Group #15777652 — subgroups

Subgroups — #15777652

Order 3328.

316 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 99 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	#13117092	C₂	1	2	yes	maximal
2	#13117089	C₂	1	2	yes	maximal, complemented
3	#13114454	C₂	1	1	yes	maximal, characteristic, complemented
4	#13114442	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₁₀₄	—	2	4	no	—
7	C₂×#88040	—	2	4	no	—
8	#11718360	C₂²	1	1	yes	characteristic
9	#11718360	C₂²	1	2	yes	—
10	#11718356	C₂²	1	1	yes	characteristic
11	#11718355	—	2	2	no	—
12	#11718343	—	2	2	no	—
13	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
14	C₈²×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₅₂	—	2	2	no	—
16	C₂×Dic₅₂	—	2	2	no	—
17	C₂×Dic₅₂	—	2	2	no	—
18	C₂×D₁₀₄	—	2	2	no	—
19	#88128	—	2	1	no	—
20	#88123	—	2	2	no	—
21	#88123	—	2	1	no	—
22	C₂×C₁₃×C₁₆	D₄	1	4	yes	—
23	C₄×C₈×C₁₃	D₄	1	2	yes	—
24	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	#88061	—	4	4	no	—
26	#88058	—	4	4	no	—
27	Dic₁₀₄	—	4	8	no	—
28	#88040	—	4	8	no	—
29	#12520	—	13	1	no	maximal, Sylow
30	C₂×D₅₂	—	4	1	no	—
31	C₂×D₅₂	—	4	2	no	—
32	C₂×Dic₂₆	—	4	2	no	—
33	C₂×Dic₂₆	—	4	1	no	—
34	C₂×C₈×C₁₃	D₈	1	4	yes	—
35	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
36	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₁₃	—	4	4	no	—
38	C₄⋊Dic₁₃	—	4	2	no	—
39	Dic₅₂	—	4	4	no	—
40	Dic₅₂	—	4	4	no	—
41	Dic₅₂	—	4	4	no	—
42	D₁₀₄	—	4	4	no	—
43	C₁₃×C₁₆	D₈	1	8	yes	—
44	#2248	—	13	2	no	—
45	#2247	—	13	2	no	—
46	#1713	—	13	1	no	—
47	#1709	—	13	1	no	—
48	#1311	D₁₃	1	1	yes	characteristic, complemented
49	C₂²×D₁₃	—	8	1	no	—
50	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
51	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
52	C₂×Dic₁₃	—	8	2	no	—
53	C₂×Dic₁₃	—	8	1	no	—
54	D₅₂	—	8	2	no	—
55	D₅₂	—	8	4	no	—
56	Dic₂₆	—	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	1	no	—
65	#494	—	13	2	no	—
66	#493	—	13	1	no	—
67	#486	—	26	2	no	—
68	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
69	C₈²	C₂×D₁₃	1	1	yes	characteristic
70	C₂²×C₁₃	#493	1	1	yes	characteristic
71	C₂×D₁₃	—	16	1	no	—
72	C₂×D₁₃	—	16	2	no	—
73	C₄×C₁₃	#493	1	2	yes	—
74	C₄×C₁₃	#510	1	4	yes	—
75	Dic₁₃	—	16	2	no	—
76	Dic₁₃	—	16	1	no	—
77	C₂×Q₁₆	—	26	2	no	—
78	C₂×Q₁₆	—	26	2	no	—
79	C₂×Q₁₆	—	26	2	no	—
80	C₂×D₈	—	26	2	no	—
81	C₄⋊Q₈	—	26	1	no	—
82	C₄⋊Q₈	—	26	2	no	—
83	#127	—	26	1	no	—
84	Q₃₂	—	52	8	no	—
85	SD₃₂	—	52	8	no	—
86	C₂×C₁₆	D₅₂	1	4	yes	—
87	#107	—	52	4	no	—
88	#102	—	52	4	no	—
89	C₄×C₈	D₅₂	1	2	yes	—
90	C₄×C₈	C₂²×D₁₃	1	1	yes	characteristic
91	C₂×C₁₃	#1712	1	1	yes	characteristic
92	C₂×C₁₃	#2251	1	2	yes	—
93	D₁₃	—	32	1	no	—
94	C₂×Q₈	—	52	1	no	—
95	C₂×Q₈	—	52	2	no	—
96	C₂×D₄	—	52	1	no	—
97	C₂×D₄	—	52	2	no	—
98	Q₁₆	—	52	4	no	—
99	Q₁₆	—	52	4	no	—
100	Q₁₆	—	52	4	no	—
101	D₈	—	52	4	no	—
102	C₂×C₈	D₁₀₄	1	4	yes	—
103	C₂×C₈	C₂×D₅₂	1	2	yes	—
104	C₄⋊C₄	—	52	2	no	—
105	C₄⋊C₄	—	52	4	no	—
106	C₄²	C₂×D₅₂	1	1	yes	characteristic
107	C₁₆	D₁₀₄	1	8	yes	—
108	C₁₃	#12520	1	1	yes	characteristic, complemented, Sylow
109	C₂³	—	104	1	no	—
110	Q₈	—	104	2	no	—
111	Q₈	—	104	2	no	—
112	Q₈	—	104	2	no	—
113	D₄	—	104	4	no	—
114	D₄	—	104	2	no	—
115	C₂×C₄	—	104	1	no	—
116	C₂×C₄	—	104	2	no	—
117	C₂×C₄	#88128	1	1	yes	characteristic
118	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
119	C₈	—	2	4	no	—
120	C₈	C₂×D₁₀₄	1	4	yes	—
121	C₂²	—	208	1	no	—
122	C₂²	—	208	2	no	—
123	C₂²	#11718356	1	1	yes	characteristic
124	C₄	—	208	1	no	—
125	C₄	—	208	2	no	—
126	C₄	#11718356	1	2	yes	—
127	C₄	#11718570	1	4	yes	—
128	C₂	—	416	1	no	—
129	C₂	#13114451	1	1	yes	characteristic
130	C₂	#13117107	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13117092

