Group #15777590 — subgroups

Subgroups — #15777590

Order 3328.

302 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 83 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	#13117086	C₂	1	1	yes	maximal, characteristic
2	#13117061	C₂	1	2	yes	maximal, complemented
3	#13114429	C₂	1	2	yes	maximal, complemented
4	#13112752	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#500	C₂²	1	2	yes	—
7	#11718749	—	2	2	no	—
8	#11718746	—	2	2	no	—
9	C₂×D₂₀₈	—	2	2	no	—
10	#11718356	C₂²	1	2	yes	—
11	C₁₃×#367	—	2	2	no	—
12	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	#11718083	—	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×#6106	—	4	4	no	—
18	C₂×D₁₀₄	—	2	2	no	—
19	C₂×D₁₀₄	—	2	2	no	—
20	#88128	—	2	2	no	—
21	C₂×C₁₃×C₁₆	D₄	1	2	yes	—
22	C₁₃×#107	—	2	2	no	—
23	C₁₃×#107	—	2	2	no	—
24	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₂×#6090	D₄	1	2	yes	—
26	#88049	—	4	4	no	—
27	C₄×#1055	D₄	1	2	yes	—
28	D₂₀₈	—	4	4	no	—
29	#12506	—	13	1	no	maximal, Sylow
30	C₂×Q₈×C₁₃	—	4	2	no	—
31	C₂×D₅₂	—	4	2	no	—
32	C₂×D₅₂	—	4	2	no	—
33	C₂×D₅₂	—	4	2	no	—
34	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
36	C₁₃×C₄⋊C₄	—	4	2	no	—
37	C₁₃×C₄⋊C₄	—	4	2	no	—
38	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
39	#6106	—	8	8	no	—
40	C₂×#1055	D₈	1	4	yes	—
41	D₁₀₄	—	4	4	no	—
42	D₁₀₄	—	4	4	no	—
43	C₁₃×C₁₆	SD₁₆	1	4	yes	complemented
44	#6090	—	2	2	no	—
45	#2254	D₁₃	1	1	yes	characteristic, complemented
46	#2246	—	13	1	no	—
47	#2240	—	13	2	no	—
48	#1706	—	13	2	no	—
49	#1311	—	13	1	no	—
50	C₂²×D₁₃	—	8	2	no	—
51	Q₈×C₁₃	—	8	4	no	—
52	C₂×C₄×C₁₃	—	8	2	no	—
53	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
54	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
55	D₅₂	—	8	2	no	—
56	D₅₂	—	8	2	no	—
57	D₅₂	—	8	4	no	—
58	D₅₂	—	8	4	no	—
59	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
60	C₈×C₁₃	#135	1	2	yes	—
61	#1055	—	2	4	no	—
62	C₂×D₁₆	—	26	2	no	—
63	#500	C₂×D₁₃	1	2	yes	—
64	#493	—	13	2	no	—
65	#492	—	26	2	no	—
66	#486	—	26	2	no	—
67	#367	—	2	2	no	—
68	#357	—	26	4	no	—
69	C₄×C₁₆	—	13	1	no	—
70	C₄×C₁₆	C₂×D₁₃	1	1	yes	characteristic
71	C₈²	—	13	1	no	—
72	C₂²×C₁₃	#493	1	1	yes	characteristic
73	C₂×D₁₃	—	16	2	no	—
74	C₂×D₁₃	—	16	2	no	—
75	C₂×D₁₃	—	16	2	no	—
76	C₄×C₁₃	—	16	2	no	—
77	C₄×C₁₃	#492	1	1	yes	characteristic
78	C₄×C₁₃	#495	1	1	yes	characteristic
79	C₄×C₁₃	#509	1	4	yes	—
80	C₂×SD₁₆	—	52	4	no	—
81	C₂×D₈	—	26	2	no	—
82	C₂×D₈	—	26	2	no	—
83	C₄⋊Q₈	—	2	2	no	—
84	#127	—	26	2	no	—
85	D₁₆	—	52	4	no	—
86	C₂×C₁₆	—	13	2	no	—
87	C₂×C₁₆	C₂²⋊D₁₃	1	2	yes	—
88	#107	—	2	2	no	—
89	#107	—	2	2	no	—
90	#102	—	52	4	no	—
91	C₄×C₈	—	13	2	no	—
92	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
93	C₂×C₁₃	#1706	1	1	yes	characteristic
94	C₂×C₁₃	#2250	1	2	yes	—
95	D₁₃	—	32	2	no	—
96	C₂×Q₈	—	4	2	no	—
97	C₂×D₄	—	52	2	no	—
98	C₂×D₄	—	52	2	no	—
99	C₂×D₄	—	52	2	no	—
100	SD₁₆	—	104	8	no	—
101	D₈	—	52	4	no	—
102	D₈	—	52	4	no	—
103	C₂×C₈	—	13	4	no	—
104	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
105	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄²	D₄×D₁₃	1	1	yes	characteristic
109	C₁₆	—	26	2	no	—
110	C₁₆	#6106	1	4	yes	complemented
111	C₁₃	#12506	1	1	yes	characteristic, complemented, Sylow
112	C₂³	—	104	2	no	—
113	Q₈	—	8	4	no	—
114	D₄	—	104	2	no	—
115	D₄	—	104	4	no	—
116	D₄	—	104	4	no	—
117	D₄	—	104	2	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	D₈×D₁₃	1	2	yes	—
120	C₂×C₄	#88194	1	1	yes	characteristic
121	C₈	—	26	4	no	—
122	C₈	#88173	1	2	yes	—
123	C₈	C₂×#6106	1	2	yes	—
124	C₂²	—	208	2	no	—
125	C₂²	—	208	2	no	—
126	C₂²	—	208	2	no	—
127	C₂²	#11718812	1	1	yes	characteristic
128	C₄	—	16	2	no	—
129	C₄	#11718575	1	4	yes	—
130	C₄	#11718746	1	1	yes	characteristic
131	C₄	#11718852	1	1	yes	characteristic
132	C₂	—	416	2	no	—
133	C₂	#13114429	1	1	yes	characteristic
134	C₂	#13117101	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13117086

