Group #16820415 — subgroups

Subgroups — #16820415

Order 3712.

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

normal subgroups only

This group has 324 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	C₂×#11745158	C₂	1	4	yes	maximal
4	C₄×C₄⋊C₄×C₂₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈	C₂²	1	2	yes	—
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	—	2	1	no	—
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	#11745158	—	2	8	no	—
16	C₄⋊#6597	—	2	2	no	—
17	C₂²×C₄×C₂₉	—	2	2	no	—
18	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₂₉	—	2	2	no	—
27	C₄⋊C₄×C₂₉	D₄	1	4	yes	—
28	C₄²×C₂₉	—	4	2	no	—
29	C₄²×C₂₉	C₂×C₄	1	2	yes	—
30	C₄²×C₂₉	D₄	1	2	yes	—
31	C₄⋊Dic₂₉	—	2	4	no	—
32	C₄⋊Dic₂₉	—	2	4	no	—
33	C₂×#6597	—	2	2	no	—
34	C₂×#6597	—	2	4	no	—
35	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₉	—	2	2	no	—
37	C₂×C₄×C₂₉	—	2	4	no	—
38	C₂×C₄×C₂₉	—	4	2	no	—
39	C₂×C₄×C₂₉	—	4	2	no	—
40	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₂₉	SD₁₆	1	2	yes	—
43	C₂×C₄×C₂₉	Q₁₆	1	2	yes	—
44	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉	#41	1	2	yes	—
48	C₂×Dic₂₉	—	2	4	no	—
49	C₂×Dic₂₉	—	4	4	no	—
50	Dic₅₈	—	2	8	no	—
51	Dic₅₈	—	4	2	no	—
52	Dic₅₈	—	4	2	no	—
53	#6597	—	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	—	4	4	no	—
62	C₄×C₂₉	#103	1	4	yes	complemented
63	C₄×C₂₉	C₄×D₄	1	2	yes	—
64	C₄×C₂₉	#124	1	1	yes	characteristic
65	C₄×C₂₉	#127	1	1	yes	characteristic
66	Dic₂₉	—	4	4	no	—
67	C₂×C₄⋊Q₈	—	29	1	no	—
68	C₂×C₄⋊C₈	—	29	1	no	—
69	C₂×#103	—	29	4	no	—
70	C₄×C₄⋊C₄	D₂₉	1	1	yes	characteristic
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	—	29	2	no	—
78	C₄⋊Q₈	—	29	4	no	—
79	C₄⋊Q₈	—	58	2	no	—
80	C₂×C₄⋊C₄	—	58	1	no	—
81	C₂×C₄⋊C₄	—	58	1	no	—
82	C₂×C₄⋊C₄	C₂×D₂₉	1	2	yes	—
83	C₂×C₄²	—	2	1	no	—
84	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
85	C₄⋊C₈	—	58	2	no	—
86	#103	—	58	8	no	—
87	#95	—	2	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	—	2	2	no	—
94	C₂²×C₄	—	58	2	no	—
95	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₂₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
98	C₂×C₈	—	58	4	no	—
99	C₂×C₈	—	58	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	58	4	no	—
102	C₄⋊C₄	—	58	4	no	—
103	C₄⋊C₄	D₁₁₆	1	4	yes	—
104	C₄²	—	4	2	no	—
105	C₄²	C₄×D₂₉	1	2	yes	—
106	C₄²	D₁₁₆	1	2	yes	—
107	C₂³	C₂×C₂²⋊D₂₉	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	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	58	4	no	—
116	C₂×C₄	—	116	4	no	—
117	C₂×C₄	#90781	1	2	yes	—
118	C₂×C₄	#90781	1	2	yes	—
119	C₂×C₄	#90783	1	2	yes	—
120	C₂×C₄	#90785	1	2	yes	—
121	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
122	C₂×C₄	C₂×D₁₁₆	1	2	yes	—
123	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
124	C₂×C₄	#90805	1	2	yes	—
125	C₈	—	116	2	no	—
126	C₂²	C₂×#90781	1	1	yes	characteristic
127	C₂²	#11745292	1	2	yes	—
128	C₂²	C₂×#90783	1	1	yes	characteristic
129	C₂²	C₂×#90785	1	1	yes	characteristic
130	C₂²	#11745313	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	4	4	no	—
133	C₄	—	116	4	no	—
134	C₄	#11745158	1	4	yes	complemented
135	C₄	C₄×D₁₁₆	1	2	yes	—
136	C₄	#11745236	1	1	yes	characteristic
137	C₄	#11745237	1	1	yes	characteristic
138	C₂	#14271065	1	1	yes	characteristic
139	C₂	C₂×#11745158	1	1	yes	characteristic
140	C₂	#14271171	1	1	yes	characteristic
141	C₂	#14271368	1	2	yes	—
142	C₂	#14271373	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745231

