Group #15774757 — subgroups

Subgroups — #15774757

Order 3328.

212 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 93 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	#13116738	C₂	1	2	yes	maximal
2	C₈×Dic₅₂	C₂	1	1	yes	maximal, characteristic
3	#13112814	C₂	1	2	yes	maximal
4	#13112754	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1367	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₅₂	C₄	1	1	yes	characteristic
7	C₈×Dic₂₆	C₂²	1	2	yes	—
8	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₈²×C₁₃	C₂²	1	1	yes	characteristic
10	#11718093	—	2	2	no	—
11	C₁₆×Dic₁₃	—	2	2	no	—
12	C₄×#6090	C₂²	1	1	yes	characteristic
13	C₄×#6090	C₂²	1	1	yes	characteristic
14	#11718050	C₄	1	1	yes	characteristic
15	#11718048	C₄	1	2	yes	—
16	C₂×Dic₅₂	C₈	1	1	yes	characteristic
17	C₄×Dic₂₆	C₂×C₄	1	2	yes	—
18	C₂×C₁₃×C₁₆	—	2	2	no	—
19	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
22	#88058	C₈	1	1	yes	characteristic
23	#88056	C₈	1	2	yes	—
24	#88054	—	2	2	no	—
25	C₈×Dic₁₃	—	2	2	no	—
26	C₂×#6090	—	2	1	no	—
27	C₂×#6090	D₄	1	2	yes	—
28	C₄⋊#1055	C₂×C₄	1	2	yes	—
29	#12059	—	13	1	no	maximal, Sylow
30	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₁₃	—	2	1	no	—
32	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
36	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊Dic₁₃	C₂×C₈	1	2	yes	—
38	#6101	—	2	2	no	—
39	C₄×Dic₁₃	—	2	2	no	—
40	C₂×#1055	—	2	2	no	—
41	Dic₅₂	C₁₆	1	4	yes	complemented
42	C₁₃×C₁₆	—	4	2	no	—
43	#6090	—	2	2	no	—
44	#6090	—	4	1	no	—
45	Q₈×C₁₆	—	13	2	no	—
46	C₈×Q₁₆	—	13	1	no	—
47	#1367	D₁₃	1	1	yes	characteristic
48	Q₈⋊C₁₆	—	13	2	no	—
49	#1312	—	13	1	no	—
50	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
52	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
53	C₂×Dic₁₃	—	2	2	no	—
54	C₂×Dic₁₃	—	2	2	no	—
55	Dic₂₆	—	2	2	no	—
56	Dic₂₆	C₂×C₁₆	1	4	yes	—
57	C₈×C₁₃	—	2	1	no	—
58	C₈×C₁₃	—	2	1	no	—
59	C₈×C₁₃	—	4	1	no	—
60	C₈×C₁₃	C₂×C₁₆	1	2	yes	—
61	C₈×C₁₃	#137	1	2	yes	—
62	#1055	—	4	2	no	—
63	C₈×Q₈	—	13	2	no	—
64	C₄×Q₁₆	—	13	1	no	—
65	C₄⋊C₁₆	—	26	2	no	—
66	C₄⋊C₁₆	C₂×D₁₃	1	2	yes	—
67	C₄×C₁₆	—	13	1	no	—
68	C₄×C₁₆	—	13	1	no	—
69	C₄×C₁₆	—	26	2	no	—
70	#335	—	13	1	no	—
71	Q₈⋊C₈	—	13	2	no	—
72	C₈²	C₂×D₁₃	1	1	yes	characteristic
73	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
74	C₄×C₁₃	—	2	1	no	—
75	C₄×C₁₃	#441	1	1	yes	characteristic
76	C₄×C₁₃	#444	1	1	yes	characteristic
77	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
78	C₄×C₁₃	#504	1	1	yes	characteristic
79	Dic₁₃	—	2	4	no	—
80	Dic₁₃	—	4	2	no	—
81	C₂×Q₁₆	—	13	1	no	—
82	C₄×Q₈	—	13	2	no	—
83	C₂×C₁₆	—	2	2	no	—
84	C₂×C₁₆	—	13	2	no	—
85	C₂×C₁₆	—	26	1	no	—
86	#107	—	13	1	no	—
87	C₄⋊C₈	—	13	2	no	—
88	C₄⋊C₈	—	26	2	no	—
89	#103	—	13	2	no	—
90	C₄×C₈	—	26	2	no	—
91	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
92	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
93	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
94	C₂×C₁₃	#1579	1	1	yes	characteristic
95	C₂×C₁₃	D₄×C₁₆	1	1	yes	characteristic
96	C₂×C₁₃	#2171	1	1	yes	characteristic
97	C₂×Q₈	—	13	2	no	—
98	Q₁₆	—	13	4	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	26	2	no	—
101	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
102	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
103	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
104	C₂×C₈	#6131	1	1	yes	characteristic
105	C₄⋊C₄	—	13	2	no	—
106	C₄⋊C₄	—	26	2	no	—
107	C₄²	—	26	2	no	—
108	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
109	C₁₆	—	4	2	no	—
110	C₁₆	—	26	2	no	—
111	C₁₆	—	52	1	no	—
112	C₁₃	#12059	1	1	yes	characteristic, complemented, Sylow
113	Q₈	—	13	4	no	—
114	Q₈	—	26	2	no	—
115	C₂×C₄	—	26	2	no	—
116	C₂×C₄	—	26	2	no	—
117	C₂×C₄	#88147	1	1	yes	characteristic
118	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
119	C₂×C₄	#88161	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	4	1	no	—
123	C₈	—	52	2	no	—
124	C₈	C₁₆×D₁₃	1	2	yes	—
125	C₈	#88169	1	2	yes	—
126	C₂²	#11718495	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	26	4	no	—
129	C₄	—	52	2	no	—
130	C₄	#11718514	1	1	yes	characteristic
131	C₄	#11718555	1	1	yes	characteristic
132	C₄	C₂×C₁₆×D₁₃	1	1	yes	characteristic
133	C₄	#11718568	1	1	yes	characteristic
134	C₂	#13113654	1	1	yes	characteristic
135	C₂	#13116655	1	1	yes	characteristic
136	C₂	#13116677	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13116738

