Group #15777606 — subgroups

Subgroups — #15777606

Order 3328.

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

normal subgroups only

This group has 306 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	#13117061	C₂	1	4	yes	maximal, complemented
2	#13114451	C₂	1	1	yes	maximal, characteristic
3	C₈×#6090	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#500	—	2	1	no	—
6	C₁₃×#500	C₂²	1	2	yes	—
7	C₁₃×#494	—	2	1	no	—
8	#11718356	—	2	1	no	—
9	#11718356	—	2	1	no	—
10	#11718356	C₂²	1	2	yes	—
11	C₈²×C₁₃	C₂²	1	1	yes	characteristic
12	#11718083	—	2	8	no	—
13	C₄×#6090	C₂²	1	2	yes	—
14	C₂×C₁₃×Q₁₆	—	4	2	no	—
15	C₁₃×C₄⋊Q₈	—	2	2	no	—
16	C₂×D₁₀₄	—	2	4	no	—
17	C₂×D₁₀₄	—	4	2	no	—
18	C₂×D₁₀₄	—	4	2	no	—
19	#88128	—	2	2	no	—
20	C₁₃×#107	—	2	4	no	—
21	C₁₃×#107	—	4	2	no	—
22	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×#6090	D₄	1	4	yes	—
26	#12509	—	13	1	no	maximal, Sylow
27	C₂×Q₈×C₁₃	—	4	2	no	—
28	C₂×D₅₂	—	4	2	no	—
29	C₂×D₅₂	—	4	2	no	—
30	C₂×D₅₂	—	4	2	no	—
31	C₁₃×Q₁₆	—	8	4	no	—
32	C₂×C₈×C₁₃	D₈	1	2	yes	—
33	C₂×C₈×C₁₃	D₈	1	2	yes	—
34	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
35	C₁₃×C₄⋊C₄	—	4	2	no	—
36	C₁₃×C₄⋊C₄	—	4	2	no	—
37	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
38	D₁₀₄	—	4	8	no	—
39	D₁₀₄	—	8	4	no	—
40	D₁₀₄	—	8	4	no	—
41	#6090	—	2	4	no	—
42	#2240	—	13	4	no	—
43	#1713	D₁₃	1	1	yes	characteristic, complemented
44	#1712	—	13	1	no	—
45	C₈×C₁₆	—	13	1	no	—
46	C₂²×D₁₃	—	8	2	no	—
47	Q₈×C₁₃	—	8	4	no	—
48	C₂×C₄×C₁₃	—	8	2	no	—
49	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
50	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
51	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
52	D₅₂	—	8	4	no	—
53	D₅₂	—	8	4	no	—
54	D₅₂	—	8	4	no	—
55	C₈×C₁₃	D₁₆	1	4	yes	—
56	C₈×C₁₃	SD₃₂	1	4	yes	—
57	C₈×C₁₃	#135	1	4	yes	—
58	#500	—	2	1	no	—
59	#500	C₂×D₁₃	1	2	yes	—
60	#494	—	2	1	no	—
61	#493	—	13	2	no	—
62	#493	—	26	1	no	—
63	#493	—	26	1	no	—
64	#357	—	26	8	no	—
65	C₄×C₁₆	—	13	2	no	—
66	C₈²	C₂×D₁₃	1	1	yes	characteristic
67	C₂²×C₁₃	#493	1	1	yes	characteristic
68	C₂×D₁₃	—	16	2	no	—
69	C₂×D₁₃	—	16	2	no	—
70	C₂×D₁₃	—	16	2	no	—
71	C₄×C₁₃	—	16	2	no	—
72	C₄×C₁₃	#495	1	2	yes	—
73	C₄×C₁₃	C₂×D₁₆	1	2	yes	—
74	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
75	C₂×Q₁₆	—	4	2	no	—
76	C₂×D₈	—	26	4	no	—
77	C₂×D₈	—	52	2	no	—
78	C₂×D₈	—	52	2	no	—
79	C₄⋊Q₈	—	2	2	no	—
80	#127	—	26	2	no	—
81	C₂×C₁₆	—	13	4	no	—
82	#107	—	2	4	no	—
83	#107	—	4	2	no	—
84	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
85	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
86	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
87	C₂×C₁₃	#1709	1	1	yes	characteristic
88	C₂×C₁₃	#2246	1	1	yes	characteristic
89	C₂×C₁₃	#2248	1	1	yes	characteristic
90	D₁₃	—	32	2	no	—
91	C₂×Q₈	—	4	2	no	—
92	C₂×D₄	—	52	2	no	—
93	C₂×D₄	—	52	2	no	—
94	C₂×D₄	—	52	2	no	—
95	Q₁₆	—	8	4	no	—
96	D₈	—	52	8	no	—
97	D₈	—	104	4	no	—
98	D₈	—	104	4	no	—
99	C₂×C₈	#6104	1	2	yes	—
100	C₂×C₈	#6104	1	2	yes	—
101	C₂×C₈	D₄×D₁₃	1	2	yes	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄²	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
105	C₁₆	—	26	4	no	—
106	C₁₃	#12509	1	1	yes	characteristic, complemented, Sylow
107	C₂³	—	104	2	no	—
108	Q₈	—	8	4	no	—
109	D₄	—	104	4	no	—
110	D₄	—	104	4	no	—
111	D₄	—	104	4	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	C₂×#6104	1	1	yes	characteristic
114	C₂×C₄	C₂×#6104	1	1	yes	characteristic
115	C₂×C₄	#88194	1	1	yes	characteristic
116	C₈	#88066	1	4	yes	—
117	C₈	#88068	1	4	yes	—
118	C₈	#88173	1	4	yes	—
119	C₂²	—	208	2	no	—
120	C₂²	—	208	2	no	—
121	C₂²	—	208	2	no	—
122	C₂²	#11718734	1	1	yes	characteristic
123	C₄	—	16	2	no	—
124	C₄	C₂×#88066	1	2	yes	—
125	C₄	C₂×#88068	1	2	yes	—
126	C₄	#11718852	1	2	yes	—
127	C₂	—	416	2	no	—
128	C₂	#13114443	1	1	yes	characteristic
129	C₂	#13117085	1	1	yes	characteristic
130	C₂	#13117093	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13117061

