Group #29591893 — subgroups

Subgroups — #29591893

Order 15616.

214 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 93 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	#21444638	C₂	1	2	yes	maximal
2	#21443885	C₂	1	1	yes	maximal, characteristic
3	#21443789	C₂	1	2	yes	maximal
4	#21443739	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2176	C₂	1	1	yes	maximal, characteristic
6	Q₈×#92418	—	2	2	no	—
7	#16839538	C₂²	1	1	yes	characteristic
8	#16839513	C₂²	1	2	yes	—
9	Q₈⋊#92418	—	2	2	no	—
10	#16839483	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₆₁	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₆₁	C₂²	1	2	yes	—
13	C₆₁×#367	C₄	1	1	yes	characteristic
14	C₆₁×#358	C₄	1	2	yes	—
15	C₆₁×#346	C₂²	1	1	yes	characteristic
16	C₂×#11758185	—	2	1	no	—
17	C₄⋊#92418	—	2	2	no	—
18	C₄⋊#92418	—	4	2	no	—
19	C₄×#92418	—	4	2	no	—
20	C₄×#92418	D₄	1	2	yes	—
21	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
22	C₄×Q₈×C₆₁	—	2	2	no	—
23	Q₃₂×C₆₁	C₈	1	4	yes	complemented
24	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
26	#107×C₆₁	C₂×C₄	1	2	yes	—
27	#103×C₆₁	—	2	2	no	—
28	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₆₁	—	2	2	no	—
30	Q₁₆×C₆₁	—	2	2	no	—
31	Q₁₆×C₆₁	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
34	C₄⋊C₄×C₆₁	—	2	2	no	—
35	C₄⋊C₄×C₆₁	—	4	2	no	—
36	C₄²×C₆₁	—	4	2	no	—
37	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂×#92418	—	2	2	no	—
39	C₂×#92418	—	4	2	no	—
40	C₁₆×C₆₁	—	2	1	no	—
41	C₁₆×C₆₁	C₂×C₈	1	2	yes	—
42	#11758185	—	4	1	no	—
43	Q₈×C₆₁	—	2	4	no	—
44	Q₈×C₆₁	—	4	2	no	—
45	C₂×C₄×C₆₁	—	4	2	no	—
46	C₂×C₄×C₆₁	—	4	2	no	—
47	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
49	C₈×C₆₁	—	2	1	no	—
50	C₈×C₆₁	C₂²×C₈	1	1	yes	characteristic
51	C₈×C₆₁	#131	1	1	yes	characteristic
52	#92418	—	4	2	no	—
53	#92418	—	8	2	no	—
54	#12066	—	61	1	no	maximal, Sylow
55	C₂²×C₆₁	#442	1	1	yes	characteristic
56	C₄×C₆₁	—	2	2	no	—
57	C₄×C₆₁	—	4	4	no	—
58	C₄×C₆₁	—	8	2	no	—
59	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
60	C₄×C₆₁	#444	1	1	yes	characteristic
61	#2176	D₆₁	1	1	yes	characteristic
62	#1579	—	61	2	no	—
63	#1368	—	61	1	no	—
64	#1334	—	61	2	no	—
65	#1313	—	61	1	no	—
66	C₂×C₆₁	#1580	1	1	yes	characteristic
67	C₂×C₆₁	#2180	1	2	yes	—
68	C₂×Q₃₂	Dic₆₁	1	1	yes	characteristic
69	C₈×Q₈	—	122	2	no	—
70	C₄×Q₁₆	C₂×D₆₁	1	2	yes	—
71	#367	Dic₆₁	1	1	yes	characteristic
72	#358	Dic₆₁	1	2	yes	—
73	#346	—	61	1	no	—
74	#346	C₂×D₆₁	1	1	yes	characteristic
75	#334	—	61	2	no	—
76	Q₈⋊C₈	—	122	2	no	—
77	#322	—	61	1	no	—
78	C₆₁	#12066	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	C₂×Dic₆₁	1	2	yes	—
80	C₄×Q₈	—	2	2	no	—
81	Q₃₂	#92418	1	4	yes	complemented
82	C₂×C₁₆	—	122	1	no	—
83	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
84	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
85	#107	C₂×Dic₆₁	1	2	yes	—
86	C₄⋊C₈	—	122	2	no	—
87	C₄⋊C₈	—	244	2	no	—
88	#103	—	2	2	no	—
89	C₄×C₈	—	61	2	no	—
90	C₄×C₈	—	244	2	no	—
91	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	C₂×#92418	1	4	yes	—
95	C₂×C₈	—	122	2	no	—
96	C₂×C₈	—	244	2	no	—
97	C₂×C₈	#11758224	1	1	yes	characteristic
98	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	4	2	no	—
102	C₄²	D₄×D₆₁	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	244	1	no	—
105	C₁₆	C₂×#92418	1	2	yes	—
106	Q₈	—	2	4	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
111	C₂×C₄	#14519801	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	488	2	no	—
114	C₈	—	244	2	no	—
115	C₈	C₂²×#92418	1	1	yes	characteristic
116	C₈	#14519784	1	1	yes	characteristic
117	C₂²	#16839874	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	4	4	no	—
120	C₄	—	8	2	no	—
121	C₄	D₄×#92418	1	1	yes	characteristic
122	C₄	#16839884	1	1	yes	characteristic
123	C₂	#21444643	1	1	yes	characteristic
124	C₂	#21447711	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21444638

