Group #16820428 — subgroups

Subgroups — #16820428

Order 3712.

292 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×Dic₅₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#6597	C₂	1	1	yes	maximal, characteristic
3	#14270917	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₂₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₂₉	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₉	C₄	1	1	yes	characteristic
9	C₂×#90779	C₄	1	2	yes	—
10	C₂×C₄×Dic₂₉	C₂²	1	2	yes	—
11	C₂²×#6597	C₂²	1	2	yes	—
12	C₄×Dic₅₈	—	2	2	no	—
13	C₄×Dic₅₈	—	2	2	no	—
14	C₄×C₈×C₂₉	—	2	2	no	—
15	C₄⋊#6597	—	2	2	no	—
16	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
19	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
20	C₂²×Dic₂₉	C₂×C₄	1	2	yes	—
21	C₂×Dic₅₈	—	2	2	no	—
22	C₂×Dic₅₈	—	2	2	no	—
23	C₂×Dic₅₈	C₈	1	4	yes	complemented
24	C₂×C₈×C₂₉	—	2	2	no	—
25	C₂×C₈×C₂₉	—	2	2	no	—
26	C₂×C₈×C₂₉	Q₈	1	4	yes	complemented
27	C₄²×C₂₉	D₄	1	2	yes	—
28	C₄²×C₂₉	D₄	1	2	yes	—
29	C₄⋊Dic₂₉	—	2	2	no	—
30	#90779	—	2	4	no	—
31	C₄×Dic₂₉	—	2	2	no	—
32	C₄×Dic₂₉	—	2	2	no	—
33	C₂×#6597	—	2	2	no	—
34	C₂×#6597	—	2	4	no	—
35	C₂³×C₂₉	C₂²×C₄	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	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₂₉	M₁₆	1	2	yes	—
41	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₉	C₂×Q₈	1	2	yes	—
44	C₂×C₄×C₂₉	#41	1	2	yes	—
45	C₂×Dic₂₉	—	2	2	no	—
46	C₂×Dic₂₉	—	2	2	no	—
47	C₂×Dic₂₉	—	2	2	no	—
48	C₂×Dic₂₉	—	2	4	no	—
49	C₂×Dic₂₉	C₂×C₈	1	4	yes	—
50	Dic₅₈	—	2	8	no	—
51	C₈×C₂₉	—	2	4	no	—
52	#6597	—	4	2	no	—
53	#1984	—	29	1	no	maximal, Sylow
54	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₉	C₄×Q₈	1	2	yes	—
56	C₂²×C₂₉	C₂²×C₈	1	1	yes	characteristic
57	C₂²×C₂₉	C₂×M₁₆	1	1	yes	characteristic
58	C₂²×C₂₉	#131	1	2	yes	—
59	C₄×C₂₉	—	2	2	no	—
60	C₄×C₂₉	C₂²⋊C₈	1	4	yes	—
61	C₄×C₂₉	C₂²⋊Q₈	1	2	yes	—
62	C₄×C₂₉	#124	1	1	yes	characteristic
63	C₄×C₂₉	C₄⋊Q₈	1	1	yes	characteristic
64	Dic₂₉	—	2	4	no	—
65	Dic₂₉	—	4	2	no	—
66	C₂×C₄×Q₈	—	29	1	no	—
67	C₂×C₄⋊C₈	—	29	1	no	—
68	C₂×C₄×C₈	D₂₉	1	1	yes	characteristic
69	#336	—	29	4	no	—
70	C₂×C₂₉	#391	1	1	yes	characteristic
