Group #15774674 — subgroups

Subgroups — #15774674

Order 3328.

216 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 75 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	#13116718	C₂	1	1	yes	maximal, characteristic
2	#13116672	C₂	1	2	yes	maximal, complemented
3	#13112808	C₂	1	2	yes	maximal, complemented
4	C₁₆×#1055	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1373	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	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	#11718107	—	2	2	no	—
12	#11718102	C₄	1	2	yes	—
13	#11718098	C₄	1	1	yes	characteristic
14	C₄×#6090	C₂²	1	1	yes	characteristic
15	C₈×#1055	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	1	yes	characteristic
24	C₂×#6090	D₄	1	1	yes	characteristic
25	#88045	—	2	2	no	—
26	C₄×#1055	—	2	1	no	—
27	Dic₁₀₄	C₈	1	1	yes	characteristic, complemented
28	#88040	C₈	1	2	yes	complemented
29	D₂₀₈	C₈	1	1	yes	characteristic, complemented
30	#88038	—	4	2	no	—
31	C₁₆×D₁₃	—	4	2	no	—
32	#12046	—	13	1	no	maximal, Sylow
33	#6127	—	2	2	no	—
34	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₁₃	#41	1	1	yes	characteristic
37	#6099	—	2	2	no	—
38	C₂×#1055	—	2	2	no	—
39	Dic₅₂	C₂×C₈	1	2	yes	—
40	D₁₀₄	C₂×C₈	1	2	yes	—
41	#6095	—	2	2	no	—
42	#6094	—	4	2	no	—
43	C₈×D₁₃	—	4	2	no	—
44	C₁₃×C₁₆	—	2	1	no	—
45	C₁₃×C₁₆	—	4	2	no	—
46	C₁₃×C₁₆	C₂×C₈	1	1	yes	characteristic
47	C₁₃×C₁₆	C₂×C₈	1	1	yes	characteristic
48	#6090	—	2	1	no	—
49	#6090	D₈	1	1	yes	characteristic
50	#6090	D₈	1	1	yes	characteristic
51	#2178	—	13	1	no	—
52	#2170	—	13	2	no	—
53	#1373	D₁₃	1	1	yes	characteristic, complemented
54	#1335	—	13	2	no	—
55	C₈×C₁₆	—	13	1	no	—
56	C₂×C₄×C₁₃	—	2	1	no	—
57	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
58	C₂²⋊D₁₃	—	4	2	no	—
59	D₅₂	—	2	2	no	—
60	C₄×D₁₃	—	4	2	no	—
61	Dic₂₆	—	2	2	no	—
62	C₈×C₁₃	C₂²×C₈	1	1	yes	characteristic
63	C₈×C₁₃	#131	1	1	yes	characteristic
64	C₈×C₁₃	C₂×D₈	1	1	yes	characteristic
65	C₈×C₁₃	#135	1	1	yes	characteristic
66	#1055	—	2	4	no	—
67	#1055	—	4	2	no	—
68	#508	—	13	1	no	—
69	#504	—	26	2	no	—
70	#443	—	13	2	no	—
71	#368	—	13	1	no	—
72	#364	C₂×D₁₃	1	2	yes	—
73	#359	—	13	2	no	—
74	#350	—	26	2	no	—
75	C₄×C₁₆	—	13	1	no	—
76	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
77	C₈²	—	13	1	no	—
78	C₂²×C₁₃	#443	1	1	yes	characteristic
79	C₂×D₁₃	—	4	2	no	—
80	C₄×C₁₃	—	2	2	no	—
81	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
82	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
83	Dic₁₃	—	4	2	no	—
84	#135	—	13	2	no	—
85	#131	—	26	2	no	—
86	Q₃₂	—	13	1	no	—
87	SD₃₂	—	13	2	no	—
88	D₁₆	—	13	1	no	—
89	M₃₂	—	2	2	no	—
90	M₃₂	—	52	2	no	—
91	C₂×C₁₆	—	13	1	no	—
92	C₂×C₁₆	—	13	1	no	—
93	C₂×C₁₆	—	52	2	no	—
94	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
95	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
96	#108	—	13	2	no	—
97	#104	—	26	2	no	—
98	C₄×C₈	—	26	1	no	—
99	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
100	C₂×C₁₃	—	2	1	no	—
101	C₂×C₁₃	C₈×D₈	1	1	yes	characteristic
102	D₁₃	—	8	2	no	—
103	#41	—	26	2	no	—
104	Q₁₆	—	13	2	no	—
105	SD₁₆	—	26	2	no	—
106	D₈	—	13	2	no	—
107	M₁₆	—	26	2	no	—
108	M₁₆	—	52	2	no	—
109	C₂×C₈	—	26	2	no	—
110	C₂×C₈	—	52	2	no	—
111	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
112	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
113	C₄²	#6131	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	4	2	no	—
116	C₁₆	—	13	1	no	—
117	C₁₆	—	13	1	no	—
118	C₁₆	—	26	1	no	—
119	C₁₆	C₈×D₁₃	1	1	yes	characteristic
120	C₁₆	C₈×D₁₃	1	1	yes	characteristic
121	C₁₃	#12046	1	1	yes	characteristic, complemented, Sylow
122	Q₈	—	26	2	no	—
123	D₄	—	26	2	no	—
124	D₄	—	52	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	52	2	no	—
127	C₂×C₄	#88147	1	1	yes	characteristic
128	C₈	—	26	4	no	—
129	C₈	—	52	2	no	—
130	C₈	C₂×C₈×D₁₃	1	1	yes	characteristic
131	C₈	#88161	1	1	yes	characteristic
132	C₈	D₈×D₁₃	1	1	yes	characteristic
133	C₈	#88173	1	1	yes	characteristic
134	C₂²	—	52	2	no	—
135	C₂²	#11718556	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	52	2	no	—
138	C₄	#11718495	1	1	yes	characteristic
139	C₄	#11718533	1	1	yes	characteristic
140	C₂	—	2	1	no	—
141	C₂	—	104	2	no	—
142	C₂	#13113630	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116718

