Group #16820231 — subgroups

Subgroups — #16820231

Order 3712.

240 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 113 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×#11745165	C₂	1	1	yes	maximal, characteristic
2	C₂×#11745163	C₂	1	1	yes	maximal, characteristic
3	#14271062	C₂	1	2	yes	maximal
4	#14270925	C₂	1	2	yes	maximal
5	C₂×C₂₉×#107	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₂₉	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₂₉	—	2	2	no	—
9	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₉	C₂²	1	1	yes	characteristic
11	C₂²×#6597	C₂²	1	1	yes	characteristic
12	C₂₉×#107	C₄	1	4	yes	complemented
13	#11745179	—	2	2	no	—
14	#11745165	—	2	2	no	—
15	#11745163	—	2	2	no	—
16	C₂²×C₄×C₂₉	—	2	2	no	—
17	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
18	C₂²×Dic₂₉	—	2	2	no	—
19	C₂²×Dic₂₉	D₄	1	2	yes	—
20	C₂×C₈×C₂₉	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₂₉	D₄	1	2	yes	—
22	C₂×C₈×C₂₉	Q₈	1	2	yes	—
23	C₄⋊C₄×C₂₉	—	2	2	no	—
24	C₄⋊C₄×C₂₉	C₂×C₄	1	4	yes	—
25	C₄⋊Dic₂₉	—	2	2	no	—
26	C₄⋊Dic₂₉	—	2	2	no	—
27	C₄⋊Dic₂₉	—	4	4	no	—
28	C₄×Dic₂₉	—	2	1	no	—
29	C₄×Dic₂₉	—	2	1	no	—
30	C₂×#6597	—	2	1	no	—
31	C₂×#6597	—	2	1	no	—
32	C₂×#6597	Q₈	1	2	yes	—
33	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₉	—	2	4	no	—
35	C₂×C₄×C₂₉	—	4	2	no	—
36	C₂×C₄×C₂₉	—	4	2	no	—
37	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₂₉	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
42	C₂×C₄×C₂₉	#41	1	1	yes	characteristic
43	C₂×Dic₂₉	—	2	2	no	—
44	C₂×Dic₂₉	—	2	2	no	—
45	C₂×Dic₂₉	—	2	2	no	—
46	C₂×Dic₂₉	—	4	2	no	—
47	C₂×Dic₂₉	—	4	2	no	—
48	C₂×Dic₂₉	—	4	2	no	—
49	C₈×C₂₉	C₄⋊C₄	1	4	yes	complemented
50	#6597	—	2	2	no	—
51	#1943	—	29	1	no	maximal, Sylow
52	C₂²×C₂₉	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₂₉	#121	1	1	yes	characteristic
54	C₂²×C₂₉	C₄⋊Q₈	1	1	yes	characteristic
55	C₂²×C₂₉	AGL(1,ℤ/8)	1	2	yes	—
56	C₂²×C₂₉	#137	1	2	yes	—
57	C₄×C₂₉	—	4	4	no	—
58	C₄×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₄×C₂₉	C₄×Q₈	1	1	yes	characteristic
60	C₄×C₂₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₄×C₂₉	#125	1	1	yes	characteristic
62	Dic₂₉	—	4	2	no	—
63	Dic₂₉	—	8	2	no	—
64	C₂×#107	D₂₉	1	1	yes	characteristic
65	C₂×#106	—	29	1	no	—
66	C₂×#97	—	29	1	no	—
67	#389	—	29	2	no	—
68	#340	—	29	2	no	—
69	C₂×C₂₉	#389	1	1	yes	characteristic
70	C₂×C₂₉	#441	1	1	yes	characteristic
71	C₂×C₂₉	#442	1	1	yes	characteristic
72	C₂×C₂₉	#468	1	2	yes	—
73	C₂×C₂₉	#501	1	2	yes	—
74	C₂²×C₈	—	29	1	no	—
75	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	29	2	no	—
77	C₂×C₄⋊C₄	—	58	2	no	—
78	C₂×C₄⋊C₄	C₂×D₂₉	1	2	yes	—
79	C₂×C₄²	—	29	1	no	—
80	#107	Dic₂₉	1	4	yes	complemented
81	#106	—	58	2	no	—
82	#97	—	58	2	no	—
83	#95	—	58	2	no	—
84	C₂₉	#1943	1	1	yes	characteristic, complemented, Sylow
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	29	2	no	—
87	C₂²×C₄	—	58	2	no	—
88	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
89	C₂×C₈	—	29	2	no	—
90	C₂×C₈	—	58	1	no	—
91	C₂×C₈	—	58	1	no	—
92	C₂×C₈	Dic₅₈	1	2	yes	—
93	C₂×C₈	D₁₁₆	1	2	yes	—
94	C₂×C₈	C₂×Dic₂₉	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	58	2	no	—
97	C₄⋊C₄	—	58	2	no	—
98	C₄⋊C₄	—	116	4	no	—
99	C₄⋊C₄	C₂×Dic₂₉	1	4	yes	—
100	C₄²	—	58	1	no	—
101	C₄²	—	58	1	no	—
102	C₂³	D₄×D₂₉	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	58	2	no	—
107	C₂×C₄	—	58	2	no	—
108	C₂×C₄	—	58	2	no	—
109	C₂×C₄	—	116	2	no	—
110	C₂×C₄	—	116	2	no	—
111	C₂×C₄	—	116	2	no	—
112	C₂×C₄	C₂×Dic₅₈	1	1	yes	characteristic
113	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
114	C₂×C₄	#90807	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₂₉	1	1	yes	characteristic
116	C₂×C₄	#90809	1	1	yes	characteristic
117	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
118	C₈	—	58	2	no	—
119	C₈	C₄⋊Dic₂₉	1	4	yes	complemented
120	C₂²	#11745250	1	1	yes	characteristic
121	C₂²	#11745257	1	1	yes	characteristic
122	C₂²	#11745273	1	2	yes	—
123	C₂²	#11745280	1	2	yes	—
124	C₂²	D₄×Dic₂₉	1	1	yes	characteristic
125	C₄	—	4	4	no	—
126	C₄	—	116	2	no	—
127	C₄	—	232	2	no	—
128	C₄	#11745252	1	1	yes	characteristic
129	C₄	#11745259	1	1	yes	characteristic
130	C₄	C₂×C₄⋊Dic₂₉	1	1	yes	characteristic
131	C₄	Q₈×Dic₂₉	1	1	yes	characteristic
132	C₂	#14271062	1	1	yes	characteristic
133	C₂	#14271270	1	1	yes	characteristic
134	C₂	#14271275	1	1	yes	characteristic
135	C₂	#14271410	1	2	yes	—
136	C₂	#14271571	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745165

