Group #29594702 — subgroups

Subgroups — #29594702

Order 15616.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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	#21448046	C₂	1	4	yes	maximal
2	#21445437	C₂	1	1	yes	maximal, characteristic
3	C₈×#11758185	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1713	C₂	1	1	yes	maximal, characteristic
5	#16840167	—	2	1	no	—
6	#16840167	C₂²	1	2	yes	—
7	#16840142	—	2	1	no	—
8	#16839566	—	2	8	no	—
9	C₄×#11758185	C₂²	1	2	yes	—
10	C₆₁×#500	—	2	1	no	—
11	C₆₁×#500	—	2	1	no	—
12	C₆₁×#494	C₂²	1	2	yes	—
13	C₈²×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₄₄	—	4	2	no	—
15	#14519777	—	2	2	no	—
16	C₂×#11758185	D₄	1	4	yes	—
17	#14519711	—	2	4	no	—
18	#14519711	—	4	2	no	—
19	C₂×Q₁₆×C₆₁	—	2	4	no	—
20	C₄⋊Q₈×C₆₁	—	2	2	no	—
21	#107×C₆₁	—	4	2	no	—
22	#107×C₆₁	—	4	2	no	—
23	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
24	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
26	C₂×Q₈×C₆₁	—	4	2	no	—
27	C₂×Dic₁₂₂	—	4	2	no	—
28	Q₁₆×C₆₁	—	4	8	no	—
29	C₂×C₈×C₆₁	D₈	1	2	yes	—
30	C₂×C₈×C₆₁	D₈	1	2	yes	—
31	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
32	C₄⋊C₄×C₆₁	—	4	2	no	—
33	C₄⋊C₄×C₆₁	—	4	2	no	—
34	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₄⋊Dic₆₁	—	4	2	no	—
36	C₄⋊Dic₆₁	—	4	2	no	—
37	Dic₂₄₄	—	8	4	no	—
38	#11758185	—	2	4	no	—
39	Q₈×C₆₁	—	8	4	no	—
40	C₂×C₄×C₆₁	—	8	2	no	—
41	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
42	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
44	C₂×Dic₆₁	—	8	2	no	—
45	Dic₁₂₂	—	8	4	no	—
46	C₈×C₆₁	SD₃₂	1	4	yes	—
47	C₈×C₆₁	Q₃₂	1	4	yes	—
48	C₈×C₆₁	#135	1	4	yes	—
49	#12510	—	61	1	no	maximal, Sylow
50	C₂²×C₆₁	#493	1	1	yes	characteristic
51	C₄×C₆₁	—	16	2	no	—
52	C₄×C₆₁	#495	1	2	yes	—
53	C₄×C₆₁	C₂×SD₃₂	1	2	yes	—
54	C₄×C₆₁	C₂×Q₃₂	1	2	yes	—
55	Dic₆₁	—	16	2	no	—
56	#2241	—	61	4	no	—
57	#1713	—	61	1	no	—
58	#1713	D₆₁	1	1	yes	characteristic
59	C₈×C₁₆	—	61	1	no	—
60	C₂×C₆₁	#1709	1	1	yes	characteristic
61	C₂×C₆₁	#2247	1	1	yes	characteristic
62	C₂×C₆₁	#2248	1	1	yes	characteristic
63	#500	—	2	1	no	—
64	#500	—	2	1	no	—
65	#500	—	61	2	no	—
66	#500	—	122	1	no	—
67	#494	—	122	1	no	—
68	#494	C₂×D₆₁	1	2	yes	—
69	#358	—	122	8	no	—
70	C₄×C₁₆	—	61	2	no	—
71	C₈²	C₂×D₆₁	1	1	yes	characteristic
72	C₆₁	#12510	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₁₆	—	2	4	no	—
74	C₂×Q₁₆	—	244	2	no	—
75	C₄⋊Q₈	—	2	2	no	—
76	C₄⋊Q₈	—	122	2	no	—
77	C₂×C₁₆	—	61	4	no	—
78	#107	—	4	2	no	—
79	#107	—	4	2	no	—
80	#107	—	122	4	no	—
81	#107	—	244	2	no	—
82	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
83	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
84	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
85	C₂×Q₈	—	4	2	no	—
86	C₂×Q₈	—	244	2	no	—
87	Q₁₆	—	4	8	no	—
88	Q₁₆	—	488	4	no	—
89	C₂×C₈	#11758199	1	2	yes	—
90	C₂×C₈	#11758199	1	2	yes	—
91	C₂×C₈	D₄×D₆₁	1	2	yes	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	244	2	no	—
95	C₄⋊C₄	—	244	2	no	—
96	C₄²	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
97	C₁₆	—	122	4	no	—
98	Q₈	—	8	4	no	—
99	Q₈	—	488	4	no	—
100	C₂×C₄	—	8	2	no	—
101	C₂×C₄	—	488	2	no	—
102	C₂×C₄	#14519773	1	1	yes	characteristic
103	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
104	C₂×C₄	C₂×#11758199	1	1	yes	characteristic
105	C₈	#14519720	1	4	yes	—
106	C₈	#14519723	1	4	yes	—
107	C₈	#14519796	1	4	yes	—
108	C₂²	#16840138	1	1	yes	characteristic
109	C₄	—	16	2	no	—
110	C₄	—	976	2	no	—
111	C₄	#16840145	1	2	yes	—
112	C₄	C₂×#14519720	1	2	yes	—
113	C₄	C₂×#14519723	1	2	yes	—
114	C₂	#21445424	1	1	yes	characteristic
115	C₂	#21448071	1	1	yes	characteristic
116	C₂	#21448076	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21448046

