Group #29601005 — subgroups

Subgroups — #29601005

Order 15616.

174 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 67 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	#21448072	C₂	1	1	yes	maximal, characteristic
2	#21443999	C₂	1	4	yes	maximal
3	#21443953	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#2253	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₄₈₈	—	2	1	no	—
6	C₂×Dic₄₈₈	C₂²	1	2	yes	—
7	#16840142	—	2	1	no	—
8	C₂×#14519683	C₂²	1	2	yes	—
9	C₆₁×#499	—	2	1	no	—
10	C₆₁×#367	—	2	1	no	—
11	C₆₁×#366	C₂²	1	2	yes	—
12	C₄×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₄₄	—	2	2	no	—
14	C₂×Dic₂₄₄	—	4	1	no	—
15	#14519777	—	4	1	no	—
16	Dic₄₈₈	—	2	4	no	—
17	Dic₄₈₈	—	4	2	no	—
18	#14519683	—	2	2	no	—
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	2	no	—
23	#106×C₆₁	—	4	1	no	—
24	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂×Dic₁₂₂	—	4	2	no	—
26	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
27	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₆₁	—	4	2	no	—
29	C₄⋊C₄×C₆₁	—	8	2	no	—
30	C₄²×C₆₁	D₈	1	1	yes	characteristic
31	C₄⋊Dic₆₁	—	8	2	no	—
32	Dic₂₄₄	—	4	2	no	—
33	Dic₂₄₄	—	4	2	no	—
34	Dic₂₄₄	—	8	2	no	—
35	C₁₆×C₆₁	SD₁₆	1	2	yes	—
36	C₁₆×C₆₁	#41	1	2	yes	—
37	C₂×C₄×C₆₁	—	8	2	no	—
38	C₂×C₄×C₆₁	D₁₆	1	2	yes	—
39	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
40	C₂×Dic₆₁	—	8	2	no	—
41	Dic₁₂₂	—	8	2	no	—
42	Dic₁₂₂	—	8	2	no	—
43	C₈×C₆₁	SD₃₂	1	2	yes	—
44	C₈×C₆₁	#124	1	1	yes	characteristic
45	C₈×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
46	#13724	—	61	1	no	maximal, Sylow
47	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
48	C₄×C₆₁	—	2	2	no	—
49	C₄×C₆₁	—	16	2	no	—
50	C₄×C₆₁	#486	1	1	yes	characteristic
51	C₄×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
52	Dic₆₁	—	16	2	no	—
53	#2253	D₆₁	1	1	yes	characteristic
54	#2247	—	61	1	no	—
55	#1416	—	61	4	no	—
56	#1397	—	61	1	no	—
57	C₂×C₆₁	#2240	1	1	yes	characteristic
58	C₂×C₆₁	#2264	1	2	yes	—
59	C₂×Q₃₂	—	61	2	no	—
60	C₂×Q₃₂	—	122	1	no	—
61	#499	—	2	1	no	—
62	#494	—	122	1	no	—
63	C₂×C₃₂	—	61	2	no	—
64	#367	—	2	1	no	—
65	#366	C₂×D₆₁	1	2	yes	—
66	C₄×C₁₆	C₂×D₆₁	1	1	yes	characteristic
67	C₆₁	#13724	1	1	yes	characteristic, complemented, Sylow
68	C₂×Q₁₆	—	122	2	no	—
69	C₂×Q₁₆	—	244	1	no	—
70	C₄⋊Q₈	—	244	1	no	—
71	#125	—	4	1	no	—
72	Q₃₂	—	122	4	no	—
73	Q₃₂	—	244	2	no	—
74	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
76	#107	—	2	2	no	—
77	#106	—	4	1	no	—
78	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
79	C₃₂	—	122	2	no	—
80	C₂×Q₈	—	244	2	no	—
81	Q₁₆	—	488	2	no	—
82	Q₁₆	—	244	2	no	—
83	Q₁₆	—	244	2	no	—
84	C₂×C₈	#11758199	1	1	yes	characteristic
85	C₂×C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
86	C₄⋊C₄	—	4	2	no	—
87	C₄⋊C₄	—	8	2	no	—
88	C₄⋊C₄	—	488	2	no	—
89	C₄²	#11758199	1	1	yes	characteristic
90	C₁₆	#11758201	1	2	yes	—
91	C₁₆	#11758226	1	2	yes	—
92	Q₈	—	488	2	no	—
93	Q₈	—	488	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	488	2	no	—
96	C₂×C₄	#14519718	1	2	yes	—
97	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
98	C₈	#14519720	1	2	yes	—
99	C₈	#14519766	1	1	yes	characteristic
100	C₈	C₂×#11758201	1	1	yes	characteristic
101	C₂²	C₂×#14519718	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	16	2	no	—
104	C₄	—	976	2	no	—
105	C₄	#16840110	1	1	yes	characteristic
106	C₄	C₂×#14519720	1	1	yes	characteristic
107	C₂	#21448044	1	1	yes	characteristic
108	C₂	#21448136	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21448072

