Group #30605824 — subgroups

Subgroups — #30605824

Order 15616.

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

normal subgroups only

This group has 232 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	#21461374	C₂	1	2	yes	maximal
2	#21460843	C₂	1	4	yes	maximal, complemented
3	C₄×#14519716	C₂	1	1	yes	maximal, characteristic
4	#16840766	—	2	2	no	—
5	#16840623	—	2	2	no	—
6	#16840623	C₂²	1	4	yes	—
7	C₂×#14519716	C₄	1	2	yes	—
8	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
9	#16839537	C₂²	1	2	yes	—
10	C₄×#11758185	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic
12	C₂×#11758213	—	4	2	no	—
13	C₄⋊#92420	—	2	4	no	—
14	C₄⋊#92420	—	4	2	no	—
15	C₂×C₈×D₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×D₆₁	Q₈	1	1	yes	characteristic
17	C₄²×D₆₁	D₄	1	1	yes	characteristic
18	#14519716	—	2	4	no	—
19	C₂×#11758185	C₂×C₄	1	2	yes	—
20	C₄×#92418	Q₈	1	1	yes	characteristic
21	C₈×Dic₆₁	D₄	1	1	yes	characteristic
22	C₈×Dic₆₁	C₂³	1	1	yes	characteristic
23	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×D₆₁	SD₁₆	1	2	yes	—
27	C₂×#92420	—	4	4	no	—
28	#11758213	—	8	4	no	—
29	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
32	C₄×Dic₆₁	SD₁₆	1	2	yes	—
33	C₄×Dic₆₁	C₂×D₄	1	1	yes	characteristic
34	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
35	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
36	C₈×D₆₁	—	2	2	no	—
37	C₈×D₆₁	—	2	2	no	—
38	C₁₆×C₆₁	M₁₆	1	4	yes	complemented
39	#11758185	—	2	2	no	—
40	C₂²×D₆₁	#106	1	1	yes	characteristic
41	C₂×C₄×C₆₁	#106	1	2	yes	—
42	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×Dic₆₁	#106	1	1	yes	characteristic
44	C₂×Dic₆₁	C₂×SD₁₆	1	2	yes	—
45	C₄×D₆₁	—	2	4	no	—
46	C₄×D₆₁	—	2	1	no	—
47	C₄×D₆₁	—	2	1	no	—
48	#92420	—	8	4	no	—
49	C₈×C₆₁	#108	1	2	yes	—
50	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
51	#92418	—	2	1	no	—
52	#92418	—	2	1	no	—
53	#12020	—	61	1	no	maximal, Sylow
54	C₂²×C₆₁	C₂×#106	1	1	yes	characteristic
55	C₂×D₆₁	—	2	1	no	—
56	C₂×D₆₁	—	2	2	no	—
57	C₄×C₆₁	#365	1	4	yes	—
58	C₄×C₆₁	#423	1	1	yes	characteristic
59	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
60	Dic₆₁	—	2	1	no	—
61	Dic₆₁	—	2	2	no	—
62	C₄×M₃₂	—	61	1	no	—
63	#1566	—	61	2	no	—
64	#1368	—	61	4	no	—
65	C₂×C₆₁	#1566	1	1	yes	characteristic
66	C₂×C₆₁	C₂×#365	1	2	yes	—
67	D₆₁	—	4	1	no	—
68	C₂×M₃₂	—	61	2	no	—
69	#423	—	122	2	no	—
70	C₂×C₄×C₈	—	61	1	no	—
71	#346	—	61	2	no	—
72	C₄×C₁₆	—	61	1	no	—
73	C₄×C₁₆	#6923	1	1	yes	characteristic
74	#334	—	61	4	no	—
75	#334	—	122	2	no	—
76	C₆₁	#12020	1	1	yes	characteristic, complemented, Sylow
77	C₂×M₁₆	—	244	2	no	—
78	C₂²×C₈	—	61	1	no	—
79	C₂²×C₈	—	61	1	no	—
80	C₂×C₄²	—	61	1	no	—
81	M₃₂	—	122	4	no	—
82	C₂×C₁₆	—	61	2	no	—
83	C₂×C₁₆	C₂×#6923	1	2	yes	—
84	C₄⋊C₈	—	122	4	no	—
85	C₄⋊C₈	—	244	2	no	—
86	C₄×C₈	—	61	1	no	—
87	C₄×C₈	—	61	1	no	—
88	C₄×C₈	—	61	1	no	—
89	C₄×C₈	C₂×#6923	1	1	yes	characteristic
90	C₂²×C₄	—	61	2	no	—
91	C₂²×C₄	—	61	1	no	—
92	M₁₆	—	488	4	no	—
93	C₂×C₈	—	61	1	no	—
94	C₂×C₈	—	61	1	no	—
95	C₂×C₈	—	122	2	no	—
96	C₂×C₈	—	122	2	no	—
97	C₂×C₈	—	244	4	no	—
98	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
99	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
100	C₄²	—	61	2	no	—
101	C₄²	—	61	1	no	—
102	C₄²	C₄⋊#6923	1	1	yes	characteristic
103	C₁₆	—	122	2	no	—
104	C₁₆	#11758213	1	4	yes	complemented
105	C₂³	—	61	1	no	—
106	C₂×C₄	—	61	1	no	—
107	C₂×C₄	—	61	2	no	—
108	C₂×C₄	—	122	1	no	—
109	C₂×C₄	—	122	1	no	—
110	C₂×C₄	—	122	4	no	—
111	C₂×C₄	#14519845	1	2	yes	—
112	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
113	C₈	—	122	1	no	—
114	C₈	—	122	1	no	—
115	C₈	—	488	4	no	—
116	C₈	#14519847	1	2	yes	—
117	C₈	C₂×#11758213	1	2	yes	—
118	C₂²	—	122	1	no	—
119	C₂²	—	122	2	no	—
120	C₂²	C₂×#14519845	1	1	yes	characteristic
121	C₄	—	122	1	no	—
122	C₄	—	122	2	no	—
123	C₄	#16840686	1	4	yes	—
124	C₄	#16840766	1	1	yes	characteristic
125	C₄	C₂×#14519847	1	1	yes	characteristic
126	C₂	—	244	1	no	—
127	C₂	#21461374	1	1	yes	characteristic
128	C₂	C₂×#16840686	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461374

