Group #15777689 — subgroups

Subgroups — #15777689

Order 3328.

274 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 87 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	#13117088	C₂	1	2	yes	maximal
2	#13117063	C₂	1	2	yes	maximal
3	#13114433	C₂	1	1	yes	maximal, characteristic
4	#13112754	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1713	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#500	—	2	1	no	—
7	C₁₃×#500	—	2	1	no	—
8	C₁₃×#494	C₂²	1	2	yes	—
9	C₂×#88069	—	2	4	no	—
10	#11718360	C₂²	1	1	yes	characteristic
11	#11718343	C₂²	1	1	yes	characteristic
12	#11718342	—	2	1	no	—
13	#11718341	—	2	1	no	—
14	C₈²×C₁₃	C₂²	1	1	yes	characteristic
15	#11718156	—	2	4	no	—
16	C₄×#6090	C₂²	1	2	yes	—
17	C₂×C₁₃×Q₁₆	—	2	2	no	—
18	C₂×C₁₃×Q₁₆	—	2	2	no	—
19	C₁₃×C₄⋊Q₈	—	2	2	no	—
20	C₂×Dic₅₂	—	2	2	no	—
21	#88123	—	2	1	no	—
22	#88123	—	2	1	no	—
23	C₁₃×#107	—	4	2	no	—
24	C₁₃×#107	—	4	2	no	—
25	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	#88069	—	4	8	no	—
29	#88058	—	2	2	no	—
30	#88057	—	4	2	no	—
31	#88056	—	4	2	no	—
32	C₂×#6090	D₄	1	4	yes	—
33	#12528	—	13	1	no	maximal, Sylow
34	C₂×Q₈×C₁₃	—	4	2	no	—
35	C₂×Dic₂₆	—	4	1	no	—
36	C₂×Dic₂₆	—	4	1	no	—
37	C₁₃×Q₁₆	—	4	4	no	—
38	C₁₃×Q₁₆	—	4	4	no	—
39	C₂×C₈×C₁₃	D₈	1	2	yes	—
40	C₂×C₈×C₁₃	D₈	1	2	yes	—
41	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
42	C₁₃×C₄⋊C₄	—	4	2	no	—
43	C₁₃×C₄⋊C₄	—	4	2	no	—
44	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
45	C₄⋊Dic₁₃	—	4	1	no	—
46	C₄⋊Dic₁₃	—	4	1	no	—
47	C₄⋊Dic₁₃	—	4	1	no	—
48	C₄⋊Dic₁₃	—	4	1	no	—
49	Dic₅₂	—	4	4	no	—
50	#6090	—	2	4	no	—
51	#2247	—	13	2	no	—
52	#2241	—	13	2	no	—
53	#1713	D₁₃	1	1	yes	characteristic
54	#1707	—	13	1	no	—
55	#1312	—	13	1	no	—
56	Q₈×C₁₃	—	8	2	no	—
57	Q₈×C₁₃	—	8	2	no	—
58	C₂×C₄×C₁₃	—	8	2	no	—
59	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
60	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
62	C₂×Dic₁₃	—	8	1	no	—
63	C₂×Dic₁₃	—	8	1	no	—
64	Dic₂₆	—	8	2	no	—
65	Dic₂₆	—	8	2	no	—
66	C₈×C₁₃	—	2	2	no	—
67	C₈×C₁₃	Q₃₂	1	4	yes	complemented
68	C₈×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
69	C₂×Q₃₂	—	26	4	no	—
70	#500	—	2	1	no	—
71	#500	—	2	1	no	—
72	#500	—	13	1	no	—
73	#498	—	26	1	no	—
74	#494	—	13	1	no	—
75	#494	C₂×D₁₃	1	2	yes	—
76	#487	—	26	1	no	—
77	#358	—	26	4	no	—
78	C₄×C₁₆	—	13	2	no	—
79	C₈²	C₂×D₁₃	1	1	yes	characteristic
80	C₂²×C₁₃	#493	1	1	yes	characteristic
81	C₄×C₁₃	—	16	2	no	—
82	C₄×C₁₃	#496	1	2	yes	—
83	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
84	C₄×C₁₃	#510	1	2	yes	—
85	Dic₁₃	—	16	1	no	—
86	Dic₁₃	—	16	1	no	—
87	C₂×Q₁₆	—	2	2	no	—
88	C₂×Q₁₆	—	2	2	no	—
89	C₂×Q₁₆	—	26	2	no	—
90	C₄⋊Q₈	—	2	2	no	—
91	C₄⋊Q₈	—	26	1	no	—
92	C₄⋊Q₈	—	26	1	no	—
93	Q₃₂	—	52	8	no	—
94	C₂×C₁₆	—	13	4	no	—
95	#107	—	4	2	no	—
96	#107	—	4	2	no	—
97	#107	—	26	2	no	—
98	#106	—	52	2	no	—
99	#103	—	52	2	no	—
100	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
101	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
102	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
103	C₂×C₁₃	#1721	1	1	yes	characteristic
104	C₂×C₁₃	#2247	1	1	yes	characteristic
105	C₂×C₁₃	#2251	1	1	yes	characteristic
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	52	1	no	—
108	C₂×Q₈	—	52	1	no	—
109	Q₁₆	—	4	4	no	—
110	Q₁₆	—	4	4	no	—
111	Q₁₆	—	52	4	no	—
112	C₂×C₈	#6104	1	2	yes	—
113	C₂×C₈	#6104	1	2	yes	—
114	C₂×C₈	D₄×D₁₃	1	2	yes	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	52	1	no	—
118	C₄⋊C₄	—	52	1	no	—
119	C₄⋊C₄	—	52	1	no	—
120	C₄⋊C₄	—	52	1	no	—
121	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
122	C₁₆	—	26	4	no	—
123	C₁₃	#12528	1	1	yes	characteristic, complemented, Sylow
124	Q₈	—	8	2	no	—
125	Q₈	—	8	2	no	—
126	Q₈	—	104	2	no	—
127	Q₈	—	104	2	no	—
128	C₂×C₄	—	8	2	no	—
129	C₂×C₄	—	104	1	no	—
130	C₂×C₄	—	104	1	no	—
131	C₂×C₄	C₂×#6104	1	1	yes	characteristic
132	C₂×C₄	C₂×#6104	1	1	yes	characteristic
133	C₂×C₄	#88194	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	#88069	1	4	yes	complemented
136	C₈	#88165	1	4	yes	—
137	C₂²	#11718734	1	1	yes	characteristic
138	C₄	—	16	2	no	—
139	C₄	—	208	1	no	—
140	C₄	—	208	1	no	—
141	C₄	#11718815	1	2	yes	—
142	C₄	#11718840	1	2	yes	—
143	C₄	C₂×#88069	1	2	yes	—
144	C₂	#13114503	1	1	yes	characteristic
145	C₂	#13117088	1	1	yes	characteristic
146	C₂	#13117105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117088

