Group #29600498 — subgroups

Subgroups — #29600498

Order 15616.

206 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 123 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×#16839596	C₂	1	1	yes	maximal, characteristic
2	#21447415	C₂	1	1	yes	maximal, characteristic
3	#21444014	C₂	1	4	yes	maximal, complemented
4	C₂²×C₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519717	C₄	1	1	yes	characteristic
6	#16839717	C₄	1	1	yes	characteristic
7	#16839596	—	2	2	no	—
8	C₂×#14519683	C₂²	1	2	yes	—
9	#16839537	C₄	1	2	yes	—
10	C₁₆×Dic₆₁	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₂×C₆₁	—	2	2	no	—
13	C₂×C₃₂×C₆₁	C₂²	1	2	yes	—
14	C₂×#11758194	C₈	1	1	yes	characteristic
15	#14519736	C₈	1	1	yes	characteristic
16	#14519717	—	2	2	no	—
17	C₂×#11758185	C₂×C₄	1	2	yes	—
18	#14519687	C₈	1	2	yes	—
19	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
20	#14519683	—	2	2	no	—
21	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
24	C₂×C₁₆×C₆₁	Q₈	1	2	yes	—
25	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
26	C₃₂×C₆₁	—	2	2	no	—
27	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈×C₆₁	M₁₆	1	2	yes	—
31	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₂²⋊Dic₆₁	C₁₆	1	2	yes	—
33	C₄⋊Dic₆₁	C₁₆	1	2	yes	—
34	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
35	#11758194	—	2	2	no	—
36	C₂×#92418	C₂×C₈	1	2	yes	—
37	C₁₆×C₆₁	C₄⋊C₄	1	2	yes	—
38	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₁₆×C₆₁	C₂×Q₈	1	1	yes	characteristic
40	#11758185	—	2	2	no	—
41	C₂³×C₆₁	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₆₁	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₆₁	M₃₂	1	2	yes	—
45	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₆₁	C₂×C₁₆	1	4	yes	—
47	C₈×C₆₁	C₄⋊C₈	1	2	yes	—
48	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
50	#92418	—	2	2	no	—
51	#13628	—	61	1	no	maximal, Sylow
52	C₂²×C₆₁	—	2	2	no	—
53	C₂²×C₆₁	C₄⋊C₁₆	1	2	yes	—
54	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
55	C₄×C₆₁	C₄⋊C₁₆	1	2	yes	—
56	C₄×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
57	C₄×C₆₁	C₂×M₃₂	1	1	yes	characteristic
58	Dic₆₁	—	2	4	no	—
59	C₂×M₆₄	—	61	1	no	—
60	C₂²×C₃₂	D₆₁	1	1	yes	characteristic
61	#2108	—	61	1	no	—
62	C₄⋊C₃₂	—	61	4	no	—
63	C₂×C₆₁	—	2	2	no	—
64	C₂×C₆₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₆₁	#2258	1	2	yes	—
66	C₂×M₃₂	—	61	1	no	—
67	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
68	#405	—	61	1	no	—
69	M₆₄	—	122	2	no	—
70	C₂×C₃₂	—	2	2	no	—
71	C₂×C₃₂	—	61	2	no	—
72	C₂×C₃₂	C₂×D₆₁	1	2	yes	—
73	#346	—	61	2	no	—
74	C₄×C₁₆	—	61	2	no	—
75	C₆₁	#13628	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	61	1	no	—
77	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
78	#117	—	61	1	no	—
79	M₃₂	—	122	2	no	—
80	C₂×C₁₆	—	61	2	no	—
81	C₂×C₁₆	Dic₁₂₂	1	2	yes	—
82	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
83	C₂×C₁₆	C₂²⋊D₆₁	1	2	yes	—
84	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
85	#97	—	61	2	no	—
86	C₄×C₈	—	61	2	no	—
87	C₃₂	—	2	2	no	—
88	C₃₂	—	122	2	no	—
89	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
90	M₁₆	—	122	2	no	—
91	C₂×C₈	—	61	2	no	—
92	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
93	C₂×C₈	#11758189	1	2	yes	—
94	C₂×C₈	#11758196	1	2	yes	—
95	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
96	C₄⋊C₄	—	61	2	no	—
97	C₂²⋊C₄	—	61	2	no	—
98	C₄²	—	61	2	no	—
99	C₁₆	—	122	2	no	—
100	C₁₆	#11758196	1	2	yes	—
101	C₁₆	C₂×Dic₁₂₂	1	1	yes	characteristic
102	C₁₆	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
103	C₂³	C₁₆×D₆₁	1	1	yes	characteristic
104	C₂×C₄	—	61	4	no	—
105	C₂×C₄	#14519706	1	2	yes	—
106	C₂×C₄	C₁₆×D₆₁	1	1	yes	characteristic
107	C₂×C₄	#14519716	1	2	yes	—
108	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
109	C₈	—	122	2	no	—
110	C₈	#14519706	1	2	yes	—
111	C₈	C₂×#11758196	1	1	yes	characteristic
112	C₈	C₂×#11758189	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	#16839580	1	2	yes	—
115	C₂²	C₂×C₁₆×D₆₁	1	1	yes	characteristic
116	C₄	—	122	4	no	—
117	C₄	#16839580	1	2	yes	—
118	C₄	C₂×#14519706	1	1	yes	characteristic
119	C₄	C₂×#14519716	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×#16839580	1	1	yes	characteristic
122	C₂	#21448112	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16839596

