Group #29601009 — subgroups

Subgroups — #29601009

Order 15616.

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

normal subgroups only

This group has 230 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	C₂×Dic₉₇₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839602	C₂	1	2	yes	maximal
3	#21448070	C₂	1	2	yes	maximal
4	C₃₂×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₆₁×Q₆₄	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₈₈	C₂²	1	2	yes	—
7	C₂×#14519723	—	2	2	no	—
8	#16840140	—	2	2	no	—
9	Dic₉₇₆	—	2	2	no	—
10	#16839602	—	2	4	no	—
11	C₂×#14519683	C₂²	1	1	yes	characteristic
12	C₁₆×Dic₆₁	C₂²	1	1	yes	characteristic
13	C₂×Q₃₂×C₆₁	C₂²	1	2	yes	—
14	C₆₁×Q₆₄	—	2	2	no	—
15	C₂×C₃₂×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₄₄	—	2	2	no	—
17	C₂×#11758202	—	4	2	no	—
18	#14519775	—	4	2	no	—
19	Dic₄₈₈	—	2	2	no	—
20	Dic₄₈₈	—	2	2	no	—
21	#14519723	—	4	4	no	—
22	C₂×#11758185	D₄	1	1	yes	characteristic
23	C₈×Dic₆₁	D₄	1	1	yes	characteristic
24	#14519683	D₄	1	2	yes	—
25	C₂×Q₁₆×C₆₁	—	2	2	no	—
26	Q₃₂×C₆₁	—	2	2	no	—
27	Q₃₂×C₆₁	—	2	2	no	—
28	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
29	C₃₂×C₆₁	D₄	1	2	yes	—
30	C₂×Q₈×C₆₁	—	4	2	no	—
31	C₂×Dic₁₂₂	—	4	2	no	—
32	Q₁₆×C₆₁	—	4	2	no	—
33	Q₁₆×C₆₁	—	4	2	no	—
34	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
35	#11758202	—	8	4	no	—
36	#11758196	—	8	4	no	—
37	C₄×Dic₆₁	D₈	1	1	yes	characteristic
38	C₂×#92418	D₈	1	1	yes	characteristic
39	Dic₂₄₄	—	4	2	no	—
40	Dic₂₄₄	—	4	2	no	—
41	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
43	#11758185	D₈	1	2	yes	—
44	Q₈×C₆₁	—	8	2	no	—
45	Q₈×C₆₁	—	8	2	no	—
46	C₂×C₄×C₆₁	—	8	2	no	—
47	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₆₁	—	8	2	no	—
49	C₂×Dic₆₁	D₁₆	1	2	yes	—
50	Dic₁₂₂	—	8	2	no	—
51	Dic₁₂₂	—	8	2	no	—
52	C₈×C₆₁	#127	1	1	yes	characteristic
53	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
54	#92418	D₁₆	1	2	yes	—
55	#13726	—	61	1	no	maximal, Sylow
56	C₂²×C₆₁	C₂×D₁₆	1	1	yes	characteristic
57	C₄×C₆₁	—	16	2	no	—