C₂

yes

maximal

#13117063

C₂

yes

maximal

#13114433

C₂

yes

maximal, characteristic

#13112754

C₂

yes

maximal, characteristic

C₁₃×#1713

C₂

yes

maximal, characteristic

C₁₃×#500

—

C₁₃×#500

—

C₁₃×#494

C₂²

yes

—

C₂×#88069

—

#11718360

C₂²

yes

characteristic

#11718343

C₂²

yes

characteristic

#11718342

—

#11718341

—

C₈²×C₁₃

C₂²

yes

characteristic

#11718156

—

C₄×#6090

C₂²

yes

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×Q₁₆

—

C₁₃×C₄⋊Q₈

—

C₂×Dic₅₂

—

#88123

—

#88123

—

C₁₃×#107

—

C₁₃×#107

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

D₄

yes

—

#12528

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×Q₁₆

—

C₁₃×Q₁₆

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

Dic₅₂

—

—

—

—

D₁₃

yes

characteristic

#1707

—

#1312

—

Q₈×C₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

Q₃₂

yes

complemented

C₈×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×Q₃₂

—

—

—

—

—

—

C₂×D₁₃

yes

—

#487

—

#358

—

C₄×C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#493

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

#496

yes

—

C₄×C₁₃

C₂×Q₃₂

yes

—

C₄×C₁₃

#510

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

—

—

—

—

—

—

100

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

101

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

102

C₄×C₈

C₂²×D₁₃

yes

characteristic

103

C₂×C₁₃

#1721

yes

characteristic

104

C₂×C₁₃

#2247

yes

characteristic

105

C₂×C₁₃

#2251

yes

characteristic

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

Q₁₆

—

110

Q₁₆

—

111

Q₁₆

—

112

C₂×C₈

#6104

yes

—

113

C₂×C₈

#6104

yes

—

114

C₂×C₈

D₄×D₁₃

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

122

C₁₆

—

123

C₁₃

#12528

yes

characteristic, complemented, Sylow

124

Q₈

—

125

Q₈

—

126

Q₈

—

104

—

127

Q₈

—

104

—

128

C₂×C₄

—

129

C₂×C₄

—

104

—

130

C₂×C₄

—

104

—

131

C₂×C₄

C₂×#6104

yes

characteristic

132

C₂×C₄

C₂×#6104

yes

characteristic

133

C₂×C₄

#88194

yes

characteristic

134

C₈

—

135

C₈

#88069

yes

complemented

136

C₈

#88165

yes

—

137

C₂²

#11718734

yes

characteristic

138

C₄

—

139

C₄

—

208

—

140

C₄

—

208

—

141

C₄

#11718815

yes

—

142

C₄

#11718840

yes

—

143

C₄

C₂×#88069

yes

—

144

C₂

#13114503

yes

characteristic

145

C₂

#13117088

yes

characteristic

146

C₂

#13117105

yes

characteristic

	subgroup	G/H
center	C₂²	#11718734
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1713	C₂
socle, prime radical	C₂²×C₁₃	#493
hypercenter	#1713	D₁₃
hypocenter	C₁₃	#12528
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

subgroup

G/H

center

C₂²

#11718734

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1713

C₂

C₂²×C₁₃

D₁₃

C₁₃

G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
67. C₈×C₁₃	Q₃₂	semidirect
123. C₁₃	#12528	semidirect
135. C₈	#88069	semidirect

H ≅ G/N

type

67. C₈×C₁₃

Q₃₂

semidirect

123. C₁₃

#12528

semidirect

135. C₈

#88069

semidirect

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

subgroup	index
#13117088	2
#13117063	2
#13114433	2
#13112754	2
C₁₃×#1713	2
#12528	13

subgroup

index

C₁₃×#1713

Subgroups — #15777689

Conjugacy classes of subgroups

Complements

Maximal subgroups