Group #16820440 — subgroups

Subgroups — #16820440

Order 3712.

292 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 125 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×#11745231	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#6597	C₂	1	1	yes	maximal, characteristic
3	#14270925	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₂₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈	—	2	1	no	—
6	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₉	—	2	1	no	—
8	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₂₉	—	2	1	no	—
10	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
11	C₂²×#6597	C₂²	1	2	yes	—
12	#11745231	—	2	2	no	—
13	#11745231	C₄	1	4	yes	complemented
14	C₂₉×#95	—	2	1	no	—
15	C₄⋊#6597	—	2	2	no	—
16	C₂²×C₄×C₂₉	—	2	2	no	—
17	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
20	C₂²×Dic₂₉	—	2	2	no	—
21	C₂×Dic₅₈	—	2	4	no	—
22	C₂×Dic₅₈	—	4	4	no	—
23	C₄⋊C₄×C₂₉	—	2	2	no	—
24	C₄⋊C₄×C₂₉	Q₈	1	4	yes	complemented
25	C₄²×C₂₉	—	4	2	no	—
26	C₄²×C₂₉	C₂×C₄	1	2	yes	—
27	C₄²×C₂₉	D₄	1	2	yes	—
28	C₄⋊Dic₂₉	—	2	4	no	—
29	C₄⋊Dic₂₉	—	2	2	no	—
30	C₄⋊Dic₂₉	C₂×C₄	1	4	yes	—
31	C₂×#6597	—	2	2	no	—
32	C₂×#6597	—	2	4	no	—
33	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₉	—	2	2	no	—
35	C₂×C₄×C₂₉	—	2	4	no	—
36	C₂×C₄×C₂₉	—	4	2	no	—
37	C₂×C₄×C₂₉	—	4	2	no	—
38	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₉	SD₁₆	1	2	yes	—
41	C₂×C₄×C₂₉	Q₁₆	1	2	yes	—
42	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₂₉	#41	1	2	yes	—
46	C₂×Dic₂₉	—	2	4	no	—
47	C₂×Dic₂₉	—	4	4	no	—
48	Dic₅₈	—	4	8	no	—
49	#6597	—	4	2	no	—
50	#1987	—	29	1	no	maximal, Sylow
51	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₂₉	C₂²⋊Q₈	1	2	yes	—
53	C₂²×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₂₉	C₂×Q₁₆	1	1	yes	characteristic
55	C₂²×C₂₉	AGL(1,ℤ/8)	1	2	yes	—
56	C₄×C₂₉	—	2	2	no	—
57	C₄×C₂₉	—	4	4	no	—
58	C₄×C₂₉	#103	1	4	yes	—
59	C₄×C₂₉	C₄×Q₈	1	2	yes	—
60	C₄×C₂₉	#124	1	1	yes	characteristic
61	C₄×C₂₉	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₂₉	—	4	4	no	—
63	C₂×C₄⋊Q₈	—	29	1	no	—
64	C₂×C₄⋊C₈	—	29	1	no	—
65	C₄×C₄⋊C₄	D₂₉	1	1	yes	characteristic
66	#340	—	29	4	no	—
67	C₂×C₂₉	#391	1	1	yes	characteristic
68	C₂×C₂₉	C₂×#103	1	1	yes	characteristic
69	C₂×C₂₉	#418	1	1	yes	characteristic
70	C₂×C₂₉	#476	1	2	yes	—
71	C₂×C₂₉	#477	1	2	yes	—
72	C₂²×Q₈	—	58	1	no	—
73	C₂²×C₈	—	29	2	no	—
74	C₄⋊Q₈	—	29	4	no	—
75	C₄⋊Q₈	—	58	2	no	—
76	C₂×C₄⋊C₄	—	29	2	no	—
77	C₂×C₄⋊C₄	—	58	1	no	—
78	C₂×C₄⋊C₄	C₂×D₂₉	1	2	yes	—
79	C₂×C₄²	—	2	1	no	—
80	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
81	C₄⋊C₈	—	58	2	no	—
82	#95	—	2	1	no	—
83	C₂₉	#1987	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₈	—	58	4	no	—
85	C₂×Q₈	—	116	4	no	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	—	58	2	no	—
88	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
89	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
91	C₂×C₈	—	58	2	no	—
92	C₂×C₈	—	58	4	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	29	4	no	—
95	C₄⋊C₄	—	58	4	no	—
96	C₄⋊C₄	—	58	2	no	—
97	C₄⋊C₄	Dic₅₈	1	4	yes	complemented
98	C₄²	—	4	2	no	—
99	C₄²	C₄×D₂₉	1	2	yes	—
100	C₄²	D₁₁₆	1	2	yes	—
101	C₂³	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
102	Q₈	—	116	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	58	4	no	—
108	C₂×C₄	—	116	4	no	—
109	C₂×C₄	#90781	1	2	yes	—
110	C₂×C₄	#90781	1	2	yes	—
111	C₂×C₄	#90783	1	2	yes	—
112	C₂×C₄	#90785	1	2	yes	—
113	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
115	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
116	C₂×C₄	#90805	1	2	yes	—
117	C₈	—	116	2	no	—
118	C₂²	#11745286	1	2	yes	—
119	C₂²	C₂×#90781	1	1	yes	characteristic
120	C₂²	C₂×#90783	1	1	yes	characteristic
121	C₂²	C₂×#90785	1	1	yes	characteristic
122	C₂²	#11745311	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	116	4	no	—
126	C₄	#11745158	1	4	yes	—
127	C₄	C₄×Dic₅₈	1	2	yes	—
128	C₄	#11745231	1	1	yes	characteristic
129	C₄	#11745237	1	1	yes	characteristic
130	C₂	#14271069	1	1	yes	characteristic
131	C₂	C₂×#11745158	1	1	yes	characteristic
132	C₂	#14271167	1	1	yes	characteristic
133	C₂	#14271453	1	2	yes	—
134	C₂	#14271458	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745231

