Group #16820760 — subgroups

Subgroups — #16820760

Order 3712.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×Dic₁₁₆	C₂	1	1	yes	maximal, characteristic
2	#14271171	C₂	1	2	yes	maximal
3	C₂×#11745172	C₂	1	2	yes	maximal
4	#14270928	C₂	1	1	yes	maximal, characteristic
5	C₂₉×#424	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₈	C₂²	1	2	yes	—
7	C₂×M₁₆×C₂₉	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₂₉	C₂²	1	1	yes	characteristic
9	C₂₉×#117	C₂²	1	1	yes	characteristic
10	C₂×#90777	C₂²	1	2	yes	—
11	C₂×Dic₁₁₆	—	2	4	no	—
12	C₂×Dic₁₁₆	—	2	2	no	—
13	C₂₉×C₄⋊C₈	—	2	2	no	—
14	#11745172	—	2	4	no	—
15	#11745158	—	2	4	no	—
16	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
17	C₂²×Dic₂₉	—	2	2	no	—
18	C₂×Dic₅₈	—	2	2	no	—
19	C₂×Dic₅₈	—	2	2	no	—
20	C₂×Dic₅₈	—	2	2	no	—
21	C₂×Dic₅₈	—	2	2	no	—
22	C₂×Dic₅₈	—	2	2	no	—
23	C₂×Dic₅₈	D₄	1	4	yes	—
24	M₁₆×C₂₉	—	2	2	no	—
25	C₂×C₈×C₂₉	—	2	2	no	—
26	C₂×C₈×C₂₉	—	2	1	no	—
27	C₂×C₈×C₂₉	D₄	1	2	yes	—
28	C₄⋊C₄×C₂₉	D₄	1	2	yes	—
29	C₂²⋊C₄×C₂₉	—	2	1	no	—
30	C₄²×C₂₉	—	2	1	no	—
31	#90777	—	2	2	no	—
32	#90777	—	2	2	no	—
33	C₂×#6597	—	2	2	no	—
34	Dic₁₁₆	—	2	8	no	—
35	C₂³×C₂₉	#41	1	1	yes	characteristic
36	C₂×C₄×C₂₉	—	2	2	no	—
37	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₉	#41	1	2	yes	—
41	C₂×Dic₂₉	—	2	4	no	—
42	C₂×Dic₂₉	—	4	2	no	—
43	C₂×Dic₂₉	—	4	2	no	—
44	Dic₅₈	—	2	4	no	—
45	Dic₅₈	—	2	4	no	—
46	Dic₅₈	—	4	2	no	—
47	Dic₅₈	—	4	2	no	—
48	C₈×C₂₉	—	2	2	no	—
49	C₈×C₂₉	—	4	1	no	—
50	#6597	—	4	2	no	—
51	#2047	—	29	1	no	maximal, Sylow
52	C₂²×C₂₉	—	2	2	no	—
53	C₂²×C₂₉	#121	1	1	yes	characteristic
54	C₂²×C₂₉	#123	1	2	yes	—
55	C₄×C₂₉	—	4	2	no	—
56	C₄×C₂₉	#120	1	1	yes	characteristic
57	C₄×C₂₉	#121	1	2	yes	—
58	C₄×C₂₉	#124	1	1	yes	characteristic
59	Dic₂₉	—	4	4	no	—
60	C₂²×Q₁₆	—	29	1	no	—
61	#424	D₂₉	1	1	yes	characteristic
62	#419	—	29	2	no	—
63	C₂×#101	—	29	2	no	—
64	#341	—	29	1	no	—
65	C₂×C₂₉	—	2	2	no	—
66	C₂×C₂₉	#394	1	1	yes	characteristic
67	C₂×C₂₉	#473	1	2	yes	—
68	C₂²×Q₈	—	29	2	no	—
69	C₂×Q₁₆	—	58	4	no	—
70	C₂×Q₁₆	—	58	2	no	—
71	C₂×M₁₆	—	29	2	no	—
72	C₂×M₁₆	C₂×D₂₉	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₂₉	1	1	yes	characteristic
74	#117	C₂×D₂₉	1	1	yes	characteristic
75	C₄⋊C₈	—	2	2	no	—
76	#103	—	58	4	no	—
77	#101	—	58	4	no	—
78	C₂₉	#2047	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₈	—	29	4	no	—
80	C₂×Q₈	—	58	2	no	—
81	C₂×Q₈	—	58	2	no	—
82	C₂×Q₈	—	58	2	no	—
83	C₂×Q₈	—	58	2	no	—
84	C₂×Q₈	—	58	2	no	—
85	C₂²×C₄	—	58	2	no	—
86	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
87	Q₁₆	—	58	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	58	2	no	—
90	M₁₆	—	58	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	58	2	no	—
94	C₂×C₈	C₂²⋊D₂₉	1	2	yes	—
95	C₄⋊C₄	D₁₁₆	1	2	yes	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	#90805	1	1	yes	characteristic
99	Q₈	—	58	4	no	—
100	Q₈	—	58	4	no	—
101	Q₈	—	116	2	no	—
102	Q₈	—	116	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	58	4	no	—
105	C₂×C₄	—	116	2	no	—
106	C₂×C₄	—	116	2	no	—
107	C₂×C₄	C₂×D₁₁₆	1	1	yes	characteristic
108	C₂×C₄	D₄×D₂₉	1	2	yes	—
109	C₂×C₄	#90809	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	116	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#11745256	1	2	yes	—
116	C₂²	#11745292	1	1	yes	characteristic
117	C₄	—	4	2	no	—
118	C₄	—	116	4	no	—
119	C₄	#11745257	1	2	yes	—
120	C₄	#11745299	1	1	yes	characteristic
121	C₄	#11745309	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#14271080	1	1	yes	characteristic
124	C₂	#14271438	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₁₁₆

