Group #30597405 — subgroups

Subgroups — #30597405

Order 15616.

294 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 96 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	#21461537	C₂	1	1	yes	maximal, characteristic
2	#21461033	C₂	1	1	yes	maximal, characteristic
3	#21460644	C₂	1	2	yes	maximal
4	#21460642	C₂	1	2	yes	maximal
5	#21444101	C₂	1	1	yes	maximal, characteristic
6	#16840780	C₂²	1	2	yes	—
7	#16840721	C₂²	1	1	yes	characteristic
8	#16840698	C₂²	1	1	yes	characteristic
9	(C₂×#6923)⋊C₈	C₄	1	4	yes	—
10	#92420⋊C₈	C₄	1	4	yes	complemented
11	#16839813	C₂²	1	2	yes	—
12	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
13	#16839484	C₄	1	2	yes	complemented
14	C₈×#92418	C₄	1	1	yes	characteristic, complemented
15	C₈²×C₆₁	C₄	1	1	yes	characteristic, complemented
16	C₂×D₆₁⋊C₈	C₂×C₄	1	2	yes	—
17	C₂×C₄×#6923	C₂×C₄	1	2	yes	—
18	#14519844	—	2	2	no	—
19	#14519834	—	2	2	no	—
20	#14519832	—	2	2	no	—
21	C₄×#92420	C₂×C₄	1	4	yes	—
22	(C₂×#6923)⋊C₄	—	2	2	no	—
23	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
24	C₄²×D₆₁	C₂³	1	1	yes	characteristic
25	#14519706	—	2	2	no	—
26	#14519689	—	2	2	no	—
27	C₄×#92418	C₂×C₄	1	2	yes	—
28	C₄×#92418	C₂×C₄	1	1	yes	characteristic
29	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×#6923	—	2	2	no	—
33	C₂×C₄×D₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×D₆₁	#41	1	2	yes	—
35	C₂×#92420	—	2	4	no	—
36	C₂×#92420	—	2	2	no	—
37	C₄×#6923	—	2	4	no	—
38	D₆₁⋊C₈	—	2	4	no	—
39	C₂×C₈×C₆₁	—	2	2	no	—
40	C₂×C₈×C₆₁	C₄²	1	2	yes	—
41	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
42	C₄×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
43	C₄×Dic₆₁	#41	1	2	yes	—
44	C₂×#92418	—	2	2	no	—
45	C₂×#92418	C₄²	1	2	yes	—
46	C₈×D₆₁	—	2	4	no	—
47	C₂²×D₆₁	#117	1	1	yes	characteristic
48	C₂×#6923	—	4	4	no	—
49	C₂×#6923	—	4	2	no	—
50	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#117	1	2	yes	—
52	C₂×Dic₆₁	#117	1	2	yes	—
53	C₂×Dic₆₁	#117	1	1	yes	characteristic
54	C₄×D₆₁	—	4	2	no	—
55	C₄×D₆₁	#131	1	4	yes	—
56	#92420	—	4	4	no	—
57	#92420	—	4	2	no	—
58	C₈×C₆₁	—	2	2	no	—
59	C₈×C₆₁	—	4	2	no	—
60	#92418	—	2	2	no	—
61	#92418	—	4	2	no	—
