Group #30594521 — subgroups

Subgroups — #30594521

Order 15616.

206 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 74 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	#21461376	C₂	1	2	yes	maximal
2	C₂×#16839580	C₂	1	1	yes	maximal, characteristic
3	#16840765	—	2	2	no	—
4	#16840626	C₄	1	4	yes	—
5	#16840625	—	2	2	no	—
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	C₂×#11758217	—	4	2	no	—
11	#14519844	—	4	2	no	—
12	C₄⋊#92420	—	2	4	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	#11758217	—	8	2	no	—
26	#11758217	—	8	2	no	—
27	C₂×#92420	—	4	4	no	—
28	C₂×C₈×C₆₁	—	2	1	no	—
29	C₂×C₈×C₆₁	—	2	1	no	—
30	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
31	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×Dic₆₁	C₂²⋊C₄	1	2	yes	—
33	C₄×Dic₆₁	SD₁₆	1	2	yes	—
34	C₂×#92418	—	4	1	no	—
35	C₂×#92418	C₄⋊C₄	1	2	yes	—
36	C₁₆×C₆₁	—	4	1	no	—
37	#11758185	—	4	1	no	—
38	C₂³×C₆₁	#107	1	1	yes	characteristic
39	C₂×C₄×C₆₁	—	2	1	no	—
40	C₂×C₄×C₆₁	—	2	1	no	—
41	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
42	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
43	C₂×Dic₆₁	—	4	1	no	—
44	C₂×Dic₆₁	#102	1	2	yes	—
45	C₂×Dic₆₁	#103	1	2	yes	—
46	C₂×Dic₆₁	D₁₆	1	1	yes	characteristic
47	C₂×Dic₆₁	SD₃₂	1	2	yes	—
48	C₂×Dic₆₁	Q₃₂	1	1	yes	characteristic
49	#92420	—	8	4	no	—
50	C₈×C₆₁	—	2	1	no	—
51	C₈×C₆₁	#100	1	1	yes	characteristic
52	C₈×C₆₁	#101	1	1	yes	characteristic
53	#92418	—	2	2	no	—
54	#7241	—	61	1	no	maximal, Sylow
55	C₂²×C₆₁	—	2	1	no	—
56	C₂²×C₆₁	—	2	1	no	—
57	C₂²×C₆₁	#340	1	1	yes	characteristic
58	C₂²×C₆₁	#366	1	1	yes	characteristic
59	C₂²×C₆₁	#367	1	1	yes	characteristic
60	C₄×C₆₁	—	2	1	no	—
61	C₄×C₆₁	#338	1	1	yes	characteristic
62	C₄×C₆₁	#343	1	1	yes	characteristic
63	Dic₆₁	—	2	2	no	—
64	Dic₆₁	#357	1	2	yes	—
65	Dic₆₁	#358	1	2	yes	—
66	C₂×C₄⋊C₁₆	—	61	1	no	—
67	#1567	—	61	2	no	—
68	C₂×C₆₁	—	2	1	no	—
69	C₂×C₆₁	—	2	1	no	—
70	C₂×C₆₁	#1292	1	1	yes	characteristic
71	C₂×C₆₁	#1380	1	1	yes	characteristic
72	C₂×C₆₁	#1385	1	1	yes	characteristic
73	C₂²×C₁₆	—	61	1	no	—
74	C₂²×C₁₆	#6923	1	1	yes	characteristic
75	#423	—	122	2	no	—
76	C₂×C₄×C₈	—	61	1	no	—
77	C₄⋊C₁₆	—	122	2	no	—
78	#335	—	61	4	no	—
79	#335	—	122	2	no	—
80	C₆₁	#7241	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₁₆	—	244	2	no	—
82	C₂²×C₈	—	61	1	no	—
83	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
84	C₂×C₄²	—	61	1	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	4	1	no	—
87	C₂×C₁₆	—	122	1	no	—
88	C₂×C₁₆	—	244	1	no	—
89	C₄⋊C₈	—	122	4	no	—
90	C₄⋊C₈	—	244	2	no	—
91	C₄×C₈	—	61	2	no	—
92	C₄×C₈	—	61	2	no	—
93	C₂²×C₄	—	61	1	no	—
94	C₂²×C₄	—	61	1	no	—
95	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
96	M₁₆	—	488	2	no	—
97	M₁₆	—	488	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	61	2	no	—
101	C₂×C₈	—	244	1	no	—
102	C₂×C₈	—	244	4	no	—
103	C₂×C₈	C₄×#6923	1	1	yes	characteristic
104	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
105	C₄²	—	61	2	no	—
106	C₄²	—	61	2	no	—
107	C₁₆	—	4	1	no	—
108	C₁₆	—	244	1	no	—
109	C₂³	#14519846	1	1	yes	characteristic
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	61	2	no	—
113	C₂×C₄	—	61	1	no	—
114	C₂×C₄	—	61	1	no	—
115	C₂×C₄	—	61	2	no	—
116	C₂×C₄	—	61	2	no	—
117	C₂×C₄	—	244	1	no	—
118	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
119	C₂×C₄	#14519845	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	122	2	no	—
122	C₈	—	488	4	no	—
123	C₈	#14519837	1	1	yes	characteristic
124	C₈	#14519838	1	1	yes	characteristic
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#16840636	1	1	yes	characteristic
128	C₂²	#16840688	1	1	yes	characteristic
129	C₂²	#16840689	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	61	2	no	—
132	C₄	—	61	2	no	—
133	C₄	—	122	2	no	—
134	C₄	#16840631	1	1	yes	characteristic
135	C₄	#16840642	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	2	1	no	—
138	C₂	#21460689	1	1	yes	characteristic
139	C₂	#21460877	1	1	yes	characteristic
140	C₂	#21460889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21461376

