Group #30608587 — subgroups

Subgroups — #30608587

Order 15616.

158 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 84 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	#21460977	C₂	1	2	yes	maximal
2	C₄⋊#14519827	C₂	1	1	yes	maximal, characteristic
3	#21460927	C₂	1	2	yes	maximal
4	C₄×#14519827	C₂	1	1	yes	maximal, characteristic
5	#21447721	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519827	C₂²	1	1	yes	characteristic
7	C₂×#14519827	C₂²	1	2	yes	—
8	C₂×#14519827	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₂₂	C₄	1	1	yes	characteristic
10	#16839580	C₄	1	2	yes	—
11	C₄×#11758185	C₂²	1	1	yes	characteristic
12	C₁₆×Dic₆₁	C₂²	1	2	yes	—
13	C₆₁×C₄⋊C₁₆	C₄	1	1	yes	characteristic
14	#14519827	—	2	1	no	—
15	#14519827	—	2	2	no	—
16	#14519827	Q₈	1	2	yes	complemented
17	C₄×Dic₁₂₂	C₈	1	1	yes	characteristic
18	C₂×#11758185	C₂×C₄	1	1	yes	characteristic
19	C₂×#11758185	C₂³	1	1	yes	characteristic
20	C₄⋊#92418	C₈	1	1	yes	characteristic
21	#14519706	C₈	1	2	yes	—
22	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₁₂₂	C₁₆	1	1	yes	characteristic
26	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
28	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
29	C₄⋊Dic₆₁	C₁₆	1	1	yes	characteristic
30	#11758196	C₁₆	1	2	yes	—
31	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
32	C₂×#92418	C₂×C₈	1	2	yes	—
33	C₁₆×C₆₁	—	2	2	no	—
34	#11758185	—	2	1	no	—
35	#11758185	C₂×Q₈	1	1	yes	characteristic
36	#11758185	#41	1	1	yes	characteristic
37	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
40	C₂×Dic₆₁	C₂×C₁₆	1	2	yes	—
41	C₂×Dic₆₁	C₂×C₁₆	1	2	yes	—
42	Dic₁₂₂	—	2	2	no	—
43	C₈×C₆₁	—	2	1	no	—
44	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
45	C₈×C₆₁	#131	1	1	yes	characteristic
46	#92418	—	2	2	no	—
47	#13660	—	61	1	no	maximal, Sylow
48	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
49	C₄×C₆₁	—	2	1	no	—
50	C₄×C₆₁	M₆₄	1	2	yes	—
51	C₄×C₆₁	C₈×Q₈	1	1	yes	characteristic
52	C₄×C₆₁	#504	1	1	yes	characteristic
53	Dic₆₁	—	2	2	no	—
54	Dic₆₁	—	2	2	no	—
55	Q₈×C₁₆	—	61	1	no	—
56	C₄⋊C₃₂	—	61	1	no	—
57	C₄⋊C₃₂	—	61	2	no	—
58	#1397	—	61	2	no	—
59	C₄×C₃₂	—	61	1	no	—
60	C₂×C₆₁	Q₈×C₁₆	1	1	yes	characteristic
61	C₂×C₆₁	C₂×M₆₄	1	1	yes	characteristic
62	C₂×C₆₁	#2258	1	1	yes	characteristic
63	C₈×Q₈	—	61	1	no	—
64	C₂×C₃₂	—	61	1	no	—
65	C₂×C₃₂	—	61	1	no	—
66	C₂×C₃₂	—	61	2	no	—
67	C₄⋊C₁₆	—	61	2	no	—
68	C₄⋊C₁₆	#6923	1	1	yes	characteristic
69	C₄×C₁₆	—	61	2	no	—
70	C₄×C₁₆	—	61	1	no	—
71	C₆₁	#13660	1	1	yes	characteristic, complemented, Sylow
72	C₄×Q₈	—	61	1	no	—
73	C₂×C₁₆	—	61	1	no	—
74	C₂×C₁₆	—	61	1	no	—
75	C₂×C₁₆	C₂×#6923	1	2	yes	—
76	C₄⋊C₈	—	61	2	no	—
77	C₄⋊C₈	—	61	1	no	—
78	C₄×C₈	—	61	2	no	—
79	C₄×C₈	C₂×#6923	1	1	yes	characteristic
80	C₃₂	—	61	2	no	—
81	C₃₂	—	122	1	no	—
82	C₃₂	—	122	2	no	—
83	C₂×Q₈	—	61	1	no	—
84	C₂×C₈	—	61	2	no	—
85	C₂×C₈	D₆₁⋊C₈	1	1	yes	characteristic
86	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
87	C₄⋊C₄	—	61	2	no	—
88	C₄⋊C₄	—	61	1	no	—
89	C₄²	—	61	2	no	—
90	C₄²	D₆₁⋊C₈	1	1	yes	characteristic
91	C₁₆	—	2	2	no	—
92	C₁₆	—	61	1	no	—
93	C₁₆	—	61	1	no	—
94	C₁₆	—	122	1	no	—
95	Q₈	—	122	2	no	—
96	C₂×C₄	—	61	2	no	—
97	C₂×C₄	—	61	2	no	—
98	C₂×C₄	D₆₁⋊C₁₆	1	1	yes	characteristic
99	C₂×C₄	D₆₁⋊C₁₆	1	1	yes	characteristic
100	C₂×C₄	C₂×D₆₁⋊C₈	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	122	2	no	—
103	C₈	Q₈×#6923	1	1	yes	characteristic
104	C₈	#14519864	1	1	yes	characteristic
105	C₂²	C₂×D₆₁⋊C₁₆	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	122	2	no	—
108	C₄	—	122	2	no	—
109	C₄	#16840695	1	2	yes	—
110	C₄	#16840811	1	1	yes	characteristic
111	C₄	#16840821	1	1	yes	characteristic
112	C₂	#21462233	1	1	yes	characteristic
113	C₂	C₂×#16840695	1	1	yes	characteristic
114	C₂	#21462243	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21460977

