Group #30605992 — subgroups

Subgroups — #30605992

Order 15616.

246 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 70 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	#21461417	C₂	1	2	yes	maximal, complemented
2	#21460843	C₂	1	1	yes	maximal, characteristic, complemented
3	#21460778	C₂	1	2	yes	maximal, complemented
4	#21460736	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447694	C₂	1	1	yes	maximal, characteristic
6	#16840790	—	2	2	no	—
7	#16840651	C₂²	1	1	yes	characteristic
8	#16840623	C₂²	1	2	yes	—
9	#16840610	—	2	2	no	—
10	#16840603	C₂²	1	1	yes	characteristic
11	C₂×D₉₇₆	C₄	1	1	yes	characteristic
12	#16839844	C₂²	1	2	yes	—
13	#16839562	C₄	1	2	yes	—
14	#16839537	C₂²	1	1	yes	characteristic
15	C₆₁×#367	C₄	1	1	yes	characteristic
16	C₂×D₆₁⋊C₈	—	4	2	no	—
17	C₂×#11758187	—	2	1	no	—
18	C₄⋊#92420	—	2	2	no	—
19	#14519834	—	4	2	no	—
20	C₄×#92420	D₄	1	2	yes	—
21	C₂×D₄₈₈	C₂×C₄	1	2	yes	—
22	#14519739	—	2	2	no	—
23	D₉₇₆	C₈	1	4	yes	complemented
24	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
25	#14519697	—	2	2	no	—
26	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
27	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
28	#107×C₆₁	C₂×C₄	1	2	yes	—
29	C₂×D₂₄₄	—	2	2	no	—
30	C₂×C₄×D₆₁	—	4	2	no	—
31	C₂×#92420	—	2	2	no	—
32	C₂×#92420	—	4	2	no	—
33	D₆₁⋊C₈	—	8	4	no	—
34	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₆₁	—	2	2	no	—
36	#11758198	—	4	2	no	—
37	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂×#92418	#41	1	1	yes	characteristic
39	D₄₈₈	—	2	2	no	—
40	D₄₈₈	C₂×C₈	1	4	yes	complemented
41	#11758187	—	4	1	no	—
42	C₁₆×C₆₁	C₂×C₈	1	2	yes	complemented
43	#11758185	—	2	1	no	—
44	C₂²×D₆₁	—	4	2	no	—
45	C₂×C₄×C₆₁	—	4	2	no	—
46	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
47	C₂×Dic₆₁	AGL(1,ℤ/8)	1	2	yes	—
48	D₂₄₄	—	2	4	no	—
49	D₂₄₄	—	4	2	no	—
50	C₄×D₆₁	—	8	2	no	—
51	C₄×D₆₁	—	8	2	no	—
52	#92420	—	4	2	no	—
53	#92420	—	8	2	no	—
54	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₆₁	#131	1	1	yes	characteristic
56	#92418	—	2	1	no	—
57	#12067	—	61	1	no	maximal, Sylow
58	C₂²×C₆₁	#442	1	1	yes	characteristic
59	C₂×D₆₁	—	4	4	no	—
60	C₂×D₆₁	—	8	2	no	—
61	C₂×D₆₁	—	8	2	no	—
62	C₄×C₆₁	—	8	2	no	—
63	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
64	C₄×C₆₁	#444	1	1	yes	characteristic
65	Dic₆₁	—	2	2	no	—
66	#2177	—	61	1	no	—
67	#1580	—	61	2	no	—
68	#1368	—	61	1	no	—
69	#1333	—	61	2	no	—
70	#1313	—	61	1	no	—
71	C₂×C₆₁	#1580	1	1	yes	characteristic
72	C₂×C₆₁	AGL(1,ℤ/16)	1	2	yes	—
73	D₆₁	—	8	4	no	—
74	C₂×D₁₆	—	61	1	no	—
75	C₄×D₈	—	61	2	no	—
76	C₈×D₄	—	122	2	no	—
77	#367	#6923	1	1	yes	characteristic
78	#357	—	61	2	no	—
79	#346	—	61	1	no	—
80	#346	—	61	1	no	—
81	#334	—	61	2	no	—
82	D₄⋊C₈	—	122	2	no	—
83	#322	—	61	1	no	—
84	C₆₁	#12067	1	1	yes	characteristic, complemented, Sylow
85	C₂×D₈	—	61	2	no	—
86	C₂²×C₈	—	244	2	no	—
87	C₄×D₄	—	122	2	no	—
88	D₁₆	—	61	4	no	—
89	C₂×C₁₆	—	61	1	no	—
90	C₂×C₁₆	—	122	1	no	—
91	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
92	#107	C₂×#6923	1	2	yes	—
93	C₄⋊C₈	—	122	2	no	—
94	#102	—	122	2	no	—
95	C₂²⋊C₈	—	244	2	no	—
96	C₄×C₈	—	61	1	no	—
97	C₄×C₈	—	61	2	no	—
98	C₂×D₄	—	122	2	no	—
99	C₂²×C₄	—	244	2	no	—
100	D₈	—	61	4	no	—
101	D₈	—	122	2	no	—
102	C₂×C₈	—	61	1	no	—
103	C₂×C₈	—	122	2	no	—
104	C₂×C₈	—	488	4	no	—
105	C₂×C₈	—	244	2	no	—
106	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₂²⋊C₄	—	244	2	no	—
109	C₄²	—	61	1	no	—
110	C₁₆	—	122	1	no	—
111	C₁₆	—	244	1	no	—
112	C₁₆	D₆₁⋊C₈	1	2	yes	complemented
113	C₂³	—	244	2	no	—
114	D₄	—	122	4	no	—
115	D₄	—	244	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	61	2	no	—
118	C₂×C₄	—	488	2	no	—
119	C₂×C₄	—	488	2	no	—
120	C₂×C₄	D₄×#6923	1	1	yes	characteristic
121	C₈	—	122	1	no	—
122	C₈	—	488	2	no	—
123	C₈	—	244	2	no	—
124	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
125	C₈	#14519865	1	1	yes	characteristic
126	C₂²	—	488	2	no	—
127	C₂²	—	488	2	no	—
128	C₂²	—	244	4	no	—
129	C₂²	#16840803	1	1	yes	characteristic
130	C₄	—	8	2	no	—
131	C₄	—	122	2	no	—
132	C₄	#16840790	1	1	yes	characteristic
133	C₄	#16840808	1	1	yes	characteristic
134	C₂	—	488	4	no	—
135	C₂	#21461417	1	1	yes	characteristic
136	C₂	#21462231	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461417

