Group #30608568 — subgroups

Subgroups — #30608568

Order 15616.

188 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 54 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	Q₃₂×#6923	C₂	1	2	yes	maximal
2	#21460981	C₂	1	1	yes	maximal, characteristic
3	#21460969	C₂	1	2	yes	maximal
4	#21460925	C₂	1	1	yes	maximal, characteristic
5	Q₆₄×D₆₁	C₂	1	1	yes	maximal, characteristic
6	Q₁₆×#6923	—	2	2	no	—
7	D₆₁⋊C₃₂	C₂²	1	1	yes	characteristic
8	#16840688	C₂²	1	2	yes	—
9	#16840675	—	2	2	no	—
10	C₁₆×#6923	C₂²	1	1	yes	characteristic
11	Q₃₂×D₆₁	C₂²	1	2	yes	—
12	Dic₉₇₆	C₄	1	1	yes	characteristic, complemented
13	#16839602	C₄	1	2	yes	complemented
14	C₃₂×D₆₁	C₂²	1	1	yes	characteristic
15	C₆₁×Q₆₄	C₄	1	1	yes	characteristic, complemented
16	Q₈×#6923	—	4	2	no	—
17	D₆₁⋊C₁₆	D₄	1	1	yes	characteristic
18	#14519846	—	2	2	no	—
19	#14519840	—	4	2	no	—
20	C₈×#6923	D₄	1	1	yes	characteristic
21	#14519827	—	2	1	no	—
22	Q₁₆×D₆₁	—	2	2	no	—
23	Dic₄₈₈	C₂×C₄	1	2	yes	—
24	#14519723	—	2	2	no	—
25	C₁₆×D₆₁	C₂³	1	1	yes	characteristic
26	#14519683	C₂×C₄	1	1	yes	characteristic
27	Q₃₂×C₆₁	C₂×C₄	1	2	yes	—
28	C₃₂×C₆₁	C₂×C₄	1	1	yes	characteristic
29	Q₈×D₆₁	—	4	2	no	—
30	C₄⋊#6923	—	4	2	no	—
31	C₄⋊#6923	—	8	2	no	—
32	C₄×#6923	—	8	2	no	—
33	C₄×#6923	D₈	1	1	yes	characteristic
34	D₆₁⋊C₈	D₈	1	1	yes	characteristic
35	Q₁₆×C₆₁	—	2	2	no	—
36	#11758202	—	4	2	no	—
37	Dic₂₄₄	—	2	2	no	—
38	C₈×D₆₁	C₂×D₄	1	1	yes	characteristic
39	#11758187	—	2	1	no	—
40	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
41	#11758185	#41	1	1	yes	characteristic
42	C₂×#6923	—	8	2	no	—
43	C₂×#6923	D₁₆	1	2	yes	—
44	Q₈×C₆₁	—	4	2	no	—
45	C₄×D₆₁	—	8	2	no	—
46	C₄×D₆₁	C₂×D₈	1	1	yes	characteristic
47	Dic₁₂₂	—	4	2	no	—
48	Dic₁₂₂	—	8	2	no	—
49	#92420	—	2	1	no	—
50	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
51	#92418	#135	1	1	yes	characteristic
