Group #16820640 — subgroups

Subgroups — #16820640

Order 3712.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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×#11745158	C₂	1	2	yes	maximal
2	#14271069	C₂	1	1	yes	maximal, characteristic
3	#14270925	C₂	1	2	yes	maximal
4	#14270918	C₂	1	1	yes	maximal, characteristic
5	C₂×C₂₉×#125	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₈	C₂²	1	1	yes	characteristic
7	C₂₉×#125	—	2	4	no	—
8	C₂×C₄⋊C₄×C₂₉	—	2	2	no	—
9	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₂₉	C₂²	1	1	yes	characteristic
12	C₂²×#6597	C₂²	1	2	yes	—
13	#11745179	—	2	2	no	—
14	#11745158	—	2	4	no	—
15	C₂²×C₄×C₂₉	—	2	2	no	—
16	C₂²×C₄×C₂₉	D₄	1	2	yes	—
17	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
18	C₂²×Dic₂₉	—	2	1	no	—
19	C₂²×Dic₂₉	—	2	1	no	—
20	C₂×Dic₅₈	—	2	2	no	—
21	C₂×Dic₅₈	—	2	2	no	—
22	C₂×Dic₅₈	—	2	1	no	—
23	C₂×Dic₅₈	Q₈	1	2	yes	—
24	C₄⋊C₄×C₂₉	—	2	8	no	—
25	C₄⋊C₄×C₂₉	—	2	2	no	—
26	C₄⋊C₄×C₂₉	D₄	1	4	yes	—
27	C₄²×C₂₉	—	2	2	no	—
28	C₄⋊Dic₂₉	—	2	2	no	—
29	C₂×#6597	—	2	2	no	—
30	C₂×#6597	—	2	2	no	—
31	C₂×#6597	—	2	2	no	—
32	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₉	—	2	2	no	—
34	C₂×C₄×C₂₉	—	2	4	no	—
35	C₂×C₄×C₂₉	—	2	2	no	—
36	C₂×C₄×C₂₉	—	2	2	no	—
37	C₂×C₄×C₂₉	—	4	2	no	—
38	C₂×C₄×C₂₉	—	4	2	no	—
39	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₂₉	#41	1	2	yes	—
42	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
43	C₂×Dic₂₉	—	2	2	no	—
44	C₂×Dic₂₉	—	4	1	no	—
45	C₂×Dic₂₉	—	4	1	no	—
46	C₂×Dic₂₉	—	4	1	no	—
47	C₂×Dic₂₉	—	4	2	no	—
48	Dic₅₈	—	2	4	no	—
49	Dic₅₈	—	4	2	no	—
50	#6597	—	4	2	no	—
51	#2026	—	29	1	no	maximal, Sylow
52	C₂²×C₂₉	#120	1	1	yes	characteristic
53	C₂²×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
54	C₂²×C₂₉	#124	1	1	yes	characteristic
55	C₂²×C₂₉	#135	1	2	yes	—
56	C₂²×C₂₉	#137	1	2	yes	—
57	C₄×C₂₉	—	4	2	no	—
58	C₄×C₂₉	—	4	4	no	—
59	C₄×C₂₉	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₂₉	#124	1	2	yes	—
61	Dic₂₉	—	4	2	no	—
62	Dic₂₉	—	8	1	no	—
63	C₂×#125	D₂₉	1	1	yes	characteristic
64	C₂×#103	—	29	2	no	—
65	#391	—	29	1	no	—
66	#340	—	29	2	no	—
67	#336	—	29	1	no	—
68	C₂×C₂₉	#393	1	1	yes	characteristic
69	C₂×C₂₉	#452	1	2	yes	—
70	C₂×C₂₉	#479	1	2	yes	—
71	C₂×C₂₉	#488	1	1	yes	characteristic
72	C₂×C₂₉	#491	1	1	yes	characteristic
73	C₂²×Q₈	—	29	1	no	—
74	C₂²×C₈	—	29	2	no	—
75	#125	—	2	4	no	—
76	C₂×C₄⋊C₄	—	2	2	no	—
77	C₂×C₄⋊C₄	—	29	1	no	—
78	C₂×C₄⋊C₄	C₂×D₂₉	1	2	yes	—
79	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
80	#103	—	58	4	no	—
81	#95	—	58	2	no	—
82	C₂₉	#2026	1	1	yes	characteristic, complemented, Sylow
83	C₂×Q₈	—	29	2	no	—
84	C₂×Q₈	—	58	1	no	—
85	C₂×Q₈	—	58	2	no	—
86	C₂×Q₈	—	58	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	58	1	no	—
89	C₂²×C₄	—	58	1	no	—
90	C₂²×C₄	C₂²⋊D₂₉	1	2	yes	—
91	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
92	C₂×C₈	—	58	2	no	—
93	C₂×C₈	—	58	2	no	—
94	C₂×C₈	—	58	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	58	2	no	—
98	C₄⋊C₄	C₂²⋊D₂₉	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
101	Q₈	—	58	4	no	—
102	Q₈	—	116	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	58	2	no	—
110	C₂×C₄	—	116	1	no	—
111	C₂×C₄	—	116	1	no	—
112	C₂×C₄	—	116	1	no	—
113	C₂×C₄	—	116	2	no	—
114	C₂×C₄	Q₈×D₂₉	1	1	yes	characteristic
115	C₂×C₄	#90809	1	2	yes	—
116	C₂×C₄	#90809	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
118	C₈	—	116	2	no	—
119	C₂²	#11745308	1	1	yes	characteristic
120	C₂²	#11745309	1	1	yes	characteristic
121	C₂²	#11745312	1	2	yes	—
122	C₂²	#11745313	1	2	yes	—
123	C₂²	#11745315	1	1	yes	characteristic
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	116	2	no	—
127	C₄	—	232	1	no	—
128	C₄	#11745308	1	2	yes	—
129	C₄	#11745309	1	2	yes	—
130	C₂	#14271074	1	1	yes	characteristic
131	C₂	#14271325	1	2	yes	—
132	C₂	#14271467	1	2	yes	—
133	C₂	#14271520	1	1	yes	characteristic
134	C₂	#14271533	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745158

