Group #30608512 — subgroups

Subgroups — #30608512

Order 15616.

198 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 124 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	C₂×#16840694	C₂	1	2	yes	maximal
2	#21460978	C₂	1	4	yes	maximal, complemented
3	#21447416	C₂	1	1	yes	maximal, characteristic
4	#16840694	—	2	2	no	—
5	#16840694	C₂²	1	4	yes	—
6	C₂×#14519827	—	2	2	no	—
7	C₂×#14519717	C₄	1	1	yes	characteristic
8	C₂²×#11758185	C₂²	1	1	yes	characteristic
9	#16839538	C₄	1	2	yes	—
10	C₄×#11758185	C₂²	1	2	yes	—
11	C₆₁×#405	C₄	1	1	yes	characteristic
12	#14519827	—	2	4	no	—
13	#14519717	—	2	2	no	—
14	C₂×#11758185	C₂×C₄	1	2	yes	—
15	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
16	C₂×#11758185	D₄	1	2	yes	—
17	C₂×#11758185	Q₈	1	2	yes	—
18	C₂×#11758185	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₆₁	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	#117×C₆₁	C₈	1	1	yes	characteristic
22	#97×C₆₁	C₈	1	2	yes	—
23	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
25	M₁₆×C₆₁	C₁₆	1	4	yes	—
26	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
30	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₂²⋊C₄×C₆₁	—	2	1	no	—
33	C₄²×C₆₁	C₂×C₈	1	2	yes	—
34	#11758185	—	2	2	no	—
35	#11758185	C₄⋊C₄	1	2	yes	—
36	#11758185	C₂×D₄	1	1	yes	characteristic
37	#11758185	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₆₁	M₃₂	1	1	yes	characteristic
39	C₂×C₄×C₆₁	—	2	2	no	—
40	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₆₁	C₂×C₁₆	1	2	yes	—
42	C₂×C₄×C₆₁	M₃₂	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
44	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
45	C₈×C₆₁	C₂×C₁₆	1	4	yes	—
46	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
48	#13630	—	61	1	no	maximal, Sylow
49	C₂²×C₆₁	—	4	1	no	—
50	C₂²×C₆₁	C₄⋊C₁₆	1	2	yes	—
51	C₂²×C₆₁	C₂×M₃₂	1	1	yes	characteristic
52	C₄×C₆₁	—	4	2	no	—
53	C₄×C₆₁	C₄⋊C₁₆	1	2	yes	—
54	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
55	C₄×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
56	C₂×M₆₄	—	61	2	no	—
57	#2108	—	61	1	no	—
58	#1422	—	61	4	no	—
59	C₂×C₆₁	—	2	1	no	—
60	C₂×C₆₁	—	2	2	no	—
61	C₂×C₆₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
62	C₂×M₃₂	—	61	1	no	—
63	C₂²×C₁₆	—	61	1	no	—
64	#405	#6923	1	1	yes	characteristic
65	M₆₄	—	61	4	no	—
66	M₆₄	—	122	2	no	—
67	C₂×C₃₂	—	122	2	no	—
68	#346	—	61	2	no	—
69	C₄×C₁₆	—	61	2	no	—
70	C₆₁	#13630	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	#92420	1	1	yes	characteristic
72	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
73	#117	#92420	1	1	yes	characteristic
74	M₃₂	—	122	2	no	—
75	C₂×C₁₆	—	61	2	no	—
76	C₂×C₁₆	—	61	2	no	—
77	C₂×C₁₆	—	61	1	no	—
78	C₂×C₁₆	—	61	1	no	—
79	C₂×C₁₆	—	61	2	no	—
80	#97	#92420	1	2	yes	—
81	C₄×C₈	C₂×#6923	1	2	yes	—
82	C₃₂	—	122	4	no	—
83	C₂²×C₄	C₂×#92420	1	1	yes	characteristic
84	M₁₆	#11758187	1	4	yes	—
85	C₂×C₈	#11758213	1	2	yes	—
86	C₂×C₈	C₄⋊#6923	1	2	yes	—
87	C₂×C₈	C₂×#92420	1	1	yes	characteristic
88	C₂×C₈	C₂×#92420	1	2	yes	—
89	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₄²	C₂×#92420	1	2	yes	—
93	C₁₆	—	61	1	no	—
94	C₁₆	—	61	2	no	—
95	C₁₆	—	61	1	no	—
96	C₁₆	—	122	2	no	—
97	C₂³	#14519851	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₄⋊#92420	1	2	yes	—
100	C₂×C₄	C₂×#11758187	1	2	yes	—
101	C₂×C₄	#14519851	1	1	yes	characteristic
102	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
103	C₈	C₄⋊#92420	1	2	yes	—
104	C₈	C₂×#11758187	1	4	yes	—
105	C₈	C₂×C₄⋊#6923	1	1	yes	characteristic
106	C₈	C₂×#11758213	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	C₄⋊#11758187	1	2	yes	—
109	C₂²	C₂×#14519851	1	1	yes	characteristic
110	C₄	—	4	2	no	—
111	C₄	C₄⋊#11758187	1	2	yes	—
112	C₄	C₂×C₄⋊#92420	1	1	yes	characteristic
113	C₄	C₂²×#11758187	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	C₂×C₄⋊#11758187	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16840694

