Group #29601045 — subgroups

Subgroups — #29601045

Order 15616.

150 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 71 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	#21444020	C₂	1	2	yes	maximal
3	#21444018	C₂	1	2	yes	maximal
4	#21443953	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2253	C₂	1	1	yes	maximal, characteristic
6	#16840167	—	2	1	no	—
7	C₂×#14519683	C₂²	1	2	yes	—
8	#16839588	—	2	1	no	—
9	#16839588	C₂²	1	2	yes	—
10	C₆₁×#499	—	2	1	no	—
11	C₆₁×#367	—	2	1	no	—
12	C₆₁×#366	C₂²	1	1	yes	characteristic
13	C₆₁×#366	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	#14519777	—	4	1	no	—
16	#14519711	—	2	2	no	—
17	#14519711	—	4	1	no	—
18	#14519683	Q₈	1	4	yes	complemented
19	#125×C₆₁	—	4	1	no	—
20	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
22	#107×C₆₁	—	2	1	no	—
23	#107×C₆₁	—	2	1	no	—
24	#106×C₆₁	—	4	1	no	—
25	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
26	C₂×Dic₁₂₂	—	8	1	no	—
27	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
28	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₆₁	—	4	1	no	—
30	C₄⋊C₄×C₆₁	—	4	1	no	—
31	C₄⋊C₄×C₆₁	—	8	1	no	—
32	C₄⋊C₄×C₆₁	—	8	1	no	—
33	C₄²×C₆₁	D₈	1	1	yes	characteristic
34	C₄⋊Dic₆₁	—	4	2	no	—
35	C₄⋊Dic₆₁	—	8	1	no	—
36	C₁₆×C₆₁	Q₁₆	1	2	yes	—
37	C₁₆×C₆₁	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₆₁	—	8	1	no	—
39	C₂×C₄×C₆₁	—	8	1	no	—
40	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
41	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
42	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
43	C₂×Dic₆₁	—	8	2	no	—
44	Dic₁₂₂	—	16	2	no	—
45	C₈×C₆₁	Q₃₂	1	2	yes	—
46	C₈×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
47	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
48	#13734	—	61	1	no	maximal, Sylow
49	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
50	C₄×C₆₁	—	2	1	no	—
51	C₄×C₆₁	—	2	1	no	—
52	C₄×C₆₁	—	16	1	no	—
53	C₄×C₆₁	—	16	1	no	—
54	C₄×C₆₁	#500	1	1	yes	characteristic
55	C₄×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
56	Dic₆₁	—	16	2	no	—
57	#2253	D₆₁	1	1	yes	characteristic
58	#2252	—	61	1	no	—
59	#1424	—	61	2	no	—
60	#1423	—	61	2	no	—
61	#1397	—	61	1	no	—
62	C₂×C₆₁	#2252	1	1	yes	characteristic
63	C₂×C₆₁	#2263	1	1	yes	characteristic
64	C₂×C₆₁	#2264	1	1	yes	characteristic
65	#500	—	122	1	no	—
66	#499	—	2	1	no	—
67	C₂×C₃₂	—	61	2	no	—
68	#367	—	2	1	no	—
69	#366	—	61	2	no	—
70	#366	—	122	1	no	—
71	#366	C₂×D₆₁	1	1	yes	characteristic
72	#366	C₂×D₆₁	1	1	yes	characteristic
73	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
74	C₆₁	#13734	1	1	yes	characteristic, complemented, Sylow
75	C₄⋊Q₈	—	244	1	no	—
76	#125	—	4	1	no	—
77	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
78	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
79	#107	—	2	1	no	—
80	#107	—	2	1	no	—
81	#107	—	122	2	no	—
82	#107	—	244	1	no	—
83	#106	—	4	1	no	—
84	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
85	C₃₂	—	61	4	no	—
86	C₂×Q₈	—	488	1	no	—
87	C₂×C₈	#11758199	1	1	yes	characteristic
88	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
89	C₄⋊C₄	—	4	1	no	—
90	C₄⋊C₄	—	4	1	no	—
91	C₄⋊C₄	—	8	1	no	—
92	C₄⋊C₄	—	8	1	no	—
93	C₄⋊C₄	—	488	1	no	—
94	C₄⋊C₄	—	244	2	no	—
95	C₄²	#11758199	1	1	yes	characteristic
96	C₁₆	#11758202	1	2	yes	—
97	C₁₆	Q₈×D₆₁	1	2	yes	—
98	Q₈	—	976	2	no	—
99	C₂×C₄	—	8	1	no	—
100	C₂×C₄	—	8	1	no	—
101	C₂×C₄	—	488	2	no	—
102	C₂×C₄	#14519718	1	1	yes	characteristic
103	C₂×C₄	#14519718	1	1	yes	characteristic
104	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
105	C₈	#14519723	1	2	yes	—
106	C₈	#14519775	1	1	yes	characteristic
107	C₈	C₂×#11758202	1	1	yes	characteristic
108	C₂²	C₂×#14519718	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	16	1	no	—
112	C₄	—	16	1	no	—
113	C₄	—	976	2	no	—
114	C₄	#16840165	1	1	yes	characteristic
115	C₄	C₂×#14519723	1	1	yes	characteristic
116	C₂	#21448092	1	1	yes	characteristic
117	C₂	#21448131	1	1	yes	characteristic
118	C₂	#21448136	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21448094

