Group #16791420 — subgroups

Subgroups — #16791420

Order 3328.

198 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 108 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	C₂×#11718249	C₂	1	2	yes	maximal
2	#13129996	C₂	1	4	yes	maximal, complemented
3	#13116435	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×M₃₂	C₄	1	1	yes	characteristic
5	#11718762	C₄	1	1	yes	characteristic
6	C₂²×#6090	C₂²	1	1	yes	characteristic
7	#11718249	—	2	2	no	—
8	#11718249	C₂²	1	4	yes	—
9	C₂×#88036	—	2	2	no	—
10	#11718094	C₄	1	2	yes	—
11	C₁₆×Dic₁₃	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
13	#88180	C₈	1	1	yes	characteristic
14	C₂×#6099	C₈	1	1	yes	characteristic
15	C₁₃×M₃₂	—	2	2	no	—
16	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
17	#88055	C₈	1	2	yes	—
18	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
19	C₂×#6090	C₂×C₄	1	1	yes	characteristic
20	C₂×#6090	D₄	1	2	yes	—
21	C₂×#6090	Q₈	1	2	yes	—
22	C₂×#6090	C₂³	1	1	yes	characteristic
23	#88036	—	2	4	no	—
24	#13630	—	13	1	no	maximal, Sylow
25	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
29	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²⋊Dic₁₃	—	2	1	no	—
31	C₄⋊Dic₁₃	—	2	1	no	—
32	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
33	#6099	C₁₆	1	4	yes	—
34	C₂×#1055	C₂×C₈	1	2	yes	—
35	C₁₃×C₁₆	—	2	2	no	—
36	#6090	C₄⋊C₄	1	2	yes	—
37	#6090	C₂×D₄	1	1	yes	characteristic
38	#6090	C₂×Q₈	1	1	yes	characteristic
39	C₂×M₆₄	—	13	2	no	—
40	#2108	—	13	1	no	—
41	#1422	—	13	4	no	—
42	C₂³×C₁₃	M₃₂	1	1	yes	characteristic
43	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
44	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
45	C₂×C₄×C₁₃	M₃₂	1	1	yes	characteristic
46	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
47	C₂×Dic₁₃	—	2	2	no	—
48	C₈×C₁₃	C₄⋊C₈	1	2	yes	—
49	C₈×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₈×C₁₃	C₂×M₁₆	1	1	yes	characteristic
51	#1055	C₂×C₁₆	1	4	yes	—
52	C₂×M₃₂	#261	1	1	yes	characteristic
53	C₂²×C₁₆	—	13	1	no	—
54	#405	—	13	1	no	—
55	M₆₄	—	13	4	no	—
56	M₆₄	—	26	2	no	—
57	C₂×C₃₂	—	26	2	no	—
58	#346	—	13	2	no	—
59	C₄×C₁₆	—	13	2	no	—
60	C₂²×C₁₃	—	4	1	no	—
61	C₂²×C₁₃	C₄⋊C₁₆	1	2	yes	—
62	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
63	C₄×C₁₃	C₄⋊C₁₆	1	2	yes	—
64	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₄×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
66	Dic₁₃	—	4	2	no	—
67	C₂×M₁₆	—	13	1	no	—
68	C₂²×C₈	C₂×#261	1	1	yes	characteristic
69	#117	—	13	1	no	—
70	M₃₂	—	2	2	no	—
71	C₂×C₁₆	—	13	2	no	—
72	C₂×C₁₆	—	13	2	no	—
73	C₂×C₁₆	—	13	1	no	—
74	C₂×C₁₆	—	13	1	no	—
75	C₂×C₁₆	C₂×#261	1	2	yes	—
76	#97	—	13	2	no	—
77	C₄×C₈	—	13	2	no	—
78	C₃₂	—	26	4	no	—
79	C₂×C₁₃	—	2	1	no	—
80	C₂×C₁₃	—	2	2	no	—
81	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
82	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
83	M₁₆	—	13	4	no	—
84	C₂×C₈	—	13	2	no	—
85	C₂×C₈	D₁₃⋊C₈	1	1	yes	characteristic
86	C₂×C₈	C₄⋊#261	1	2	yes	—
87	C₂×C₈	#6122	1	2	yes	—
88	C₂×C₈	C₂²×#261	1	1	yes	characteristic
89	C₄⋊C₄	—	26	1	no	—
90	C₂²⋊C₄	—	26	1	no	—
91	C₄²	—	13	2	no	—
92	C₁₆	—	2	2	no	—
93	C₁₆	—	13	1	no	—
94	C₁₆	—	13	2	no	—
95	C₁₆	—	13	1	no	—
96	C₁₃	#13630	1	1	yes	characteristic, complemented, Sylow
97	C₂³	#88098	1	1	yes	characteristic
98	C₂×C₄	—	26	2	no	—
99	C₂×C₄	D₁₃⋊C₁₆	1	2	yes	—
100	C₂×C₄	#88098	1	1	yes	characteristic
101	C₂×C₄	#88112	1	2	yes	—
102	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
103	C₈	—	13	4	no	—
104	C₈	#88112	1	2	yes	—
105	C₈	C₂×C₄⋊#261	1	1	yes	characteristic
106	C₈	C₂×#6122	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	#11718260	1	2	yes	—
109	C₂²	C₂×#88098	1	1	yes	characteristic
110	C₄	—	52	2	no	—
111	C₄	#11718260	1	2	yes	—
112	C₄	C₂×D₁₃⋊C₁₆	1	1	yes	characteristic
113	C₄	C₂×#88112	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	C₂×#11718260	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11718249

