Group #16820414 — subgroups

Subgroups — #16820414

Order 3712.

324 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 121 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×#11745164	C₂	1	4	yes	maximal
3	C₄×C₄⋊Dic₂₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈	C₂²	1	2	yes	—
6	C₂²×C₈×C₂₉	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₉	—	2	1	no	—
9	C₂×C₄⋊Dic₂₉	—	2	1	no	—
10	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
11	C₂×C₄×Dic₂₉	—	2	1	no	—
12	#11745231	—	2	2	no	—
13	#11745231	C₄	1	4	yes	complemented
14	C₂₉×C₄⋊C₈	—	2	2	no	—
15	#11745179	—	2	1	no	—
16	#11745164	—	2	8	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	C₂²×Dic₂₉	—	2	2	no	—
22	C₂×Dic₅₈	—	2	4	no	—
23	C₂×Dic₅₈	—	2	4	no	—
24	C₂×Dic₅₈	—	2	2	no	—
25	C₂×Dic₅₈	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₂₉	—	2	4	no	—
27	C₂×C₈×C₂₉	—	2	2	no	—
28	C₄²×C₂₉	C₂×C₄	1	2	yes	—
29	C₄²×C₂₉	D₄	1	2	yes	—
30	C₄⋊Dic₂₉	—	2	4	no	—
31	C₄⋊Dic₂₉	—	2	4	no	—
32	C₄⋊Dic₂₉	—	2	2	no	—
33	C₄⋊Dic₂₉	D₄	1	4	yes	—
34	C₄×Dic₂₉	—	4	2	no	—
35	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₉	—	2	2	no	—
37	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₉	SD₁₆	1	2	yes	—
40	C₂×C₄×C₂₉	Q₁₆	1	2	yes	—
41	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₉	#41	1	2	yes	—
45	C₂×Dic₂₉	—	2	4	no	—
46	C₂×Dic₂₉	—	2	4	no	—
47	C₂×Dic₂₉	—	4	4	no	—
48	C₂×Dic₂₉	—	4	2	no	—
49	C₂×Dic₂₉	—	4	2	no	—
50	Dic₅₈	—	2	8	no	—
51	Dic₅₈	—	4	2	no	—
52	Dic₅₈	—	4	2	no	—
53	C₈×C₂₉	—	4	2	no	—
54	#1981	—	29	1	no	maximal, Sylow
55	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₉	#121	1	2	yes	—
57	C₂²×C₂₉	C₂×SD₁₆	1	1	yes	characteristic
58	C₂²×C₂₉	C₂×Q₁₆	1	1	yes	characteristic
59	C₂²×C₂₉	#137	1	2	yes	—
60	C₄×C₂₉	—	2	2	no	—
61	C₄×C₂₉	#103	1	4	yes	complemented
62	C₄×C₂₉	C₄×D₄	1	2	yes	—
63	C₄×C₂₉	#124	1	1	yes	characteristic
64	C₄×C₂₉	#127	1	1	yes	characteristic
65	Dic₂₉	—	4	4	no	—
66	Dic₂₉	—	4	4	no	—
67	C₂×C₄⋊Q₈	—	29	1	no	—
68	C₂×C₄⋊C₈	D₂₉	1	1	yes	characteristic
69	C₂×#103	—	29	4	no	—
70	C₄×C₄⋊C₄	—	29	1	no	—
71	C₂×C₂₉	#390	1	1	yes	characteristic
72	C₂×C₂₉	C₂×#103	1	1	yes	characteristic
73	C₂×C₂₉	#419	1	1	yes	characteristic
74	C₂×C₂₉	#461	1	2	yes	—
75	C₂×C₂₉	#462	1	2	yes	—
76	C₂²×Q₈	—	29	2	no	—
77	C₂²×C₈	C₂×D₂₉	1	2	yes	—
78	C₄⋊Q₈	—	29	4	no	—
79	C₄⋊Q₈	—	58	2	no	—
80	C₂×C₄⋊C₄	—	29	2	no	—
81	C₂×C₄⋊C₄	—	58	1	no	—
82	C₂×C₄⋊C₄	—	58	1	no	—
83	C₂×C₄²	—	58	1	no	—
84	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
85	C₄⋊C₈	—	2	2	no	—
86	#103	—	58	8	no	—
87	#95	—	58	1	no	—
88	C₂₉	#1981	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	29	4	no	—
90	C₂×Q₈	—	58	2	no	—
91	C₂×Q₈	—	58	4	no	—
92	C₂×Q₈	—	58	4	no	—
93	C₂²×C₄	—	58	2	no	—
94	C₂²×C₄	—	58	2	no	—
95	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
96	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
98	C₂×C₈	—	2	4	no	—
99	C₂×C₈	—	2	2	no	—
100	C₄⋊C₄	—	29	4	no	—
101	C₄⋊C₄	—	58	4	no	—
102	C₄⋊C₄	—	58	4	no	—
103	C₄⋊C₄	—	58	2	no	—
104	C₄²	—	116	2	no	—
105	C₄²	C₄×D₂₉	1	2	yes	—
106	C₄²	C₂²⋊D₂₉	1	2	yes	—
107	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
108	Q₈	—	58	8	no	—
109	Q₈	—	116	2	no	—
110	Q₈	—	116	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	58	4	no	—
113	C₂×C₄	—	58	4	no	—
114	C₂×C₄	—	116	4	no	—
115	C₂×C₄	—	116	2	no	—
116	C₂×C₄	—	116	2	no	—
117	C₂×C₄	#90773	1	2	yes	—
118	C₂×C₄	Dic₁₁₆	1	2	yes	—
119	C₂×C₄	#90781	1	2	yes	—
120	C₂×C₄	#90781	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
122	C₂×C₄	D₄×D₂₉	1	2	yes	—
123	C₂×C₄	#90809	1	2	yes	—
124	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
125	C₈	—	4	2	no	—
126	C₂²	#11745257	1	2	yes	—
127	C₂²	C₂×#90773	1	1	yes	characteristic
128	C₂²	C₂×Dic₁₁₆	1	1	yes	characteristic
129	C₂²	#11745271	1	2	yes	—
130	C₂²	C₂×#90781	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	116	4	no	—
133	C₄	—	116	4	no	—
134	C₄	#11745164	1	4	yes	complemented
135	C₄	#11745255	1	2	yes	—
136	C₄	#11745302	1	1	yes	characteristic
137	C₄	#11745309	1	1	yes	characteristic
138	C₂	#14271064	1	1	yes	characteristic
139	C₂	C₂×#11745164	1	1	yes	characteristic
140	C₂	#14271173	1	1	yes	characteristic
141	C₂	#14271369	1	2	yes	—
142	C₂	#14271374	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745231

