Group #29567430 — subgroups

Subgroups — #29567430

Order 15616.

190 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 47 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	#21443765	C₂	1	2	yes	maximal
2	C₆₁×#1844	C₂	1	1	yes	maximal, characteristic
3	#16839488	—	2	2	no	—
4	C₂×#106×C₆₁	—	2	2	no	—
5	C₂×C₄×C₈×C₆₁	C₄	1	1	yes	characteristic
6	C₆₁×#384	C₄	1	1	yes	characteristic
7	C₆₁×#384	C₂²	1	1	yes	characteristic
8	C₂²⋊#92418	—	4	2	no	—
9	C₂²×C₈×C₆₁	—	2	1	no	—
10	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
11	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
12	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
13	C₂×C₄⋊C₄×C₆₁	D₄	1	2	yes	—
14	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
15	#106×C₆₁	—	4	4	no	—
16	C₄×C₈×C₆₁	—	2	1	no	—
17	C₄×C₈×C₆₁	C₈	1	2	yes	complemented
18	C₂²×C₄×C₆₁	—	2	1	no	—
19	C₂²×C₄×C₆₁	—	2	2	no	—
20	C₂²×C₄×C₆₁	—	4	1	no	—
21	C₂²×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₈×C₆₁	—	2	2	no	—
23	C₂×C₈×C₆₁	—	2	2	no	—
24	C₂×C₈×C₆₁	—	2	2	no	—
25	C₄⋊C₄×C₆₁	—	4	2	no	—
26	C₄⋊C₄×C₆₁	—	4	2	no	—
27	C₄⋊C₄×C₆₁	—	8	2	no	—
28	C₄⋊C₄×C₆₁	—	8	2	no	—
29	C₄²×C₆₁	—	2	1	no	—
30	C₄²×C₆₁	C₂×C₈	1	1	yes	characteristic
31	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
32	C₂×#92418	—	8	2	no	—
33	C₂³×C₆₁	#99	1	1	yes	characteristic
34	C₂×C₄×C₆₁	—	2	2	no	—
35	C₂×C₄×C₆₁	—	2	2	no	—
36	C₂×C₄×C₆₁	—	2	1	no	—
37	C₂×C₄×C₆₁	—	2	1	no	—
38	C₂×C₄×C₆₁	—	2	1	no	—
39	C₂×C₄×C₆₁	—	2	1	no	—
40	C₂×C₄×C₆₁	—	4	2	no	—
41	C₂×C₄×C₆₁	—	8	2	no	—
42	C₂×C₄×C₆₁	—	8	2	no	—
43	C₂×C₄×C₆₁	—	8	1	no	—
44	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₆₁	#101	1	1	yes	characteristic
46	C₈×C₆₁	—	2	4	no	—
47	#92418	—	16	2	no	—
48	#7371	—	61	1	no	maximal, Sylow
49	C₂²×C₆₁	—	2	1	no	—
50	C₂²×C₆₁	—	2	1	no	—
51	C₂²×C₆₁	#324	1	1	yes	characteristic
52	C₂²×C₆₁	#353	1	1	yes	characteristic
53	C₂²×C₆₁	#356	1	1	yes	characteristic
54	C₄×C₆₁	—	2	2	no	—
55	C₄×C₆₁	—	2	2	no	—
56	C₄×C₆₁	—	2	1	no	—
57	C₄×C₆₁	—	2	1	no	—
58	C₄×C₆₁	—	8	2	no	—
59	C₄×C₆₁	—	16	1	no	—
60	#1844	D₆₁	1	1	yes	characteristic
61	#1325	—	61	2	no	—
62	C₂×C₆₁	—	2	1	no	—
63	C₂×C₆₁	—	2	1	no	—
64	C₂×C₆₁	#1325	1	1	yes	characteristic
65	C₂×C₆₁	#1409	1	1	yes	characteristic
66	C₂×C₆₁	#1410	1	1	yes	characteristic
67	C₂×#106	—	2	2	no	—
68	C₂×C₄×C₈	Dic₆₁	1	1	yes	characteristic
69	#384	Dic₆₁	1	1	yes	characteristic
70	#384	C₂×D₆₁	1	1	yes	characteristic
71	#324	—	122	2	no	—
72	C₆₁	#7371	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	2	1	no	—
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄⋊C₄	—	4	1	no	—
76	C₂×C₄⋊C₄	—	4	1	no	—
77	C₂×C₄⋊C₄	C₂²⋊D₆₁	1	2	yes	—
78	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
79	#106	—	4	4	no	—
80	C₂²⋊C₈	—	244	2	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	#92418	1	2	yes	complemented
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	4	1	no	—
86	C₂²×C₄	C₂²⋊Dic₆₁	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	488	2	no	—
91	C₄⋊C₄	—	4	2	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	8	2	no	—
94	C₄⋊C₄	—	8	2	no	—
95	C₄²	—	2	1	no	—
96	C₄²	C₂×#92418	1	1	yes	characteristic
97	C₄²	#11758194	1	1	yes	characteristic
98	C₂³	#14519692	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	8	2	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	8	1	no	—
109	C₂×C₄	C₂²⋊#92418	1	1	yes	characteristic
110	C₂×C₄	#14519696	1	1	yes	characteristic
111	C₈	—	2	4	no	—
112	C₈	—	976	2	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	#16839488	1	1	yes	characteristic
116	C₂²	#16839555	1	1	yes	characteristic
117	C₂²	#16839561	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	8	2	no	—
123	C₄	—	16	1	no	—
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#21443765	1	1	yes	characteristic
127	C₂	#21443984	1	1	yes	characteristic
128	C₂	#21443986	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21443765