C₂

yes

maximal, complemented

#21460843

C₂

yes

maximal, characteristic, complemented

#21460778

C₂

yes

maximal, complemented

#21460736

C₂

yes

maximal, characteristic, complemented

#21447694

C₂

yes

maximal, characteristic

#16840790

—

#16840651

C₂²

yes

characteristic

#16840623

C₂²

yes

—

#16840610

—

#16840603

C₂²

yes

characteristic

C₂×D₉₇₆

C₄

yes

characteristic

#16839844

C₂²

yes

—

#16839562

C₄

yes

—

#16839537

C₂²

yes

characteristic

C₆₁×#367

C₄

yes

characteristic

C₂×D₆₁⋊C₈

—

—

—

—

D₄

yes

—

C₂×D₄₈₈

C₂×C₄

yes

—

#14519739

—

D₉₇₆

C₈

yes

complemented

C₂×#11758185

C₂×C₄

yes

characteristic

#14519697

—

C₈×Dic₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

#107×C₆₁

C₂×C₄

yes

—

C₂×D₂₄₄

—

C₂×C₄×D₆₁

—

C₂×#92420

—

C₂×#92420

—

D₆₁⋊C₈

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₆₁

—

#11758198

—

C₄×Dic₆₁

C₂×D₄

yes

yes

D₄₈₈

—

D₄₈₈

C₂×C₈

yes

complemented

#11758187

—

C₁₆×C₆₁

C₂×C₈

yes

complemented

#11758185

—

C₂²×D₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×Dic₆₁

AGL(1,ℤ/8)

