Group #29573785 — subgroups

Subgroups — #29573785

Order 15616.

422 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 203 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	#21445976	C₂	1	1	yes	maximal, characteristic
2	#21445946	C₂	1	4	yes	maximal
3	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
4	C₆₁×#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519709	—	2	2	no	—
6	C₂×C₄×#92418	C₂²	1	2	yes	—
7	#16839632	C₂²	1	2	yes	—
8	#16839524	—	2	8	no	—
9	C₈×#92418	C₄	1	4	yes	complemented
10	C₂×#106×C₆₁	—	2	2	no	—
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₆₁×#384	C₂²	1	2	yes	—
13	C₂²×#92418	D₄	1	4	yes	—
14	C₂×C₄⋊Dic₆₁	—	2	4	no	—
15	C₂×C₄⋊Dic₆₁	—	2	2	no	—
16	#14519709	—	4	4	no	—
17	C₄×#92418	C₂×C₄	1	4	yes	—
18	C₄×#92418	Q₈	1	4	yes	—
19	C₂²×C₈×C₆₁	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₆₁	—	2	4	no	—
21	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
22	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
23	#106×C₆₁	—	4	4	no	—
24	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₁	Q₈	1	2	yes	—
26	C₂²×C₄×C₆₁	—	4	2	no	—
27	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
29	C₂²×Dic₆₁	—	4	2	no	—
30	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
32	C₂×C₈×C₆₁	SD₁₆	1	4	yes	—
33	C₄⋊C₄×C₆₁	—	4	4	no	—
34	C₄⋊C₄×C₆₁	—	4	8	no	—
35	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₆₁	C₂×Q₈	1	2	yes	—
38	C₄⋊Dic₆₁	—	4	4	no	—
39	C₄⋊Dic₆₁	—	4	8	no	—
40	C₂×#92418	—	2	8	no	—
41	C₂×#92418	C₄⋊C₄	1	8	yes	—
42	C₂³×C₆₁	#127	1	1	yes	characteristic
43	C₂×C₄×C₆₁	—	8	2	no	—
44	C₂×C₄×C₆₁	—	8	4	no	—
45	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
49	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
50	C₂×C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
51	C₂×C₄×C₆₁	#135	1	8	yes	—
52	C₂×Dic₆₁	—	8	2	no	—
53	C₂×Dic₆₁	—	8	4	no	—
54	C₈×C₆₁	#106	1	8	yes	—
55	#92418	—	2	8	no	—
56	#8815	—	61	1	no	maximal, Sylow
57	C₂²×C₆₁	#384	1	1	yes	characteristic
58	C₂²×C₆₁	#492	1	1	yes	characteristic
59	C₂²×C₆₁	#492	1	1	yes	characteristic
60	C₂²×C₆₁	#495	1	4	yes	—
61	C₄×C₆₁	—	16	2	no	—
62	C₄×C₆₁	C₂×#106	1	2	yes	—
63	C₄×C₆₁	#427	1	4	yes	—
64	C₄×C₆₁	#498	1	2	yes	—
65	C₄×C₆₁	#499	1	4	yes	—
66	Dic₆₁	—	16	2	no	—
67	#1844	—	61	1	no	—
68	#1844	D₆₁	1	1	yes	characteristic
69	#1834	—	61	4	no	—
70	C₂×C₈²	—	61	1	no	—
71	C₂×C₆₁	#1708	1	2	yes	—
72	C₂×C₆₁	#1708	1	2	yes	—
73	C₂×C₆₁	#1844	1	1	yes	characteristic
74	C₂×C₆₁	#1846	1	2	yes	—
75	C₂×#106	—	2	2	no	—
76	C₂×#106	—	122	2	no	—
77	C₂×C₄×C₈	—	61	2	no	—
78	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
79	#384	—	61	2	no	—
80	#384	C₂×D₆₁	1	2	yes	—
81	#340	—	122	8	no	—
82	C₈²	—	61	4	no	—
83	C₆₁	#8815	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	61	4	no	—
85	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
86	C₂×C₄⋊C₄	—	2	4	no	—
87	C₂×C₄⋊C₄	—	2	2	no	—
88	C₂×C₄⋊C₄	—	122	4	no	—
89	C₂×C₄⋊C₄	—	122	2	no	—
90	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
91	#106	—	4	4	no	—
92	#106	—	244	4	no	—
93	C₄×C₈	—	61	4	no	—
94	C₄×C₈	—	61	4	no	—
95	C₄×C₈	Dic₁₂₂	1	2	yes	—
96	C₄×C₈	C₄×D₆₁	1	2	yes	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	—	244	2	no	—
99	C₂²×C₄	D₄×D₆₁	1	2	yes	—
100	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
101	C₂×C₈	—	61	8	no	—
102	C₂×C₈	—	122	8	no	—
103	C₂×C₈	#11758196	1	4	yes	—
104	C₂×C₈	#11758200	1	4	yes	—
105	C₂×C₈	#11758201	1	4	yes	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	4	8	no	—
108	C₄⋊C₄	—	244	4	no	—
109	C₄⋊C₄	—	244	8	no	—
110	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
111	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
112	C₄²	Q₈×D₆₁	1	2	yes	—
113	C₂³	#14519773	1	1	yes	characteristic
114	C₂×C₄	—	8	2	no	—
115	C₂×C₄	—	8	4	no	—
116	C₂×C₄	—	488	2	no	—
117	C₂×C₄	—	488	4	no	—
118	C₂×C₄	C₄⋊C₄×D₆₁	1	2	yes	—
119	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
120	C₂×C₄	#14519775	1	1	yes	characteristic
121	C₂×C₄	#14519776	1	2	yes	—
122	C₂×C₄	C₂×#11758201	1	2	yes	—
123	C₂×C₄	C₂×#11758200	1	2	yes	—
124	C₂×C₄	#14519798	1	8	yes	—
125	C₈	—	122	8	no	—
126	C₈	#14519708	1	8	yes	—
127	C₂²	#16839631	1	1	yes	characteristic
128	C₂²	#16840134	1	1	yes	characteristic
129	C₂²	#16840136	1	1	yes	characteristic
130	C₂²	#16840143	1	4	yes	—
131	C₄	—	16	2	no	—
132	C₄	—	976	2	no	—
133	C₄	C₂×#14519708	1	2	yes	—
134	C₄	#16839797	1	4	yes	—
135	C₄	#16840158	1	2	yes	—
136	C₄	#16840162	1	4	yes	—
137	C₂	#21445418	1	2	yes	—
138	C₂	#21445421	1	2	yes	—
139	C₂	#21445975	1	1	yes	characteristic
140	C₂	#21445980	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21445976

