Group #15777728 — subgroups

Subgroups — #15777728

Order 3328.

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

normal subgroups only

This group has 250 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	#13117116	C₂	1	2	yes	maximal
2	#13117065	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	C₂²	1	2	yes	—
8	C₁₃×#494	—	2	1	no	—
9	#11718360	C₂²	1	1	yes	characteristic
10	#11718343	C₂²	1	1	yes	characteristic
11	#11718342	—	2	1	no	—
12	#11718341	—	2	1	no	—
13	C₈²×C₁₃	C₂²	1	1	yes	characteristic
14	#11718084	—	2	4	no	—
15	#11718082	—	2	4	no	—
16	C₄×#6090	C₂²	1	2	yes	—
17	C₂×C₁₃×Q₁₆	—	4	2	no	—
18	C₁₃×C₄⋊Q₈	—	2	2	no	—
19	C₂×Dic₅₂	—	2	2	no	—
20	#88123	—	2	1	no	—
21	#88123	—	2	1	no	—
22	C₁₃×#107	—	2	2	no	—
23	C₁₃×#107	—	2	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	#88058	—	2	2	no	—
29	#88057	—	4	2	no	—
30	#88056	—	4	2	no	—
31	C₂×#6090	D₄	1	4	yes	—
32	#12535	—	13	1	no	maximal, Sylow
33	C₂×Q₈×C₁₃	—	4	2	no	—
34	C₂×Dic₂₆	—	4	1	no	—
35	C₂×Dic₂₆	—	4	1	no	—
36	C₁₃×Q₁₆	—	8	4	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	C₁₃×C₄⋊C₄	—	4	2	no	—
41	C₁₃×C₄⋊C₄	—	4	2	no	—
42	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₁₃	—	4	1	no	—
44	C₄⋊Dic₁₃	—	4	1	no	—
45	C₄⋊Dic₁₃	—	4	1	no	—
46	C₄⋊Dic₁₃	—	4	1	no	—
47	Dic₅₂	—	4	4	no	—
48	#6090	—	2	4	no	—
49	#2254	—	13	2	no	—
50	#2241	—	13	2	no	—
51	#1713	D₁₃	1	1	yes	characteristic
52	#1707	—	13	1	no	—
53	#1312	—	13	1	no	—
54	Q₈×C₁₃	—	8	4	no	—
55	C₂×C₄×C₁₃	—	8	2	no	—
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	1	no	—
61	Dic₂₆	—	8	2	no	—
62	Dic₂₆	—	8	2	no	—
63	C₈×C₁₃	—	2	2	no	—
64	C₈×C₁₃	SD₃₂	1	4	yes	—
65	C₈×C₁₃	#137	1	4	yes	—
66	#500	—	2	1	no	—
67	#500	—	13	1	no	—
68	#500	C₂×D₁₃	1	2	yes	—
69	#498	—	26	1	no	—
70	#494	—	2	1	no	—
71	#494	—	13	1	no	—
72	#487	—	26	1	no	—
73	#367	—	26	4	no	—
74	#358	—	26	4	no	—
75	C₄×C₁₆	—	13	2	no	—
76	C₈²	C₂×D₁₃	1	1	yes	characteristic
77	C₂²×C₁₃	#493	1	1	yes	characteristic
78	C₄×C₁₃	—	16	2	no	—
79	C₄×C₁₃	#497	1	2	yes	—
80	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
81	C₄×C₁₃	#510	1	2	yes	—
82	Dic₁₃	—	16	1	no	—
83	Dic₁₃	—	16	1	no	—
84	C₂×Q₁₆	—	4	2	no	—
85	C₂×Q₁₆	—	26	2	no	—
86	C₄⋊Q₈	—	2	2	no	—
87	C₄⋊Q₈	—	26	1	no	—
88	C₄⋊Q₈	—	26	1	no	—
89	C₂×C₁₆	—	13	4	no	—
90	#107	—	2	2	no	—
91	#107	—	2	2	no	—
92	#107	—	4	2	no	—
93	#107	—	26	2	no	—
94	#106	—	52	2	no	—
95	#103	—	52	2	no	—
96	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
97	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
98	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
99	C₂×C₁₃	#1722	1	1	yes	characteristic
100	C₂×C₁₃	#2248	1	1	yes	characteristic
101	C₂×C₁₃	#2251	1	1	yes	characteristic
102	C₂×Q₈	—	4	2	no	—
103	C₂×Q₈	—	52	1	no	—
104	C₂×Q₈	—	52	1	no	—
105	Q₁₆	—	8	4	no	—
106	Q₁₆	—	52	4	no	—
107	C₂×C₈	#6104	1	2	yes	—
108	C₂×C₈	#6104	1	2	yes	—
109	C₂×C₈	D₄×D₁₃	1	2	yes	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	52	1	no	—
113	C₄⋊C₄	—	52	1	no	—
114	C₄⋊C₄	—	52	1	no	—
115	C₄⋊C₄	—	52	1	no	—
116	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
117	C₁₆	—	26	4	no	—
118	C₁₃	#12535	1	1	yes	characteristic, complemented, Sylow
119	Q₈	—	8	4	no	—
120	Q₈	—	104	2	no	—
121	Q₈	—	104	2	no	—
122	C₂×C₄	—	8	2	no	—
123	C₂×C₄	—	104	1	no	—
124	C₂×C₄	—	104	1	no	—
125	C₂×C₄	C₂×#6104	1	1	yes	characteristic
126	C₂×C₄	C₂×#6104	1	1	yes	characteristic
127	C₂×C₄	#88194	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	#88067	1	4	yes	—
130	C₈	#88172	1	4	yes	—
131	C₂²	#11718734	1	1	yes	characteristic
132	C₄	—	16	2	no	—
133	C₄	—	208	1	no	—
134	C₄	—	208	1	no	—
135	C₄	C₂×#88067	1	2	yes	—
136	C₄	#11718840	1	2	yes	—
137	C₄	#11718851	1	2	yes	—
138	C₂	#13114508	1	1	yes	characteristic
139	C₂	#13117091	1	1	yes	characteristic
140	C₂	#13117105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117116

