Group #29566670 — subgroups

Subgroups — #29566670

Order 15616.

140 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₈×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
2	#21443737	C₂	1	1	yes	maximal, characteristic
3	C₆₁×#1367	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₂₂	—	2	1	no	—
5	C₄×Dic₂₄₄	C₄	1	1	yes	characteristic
6	C₄×#11758185	—	2	1	no	—
7	#16839515	C₄	1	1	yes	characteristic
8	#16839494	—	2	1	no	—
9	C₆₁×C₄⋊C₁₆	—	2	1	no	—
10	C₈²×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₄₄	C₈	1	1	yes	characteristic
12	C₄×Dic₁₂₂	—	2	1	no	—
13	C₂×#11758185	—	2	2	no	—
14	#14519711	C₈	1	1	yes	characteristic
15	C₄⋊#92418	—	2	1	no	—
16	#14519706	—	4	1	no	—
17	#14519702	—	2	1	no	—
18	C₈×Dic₆₁	—	4	1	no	—
19	C₂×C₁₆×C₆₁	—	4	1	no	—
20	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
22	C₄×C₈×C₆₁	D₄	1	1	yes	characteristic
23	C₂×Dic₁₂₂	—	2	1	no	—
24	C₂×C₈×C₆₁	—	2	1	no	—
25	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
26	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
28	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
29	C₄²×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₄⋊Dic₆₁	—	2	1	no	—
31	#11758196	—	4	1	no	—
32	C₄×Dic₆₁	—	4	1	no	—
33	C₂×#92418	—	4	1	no	—
34	Dic₂₄₄	—	2	1	no	—
35	Dic₂₄₄	C₁₆	1	2	yes	complemented
36	C₁₆×C₆₁	—	8	1	no	—
37	#11758185	—	4	2	no	—
38	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
39	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
40	C₂×C₄×C₆₁	#104	1	1	yes	characteristic
41	C₂×Dic₆₁	—	4	1	no	—
42	C₂×Dic₆₁	—	4	1	no	—
43	Dic₁₂₂	—	2	2	no	—
44	Dic₁₂₂	—	4	1	no	—
45	C₈×C₆₁	—	2	2	no	—
46	C₈×C₆₁	—	2	1	no	—
47	C₈×C₆₁	—	2	1	no	—
48	C₈×C₆₁	—	2	1	no	—
49	C₈×C₆₁	C₂×C₁₆	1	1	yes	characteristic
50	C₈×C₆₁	M₃₂	1	1	yes	characteristic
51	#92418	—	8	1	no	—
52	#7071	—	61	1	no	maximal, Sylow
53	C₂²×C₆₁	D₄⋊C₈	1	1	yes	characteristic
54	C₄×C₆₁	—	2	1	no	—
55	C₄×C₆₁	C₂²⋊C₁₆	1	1	yes	characteristic
56	C₄×C₆₁	#350	1	1	yes	characteristic
57	C₄×C₆₁	#360	1	1	yes	characteristic
58	C₄×C₆₁	#362	1	1	yes	characteristic
59	Dic₆₁	—	4	2	no	—
60	Dic₆₁	—	8	1	no	—
61	C₈×Q₁₆	—	61	1	no	—
62	#1367	D₆₁	1	1	yes	characteristic
63	#1312	—	61	1	no	—
64	C₂×C₆₁	D₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₆₁	#1334	1	1	yes	characteristic
66	C₂×C₆₁	#1336	1	1	yes	characteristic
67	C₈×Q₈	—	122	1	no	—
68	C₄×Q₁₆	—	61	1	no	—
69	C₄⋊C₁₆	—	2	1	no	—
70	C₄×C₁₆	—	122	1	no	—
71	#335	—	61	1	no	—
72	Q₈⋊C₈	—	122	1	no	—
73	C₈²	C₂×D₆₁	1	1	yes	characteristic
74	C₆₁	#7071	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₁₆	—	61	1	no	—
76	C₄×Q₈	—	122	1	no	—
77	C₂×C₁₆	—	4	1	no	—
78	C₂×C₁₆	—	122	2	no	—
79	#107	—	61	1	no	—
80	C₄⋊C₈	—	122	1	no	—
81	C₄⋊C₈	—	244	1	no	—
82	#103	—	122	1	no	—
83	C₄×C₈	—	244	1	no	—
84	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
85	C₄×C₈	D₂₄₄	1	1	yes	characteristic
86	C₄×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
87	C₂×Q₈	—	122	1	no	—
88	Q₁₆	—	61	2	no	—
89	Q₁₆	—	122	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	244	1	no	—
92	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
93	C₂×C₈	#11758189	1	1	yes	characteristic
94	C₂×C₈	#11758199	1	1	yes	characteristic
95	C₂×C₈	#11758201	1	1	yes	characteristic
96	C₄⋊C₄	—	122	1	no	—
97	C₄⋊C₄	—	244	1	no	—
98	C₄²	—	244	1	no	—
99	C₄²	#11758198	1	1	yes	characteristic
100	C₁₆	—	8	1	no	—
101	C₁₆	—	244	2	no	—
102	Q₈	—	122	2	no	—
103	Q₈	—	244	1	no	—
104	C₂×C₄	—	244	1	no	—
105	C₂×C₄	—	244	1	no	—
106	C₂×C₄	#14519689	1	1	yes	characteristic
107	C₂×C₄	#14519697	1	1	yes	characteristic
108	C₂×C₄	#14519703	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	1	no	—
111	C₈	—	2	1	no	—
112	C₈	—	2	1	no	—
113	C₈	—	488	1	no	—
114	C₈	C₁₆×D₆₁	1	1	yes	characteristic
115	C₈	#14519716	1	1	yes	characteristic
116	C₂²	#16839489	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	488	1	no	—
119	C₄	—	244	2	no	—
120	C₄	#16839541	1	1	yes	characteristic
121	C₄	#16839547	1	1	yes	characteristic
122	C₄	#16839571	1	1	yes	characteristic
123	C₄	#16839577	1	1	yes	characteristic
124	C₂	#21443773	1	1	yes	characteristic
125	C₂	#21443788	1	1	yes	characteristic
126	C₂	#21443794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₂₄₄

