Group #15765996 — subgroups

Subgroups — #15765996

Order 3328.

476 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 93 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	#13115659	C₂	1	3	yes	maximal
2	#13112717	C₂	1	3	yes	maximal, complemented
3	C₁₃×#2845	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88128	C₂²	1	1	yes	characteristic
5	C₁₃×#398	—	2	6	no	—
6	C₁₃×#382	C₂²	1	3	yes	—
7	C₂×#88049	—	2	6	no	—
8	C₂×#88043	C₂²	1	3	yes	—
9	C₂²×D₅₂	—	2	3	no	—
10	C₂×C₁₃×C₄⋊C₄	—	2	6	no	—
11	C₂×C₁₃×C₄⋊C₄	D₄	1	6	yes	—
12	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
13	C₂²×#1055	—	2	3	no	—
14	#88128	—	2	3	no	—
15	#88128	Q₈	1	2	yes	—
16	C₁₃×#95	—	2	2	no	—
17	C₁₃×#95	—	2	6	no	—
18	#88049	—	4	12	no	—
19	#88043	—	2	6	no	—
20	#10360	—	13	1	no	maximal, Sylow
21	C₂³×D₁₃	—	4	1	no	—
22	C₂²×C₄×C₁₃	—	2	6	no	—
23	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
24	C₂×D₅₂	—	2	6	no	—
25	C₂×D₅₂	—	4	6	no	—
26	C₂×D₅₂	—	4	6	no	—
27	C₂×D₅₂	—	4	3	no	—
28	C₁₃×C₄⋊C₄	—	4	6	no	—
29	C₁₃×C₄⋊C₄	—	8	6	no	—
30	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
31	C₄²×C₁₃	#41	1	3	yes	—
32	C₂×#1055	—	2	6	no	—
33	C₂×#1055	—	4	3	no	—
34	C₂×#1055	—	4	3	no	—
35	#2845	D₁₃	1	1	yes	characteristic, complemented
36	#1961	—	13	3	no	—
37	#1295	—	13	3	no	—
38	C₂³×C₁₃	#120	1	1	yes	characteristic
39	C₂²×D₁₃	—	4	2	no	—
40	C₂²×D₁₃	—	8	3	no	—
41	C₂²×D₁₃	—	8	3	no	—
42	C₂×C₄×C₁₃	—	2	3	no	—
43	C₂×C₄×C₁₃	—	2	3	no	—
44	C₂×C₄×C₁₃	—	4	6	no	—
45	C₂×C₄×C₁₃	—	4	6	no	—
46	C₂×C₄×C₁₃	—	4	6	no	—
47	C₂×C₄×C₁₃	C₂²⋊Q₈	1	3	yes	—
48	C₂×C₄×C₁₃	#124	1	3	yes	—
49	D₅₂	—	4	12	no	—
50	D₅₂	—	8	6	no	—
51	#1055	—	4	6	no	—
52	C₂×#127	—	13	1	no	—
53	C₂×#102	—	26	6	no	—
54	C₂×#97	—	13	3	no	—
55	#398	—	2	6	no	—
56	#382	C₂×D₁₃	1	3	yes	—
57	C₂²×C₁₃	#393	1	1	yes	characteristic
58	C₂²×C₁₃	#453	1	3	yes	—
59	C₂²×C₁₃	#454	1	3	yes	—
60	C₂×D₁₃	—	8	6	no	—
61	C₂×D₁₃	—	16	3	no	—
62	C₂×D₁₃	—	16	1	no	—
63	C₄×C₁₃	—	2	3	no	—
64	C₄×C₁₃	—	2	3	no	—
65	C₄×C₁₃	—	8	6	no	—
66	C₂²×D₄	—	26	3	no	—
67	C₂²×C₈	—	26	3	no	—
68	#127	—	13	2	no	—
69	#127	—	26	3	no	—
70	C₂×C₄⋊C₄	—	2	6	no	—
71	C₂×C₄⋊C₄	C₂²⋊D₁₃	1	6	yes	—
72	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
73	#102	—	52	12	no	—
74	#97	—	26	6	no	—
75	#95	—	2	2	no	—
76	#95	—	2	6	no	—
77	C₂×C₁₃	#1656	1	3	yes	—
78	C₂×C₁₃	#1658	1	1	yes	characteristic
79	C₂×C₁₃	#2027	1	3	yes	—
80	D₁₃	—	16	2	no	—
81	C₂⁴	—	52	1	no	—
82	C₂×D₄	—	26	6	no	—
83	C₂×D₄	—	52	6	no	—
84	C₂×D₄	—	52	6	no	—
85	C₂×D₄	—	52	3	no	—
86	C₂²×C₄	—	2	6	no	—
87	C₂²×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
88	C₂×C₈	—	26	6	no	—
89	C₂×C₈	—	52	3	no	—
90	C₂×C₈	—	52	3	no	—
91	C₄⋊C₄	—	4	6	no	—
92	C₄⋊C₄	—	8	6	no	—
93	C₄²	Q₈×D₁₃	1	1	yes	characteristic
94	C₄²	#6131	1	3	yes	—
95	C₁₃	#10360	1	1	yes	characteristic, complemented, Sylow
96	C₂³	—	52	2	no	—
97	C₂³	—	104	3	no	—
98	C₂³	—	104	3	no	—
99	C₂³	#88207	1	1	yes	characteristic
100	D₄	—	52	12	no	—
101	D₄	—	104	6	no	—
102	C₂×C₄	—	2	3	no	—
103	C₂×C₄	—	2	3	no	—
104	C₂×C₄	—	4	6	no	—
105	C₂×C₄	—	4	6	no	—
106	C₂×C₄	—	4	6	no	—
107	C₂×C₄	#88200	1	3	yes	—
108	C₂×C₄	#88201	1	3	yes	—
109	C₈	—	52	6	no	—
110	C₂²	—	104	6	no	—
111	C₂²	—	208	3	no	—
112	C₂²	—	208	1	no	—
113	C₂²	#11718892	1	1	yes	characteristic
114	C₂²	#11718902	1	3	yes	—
115	C₂²	#11718906	1	3	yes	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	8	6	no	—
119	C₂	—	208	2	no	—
120	C₂	#13114129	1	3	yes	—
121	C₂	#13114138	1	1	yes	characteristic
122	C₂	#13115989	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115659