C₂

yes

maximal

#14271069

C₂

yes

maximal, characteristic

#14270925

C₂

yes

maximal

#14270918

C₂

yes

maximal, characteristic

C₂×C₂₉×#125

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₂²

yes

characteristic

C₂₉×#125

—

C₂×C₄⋊C₄×C₂₉

—

C₂×C₄⋊C₄×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

C₂²

yes

characteristic

C₂²×#6597

C₂²

yes

—

#11745179

—

#11745158

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

—

C₂²×Dic₂₉

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

Q₈

yes

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

D₄

yes

—

C₄²×C₂₉

—

C₄⋊Dic₂₉

—

C₂×#6597

—

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

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₉

#41

yes

—

C₂×C₄×C₂₉

#41

yes

characteristic

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

—

—

C₂²×C₂₉

yes

C₂²×C₂₉

C₂²⋊Q₈

yes

C₂²×C₂₉

yes

C₂²×C₂₉

yes

—

C₂²×C₂₉

#137

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₂²⋊Q₈

yes

—

C₄×C₂₉

#124

yes

—

Dic₂₉

—

Dic₂₉

—

C₂×#125

D₂₉

yes

—

—

—

—

C₂×C₂₉

#393

yes

characteristic

C₂×C₂₉

#452

yes

—

C₂×C₂₉

#479

yes

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂²×Q₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₉

yes

—

C₂×C₄²

C₂×D₂₉

yes

characteristic

#103

—

#95

—

C₂₉

#2026

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₉

yes

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₂₉

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₂₉

yes

characteristic

101

Q₈

—

102

Q₈

—

116

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

116

—

111

C₂×C₄

—

116

—

112

C₂×C₄

—

116

—

113

C₂×C₄

—

116

—

114

C₂×C₄

Q₈×D₂₉

yes

characteristic

115

C₂×C₄

#90809

yes

—

116

C₂×C₄

#90809

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

118

C₈

—

116

—

119

C₂²

#11745308

yes

characteristic

120

C₂²

#11745309

yes

characteristic

121

C₂²

#11745312

yes

—

122

C₂²

#11745313

yes

—

123

C₂²

#11745315

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

116

—

127

C₄

—

232

—

128

C₄

#11745308

yes

—

129

C₄

#11745309

yes

—

130

C₂

#14271074

yes

characteristic

131

C₂

#14271325

yes

—

132

C₂

#14271467

yes

—

133

C₂

#14271520

yes

characteristic

134

C₂

#14271533

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₉
commutator, cosocle	C₂²×C₄×C₂₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂×C₂₉×#125	C₂
socle, prime radical	C₂³×C₂₉	C₂×D₄
hypercenter	C₂×#125	D₂₉
hypocenter	C₂₉	#2026
LCS	G ▷ C₂²×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₉

commutator, cosocle

C₂²×C₄×C₂₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂×C₂₉×#125

C₂

socle, prime radical

C₂³×C₂₉

C₂×D₄

D₂₉

C₂₉

G ▷ C₂²×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
82. C₂₉	#2026	semidirect

H ≅ G/N

type

82. C₂₉

#2026

semidirect

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

subgroup	index
C₂×#11745158	2
#14271069	2
#14270925	2
#14270918	2
C₂×C₂₉×#125	2
#2026	29

subgroup

index

C₂×C₂₉×#125

Subgroups — #16820640

Conjugacy classes of subgroups

Complements

Maximal subgroups