Group #30605998 — subgroups

Subgroups — #30605998

Order 15616.

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

normal subgroups only

This group has 204 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	#21462211	C₂	1	2	yes	maximal, complemented
2	#21460846	C₂	1	1	yes	maximal, characteristic, complemented
3	#21460784	C₂	1	2	yes	maximal, complemented
4	#21460736	C₂	1	1	yes	maximal, characteristic, complemented
5	#21447704	C₂	1	1	yes	maximal, characteristic
6	#16840821	—	2	2	no	—
7	#16840684	C₂²	1	2	yes	—
8	#16840661	—	2	2	no	—
9	#16840651	C₂²	1	1	yes	characteristic
10	#16840603	C₂²	1	1	yes	characteristic
11	#16840194	C₄	1	1	yes	characteristic
12	#16839878	C₂²	1	2	yes	—
13	#16839568	C₄	1	2	yes	—
14	#16839537	C₂²	1	1	yes	characteristic
15	C₆₁×#368	C₄	1	1	yes	characteristic
16	#14519852	—	4	2	no	—
17	#14519851	—	2	2	no	—
18	D₆₁⋊C₁₆	—	4	2	no	—
19	C₂×#11758187	D₄	1	2	yes	—
20	C₄×#92420	—	2	1	no	—
21	#14519799	C₂×C₄	1	2	yes	—
22	#14519785	—	2	2	no	—
23	Dic₄₈₈	C₈	1	1	yes	characteristic, complemented
24	#14519722	C₈	1	2	yes	complemented
25	D₉₇₆	C₈	1	1	yes	characteristic, complemented
26	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
27	#14519703	—	2	2	no	—
28	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
29	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
30	#108×C₆₁	C₂×C₄	1	2	yes	—
31	#11758222	—	2	2	no	—
32	C₂×#92420	—	4	1	no	—
33	M₁₆×C₆₁	—	2	2	no	—
34	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
35	C₄×Dic₆₁	#41	1	1	yes	characteristic
36	C₂×#92418	C₂×D₄	1	1	yes	characteristic
37	Dic₂₄₄	C₂×C₈	1	2	yes	—
38	D₄₈₈	C₂×C₈	1	2	yes	—
39	#11758190	—	2	2	no	—
40	#11758189	—	4	2	no	—
41	C₈×D₆₁	—	4	2	no	—
42	#11758187	—	2	2	no	—
43	#11758187	—	4	2	no	—
44	C₁₆×C₆₁	C₂×C₈	1	1	yes	characteristic
45	C₁₆×C₆₁	C₂×C₈	1	1	yes	characteristic
46	#11758185	—	2	1	no	—
47	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
48	C₂²⋊D₆₁	—	4	2	no	—
49	C₂×Dic₆₁	—	2	1	no	—
50	D₂₄₄	—	2	2	no	—
51	C₄×D₆₁	—	4	2	no	—
52	Dic₁₂₂	—	2	2	no	—
53	#92420	—	8	1	no	—
54	C₈×C₆₁	—	4	2	no	—
55	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₆₁	#131	1	1	yes	characteristic