C₂

yes

maximal

#21443885

C₂

yes

maximal, characteristic

#21443789

C₂

yes

maximal

#21443739

C₂

yes

maximal, characteristic

C₆₁×#2176

C₂

yes

maximal, characteristic

Q₈×#92418

—

#16839538

C₂²

yes

characteristic

#16839513

C₂²

yes

—

Q₈⋊#92418

—

#16839483

C₂²

yes

characteristic

C₂×Q₃₂×C₆₁

C₄

yes

characteristic

C₄×Q₁₆×C₆₁

C₂²

yes

—

C₆₁×#367

C₄

yes

characteristic

C₆₁×#358

C₄

yes

—

C₆₁×#346

C₂²

yes

—

—

—

—

D₄

yes

—

C₂×Q₁₆×C₆₁

C₂×C₄

yes

—

C₄×Q₈×C₆₁

—

Q₃₂×C₆₁

C₈

yes

complemented

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

#107×C₆₁

C₂×C₄

yes

—

#103×C₆₁

—

C₄×C₈×C₆₁

C₂³

yes

characteristic

C₂×Q₈×C₆₁

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

C₂×C₈

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₁

#41

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×D₄

yes

characteristic

C₂×#92418

—

C₂×#92418

—

C₁₆×C₆₁

—

C₁₆×C₆₁

C₂×C₈

yes

—

#11758185

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄×D₄

yes

characteristic

C₂×C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₈×C₆₁

—

C₈×C₆₁

C₂²×C₈

yes

C₈×C₆₁

yes

—

—

—

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₈×D₄

yes

C₄×C₆₁

yes

D₆₁

yes

—

—

—

—

C₂×C₆₁

#1580

yes

characteristic

C₂×C₆₁

#2180

yes

—

C₂×Q₃₂

Dic₆₁

yes

characteristic

C₈×Q₈

—

122

—

C₄×Q₁₆

C₂×D₆₁

yes

—

#367

Dic₆₁

yes

characteristic

#358

Dic₆₁

yes

—

#346

—

#346

C₂×D₆₁

yes

characteristic

#334

—

Q₈⋊C₈

—

122

—

#322

—

C₆₁

#12066

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×Dic₆₁

yes

—

C₄×Q₈

—

Q₃₂

#92418

yes

complemented

C₂×C₁₆

—

122

—

C₂×C₁₆

C₂×Dic₆₁

yes

characteristic

C₂×C₁₆

C₂×Dic₆₁

yes

characteristic

#107

C₂×Dic₆₁

yes

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

244

—

#103

—

C₄×C₈

—

C₄×C₈

—

244

—

C₄×C₈

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

C₂×#92418

yes

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

#11758224

yes

characteristic

C₂×C₈

C₂²×Dic₆₁

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

D₄×D₆₁

yes

characteristic

103

C₁₆

—

104

C₁₆

—

244

—

105

C₁₆

C₂×#92418

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

D₄×Dic₆₁

yes

characteristic

111

C₂×C₄

#14519801

yes

—

112

C₈

—

113

C₈

—

488

—

114

C₈

—

244

—

115

C₈

C₂²×#92418

yes

characteristic

116

C₈

#14519784

yes

characteristic

117

C₂²

#16839874

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

D₄×#92418

yes

characteristic

122

C₄

#16839884

yes

characteristic

123

C₂

#21444643

yes

characteristic

124

C₂

#21447711

yes

—

	subgroup	G/H
center	C₂²	#16839874
commutator	C₈×C₆₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#2176	C₂
socle, prime radical	C₂²×C₆₁	#442
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#2176	D₆₁
hypocenter	C₆₁	#12066
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

subgroup

G/H

center

C₂²

#16839874

commutator

C₈×C₆₁

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₆₁

Fitting

C₆₁×#2176

C₂

socle, prime radical

C₂²×C₆₁

#442

cosocle

C₄×C₈×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

N	H ≅ G/N	type
23. Q₃₂×C₆₁	C₈	semidirect
78. C₆₁	#12066	semidirect
81. Q₃₂	#92418	semidirect

H ≅ G/N

type

23. Q₃₂×C₆₁

C₈

semidirect

78. C₆₁

#12066

semidirect

81. Q₃₂

#92418

semidirect

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

subgroup	index
#21444638	2
#21443885	2
#21443789	2
#21443739	2
C₆₁×#2176	2
#12066	61

subgroup

index

C₆₁×#2176

Subgroups — #29591893

Conjugacy classes of subgroups

Complements

Maximal subgroups