52	#13653	—	61	1	no	maximal, Sylow
53	C₂×D₆₁	C₂×D₁₆	1	1	yes	characteristic
54	#6923	—	2	2	no	—
55	#6923	—	16	2	no	—
56	C₄×C₆₁	—	8	2	no	—
57	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
58	Dic₆₁	—	8	2	no	—
59	Dic₆₁	#508	1	1	yes	characteristic
60	C₂×Q₆₄	—	61	1	no	—
61	C₄×Q₃₂	—	61	2	no	—
62	#1424	—	61	1	no	—
63	#1416	—	61	2	no	—
64	#1397	—	61	1	no	—
65	C₂×C₆₁	C₄×D₁₆	1	1	yes	characteristic
66	D₆₁	#2264	1	2	yes	—
67	C₂×Q₃₂	—	61	2	no	—
68	C₄×Q₁₆	—	122	2	no	—
69	Q₆₄	—	61	2	no	—
70	Q₆₄	—	61	1	no	—
71	Q₆₄	#6923	1	1	yes	characteristic, complemented
72	C₂×C₃₂	—	61	1	no	—
73	C₂×C₃₂	—	61	1	no	—
74	#366	—	61	2	no	—
75	#358	—	122	2	no	—
76	C₄×C₁₆	—	61	1	no	—
77	C₆₁	#13653	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	122	2	no	—
79	C₄×Q₈	—	244	2	no	—
80	Q₃₂	—	61	2	no	—
81	Q₃₂	—	122	2	no	—
82	Q₃₂	C₂×#6923	1	2	yes	—
83	C₂×C₁₆	—	61	1	no	—
84	C₂×C₁₆	—	61	1	no	—
85	#107	—	122	2	no	—
86	#103	—	244	2	no	—
87	C₄×C₈	—	61	1	no	—
88	C₃₂	—	61	1	no	—
89	C₃₂	—	122	1	no	—
90	C₃₂	C₂×#6923	1	1	yes	characteristic
91	C₂×Q₈	—	244	2	no	—
92	Q₁₆	—	2	2	no	—
93	Q₁₆	—	122	2	no	—
94	Q₁₆	—	244	2	no	—
95	C₂×C₈	—	61	1	no	—
96	C₂×C₈	—	61	1	no	—
97	C₄⋊C₄	—	488	2	no	—
98	C₄⋊C₄	—	244	2	no	—
99	C₄²	—	61	1	no	—
100	C₄²	—	488	2	no	—
101	C₁₆	—	61	1	no	—
102	C₁₆	—	122	1	no	—
103	C₁₆	C₂²×#6923	1	1	yes	characteristic
104	Q₈	—	4	2	no	—
105	Q₈	—	488	2	no	—
106	Q₈	—	244	2	no	—
107	C₂×C₄	—	61	1	no	—
108	C₂×C₄	—	61	2	no	—
109	C₂×C₄	—	488	2	no	—
110	C₂×C₄	—	488	2	no	—
111	C₈	—	61	1	no	—
112	C₈	—	122	1	no	—
113	C₈	D₄×#6923	1	1	yes	characteristic
114	C₂²	—	61	1	no	—
115	C₄	—	8	2	no	—
116	C₄	—	61	1	no	—
117	C₄	—	122	2	no	—
118	C₄	—	976	2	no	—
119	C₄	—	488	2	no	—
120	C₄	D₈×#6923	1	1	yes	characteristic
121	C₂	—	61	2	no	—
122	C₂	D₁₆×#6923	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₃₂×#6923