71	C₂×C₂₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
72	C₂×C₂₉	#408	1	1	yes	characteristic
73	C₂×C₂₉	C₈×Q₈	1	2	yes	—
74	C₂×C₂₉	#446	1	2	yes	—
75	C₂²×Q₈	—	29	1	no	—
76	C₂²×C₈	—	29	2	no	—
77	C₂²×C₈	C₂×D₂₉	1	2	yes	—
78	C₄×Q₈	—	58	2	no	—
79	C₄×Q₈	—	58	2	no	—
80	C₂×C₄⋊C₄	—	29	2	no	—
81	C₂×C₄⋊C₄	—	29	1	no	—
82	C₂×C₄²	—	29	2	no	—
83	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
84	C₄⋊C₈	—	58	2	no	—
85	C₄×C₈	—	2	2	no	—
86	C₂₉	#1984	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₈	—	29	4	no	—
88	C₂×Q₈	—	58	2	no	—
89	C₂×Q₈	—	58	2	no	—
90	C₂²×C₄	—	29	2	no	—
91	C₂²×C₄	—	29	2	no	—
92	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₂₉	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	58	2	no	—
98	C₂×C₈	—	58	4	no	—
99	C₂×C₈	Dic₅₈	1	4	yes	complemented
100	C₄⋊C₄	—	58	4	no	—
101	C₄⋊C₄	—	58	2	no	—
102	C₄²	—	58	2	no	—
103	C₄²	—	58	2	no	—
104	C₄²	D₁₁₆	1	2	yes	—
105	C₄²	C₂²⋊D₂₉	1	2	yes	—
106	C₂³	C₂×C₄×D₂₉	1	1	yes	characteristic
107	Q₈	—	58	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	29	4	no	—
110	C₂×C₄	—	58	2	no	—
111	C₂×C₄	—	58	2	no	—
112	C₂×C₄	—	58	2	no	—
113	C₂×C₄	—	58	4	no	—
114	C₂×C₄	C₈×D₂₉	1	2	yes	—
115	C₂×C₄	#90772	1	2	yes	—
116	C₂×C₄	#90781	1	2	yes	—
117	C₂×C₄	#90781	1	2	yes	—
118	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
119	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
120	C₂×C₄	#90805	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
122	C₈	—	2	4	no	—
123	C₈	—	116	2	no	—
124	C₂²	C₄×Dic₅₈	1	2	yes	—
125	C₂²	C₂×C₈×D₂₉	1	1	yes	characteristic
126	C₂²	C₂×#90772	1	1	yes	characteristic
127	C₂²	#11745263	1	2	yes	—
128	C₂²	C₂×#90781	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	58	4	no	—
131	C₄	—	116	2	no	—
132	C₄	#11745168	1	4	yes	—
133	C₄	#11745231	1	1	yes	characteristic
134	C₄	#11745237	1	1	yes	characteristic
135	C₄	#11745286	1	2	yes	—
136	C₂	#14271069	1	1	yes	characteristic
137	C₂	C₂×#11745168	1	1	yes	characteristic
138	C₂	#14271129	1	1	yes	characteristic
139	C₂	C₈×Dic₅₈	1	2	yes	—
140	C₂	#14271293	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₅₈

