Group #15774808 — subgroups

Subgroups — #15774808

Order 3328.

204 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 73 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	#13116723	C₂	1	1	yes	maximal, characteristic
2	#13116677	C₂	1	2	yes	maximal, complemented
3	#13112811	C₂	1	2	yes	maximal, complemented
4	#13112756	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1369	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718568	—	2	2	no	—
7	#11718565	C₄	1	1	yes	characteristic
8	#11718361	C₂²	1	2	yes	—
9	C₁₃×#364	C₂²	1	2	yes	—
10	C₁₃×#346	C₂²	1	1	yes	characteristic
11	#11718107	—	2	2	no	—
12	#11718102	C₄	1	2	yes	—
13	#11718098	C₄	1	1	yes	characteristic
14	#11718053	C₂²	1	1	yes	characteristic
15	#11718047	C₂²	1	1	yes	characteristic
16	#88158	C₂×C₄	1	2	yes	—
17	#88155	—	2	2	no	—
18	C₁₃×M₃₂	—	2	2	no	—
19	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	#88059	C₂×C₄	1	2	yes	—
23	C₂×#6090	D₄	1	2	yes	—
24	#88045	—	2	2	no	—
25	C₄×#1055	—	2	1	no	—
26	Dic₁₀₄	C₈	1	1	yes	characteristic, complemented
27	#88040	C₈	1	2	yes	complemented
28	D₂₀₈	C₈	1	1	yes	characteristic, complemented
29	#88038	—	4	2	no	—
30	C₁₆×D₁₃	—	4	2	no	—
31	#12068	—	13	1	no	maximal, Sylow
32	#6127	—	2	2	no	—
33	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₁₃	#41	1	1	yes	characteristic
36	#6099	—	2	2	no	—
37	C₂×#1055	—	4	1	no	—
38	Dic₅₂	C₂×C₈	1	2	yes	—
39	D₁₀₄	C₂×C₈	1	2	yes	—
40	#6095	—	2	2	no	—
41	#6094	—	4	2	no	—
42	C₈×D₁₃	—	4	2	no	—
43	C₁₃×C₁₆	—	2	1	no	—
44	C₁₃×C₁₆	—	4	2	no	—
45	C₁₃×C₁₆	C₂×C₈	1	1	yes	characteristic
46	C₁₃×C₁₆	C₂×C₈	1	1	yes	characteristic
47	#6090	—	2	2	no	—
48	#2179	—	13	1	no	—
49	#2171	—	13	2	no	—
50	#1369	D₁₃	1	1	yes	characteristic, complemented
51	#1336	—	13	2	no	—
52	#1313	—	13	1	no	—
53	C₂×C₄×C₁₃	—	2	1	no	—
54	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
55	C₂²⋊D₁₃	—	4	2	no	—
56	D₅₂	—	2	2	no	—
57	C₄×D₁₃	—	4	2	no	—
58	Dic₂₆	—	2	2	no	—
59	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
60	C₈×C₁₃	#131	1	1	yes	characteristic
61	C₈×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
62	#1055	—	4	2	no	—
63	#1055	—	8	1	no	—
64	#508	—	13	1	no	—
65	#504	—	26	2	no	—
66	#443	—	13	2	no	—
67	#368	—	13	1	no	—
68	#364	C₂×D₁₃	1	2	yes	—
69	#359	—	13	2	no	—
70	#350	—	26	2	no	—
71	#346	—	13	1	no	—
72	#346	C₂×D₁₃	1	1	yes	characteristic
73	#322	—	13	1	no	—
74	C₂²×C₁₃	#444	1	1	yes	characteristic
75	C₂×D₁₃	—	4	2	no	—
76	C₄×C₁₃	—	4	1	no	—
77	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
78	C₄×C₁₃	#442	1	1	yes	characteristic
79	Dic₁₃	—	4	2	no	—
80	#135	—	13	2	no	—
81	#131	—	26	2	no	—
82	Q₃₂	—	13	1	no	—
83	SD₃₂	—	13	2	no	—
84	D₁₆	—	13	1	no	—
85	M₃₂	—	2	2	no	—
86	M₃₂	—	52	2	no	—
87	C₂×C₁₆	—	13	2	no	—
88	C₂×C₁₆	—	52	2	no	—
89	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
90	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
91	#108	—	13	2	no	—
92	#104	—	26	2	no	—
93	C₄×C₈	—	26	1	no	—
94	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
95	C₂×C₁₃	—	2	1	no	—
96	C₂×C₁₃	#1580	1	1	yes	characteristic
97	D₁₃	—	8	2	no	—
98	#41	—	26	2	no	—
99	Q₁₆	—	13	2	no	—
100	SD₁₆	—	26	2	no	—
101	D₈	—	13	2	no	—
102	M₁₆	—	26	2	no	—
103	M₁₆	—	52	2	no	—
104	C₂×C₈	—	52	2	no	—
105	C₂×C₈	—	52	1	no	—
106	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
107	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
108	C₄²	#6131	1	1	yes	characteristic
109	C₁₆	—	2	1	no	—
110	C₁₆	—	4	2	no	—
111	C₁₆	—	26	2	no	—
112	C₁₆	C₈×D₁₃	1	1	yes	characteristic
113	C₁₆	C₈×D₁₃	1	1	yes	characteristic
114	C₁₃	#12068	1	1	yes	characteristic, complemented, Sylow
115	Q₈	—	26	2	no	—
116	D₄	—	26	2	no	—
117	D₄	—	52	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	52	2	no	—
120	C₂×C₄	#88147	1	1	yes	characteristic
121	C₈	—	52	2	no	—
122	C₈	—	104	1	no	—
123	C₈	C₂×C₈×D₁₃	1	1	yes	characteristic
124	C₈	#88161	1	1	yes	characteristic
125	C₈	#88168	1	2	yes	—
126	C₂²	—	52	2	no	—
127	C₂²	#11718555	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	52	2	no	—
130	C₄	#11718495	1	1	yes	characteristic
131	C₄	#11718530	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	104	2	no	—
134	C₂	#13113659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116723

