Group #15774805 — subgroups

Subgroups — #15774805

Order 3328.

246 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 79 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	#13116713	C₂	1	1	yes	maximal, characteristic
2	#13113661	C₂	1	2	yes	maximal, complemented
3	#13112902	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112804	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₀₈	C₄	1	1	yes	characteristic
7	C₄×D₁₀₄	C₂²	1	2	yes	—
8	C₈×D₅₂	—	2	2	no	—
9	C₁₃×#346	C₂²	1	1	yes	characteristic
10	C₁₃×#346	C₂²	1	1	yes	characteristic
11	C₁₃×#322	C₂²	1	1	yes	characteristic
12	#11718101	C₄	1	2	yes	—
13	#11718097	C₄	1	1	yes	characteristic
14	#11718051	—	2	2	no	—
15	#11718049	C₂²	1	2	yes	—
16	C₂×D₁₀₄	C₂×C₄	1	2	yes	—
17	C₂×C₈×D₁₃	—	4	2	no	—
18	C₄×D₅₂	—	2	2	no	—
19	C₂×C₁₃×C₁₆	—	2	1	no	—
20	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₁₃	D₄	1	2	yes	—
23	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
24	#88061	—	2	2	no	—
25	#88060	—	4	2	no	—
26	#88058	C₂×C₄	1	2	yes	—
27	C₄⋊#1055	—	2	2	no	—
28	D₂₀₈	C₈	1	4	yes	complemented
29	#12067	—	13	1	no	maximal, Sylow
30	C₂×D₅₂	—	2	2	no	—
31	C₂×C₄×D₁₃	—	4	2	no	—
32	C₂×C₈×C₁₃	—	2	2	no	—
33	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
35	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
36	#6103	—	4	2	no	—
37	C₄⋊Dic₁₃	—	2	2	no	—
38	C₂×#1055	—	4	2	no	—
39	D₁₀₄	—	2	2	no	—
40	D₁₀₄	C₂×C₈	1	4	yes	complemented
41	C₈×D₁₃	—	8	4	no	—
42	C₁₃×C₁₆	—	2	1	no	—
43	C₁₃×C₁₆	—	4	1	no	—
44	C₁₃×C₁₆	C₂×C₈	1	2	yes	complemented
45	#2177	—	13	1	no	—
46	#1580	—	13	2	no	—
47	#1368	—	13	1	no	—
48	#1333	—	13	2	no	—
49	#1313	D₁₃	1	1	yes	characteristic, complemented
50	C₂²×D₁₃	—	4	2	no	—
51	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
53	C₂×Dic₁₃	—	4	2	no	—
54	D₅₂	—	2	4	no	—
55	D₅₂	—	4	2	no	—
56	C₄×D₁₃	—	8	2	no	—
57	C₄×D₁₃	—	8	2	no	—
58	C₈×C₁₃	—	2	1	no	—
59	C₈×C₁₃	—	4	2	no	—
60	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
61	C₈×C₁₃	#131	1	1	yes	characteristic
62	#1055	—	8	2	no	—
63	C₂×D₁₆	—	13	1	no	—
64	C₄×D₈	—	13	2	no	—
65	C₈×D₄	—	26	2	no	—
66	#367	—	13	1	no	—
67	#357	—	13	2	no	—
68	#346	C₂×D₁₃	1	1	yes	characteristic
69	#346	C₂×D₁₃	1	1	yes	characteristic
70	#334	—	13	2	no	—
71	D₄⋊C₈	—	26	2	no	—
72	#322	C₂×D₁₃	1	1	yes	characteristic
73	C₂²×C₁₃	#442	1	1	yes	characteristic
74	C₂×D₁₃	—	4	4	no	—
75	C₂×D₁₃	—	8	2	no	—
76	C₂×D₁₃	—	8	2	no	—
77	C₄×C₁₃	—	2	2	no	—
78	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
79	C₄×C₁₃	#444	1	1	yes	characteristic
80	Dic₁₃	—	8	2	no	—
81	C₂×D₈	—	13	2	no	—
82	C₂²×C₈	—	52	2	no	—
83	C₄×D₄	—	26	2	no	—
84	D₁₆	—	13	4	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
87	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
88	#107	—	13	2	no	—
89	C₄⋊C₈	—	26	2	no	—
90	#102	—	26	2	no	—
91	C₂²⋊C₈	—	52	2	no	—
92	C₄×C₈	D₅₂	1	2	yes	—
93	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
94	C₂×C₁₃	#1580	1	1	yes	characteristic
95	C₂×C₁₃	AGL(1,ℤ/16)	1	2	yes	—
96	D₁₃	—	8	4	no	—
97	C₂×D₄	—	26	2	no	—
98	C₂²×C₄	—	52	2	no	—
99	D₈	—	13	4	no	—
100	D₈	—	26	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	52	2	no	—
103	C₂×C₈	—	104	4	no	—
104	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
105	C₂×C₈	#6127	1	1	yes	characteristic
106	C₄⋊C₄	—	26	2	no	—
107	C₂²⋊C₄	—	52	2	no	—
108	C₄²	C₂×D₅₂	1	1	yes	characteristic
109	C₁₆	—	2	1	no	—
110	C₁₆	—	4	1	no	—
111	C₁₆	C₈×D₁₃	1	2	yes	complemented
112	C₁₃	#12067	1	1	yes	characteristic, complemented, Sylow
113	C₂³	—	52	2	no	—
114	D₄	—	26	4	no	—
115	D₄	—	52	2	no	—
116	C₂×C₄	—	52	2	no	—
117	C₂×C₄	—	104	2	no	—
118	C₂×C₄	—	104	2	no	—
119	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
120	C₂×C₄	#88162	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	4	2	no	—
123	C₈	—	104	2	no	—
124	C₈	C₂×C₈×D₁₃	1	1	yes	characteristic
125	C₈	#88155	1	1	yes	characteristic
126	C₂²	—	52	4	no	—
127	C₂²	—	104	2	no	—
128	C₂²	—	104	2	no	—
129	C₂²	#11718372	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	104	2	no	—
132	C₄	C₈×D₅₂	1	1	yes	characteristic
133	C₄	#11718378	1	1	yes	characteristic
134	C₂	—	104	4	no	—
135	C₂	#13113661	1	1	yes	characteristic
136	C₂	#13116736	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13116713