C₂

yes

maximal

#13117089

C₂

yes

maximal, complemented

#13114454

C₂

yes

maximal, characteristic, complemented

#13114442

C₂

yes

maximal, characteristic

C₁₃×#1311

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₀₄

—

C₂×#88040

—

#11718360

C₂²

yes

characteristic

#11718360

C₂²

yes

—

#11718356

C₂²

yes

characteristic

#11718355

—

#11718343

—

C₄×C₁₃×C₁₆

C₂²

yes

—

C₈²×C₁₃

C₂²

yes

characteristic

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×Dic₅₂

—

C₂×D₁₀₄

—

#88128

—

#88123

—

#88123

—

C₂×C₁₃×C₁₆

D₄

yes

—

C₄×C₈×C₁₃

D₄

yes

—

C₄×C₈×C₁₃

C₂³

yes

characteristic

#88061

—

#88058

—

Dic₁₀₄

—

#88040

—

#12520

—

maximal, Sylow

C₂×D₅₂

—

C₂×D₅₂

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₂×C₈×C₁₃

D₈

yes

—

C₂×C₈×C₁₃

C₂×D₄

yes

—

C₄²×C₁₃

C₂×D₄

yes

characteristic

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

Dic₅₂

—

Dic₅₂

—

Dic₅₂

—

D₁₀₄

—

C₁₃×C₁₆

D₈

yes

—

—

—

—

—

D₁₃

yes

characteristic, complemented

C₂²×D₁₃

—

C₂×C₄×C₁₃

#127

yes

characteristic

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

—

C₈×C₁₃

—

C₈×C₁₃

C₂×D₈

yes

—

C₂×Q₃₂

—

C₂×SD₃₂

—

—

—

—

—

—

C₄×C₁₆

C₂×D₁₃

yes

—

C₈²

C₂×D₁₃

yes

characteristic

C₂²×C₁₃

#493

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

#493

yes

—

C₄×C₁₃

#510

yes

—

Dic₁₃

—

Dic₁₃

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

C₂×C₁₆

D₅₂

yes

—

#107

—

#102

—

C₄×C₈

D₅₂

yes

—

C₄×C₈

C₂²×D₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

D₁₃

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

100

Q₁₆

—

101

D₈

—

102

C₂×C₈

D₁₀₄

yes

—

103

C₂×C₈

C₂×D₅₂

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

C₂×D₅₂

yes

characteristic

107

C₁₆

D₁₀₄

yes

—

108

C₁₃

#12520

yes

characteristic, complemented, Sylow

109

C₂³

—

104

—

110

Q₈

—

104

—

111

Q₈

—

104

—

112

Q₈

—

104

—

113

D₄

—

104

—

114

D₄

—

104

—

115

C₂×C₄

—

104

—

116

C₂×C₄

—

104

—

117

C₂×C₄

#88128

yes

characteristic

118

C₂×C₄

C₂×D₁₀₄

yes

—

119

C₈

—

120

C₈

C₂×D₁₀₄

yes

—

121

C₂²

—

208

—

122

C₂²

—

208

—

123

C₂²

#11718356

yes

characteristic

124

C₄

—

208

—

125

C₄

—

208

—

126

C₄

#11718356

yes

—

127

C₄

#11718570

yes

—

128

C₂

—

416

—

129

C₂

#13114451

yes

characteristic

130

C₂

#13117107

yes

—

	subgroup	G/H
center	C₂²	#11718356
commutator, cosocle	C₄×C₈×C₁₃	C₂³
Frattini	C₄×C₈	C₂²×D₁₃
Fitting	C₁₃×#1311	C₂
socle	C₂²×C₁₃	#493
prime radical	#11718356	C₂²
hypercenter	#1311	D₁₃
hypocenter	C₁₃	#12520
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#11718356

commutator, cosocle

C₄×C₈×C₁₃

C₂³

Frattini

C₄×C₈

C₂²×D₁₃

Fitting

C₁₃×#1311

C₂

C₂²×C₁₃

C₂²

D₁₃

C₁₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
2. #13117089	C₂	semidirect
3. #13114454	C₂	semidirect
5. C₁₃×#1311	C₂	semidirect
48. #1311	D₁₃	semidirect
108. C₁₃	#12520	semidirect

H ≅ G/N

type

2. #13117089

C₂

semidirect

3. #13114454

C₂

semidirect

5. C₁₃×#1311

C₂

semidirect

48. #1311

D₁₃

semidirect

108. C₁₃

#12520

semidirect

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

subgroup	index
#13117092	2
#13117089	2
#13114454	2
#13114442	2
C₁₃×#1311	2
#12520	13

subgroup

index

C₁₃×#1311

Subgroups — #15777652

Conjugacy classes of subgroups

Complements

Maximal subgroups