C₂

yes

maximal, characteristic

C₂×#11745163

C₂

yes

maximal, characteristic

#14271062

C₂

yes

maximal

#14270925

C₂

yes

maximal

C₂×C₂₉×#107

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₉

C₂²

yes

—

C₂×C₄⋊Dic₂₉

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

C₂×C₄×Dic₂₉

C₂²

yes

characteristic

C₂²×#6597

C₂²

yes

characteristic

C₂₉×#107

C₄

yes

—

—

—

C₂²×C₄×C₂₉

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

—

C₂²×Dic₂₉

D₄

yes

—

C₂×C₈×C₂₉

C₂×C₄

yes

—

C₂×C₈×C₂₉

D₄

yes

—

C₂×C₈×C₂₉

Q₈

yes

—

C₄⋊C₄×C₂₉

—

C₄⋊C₄×C₂₉

C₂×C₄

yes

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄×Dic₂₉

—

C₄×Dic₂₉

—

C₂×#6597

—

C₂×#6597

—

C₂×#6597

Q₈

yes

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₉

C₂×Q₈

yes

C₂×C₄×C₂₉

yes

C₂×C₄×C₂₉

yes

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₈×C₂₉

C₄⋊C₄

yes

—

—

C₂²×C₂₉

C₄×D₄

yes

characteristic

C₂²×C₂₉

#121

yes

characteristic

C₂²×C₂₉

C₄⋊Q₈

yes

characteristic

C₂²×C₂₉

AGL(1,ℤ/8)