C₂

yes

maximal

#21460981

C₂

yes

maximal, characteristic

#21460969

C₂

yes

maximal

#21460925

C₂

yes

maximal, characteristic

Q₆₄×D₆₁

C₂

yes

maximal, characteristic

Q₁₆×#6923

—

D₆₁⋊C₃₂

C₂²

yes

characteristic

#16840688

C₂²

yes

—

#16840675

—

C₁₆×#6923

C₂²

yes

characteristic

Q₃₂×D₆₁

C₂²

yes

—

Dic₉₇₆

C₄

yes

characteristic, complemented

#16839602

C₄

yes

complemented

C₃₂×D₆₁

C₂²

yes

characteristic

C₆₁×Q₆₄

C₄

yes

characteristic, complemented

Q₈×#6923

—

D₆₁⋊C₁₆

D₄

yes

—

—

D₄

yes

characteristic

#14519827

—

Q₁₆×D₆₁

—

Dic₄₈₈

C₂×C₄

yes

—

#14519723

—

C₁₆×D₆₁

C₂³

yes

characteristic

#14519683

C₂×C₄

yes

characteristic

Q₃₂×C₆₁

C₂×C₄

yes

—

C₃₂×C₆₁

C₂×C₄

yes

characteristic

Q₈×D₆₁

—

—

—

—

D₈

yes

characteristic

D₆₁⋊C₈

D₈

yes

characteristic

Q₁₆×C₆₁

—

#11758202

—

Dic₂₄₄

—

C₈×D₆₁

C₂×D₄

yes

characteristic

#11758187

—

C₁₆×C₆₁

C₂²×C₄

yes

yes

—

D₁₆

yes

—

Q₈×C₆₁

—

C₄×D₆₁

—

C₄×D₆₁

C₂×D₈

yes

characteristic

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

C₈×C₆₁

C₄×D₄

yes

yes

—

C₂×D₆₁

C₂×D₁₆

yes

characteristic

#6923

—

#6923

—

C₄×C₆₁

—

C₄×C₆₁

C₄×D₈

yes

characteristic

Dic₆₁

—

Dic₆₁

#508

yes

characteristic

C₂×Q₆₄

—

C₄×Q₃₂

—

#1424

—

#1416

—

#1397

—

C₂×C₆₁

C₄×D₁₆

yes

characteristic

D₆₁

#2264

yes

—

C₂×Q₃₂

—

C₄×Q₁₆

—

122

—

Q₆₄

—

Q₆₄

—

Q₆₄

#6923

yes

characteristic, complemented

C₂×C₃₂

—

C₂×C₃₂

—

#366

—

#358

—

122

—

C₄×C₁₆

—

C₆₁

#13653

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₄×Q₈

—

244

—

Q₃₂

—

Q₃₂

—

122

—

Q₃₂

C₂×#6923

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

122

—

#103

—

244

—

C₄×C₈

—

C₃₂

—

C₃₂

—

122

—

C₃₂

C₂×#6923

yes

characteristic

C₂×Q₈

—

244

—

Q₁₆

—

Q₁₆

—

122

—

Q₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

488

—

C₄⋊C₄

—

244

—

C₄²

—

100

C₄²

—

488

—

101

C₁₆

—

102

C₁₆

—

122

—

103

C₁₆

C₂²×#6923

yes

characteristic

104

Q₈

—

105

Q₈

—

488

—

106

Q₈

—

244

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

488

—

110

C₂×C₄

—

488

—

111

C₈

—

112

C₈

—

122

—

113

C₈

D₄×#6923

yes

characteristic

114

C₂²

—

115

C₄

—

116

C₄

—

117

C₄

—

122

—

118

C₄

—

976

—

119

C₄

—

488

—

120

C₄

D₈×#6923

yes

characteristic

121

C₂

—

122

C₂

D₁₆×#6923

yes

characteristic

	subgroup	G/H
center	C₂	D₁₆×#6923
commutator	C₁₆×C₆₁	C₂²×C₄
Frattini	C₁₆	C₂²×#6923
Fitting	C₆₁×Q₆₄	C₄
socle	C₂×C₆₁	C₄×D₁₆
prime radical	C₂×D₆₁	C₂×D₁₆
cosocle	C₁₆×D₆₁	C₂³
hypercenter	Q₆₄	#6923
hypocenter	C₆₁	#13653
LCS	G ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

subgroup

G/H

center

C₂

D₁₆×#6923

commutator

C₁₆×C₆₁

C₂²×C₄

Frattini

C₁₆

C₂²×#6923

Fitting

C₆₁×Q₆₄

C₄

socle

C₂×C₆₁

C₄×D₁₆

prime radical

C₂×D₆₁

C₂×D₁₆

cosocle

C₁₆×D₆₁

C₂³

Q₆₄

C₆₁

G ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

N	H ≅ G/N	type
12. Dic₉₇₆	C₄	semidirect
13. #16839602	C₄	semidirect
15. C₆₁×Q₆₄	C₄	semidirect
71. Q₆₄	#6923	semidirect
77. C₆₁	#13653	semidirect

H ≅ G/N

type

12. Dic₉₇₆

C₄

semidirect

13. #16839602

C₄

semidirect

15. C₆₁×Q₆₄

C₄

semidirect

71. Q₆₄

#6923

semidirect

77. C₆₁

#13653

semidirect

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

subgroup	index
Q₃₂×#6923	2
#21460981	2
#21460969	2
#21460925	2
Q₆₄×D₆₁	2
#13653	61

subgroup

index

Q₃₂×#6923

Q₆₄×D₆₁

Subgroups — #30608568

Conjugacy classes of subgroups

Complements

Maximal subgroups