Group #15777653 — subgroups

Subgroups — #15777653

Order 3328.

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

normal subgroups only

This group has 316 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	#13117091	C₂	1	2	yes	maximal
2	#13117088	C₂	1	2	yes	maximal, complemented
3	#13114454	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112753	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1709	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#494	C₂²	1	1	yes	characteristic
7	C₁₃×#493	C₂²	1	1	yes	characteristic
8	C₁₃×#486	—	2	2	no	—
9	C₂×#88069	—	2	4	no	—
10	C₂×#88067	—	2	4	no	—
11	#11718360	C₂²	1	2	yes	—
12	#11718343	—	2	2	no	—
13	C₈²×C₁₃	C₂²	1	1	yes	characteristic
14	C₄×#6090	C₂²	1	2	yes	—
15	C₂×C₁₃×Q₁₆	—	2	2	no	—
16	C₂×C₁₃×D₈	—	2	2	no	—
17	C₁₃×C₄⋊Q₈	—	2	1	no	—
18	C₁₃×#127	—	2	1	no	—
19	C₂×Dic₅₂	—	2	2	no	—
20	C₂×Dic₅₂	—	2	2	no	—
21	#88123	—	2	2	no	—
22	C₁₃×#102	—	4	4	no	—
23	C₄×C₈×C₁₃	D₄	1	2	yes	—
24	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	#88069	—	4	8	no	—
26	#88067	—	4	8	no	—
27	#88058	—	4	4	no	—
28	C₂×#6090	D₄	1	4	yes	—
29	#12520	—	13	1	no	maximal, Sylow
30	C₂×Q₈×C₁₃	—	4	1	no	—
31	C₂×D₄×C₁₃	—	4	1	no	—
32	C₂×D₄×C₁₃	—	4	2	no	—
33	C₂×Dic₂₆	—	4	2	no	—
34	C₁₃×Q₁₆	—	4	4	no	—
35	C₁₃×D₈	—	4	4	no	—
36	C₂×C₈×C₁₃	D₈	1	4	yes	—
37	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
38	C₁₃×C₄⋊C₄	—	4	2	no	—
39	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
40	C₄⋊Dic₁₃	—	4	4	no	—
41	Dic₅₂	—	4	4	no	—
42	Dic₅₂	—	4	4	no	—
43	#6090	D₈	1	8	yes	—
44	#2248	—	13	2	no	—
45	#2247	—	13	2	no	—
46	#1713	—	13	1	no	—
47	#1709	D₁₃	1	1	yes	characteristic
48	#1311	—	13	1	no	—
49	C₂³×C₁₃	—	8	1	no	—
50	Q₈×C₁₃	—	8	2	no	—
51	D₄×C₁₃	—	8	2	no	—
52	D₄×C₁₃	—	8	4	no	—
53	C₂×C₄×C₁₃	—	8	1	no	—
54	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
55	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
56	C₂×Dic₁₃	—	8	2	no	—
57	Dic₂₆	—	8	2	no	—
58	Dic₂₆	—	8	2	no	—
59	C₈×C₁₃	—	2	4	no	—
60	C₈×C₁₃	C₂×D₈	1	4	yes	—
61	C₂×Q₃₂	—	26	4	no	—
62	C₂×SD₃₂	—	26	4	no	—
63	#500	—	26	2	no	—
64	#494	—	13	2	no	—
65	#494	C₂×D₁₃	1	1	yes	characteristic
66	#493	C₂×D₁₃	1	1	yes	characteristic
67	#486	—	2	2	no	—
68	C₄×C₁₆	—	13	2	no	—
69	C₈²	C₂×D₁₃	1	1	yes	characteristic
70	C₂²×C₁₃	—	16	1	no	—
71	C₂²×C₁₃	—	16	2	no	—
72	C₂²×C₁₃	#493	1	1	yes	characteristic
73	C₄×C₁₃	—	16	1	no	—
74	C₄×C₁₃	#493	1	2	yes	—
75	C₄×C₁₃	#510	1	4	yes	—
76	Dic₁₃	—	16	2	no	—
77	C₂×Q₁₆	—	2	2	no	—
78	C₂×Q₁₆	—	26	2	no	—
79	C₂×Q₁₆	—	26	2	no	—
80	C₂×D₈	—	2	2	no	—
81	C₄⋊Q₈	—	2	1	no	—
82	C₄⋊Q₈	—	26	2	no	—
83	#127	—	2	1	no	—
84	Q₃₂	—	52	8	no	—
85	SD₃₂	—	52	8	no	—
86	C₂×C₁₆	—	13	4	no	—
87	#107	—	52	4	no	—
88	#102	—	4	4	no	—
89	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
90	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
91	C₂×C₁₃	—	32	1	no	—
92	C₂×C₁₃	#1712	1	1	yes	characteristic
93	C₂×C₁₃	#2251	1	2	yes	—
94	C₂×Q₈	—	4	1	no	—
95	C₂×Q₈	—	52	2	no	—
96	C₂×D₄	—	4	2	no	—
97	C₂×D₄	—	4	1	no	—
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	52	4	no	—
100	Q₁₆	—	52	4	no	—
101	D₈	—	4	4	no	—
102	C₂×C₈	#6104	1	4	yes	—
103	C₂×C₈	D₄×D₁₃	1	2	yes	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	52	4	no	—
106	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
107	C₁₆	—	13	8	no	—
108	C₁₃	#12520	1	1	yes	characteristic, complemented, Sylow
109	C₂³	—	8	1	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	104	2	no	—
112	Q₈	—	104	2	no	—
113	D₄	—	8	2	no	—
114	D₄	—	8	4	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	—	104	2	no	—
117	C₂×C₄	C₂×#6104	1	2	yes	—
118	C₂×C₄	#88194	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	D₈×D₁₃	1	4	yes	—
121	C₂²	—	16	1	no	—
122	C₂²	—	16	2	no	—
123	C₂²	#11718734	1	1	yes	characteristic
124	C₄	—	16	1	no	—
125	C₄	—	208	2	no	—
126	C₄	#11718812	1	2	yes	—
127	C₄	#11718856	1	4	yes	—
128	C₂	—	32	1	no	—
129	C₂	#13114450	1	1	yes	characteristic
130	C₂	#13117108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13117091