C₂

yes

maximal, characteristic

#13113661

C₂

yes

maximal, complemented

#13112902

C₂

yes

maximal, characteristic, complemented

#13112804

C₂

yes

maximal, complemented

C₁₃×#1313

C₂

yes

maximal, characteristic, complemented

C₂×D₂₀₈

C₄

yes

characteristic

C₄×D₁₀₄

C₂²

yes

—

C₈×D₅₂

—

C₁₃×#346

C₂²

yes

characteristic

C₁₃×#346

C₂²

yes

characteristic

C₁₃×#322

C₂²

yes

characteristic

#11718101

C₄

yes

—

#11718097

C₄

yes

characteristic

#11718051

—

#11718049

C₂²

yes

—

C₂×D₁₀₄

C₂×C₄

yes

—

C₂×C₈×D₁₃

—

C₄×D₅₂

—

C₂×C₁₃×C₁₆

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

—

C₂×C₄

yes

—

C₄⋊#1055

—

D₂₀₈

C₈

yes

complemented

#12067

—

maximal, Sylow

C₂×D₅₂

—

C₂×C₄×D₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₄²×C₁₃

C₂×D₄

yes

characteristic

#6103

—

C₄⋊Dic₁₃

—

C₂×#1055

—

D₁₀₄

—

D₁₀₄

C₂×C₈

yes

complemented

C₈×D₁₃

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₁₃×C₁₆

C₂×C₈

yes

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×D₁₃

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

C₄×D₁₃

—

C₄×D₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂²×C₈

yes

C₈×C₁₃

yes

—

C₂×D₁₆

—

C₄×D₈

—

C₈×D₄

—

#367

—

#357

—

#346

C₂×D₁₃

yes

characteristic

#346

C₂×D₁₃

yes

characteristic

#334

—

D₄⋊C₈

—

#322

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#442

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

#444

yes

characteristic

Dic₁₃

—

C₂×D₈

—

C₂²×C₈

—

C₄×D₄

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₁₃

yes

characteristic

C₂×C₁₆

C₄×D₁₃

yes

characteristic

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

D₅₂

yes

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

D₁₃

—

C₂×D₄

—

C₂²×C₄

—

D₈

—

100

D₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

—

104

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

105

C₂×C₈

#6127

yes

characteristic

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

C₂×D₅₂

yes

characteristic

109

C₁₆

—

110

C₁₆

—

111

C₁₆

C₈×D₁₃

yes

complemented

112

C₁₃

#12067

yes

characteristic, complemented, Sylow

113

C₂³

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

104

—

118

C₂×C₄

—

104

—

119

C₂×C₄

C₄×D₅₂

yes

characteristic

120

C₂×C₄

#88162

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

104

—

124

C₈

C₂×C₈×D₁₃

yes

characteristic

125

C₈

#88155

yes

characteristic

126

C₂²

—

127

C₂²

—

104

—

128

C₂²

—

104

—

129

C₂²

#11718372

yes

characteristic

130

C₄

—

131

C₄

—

104

—

132

C₄

C₈×D₅₂

yes

characteristic

133

C₄

#11718378

yes

characteristic

134

C₂

—

104

—

135

C₂

#13113661

yes

characteristic

136

C₂

#13116736

yes

—

	subgroup	G/H
center	C₂²	#11718372
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1313	C₂
socle	C₂²×C₁₃	#442
prime radical	C₂×D₂₀₈	C₄
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#1313	D₁₃
hypocenter	C₁₃	#12067
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

subgroup

G/H

center

C₂²

#11718372

commutator

C₈×C₁₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1313

C₂

socle

C₂²×C₁₃

#442

prime radical

C₂×D₂₀₈

C₄

cosocle

C₄×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322 ◁ #1313

N	H ≅ G/N	type
2. #13113661	C₂	semidirect
3. #13112902	C₂	semidirect
4. #13112804	C₂	semidirect
5. C₁₃×#1313	C₂	semidirect
28. D₂₀₈	C₈	semidirect
40. D₁₀₄	C₂×C₈	semidirect
44. C₁₃×C₁₆	C₂×C₈	semidirect
49. #1313	D₁₃	semidirect
111. C₁₆	C₈×D₁₃	semidirect
112. C₁₃	#12067	semidirect

H ≅ G/N

type

2. #13113661

C₂

semidirect

3. #13112902

C₂

semidirect

4. #13112804

C₂

semidirect

5. C₁₃×#1313

C₂

semidirect

28. D₂₀₈

C₈

semidirect

40. D₁₀₄

C₂×C₈

semidirect

44. C₁₃×C₁₆

C₂×C₈

semidirect

49. #1313

D₁₃

semidirect

111. C₁₆

C₈×D₁₃

semidirect

112. C₁₃

#12067

semidirect

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

subgroup	index
#13116713	2
#13113661	2
#13112902	2
#13112804	2
C₁₃×#1313	2
#12067	13

subgroup

index

C₁₃×#1313

Subgroups — #15774805

Conjugacy classes of subgroups

Complements

Maximal subgroups