Group #15777655 — subgroups

Subgroups — #15777655

Order 3328.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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	#13117111	C₂	1	1	yes	maximal, characteristic
2	#13117109	C₂	1	2	yes	maximal
3	#13114453	C₂	1	2	yes	maximal
4	C₁₆×#1055	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#500	C₂²	1	2	yes	—
7	#11718753	—	2	2	no	—
8	#11718739	—	2	2	no	—
9	#11718343	C₂²	1	2	yes	—
10	C₁₃×#366	—	2	2	no	—
11	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	#11718096	—	2	2	no	—
13	#11718081	—	2	4	no	—
14	C₄×#6090	C₂²	1	1	yes	characteristic
15	C₈×#1055	C₂²	1	1	yes	characteristic
16	C₁₃×C₄⋊Q₈	—	2	2	no	—
17	C₂×#6107	—	4	4	no	—
18	#88123	—	2	2	no	—
19	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
20	C₁₃×#107	—	2	2	no	—
21	C₁₃×#107	—	2	2	no	—
22	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	#88058	—	2	2	no	—
24	#88058	—	2	2	no	—
25	C₂×#6090	D₄	1	2	yes	—
26	#88047	—	4	4	no	—
27	C₄×#1055	D₄	1	1	yes	characteristic
28	C₄×#1055	D₄	1	1	yes	characteristic
29	#12521	—	13	1	no	maximal, Sylow
30	C₂×Q₈×C₁₃	—	4	2	no	—
31	C₂×Dic₂₆	—	4	2	no	—
32	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
34	C₁₃×C₄⋊C₄	—	4	2	no	—
35	C₁₃×C₄⋊C₄	—	4	2	no	—
36	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
37	#6107	—	8	8	no	—
38	C₄⋊Dic₁₃	—	4	2	no	—
39	C₄⋊Dic₁₃	—	4	2	no	—
40	C₂×#1055	D₈	1	2	yes	—
41	C₂×#1055	D₈	1	2	yes	—
42	C₁₃×C₁₆	Q₁₆	1	4	yes	complemented
43	#6090	Q₁₆	1	4	yes	complemented
44	#2252	—	13	2	no	—
45	#2252	—	13	1	no	—
46	#2252	D₁₃	1	1	yes	characteristic
47	#1713	—	13	2	no	—
48	C₈×C₁₆	—	13	1	no	—
49	Q₈×C₁₃	—	8	4	no	—
50	C₂×C₄×C₁₃	—	8	2	no	—
51	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
52	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
54	C₂×Dic₁₃	—	8	2	no	—
55	Dic₂₆	—	8	4	no	—
56	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
57	C₈×C₁₃	C₂×Q₁₆	1	2	yes	—
58	#1055	D₁₆	1	4	yes	—
59	#1055	Q₃₂	1	4	yes	—
60	#500	—	13	2	no	—
61	#500	—	26	2	no	—
62	#500	C₂×D₁₃	1	2	yes	—
63	#494	—	26	2	no	—
64	#366	—	2	2	no	—
65	#366	—	26	4	no	—
66	#366	—	26	2	no	—
67	C₄×C₁₆	—	13	1	no	—
68	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
69	C₈²	—	13	1	no	—
70	C₂²×C₁₃	#493	1	1	yes	characteristic
71	C₄×C₁₃	—	16	2	no	—
72	C₄×C₁₃	#494	1	1	yes	characteristic
73	C₄×C₁₃	#494	1	1	yes	characteristic
74	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
75	C₄×C₁₃	C₂×Q₃₂	1	2	yes	—
76	Dic₁₃	—	16	2	no	—
77	C₂×Q₁₆	—	52	4	no	—
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	—	26	2	no	—
80	C₂×C₁₆	—	13	2	no	—
81	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
82	#107	—	2	2	no	—
83	#107	—	2	2	no	—
84	#107	—	26	2	no	—
85	#107	—	26	2	no	—
86	#107	—	52	4	no	—
87	C₄×C₈	—	13	1	no	—
88	C₄×C₈	—	13	1	no	—
89	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
90	C₂×C₁₃	#1713	1	1	yes	characteristic
91	C₂×C₁₃	#2246	1	1	yes	characteristic
92	C₂×C₁₃	#2247	1	1	yes	characteristic
93	C₂×Q₈	—	4	2	no	—
94	C₂×Q₈	—	52	2	no	—
95	Q₁₆	—	104	8	no	—
96	C₂×C₈	—	13	2	no	—
97	C₂×C₈	—	13	2	no	—
98	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
99	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	52	2	no	—
103	C₄⋊C₄	—	52	2	no	—
104	C₄²	D₄×D₁₃	1	1	yes	characteristic
105	C₁₆	—	13	4	no	—
106	C₁₆	#6107	1	4	yes	complemented
107	C₁₃	#12521	1	1	yes	characteristic, complemented, Sylow
108	Q₈	—	8	4	no	—
109	Q₈	—	104	4	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	104	2	no	—
112	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
113	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
114	C₂×C₄	#88194	1	1	yes	characteristic
115	C₈	—	13	4	no	—
116	C₈	—	13	4	no	—
117	C₈	Q₁₆×D₁₃	1	2	yes	—
118	C₈	C₂×#6107	1	2	yes	—
119	C₂²	#11718812	1	1	yes	characteristic
120	C₄	—	16	2	no	—
121	C₄	—	208	2	no	—
122	C₄	D₁₃×D₁₆	1	2	yes	—
123	C₄	D₁₃×Q₃₂	1	2	yes	—
124	C₄	#11718753	1	1	yes	characteristic
125	C₄	#11718842	1	1	yes	characteristic
126	C₂	#13114453	1	1	yes	characteristic
127	C₂	#13117084	1	1	yes	characteristic
128	C₂	#13117087	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117111

