Group #15777688 — subgroups

Subgroups — #15777688

Order 3328.

274 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 91 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	#13117089	C₂	1	2	yes	maximal
2	#13117064	C₂	1	2	yes	maximal
3	#13114454	C₂	1	1	yes	maximal, characteristic
4	#13114433	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1312	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₁₀₄	—	2	4	no	—
7	#11718360	C₂²	1	1	yes	characteristic
8	#11718360	C₂²	1	2	yes	—
9	#11718343	—	2	1	no	—
10	#11718343	—	2	1	no	—
11	#11718343	C₂²	1	1	yes	characteristic
12	#11718342	—	2	1	no	—
13	#11718341	—	2	1	no	—
14	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
15	C₈²×C₁₃	C₂²	1	1	yes	characteristic
16	#11718095	—	2	4	no	—
17	C₂×Dic₅₂	—	2	2	no	—
18	C₂×Dic₅₂	—	2	2	no	—
19	C₂×Dic₅₂	—	2	2	no	—
20	#88123	—	2	1	no	—
21	#88123	—	2	1	no	—
22	#88123	—	2	2	no	—
23	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
24	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	#88058	—	2	2	no	—
28	#88058	—	4	2	no	—
29	#88058	—	4	2	no	—
30	#88057	—	4	2	no	—
31	#88056	—	4	2	no	—
32	Dic₁₀₄	—	4	8	no	—
33	#12528	—	13	1	no	maximal, Sylow
34	C₂×Dic₂₆	—	4	1	no	—
35	C₂×Dic₂₆	—	4	2	no	—
36	C₂×Dic₂₆	—	4	1	no	—
37	C₂×C₈×C₁₃	D₈	1	2	yes	—
38	C₂×C₈×C₁₃	D₈	1	2	yes	—
39	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
40	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
41	C₄⋊Dic₁₃	—	4	1	no	—
42	C₄⋊Dic₁₃	—	4	1	no	—
43	C₄⋊Dic₁₃	—	4	1	no	—
44	C₄⋊Dic₁₃	—	4	1	no	—
45	C₄⋊Dic₁₃	—	4	2	no	—
46	C₄⋊Dic₁₃	—	4	2	no	—
47	Dic₅₂	—	4	4	no	—
48	Dic₅₂	—	4	4	no	—
49	Dic₅₂	—	4	4	no	—
50	C₁₃×C₁₆	—	2	4	no	—
51	#2247	—	13	2	no	—
52	#2241	—	13	2	no	—
53	#1713	—	13	1	no	—
54	#1707	—	13	1	no	—
55	#1312	D₁₃	1	1	yes	characteristic
56	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
59	C₂×Dic₁₃	—	8	1	no	—
60	C₂×Dic₁₃	—	8	2	no	—
61	C₂×Dic₁₃	—	8	1	no	—
62	Dic₂₆	—	8	2	no	—
63	Dic₂₆	—	8	2	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	—	13	1	no	—
71	#500	—	26	1	no	—
72	#500	—	26	1	no	—
73	#498	—	26	1	no	—
74	#494	—	13	1	no	—
75	#494	—	13	2	no	—
76	#487	—	26	1	no	—
77	#358	—	26	4	no	—
78	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
79	C₈²	C₂×D₁₃	1	1	yes	characteristic
80	C₂²×C₁₃	#493	1	1	yes	characteristic
81	C₄×C₁₃	#496	1	2	yes	—
82	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
83	C₄×C₁₃	#510	1	2	yes	—
84	Dic₁₃	—	16	2	no	—
85	Dic₁₃	—	16	1	no	—
86	Dic₁₃	—	16	1	no	—
87	C₂×Q₁₆	—	26	2	no	—
88	C₂×Q₁₆	—	26	2	no	—
89	C₂×Q₁₆	—	26	2	no	—
90	C₄⋊Q₈	—	26	1	no	—
91	C₄⋊Q₈	—	26	1	no	—
92	C₄⋊Q₈	—	26	2	no	—
93	Q₃₂	—	52	8	no	—
94	C₂×C₁₆	D₅₂	1	4	yes	—
95	#107	—	26	2	no	—
96	#107	—	52	2	no	—
97	#107	—	52	2	no	—
98	#106	—	52	2	no	—
99	#103	—	52	2	no	—
100	C₄×C₈	D₅₂	1	1	yes	characteristic
101	C₄×C₈	D₅₂	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	—	52	1	no	—
107	C₂×Q₈	—	52	2	no	—
108	C₂×Q₈	—	52	1	no	—
109	Q₁₆	—	52	4	no	—
110	Q₁₆	—	52	4	no	—
111	Q₁₆	—	52	4	no	—
112	C₂×C₈	D₁₀₄	1	2	yes	—
113	C₂×C₈	D₁₀₄	1	2	yes	—
114	C₂×C₈	C₂×D₅₂	1	2	yes	—
115	C₄⋊C₄	—	52	1	no	—
116	C₄⋊C₄	—	52	1	no	—
117	C₄⋊C₄	—	52	1	no	—
118	C₄⋊C₄	—	52	1	no	—
119	C₄⋊C₄	—	52	2	no	—
120	C₄⋊C₄	—	52	2	no	—
121	C₄²	C₂×D₅₂	1	1	yes	characteristic
122	C₁₆	—	2	4	no	—
123	C₁₃	#12528	1	1	yes	characteristic, complemented, Sylow
124	Q₈	—	104	2	no	—
125	Q₈	—	104	2	no	—
126	Q₈	—	104	2	no	—
127	Q₈	—	104	2	no	—
128	C₂×C₄	—	104	1	no	—
129	C₂×C₄	—	104	2	no	—
130	C₂×C₄	—	104	1	no	—
131	C₂×C₄	#88128	1	1	yes	characteristic
132	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
133	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	Dic₁₀₄	1	4	yes	complemented
136	C₈	#88162	1	4	yes	—
137	C₂²	#11718356	1	1	yes	characteristic
138	C₄	—	208	1	no	—
139	C₄	—	208	2	no	—
140	C₄	—	208	1	no	—
141	C₄	#11718373	1	2	yes	—
142	C₄	C₂×Dic₁₀₄	1	2	yes	—
143	C₄	#11718570	1	2	yes	—
144	C₂	#13114502	1	1	yes	characteristic
145	C₂	#13117089	1	1	yes	characteristic
146	C₂	#13117107	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117089