C₂

yes

maximal

C₄⋊#14519827

C₂

yes

maximal, characteristic

#21460927

C₂

yes

maximal

C₄×#14519827

C₂

yes

maximal, characteristic

#21447721

C₂

yes

maximal, characteristic

C₂×#14519827

C₂²

yes

characteristic

C₂×#14519827

C₂²

yes

—

C₂×#14519827

C₂²

yes

characteristic

C₈×Dic₁₂₂

C₄

yes

characteristic

#16839580

C₄

yes

—

C₄×#11758185

C₂²

yes

characteristic

C₁₆×Dic₆₁

C₂²

yes

—

C₆₁×C₄⋊C₁₆

C₄

yes

—

—

Q₈

yes

complemented

C₄×Dic₁₂₂

C₈

yes

characteristic

C₂×#11758185

C₂×C₄

yes

characteristic

C₂×#11758185

C₂³

yes

characteristic

C₄⋊#92418

C₈

yes

characteristic

#14519706

C₈

yes

—

C₈×Dic₆₁

C₂×C₄

yes

—

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×Dic₁₂₂

C₁₆

yes

characteristic

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄⋊Dic₆₁

C₁₆

yes

characteristic

#11758196

C₁₆

yes

—

C₄×Dic₆₁

C₂×C₈

yes

—

C₂×#92418

C₂×C₈

yes

—

C₁₆×C₆₁

—

#11758185

—

#11758185

C₂×Q₈

yes

yes

C₂×C₄×C₆₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×Dic₆₁

C₂×C₁₆

yes

—

C₂×Dic₆₁

C₂×C₁₆

yes

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₄×Q₈

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

C₂²×C₁₆

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

M₆₄

yes

—

C₄×C₆₁

C₈×Q₈

yes

characteristic

C₄×C₆₁

#504

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

—

#1397

—

C₄×C₃₂

—

C₂×C₆₁

Q₈×C₁₆

yes

characteristic

C₂×C₆₁

C₂×M₆₄

yes

characteristic

C₂×C₆₁

#2258

yes

characteristic

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₄⋊C₁₆

—

C₄⋊C₁₆

#6923

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

C₆₁

#13660

yes

characteristic, complemented, Sylow

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#6923

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6923

yes

characteristic

C₃₂

—

C₃₂

—

122

—

C₃₂

—

122

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

D₆₁⋊C₈

yes

characteristic

C₂×C₈

C₂²×#6923

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

D₆₁⋊C₈

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

122

—

Q₈

—

122

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₆₁⋊C₁₆

yes

characteristic

C₂×C₄

D₆₁⋊C₁₆

yes

characteristic

100

C₂×C₄

C₂×D₆₁⋊C₈

yes

characteristic

101

C₈

—

102

C₈

—

122

—

103

C₈

Q₈×#6923

yes

characteristic

104

C₈

#14519864

yes

characteristic

105

C₂²

C₂×D₆₁⋊C₁₆

yes

characteristic

106

C₄

—

107

C₄

—

122

—

108

C₄

—

122

—

109

C₄

#16840695

yes

—

110

C₄

#16840811

yes

characteristic

111

C₄

#16840821

yes

characteristic

112

C₂

#21462233

yes

characteristic

113

C₂

C₂×#16840695

yes

characteristic

114

C₂

#21462243

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6923
commutator, socle, prime radical	C₂²×C₆₁	C₂²×C₁₆
Fitting	C₆₁×C₄⋊C₁₆	C₄
cosocle	C₂×#11758185	C₂³
hypercenter	C₄⋊C₁₆	#6923
hypocenter	C₆₁	#13660
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#6923

commutator, socle, prime radical

C₂²×C₆₁

C₂²×C₁₆

Fitting

C₆₁×C₄⋊C₁₆

C₄

C₂³

C₄⋊C₁₆

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
16. #14519827	Q₈	semidirect
71. C₆₁	#13660	semidirect

H ≅ G/N

type

16. #14519827

Q₈

semidirect

71. C₆₁

#13660

semidirect

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

subgroup	index
#21460977	2
C₄⋊#14519827	2
#21460927	2
C₄×#14519827	2
#21447721	2
#13660	61

subgroup

index

Subgroups — #30608587

Conjugacy classes of subgroups

Complements

Maximal subgroups