C₂

yes

maximal, characteristic

#21443999

C₂

yes

maximal

#21443953

C₂

yes

maximal, characteristic

C₆₁×#2253

C₂

yes

maximal, characteristic

C₂×Dic₄₈₈

—

C₂×Dic₄₈₈

C₂²

yes

—

#16840142

—

C₂×#14519683

C₂²

yes

—

C₆₁×#499

—

C₆₁×#367

—

C₆₁×#366

C₂²

yes

—

C₄×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

C₂×Dic₂₄₄

—

#14519777

—

Dic₄₈₈

—

Dic₄₈₈

—

#14519683

—

#125×C₆₁

—

C₂×C₁₆×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂³

yes

characteristic

#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

—

C₄²×C₆₁

D₈

yes

characteristic

C₄⋊Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

—

Dic₂₄₄

—

C₁₆×C₆₁

SD₁₆

yes

—

C₁₆×C₆₁

#41

yes

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

D₁₆

yes

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

SD₃₂

yes

—

C₈×C₆₁

#124

yes

characteristic

C₈×C₆₁

C₂×SD₁₆

yes

characteristic

#13724

—

maximal, Sylow

C₂²×C₆₁

C₂×D₁₆

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#486

yes

characteristic

C₄×C₆₁

C₂×SD₃₂

yes

characteristic

Dic₆₁

—

#2253

D₆₁

yes

—

—

—

C₂×C₆₁

#2240

yes

characteristic

C₂×C₆₁

#2264

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

122

—

#499

—

#494

—

122

—

C₂×C₃₂

—

#367

—

#366

C₂×D₆₁

yes

—

C₄×C₁₆

C₂×D₆₁

yes

characteristic

C₆₁

#13724

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

244

—

C₄⋊Q₈

—

244

—

#125

—

Q₃₂

—

122

—

Q₃₂

—

244

—

C₂×C₁₆

C₂²⋊D₆₁

yes

characteristic

C₂×C₁₆

C₂²×D₆₁

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

C₃₂

—

122

—

C₂×Q₈

—

244

—

Q₁₆

—

488

—

Q₁₆

—

244

—

Q₁₆

—

244

—

C₂×C₈

#11758199

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

488

—

C₄²

#11758199

yes

characteristic

C₁₆

#11758201

yes

—

C₁₆

#11758226

yes

—

Q₈

—

488

—

Q₈

—

488

—

C₂×C₄

—

C₂×C₄

—

488

—

C₂×C₄

#14519718

yes

—

C₂×C₄

C₂×#11758199

yes

characteristic

C₈

#14519720

yes

—

C₈

#14519766

yes

characteristic

100

C₈

C₂×#11758201

yes

characteristic

101

C₂²

C₂×#14519718

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

976

—

105

C₄

#16840110

yes

characteristic

106

C₄

C₂×#14519720

yes

characteristic

107

C₂

#21448044

yes

characteristic

108

C₂

#21448136

yes

—

	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	#13724
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
67. C₆₁	#13724	semidirect

H ≅ G/N

type

67. C₆₁

#13724

semidirect

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

subgroup	index
#21448072	2
#21443999	2
#21443953	2
C₆₁×#2253	2
#13724	61

subgroup

index

C₆₁×#2253

Subgroups — #29601005

Conjugacy classes of subgroups

Complements

Maximal subgroups