yes

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

#92420

—

#92420

—

C₈×C₆₁

C₂²×C₈

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₈×D₄

yes

characteristic

C₄×C₆₁

#444

yes

characteristic

Dic₆₁

—

—

—

—

—

—

C₂×C₆₁

#1580

yes

characteristic

C₂×C₆₁

AGL(1,ℤ/16)

yes

—

D₆₁

—

C₂×D₁₆

—

C₄×D₈

—

C₈×D₄

—

122

—

yes

—

—

—

—

D₄⋊C₈

—

122

—

#322

—

C₆₁

#12067

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

244

—

C₄×D₄

—

122

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

yes

yes

—

C₄⋊C₈

—

122

—

#102

—

122

—

C₂²⋊C₈

—

244

—

C₄×C₈

—

C₄×C₈

—

C₂×D₄

—

122

—

C₂²×C₄

—

244

—

100

D₈

—

101

D₈

—

122

—

102

C₂×C₈

—

103

C₂×C₈

—

122

—

104

C₂×C₈

—

488

—

105

C₂×C₈

—

244

—

106

C₂×C₈

C₂²×#6923

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

244

—

109

C₄²

—

110

C₁₆

—

122

—

111

C₁₆

—

244

—

112

C₁₆

D₆₁⋊C₈

yes

complemented

113

C₂³

—

244

—

114

D₄

—

122

—

115

D₄

—

244

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

488

—

119

C₂×C₄

—

488

—

120

C₂×C₄

D₄×#6923

yes

characteristic

121

C₈

—

122

—

122

C₈

—

488

—

123

C₈

—

244

—

124

C₈

C₂×D₆₁⋊C₈

yes

characteristic

125

C₈

#14519865

yes

characteristic

126

C₂²

—

488

—

127

C₂²

—

488

—

128

C₂²

—

244

—

129

C₂²

#16840803

yes

characteristic

130

C₄

—

131

C₄

—

122

—

132

C₄

#16840790

yes

characteristic

133

C₄

#16840808

yes

characteristic

134

C₂

—

488

—

135

C₂

#21461417

yes

characteristic

136

C₂

#21462231

yes

—

	subgroup	G/H
center	C₂²	#16840803
commutator	C₈×C₆₁	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#367	C₄
socle	C₂²×C₆₁	#442
prime radical	C₂×D₉₇₆	C₄
cosocle	C₈×Dic₆₁	C₂³
hypercenter	#367	#6923
hypocenter	C₆₁	#12067
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

#16840803

commutator

C₈×C₆₁

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₆₁×#367

C₄

socle

C₂²×C₆₁

#442

prime radical

C₂×D₉₇₆

C₄

cosocle

C₈×Dic₆₁

C₂³

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
1. #21461417	C₂	semidirect
2. #21460843	C₂	semidirect
3. #21460778	C₂	semidirect
4. #21460736	C₂	semidirect
23. D₉₇₆	C₈	semidirect
40. D₄₈₈	C₂×C₈	semidirect
42. C₁₆×C₆₁	C₂×C₈	semidirect
84. C₆₁	#12067	semidirect
112. C₁₆	D₆₁⋊C₈	semidirect

H ≅ G/N

type

1. #21461417

C₂

semidirect

2. #21460843

C₂

semidirect

3. #21460778

C₂

semidirect

4. #21460736

C₂

semidirect

23. D₉₇₆

C₈

semidirect

40. D₄₈₈

C₂×C₈

semidirect

42. C₁₆×C₆₁

C₂×C₈

semidirect

84. C₆₁

#12067

semidirect

112. C₁₆

D₆₁⋊C₈

semidirect

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

subgroup	index
#21461417	2
#21460843	2
#21460778	2
#21460736	2
#21447694	2
#12067	61

subgroup

index

Subgroups — #30605992

Conjugacy classes of subgroups

Complements

Maximal subgroups