C₂

yes

maximal, characteristic

C₂×C₄⋊#6597

C₂

yes

maximal, characteristic

#14270917

C₂

yes

maximal

C₂×C₄×C₈×C₂₉

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₄

yes

characteristic

C₂²×C₈×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

C₄

yes

characteristic

C₂×#90779

C₄

yes

—

C₂×C₄×Dic₂₉

C₂²

yes

—

C₂²×#6597

C₂²

yes

—

C₄×Dic₅₈

—

C₄×Dic₅₈

—

C₄×C₈×C₂₉

—

C₄⋊#6597

—

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

C₂×C₄

yes

—

C₂²×Dic₂₉

C₂×C₄

yes

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

C₈

yes

complemented

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

Q₈

yes

complemented

C₄²×C₂₉

D₄

yes

—

C₄²×C₂₉

D₄

yes

—

C₄⋊Dic₂₉

—

#90779

—

C₄×Dic₂₉

—

C₄×Dic₂₉

—

C₂×#6597

—

C₂×#6597

—

C₂³×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂×C₈

yes

—

C₂×C₄×C₂₉

M₁₆

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×Q₈

yes

—

C₂×C₄×C₂₉

#41

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

C₂×C₈

yes

—

Dic₅₈

—

C₈×C₂₉

—

#6597

—

#1984

—

maximal, Sylow

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

C₄×Q₈

yes

—

C₂²×C₂₉

C₂²×C₈

yes

characteristic

C₂²×C₂₉

C₂×M₁₆

yes

characteristic

C₂²×C₂₉

#131

yes

—

C₄×C₂₉

—

C₄×C₂₉

C₂²⋊C₈

yes

—

C₄×C₂₉

C₂²⋊Q₈

yes

—

C₄×C₂₉

#124

yes

characteristic

C₄×C₂₉

C₄⋊Q₈

yes

characteristic

Dic₂₉

—

Dic₂₉

—

C₂×C₄×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₂₉

yes

characteristic

#336

—

C₂×C₂₉

#391

yes

characteristic

C₂×C₂₉

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₉

#408

yes

characteristic

C₂×C₂₉

C₈×Q₈

yes

—

C₂×C₂₉

#446

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₉

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₂₉

#1984

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₂₉

yes

characteristic

C₂²×C₄

C₄×D₂₉

yes

characteristic

C₂²×C₄

C₂²×D₂₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₅₈

yes

complemented

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

D₁₁₆

yes

—

105

C₄²

C₂²⋊D₂₉

yes

—

106

C₂³

C₂×C₄×D₂₉

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₈×D₂₉

yes

—

115

C₂×C₄

#90772

yes

—

116

C₂×C₄

#90781

yes

—

117

C₂×C₄

#90781

yes

—

118

C₂×C₄

C₂×Dic₅₈

yes

—

119

C₂×C₄

C₂×D₁₁₆

yes

characteristic

120

C₂×C₄

#90805

yes

—

121

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

122

C₈

—

123

C₈

—

116

—

124

C₂²

C₄×Dic₅₈

yes

—

125

C₂²

C₂×C₈×D₂₉

yes

characteristic

126

C₂²

C₂×#90772

yes

characteristic

127

C₂²

#11745263

yes

—

128

C₂²

C₂×#90781

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

116

—

132

C₄

#11745168

yes

—

133

C₄

#11745231

yes

characteristic

134

C₄

#11745237

yes

characteristic

135

C₄

#11745286

yes

—

136

C₂

#14271069

yes

characteristic

137

C₂

C₂×#11745168

yes

characteristic

138

C₂

#14271129

yes

characteristic

139

C₂

C₈×Dic₅₈

yes

—

140

C₂

#14271293

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₉
commutator	C₂²×C₂₉	C₂²×C₈
Fitting	C₂×C₄×C₈×C₂₉	C₂
socle, prime radical	C₂³×C₂₉	C₂²×C₄
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	C₂×C₄×C₈	D₂₉
hypocenter	C₂₉	#1984
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₉

commutator

C₂²×C₂₉

C₂²×C₈

Fitting

C₂×C₄×C₈×C₂₉

C₂

socle, prime radical

C₂³×C₂₉

C₂²×C₄

cosocle

C₂²×C₄×C₂₉

C₂³

hypercenter

C₂×C₄×C₈

D₂₉

C₂₉

G ▷ C₂²×C₂₉ ▷ C₂₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
23. C₂×Dic₅₈	C₈	semidirect
26. C₂×C₈×C₂₉	Q₈	semidirect
86. C₂₉	#1984	semidirect
99. C₂×C₈	Dic₅₈	semidirect

H ≅ G/N

type

23. C₂×Dic₅₈

C₈

semidirect

26. C₂×C₈×C₂₉

Q₈

semidirect

86. C₂₉

#1984

semidirect

99. C₂×C₈

Dic₅₈

semidirect

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

subgroup	index
C₂×C₄×Dic₅₈	2
C₂×C₄⋊#6597	2
#14270917	2
C₂×C₄×C₈×C₂₉	2
#1984	29

subgroup

index

C₂×C₄×Dic₅₈

C₂×C₄⋊#6597

#14270917

C₂×C₄×C₈×C₂₉

#1984

Subgroups — #16820428

Conjugacy classes of subgroups

Complements

Maximal subgroups