Group #15767969 — subgroups

Subgroups — #15767969

Order 3328.

526 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 165 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	#13115642	C₂	1	3	yes	maximal
2	#13114996	C₂	1	3	yes	maximal, complemented
3	C₁₃×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#88128	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₈×C₁₃	C₂²	1	3	yes	—
6	C₂×#88061	—	2	6	no	—
7	C₂×#88058	—	2	6	no	—
8	#11718596	C₂²	1	3	yes	—
9	C₂²×D₅₂	—	2	3	no	—
10	C₂²×C₈×C₁₃	D₄	1	6	yes	—
11	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₃	—	2	6	no	—
13	C₂×C₄⋊Dic₁₃	—	2	3	no	—
14	#88128	—	2	3	no	—
15	#88128	Q₈	1	2	yes	—
16	C₄×C₈×C₁₃	—	2	6	no	—
17	#88061	—	4	12	no	—
18	#88058	—	4	12	no	—
19	#10673	—	13	1	no	maximal, Sylow
20	C₂³×D₁₃	—	4	1	no	—
21	C₂²×C₄×C₁₃	C₂×D₄	1	3	yes	—
22	C₂²×Dic₁₃	—	4	3	no	—
23	C₂×D₅₂	—	2	6	no	—
24	C₂×D₅₂	—	4	6	no	—
25	C₂×D₅₂	—	4	6	no	—
26	C₂×D₅₂	—	4	3	no	—
27	C₂×C₈×C₁₃	—	2	6	no	—
28	C₂×C₈×C₁₃	—	2	6	no	—
29	C₂×C₈×C₁₃	D₈	1	12	yes	—
30	C₄²×C₁₃	C₂×Q₈	1	1	yes	characteristic
31	C₄²×C₁₃	#41	1	3	yes	—
32	C₄⋊Dic₁₃	—	4	12	no	—
33	C₄⋊Dic₁₃	—	4	6	no	—
34	#1957	—	13	3	no	—
35	#1845	—	13	3	no	—
36	#1273	D₁₃	1	1	yes	characteristic, complemented
37	C₂³×C₁₃	#127	1	1	yes	characteristic
38	C₂²×D₁₃	—	4	2	no	—
39	C₂²×D₁₃	—	8	3	no	—
40	C₂²×D₁₃	—	8	3	no	—
41	C₂×C₄×C₁₃	C₂²⋊Q₈	1	3	yes	—
42	C₂×C₄×C₁₃	#123	1	3	yes	—
43	C₂×C₄×C₁₃	C₂×D₈	1	6	yes	—
44	C₂×C₄×C₁₃	#137	1	6	yes	—
45	C₂×Dic₁₃	—	8	6	no	—
46	C₂×Dic₁₃	—	8	3	no	—
47	D₅₂	—	4	12	no	—
48	D₅₂	—	8	6	no	—
49	C₈×C₁₃	—	2	12	no	—
50	C₂×#127	—	13	1	no	—
51	C₂×#107	—	26	6	no	—
52	C₂×#102	—	26	6	no	—
53	C₂×C₄×C₈	C₂×D₁₃	1	3	yes	—
54	#384	—	13	3	no	—
55	C₂²×C₁₃	#399	1	1	yes	characteristic
56	C₂²×C₁₃	#493	1	3	yes	—
57	C₂²×C₁₃	#497	1	3	yes	—
58	C₂×D₁₃	—	8	6	no	—
59	C₂×D₁₃	—	16	1	no	—
60	C₂×D₁₃	—	16	3	no	—
61	C₄×C₁₃	#474	1	6	yes	—
62	C₄×C₁₃	#480	1	6	yes	—
63	Dic₁₃	—	16	3	no	—
64	C₂²×D₄	—	26	3	no	—
65	C₂²×C₈	D₅₂	1	6	yes	—
66	#127	—	13	2	no	—
67	#127	—	26	3	no	—
68	C₂×C₄⋊C₄	—	26	3	no	—
69	C₂×C₄⋊C₄	—	26	6	no	—
70	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
71	#107	—	52	12	no	—
72	#102	—	52	12	no	—
73	C₄×C₈	—	2	6	no	—
74	C₂×C₁₃	#1712	1	1	yes	characteristic
75	C₂×C₁₃	#1722	1	3	yes	—
76	C₂×C₁₃	#2093	1	3	yes	—
77	D₁₃	—	16	2	no	—
78	C₂⁴	—	52	1	no	—
79	C₂×D₄	—	26	6	no	—
80	C₂×D₄	—	52	3	no	—
81	C₂×D₄	—	52	6	no	—
82	C₂×D₄	—	52	6	no	—
83	C₂²×C₄	—	52	3	no	—
84	C₂²×C₄	C₂×D₅₂	1	3	yes	—
85	C₂×C₈	—	2	6	no	—
86	C₂×C₈	—	2	6	no	—
87	C₂×C₈	D₁₀₄	1	12	yes	—
88	C₄⋊C₄	—	52	6	no	—
89	C₄⋊C₄	—	52	12	no	—
90	C₄²	#6129	1	3	yes	—
91	C₄²	Q₈×D₁₃	1	1	yes	characteristic
92	C₁₃	#10673	1	1	yes	characteristic, complemented, Sylow
93	C₂³	—	52	2	no	—
94	C₂³	—	104	3	no	—
95	C₂³	—	104	3	no	—
96	C₂³	#88128	1	1	yes	characteristic
97	D₄	—	52	12	no	—
98	D₄	—	104	6	no	—
99	C₂×C₄	—	104	3	no	—
100	C₂×C₄	—	104	6	no	—
101	C₂×C₄	#88140	1	3	yes	—
102	C₂×C₄	#88151	1	3	yes	—
103	C₂×C₄	C₂×D₁₀₄	1	6	yes	—
104	C₂×C₄	#88163	1	6	yes	—
105	C₈	—	2	12	no	—
106	C₂²	—	104	6	no	—
107	C₂²	—	208	1	no	—
108	C₂²	—	208	3	no	—
109	C₂²	#11718338	1	1	yes	characteristic
110	C₂²	#11718356	1	3	yes	—
111	C₂²	#11718376	1	3	yes	—
112	C₄	—	208	3	no	—
113	C₄	#11718397	1	6	yes	—
114	C₄	#11718507	1	6	yes	—
115	C₂	—	208	2	no	—
116	C₂	#13114451	1	1	yes	characteristic
117	C₂	#13114507	1	3	yes	—
118	C₂	#13116363	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13115642