57	#92418	AGL(1,ℤ/8)	1	2	yes	—
58	#12068	—	61	1	no	maximal, Sylow
59	C₂²×C₆₁	#444	1	1	yes	characteristic
60	C₂×D₆₁	—	4	2	no	—
61	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
62	C₄×C₆₁	#442	1	1	yes	characteristic
63	Dic₆₁	—	4	1	no	—
64	Dic₆₁	—	4	2	no	—
65	#2179	—	61	1	no	—
66	#2171	—	61	2	no	—
67	#1369	—	61	1	no	—
68	#1336	—	61	2	no	—
69	#1313	—	61	1	no	—
70	C₂×C₆₁	—	2	1	no	—
71	C₂×C₆₁	#1580	1	1	yes	characteristic
72	D₆₁	—	8	2	no	—
73	#508	—	61	1	no	—
74	#504	—	122	2	no	—
75	#443	—	61	2	no	—
76	#368	#6923	1	1	yes	characteristic
77	#364	—	61	2	no	—
78	#359	—	61	2	no	—
79	#350	—	122	2	no	—
80	#346	—	61	1	no	—
81	#346	—	61	1	no	—
82	#322	—	61	1	no	—
83	C₆₁	#12068	1	1	yes	characteristic, complemented, Sylow
84	#135	—	61	2	no	—
85	#131	—	122	2	no	—
86	Q₃₂	—	61	1	no	—
87	SD₃₂	—	61	2	no	—
88	D₁₆	—	61	1	no	—
89	M₃₂	—	122	2	no	—
90	M₃₂	—	244	2	no	—
91	C₂×C₁₆	—	61	2	no	—
92	C₂×C₁₆	—	61	1	no	—
93	C₂×C₁₆	—	244	2	no	—
94	C₂×C₁₆	C₂×#6923	1	1	yes	characteristic
95	#108	C₂×#6923	1	2	yes	—
96	#104	—	122	2	no	—
97	C₄×C₈	—	61	1	no	—
98	C₄×C₈	—	122	1	no	—
99	#41	—	122	2	no	—
100	Q₁₆	—	61	2	no	—
101	SD₁₆	—	122	2	no	—
102	D₈	—	61	2	no	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	244	2	no	—
105	C₂×C₈	—	61	1	no	—
106	C₂×C₈	—	244	2	no	—
107	C₂×C₈	—	244	1	no	—
108	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
109	C₄²	—	61	1	no	—
110	C₁₆	—	122	2	no	—
111	C₁₆	—	122	1	no	—
112	C₁₆	—	244	2	no	—
113	C₁₆	D₆₁⋊C₈	1	1	yes	characteristic
114	C₁₆	D₆₁⋊C₈	1	1	yes	characteristic
115	Q₈	—	122	2	no	—
116	D₄	—	122	2	no	—
117	D₄	—	244	2	no	—
118	C₂×C₄	—	122	1	no	—
119	C₂×C₄	—	244	2	no	—
120	C₂×C₄	D₄×#6923	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₈	—	61	2	no	—
123	C₈	—	488	1	no	—
124	C₈	C₂×D₆₁⋊C₈	1	1	yes	characteristic
125	C₈	#14519865	1	1	yes	characteristic
126	C₂²	—	244	2	no	—
127	C₂²	#16840808	1	1	yes	characteristic
128	C₄	—	244	1	no	—
129	C₄	—	244	2	no	—
130	C₄	#16840790	1	1	yes	characteristic
131	C₄	#16840803	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	488	2	no	—
134	C₂	#21461417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21462211