yes

—

C₂²×C₂₉

#137

yes

—

C₄×C₂₉

—

C₄×C₂₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₉

C₄×Q₈

yes

characteristic

C₄×C₂₉

C₂²⋊Q₈

yes

characteristic

C₄×C₂₉

#125

yes

characteristic

Dic₂₉

—

Dic₂₉

—

C₂×#107

D₂₉

yes

characteristic

C₂×#106

—

C₂×#97

—

#389

—

#340

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

—

C₂×C₂₉

#501

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₂₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₉

yes

—

C₂×C₄²

—

#107

Dic₂₉

yes

—

—

—

C₂₉

#1943

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₅₈

yes

—

C₂×C₈

D₁₁₆

yes

—

C₂×C₈

C₂×Dic₂₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

116

—

C₄⋊C₄

C₂×Dic₂₉

yes

—

100

C₄²

—

101

C₄²

—

102

C₂³

D₄×D₂₉

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

116

—

110

C₂×C₄

—

116

—

111

C₂×C₄

—

116

—

112

C₂×C₄

C₂×Dic₅₈

yes

characteristic

113

C₂×C₄

C₂×D₁₁₆

yes

characteristic

114

C₂×C₄

#90807

yes

characteristic

115

C₂×C₄

Q₈×D₂₉

yes

characteristic

116

C₂×C₄

#90809

yes

characteristic

117

C₂×C₄

C₂²×Dic₂₉

yes

characteristic

118

C₈

—

119

C₈

C₄⋊Dic₂₉

yes

complemented

120

C₂²

#11745250

yes

characteristic

121

C₂²

#11745257

yes

characteristic

122

C₂²

#11745273

yes

—

123

C₂²

#11745280

yes

—

124

C₂²

D₄×Dic₂₉

yes

characteristic

125

C₄

—

126

C₄

—

116

—

127

C₄

—

232

—

128

C₄

#11745252

yes

characteristic

129

C₄

#11745259

yes

characteristic

130

C₄

C₂×C₄⋊Dic₂₉

yes

characteristic

131

C₄

Q₈×Dic₂₉

yes

characteristic

132

C₂

#14271062

yes

characteristic

133

C₂

#14271270

yes

characteristic

134

C₂

#14271275

yes

characteristic

135

C₂

#14271410

yes

—

136

C₂

#14271571

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₉
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂×C₂₉×#107	C₂
socle, prime radical	C₂³×C₂₉	C₂×D₄
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	C₂×#107	D₂₉
hypocenter	C₂₉	#1943
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₂₉

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂×C₂₉×#107

C₂

socle, prime radical

C₂³×C₂₉

C₂×D₄

cosocle

C₂²×C₄×C₂₉

C₂³

D₂₉

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
12. C₂₉×#107	C₄	semidirect
49. C₈×C₂₉	C₄⋊C₄	semidirect
80. #107	Dic₂₉	semidirect
84. C₂₉	#1943	semidirect
119. C₈	C₄⋊Dic₂₉	semidirect

H ≅ G/N

type

12. C₂₉×#107

C₄

semidirect

49. C₈×C₂₉

C₄⋊C₄

semidirect

80. #107

Dic₂₉

semidirect

84. C₂₉

#1943

semidirect

119. C₈

C₄⋊Dic₂₉

semidirect

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

subgroup	index
C₂×#11745165	2
C₂×#11745163	2
#14271062	2
#14270925	2
C₂×C₂₉×#107	2
#1943	29

subgroup

index

C₂×C₂₉×#107

Subgroups — #16820231

Conjugacy classes of subgroups

Complements

Maximal subgroups