62	#8142	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	#376	1	1	yes	characteristic
64	C₂×D₆₁	—	2	2	no	—
65	C₂×D₆₁	#433	1	2	yes	—
66	#6923	—	8	2	no	—
67	C₄×C₆₁	—	2	2	no	—
68	C₄×C₆₁	#405	1	2	yes	—
69	Dic₆₁	—	2	2	no	—
70	Dic₆₁	#433	1	2	yes	—
71	#1746	—	61	1	no	—
72	#1452	—	61	1	no	—
73	#1450	—	61	1	no	—
74	#1275	—	61	2	no	—
75	#1274	—	61	2	no	—
76	C₂×C₆₁	#1746	1	1	yes	characteristic
77	C₂×C₆₁	#1747	1	2	yes	—
78	D₆₁	—	2	2	no	—
79	#431	—	61	2	no	—
80	#431	—	61	2	no	—
81	C₂×#97	—	61	1	no	—
82	C₂×C₄×C₈	—	61	1	no	—
83	#376	—	61	1	no	—
84	#336	—	61	4	no	—
85	#322	—	61	4	no	—
86	#322	—	61	2	no	—
87	C₈²	—	61	1	no	—
88	C₈²	#6923	1	1	yes	characteristic, complemented
89	C₆₁	#8142	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	61	2	no	—
91	C₂²×C₈	—	61	2	no	—
92	C₂×C₄²	—	61	1	no	—
93	C₂×C₄²	—	61	2	no	—
94	C₄⋊C₈	—	122	2	no	—
95	C₄⋊C₈	—	122	2	no	—
96	C₂²⋊C₈	—	122	2	no	—
97	C₂²⋊C₈	—	122	2	no	—
98	#97	—	122	2	no	—
99	C₄×C₈	—	61	2	no	—
100	C₄×C₈	—	61	4	no	—
101	C₄×C₈	—	61	1	no	—
102	C₄×C₈	—	61	2	no	—
103	C₄×C₈	C₂×#6923	1	2	yes	—
104	C₄×C₈	C₂×#6923	1	1	yes	characteristic
105	#95	—	122	2	no	—
106	C₂²×C₄	—	61	1	no	—
107	C₂²×C₄	—	61	2	no	—
108	C₂²×C₄	—	122	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	61	2	no	—
111	C₂×C₈	—	122	4	no	—
112	C₂×C₈	—	122	4	no	—
113	C₂×C₈	—	122	2	no	—
114	C₂×C₈	—	122	4	no	—
115	C₂×C₈	—	122	2	no	—
116	C₂×C₈	C₄×#6923	1	2	yes	—
117	C₄²	—	61	1	no	—
118	C₄²	—	61	2	no	—
119	C₄²	—	122	4	no	—
120	C₄²	C₂²×#6923	1	1	yes	characteristic
121	C₂³	—	61	1	no	—
122	C₂×C₄	—	61	1	no	—
123	C₂×C₄	—	61	4	no	—
124	C₂×C₄	—	61	2	no	—
125	C₂×C₄	—	244	2	no	—
126	C₂×C₄	—	244	2	no	—
127	C₂×C₄	—	244	4	no	—
128	C₂×C₄	C₂×C₄×#6923	1	1	yes	characteristic
129	C₂×C₄	#14519856	1	2	yes	—
130	C₈	—	2	2	no	—
131	C₈	—	4	2	no	—
132	C₈	—	122	2	no	—
133	C₈	—	244	2	no	—
134	C₈	—	244	2	no	—
135	C₈	—	244	4	no	—
136	C₂²	—	61	2	no	—
137	C₂²	—	122	2	no	—
138	C₂²	#16840698	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	61	2	no	—
141	C₄	—	122	2	no	—
142	C₄	—	488	2	no	—
143	C₄	#16840725	1	2	yes	—
144	C₂	—	122	2	no	—
145	C₂	#21461537	1	1	yes	characteristic
146	C₂	#21461540	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461537