C₂

yes

maximal

#21460843

C₂

yes

maximal, complemented

C₄×#14519716

C₂

yes

maximal, characteristic

#16840766

—

#16840623

—

#16840623

C₂²

yes

—

C₂×#14519716

C₄

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

#16839537

C₂²

yes

—

C₄×#11758185

C₄

yes

characteristic

C₄×C₁₆×C₆₁

C₄

yes

—

—

—

C₂×C₈×D₆₁

C₂×C₄

yes

characteristic

C₂×C₈×D₆₁

Q₈

yes

characteristic

C₄²×D₆₁

D₄

yes

characteristic

#14519716

—

C₂×#11758185

C₂×C₄

yes

—

C₄×#92418

Q₈

yes

characteristic

C₈×Dic₆₁

D₄

yes

characteristic

C₈×Dic₆₁

C₂³

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×D₆₁

SD₁₆

yes

—

C₂×#92420

—

#11758213

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

SD₁₆

yes

—

C₄×Dic₆₁

C₂×D₄

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

C₂×Q₈

yes

characteristic

C₈×D₆₁

—

C₈×D₆₁

—

C₁₆×C₆₁

M₁₆

yes

complemented

#11758185

—

C₂²×D₆₁

#106

yes

characteristic

C₂×C₄×C₆₁

#106

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

#106

yes

characteristic

C₂×Dic₆₁

C₂×SD₁₆

yes

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

#92420

—

C₈×C₆₁

#108

yes

—

C₈×C₆₁

C₂×M₁₆

yes

—

—

—

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

#365

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

C₄×M₃₂

—

#1566

—

#1368

—

C₂×C₆₁

#1566

yes

characteristic

C₂×C₆₁

C₂×#365

yes

—

D₆₁

—

C₂×M₃₂

—

#423

—

122

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

yes

—

—

122

—

C₆₁

#12020

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

244

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

122

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#6923

yes

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

244

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6923

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

488

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

C₄⋊#6923

yes

characteristic

C₂×C₈

C₂²×#6923

yes

characteristic

100

C₄²

—

101

C₄²

—

102

C₄²

C₄⋊#6923

yes

characteristic

103

C₁₆

—

122

—

104

C₁₆

#11758213

yes

complemented

105

C₂³

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

122

—

109

C₂×C₄

—

122

—

110

C₂×C₄

—

122

—

111

C₂×C₄

#14519845

yes

—

112

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

113

C₈

—

122

—

114

C₈

—

122

—

115

C₈

—

488

—

116

C₈

#14519847

yes

—

117

C₈

C₂×#11758213

yes

—

118

C₂²

—

122

—

119

C₂²

—

122

—

120

C₂²

C₂×#14519845

yes

characteristic

121

C₄

—

122

—

122

C₄

—

122

—

123

C₄

#16840686

yes

—

124

C₄

#16840766

yes

characteristic

125

C₄

C₂×#14519847

yes

characteristic

126

C₂

—

244

—

127

C₂

#21461374

yes

characteristic

128

C₂

C₂×#16840686

yes

—

	subgroup	G/H
center	C₂²	C₂×#14519845
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₄×C₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂×#106
prime radical	C₂²×D₆₁	#106
cosocle	C₈×Dic₆₁	C₂³
hypercenter	C₄×C₁₆	#6923
hypocenter	C₆₁	#12020
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#14519845

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₄×C₁₆×C₆₁

C₄

C₂²×C₆₁

C₂²×D₆₁

C₈×Dic₆₁

C₂³

C₄×C₁₆

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. #21460843	C₂	semidirect
38. C₁₆×C₆₁	M₁₆	semidirect
76. C₆₁	#12020	semidirect
104. C₁₆	#11758213	semidirect

H ≅ G/N

type

2. #21460843

C₂

semidirect

38. C₁₆×C₆₁

M₁₆

semidirect

76. C₆₁

#12020

semidirect

104. C₁₆

#11758213

semidirect

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

subgroup	index
#21461374	2
#21460843	2
C₄×#14519716	2
#12020	61

subgroup

index

Subgroups — #30605824

Conjugacy classes of subgroups

Complements

Maximal subgroups