C₂

yes

maximal

C₂×#16839580

C₂

yes

maximal, characteristic

#16840765

—

#16840626

C₄

yes

—

#16840625

—

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

—

—

—

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

#102

yes

—

C₂×Dic₆₁

#103

yes

—

C₂×Dic₆₁

D₁₆

yes

characteristic

C₂×Dic₆₁

SD₃₂

yes

—

C₂×Dic₆₁

Q₃₂

yes

characteristic

#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

#357

yes

—

Dic₆₁

#358

yes

—

C₂×C₄⋊C₁₆

—

#1567

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂²×C₁₆

—

C₂²×C₁₆

#6923

yes

characteristic

#423

—

122

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

122

—

#335

—

#335

—

122

—

C₆₁

#7241

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

244

—

C₂²×C₈

—

C₂²×C₈

C₂×#6923

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

—

244

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

244

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄⋊#6923

yes

characteristic

M₁₆

—

488

—

M₁₆

—

488

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

244

—

102

C₂×C₈

—

244

—

103

C₂×C₈

C₄×#6923

yes

characteristic

104

C₂×C₈

C₂²⋊#6923

yes

characteristic

105

C₄²

—

106

C₄²

—

107

C₁₆

—

108

C₁₆

—

244

—

109

C₂³

#14519846

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

244

—

118

C₂×C₄

(C₂×#6923)⋊C₄

yes

characteristic

119

C₂×C₄

#14519845

yes

characteristic

120

C₈

—

121

C₈

—

122

—

122

C₈

—

488

—

123

C₈

#14519837

yes

characteristic

124

C₈

#14519838

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

#16840636

yes

characteristic

128

C₂²

#16840688

yes

characteristic

129

C₂²

#16840689

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

122

—

134

C₄

#16840631

yes

characteristic

135

C₄

#16840642

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

#21460689

yes

characteristic

139

C₂

#21460877

yes

characteristic

140

C₂

#21460889

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	#7241
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
80. C₆₁	#7241	semidirect

H ≅ G/N

type

80. C₆₁

#7241

semidirect

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

subgroup	index
#21461376	2
C₂×#16839580	2
#7241	61

subgroup

index

#21461376

C₂×#16839580

#7241

Subgroups — #30594521

Conjugacy classes of subgroups

Complements

Maximal subgroups