C₂

yes

maximal, characteristic

#21444020

C₂

yes

maximal

#21444018

C₂

yes

maximal

#21443953

C₂

yes

maximal, characteristic

C₆₁×#2253

C₂

yes

maximal, characteristic

#16840167

—

C₂×#14519683

C₂²

yes

—

#16839588

—

#16839588

C₂²

yes

—

C₆₁×#499

—

C₆₁×#367

—

C₆₁×#366

C₂²

yes

characteristic

C₆₁×#366

C₂²

yes

characteristic

C₄×C₁₆×C₆₁

C₂²

yes

—

—

—

Q₈

yes

complemented

#125×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂³

yes

characteristic

#107×C₆₁

—

#107×C₆₁

—

#106×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

D₈

yes

characteristic

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

D₈

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₁₆×C₆₁

Q₁₆

yes

—

C₁₆×C₆₁

C₂×Q₈

yes

—

C₂×C₄×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

Q₃₂

yes

—

C₈×C₆₁

C₄⋊Q₈

yes

characteristic

C₈×C₆₁

C₂×Q₁₆

yes

characteristic

#13734

—

maximal, Sylow

C₂²×C₆₁

C₂×D₁₆

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#500

yes

characteristic

C₄×C₆₁

C₂×Q₃₂

yes

characteristic

Dic₆₁

—

#2253

D₆₁

yes

—

—

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

122

—

#499

—

C₂×C₃₂

—

#367

—

#366

—

#366

—

122

—

#366

C₂×D₆₁

yes

characteristic

#366

C₂×D₆₁

yes

characteristic

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₆₁

#13734

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

244

—

#125

—

C₂×C₁₆

C₂²⋊D₆₁

yes

characteristic

C₂×C₁₆

C₂²×D₆₁

yes

—

—

—

122

—

#107

—

244

—

#106

—

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

C₃₂

—

C₂×Q₈

—

488

—

C₂×C₈

#11758199

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

488

—

C₄⋊C₄

—

244

—

C₄²

#11758199

yes

characteristic

C₁₆

#11758202

yes

—

C₁₆

Q₈×D₆₁

yes

—

Q₈

—

976

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

488

—

102

C₂×C₄

#14519718

yes

characteristic

103

C₂×C₄

#14519718

yes

characteristic

104

C₂×C₄

C₂×#11758199

yes

characteristic

105

C₈

#14519723

yes

—

106

C₈

#14519775

yes

characteristic

107

C₈

C₂×#11758202

yes

characteristic

108

C₂²

C₂×#14519718

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

976

—

114

C₄

#16840165

yes

characteristic

115

C₄

C₂×#14519723

yes

characteristic

116

C₂

#21448092

yes

characteristic

117

C₂

#21448131

yes

characteristic

118

C₂

#21448136

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14519718
commutator, cosocle	C₂×C₁₆×C₆₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₆₁×#2253	C₂
socle, prime radical	C₂²×C₆₁	C₂×D₁₆
hypercenter	#2253	D₆₁
hypocenter	C₆₁	#13734
LCS	G ▷ C₂×C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2253

subgroup

G/H

center

C₂²

C₂×#14519718

commutator, cosocle

C₂×C₁₆×C₆₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₆₁

Fitting

C₆₁×#2253

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 ◁ #2253

N	H ≅ G/N	type
18. #14519683	Q₈	semidirect
74. C₆₁	#13734	semidirect

H ≅ G/N

type

18. #14519683

Q₈

semidirect

74. C₆₁

#13734

semidirect

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

subgroup	index
#21448094	2
#21444020	2
#21444018	2
#21443953	2
C₆₁×#2253	2
#13734	61

subgroup

index

C₆₁×#2253

Subgroups — #29601045

Conjugacy classes of subgroups

Complements

Maximal subgroups