58	C₄×C₆₁	#493	1	1	yes	characteristic
59	C₄×C₆₁	C₂×D₁₆	1	1	yes	characteristic
60	Dic₆₁	—	16	2	no	—
61	Dic₆₁	Q₆₄	1	4	yes	complemented
62	C₂×Q₆₄	—	61	2	no	—
63	C₂×Q₆₄	—	61	1	no	—
64	C₂×Q₆₄	D₆₁	1	1	yes	characteristic
65	#2247	—	61	2	no	—
66	C₄×C₃₂	—	61	1	no	—
67	C₂×C₆₁	#2246	1	1	yes	characteristic
68	C₂×C₆₁	C₂×Q₆₄	1	2	yes	—
69	C₂×Q₃₂	—	61	2	no	—
70	C₂×Q₃₂	—	122	2	no	—
71	C₂×Q₃₂	C₂×D₆₁	1	2	yes	—
72	#494	—	122	2	no	—
73	Q₆₄	—	2	2	no	—
74	Q₆₄	—	122	4	no	—
75	Q₆₄	—	122	2	no	—
76	C₂×C₃₂	—	61	1	no	—
77	C₂×C₃₂	C₂×D₆₁	1	1	yes	characteristic
78	C₄×C₁₆	—	61	1	no	—
79	C₆₁	#13726	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	2	2	no	—
81	C₂×Q₁₆	—	122	2	no	—
82	C₂×Q₁₆	—	244	2	no	—
83	C₄⋊Q₈	—	244	2	no	—
84	Q₃₂	—	2	2	no	—
85	Q₃₂	—	2	2	no	—
86	Q₃₂	—	122	2	no	—
87	Q₃₂	—	122	2	no	—
88	Q₃₂	—	244	4	no	—
89	C₂×C₁₆	—	61	1	no	—
90	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
91	C₄×C₈	—	61	1	no	—
92	C₃₂	—	61	2	no	—
93	C₃₂	C₂²⋊D₆₁	1	2	yes	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	244	2	no	—
96	Q₁₆	—	4	2	no	—
97	Q₁₆	—	4	2	no	—
98	Q₁₆	—	488	4	no	—
99	Q₁₆	—	244	2	no	—
100	Q₁₆	—	244	2	no	—
101	C₂×C₈	—	61	1	no	—
102	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
103	C₄⋊C₄	—	488	4	no	—
104	C₄²	—	61	1	no	—
105	C₁₆	—	61	2	no	—
106	C₁₆	D₄×D₆₁	1	1	yes	characteristic
107	C₁₆	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
108	Q₈	—	8	2	no	—
109	Q₈	—	8	2	no	—
110	Q₈	—	488	2	no	—
111	Q₈	—	488	2	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	—	61	2	no	—
114	C₂×C₄	—	488	2	no	—
115	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
116	C₈	—	61	2	no	—
117	C₈	#14519773	1	1	yes	characteristic
118	C₈	D₈×D₆₁	1	1	yes	characteristic
119	C₂²	D₁₆×D₆₁	1	1	yes	characteristic
120	C₄	—	16	2	no	—
121	C₄	—	61	4	no	—
122	C₄	—	976	2	no	—
123	C₄	#16840137	1	1	yes	characteristic
124	C₄	D₁₆×D₆₁	1	1	yes	characteristic
125	C₂	#21448067	1	1	yes	characteristic
126	C₂	Q₆₄×D₆₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₉₇₆