C₂

yes

maximal, characteristic

#13117061

C₂

yes

maximal, complemented

#13114429

C₂

yes

maximal, complemented

#13112752

C₂

yes

maximal, characteristic, complemented

C₁₃×#2254

C₂

yes

maximal, characteristic, complemented

C₁₃×#500

C₂²

yes

—

#11718749

—

#11718746

—

C₂×D₂₀₈

—

#11718356

C₂²

yes

—

C₁₃×#367

—

C₄×C₁₃×C₁₆

C₂²

yes

characteristic

#11718083

—

C₄×#6090

C₂²

yes

characteristic

C₈×#1055

C₂²

yes

characteristic

C₁₃×C₄⋊Q₈

—

C₂×#6106

—

C₂×D₁₀₄

—

C₂×D₁₀₄

—

#88128

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₁₃×#107

—

C₁₃×#107

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

C₂×#6090

D₄

yes

—

#88049

—

C₄×#1055

D₄

yes

—

D₂₀₈

—

#12506

—

maximal, Sylow

C₂×Q₈×C₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

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

#6106

—

C₂×#1055

D₈

yes

—

D₁₀₄

—

D₁₀₄

—

C₁₃×C₁₆

SD₁₆

yes

complemented

#6090

—

#2254

D₁₃

yes

characteristic, complemented

—

—

—

—

C₂²×D₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

D₅₂

—

D₅₂

—

D₅₂

—

D₅₂

—

C₈×C₁₃

C₂×SD₁₆

yes

—

C₈×C₁₃

#135

yes

—

#1055

—

C₂×D₁₆

—

#500

C₂×D₁₃

yes

—

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₁₃

yes

characteristic

C₈²

—

C₂²×C₁₃

#493

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

yes

C₄×C₁₃

yes

C₄×C₁₃

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₁₃

yes

—

#107

—

#107

—

#102

—

C₄×C₈

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

D₁₃

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

100

SD₁₆

—

104

—

101

D₈

—

102

D₈

—

103

C₂×C₈

—

104

C₂×C₈

D₄×D₁₃

yes

characteristic

105

C₂×C₈

C₂×C₂²⋊D₁₃

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

D₄×D₁₃

yes

characteristic

109

C₁₆

—

110

C₁₆

#6106

yes

complemented

111

C₁₃

#12506

yes

characteristic, complemented, Sylow

112

C₂³

—

104

—

113

Q₈

—

114

D₄

—

104

—

115

D₄

—

104

—

116

D₄

—

104

—

117

D₄

—

104

—

118

C₂×C₄

—

119

C₂×C₄

D₈×D₁₃

yes

—

120

C₂×C₄

#88194

yes

characteristic

121

C₈

—

122

C₈

#88173

yes

—

123

C₈

C₂×#6106

yes

—

124

C₂²

—

208

—

125

C₂²

—

208

—

126

C₂²

—

208

—

127

C₂²

#11718812

yes

characteristic

128

C₄

—

129

C₄

#11718575

yes

—

130

C₄

#11718746

yes

characteristic

131

C₄

#11718852

yes

characteristic

132

C₂

—

416

—

133

C₂

#13114429

yes

characteristic

134

C₂

#13117101

yes

—

	subgroup	G/H
center	C₂²	#11718812
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#2254	C₂
socle	C₂²×C₁₃	#493
prime radical	#13117086	C₂
hypercenter	#2254	D₁₃
hypocenter	C₁₃	#12506
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#11718812

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#2254

C₂

C₂²×C₁₃

C₂

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
2. #13117061	C₂	semidirect
3. #13114429	C₂	semidirect
4. #13112752	C₂	semidirect
5. C₁₃×#2254	C₂	semidirect
43. C₁₃×C₁₆	SD₁₆	semidirect
45. #2254	D₁₃	semidirect
110. C₁₆	#6106	semidirect
111. C₁₃	#12506	semidirect

H ≅ G/N

type

2. #13117061

C₂

semidirect

3. #13114429

C₂

semidirect

4. #13112752

C₂

semidirect

5. C₁₃×#2254

C₂

semidirect

43. C₁₃×C₁₆

SD₁₆

semidirect

45. #2254

D₁₃

semidirect

110. C₁₆

#6106

semidirect

111. C₁₃

#12506

semidirect

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

subgroup	index
#13117086	2
#13117061	2
#13114429	2
#13112752	2
C₁₃×#2254	2
#12506	13

subgroup

index

C₁₃×#2254

Subgroups — #15777590

Conjugacy classes of subgroups

Complements

Maximal subgroups