C₂

yes

maximal, characteristic

C₂×C₄⋊#6597

C₂

yes

maximal, characteristic

C₂×#11745158

C₂

yes

maximal

C₄×C₄⋊C₄×C₂₉

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₂²

yes

—

C₂×C₄⋊C₄×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

—

C₂×C₄⋊Dic₂₉

—

C₂²×#6597

C₂²

yes

—

#11745231

—

#11745231

C₄

yes

complemented

C₂₉×#95

—

#11745158

—

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

—

C₂×Dic₅₈

—

C₂×Dic₅₈

C₂×C₄

yes

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

D₄

yes

—

C₄²×C₂₉

—

C₄²×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

D₄

yes

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

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

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

#41

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

#6597

—

#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

—

C₄×C₂₉

#103

yes

complemented

C₄×C₂₉

C₄×D₄

yes

—

C₄×C₂₉

yes

C₄×C₂₉

yes

Dic₂₉

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₄×C₄⋊C₄

D₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

—

C₂×C₂₉

#462

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

—

#103

—

#95

—

C₂₉

#1981

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

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

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

D₁₁₆

yes

—

104

C₄²

—

105

C₄²

C₄×D₂₉

yes

—

106

C₄²

D₁₁₆

yes

—

107

C₂³

C₂×C₂²⋊D₂₉

yes

characteristic

108

Q₈

—

109

Q₈

—

116

—

110

Q₈

—

116

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

116

—

117

C₂×C₄

#90781

yes

—

118

C₂×C₄

#90781

yes

—

119

C₂×C₄

#90783

yes

—

120

C₂×C₄

#90785

yes

—

121

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

122

C₂×C₄

C₂×D₁₁₆

yes

—

123

C₂×C₄

C₂×D₁₁₆

yes

characteristic

124

C₂×C₄

#90805

yes

—

125

C₈

—

116

—

126

C₂²

C₂×#90781

yes

characteristic

127

C₂²

#11745292

yes

—

128

C₂²

C₂×#90783

yes

characteristic

129

C₂²

C₂×#90785

yes

characteristic

130

C₂²

#11745313

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

116

—

134

C₄

#11745158

yes

complemented

135

C₄

C₄×D₁₁₆

yes

—

136

C₄

#11745236

yes

characteristic

137

C₄

#11745237

yes

characteristic

138

C₂

#14271065

yes

characteristic

139

C₂

C₂×#11745158

yes

characteristic

140

C₂

#14271171

yes

characteristic

141

C₂

#14271368

yes

—

142

C₂

#14271373

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	#1981
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

#1981

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
62. C₄×C₂₉	#103	semidirect
88. C₂₉	#1981	semidirect
134. C₄	#11745158	semidirect

H ≅ G/N

type

13. #11745231

C₄

semidirect

62. C₄×C₂₉

#103

semidirect

88. C₂₉

#1981

semidirect

134. C₄

#11745158

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
C₂×#11745158	2
C₄×C₄⋊C₄×C₂₉	2
#1981	29

subgroup

index

C₂×#11745231

C₂×C₄⋊#6597

C₂×#11745158

C₄×C₄⋊C₄×C₂₉

#1981

Subgroups — #16820415

Conjugacy classes of subgroups

Complements

Maximal subgroups