	—
51	#92418	—	2	2	no	—
52	#92418	—	2	2	no	—
53	#7266	—	61	1	no	maximal, Sylow
54	C₂²×C₆₁	—	2	1	no	—
55	C₂²×C₆₁	—	2	1	no	—
56	C₂²×C₆₁	#329	1	1	yes	characteristic
57	C₂²×C₆₁	#329	1	1	yes	characteristic
58	C₂²×C₆₁	#343	1	1	yes	characteristic
59	C₄×C₆₁	—	4	1	no	—
60	C₄×C₆₁	—	4	1	no	—
61	C₄×C₆₁	—	16	1	no	—
62	C₄×C₆₁	#330	1	2	yes	—
63	C₄×C₆₁	#339	1	2	yes	—
64	Dic₆₁	—	16	1	no	—
65	C₂×#347	—	61	2	no	—
66	#1836	—	61	1	no	—
67	C₂×C₆₁	—	2	1	no	—
68	C₂×C₆₁	—	2	1	no	—
69	C₂×C₆₁	#1295	1	1	yes	characteristic
70	C₂×C₆₁	#1355	1	1	yes	characteristic
71	C₂×C₆₁	#1356	1	1	yes	characteristic
72	C₂×M₃₂	—	122	2	no	—
73	C₂×#97	—	61	1	no	—
74	#384	—	61	1	no	—
75	#382	#6923	1	1	yes	characteristic
76	#347	—	61	4	no	—
77	#347	—	122	2	no	—
78	#340	—	122	2	no	—
79	C₆₁	#7266	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	—	61	2	no	—
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	122	1	no	—
83	C₂×C₄⋊C₄	—	244	1	no	—
84	C₂×C₄²	C₂×#6923	1	1	yes	characteristic
85	M₃₂	—	122	4	no	—
86	M₃₂	—	122	4	no	—
87	C₂×C₁₆	—	122	4	no	—
88	#97	—	61	2	no	—
89	#97	—	61	1	no	—
90	#97	—	61	1	no	—
91	#95	—	4	1	no	—
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	—	244	1	no	—
95	C₂²×C₄	C₂²⋊#6923	1	1	yes	characteristic
96	C₂×C₈	—	61	2	no	—
97	C₂×C₈	—	61	2	no	—
98	C₂×C₈	—	122	2	no	—
99	C₂×C₈	—	122	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	488	2	no	—
102	C₄⋊C₄	—	244	2	no	—
103	C₄²	—	2	1	no	—
104	C₄²	C₄×#6923	1	1	yes	characteristic
105	C₄²	C₄⋊#6923	1	1	yes	characteristic
106	C₁₆	—	122	8	no	—
107	C₂³	#14519837	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	8	1	no	—
113	C₂×C₄	—	8	1	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	—	488	2	no	—
116	C₂×C₄	—	488	1	no	—
117	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
118	C₂×C₄	#14519838	1	1	yes	characteristic
119	C₂×C₄	#14519841	1	2	yes	—
120	C₂×C₄	#14519842	1	2	yes	—
121	C₈	—	122	2	no	—
122	C₈	—	122	2	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	#16840617	1	1	yes	characteristic
126	C₂²	#16840618	1	1	yes	characteristic
127	C₂²	#16840642	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	16	1	no	—
131	C₄	—	976	1	no	—
132	C₄	#16840619	1	2	yes	—
133	C₄	#16840632	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#21460695	1	1	yes	characteristic
137	C₂	#21460823	1	1	yes	characteristic
138	C₂	#21460826	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840654

C₂

yes

maximal

#21445952

C₂

yes

maximal, characteristic

C₂×#14519851

—

#16840654

—

#16840654

C₄

yes

complemented

C₂×#14519688

C₂²

yes

characteristic

#16839632

C₄

yes

characteristic

#16839524

—

C₆₁×#382

C₄

yes

—

—

—

C₂²×#92418

D₄

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

#14519688

C₂×C₄

yes

—

#14519688

D₄

yes

characteristic

#14519688

Q₈

yes

characteristic

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

#95×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂²×Dic₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₄²

yes

characteristic

C₄²×C₆₁

C₄⋊C₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₂²⋊C₄

yes

—

C₂×#92418

C₄⋊C₄

yes

—

#11758187

—

C₂³×C₆₁

#100

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

—

C₂×C₄×C₆₁

#104

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

—

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#330

yes

—

C₄×C₆₁

#339

yes

—

Dic₆₁

—

C₂×#347

—

#1836

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×M₃₂

—

122

—

C₂×#97

—

—

yes

—

—

122

—

#340

—

122

—

C₆₁

#7266

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

244

—

C₂×C₄²

C₂×#6923

yes

characteristic

M₃₂

—

122

—

M₃₂

—

122

—

C₂×C₁₆

—

122

—

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

244

—

C₂²×C₄

C₂²⋊#6923

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

488

—

102

C₄⋊C₄

—

244

—

103

C₄²

—

104

C₄²

C₄×#6923

yes

characteristic

105

C₄²

C₄⋊#6923

yes

characteristic

106

C₁₆

—

122

—

107

C₂³

#14519837

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

488

—

116

C₂×C₄

—

488

—

117

C₂×C₄

(C₂×#6923)⋊C₄

yes

characteristic

118

C₂×C₄

#14519838

yes

characteristic

119

C₂×C₄

#14519841

yes

—

120

C₂×C₄

#14519842

yes

—

121

C₈

—

122

—

122

C₈

—

122

—

123

C₂²

—

124

C₂²

—

125

C₂²

#16840617

yes

characteristic

126

C₂²

#16840618

yes

characteristic

127

C₂²

#16840642

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

976

—

132

C₄

#16840619

yes

—

133

C₄

#16840632

yes

—

134

C₂

—

135

C₂

—

136

C₂

#21460695

yes

characteristic

137

C₂

#21460823

yes

characteristic

138

C₂

#21460826

yes

characteristic

	subgroup	G/H
center	C₂²	#16840618
commutator	C₄²×C₆₁	C₄²
Frattini	C₂×C₄²	C₂×#6923
Fitting	C₆₁×#382	C₄
socle	C₂²×C₆₁	#329
prime radical	C₂³×C₆₁	#100
cosocle	C₂×#14519688	C₂²
hypercenter	#382	#6923
hypocenter	C₆₁	#7266
LCS	G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂²

#16840618

commutator

C₄²×C₆₁

C₄²

Frattini

C₂×C₄²

C₂×#6923

Fitting

C₆₁×#382

C₄

C₂²×C₆₁

C₂³×C₆₁

C₂²

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
5. #16840654	C₄	semidirect
79. C₆₁	#7266	semidirect

H ≅ G/N

type

5. #16840654

C₄

semidirect

79. C₆₁

#7266

semidirect

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

subgroup	index
C₂×#16840654	2
#21445952	2
#7266	61

subgroup

index

C₂×#16840654

#21445952

#7266

Subgroups — #30594602

Conjugacy classes of subgroups

Complements

Maximal subgroups