Group #16820465 — subgroups

Subgroups — #16820465

Order 3712.

276 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 109 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	#14271029	C₂	1	1	yes	maximal, characteristic
3	#14270926	C₂	1	4	yes	maximal
4	C₂×C₂₉×#97	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈	—	2	1	no	—
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	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₉	C₂²	1	2	yes	—
11	#11745231	—	2	2	no	—
12	#11745231	C₄	1	4	yes	complemented
13	C₂₉×#97	—	2	2	no	—
14	#11745179	—	2	1	no	—
15	#11745179	—	2	1	no	—
16	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₉	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
19	C₂²×Dic₂₉	—	2	2	no	—
20	C₂²×Dic₂₉	—	2	2	no	—
21	C₂×Dic₅₈	—	2	4	no	—
22	C₂×Dic₅₈	—	4	4	no	—
23	C₂×C₈×C₂₉	—	2	2	no	—
24	C₂×C₈×C₂₉	—	2	2	no	—
25	C₂×C₈×C₂₉	Q₈	1	4	yes	complemented
26	C₄²×C₂₉	C₂×C₄	1	2	yes	—
27	C₄²×C₂₉	D₄	1	2	yes	—
28	C₄⋊Dic₂₉	—	2	4	no	—
29	C₄⋊Dic₂₉	—	2	2	no	—
30	C₄⋊Dic₂₉	—	2	4	no	—
31	C₄⋊Dic₂₉	C₂×C₄	1	4	yes	—
32	C₂³×C₂₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₉	—	2	2	no	—
34	C₂×C₄×C₂₉	—	2	2	no	—
35	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₂₉	C₂²⋊C₄	1	2	yes	—
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	2	yes	—
40	C₂×C₄×C₂₉	#41	1	2	yes	—
41	C₂×Dic₂₉	—	2	4	no	—
42	C₂×Dic₂₉	—	4	4	no	—
43	C₂×Dic₂₉	—	4	2	no	—
44	C₂×Dic₂₉	—	4	2	no	—
45	C₂×Dic₂₉	—	4	2	no	—
46	Dic₅₈	—	4	8	no	—
47	C₈×C₂₉	—	2	4	no	—
48	#1993	—	29	1	no	maximal, Sylow
49	C₂²×C₂₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₂₉	#124	1	1	yes	characteristic
51	C₂²×C₂₉	C₄⋊Q₈	1	1	yes	characteristic
52	C₂²×C₂₉	AGL(1,ℤ/8)	1	2	yes	—
53	C₂²×C₂₉	#137	1	2	yes	—
54	C₄×C₂₉	—	2	2	no	—
55	C₄×C₂₉	—	2	2	no	—
56	C₄×C₂₉	C₄×Q₈	1	2	yes	—
57	C₄×C₂₉	C₂²⋊Q₈	1	2	yes	—
58	Dic₂₉	—	4	4	no	—
59	Dic₂₉	—	8	2	no	—
60	C₂×C₄⋊Q₈	—	29	1	no	—
61	C₂×#97	D₂₉	1	1	yes	characteristic
62	#382	—	29	1	no	—
63	#340	—	29	4	no	—
64	C₂×C₂₉	#391	1	1	yes	characteristic
65	C₂×C₂₉	#418	1	1	yes	characteristic
66	C₂×C₂₉	#419	1	1	yes	characteristic
67	C₂×C₂₉	#489	1	2	yes	—
68	C₂×C₂₉	#501	1	2	yes	—
69	C₂²×Q₈	—	58	1	no	—
70	C₂²×C₈	C₂×D₂₉	1	2	yes	—
71	C₄⋊Q₈	—	29	4	no	—
72	C₄⋊Q₈	—	58	2	no	—
73	C₂×C₄⋊C₄	—	29	2	no	—
74	C₂×C₄⋊C₄	—	29	2	no	—
75	C₂×C₄⋊C₄	—	58	1	no	—
76	C₂×C₄²	C₂×D₂₉	1	1	yes	characteristic
77	#97	—	2	2	no	—
78	#95	—	58	1	no	—
79	#95	—	58	1	no	—
80	C₂₉	#1993	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₈	—	58	4	no	—
82	C₂×Q₈	—	116	4	no	—
83	C₂²×C₄	—	58	2	no	—
84	C₂²×C₄	—	58	2	no	—
85	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
86	C₂²×C₄	D₁₁₆	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	Dic₅₈	1	4	yes	complemented
91	C₄⋊C₄	—	29	4	no	—
92	C₄⋊C₄	—	58	4	no	—
93	C₄⋊C₄	—	58	2	no	—
94	C₄⋊C₄	—	58	4	no	—
95	C₄²	C₄×D₂₉	1	2	yes	—
96	C₄²	C₂²⋊D₂₉	1	2	yes	—
97	C₂³	C₂×D₁₁₆	1	1	yes	characteristic
98	Q₈	—	116	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	58	4	no	—
102	C₂×C₄	—	116	4	no	—
103	C₂×C₄	—	116	2	no	—
104	C₂×C₄	—	116	2	no	—
105	C₂×C₄	—	116	2	no	—
106	C₂×C₄	#90781	1	2	yes	—
107	C₂×C₄	#90781	1	2	yes	—
108	C₂×C₄	C₂×Dic₅₈	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
110	C₂×C₄	#90805	1	2	yes	—
111	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
112	C₈	—	2	4	no	—
113	C₂²	#11745231	1	1	yes	characteristic
114	C₂²	#11745237	1	1	yes	characteristic
115	C₂²	#11745270	1	2	yes	—
116	C₂²	#11745271	1	2	yes	—
117	C₂²	C₂×#90781	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	116	4	no	—
121	C₄	—	232	2	no	—
122	C₄	C₄×Dic₅₈	1	2	yes	—
123	C₄	#11745286	1	2	yes	—
124	C₂	#14271069	1	1	yes	characteristic
125	C₂	#14271169	1	1	yes	characteristic
126	C₂	#14271173	1	1	yes	characteristic
127	C₂	#14271523	1	2	yes	—
128	C₂	#14271572	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11745231