C₂

yes

maximal, characteristic

C₂×C₄⋊#6597

C₂

yes

maximal, characteristic

#14270925

C₂

yes

maximal

C₄×C₄⋊C₄×C₂₉

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

—

C₂×C₄⋊C₄×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

C₂²×#6597

C₂²

yes

—

#11745231

—

#11745231

C₄

yes

complemented

C₂₉×#95

—

C₄⋊#6597

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

Q₈

yes

complemented

C₄²×C₂₉

—

C₄²×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

D₄

yes

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

C₂×#6597

—

C₂×#6597

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

SD₁₆

yes

—

C₂×C₄×C₂₉

Q₁₆

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×Q₈

yes

—

C₂×C₄×C₂₉

#41

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

#6597

—

#1987

—

maximal, Sylow

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

C₂²⋊Q₈

yes

—

C₂²×C₂₉

C₂×SD₁₆

yes

characteristic

C₂²×C₂₉

C₂×Q₁₆

yes

characteristic

C₂²×C₂₉

AGL(1,ℤ/8)

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

#103

yes

—

C₄×C₂₉

C₄×Q₈

yes

—

C₄×C₂₉

#124

yes

characteristic

C₄×C₂₉

C₄⋊Q₈

yes

characteristic

Dic₂₉

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

D₂₉

yes

characteristic

#340

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

—

C₂×C₂₉

#477

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

#95

—

C₂₉

#1987

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

116

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₉

yes

characteristic

C₂²×C₄

C₂²⋊D₂₉

yes

characteristic

C₂²×C₄

C₂²×D₂₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Dic₅₈

yes

complemented

C₄²

—

C₄²

C₄×D₂₉

yes

—

100

C₄²

D₁₁₆

yes

—

101

C₂³

C₂×C₂²⋊D₂₉

yes

characteristic

102

Q₈

—

116

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

116

—

109

C₂×C₄

#90781

yes

—

110

C₂×C₄

#90781

yes

—

111

C₂×C₄

#90783

yes

—

112

C₂×C₄

#90785

yes

—

113

C₂×C₄

C₂×Dic₅₈

yes

—

114

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

115

C₂×C₄

C₂×D₁₁₆

yes

characteristic

116

C₂×C₄

#90805

yes

—

117

C₈

—

116

—

118

C₂²

#11745286

yes

—

119

C₂²

C₂×#90781

yes

characteristic

120

C₂²

C₂×#90783

yes

characteristic

121

C₂²

C₂×#90785

yes

characteristic

122

C₂²

#11745311

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

116

—

126

C₄

#11745158

yes

—

127

C₄

C₄×Dic₅₈

yes

—

128

C₄

#11745231

yes

characteristic

129

C₄

#11745237

yes

characteristic

130

C₂

#14271069

yes

characteristic

131

C₂

C₂×#11745158

yes

characteristic

132

C₂

#14271167

yes

characteristic

133

C₂

#14271453

yes

—

134

C₂

#14271458

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₉
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₄×C₄⋊C₄×C₂₉	C₂
socle, prime radical	C₂³×C₂₉	C₂×D₄
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₂₉
hypocenter	C₂₉	#1987
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₉

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₄×C₄⋊C₄×C₂₉

C₂

socle, prime radical

C₂³×C₂₉

C₂×D₄

cosocle

C₂²×C₄×C₂₉

C₂³

hypercenter

C₄×C₄⋊C₄

D₂₉

hypocenter

C₂₉

#1987

LCS

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
13. #11745231	C₄	semidirect
24. C₄⋊C₄×C₂₉	Q₈	semidirect
83. C₂₉	#1987	semidirect
97. C₄⋊C₄	Dic₅₈	semidirect

H ≅ G/N

type

13. #11745231

C₄

semidirect

24. C₄⋊C₄×C₂₉

Q₈

semidirect

83. C₂₉

#1987

semidirect

97. C₄⋊C₄

Dic₅₈

semidirect

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

subgroup	index
C₂×#11745231	2
C₂×C₄⋊#6597	2
#14270925	2
C₄×C₄⋊C₄×C₂₉	2
#1987	29

subgroup

index

C₂×#11745231

C₂×C₄⋊#6597

#14270925

C₄×C₄⋊C₄×C₂₉

#1987

Subgroups — #16820440

Conjugacy classes of subgroups

Complements

Maximal subgroups