Group #16819712 — subgroups

Subgroups — #16819712

Order 3712.

252 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 89 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	#14271211	C₂	1	2	yes	maximal, complemented
2	#14271147	C₂	1	1	yes	maximal, characteristic
3	#14271130	C₂	1	1	yes	maximal, characteristic
4	#14270927	C₂	1	2	yes	maximal, complemented
5	C₂₉×#568	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₂₉	C₂²	1	1	yes	characteristic
7	C₂₉×#131	—	2	4	no	—
8	C₂×M₁₆×C₂₉	—	2	2	no	—
9	C₂×M₁₆×C₂₉	C₂²	1	2	yes	—
10	C₂×#90777	C₂²	1	1	yes	characteristic
11	C₂×#90777	C₂²	1	2	yes	—
12	C₂²×#6597	C₂²	1	1	yes	characteristic
13	#11745184	C₄	1	2	yes	—
14	#11745181	C₄	1	2	yes	—
15	C₂²⋊#6597	—	2	2	no	—
16	#11745170	—	2	4	no	—
17	#41×C₂₉	—	2	2	no	—
18	#41×C₂₉	—	2	2	no	—
19	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₂₉	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₂₉	Q₈	1	2	yes	—
22	C₂²×C₄×C₂₉	D₄	1	2	yes	—
23	C₂²×C₄×C₂₉	C₂³	1	1	yes	characteristic
24	M₁₆×C₂₉	—	2	4	no	—
25	M₁₆×C₂₉	—	2	2	no	—
26	M₁₆×C₂₉	D₄	1	4	yes	—
27	C₂×C₈×C₂₉	—	2	4	no	—
28	C₂×C₈×C₂₉	—	2	2	no	—
29	#90777	—	2	2	no	—
30	#90777	—	2	2	no	—
31	#90777	C₂×C₄	1	4	yes	—
32	C₂×#6597	—	2	2	no	—
33	C₂×#6597	—	2	1	no	—
34	C₂×#6597	—	2	1	no	—
35	C₂×#6597	—	2	2	no	—
36	C₂³×C₂₉	C₄⋊C₄	1	2	yes	—
37	C₂³×C₂₉	C₂×Q₈	1	1	yes	characteristic
38	Q₈×C₂₉	—	2	2	no	—
39	D₄×C₂₉	—	2	4	no	—
40	D₄×C₂₉	—	2	2	no	—
41	C₂×C₄×C₂₉	—	2	2	no	—
42	C₂×C₄×C₂₉	—	2	2	no	—
43	C₂×C₄×C₂₉	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉	#41	1	2	yes	—
48	C₈×C₂₉	—	2	4	no	—
49	#6597	—	2	4	no	—
50	#6597	—	4	1	no	—
51	#6597	—	4	1	no	—
52	#1858	—	29	1	no	maximal, Sylow
53	C₂²×C₂₉	—	2	2	no	—
54	C₂²×C₂₉	—	4	1	no	—
55	C₂²×C₂₉	—	4	2	no	—
56	C₂²×C₂₉	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₂₉	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₂₉	—	2	2	no	—
59	C₄×C₂₉	C₄×D₄	1	2	yes	—
60	C₄×C₂₉	#120	1	1	yes	characteristic
61	C₄×C₂₉	#123	1	1	yes	characteristic
62	#568	D₂₉	1	1	yes	characteristic
63	#430	—	29	2	no	—
64	#413	—	29	1	no	—
65	#408	—	29	1	no	—
66	#341	—	29	2	no	—
67	C₂×C₂₉	—	2	1	no	—
68	C₂×C₂₉	—	2	2	no	—
69	C₂×C₂₉	—	4	2	no	—
70	C₂×C₂₉	#385	1	1	yes	characteristic
71	C₂×#41	C₂×D₂₉	1	1	yes	characteristic
72	#131	—	2	4	no	—
73	C₂×M₁₆	—	2	2	no	—
74	C₂×M₁₆	—	29	2	no	—
75	C₂×M₁₆	—	29	1	no	—
76	C₂×M₁₆	C₂×D₂₉	1	2	yes	—
77	C₂²×C₈	—	29	1	no	—
78	#108	—	58	4	no	—
79	#101	—	29	2	no	—
80	#100	—	29	2	no	—
81	C₂²⋊C₈	—	58	2	no	—
82	C₂₉	#1858	1	1	yes	characteristic, complemented, Sylow
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₄×D₂₉	1	1	yes	characteristic
86	C₂×D₄	Dic₅₈	1	2	yes	—
87	C₂×D₄	C₄×D₂₉	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₂₉	1	2	yes	—
89	C₂²×C₄	C₂²×D₂₉	1	1	yes	characteristic
90	M₁₆	—	2	4	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	29	4	no	—
93	M₁₆	—	58	2	no	—
94	M₁₆	—	58	2	no	—
95	M₁₆	C₂²⋊D₂₉	1	4	yes	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	58	2	no	—
99	C₂×C₈	—	58	2	no	—
100	C₂×C₈	—	58	1	no	—
101	C₂×C₈	—	58	1	no	—
102	C₂³	#90779	1	2	yes	—
103	C₂³	C₂×Dic₅₈	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	#90779	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₂₉	1	1	yes	characteristic
111	C₂×C₄	#90805	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₂₉	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₂₉	1	1	yes	characteristic
114	C₈	—	2	4	no	—
115	C₈	—	58	4	no	—
116	C₈	—	116	1	no	—
117	C₈	—	116	1	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×#90779	1	1	yes	characteristic
122	C₂²	#11745286	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	C₄×C₂²⋊D₂₉	1	2	yes	—
125	C₄	#11745291	1	1	yes	characteristic
126	C₄	#11745315	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#14271036	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14271211