C₂

yes

maximal, complemented

#13114451

C₂

yes

maximal, characteristic

C₈×#6090

C₂

yes

maximal, characteristic, complemented

C₁₃×#1713

C₂

yes

maximal, characteristic, complemented

C₁₃×#500

—

C₁₃×#500

C₂²

yes

—

C₁₃×#494

—

#11718356

—

#11718356

—

#11718356

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

#11718083

—

C₄×#6090

C₂²

yes

—

C₂×C₁₃×Q₁₆

—

C₁₃×C₄⋊Q₈

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

#88128

—

C₁₃×#107

—

C₁₃×#107

—

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

D₄

yes

characteristic

C₄×C₈×C₁₃

C₂³

yes

characteristic

C₂×#6090

D₄

yes

—

#12509

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

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

D₁₀₄

—

D₁₀₄

—

D₁₀₄

—

#6090

—

#2240

—

#1713

D₁₃

yes

characteristic, complemented

#1712

—

C₈×C₁₆

—

C₂²×D₁₃

—

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

D₅₂

—

D₅₂

—

D₅₂

—

C₈×C₁₃

D₁₆

yes

—

C₈×C₁₃

SD₃₂

yes

—

C₈×C₁₃

#135

yes

—

#500

—

#500

C₂×D₁₃

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#493

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#495

yes

—

C₄×C₁₃

C₂×D₁₆

yes

—

C₄×C₁₃

C₂×SD₃₂

yes

—

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#107

—

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

C₄×C₈

C₂²⋊D₁₃

yes

characteristic

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

D₁₃

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

D₈

—

D₈

—

104

—

D₈

—

104

—

C₂×C₈

#6104

yes

—

100

C₂×C₈

#6104

yes

—

101

C₂×C₈

D₄×D₁₃

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₂×C₂²⋊D₁₃

yes

characteristic

105

C₁₆

—

106

C₁₃

#12509

yes

characteristic, complemented, Sylow

107

C₂³

—

104

—

108

Q₈

—

109

D₄

—

104

—

110

D₄

—

104

—

111

D₄

—

104

—

112

C₂×C₄

—

113

C₂×C₄

C₂×#6104

yes

characteristic

114

C₂×C₄

C₂×#6104

yes

characteristic

115

C₂×C₄

#88194

yes

characteristic

116

C₈

#88066

yes

—

117

C₈

#88068

yes

—

118

C₈

#88173

yes

—

119

C₂²

—

208

—

120

C₂²

—

208

—

121

C₂²

—

208

—

122

C₂²

#11718734

yes

characteristic

123

C₄

—

124

C₄

C₂×#88066

yes

—

125

C₄

C₂×#88068

yes

—

126

C₄

#11718852

yes

—

127

C₂

—

416

—

128

C₂

#13114443

yes

characteristic

129

C₂

#13117085

yes

characteristic

130

C₂

#13117093

yes

characteristic

	subgroup	G/H
center	C₂²	#11718734
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1713	C₂
socle	C₂²×C₁₃	#493
prime radical	#13114451	C₂
hypercenter	#1713	D₁₃
hypocenter	C₁₃	#12509
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

C₂

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

N	H ≅ G/N	type
1. #13117061	C₂	semidirect
3. C₈×#6090	C₂	semidirect
4. C₁₃×#1713	C₂	semidirect
43. #1713	D₁₃	semidirect
106. C₁₃	#12509	semidirect

H ≅ G/N

type

1. #13117061

C₂

semidirect

3. C₈×#6090

C₂

semidirect

4. C₁₃×#1713

C₂

semidirect

43. #1713

D₁₃

semidirect

106. C₁₃

#12509

semidirect

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

subgroup	index
#13117061	2
#13114451	2
C₈×#6090	2
C₁₃×#1713	2
#12509	13

subgroup

index

C₁₃×#1713

Subgroups — #15777606

Conjugacy classes of subgroups

Complements

Maximal subgroups