C₂

yes

maximal, characteristic

#21443737

C₂

yes

maximal, characteristic

C₆₁×#1367

C₂

yes

maximal, characteristic

C₈×Dic₁₂₂

—

C₄×Dic₂₄₄

C₄

yes

characteristic

C₄×#11758185

—

#16839515

C₄

yes

characteristic

#16839494

—

C₆₁×C₄⋊C₁₆

—

C₈²×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

C₈

yes

characteristic

C₄×Dic₁₂₂

—

C₂×#11758185

—

#14519711

C₈

yes

—

—

—

C₈×Dic₆₁

—

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₄×C₈×C₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

M₁₆

yes

characteristic

C₂×C₈×C₆₁

D₈

yes

characteristic

C₂×C₈×C₆₁

SD₁₆

yes

characteristic

C₄²×C₆₁

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

Dic₂₄₄

—

Dic₂₄₄

C₁₆

yes

complemented

C₁₆×C₆₁

—

#11758185

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×C₁₆

yes

characteristic

C₈×C₆₁

M₃₂

yes

—

—

C₂²×C₆₁

D₄⋊C₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊C₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

C₈×Q₁₆

—

#1367

D₆₁

yes

characteristic

#1312

—

C₂×C₆₁

D₄⋊C₁₆

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₈×Q₈

—

122

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

122

—

#335

—

Q₈⋊C₈

—

122

—

C₈²

C₂×D₆₁

yes

characteristic

C₆₁

#7071

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

122

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

#107

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

244

—

#103

—

122

—

C₄×C₈

—

244

—

C₄×C₈

C₄×D₆₁

yes

characteristic

C₄×C₈

D₂₄₄

yes

characteristic

C₄×C₈

C₂²⋊D₆₁

yes

characteristic

C₂×Q₈

—

122

—

Q₁₆

—

Q₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

C₈×D₆₁

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₄²

—

244

—

C₄²

#11758198

yes

characteristic

100

C₁₆

—

101

C₁₆

—

244

—

102

Q₈

—

122

—

103

Q₈

—

244

—

104

C₂×C₄

—

244

—

105

C₂×C₄

—

244

—

106

C₂×C₄

#14519689

yes

characteristic

107

C₂×C₄

#14519697

yes

characteristic

108

C₂×C₄

#14519703

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

488

—

114

C₈

C₁₆×D₆₁

yes

characteristic

115

C₈

#14519716

yes

characteristic

116

C₂²

#16839489

yes

characteristic

117

C₄

—

118

C₄

—

488

—

119

C₄

—

244

—

120

C₄

#16839541

yes

characteristic

121

C₄

#16839547

yes

characteristic

122

C₄

#16839571

yes

characteristic

123

C₄

#16839577

yes

characteristic

124

C₂

#21443773

yes

characteristic

125

C₂

#21443788

yes

characteristic

126

C₂

#21443794

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14519697
commutator	C₈×C₆₁	C₂×C₁₆
Frattini	C₈²	C₂×D₆₁
Fitting	C₆₁×#1367	C₂
socle, prime radical	C₂²×C₆₁	D₄⋊C₈
cosocle	C₈²×C₆₁	C₂²
hypercenter	#1367	D₆₁
hypocenter	C₆₁	#7071
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂×C₄

#14519697

commutator

C₈×C₆₁

C₂×C₁₆

Frattini

C₈²

C₂×D₆₁

Fitting

C₆₁×#1367

C₂

socle, prime radical

C₂²×C₆₁

D₄⋊C₈

cosocle

C₈²×C₆₁

C₂²

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

N	H ≅ G/N	type
35. Dic₂₄₄	C₁₆	semidirect
74. C₆₁	#7071	semidirect

H ≅ G/N

type

35. Dic₂₄₄

C₁₆

semidirect

74. C₆₁

#7071

semidirect

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

subgroup	index
C₈×Dic₂₄₄	2
#21443737	2
C₆₁×#1367	2
#7071	61

subgroup

index

C₈×Dic₂₄₄

#21443737

C₆₁×#1367

#7071

Subgroups — #29566670

Conjugacy classes of subgroups

Complements

Maximal subgroups