C₂

yes

maximal

#13114996

C₂

yes

maximal, complemented

C₁₃×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#88128

C₂²

yes

characteristic

C₂×C₄×C₈×C₁₃

C₂²

yes

—

C₂×#88061

—

C₂×#88058

—

#11718596

C₂²

yes

—

C₂²×D₅₂

—

C₂²×C₈×C₁₃

D₄

yes

—

C₂×C₄²×C₁₃

C₂³

yes

characteristic

C₂×C₄⋊Dic₁₃

—

C₂×C₄⋊Dic₁₃

—

#88128

—

#88128

Q₈

yes

—

C₄×C₈×C₁₃

—

—

—

—

C₂³×D₁₃

—

C₂²×C₄×C₁₃

C₂×D₄

yes

—

C₂²×Dic₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

—

C₂×C₈×C₁₃

D₈

yes

—

C₄²×C₁₃

C₂×Q₈

yes

characteristic

C₄²×C₁₃

#41

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#1957

—

#1845

—

#1273

D₁₃

yes

characteristic, complemented

C₂³×C₁₃

#127

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂×C₄×C₁₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₁₃

#123

yes

—

C₂×C₄×C₁₃

C₂×D₈

yes

—

C₂×C₄×C₁₃

#137

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

D₅₂

—

D₅₂

—

C₈×C₁₃

—

C₂×#127

—

C₂×#107

—

C₂×#102

—

C₂×C₄×C₈

C₂×D₁₃

yes

—

#384

—

C₂²×C₁₃

#399

yes

characteristic

C₂²×C₁₃

#493

yes

—

C₂²×C₁₃

#497

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

#474

yes

—

C₄×C₁₃

#480

yes

—

Dic₁₃

—

C₂²×D₄

—

C₂²×C₈

D₅₂

yes

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₁₃

yes

characteristic

#107

—

#102

—

C₄×C₈

—

C₂×C₁₃

#1712

yes

characteristic

C₂×C₁₃

#1722

yes

—

C₂×C₁₃

#2093

yes

—

D₁₃

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₅₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₀₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

#6129

yes

—

C₄²

Q₈×D₁₃

yes

characteristic

C₁₃

#10673

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

104

—

C₂³

—

104

—

C₂³

#88128

yes

characteristic

D₄

—

D₄

—

104

—

C₂×C₄

—

104

—

100

C₂×C₄

—

104

—

101

C₂×C₄

#88140

yes

—

102

C₂×C₄

#88151

yes

—

103

C₂×C₄

C₂×D₁₀₄

yes

—

104

C₂×C₄

#88163

yes

—

105

C₈

—

106

C₂²

—

104

—

107

C₂²

—

208

—

108

C₂²

—

208

—

109

C₂²

#11718338

yes

characteristic

110

C₂²

#11718356

yes

—

111

C₂²

#11718376

yes

—

112

C₄

—

208

—

113

C₄

#11718397

yes

—

114

C₄

#11718507

yes

—

115

C₂

—

208

—

116

C₂

#13114451

yes

characteristic

117

C₂

#13114507

yes

—

118

C₂

#13116363

yes

—

	subgroup	G/H
center	C₂³	#88128
commutator, cosocle	C₂×C₄²×C₁₃	C₂³
Frattini	C₂×C₄²	C₂²×D₁₃
Fitting	C₁₃×#1273	C₂
socle	C₂³×C₁₃	#127
prime radical	C₂×#88128	C₂²
hypercenter	#1273	D₁₃
hypocenter	C₁₃	#10673
LCS	G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#88128

commutator, cosocle

C₂×C₄²×C₁₃

C₂³

Frattini

C₂×C₄²

C₂²×D₁₃

Fitting

C₁₃×#1273

C₂

C₂³×C₁₃

C₂²

D₁₃

C₁₃

G ▷ C₂×C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #13114996	C₂	semidirect
3. C₁₃×#1273	C₂	semidirect
36. #1273	D₁₃	semidirect
92. C₁₃	#10673	semidirect

H ≅ G/N

type

2. #13114996

C₂

semidirect

3. C₁₃×#1273

C₂

semidirect

36. #1273

D₁₃

semidirect

92. C₁₃

#10673

semidirect

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

subgroup	index
#13115642	2
#13114996	2
C₁₃×#1273	2
#10673	13

subgroup

index

#13115642

#13114996

C₁₃×#1273

#10673

Subgroups — #15767969

Conjugacy classes of subgroups

Complements

Maximal subgroups