C₂

yes

maximal, characteristic

#14271171

C₂

yes

maximal

C₂×#11745172

C₂

yes

maximal

#14270928

C₂

yes

maximal, characteristic

C₂₉×#424

C₂

yes

maximal, characteristic

C₂²×Dic₅₈

C₂²

yes

—

C₂×M₁₆×C₂₉

C₂²

yes

characteristic

C₂²×C₈×C₂₉

C₂²

yes

characteristic

C₂₉×#117

C₂²

yes

characteristic

C₂×#90777

C₂²

yes

—

C₂×Dic₁₁₆

—

C₂×Dic₁₁₆

—

C₂₉×C₄⋊C₈

—

#11745172

—

#11745158

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

C₂²×Dic₂₉

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

—

C₂×Dic₅₈

D₄

yes

—

M₁₆×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

D₄

yes

—

C₄⋊C₄×C₂₉

D₄

yes

—

C₂²⋊C₄×C₂₉

—

C₄²×C₂₉

—

#90777

—

#90777

—

C₂×#6597

—

Dic₁₁₆

—

C₂³×C₂₉

#41

yes

characteristic

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

#41

yes

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

C₂×Dic₂₉

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

Dic₅₈

—

C₈×C₂₉

—

C₈×C₂₉

—

#6597

—

#2047

—

maximal, Sylow

C₂²×C₂₉

—

C₂²×C₂₉

#121

yes

characteristic

C₂²×C₂₉

#123

yes

—

C₄×C₂₉

—

C₄×C₂₉

#120

yes

characteristic

C₄×C₂₉

#121

yes

—

C₄×C₂₉

#124

yes

characteristic

Dic₂₉

—

C₂²×Q₁₆

—

#424

D₂₉

yes

—

—

—

C₂×C₂₉

—

C₂×C₂₉

#394

yes

characteristic

C₂×C₂₉

#473

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₉

yes

characteristic

C₂²×C₈

C₂×D₂₉

yes

characteristic

#117

C₂×D₂₉

yes

characteristic

C₄⋊C₈

—

#103

—

#101

—

C₂₉

#2047

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₂₉

yes

—

C₄⋊C₄

D₁₁₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#90805

yes

characteristic

Q₈

—

100

Q₈

—

101

Q₈

—

116

—

102

Q₈

—

116

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

116

—

106

C₂×C₄

—

116

—

107

C₂×C₄

C₂×D₁₁₆

yes

characteristic

108

C₂×C₄

D₄×D₂₉

yes

—

109

C₂×C₄

#90809

yes

—

110

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

116

—

114

C₂²

—

115

C₂²

#11745256

yes

—

116

C₂²

#11745292

yes

characteristic

117

C₄

—

118

C₄

—

116

—

119

C₄

#11745257

yes

—

120

C₄

#11745299

yes

characteristic

121

C₄

#11745309

yes

characteristic

122

C₂

—

123

C₂

#14271080

yes

characteristic

124

C₂

#14271438

yes

—

	subgroup	G/H
center	C₂²	#11745292
commutator, cosocle	C₂²×C₄×C₂₉	C₂³
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂₉×#424	C₂
socle	C₂²×C₂₉	#121
prime radical	C₂³×C₂₉	#41
hypercenter	#424	D₂₉
hypocenter	C₂₉	#2047
LCS	G ▷ C₂²×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#11745292

commutator, cosocle

C₂²×C₄×C₂₉

C₂³

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂₉×#424

C₂

C₂²×C₂₉

C₂³×C₂₉

D₂₉

C₂₉

G ▷ C₂²×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
78. C₂₉	#2047	semidirect

H ≅ G/N

type

78. C₂₉

#2047

semidirect

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

subgroup	index
C₂²×Dic₁₁₆	2
#14271171	2
C₂×#11745172	2
#14270928	2
C₂₉×#424	2
#2047	29

subgroup

index

C₂²×Dic₁₁₆

C₂₉×#424

Subgroups — #16820760

Conjugacy classes of subgroups

Complements

Maximal subgroups