C₂

yes

maximal

#21460978

C₂

yes

maximal, complemented

#21447416

C₂

yes

maximal, characteristic

#16840694

—

#16840694

C₂²

yes

—

C₂×#14519827

—

C₂×#14519717

C₄

yes

characteristic

C₂²×#11758185

C₂²

yes

characteristic

#16839538

C₄

yes

—

C₄×#11758185

C₂²

yes

—

C₆₁×#405

C₄

yes

—

—

C₂×C₄

yes

—

C₂×#11758185

C₂×C₄

yes

characteristic

C₂×#11758185

D₄

yes

—

C₂×#11758185

Q₈

yes

—

C₂×#11758185

C₂³

yes

characteristic

C₂×M₁₆×C₆₁

C₈

yes

characteristic

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

#117×C₆₁

C₈

yes

characteristic

#97×C₆₁

C₈

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

M₁₆×C₆₁

C₁₆

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

M₁₆

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₈

yes

—

#11758185

—

#11758185

C₄⋊C₄

yes

—

#11758185

C₂×D₄

yes

characteristic

#11758185

C₂×Q₈

yes

characteristic

C₂³×C₆₁

M₃₂

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₁₆

yes

—

C₂×C₄×C₆₁

M₃₂

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₈×C₆₁

C₄⋊C₈

yes

—

C₈×C₆₁

C₂×C₁₆

yes

—

C₈×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₆₁

C₂×M₁₆

yes

characteristic

#13630

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

C₄⋊C₁₆

yes

—

C₂²×C₆₁

C₂×M₃₂

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄⋊C₁₆

yes

—

C₄×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₆₁

C₂²×C₁₆

yes

characteristic

C₂×M₆₄

—

#2108

—

#1422

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄⋊C₁₆

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#405

#6923

yes

characteristic

M₆₄

—

M₆₄

—

122

—

C₂×C₃₂

—

122

—

#346

—

C₄×C₁₆

—

C₆₁

#13630

yes

characteristic, complemented, Sylow

C₂×M₁₆

yes

C₂²×C₈

yes

yes

M₃₂

—

122

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

#92420

yes

—

C₄×C₈

C₂×#6923

yes

—

C₃₂

—

122

—

C₂²×C₄

C₂×#92420

yes

characteristic

M₁₆

#11758187

yes

—

C₂×C₈

#11758213

yes

—

C₂×C₈

C₄⋊#6923

yes

—

C₂×C₈

C₂×#92420

yes

characteristic

C₂×C₈

C₂×#92420

yes

—

C₂×C₈

C₂²×#6923

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×#92420

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

122

—

C₂³

#14519851

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄⋊#92420

yes

—

100

C₂×C₄

C₂×#11758187

yes

—

101

C₂×C₄

#14519851

yes

characteristic

102

C₂×C₄

C₂²×#92420

yes

characteristic

103

C₈

C₄⋊#92420

yes

—

104

C₈

C₂×#11758187

yes

—

105

C₈

C₂×C₄⋊#6923

yes

characteristic

106

C₈

C₂×#11758213

yes

characteristic

107

C₂²

—

108

C₂²

C₄⋊#11758187

yes

—

109

C₂²

C₂×#14519851

yes

characteristic

110

C₄

—

111

C₄

C₄⋊#11758187

yes

—

112

C₄

C₂×C₄⋊#92420

yes

characteristic

113

C₄

C₂²×#11758187

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

C₂×C₄⋊#11758187

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₄⋊#6923
commutator	C₄×C₆₁	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₆₁×#405	C₄
socle	C₂×C₆₁	C₂×C₄⋊C₁₆
prime radical	C₂³×C₆₁	M₃₂
cosocle	C₂×#11758185	C₂³
hypercenter	#405	#6923
hypocenter	C₆₁	#13630
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #405

subgroup

G/H

center

C₈

C₂×C₄⋊#6923

commutator

C₄×C₆₁

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₆₁×#405

C₄

socle

C₂×C₆₁

C₂×C₄⋊C₁₆

prime radical

C₂³×C₆₁

M₃₂

C₂³

C₆₁

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #405

N	H ≅ G/N	type
2. #21460978	C₂	semidirect
70. C₆₁	#13630	semidirect

H ≅ G/N

type

2. #21460978

C₂

semidirect

70. C₆₁

#13630

semidirect

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

subgroup	index
C₂×#16840694	2
#21460978	2
#21447416	2
#13630	61

subgroup

index

Subgroups — #30608512

Conjugacy classes of subgroups

Complements

Maximal subgroups