C₂

yes

maximal, characteristic

#13116672

C₂

yes

maximal, complemented

#13112808

C₂

yes

maximal, complemented

C₁₆×#1055

C₂

yes

maximal, characteristic, complemented

C₁₃×#1373

C₂

yes

maximal, characteristic, complemented

#11718568

—

#11718565

C₄

yes

characteristic

#11718361

C₂²

yes

—

C₁₃×#364

C₂²

yes

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

#11718107

—

#11718102

C₄

yes

—

#11718098

C₄

yes

characteristic

C₄×#6090

C₂²

yes

characteristic

C₈×#1055

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

characteristic

C₂×#6090

D₄

yes

characteristic

#88045

—

C₄×#1055

—

Dic₁₀₄

C₈

yes

characteristic, complemented

#88040

C₈

yes

complemented

D₂₀₈

C₈

yes

characteristic, complemented

#88038

—

C₁₆×D₁₃

—

#12046

—

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

characteristic

#6090

—

#6090

D₈

yes

characteristic

#6090

D₈

yes

—

—

D₁₃

yes

characteristic, complemented

#1335

—

C₈×C₁₆

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂²⋊D₁₃

—

D₅₂

—

C₄×D₁₃

—

Dic₂₆

—

C₈×C₁₃

C₂²×C₈

yes

characteristic

C₈×C₁₃

#131

yes

characteristic

C₈×C₁₃

C₂×D₈

yes

C₈×C₁₃

yes

—

—

—

—

—

—

C₂×D₁₃

yes

—

#359

—

#350

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₈²

—

C₂²×C₁₃

#443

yes

characteristic

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

C₄×D₈

yes

characteristic

Dic₁₃

—

#135

—

#131

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

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

100

C₂×C₁₃

—

101

C₂×C₁₃

C₈×D₈

yes

characteristic

102

D₁₃

—

103

#41

—

104

Q₁₆

—

105

SD₁₆

—

106

D₈

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

C₂×C₄×D₁₃

yes

characteristic

112

C₂×C₈

D₄×D₁₃

yes

characteristic

113

C₄²

#6131

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

119

C₁₆

C₈×D₁₃

yes

characteristic

120

C₁₆

C₈×D₁₃

yes

characteristic

121

C₁₃

#12046

yes

characteristic, complemented, Sylow

122

Q₈

—

123

D₄

—

124

D₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

#88147

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

C₂×C₈×D₁₃

yes

characteristic

131

C₈

#88161

yes

characteristic

132

C₈

D₈×D₁₃

yes

characteristic

133

C₈

#88173

yes

characteristic

134

C₂²

—

135

C₂²

#11718556

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

#11718495

yes

characteristic

139

C₄

#11718533

yes

characteristic

140

C₂

—

141

C₂

—

104

—

142

C₂

#13113630

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₁₃
commutator	C₈×C₁₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1373	C₂
socle	C₂×C₁₃	C₈×D₈
prime radical	#11718565	C₄
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#1373	D₁₃
hypocenter	C₁₃	#12046
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₈

D₈×D₁₃

commutator

C₈×C₁₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1373

C₂

socle

C₂×C₁₃

C₈×D₈

prime radical

#11718565

C₄

cosocle

C₄×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
2. #13116672	C₂	semidirect
3. #13112808	C₂	semidirect
4. C₁₆×#1055	C₂	semidirect
5. C₁₃×#1373	C₂	semidirect
27. Dic₁₀₄	C₈	semidirect
28. #88040	C₈	semidirect
29. D₂₀₈	C₈	semidirect
53. #1373	D₁₃	semidirect
121. C₁₃	#12046	semidirect

H ≅ G/N

type

2. #13116672

C₂

semidirect

3. #13112808

C₂

semidirect

4. C₁₆×#1055

C₂

semidirect

5. C₁₃×#1373

C₂

semidirect

27. Dic₁₀₄

C₈

semidirect

28. #88040

C₈

semidirect

29. D₂₀₈

C₈

semidirect

53. #1373

D₁₃

semidirect

121. C₁₃

#12046

semidirect

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

subgroup	index
#13116718	2
#13116672	2
#13112808	2
C₁₆×#1055	2
C₁₃×#1373	2
#12046	13

subgroup

index

#13116718

#13116672

#13112808

C₁₆×#1055

C₁₃×#1373

#12046

Subgroups — #15774674

Conjugacy classes of subgroups

Complements

Maximal subgroups