Group #29600985 — subgroups

Subgroups — #29600985

Order 15616.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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	#21448094	C₂	1	1	yes	maximal, characteristic
2	#21444000	C₂	1	4	yes	maximal
3	C₄×#14519683	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2247	C₂	1	1	yes	maximal, characteristic
5	#16840167	—	2	1	no	—
6	C₂×#14519683	C₂²	1	2	yes	—
7	#16839588	—	2	1	no	—
8	#16839588	C₂²	1	2	yes	—
9	C₂×Q₃₂×C₆₁	—	2	1	no	—
10	C₂×Q₃₂×C₆₁	C₂²	1	2	yes	—
11	C₆₁×#494	—	2	1	no	—
12	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	#14519777	—	4	1	no	—
14	#14519711	—	2	2	no	—
15	#14519711	—	4	1	no	—
16	#14519683	—	2	2	no	—
17	C₂×Q₁₆×C₆₁	—	2	2	no	—
18	C₂×Q₁₆×C₆₁	—	4	1	no	—
19	C₄⋊Q₈×C₆₁	—	4	1	no	—
20	Q₃₂×C₆₁	—	2	4	no	—
21	Q₃₂×C₆₁	—	4	2	no	—
22	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂×Q₈×C₆₁	—	4	2	no	—
26	C₂×Dic₁₂₂	—	8	1	no	—
27	Q₁₆×C₆₁	—	4	2	no	—
28	Q₁₆×C₆₁	—	4	2	no	—
29	Q₁₆×C₆₁	—	8	2	no	—
30	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
31	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₄⋊C₄×C₆₁	—	8	1	no	—
33	C₄⋊C₄×C₆₁	—	8	1	no	—
34	C₄²×C₆₁	D₈	1	1	yes	characteristic
35	C₄⋊Dic₆₁	—	4	2	no	—
36	C₄⋊Dic₆₁	—	8	1	no	—
37	C₁₆×C₆₁	SD₁₆	1	2	yes	—
38	C₁₆×C₆₁	#41	1	2	yes	—
39	Q₈×C₆₁	—	8	2	no	—
40	Q₈×C₆₁	—	8	2	no	—
41	C₂×C₄×C₆₁	—	8	2	no	—
42	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
43	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
45	C₂×Dic₆₁	—	8	2	no	—
46	Dic₁₂₂	—	16	2	no	—
47	C₈×C₆₁	SD₃₂	1	2	yes	—
48	C₈×C₆₁	#124	1	1	yes	characteristic
49	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
50	#13720	—	61	1	no	maximal, Sylow
51	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
52	C₄×C₆₁	—	16	2	no	—
53	C₄×C₆₁	SD₆₄	1	2	yes	—
54	C₄×C₆₁	Q₆₄	1	2	yes	—
55	C₄×C₆₁	#486	1	1	yes	characteristic
56	C₄×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
57	Dic₆₁	—	16	2	no	—
58	#2252	—	61	1	no	—
59	#2247	D₆₁	1	1	yes	characteristic
60	#1416	—	61	4	no	—
61	C₄×C₃₂	—	61	1	no	—
62	C₂×C₆₁	#2240	1	1	yes	characteristic
63	C₂×C₆₁	C₂×SD₆₄	1	1	yes	characteristic
64	C₂×C₆₁	C₂×Q₆₄	1	1	yes	characteristic
65	C₂×Q₃₂	—	2	1	no	—
66	C₂×Q₃₂	C₂×D₆₁	1	2	yes	—
67	#500	—	122	1	no	—
68	#494	—	2	1	no	—
69	C₂×C₃₂	—	61	2	no	—
70	#366	—	61	2	no	—
71	#366	—	122	1	no	—
72	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
73	C₆₁	#13720	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₁₆	—	2	2	no	—
75	C₂×Q₁₆	—	4	1	no	—
76	C₄⋊Q₈	—	4	1	no	—
77	C₄⋊Q₈	—	244	1	no	—
78	Q₃₂	—	2	4	no	—
79	Q₃₂	—	4	2	no	—
80	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
82	#107	—	122	2	no	—
83	#107	—	244	1	no	—
84	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
85	C₃₂	—	122	2	no	—
86	C₂×Q₈	—	4	2	no	—
87	C₂×Q₈	—	488	1	no	—
88	Q₁₆	—	4	2	no	—
89	Q₁₆	—	4	2	no	—
90	Q₁₆	—	8	2	no	—
91	C₂×C₈	#11758199	1	1	yes	characteristic
92	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
93	C₄⋊C₄	—	8	1	no	—
94	C₄⋊C₄	—	8	1	no	—
95	C₄⋊C₄	—	488	1	no	—
96	C₄⋊C₄	—	244	2	no	—
97	C₄²	#11758199	1	1	yes	characteristic
98	C₁₆	#11758200	1	2	yes	—
99	C₁₆	#11758224	1	2	yes	—
100	Q₈	—	8	2	no	—
101	Q₈	—	8	2	no	—
102	Q₈	—	976	2	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	—	488	2	no	—
105	C₂×C₄	#14519718	1	1	yes	characteristic
106	C₂×C₄	#14519718	1	1	yes	characteristic
107	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
108	C₈	#14519721	1	2	yes	—
109	C₈	#14519764	1	1	yes	characteristic
110	C₈	C₂×#11758200	1	1	yes	characteristic
111	C₂²	C₂×#14519718	1	1	yes	characteristic
112	C₄	—	16	2	no	—
113	C₄	—	976	2	no	—
114	C₄	#16839599	1	2	yes	—
115	C₄	#16839602	1	2	yes	—
116	C₄	#16840108	1	1	yes	characteristic
117	C₄	C₂×#14519721	1	1	yes	characteristic
118	C₂	#21448042	1	1	yes	characteristic
119	C₂	C₂×#16839599	1	1	yes	characteristic
120	C₂	C₂×#16839602	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21448094