C₂

yes

maximal, characteristic

C₂×#11745164

C₂

yes

maximal

C₄×C₄⋊Dic₂₉

C₂

yes

maximal, characteristic

C₂×C₂₉×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₂²

yes

—

C₂²×C₈×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

—

C₂×C₄⋊Dic₂₉

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

C₂×C₄×Dic₂₉

—

#11745231

—

#11745231

C₄

yes

complemented

C₂₉×C₄⋊C₈

—

#11745179

—

#11745164

—

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

D₄

yes

characteristic

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

—

C₂²×Dic₂₉

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

C₂×C₄

yes

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₄²×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

D₄

yes

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

D₄

yes

—

C₄×Dic₂₉

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

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

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

#41

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

C₈×C₂₉

—

#1981

—

maximal, Sylow

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

#121

yes

—

C₂²×C₂₉

C₂×SD₁₆

yes

characteristic

C₂²×C₂₉

C₂×Q₁₆

yes

characteristic

C₂²×C₂₉

#137

yes

—

C₄×C₂₉

—

C₄×C₂₉

#103

yes

complemented

C₄×C₂₉

C₄×D₄

yes

—

C₄×C₂₉

yes

C₄×C₂₉

yes

Dic₂₉

—

Dic₂₉

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

D₂₉

yes

characteristic

C₂×#103

—

C₄×C₄⋊C₄

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

—

C₂×C₂₉

#462

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₂₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

#103

—

#95

—

C₂₉

#1981

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₁₆

yes

characteristic

C₂²×C₄

D₁₁₆

yes

characteristic

C₂²×C₄

C₂²×D₂₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

—

116

—

105

C₄²

C₄×D₂₉

yes

—

106

C₄²

C₂²⋊D₂₉

yes

—

107

C₂³

C₂×D₁₁₆

yes

characteristic

108

Q₈

—

109

Q₈

—

116

—

110

Q₈

—

116

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

116

—

115

C₂×C₄

—

116

—

116

C₂×C₄

—

116

—

117

C₂×C₄

#90773

yes

—

118

C₂×C₄

Dic₁₁₆

yes

—

119

C₂×C₄

#90781

yes

—

120

C₂×C₄

#90781

yes

—

121

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

122

C₂×C₄

D₄×D₂₉

yes

—

123

C₂×C₄

#90809

yes

—

124

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

125

C₈

—

126

C₂²

#11745257

yes

—

127

C₂²

C₂×#90773

yes

characteristic

128

C₂²

C₂×Dic₁₁₆

yes

characteristic

129

C₂²

#11745271

yes

—

130

C₂²

C₂×#90781

yes

characteristic

131

C₄

—

132

C₄

—

116

—

133

C₄

—

116

—

134

C₄

#11745164

yes

complemented

135

C₄

#11745255

yes

—

136

C₄

#11745302

yes

characteristic

137

C₄

#11745309

yes

characteristic

138

C₂

#14271064

yes

characteristic

139

C₂

C₂×#11745164

yes

characteristic

140

C₂

#14271173

yes

characteristic

141

C₂

#14271369

yes

—

142

C₂

#14271374

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₁₆
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂×C₂₉×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₂₉	C₂×D₄
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₉
hypocenter	C₂₉	#1981
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₁₆

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂×C₂₉×C₄⋊C₈

C₂

socle, prime radical

C₂³×C₂₉

C₂×D₄

cosocle

C₂²×C₄×C₂₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₉

hypocenter

C₂₉

#1981

LCS

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
13. #11745231	C₄	semidirect
61. C₄×C₂₉	#103	semidirect
88. C₂₉	#1981	semidirect
134. C₄	#11745164	semidirect

H ≅ G/N

type

13. #11745231

C₄

semidirect

61. C₄×C₂₉

#103

semidirect

88. C₂₉

#1981

semidirect

134. C₄

#11745164

semidirect

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

subgroup	index
C₂×#11745231	2
C₂×#11745164	2
C₄×C₄⋊Dic₂₉	2
C₂×C₂₉×C₄⋊C₈	2
#1981	29

subgroup

index

C₂×#11745231

C₂×#11745164

C₄×C₄⋊Dic₂₉

C₂×C₂₉×C₄⋊C₈

#1981

Subgroups — #16820414

Conjugacy classes of subgroups

Complements

Maximal subgroups