C₂

yes

maximal

#13112717

C₂

yes

maximal, complemented

C₁₃×#2845

C₂

yes

maximal, characteristic, complemented

C₂×#88128

C₂²

yes

characteristic

C₁₃×#398

—

C₁₃×#382

C₂²

yes

—

C₂×#88049

—

C₂×#88043

C₂²

yes

—

C₂²×D₅₂

—

C₂×C₁₃×C₄⋊C₄

—

C₂×C₁₃×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂²×#1055

—

#88128

—

#88128

Q₈

yes

—

C₁₃×#95

—

C₁₃×#95

—

—

—

—

C₂³×D₁₃

—

C₂²×C₄×C₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

#41

yes

—

—

—

—

D₁₃

yes

characteristic, complemented

#1961

—

#1295

—

C₂³×C₁₃

#120

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

#124

yes

—

D₅₂

—

D₅₂

—

#1055

—

C₂×#127

—

C₂×#102

—

C₂×#97

—

#398

—

#382

C₂×D₁₃

yes

—

C₂²×C₁₃

#393

yes

characteristic

C₂²×C₁₃

#453

yes

—

C₂²×C₁₃

#454

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₁₃

yes

—

C₂×C₄²

C₂²×D₁₃

yes

—

—

—

—

C₂×C₁₃

#1656

yes

—

C₂×C₁₃

#1658

yes

characteristic

C₂×C₁₃

#2027

yes

—

D₁₃

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₁₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₁₃

yes

characteristic

C₄²

#6131

yes

—

C₁₃

#10360

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

104

—

C₂³

—

104

—

C₂³

#88207

yes

characteristic

100

D₄

—

101

D₄

—

104

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#88200

yes

—

108

C₂×C₄

#88201

yes

—

109

C₈

—

110

C₂²

—

104

—

111

C₂²

—

208

—

112

C₂²

—

208

—

113

C₂²

#11718892

yes

characteristic

114

C₂²

#11718902

yes

—

115

C₂²

#11718906

yes

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₂

—

208

—

120

C₂

#13114129

yes

—

121

C₂

#13114138

yes

characteristic

122

C₂

#13115989

yes

—

	subgroup	G/H
center	C₂³	#88207
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#2845	C₂
socle	C₂³×C₁₃	#120
prime radical	C₂×#88128	C₂²
hypercenter	#2845	D₁₃
hypocenter	C₁₃	#10360
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#88207

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#2845

C₂

C₂³×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
2. #13112717	C₂	semidirect
3. C₁₃×#2845	C₂	semidirect
35. #2845	D₁₃	semidirect
95. C₁₃	#10360	semidirect

H ≅ G/N

type

2. #13112717

C₂

semidirect

3. C₁₃×#2845

C₂

semidirect

35. #2845

D₁₃

semidirect

95. C₁₃

#10360

semidirect

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

subgroup	index
#13115659	2
#13112717	2
C₁₃×#2845	2
#10360	13

subgroup

index

#13115659

#13112717

C₁₃×#2845

#10360

Subgroups — #15765996

Conjugacy classes of subgroups

Complements

Maximal subgroups