C₂

yes

maximal, characteristic

#21444000

C₂

yes

maximal

C₄×#14519683

C₂

yes

maximal, characteristic

C₆₁×#2247

C₂

yes

maximal, characteristic

#16840167

—

C₂×#14519683

C₂²

yes

—

#16839588

—

#16839588

C₂²

yes

—

C₂×Q₃₂×C₆₁

—

C₂×Q₃₂×C₆₁

C₂²

yes

—

C₆₁×#494

—

C₄×C₁₆×C₆₁

C₂²

yes

—

—

—

—

C₂×Q₁₆×C₆₁

—

C₂×Q₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

—

Q₃₂×C₆₁

—

Q₃₂×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂³

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Dic₁₂₂

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

D₈

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₁₆×C₆₁

SD₁₆

yes

—

C₁₆×C₆₁

#41

yes

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

D₁₆

yes

characteristic

C₂×C₄×C₆₁

D₁₆

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

SD₃₂

yes

—

C₈×C₆₁

#124

yes

characteristic

C₈×C₆₁

C₂×SD₁₆

yes

characteristic

#13720

—

maximal, Sylow

C₂²×C₆₁

C₂×D₁₆

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

SD₆₄

yes

—

C₄×C₆₁

Q₆₄

yes

—

C₄×C₆₁

#486

yes

characteristic

C₄×C₆₁

C₂×SD₃₂

yes

characteristic

Dic₆₁

—

#2252

—

#2247

D₆₁

yes

characteristic

#1416

—

C₄×C₃₂

—

C₂×C₆₁

#2240

yes

characteristic

C₂×C₆₁

C₂×SD₆₄

yes

characteristic

C₂×C₆₁

C₂×Q₆₄

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₆₁

yes

—

#500

—

122

—

#494

—

C₂×C₃₂

—

#366

—

#366

—

122

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₆₁

#13720

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

244

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

C₂²⋊D₆₁

yes

characteristic

C₂×C₁₆

C₂²×D₆₁

yes

characteristic

#107

—

122

—

#107

—

244

—

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

C₃₂

—

122

—

C₂×Q₈

—

C₂×Q₈

—

488

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#11758199

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

488

—

C₄⋊C₄

—

244

—

C₄²

#11758199

yes

characteristic

C₁₆

#11758200

yes

—

C₁₆

#11758224

yes

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

976

—

103

C₂×C₄

—

104

C₂×C₄

—

488

—

105

C₂×C₄

#14519718

yes

characteristic

106

C₂×C₄

#14519718

yes

characteristic

107

C₂×C₄

C₂×#11758199

yes

characteristic

108

C₈

#14519721

yes

—

109

C₈

#14519764

yes

characteristic

110

C₈

C₂×#11758200

yes

characteristic

111

C₂²

C₂×#14519718

yes

characteristic

112

C₄

—

113

C₄

—

976

—

114

C₄

#16839599

yes

—

115

C₄

#16839602

yes

—

116

C₄

#16840108

yes

characteristic

117

C₄

C₂×#14519721

yes

characteristic

118

C₂

#21448042

yes

characteristic

119

C₂

C₂×#16839599

yes

characteristic

120

C₂

C₂×#16839602

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14519718
commutator, cosocle	C₂×C₁₆×C₆₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₆₁×#2247	C₂
socle, prime radical	C₂²×C₆₁	C₂×D₁₆
hypercenter	#2247	D₆₁
hypocenter	C₆₁	#13720
LCS	G ▷ C₂×C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

subgroup

G/H

center

C₂²

C₂×#14519718

commutator, cosocle

C₂×C₁₆×C₆₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₆₁

Fitting

C₆₁×#2247

C₂

socle, prime radical

C₂²×C₆₁

C₂×D₁₆

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

N	H ≅ G/N	type
73. C₆₁	#13720	semidirect

H ≅ G/N

type

73. C₆₁

#13720

semidirect

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

subgroup	index
#21448094	2
#21444000	2
C₄×#14519683	2
C₆₁×#2247	2
#13720	61

subgroup

index

C₆₁×#2247

Subgroups — #29600985

Conjugacy classes of subgroups

Complements

Maximal subgroups