C₂

yes

maximal

#13117065

C₂

yes

maximal

#13114433

C₂

yes

maximal, characteristic

#13112754

C₂

yes

maximal, characteristic

C₁₃×#1713

C₂

yes

maximal, characteristic

C₁₃×#500

—

C₁₃×#500

C₂²

yes

—

C₁₃×#494

—

#11718360

C₂²

yes

characteristic

#11718343

C₂²

yes

characteristic

#11718342

—

#11718341

—

C₈²×C₁₃

C₂²

yes

—

—

C₂²

yes

—

C₂×C₁₃×Q₁₆

—

C₁₃×C₄⋊Q₈

—

C₂×Dic₅₂

—

#88123

—

#88123

—

C₁₃×#107

—

C₁₃×#107

—

C₁₃×#107

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

—

—

—

D₄

yes

—

#12535

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

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

—

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

SD₃₂

yes

—

C₈×C₁₃

#137

yes

—

#500

—

#500

—

#500

C₂×D₁₃

yes

—

—

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#493

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

#497

yes

—

C₄×C₁₃

C₂×SD₃₂

yes

—

C₄×C₁₃

#510

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

—

—

—

—

—

—

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

C₄×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₃

#1722

yes

characteristic

100

C₂×C₁₃

#2248

yes

characteristic

101

C₂×C₁₃

#2251

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

Q₁₆

—

106

Q₁₆

—

107

C₂×C₈

#6104

yes

—

108

C₂×C₈

#6104

yes

—

109

C₂×C₈

D₄×D₁₃

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

117

C₁₆

—

118

C₁₃

#12535

yes

characteristic, complemented, Sylow

119

Q₈

—

120

Q₈

—

104

—

121

Q₈

—

104

—

122

C₂×C₄

—

123

C₂×C₄

—

104

—

124

C₂×C₄

—

104

—

125

C₂×C₄

C₂×#6104

yes

characteristic

126

C₂×C₄

C₂×#6104

yes

characteristic

127

C₂×C₄

#88194

yes

characteristic

128

C₈

—

129

C₈

#88067

yes

—

130

C₈

#88172

yes

—

131

C₂²

#11718734

yes

characteristic

132

C₄

—

133

C₄

—

208

—

134

C₄

—

208

—

135

C₄

C₂×#88067

yes

—

136

C₄

#11718840

yes

—

137

C₄

#11718851

yes

—

138

C₂

#13114508

yes

characteristic

139

C₂

#13117091

yes

characteristic

140

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	#12535
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
118. C₁₃	#12535	semidirect

H ≅ G/N

type

118. C₁₃

#12535

semidirect

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

subgroup	index
#13117116	2
#13117065	2
#13114433	2
#13112754	2
C₁₃×#1713	2
#12535	13

subgroup

index

C₁₃×#1713

Subgroups — #15777728

Conjugacy classes of subgroups

Complements

Maximal subgroups