C₂

yes

maximal

#13129996

C₂

yes

maximal, complemented

#13116435

C₂

yes

maximal, characteristic

C₂×C₁₃×M₃₂

C₄

yes

characteristic

#11718762

C₄

yes

characteristic

C₂²×#6090

C₂²

yes

characteristic

#11718249

—

#11718249

C₂²

yes

—

C₂×#88036

—

#11718094

C₄

yes

—

C₁₆×Dic₁₃

C₂²

yes

—

C₂²×C₈×C₁₃

C₂×C₄

yes

characteristic

#88180

C₈

yes

characteristic

C₂×#6099

C₈

yes

characteristic

C₁₃×M₃₂

—

C₂×C₁₃×C₁₆

C₂×C₄

yes

—

#88055

C₈

yes

—

C₈×Dic₁₃

C₂×C₄

yes

—

C₂×#6090

C₂×C₄

yes

characteristic

C₂×#6090

D₄

yes

—

C₂×#6090

Q₈

yes

—

C₂×#6090

C₂³

yes

—

—

C₂²×C₄×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

C₄⋊C₄

yes

—

C₂×C₈×C₁₃

C₂×C₈

yes

characteristic

C₂×C₈×C₁₃

M₁₆

yes

—

C₂×C₈×C₁₃

C₂²×C₄

yes

characteristic

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

C₂×C₈

yes

—

#6099

C₁₆

yes

—

C₂×#1055

C₂×C₈

yes

—

C₁₃×C₁₆

—

#6090

C₄⋊C₄

yes

—

#6090

C₂×D₄

yes

characteristic

#6090

C₂×Q₈

yes

characteristic

C₂×M₆₄

—

#2108

—

#1422

—

C₂³×C₁₃

M₃₂

yes

characteristic

C₂×C₄×C₁₃

C₄⋊C₈

yes

—

C₂×C₄×C₁₃

C₂×C₁₆

yes

—

C₂×C₄×C₁₃

M₃₂

yes

characteristic

C₂×C₄×C₁₃

C₂²×C₈

yes

characteristic

C₂×Dic₁₃

—

C₈×C₁₃

C₄⋊C₈

yes

—

C₈×C₁₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₁₃

C₂×M₁₆

yes

characteristic

#1055

C₂×C₁₆

yes

—

C₂×M₃₂

#261

yes

characteristic

C₂²×C₁₆

—

#405

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

#346

—

C₄×C₁₆

—

C₂²×C₁₃

—

C₂²×C₁₃

C₄⋊C₁₆

yes

—

C₂²×C₁₃

C₂×M₃₂

yes

characteristic

C₄×C₁₃

C₄⋊C₁₆

yes

—

C₄×C₁₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₁₃

C₂²×C₁₆

yes

characteristic

Dic₁₃

—

C₂×M₁₆

—

C₂²×C₈

C₂×#261

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#261

yes

—

#97

—

C₄×C₈

—

C₃₂

—

C₂×C₁₃

—

C₂×C₁₃

—

C₂×C₁₃

C₂×C₄⋊C₁₆

yes

characteristic

C₂²×C₄

D₁₃⋊C₈

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

D₁₃⋊C₈

yes

characteristic

C₂×C₈

C₄⋊#261

yes

—

C₂×C₈

#6122

yes

—

C₂×C₈

C₂²×#261

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₃

#13630

yes

characteristic, complemented, Sylow

C₂³

#88098

yes

characteristic

C₂×C₄

—

C₂×C₄

D₁₃⋊C₁₆

yes

—

100

C₂×C₄

#88098

yes

characteristic

101

C₂×C₄

#88112

yes

—

102

C₂×C₄

C₂×D₁₃⋊C₈

yes

characteristic

103

C₈

—

104

C₈

#88112

yes

—

105

C₈

C₂×C₄⋊#261

yes

characteristic

106

C₈

C₂×#6122

yes

characteristic

107

C₂²

—

108

C₂²

#11718260

yes

—

109

C₂²

C₂×#88098

yes

characteristic

110

C₄

—

111

C₄

#11718260

yes

—

112

C₄

C₂×D₁₃⋊C₁₆

yes

characteristic

113

C₄

C₂×#88112

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

C₂×#11718260

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₄⋊#261
commutator	C₄×C₁₃	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#261
Fitting	C₂×C₁₃×M₃₂	C₄
socle	C₂×C₁₃	C₂×C₄⋊C₁₆
prime radical	C₂³×C₁₃	M₃₂
cosocle	C₂×#6090	C₂³
hypercenter	C₂×M₃₂	#261
hypocenter	C₁₃	#13630
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₈

C₂×C₄⋊#261

commutator

C₄×C₁₃

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×#261

Fitting

C₂×C₁₃×M₃₂

C₄

socle

C₂×C₁₃

C₂×C₄⋊C₁₆

prime radical

C₂³×C₁₃

M₃₂

C₂³

C₂×M₃₂

C₁₃

G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. #13129996	C₂	semidirect
96. C₁₃	#13630	semidirect

H ≅ G/N

type

2. #13129996

C₂

semidirect

96. C₁₃

#13630

semidirect

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

subgroup	index
C₂×#11718249	2
#13129996	2
#13116435	2
#13630	13

subgroup

index

Subgroups — #16791420

Conjugacy classes of subgroups

Complements

Maximal subgroups