C₂

yes

maximal

#13117064

C₂

yes

maximal

#13114454

C₂

yes

maximal, characteristic

#13114433

C₂

yes

maximal, characteristic

C₁₃×#1312

C₂

yes

maximal, characteristic

C₂×Dic₁₀₄

—

#11718360

C₂²

yes

characteristic

#11718360

C₂²

yes

—

#11718343

—

#11718343

—

#11718343

C₂²

yes

characteristic

#11718342

—

#11718341

—

C₄×C₁₃×C₁₆

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

#11718095

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

#88123

—

#88123

—

#88123

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

—

—

Dic₁₀₄

—

#12528

—

maximal, Sylow

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

C₁₃×C₁₆

—

—

—

—

—

D₁₃

yes

characteristic

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

characteristic

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

Q₃₂

yes

complemented

C₈×C₁₃

AGL(1,ℤ/8)

yes

—

C₂×Q₃₂

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₃

yes

—

C₈²

C₂×D₁₃

yes

C₂²×C₁₃

yes

C₄×C₁₃

yes

—

C₄×C₁₃

C₂×Q₃₂

yes

—

C₄×C₁₃

#510

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

C₂×C₁₆

D₅₂

yes

—

—

—

—

—

—

100

C₄×C₈

D₅₂

yes

characteristic

101

C₄×C₈

D₅₂

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

D₁₀₄

yes

—

113

C₂×C₈

D₁₀₄

yes

—

114

C₂×C₈

C₂×D₅₂

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

C₂×D₅₂

yes

characteristic

122

C₁₆

—

123

C₁₃

#12528

yes

characteristic, complemented, Sylow

124

Q₈

—

104

—

125

Q₈

—

104

—

126

Q₈

—

104

—

127

Q₈

—

104

—

128

C₂×C₄

—

104

—

129

C₂×C₄

—

104

—

130

C₂×C₄

—

104

—

131

C₂×C₄

#88128

yes

characteristic

132

C₂×C₄

C₂×D₁₀₄

yes

characteristic

133

C₂×C₄

C₂×D₁₀₄

yes

characteristic

134

C₈

—

135

C₈

Dic₁₀₄

yes

complemented

136

C₈

#88162

yes

—

137

C₂²

#11718356

yes

characteristic

138

C₄

—

208

—

139

C₄

—

208

—

140

C₄

—

208

—

141

C₄

#11718373

yes

—

142

C₄

C₂×Dic₁₀₄

yes

—

143

C₄

#11718570

yes

—

144

C₂

#13114502

yes

characteristic

145

C₂

#13117089

yes

characteristic

146

C₂

#13117107

yes

characteristic

	subgroup	G/H
center	C₂²	#11718356
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1312	C₂
socle, prime radical	C₂²×C₁₃	#493
hypercenter	#1312	D₁₃
hypocenter	C₁₃	#12528
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#11718356

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1312

C₂

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
67. C₈×C₁₃	Q₃₂	semidirect
123. C₁₃	#12528	semidirect
135. C₈	Dic₁₀₄	semidirect

H ≅ G/N

type

67. C₈×C₁₃

Q₃₂

semidirect

123. C₁₃

#12528

semidirect

135. C₈

Dic₁₀₄

semidirect

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

subgroup	index
#13117089	2
#13117064	2
#13114454	2
#13114433	2
C₁₃×#1312	2
#12528	13

subgroup

index

C₁₃×#1312

Subgroups — #15777688

Conjugacy classes of subgroups

Complements

Maximal subgroups