C₂

yes

maximal

#13117088

C₂

yes

maximal, complemented

#13114454

C₂

yes

maximal, characteristic, complemented

#13112753

C₂

yes

maximal, characteristic, complemented

C₁₃×#1709

C₂

yes

maximal, characteristic

C₁₃×#494

C₂²

yes

characteristic

C₁₃×#493

C₂²

yes

characteristic

C₁₃×#486

—

C₂×#88069

—

C₂×#88067

—

#11718360

C₂²

yes

—

#11718343

—

C₈²×C₁₃

C₂²

yes

characteristic

C₄×#6090

C₂²

yes

—

C₂×C₁₃×Q₁₆

—

C₂×C₁₃×D₈

—

C₁₃×C₄⋊Q₈

—

C₁₃×#127

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

#88123

—

C₁₃×#102

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

—

—

—

D₄

yes

—

#12520

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×Dic₂₆

—

C₁₃×Q₁₆

—

C₁₃×D₈

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

Dic₅₂

—

Dic₅₂

—

#6090

D₈

yes

—

—

—

—

D₁₃

yes

characteristic

#1311

—

C₂³×C₁₃

—

Q₈×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×Dic₁₃

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

C₂×D₈

yes

—

C₂×Q₃₂

—

C₂×SD₃₂

—

#500

—

#494

—

#494

C₂×D₁₃

yes

characteristic

#493

C₂×D₁₃

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#493

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

#493

yes

—

C₄×C₁₃

#510

yes

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

C₂×C₁₆

—

#107

—

#102

—

C₄×C₈

C₂²⋊D₁₃

yes

—

C₄×C₈

C₂²×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

#1712

yes

characteristic

C₂×C₁₃

#2251

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

100

Q₁₆

—

101

D₈

—

102

C₂×C₈

#6104

yes

—

103

C₂×C₈

D₄×D₁₃

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

107

C₁₆

—

108

C₁₃

#12520

yes

characteristic, complemented, Sylow

109

C₂³

—

110

Q₈

—

111

Q₈

—

104

—

112

Q₈

—

104

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

104

—

117

C₂×C₄

C₂×#6104

yes

—

118

C₂×C₄

#88194

yes

characteristic

119

C₈

—

120

C₈

D₈×D₁₃

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

#11718734

yes

characteristic

124

C₄

—

125

C₄

—

208

—

126

C₄

#11718812

yes

—

127

C₄

#11718856

yes

—

128

C₂

—

129

C₂

#13114450

yes

characteristic

130

C₂

#13117108

yes

—

	subgroup	G/H
center	C₂²	#11718734
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1709	C₂
socle	C₂²×C₁₃	#493
prime radical	C₁₃×#493	C₂²
hypercenter	#1709	D₁₃
hypocenter	C₁₃	#12520
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#11718734

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1709

C₂

socle

C₂²×C₁₃

#493

prime radical

C₁₃×#493

C₂²

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
2. #13117088	C₂	semidirect
3. #13114454	C₂	semidirect
4. #13112753	C₂	semidirect
108. C₁₃	#12520	semidirect

H ≅ G/N

type

2. #13117088

C₂

semidirect

3. #13114454

C₂

semidirect

4. #13112753

C₂

semidirect

108. C₁₃

#12520

semidirect

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

subgroup	index
#13117091	2
#13117088	2
#13114454	2
#13112753	2
C₁₃×#1709	2
#12520	13

subgroup

index

C₁₃×#1709

Subgroups — #15777653

Conjugacy classes of subgroups

Complements

Maximal subgroups