C₂

yes

maximal, characteristic

#21445946

C₂

yes

maximal

C₂×C₈×#92418

C₂

yes

maximal, characteristic

C₆₁×#1844

C₂

yes

maximal, characteristic

C₂×#14519709

—

C₂×C₄×#92418

C₂²

yes

—

#16839632

C₂²

yes

—

#16839524

—

C₈×#92418

C₄

yes

complemented

C₂×#106×C₆₁

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×#384

C₂²

yes

—

C₂²×#92418

D₄

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

—

#14519709

—

C₄×#92418

C₂×C₄

yes

—

C₄×#92418

Q₈

yes

—

C₂²×C₈×C₆₁

D₄

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

C₂³

yes

characteristic

#106×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×D₄

yes

—

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₂×C₈×C₆₁

SD₁₆

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

characteristic

C₄²×C₆₁

C₂×Q₈

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

C₄⋊C₄

yes

—

C₂³×C₆₁

#127

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₄⋊Q₈

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

C₂×SD₁₆

yes

—

C₂×C₄×C₆₁

#135

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

#106

yes

—

—

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#106

yes

—

C₄×C₆₁

#427

yes

—

C₄×C₆₁

#498

yes

—

C₄×C₆₁

#499

yes

—

Dic₆₁

—

#1844

—

#1844

D₆₁

yes

characteristic

#1834

—

C₂×C₈²

—

C₂×C₆₁

#1708

yes

—

C₂×C₆₁

#1708

yes

—

C₂×C₆₁

#1844

yes

characteristic

C₂×C₆₁

#1846

yes

—

C₂×#106

—

C₂×#106

—

122

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

#384

—

#384

C₂×D₆₁

yes

—

#340

—

122

—

C₈²

—

C₆₁

#8815

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₆₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂²×D₆₁

yes

characteristic

#106

—

#106

—

244

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₂₂

yes

—

C₄×C₈

C₄×D₆₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

244

—

C₂²×C₄

D₄×D₆₁

yes

—

100

C₂²×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

122

—

103

C₂×C₈

#11758196

yes

—

104

C₂×C₈

#11758200

yes

—

105

C₂×C₈

#11758201

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

244

—

109

C₄⋊C₄

—

244

—

110

C₄²

C₂×Dic₁₂₂

yes

characteristic

111

C₄²

C₂×C₄×D₆₁

yes

characteristic

112

C₄²

Q₈×D₆₁

yes

—

113

C₂³

#14519773

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

488

—

117

C₂×C₄

—

488

—

118

C₂×C₄

C₄⋊C₄×D₆₁

yes

—

119

C₂×C₄

C₂×#11758196

yes

characteristic

120

C₂×C₄

#14519775

yes

characteristic

121

C₂×C₄

#14519776

yes

—

122

C₂×C₄

C₂×#11758201

yes

—

123

C₂×C₄

C₂×#11758200

yes

—

124

C₂×C₄

#14519798

yes

—

125

C₈

—

122

—

126

C₈

#14519708

yes

—

127

C₂²

#16839631

yes

characteristic

128

C₂²

#16840134

yes

characteristic

129

C₂²

#16840136

yes

characteristic

130

C₂²

#16840143

yes

—

131

C₄

—

132

C₄

—

976

—

133

C₄

C₂×#14519708

yes

—

134

C₄

#16839797

yes

—

135

C₄

#16840158

yes

—

136

C₄

#16840162

yes

—

137

C₂

#21445418

yes

—

138

C₂

#21445421

yes

—

139

C₂

#21445975

yes

characteristic

140

C₂

#21445980

yes

—

	subgroup	G/H
center	C₂³	#14519773
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1844	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1844	D₆₁
hypocenter	C₆₁	#8815
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

subgroup

G/H

center

C₂³

#14519773

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₁

Fitting

C₆₁×#1844

C₂

socle, prime radical

C₂³×C₆₁

#127

cosocle

C₂×C₄²×C₆₁

C₂³

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

N	H ≅ G/N	type
9. C₈×#92418	C₄	semidirect
83. C₆₁	#8815	semidirect

H ≅ G/N

type

9. C₈×#92418

C₄

semidirect

83. C₆₁

#8815

semidirect

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

subgroup	index
#21445976	2
#21445946	2
C₂×C₈×#92418	2
C₆₁×#1844	2
#8815	61

subgroup

index

#21445976

#21445946

C₂×C₈×#92418

C₆₁×#1844

#8815

Subgroups — #29573785

Conjugacy classes of subgroups

Complements

Maximal subgroups