C₂

yes

maximal, characteristic

#21447415

C₂

yes

maximal, characteristic

#21444014

C₂

yes

maximal, complemented

C₂²×C₃₂×C₆₁

C₂

yes

maximal, characteristic

C₂×#14519717

C₄

yes

characteristic

#16839717

C₄

yes

characteristic

#16839596

—

C₂×#14519683

C₂²

yes

—

#16839537

C₄

yes

—

C₁₆×Dic₆₁

C₂²

yes

—

C₂²×C₁₆×C₆₁

C₂²

yes

characteristic

C₂×C₃₂×C₆₁

—

C₂×C₃₂×C₆₁

C₂²

yes

—

C₂×#11758194

C₈

yes

characteristic

#14519736

C₈

yes

characteristic

#14519717

—

C₂×#11758185

C₂×C₄

yes

—

#14519687

C₈

yes

—

C₈×Dic₆₁

C₂×C₄

yes

—

#14519683

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₁

D₄

yes

—

C₂×C₁₆×C₆₁

Q₈

yes

—

C₂×C₁₆×C₆₁

C₂³

yes

characteristic

C₃₂×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₂×C₈

yes

characteristic

C₂×C₈×C₆₁

M₁₆

yes

—

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₂²⋊Dic₆₁

C₁₆

yes

—

C₄⋊Dic₆₁

C₁₆

yes

—

C₄×Dic₆₁

C₂×C₈

yes

—

#11758194

—

C₂×#92418

C₂×C₈

yes

—

C₁₆×C₆₁

C₄⋊C₄

yes

—

C₁₆×C₆₁

C₂×D₄

yes

characteristic

C₁₆×C₆₁

C₂×Q₈

yes

characteristic

#11758185

—

C₂³×C₆₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₈

yes

—

C₂×C₄×C₆₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₆₁

M₃₂

yes

—

C₂×C₄×C₆₁

C₂²×C₈

yes

characteristic

C₂×Dic₆₁

C₂×C₁₆

yes

—

C₈×C₆₁

C₄⋊C₈

yes

—

C₈×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₈×C₆₁

C₂×M₁₆

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄⋊C₁₆

yes

—

C₂²×C₆₁

C₂²×C₁₆

yes

characteristic

C₄×C₆₁

C₄⋊C₁₆

yes

—

C₄×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₄×C₆₁

C₂×M₃₂

yes

characteristic

Dic₆₁

—

C₂×M₆₄

—

C₂²×C₃₂

D₆₁

yes

characteristic

#2108

—

C₄⋊C₃₂

—

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₆₁

#2258

yes

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₆₁

yes

characteristic

#405

—

M₆₄

—

122

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₆₁

yes

—

#346

—

C₄×C₁₆

—

C₆₁

#13628

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₄×D₆₁

yes

characteristic

#117

—

M₃₂

—

122

—

C₂×C₁₆

—

C₂×C₁₆

Dic₁₂₂

yes

—

C₂×C₁₆

C₄×D₆₁

yes

characteristic

C₂×C₁₆

C₂²⋊D₆₁

yes

—

C₂×C₁₆

C₂²×D₆₁

yes

characteristic

#97

—

C₄×C₈

—

C₃₂

—

C₃₂

—

122

—

C₂²×C₄

C₈×D₆₁

yes

characteristic

M₁₆

—

122

—

C₂×C₈

—

C₂×C₈

C₈×D₆₁

yes

characteristic

C₂×C₈

#11758189

yes

—

C₂×C₈

#11758196

yes

—

C₂×C₈

C₂×C₄×D₆₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

122

—

100

C₁₆

#11758196

yes

—

101

C₁₆

C₂×Dic₁₂₂

yes

characteristic

102

C₁₆

C₂×C₂²⋊D₆₁

yes

characteristic

103

C₂³

C₁₆×D₆₁

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#14519706

yes

—

106

C₂×C₄

C₁₆×D₆₁

yes

characteristic

107

C₂×C₄

#14519716

yes

—

108

C₂×C₄

C₂×C₈×D₆₁

yes

characteristic

109

C₈

—

122

—

110

C₈

#14519706

yes

—

111

C₈

C₂×#11758196

yes

characteristic

112

C₈

C₂×#11758189

yes

characteristic

113

C₂²

—

114

C₂²

#16839580

yes

—

115

C₂²

C₂×C₁₆×D₆₁

yes

characteristic

116

C₄

—

122

—

117

C₄

#16839580

yes

—

118

C₄

C₂×#14519706

yes

characteristic

119

C₄

C₂×#14519716

yes

characteristic

120

C₂

—

121

C₂

C₂×#16839580

yes

characteristic

122

C₂

#21448112

yes

—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₆₁
commutator, socle	C₂²×C₆₁	C₂²×C₁₆
Fitting	C₂²×C₃₂×C₆₁	C₂
prime radical	C₂³×C₆₁	C₂×C₁₆
cosocle	C₂×C₁₆×C₆₁	C₂³
hypercenter	C₂²×C₃₂	D₆₁
hypocenter	C₆₁	#13628
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₆₁

commutator, socle

C₂²×C₆₁

C₂²×C₁₆

Fitting

C₂²×C₃₂×C₆₁

C₂

prime radical

C₂³×C₆₁

C₂×C₁₆

cosocle

C₂×C₁₆×C₆₁

C₂³

hypercenter

C₂²×C₃₂

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂

N	H ≅ G/N	type
3. #21444014	C₂	semidirect
75. C₆₁	#13628	semidirect

H ≅ G/N

type

3. #21444014

C₂

semidirect

75. C₆₁

#13628

semidirect

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

subgroup	index
C₂×#16839596	2
#21447415	2
#21444014	2
C₂²×C₃₂×C₆₁	2
#13628	61

subgroup

index

C₂²×C₃₂×C₆₁

Subgroups — #29600498

Conjugacy classes of subgroups

Complements

Maximal subgroups