C₂

yes

maximal, complemented

#14271147

C₂

yes

maximal, characteristic

#14271130

C₂

yes

maximal, characteristic

#14270927

C₂

yes

maximal, complemented

C₂₉×#568

C₂

yes

maximal, characteristic

C₂×#41×C₂₉

C₂²

yes

characteristic

C₂₉×#131

—

C₂×M₁₆×C₂₉

—

C₂×M₁₆×C₂₉

C₂²

yes

—

C₂×#90777

C₂²

yes

characteristic

C₂×#90777

C₂²

yes

—

C₂²×#6597

C₂²

yes

characteristic

#11745184

C₄

yes

—

#11745181

C₄

yes

—

C₂²⋊#6597

—

#11745170

—

#41×C₂₉

—

#41×C₂₉

—

C₂×Q₈×C₂₉

C₂×C₄

yes

characteristic

C₂×D₄×C₂₉

C₂×C₄

yes

characteristic

C₂×D₄×C₂₉

Q₈

yes

—

C₂²×C₄×C₂₉

D₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

characteristic

M₁₆×C₂₉

—

M₁₆×C₂₉

—

M₁₆×C₂₉

D₄

yes

—

C₂×C₈×C₂₉

—

C₂×C₈×C₂₉

—

#90777

—

#90777

—

#90777

C₂×C₄

yes

—

—

—

—

—

C₂³×C₂₉

C₄⋊C₄

yes

—

C₂³×C₂₉

C₂×Q₈

yes

characteristic

Q₈×C₂₉

—

D₄×C₂₉

—

D₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂×D₄

yes

—

C₂×C₄×C₂₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉

#41

yes

—

C₈×C₂₉

—

—

—

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₉

C₂²⋊Q₈

yes

—

C₄×C₂₉

—

C₄×C₂₉

C₄×D₄

yes

—

C₄×C₂₉

yes

C₄×C₂₉

yes

D₂₉

yes

—

—

—

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

—

C₂×C₂₉

#385

yes

characteristic

C₂×#41

C₂×D₂₉

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₉

yes

—

C₂²×C₈

—

#108

—

#101

—

#100

—

C₂²⋊C₈

—

C₂₉

#1858

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

C₄×D₂₉

yes

characteristic

C₂×D₄

Dic₅₈

yes

—

C₂×D₄

C₄×D₂₉

yes

characteristic

C₂²×C₄

C₂²⋊D₂₉

yes

—

C₂²×C₄

C₂²×D₂₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂²⋊D₂₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂³

#90779

yes

—

103

C₂³

C₂×Dic₅₈

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#90779

yes

—

110

C₂×C₄

C₂×C₄×D₂₉

yes

characteristic

111

C₂×C₄

#90805

yes

—

112

C₂×C₄

C₂×C₂²⋊D₂₉

yes

—

113

C₂×C₄

C₂×C₂²⋊D₂₉

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

116

—

117

C₈

—

116

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×#90779

yes

characteristic

122

C₂²

#11745286

yes

—

123

C₄

—

124

C₄

C₄×C₂²⋊D₂₉

yes

—

125

C₄

#11745291

yes

characteristic

126

C₄

#11745315

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#14271036

yes

characteristic

	subgroup	G/H
center	C₄	#11745315
commutator	C₂×C₄×C₂₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉
Fitting	C₂₉×#568	C₂
socle	C₂×C₂₉	#385
prime radical	C₂×#41×C₂₉	C₂²
cosocle	C₂²×C₄×C₂₉	C₂³
hypercenter	#568	D₂₉
hypocenter	C₂₉	#1858
LCS	G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₄

#11745315

commutator

C₂×C₄×C₂₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₉

Fitting

C₂₉×#568

C₂

socle

C₂×C₂₉

#385

prime radical

C₂×#41×C₂₉

C₂²

cosocle

C₂²×C₄×C₂₉

C₂³

D₂₉

C₂₉

G ▷ C₂×C₄×C₂₉ ▷ C₂×C₂₉ ▷ C₂₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
1. #14271211	C₂	semidirect
4. #14270927	C₂	semidirect
82. C₂₉	#1858	semidirect

H ≅ G/N

type

1. #14271211

C₂

semidirect

4. #14270927

C₂

semidirect

82. C₂₉

#1858

semidirect

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

subgroup	index
#14271211	2
#14271147	2
#14271130	2
#14270927	2
C₂₉×#568	2
#1858	29

subgroup

index

C₂₉×#568

Subgroups — #16819712

Conjugacy classes of subgroups

Complements

Maximal subgroups