C₂

yes

maximal, complemented

#21460846

C₂

yes

maximal, characteristic, complemented

#21460784

C₂

yes

maximal, complemented

#21460736

C₂

yes

maximal, characteristic, complemented

#21447704

C₂

yes

maximal, characteristic

#16840821

—

#16840684

C₂²

yes

—

#16840661

—

#16840651

C₂²

yes

characteristic

#16840603

C₂²

yes

characteristic

#16840194

C₄

yes

characteristic

#16839878

C₂²

yes

—

#16839568

C₄

yes

—

#16839537

C₂²

yes

characteristic

C₆₁×#368

C₄

yes

characteristic

#14519852

—

#14519851

—

D₆₁⋊C₁₆

—

C₂×#11758187

D₄

yes

—

C₄×#92420

—

#14519799

C₂×C₄

yes

—

#14519785

—

Dic₄₈₈

C₈

yes

characteristic, complemented

#14519722

C₈

yes

complemented

D₉₇₆

C₈

yes

characteristic, complemented

C₂×#11758185

C₂×C₄

yes

characteristic

#14519703

—

C₈×Dic₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

#108×C₆₁

C₂×C₄

yes

—

#11758222

—

C₂×#92420

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂²×C₄

yes

C₄×Dic₆₁

yes

C₂×D₄

yes

characteristic

Dic₂₄₄

C₂×C₈

yes

—

D₄₈₈

C₂×C₈

yes

—

#11758190

—

#11758189

—

C₈×D₆₁

—

#11758187

—

#11758187

—

C₁₆×C₆₁

C₂×C₈

yes

characteristic

C₁₆×C₆₁

C₂×C₈

yes

characteristic

#11758185

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

C₂²×C₈

yes

C₈×C₆₁

yes

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₄×C₆₁

C₈×D₄

yes

characteristic

C₄×C₆₁

#442

yes

characteristic

Dic₆₁

—

Dic₆₁

—

—

—

—

—

—

C₂×C₆₁

—

C₂×C₆₁

#1580

yes

characteristic

D₆₁

—

#508

—

#504

—

122

—

—

yes

—

—

—

122

—

#346

—

#346

—

#322

—

C₆₁

#12068

yes

characteristic, complemented, Sylow

#135

—

#131

—

122

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

122

—

M₃₂

—

244

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

244

—

C₂×C₁₆

yes

yes

—

#104

—

122

—

C₄×C₈

—

C₄×C₈

—

122

—

#41

—

122

—

100

Q₁₆

—

101

SD₁₆

—

122

—

102

D₈

—

103

M₁₆

—

104

M₁₆

—

244

—

105

C₂×C₈

—

106

C₂×C₈

—

244

—

107

C₂×C₈

—

244

—

108

C₂×C₈

C₂²×#6923

yes

characteristic

109

C₄²

—

110

C₁₆

—

122

—

111

C₁₆

—

122

—

112

C₁₆

—

244

—

113

C₁₆

D₆₁⋊C₈

yes

characteristic

114

C₁₆

D₆₁⋊C₈

yes

characteristic

115

Q₈

—

122

—

116

D₄

—

122

—

117

D₄

—

244

—

118

C₂×C₄

—

122

—

119

C₂×C₄

—

244

—

120

C₂×C₄

D₄×#6923

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

488

—

124

C₈

C₂×D₆₁⋊C₈

yes

characteristic

125

C₈

#14519865

yes

characteristic

126

C₂²

—

244

—

127

C₂²

#16840808

yes

characteristic

128

C₄

—

244

—

129

C₄

—

244

—

130

C₄

#16840790

yes

characteristic

131

C₄

#16840803

yes

characteristic

132

C₂

—

133

C₂

—

488

—

134

C₂

#21461417

yes

characteristic

	subgroup	G/H
center	C₄	#16840803
commutator	C₈×C₆₁	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#368	C₄
socle	C₂×C₆₁	#1580
prime radical	#16840194	C₄
cosocle	C₈×Dic₆₁	C₂³
hypercenter	#368	#6923
hypocenter	C₆₁	#12068
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

#16840803

commutator

C₈×C₆₁

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₆₁×#368

C₄

C₂×C₆₁

C₄

C₈×Dic₆₁

C₂³

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
1. #21462211	C₂	semidirect
2. #21460846	C₂	semidirect
3. #21460784	C₂	semidirect
4. #21460736	C₂	semidirect
23. Dic₄₈₈	C₈	semidirect
24. #14519722	C₈	semidirect
25. D₉₇₆	C₈	semidirect
83. C₆₁	#12068	semidirect

H ≅ G/N

type

1. #21462211

C₂

semidirect

2. #21460846

C₂

semidirect

3. #21460784

C₂

semidirect

4. #21460736

C₂

semidirect

23. Dic₄₈₈

C₈

semidirect

24. #14519722

C₈

semidirect

25. D₉₇₆

C₈

semidirect

83. C₆₁

#12068

semidirect

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

subgroup	index
#21462211	2
#21460846	2
#21460784	2
#21460736	2
#21447704	2
#12068	61

subgroup

index

Subgroups — #30605998

Conjugacy classes of subgroups

Complements

Maximal subgroups