C₂

yes

maximal, characteristic

#14271029

C₂

yes

maximal, characteristic

#14270926

C₂

yes

maximal

C₂×C₂₉×#97

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

—

C₂²×C₈×C₂₉

C₂²

yes

—

C₂×C₄²×C₂₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₉

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

C₂×C₄⋊Dic₂₉

C₂²

yes

—

#11745231

—

#11745231

C₄

yes

complemented

C₂₉×#97

—

#11745179

—

#11745179

—

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×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

Q₈

yes

complemented

C₄²×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

D₄

yes

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

—

C₄⋊Dic₂₉

C₂×C₄

yes

—

C₂³×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

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

—

Dic₅₈

—

C₈×C₂₉

—

#1993

—

maximal, Sylow

C₂²×C₂₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉

#124

yes

characteristic

C₂²×C₂₉

C₄⋊Q₈

yes

characteristic

C₂²×C₂₉

AGL(1,ℤ/8)

yes

—

C₂²×C₂₉

#137

yes

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₄×Q₈

yes

—

C₄×C₂₉

C₂²⋊Q₈

yes

—

Dic₂₉

—

Dic₂₉

—

C₂×C₄⋊Q₈

—

C₂×#97

D₂₉

yes

characteristic

#382

—

#340

—

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

C₂×C₂₉

yes

—

C₂×C₂₉

#501

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₂₉

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₉

yes

—

—

—

C₂₉

#1993

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

116

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₁₆

yes

characteristic

C₂²×C₄

D₁₁₆

yes

characteristic

C₂²×C₄

C₂²×D₂₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₅₈

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₂₉

yes

—

C₄²

C₂²⋊D₂₉

yes

—

C₂³

C₂×D₁₁₆

yes

characteristic

Q₈

—

116

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

116

—

103

C₂×C₄

—

116

—

104

C₂×C₄

—

116

—

105

C₂×C₄

—

116

—

106

C₂×C₄

#90781

yes

—

107

C₂×C₄

#90781

yes

—

108

C₂×C₄

C₂×Dic₅₈

yes

—

109

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

110

C₂×C₄

#90805

yes

—

111

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

112

C₈

—

113

C₂²

#11745231

yes

characteristic

114

C₂²

#11745237

yes

characteristic

115

C₂²

#11745270

yes

—

116

C₂²

#11745271

yes

—

117

C₂²

C₂×#90781

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

116

—

121

C₄

—

232

—

122

C₄

C₄×Dic₅₈

yes

—

123

C₄

#11745286

yes

—

124

C₂

#14271069

yes

characteristic

125

C₂

#14271169

yes

characteristic

126

C₂

#14271173

yes

characteristic

127

C₂

#14271523

yes

—

128

C₂

#14271572

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₁₆
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂×C₂₉×#97	C₂
socle, prime radical	C₂³×C₂₉	C₂×D₄
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	C₂×#97	D₂₉
hypocenter	C₂₉	#1993
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₁₆

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂×C₂₉×#97

C₂

socle, prime radical

C₂³×C₂₉

C₂×D₄

cosocle

C₂²×C₄×C₂₉

C₂³

hypercenter

C₂×#97

D₂₉

hypocenter

C₂₉

#1993

LCS

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
12. #11745231	C₄	semidirect
25. C₂×C₈×C₂₉	Q₈	semidirect
80. C₂₉	#1993	semidirect
90. C₂×C₈	Dic₅₈	semidirect

H ≅ G/N

type

12. #11745231

C₄

semidirect

25. C₂×C₈×C₂₉

Q₈

semidirect

80. C₂₉

#1993

semidirect

90. C₂×C₈

Dic₅₈

semidirect

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

subgroup	index
C₂×#11745231	2
#14271029	2
#14270926	2
C₂×C₂₉×#97	2
#1993	29

subgroup

index

C₂×C₂₉×#97

Subgroups — #16820465

Conjugacy classes of subgroups

Complements

Maximal subgroups