C₂

yes

maximal, characteristic

C₂×#16839602

C₂

yes

maximal

#21448070

C₂

yes

maximal

C₃₂×Dic₆₁

C₂

yes

maximal, characteristic

C₂×C₆₁×Q₆₄

C₂

yes

maximal, characteristic

C₂×Dic₄₈₈

C₂²

yes

—

C₂×#14519723

—

#16840140

—

Dic₉₇₆

—

#16839602

—

C₂×#14519683

C₂²

yes

characteristic

C₁₆×Dic₆₁

C₂²

yes

characteristic

C₂×Q₃₂×C₆₁

C₂²

yes

—

C₆₁×Q₆₄

—

C₂×C₃₂×C₆₁

C₂²

yes

characteristic

C₂×Dic₂₄₄

—

C₂×#11758202

—

#14519775

—

Dic₄₈₈

—

Dic₄₈₈

—

#14519723

—

C₂×#11758185

D₄

yes

characteristic

C₈×Dic₆₁

D₄

yes

characteristic

#14519683

D₄

yes

—

C₂×Q₁₆×C₆₁

—

Q₃₂×C₆₁

—

Q₃₂×C₆₁

—

C₂×C₁₆×C₆₁

C₂³

yes

characteristic

C₃₂×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

—

C₂×Dic₁₂₂

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂×D₄

yes

characteristic

#11758202

—

#11758196

—

C₄×Dic₆₁

D₈

yes

characteristic

C₂×#92418

D₈

yes

characteristic

Dic₂₄₄

—

Dic₂₄₄

—

C₁₆×C₆₁

C₂×D₄

yes

characteristic

C₁₆×C₆₁

C₂×D₄

yes

characteristic

#11758185

D₈

yes

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₈

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

D₁₆

yes

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

#127

yes

characteristic

C₈×C₆₁

C₂×D₈

yes

characteristic

#92418

D₁₆

yes

—

#13726

—

maximal, Sylow

C₂²×C₆₁

C₂×D₁₆

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

#493

yes

characteristic

C₄×C₆₁

C₂×D₁₆

yes

characteristic

Dic₆₁

—

Dic₆₁

Q₆₄

yes

complemented

C₂×Q₆₄

—

C₂×Q₆₄

—

C₂×Q₆₄

D₆₁

yes

characteristic

#2247

—

C₄×C₃₂

—

C₂×C₆₁

#2246

yes

characteristic

C₂×C₆₁

C₂×Q₆₄

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

122

—

C₂×Q₃₂

C₂×D₆₁

yes

—

#494

—

122

—

Q₆₄

—

Q₆₄

—

122

—

Q₆₄

—

122

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₆₁

yes

characteristic

C₄×C₁₆

—

C₆₁

#13726

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

122

—

C₂×Q₁₆

—

244

—

C₄⋊Q₈

—

244

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

122

—

Q₃₂

—

122

—

Q₃₂

—

244

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₆₁

yes

characteristic

C₄×C₈

—

C₃₂

—

C₃₂

C₂²⋊D₆₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

244

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

488

—

Q₁₆

—

244

—

100

Q₁₆

—

244

—

101

C₂×C₈

—

102

C₂×C₈

D₄×D₆₁

yes

characteristic

103

C₄⋊C₄

—

488

—

104

C₄²

—

105

C₁₆

—

106

C₁₆

D₄×D₆₁

yes

characteristic

107

C₁₆

C₂×C₂²⋊D₆₁

yes

characteristic

108

Q₈

—

109

Q₈

—

110

Q₈

—

488

—

111

Q₈

—

488

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

488

—

115

C₂×C₄

D₈×D₆₁

yes

characteristic

116

C₈

—

117

C₈

#14519773

yes

characteristic

118

C₈

D₈×D₆₁

yes

characteristic

119

C₂²

D₁₆×D₆₁

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

976

—

123

C₄

#16840137

yes

characteristic

124

C₄

D₁₆×D₆₁

yes

characteristic

125

C₂

#21448067

yes

characteristic

126

C₂

Q₆₄×D₆₁

yes

—

	subgroup	G/H
center	C₂²	D₁₆×D₆₁
commutator, cosocle	C₂×C₁₆×C₆₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₂×C₆₁×Q₆₄	C₂
socle, prime radical	C₂²×C₆₁	C₂×D₁₆
hypercenter	C₂×Q₆₄	D₆₁
hypocenter	C₆₁	#13726
LCS	G ▷ C₂×C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

subgroup

G/H

center

C₂²

D₁₆×D₆₁

commutator, cosocle

C₂×C₁₆×C₆₁

C₂³

Frattini

C₂×C₁₆

C₂²×D₆₁

Fitting

C₂×C₆₁×Q₆₄

C₂

socle, prime radical

C₂²×C₆₁

C₂×D₁₆

hypercenter

C₂×Q₆₄

D₆₁

hypocenter

C₆₁

#13726

LCS

G ▷ C₂×C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×Q₆₄

N	H ≅ G/N	type
61. Dic₆₁	Q₆₄	semidirect
79. C₆₁	#13726	semidirect

H ≅ G/N

type

61. Dic₆₁

Q₆₄

semidirect

79. C₆₁

#13726

semidirect

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

subgroup	index
C₂×Dic₉₇₆	2
C₂×#16839602	2
#21448070	2
C₃₂×Dic₆₁	2
C₂×C₆₁×Q₆₄	2
#13726	61

subgroup

index

C₂×Dic₉₇₆

C₂×#16839602

#21448070

C₃₂×Dic₆₁

C₂×C₆₁×Q₆₄

#13726

Subgroups — #29601009

Conjugacy classes of subgroups

Complements

Maximal subgroups