C₂

yes

maximal, characteristic

#21461033

C₂

yes

maximal, characteristic

#21460644

C₂

yes

maximal

#21460642

C₂

yes

maximal

#21444101

C₂

yes

maximal, characteristic

#16840780

C₂²

yes

—

#16840721

C₂²

yes

characteristic

#16840698

C₂²

yes

characteristic

(C₂×#6923)⋊C₈

C₄

yes

—

#92420⋊C₈

C₄

yes

complemented

#16839813

C₂²

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839484

C₄

yes

complemented

C₈×#92418

C₄

yes

characteristic, complemented

C₈²×C₆₁

C₄

yes

characteristic, complemented

C₂×D₆₁⋊C₈

C₂×C₄

yes

—

C₂×C₄×#6923

C₂×C₄

yes

—

—

—

—

C₂×C₄

yes

—

(C₂×#6923)⋊C₄

—

C₂×C₈×D₆₁

C₂×C₄

yes

—

C₄²×D₆₁

C₂³

yes

—

—

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×#6923

—

C₂×C₄×D₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×D₆₁

#41

yes

—

C₂×#92420

—

C₂×#92420

—

C₄×#6923

—

D₆₁⋊C₈

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

#41

yes

—

C₂×#92418

—

C₂×#92418

C₄²

yes

—

C₈×D₆₁

—

C₂²×D₆₁

yes

—

—

C₂×C₄×C₆₁

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₁

#117

yes

—

C₂×Dic₆₁

#117

yes

—

C₂×Dic₆₁

#117

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

#131

yes

—

#92420

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

—

—

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

#433

yes

—

#6923

—

C₄×C₆₁

—

C₄×C₆₁

#405

yes

—

Dic₆₁

—

Dic₆₁

#433

yes

—

—

—

—

—

—

C₂×C₆₁

#1746

yes

characteristic

C₂×C₆₁

#1747

yes

—

D₆₁

—

#431

—

#431

—

C₂×#97

—

C₂×C₄×C₈

—

—

—

—

—

C₈²

—

C₈²

#6923

yes

characteristic, complemented

C₆₁

#8142

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

#97

—

122

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂×#6923

yes

—

104

C₄×C₈

C₂×#6923

yes

characteristic

105

#95

—

122

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

122

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

122

—

112

C₂×C₈

—

122

—

113

C₂×C₈

—

122

—

114

C₂×C₈

—

122

—

115

C₂×C₈

—

122

—

116

C₂×C₈

C₄×#6923

yes

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

122

—

120

C₄²

C₂²×#6923

yes

characteristic

121

C₂³

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

244

—

126

C₂×C₄

—

244

—

127

C₂×C₄

—

244

—

128

C₂×C₄

C₂×C₄×#6923

yes

characteristic

129

C₂×C₄

#14519856

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

122

—

133

C₈

—

244

—

134

C₈

—

244

—

135

C₈

—

244

—

136

C₂²

—

137

C₂²

—

122

—

138

C₂²

#16840698

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

122

—

142

C₄

—

488

—

143

C₄

#16840725

yes

—

144

C₂

—

122

—

145

C₂

#21461537

yes

characteristic

146

C₂

#21461540

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6923
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₄²	C₂²×#6923
Fitting	C₈²×C₆₁	C₄
socle	C₂²×C₆₁	#376
prime radical	C₂²×D₆₁	#117
cosocle	C₄²×D₆₁	C₂³
hypercenter	C₈²	#6923
hypocenter	C₆₁	#8142
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈²

subgroup

G/H

center

C₂×C₄

C₂×C₄×#6923

commutator

C₂×C₄×C₆₁

C₂×C₄²

Frattini

C₄²

C₂²×#6923

Fitting

C₈²×C₆₁

C₄

C₂²×C₆₁

C₂²×D₆₁

C₄²×D₆₁

C₂³

C₈²

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ C₈²

N	H ≅ G/N	type
10. #92420⋊C₈	C₄	semidirect
13. #16839484	C₄	semidirect
14. C₈×#92418	C₄	semidirect
15. C₈²×C₆₁	C₄	semidirect
88. C₈²	#6923	semidirect
89. C₆₁	#8142	semidirect

H ≅ G/N

type

10. #92420⋊C₈

C₄

semidirect

13. #16839484

C₄

semidirect

14. C₈×#92418

C₄

semidirect

15. C₈²×C₆₁

C₄

semidirect

88. C₈²

#6923

semidirect

89. C₆₁

#8142

semidirect

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

subgroup	index
#21461537	2
#21461033	2
#21460644	2
#21460642	2
#21444101	2
#8142	61

subgroup

index

Subgroups — #30597405

Conjugacy classes of subgroups

Complements

Maximal subgroups