Group #15777664 — subgroups

Subgroups — #15777664

Order 3328.

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

normal subgroups only

This group has 246 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	#13117109	C₂	1	2	yes	maximal
3	#13114446	C₂	1	1	yes	maximal, characteristic
4	#13112753	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	#11718343	C₂²	1	1	yes	characteristic
10	#11718343	C₂²	1	1	yes	characteristic
11	#11718342	—	2	1	no	—
12	#11718342	—	2	1	no	—
13	C₈²×C₁₃	C₂²	1	1	yes	characteristic
14	#11718082	—	2	4	no	—
15	#11718081	—	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	#88123	—	2	1	no	—
20	#88123	—	2	1	no	—
21	C₁₃×#107	—	2	2	no	—
22	C₁₃×#107	—	2	2	no	—
23	C₁₃×#107	—	4	2	no	—
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	—	2	2	no	—
29	#88056	—	4	2	no	—
30	#88056	—	4	2	no	—
31	C₂×#6090	D₄	1	4	yes	—
32	#12524	—	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	#6090	Q₁₆	1	8	yes	complemented
48	#2254	—	13	2	no	—
49	#2252	—	13	2	no	—
50	#1713	D₁₃	1	1	yes	characteristic
51	#1710	—	13	1	no	—
52	#1311	—	13	1	no	—
53	Q₈×C₁₃	—	8	4	no	—
54	C₂×C₄×C₁₃	—	8	2	no	—
55	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
56	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
58	C₂×Dic₁₃	—	8	1	no	—
59	C₂×Dic₁₃	—	8	1	no	—
60	Dic₂₆	—	8	2	no	—
61	Dic₂₆	—	8	2	no	—
62	C₈×C₁₃	—	2	2	no	—
63	C₈×C₁₃	—	2	2	no	—
64	C₈×C₁₃	C₂×Q₁₆	1	4	yes	—
65	#500	—	2	1	no	—
66	#500	—	13	1	no	—
67	#500	—	13	1	no	—
68	#500	C₂×D₁₃	1	2	yes	—
69	#494	—	2	1	no	—
70	#487	—	26	1	no	—
71	#487	—	26	1	no	—
72	#367	—	26	4	no	—
73	#366	—	26	4	no	—
74	C₄×C₁₆	—	13	2	no	—
75	C₈²	C₂×D₁₃	1	1	yes	characteristic
76	C₂²×C₁₃	#493	1	1	yes	characteristic
77	C₄×C₁₃	—	16	2	no	—
78	C₄×C₁₃	#494	1	2	yes	—
79	C₄×C₁₃	#509	1	2	yes	—
80	C₄×C₁₃	#510	1	2	yes	—
81	Dic₁₃	—	16	1	no	—
82	Dic₁₃	—	16	1	no	—
83	C₂×Q₁₆	—	4	2	no	—
84	C₄⋊Q₈	—	2	2	no	—
85	C₄⋊Q₈	—	26	1	no	—
86	C₄⋊Q₈	—	26	1	no	—
87	C₂×C₁₆	—	13	4	no	—
88	#107	—	2	2	no	—
89	#107	—	2	2	no	—
90	#107	—	4	2	no	—
91	#107	—	26	2	no	—
92	#107	—	26	2	no	—
93	#103	—	52	2	no	—
94	#103	—	52	2	no	—
95	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
96	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
98	C₂×C₁₃	#1713	1	1	yes	characteristic
99	C₂×C₁₃	#2250	1	1	yes	characteristic
100	C₂×C₁₃	#2251	1	1	yes	characteristic
101	C₂×Q₈	—	4	2	no	—
102	C₂×Q₈	—	52	1	no	—
103	C₂×Q₈	—	52	1	no	—
104	Q₁₆	—	8	4	no	—
105	C₂×C₈	#6104	1	2	yes	—
106	C₂×C₈	#6104	1	2	yes	—
107	C₂×C₈	D₄×D₁₃	1	2	yes	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	52	1	no	—
111	C₄⋊C₄	—	52	1	no	—
112	C₄⋊C₄	—	52	1	no	—
113	C₄⋊C₄	—	52	1	no	—
114	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
115	C₁₆	—	13	8	no	—
116	C₁₃	#12524	1	1	yes	characteristic, complemented, Sylow
117	Q₈	—	8	4	no	—
118	Q₈	—	104	2	no	—
119	Q₈	—	104	2	no	—
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	—	104	1	no	—
122	C₂×C₄	—	104	1	no	—
123	C₂×C₄	C₂×#6104	1	1	yes	characteristic
124	C₂×C₄	C₂×#6104	1	1	yes	characteristic
125	C₂×C₄	#88194	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₈	Q₁₆×D₁₃	1	4	yes	—
129	C₂²	#11718734	1	1	yes	characteristic
130	C₄	—	16	2	no	—
131	C₄	—	208	1	no	—
132	C₄	—	208	1	no	—
133	C₄	#11718808	1	2	yes	—
134	C₄	#11718840	1	2	yes	—
135	C₄	#11718842	1	2	yes	—
136	C₂	#13114452	1	1	yes	characteristic
137	C₂	#13117100	1	1	yes	characteristic
138	C₂	#13117105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117116