C₂

yes

maximal

C₆₁×#1844

C₂

yes

maximal, characteristic

#16839488

—

C₂×#106×C₆₁

—

C₂×C₄×C₈×C₆₁

C₄

yes

characteristic

C₆₁×#384

C₄

yes

characteristic

C₆₁×#384

C₂²

yes

characteristic

C₂²⋊#92418

—

C₂²×C₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

D₄

yes

—

C₂×C₄²×C₆₁

C₂×C₄

yes

characteristic

#106×C₆₁

—

C₄×C₈×C₆₁

—

C₄×C₈×C₆₁

C₈

yes

complemented

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₈

yes

characteristic

C₄²×C₆₁

M₁₆

yes

characteristic

C₂×#92418

—

C₂³×C₆₁

#99

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₆₁

#101

yes

characteristic

C₈×C₆₁

—

#92418

—

#7371

—

maximal, Sylow

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

#1844

D₆₁

yes

characteristic

#1325

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₄×C₈

Dic₆₁

yes

characteristic

#384

Dic₆₁

yes

characteristic

#384

C₂×D₆₁

yes

characteristic

#324

—

122

—

C₆₁

#7371

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₆₁

yes

—

C₂×C₄²

C₂×Dic₆₁

yes

characteristic

#106

—

C₂²⋊C₈

—

244

—

C₄×C₈

—

C₄×C₈

#92418

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

488

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

yes

C₄²

yes

C₂³

yes

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊#92418

yes

characteristic

110

C₂×C₄

#14519696

yes

characteristic

111

C₈

—

112

C₈

—

976

—

113

C₂²

—

114

C₂²

—

115

C₂²

#16839488

yes

characteristic

116

C₂²

#16839555

yes

characteristic

117

C₂²

#16839561

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₂

—

125

C₂

—

126

C₂

#21443765

yes

characteristic

127

C₂

#21443984

yes

characteristic

128

C₂

#21443986

yes

characteristic

	subgroup	G/H
center	C₂²	#16839555
commutator	C₄²×C₆₁	C₂×C₈
Frattini	#384	C₂×D₆₁
Fitting	C₆₁×#1844	C₂
socle	C₂²×C₆₁	#353
prime radical	C₂³×C₆₁	#99
cosocle	C₆₁×#384	C₂²
hypercenter	#1844	D₆₁
hypocenter	C₆₁	#7371
LCS	G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1844

subgroup

G/H

center

C₂²

#16839555

commutator

C₄²×C₆₁

C₂×C₈

Frattini

#384

C₂×D₆₁

Fitting

C₆₁×#1844

C₂

C₂²×C₆₁

C₂³×C₆₁

C₆₁×#384

C₂²

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1844

N	H ≅ G/N	type
17. C₄×C₈×C₆₁	C₈	semidirect
72. C₆₁	#7371	semidirect
82. C₄×C₈	#92418	semidirect

H ≅ G/N

type

17. C₄×C₈×C₆₁

C₈

semidirect

72. C₆₁

#7371

semidirect

82. C₄×C₈

#92418

semidirect

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

subgroup	index
#21443765	2
C₆₁×#1844	2
#7371	61

subgroup

index

#21443765

C₆₁×#1844

#7371

Subgroups — #29567430

Conjugacy classes of subgroups

Complements

Maximal subgroups