C₂

yes

maximal

C₈×Dic₅₂

C₂

yes

maximal, characteristic

#13112814

C₂

yes

maximal

#13112754

C₂

yes

maximal, characteristic

C₁₃×#1367

C₂

yes

maximal, characteristic

C₄×Dic₅₂

C₄

yes

characteristic

C₈×Dic₂₆

C₂²

yes

—

C₁₃×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

#11718093

—

C₁₆×Dic₁₃

—

C₄×#6090

C₂²

yes

characteristic

C₄×#6090

C₂²

yes

characteristic

#11718050

C₄

yes

characteristic

#11718048

C₄

yes

—

C₂×Dic₅₂

C₈

yes

characteristic

C₄×Dic₂₆

C₂×C₄

yes

—

C₂×C₁₃×C₁₆

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88058

C₈

yes

characteristic

#88056

C₈

yes

—

#88054

—

C₈×Dic₁₃

—

C₂×#6090

—

C₂×#6090

D₄

yes

—

C₄⋊#1055

C₂×C₄

yes

—

#12059

—

maximal, Sylow

C₂×Dic₂₆

C₂×C₈

yes

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

#41

yes

characteristic

C₄²×C₁₃

C₂²×C₄

yes

characteristic

C₄⋊Dic₁₃

C₂×C₈

yes

—

#6101

—

C₄×Dic₁₃

—

C₂×#1055

—

Dic₅₂

C₁₆

yes

complemented

C₁₃×C₁₆

—

#6090

—

#6090

—

Q₈×C₁₆

—

C₈×Q₁₆

—

#1367

D₁₃

yes

characteristic

Q₈⋊C₁₆

—

#1312

—

C₂×C₄×C₁₃

C₄×D₄

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₁₃

#131

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

C₂×C₁₆

yes

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

—

C₈×C₁₃

C₂×C₁₆

yes

—

C₈×C₁₃

#137

yes

—

#1055

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×D₁₃

yes

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

C₈×D₄

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂²×C₁₆

yes

characteristic

C₄×C₁₃

#504

yes

characteristic

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

C₄×D₁₃

yes

characteristic

C₄×C₈

C₄×D₁₃

yes

characteristic

C₄×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₃

#1579

yes

characteristic

C₂×C₁₃

D₄×C₁₆

yes

characteristic

C₂×C₁₃

#2171

yes

characteristic

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₈×D₁₃

yes

characteristic

102

C₂×C₈

C₈×D₁₃

yes

characteristic

103

C₂×C₈

D₄×D₁₃

yes

characteristic

104

C₂×C₈

#6131

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

C₂×C₄×D₁₃

yes

characteristic

109

C₁₆

—

110

C₁₆

—

111

C₁₆

—

112

C₁₃

#12059

yes

characteristic, complemented, Sylow

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#88147

yes

characteristic

118

C₂×C₄

C₂×C₈×D₁₃

yes

characteristic

119

C₂×C₄

#88161

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

C₁₆×D₁₃

yes

—

125

C₈

#88169

yes

—

126

C₂²

#11718495

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#11718514

yes

characteristic

131

C₄

#11718555

yes

characteristic

132

C₄

C₂×C₁₆×D₁₃

yes

characteristic

133

C₄

#11718568

yes

characteristic

134

C₂

#13113654

yes

characteristic

135

C₂

#13116655

yes

characteristic

136

C₂

#13116677

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₄×C₁₃	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1367	C₂
socle, prime radical	C₂²×C₁₃	C₈×D₄
cosocle	C₄×C₈×C₁₃	C₂³
hypercenter	#1367	D₁₃
hypocenter	C₁₃	#12059
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

subgroup

G/H

center

C₂×C₈

D₄×D₁₃

commutator

C₄×C₁₃

C₂²×C₁₆

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1367

C₂

socle, prime radical

C₂²×C₁₃

C₈×D₄

cosocle

C₄×C₈×C₁₃

C₂³

D₁₃

C₁₃

G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1367

N	H ≅ G/N	type
41. Dic₅₂	C₁₆	semidirect
112. C₁₃	#12059	semidirect

H ≅ G/N

type

41. Dic₅₂

C₁₆

semidirect

112. C₁₃

#12059

semidirect

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

subgroup	index
#13116738	2
C₈×Dic₅₂	2
#13112814	2
#13112754	2
C₁₃×#1367	2
#12059	13

subgroup

index

C₈×Dic₅₂

C₁₃×#1367

Subgroups — #15774757

Conjugacy classes of subgroups

Complements

Maximal subgroups