C₂

yes

maximal

#13117109

C₂

yes

maximal

#13114446

C₂

yes

maximal, characteristic

#13112753

C₂

yes

maximal, characteristic

C₁₃×#1713

C₂

yes

maximal, characteristic

C₁₃×#500

—

C₁₃×#500

C₂²

yes

—

C₁₃×#494

—

#11718343

C₂²

yes

characteristic

#11718343

C₂²

yes

characteristic

#11718342

—

#11718342

—

C₈²×C₁₃

C₂²

yes

—

—

C₂²

yes

—

C₂×C₁₃×Q₁₆

—

C₁₃×C₄⋊Q₈

—

#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

—

#12524

—

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

—

#6090

Q₁₆

yes

—

—

D₁₃

yes

characteristic

#1710

—

#1311

—

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

—

C₈×C₁₃

C₂×Q₁₆

yes

—

—

—

—

C₂×D₁₃

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#493

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

#494

yes

—

C₄×C₁₃

#509

yes

—

C₄×C₁₃

#510

yes

—

Dic₁₃

—

Dic₁₃

—

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

C₂×C₁₃

yes

C₂×C₁₃

yes

100

C₂×C₁₃

#2251

yes

characteristic

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

Q₁₆

—

105

C₂×C₈

#6104

yes

—

106

C₂×C₈

#6104

yes

—

107

C₂×C₈

D₄×D₁₃

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

115

C₁₆

—

116

C₁₃

#12524

yes

characteristic, complemented, Sylow

117

Q₈

—

118

Q₈

—

104

—

119

Q₈

—

104

—

120

C₂×C₄

—

121

C₂×C₄

—

104

—

122

C₂×C₄

—

104

—

123

C₂×C₄

C₂×#6104

yes

characteristic

124

C₂×C₄

C₂×#6104

yes

characteristic

125

C₂×C₄

#88194

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

Q₁₆×D₁₃

yes

—

129

C₂²

#11718734

yes

characteristic

130

C₄

—

131

C₄

—

208

—

132

C₄

—

208

—

133

C₄

#11718808

yes

—

134

C₄

#11718840

yes

—

135

C₄

#11718842

yes

—

136

C₂

#13114452

yes

characteristic

137

C₂

#13117100

yes

characteristic

138

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	#12524
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
47. #6090	Q₁₆	semidirect
116. C₁₃	#12524	semidirect

H ≅ G/N

type

47. #6090

Q₁₆

semidirect

116. C₁₃

#12524

semidirect

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

subgroup	index
#13117116	2
#13117109	2
#13114446	2
#13112753	2
C₁₃×#1713	2
#12524	13

subgroup

index

C₁₃×#1713

Subgroups — #15777664

Conjugacy classes of subgroups

Complements

Maximal subgroups