C₂

yes

maximal

#21445437

C₂

yes

maximal, characteristic

C₈×#11758185

C₂

yes

maximal, characteristic

C₆₁×#1713

C₂

yes

maximal, characteristic

#16840167

—

#16840167

C₂²

yes

—

#16840142

—

#16839566

—

C₄×#11758185

C₂²

yes

—

C₆₁×#500

—

C₆₁×#500

—

C₆₁×#494

C₂²

yes

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

#14519777

—

C₂×#11758185

D₄

yes

—

#14519711

—

#14519711

—

C₂×Q₁₆×C₆₁

—

C₄⋊Q₈×C₆₁

—

#107×C₆₁

—

#107×C₆₁

—

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Dic₁₂₂

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

D₈

yes

—

C₂×C₈×C₆₁

C₂×D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

Dic₂₄₄

—

#11758185

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

SD₃₂

yes

—

C₈×C₆₁

Q₃₂

yes

—

C₈×C₆₁

#135

yes

—

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

#495

yes

—

C₄×C₆₁

C₂×SD₃₂

yes

—

C₄×C₆₁

C₂×Q₃₂

yes

—

Dic₆₁

—

#2241

—

#1713

—

#1713

D₆₁

yes

characteristic

C₈×C₁₆

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

—

—

122

—

#494

—

122

—

#494

C₂×D₆₁

yes

—

#358

—

122

—

C₄×C₁₆

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#12510

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

244

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₂×C₁₆

—

#107

—

#107

—

#107

—

122

—

#107

—

244

—

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

244

—

Q₁₆

—

Q₁₆

—

488

—

C₂×C₈

#11758199

yes

—

C₂×C₈

#11758199

yes

—

C₂×C₈

D₄×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄²

C₂×C₂²⋊D₆₁

yes

characteristic

C₁₆

—

122

—

Q₈

—

Q₈

—

488

—

100

C₂×C₄

—

101

C₂×C₄

—

488

—

102

C₂×C₄

#14519773

yes

characteristic

103

C₂×C₄

C₂×#11758199

yes

characteristic

104

C₂×C₄

C₂×#11758199

yes

characteristic

105

C₈

#14519720

yes

—

106

C₈

#14519723

yes

—

107

C₈

#14519796

yes

—

108

C₂²

#16840138

yes

characteristic

109

C₄

—

110

C₄

—

976

—

111

C₄

#16840145

yes

—

112

C₄

C₂×#14519720

yes

—

113

C₄

C₂×#14519723

yes

—

114

C₂

#21445424

yes

characteristic

115

C₂

#21448071

yes

characteristic

116

C₂

#21448076

yes

characteristic

	subgroup	G/H
center	C₂²	#16840138
commutator, cosocle	C₄×C₈×C₆₁	C₂³
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1713	C₂
socle, prime radical	C₂²×C₆₁	#493
hypercenter	#1713	D₆₁
hypocenter	C₆₁	#12510
LCS	G ▷ C₄×C₈×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#16840138

commutator, cosocle

C₄×C₈×C₆₁

C₂³

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#1713

C₂

C₂²×C₆₁

D₆₁

C₆₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
72. C₆₁	#12510	semidirect

H ≅ G/N

type

72. C₆₁

#12510

semidirect

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

subgroup	index
#21448046	2
#21445437	2
C₈×#11758185	2
C₆₁×#1713	2
#12510	61

subgroup

index

C₆₁×#1713

Subgroups — #29594702

Conjugacy classes of subgroups

Complements

Maximal subgroups