C₂

yes

maximal, characteristic

#13116677

C₂

yes

maximal, complemented

#13112811

C₂

yes

maximal, complemented

#13112756

C₂

yes

maximal, characteristic, complemented

C₁₃×#1369

C₂

yes

maximal, characteristic, complemented

#11718568

—

#11718565

C₄

yes

characteristic

#11718361

C₂²

yes

—

C₁₃×#364

C₂²

yes

—

C₁₃×#346

C₂²

yes

characteristic

#11718107

—

#11718102

C₄

yes

—

#11718098

C₄

yes

characteristic

#11718053

C₂²

yes

characteristic

#11718047

C₂²

yes

characteristic

#88158

C₂×C₄

yes

—

#88155

—

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₁₃×C₁₆

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88059

C₂×C₄

yes

—

C₂×#6090

D₄

yes

—

#88045

—

C₄×#1055

—

Dic₁₀₄

C₈

yes

characteristic, complemented

#88040

C₈

yes

complemented

D₂₀₈

C₈

yes

characteristic, complemented

#88038

—

C₁₆×D₁₃

—

#12068

—

maximal, Sylow

#6127

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

C₄²×C₁₃

yes

—

—

Dic₅₂

C₂×C₈

yes

—

D₁₀₄

C₂×C₈

yes

—

#6095

—

#6094

—

C₈×D₁₃

—

C₁₃×C₁₆

—

C₁₃×C₁₆

—

C₁₃×C₁₆

C₂×C₈

yes

characteristic

C₁₃×C₁₆

C₂×C₈

yes

—

—

—

D₁₃

yes

characteristic, complemented

#1336

—

#1313

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂²⋊D₁₃

—

D₅₂

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

C₂²×C₈

yes

C₈×C₁₃

yes

C₈×C₁₃

yes

—

—

—

—

—

—

—

C₂×D₁₃

yes

—

—

—

—

C₂×D₁₃

yes

characteristic

#322

—

C₂²×C₁₃

#444

yes

characteristic

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

#442

yes

characteristic

Dic₁₃

—

#135

—

#131

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₁₃

yes

characteristic

C₂×C₁₆

C₄×D₁₃

yes

characteristic

#108

—

#104

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

#1580

yes

characteristic

D₁₃

—

#41

—

Q₁₆

—

100

SD₁₆

—

101

D₈

—

102

M₁₆

—

103

M₁₆

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

107

C₂×C₈

D₄×D₁₃

yes

characteristic

108

C₄²

#6131

yes

characteristic

109

C₁₆

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

C₈×D₁₃

yes

characteristic

113

C₁₆

C₈×D₁₃

yes

characteristic

114

C₁₃

#12068

yes

characteristic, complemented, Sylow

115

Q₈

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#88147

yes

characteristic

121

C₈

—

122

C₈

—

104

—

123

C₈

C₂×C₈×D₁₃

yes

characteristic

124

C₈

#88161

yes

characteristic

125

C₈

#88168

yes

—

126

C₂²

—

127

C₂²

#11718555

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

#11718495

yes

characteristic

131

C₄

#11718530

yes

characteristic

132

C₂

—

133

C₂

—

104

—

134

C₂

#13113659

yes

characteristic

	subgroup	G/H
center	C₄	#11718530
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1369	C₂
socle	C₂×C₁₃	#1580
prime radical	#11718565	C₄
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#1369	D₁₃
hypocenter	C₁₃	#12068
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

subgroup

G/H

center

C₄

#11718530

commutator

C₈×C₁₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1369

C₂

C₂×C₁₃

C₄

C₄×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

N	H ≅ G/N	type
2. #13116677	C₂	semidirect
3. #13112811	C₂	semidirect
4. #13112756	C₂	semidirect
5. C₁₃×#1369	C₂	semidirect
26. Dic₁₀₄	C₈	semidirect
27. #88040	C₈	semidirect
28. D₂₀₈	C₈	semidirect
50. #1369	D₁₃	semidirect
114. C₁₃	#12068	semidirect

H ≅ G/N

type

2. #13116677

C₂

semidirect

3. #13112811

C₂

semidirect

4. #13112756

C₂

semidirect

5. C₁₃×#1369

C₂

semidirect

26. Dic₁₀₄

C₈

semidirect

27. #88040

C₈

semidirect

28. D₂₀₈

C₈

semidirect

50. #1369

D₁₃

semidirect

114. C₁₃

#12068

semidirect

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

subgroup	index
#13116723	2
#13116677	2
#13112811	2
#13112756	2
C₁₃×#1369	2
#12068	13

subgroup

index

C₁₃×#1369

Subgroups — #15774808

Conjugacy classes of subgroups

Complements

Maximal subgroups