C₂

yes

maximal, characteristic

#13117109

C₂

yes

maximal

#13114453

C₂

yes

maximal

C₁₆×#1055

C₂

yes

maximal, characteristic

C₁₃×#2252

C₂

yes

maximal, characteristic

C₁₃×#500

C₂²

yes

—

#11718753

—

#11718739

—

#11718343

C₂²

yes

—

C₁₃×#366

—

C₄×C₁₃×C₁₆

C₂²

yes

—

—

C₂²

yes

characteristic

C₈×#1055

C₂²

yes

characteristic

C₁₃×C₄⋊Q₈

—

C₂×#6107

—

#88123

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₁₃×#107

—

C₁₃×#107

—

C₄×C₈×C₁₃

C₂³

yes

—

—

D₄

yes

—

#88047

—

C₄×#1055

D₄

yes

characteristic

C₄×#1055

D₄

yes

characteristic

#12521

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃

C₂×D₄

yes

characteristic

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

#6107

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₂×#1055

D₈

yes

—

C₂×#1055

D₈

yes

—

C₁₃×C₁₆

Q₁₆

yes

complemented

#6090

Q₁₆

yes

—

—

D₁₃

yes

characteristic

#1713

—

C₈×C₁₆

—

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

—

Dic₂₆

—

C₈×C₁₃

C₂×Q₁₆

yes

—

C₈×C₁₃

C₂×Q₁₆

yes

—

#1055

D₁₆

yes

—

#1055

Q₃₂

yes

—

#500

—

#500

—

#500

C₂×D₁₃

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₈²

—

C₂²×C₁₃

#493

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

C₂×D₁₆

yes

—

C₄×C₁₃

C₂×Q₃₂

yes

—

Dic₁₃

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₃

yes

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

104

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₁₃

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

D₄×D₁₃

yes

characteristic

105

C₁₆

—

106

C₁₆

#6107

yes

complemented

107

C₁₃

#12521

yes

characteristic, complemented, Sylow

108

Q₈

—

109

Q₈

—

104

—

110

C₂×C₄

—

111

C₂×C₄

—

104

—

112

C₂×C₄

D₈×D₁₃

yes

characteristic

113

C₂×C₄

D₈×D₁₃

yes

characteristic

114

C₂×C₄

#88194

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

Q₁₆×D₁₃

yes

—

118

C₈

C₂×#6107

yes

—

119

C₂²

#11718812

yes

characteristic

120

C₄

—

121

C₄

—

208

—

122

C₄

D₁₃×D₁₆

yes

—

123

C₄

D₁₃×Q₃₂

yes

—

124

C₄

#11718753

yes

characteristic

125

C₄

#11718842

yes

characteristic

126

C₂

#13114453

yes

characteristic

127

C₂

#13117084

yes

characteristic

128

C₂

#13117087

yes

characteristic

	subgroup	G/H
center	C₂²	#11718812
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2252	C₂
socle, prime radical	C₂²×C₁₃	#493
hypercenter	#2252	D₁₃
hypocenter	C₁₃	#12521
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#11718812

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#2252

C₂

C₂²×C₁₃

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
42. C₁₃×C₁₆	Q₁₆	semidirect
43. #6090	Q₁₆	semidirect
106. C₁₆	#6107	semidirect
107. C₁₃	#12521	semidirect

H ≅ G/N

type

42. C₁₃×C₁₆

Q₁₆

semidirect

43. #6090

Q₁₆

semidirect

106. C₁₆

#6107

semidirect

107. C₁₃

#12521

semidirect

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

subgroup	index
#13117111	2
#13117109	2
#13114453	2
C₁₆×#1055	2
C₁₃×#2252	2
#12521	13

subgroup

index

#13117111

#13117109

#13114453

C₁₆×#1055

C₁₃×#2252

#12521

Subgroups — #15777655

Conjugacy classes of subgroups

Complements

Maximal subgroups