Group #16777002 — subgroups

Subgroups — #16777002

Order 3328.

186 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 46 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	#13129866	C₂	1	2	yes	maximal, complemented
2	#13114450	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#493	C₄	1	1	yes	characteristic
4	#11718734	—	2	2	no	—
5	#11718356	C₄	1	1	yes	characteristic
6	C₄⋊#6092	—	2	2	no	—
7	C₈×#1055	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×D₈	—	4	1	no	—
9	C₁₃×#127	—	2	1	no	—
10	C₂×#6104	—	4	2	no	—
11	C₂×#6104	—	4	2	no	—
12	C₂×D₁₀₄	—	4	1	no	—
13	#88128	—	2	1	no	—
14	C₂×#6092	—	4	2	no	—
15	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₄×#1055	D₄	1	2	yes	—
17	#7096	—	13	1	no	maximal, Sylow
18	C₂×D₄×C₁₃	—	4	1	no	—
19	C₂×D₄×C₁₃	—	4	2	no	—
20	C₂×D₅₂	—	4	1	no	—
21	C₂×D₅₂	—	4	2	no	—
22	C₁₃×D₈	—	8	2	no	—
23	C₂×C₈×C₁₃	—	2	1	no	—
24	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
25	#6104	—	8	4	no	—
26	#6104	—	8	2	no	—
27	#6104	—	8	2	no	—
28	C₂×#1055	D₈	1	2	yes	—
29	C₂×#1055	SD₁₆	1	2	yes	—
30	D₁₀₄	—	8	2	no	—
31	#6092	—	8	2	no	—
32	#1712	—	13	1	no	—
33	#1370	—	13	2	no	—
34	C₂³×C₁₃	—	8	1	no	—
35	C₂²×D₁₃	—	8	1	no	—
36	D₄×C₁₃	—	8	2	no	—
37	D₄×C₁₃	—	8	2	no	—
38	D₄×C₁₃	—	8	2	no	—
39	C₂×C₄×C₁₃	#100	1	1	yes	characteristic
40	C₂×C₄×C₁₃	#102	1	2	yes	—
41	D₅₂	—	8	2	no	—
42	D₅₂	—	8	2	no	—
43	D₅₂	—	8	2	no	—
44	C₈×C₁₃	—	2	2	no	—
45	#1055	—	2	2	no	—
46	#1055	D₁₆	1	2	yes	—
47	#1055	SD₃₂	1	2	yes	—
48	#493	—	13	1	no	—
49	#493	—	26	2	no	—
50	#493	#261	1	1	yes	characteristic
51	C₄⋊C₁₆	—	26	2	no	—
52	C₈²	—	13	1	no	—
53	C₂²×C₁₃	—	16	1	no	—
54	C₂²×C₁₃	—	16	2	no	—
55	C₂²×C₁₃	#331	1	1	yes	characteristic
56	C₂×D₁₃	—	16	1	no	—
57	C₂×D₁₃	—	16	2	no	—
58	C₄×C₁₃	—	2	1	no	—
59	C₄×C₁₃	#357	1	2	yes	—
60	C₄×C₁₃	#361	1	2	yes	—
61	C₂×D₈	—	4	1	no	—
62	C₂×D₈	—	52	1	no	—
63	C₂×D₈	—	52	2	no	—
64	C₂×D₈	—	52	2	no	—
65	#127	—	2	1	no	—
66	#127	—	26	1	no	—
67	C₂×C₁₆	—	52	2	no	—
68	C₄×C₈	—	13	2	no	—
69	C₄×C₈	C₂×#261	1	1	yes	characteristic
70	C₂×C₁₃	—	32	1	no	—
71	C₂×C₁₃	#1357	1	1	yes	characteristic
72	C₂×C₁₃	#1361	1	2	yes	—
73	D₁₃	—	32	1	no	—
74	C₂×D₄	—	4	2	no	—
75	C₂×D₄	—	4	1	no	—
76	C₂×D₄	—	52	2	no	—
77	C₂×D₄	—	52	1	no	—
78	D₈	—	8	2	no	—
79	D₈	—	104	2	no	—
80	D₈	—	104	2	no	—
81	D₈	—	104	2	no	—
82	D₈	—	104	4	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	13	2	no	—
85	C₂×C₈	—	13	2	no	—
86	C₄²	C₂²⋊#261	1	1	yes	characteristic
87	C₁₆	—	104	2	no	—
88	C₁₃	#7096	1	1	yes	characteristic, complemented, Sylow
89	C₂³	—	8	1	no	—
90	C₂³	—	104	1	no	—
91	D₄	—	8	2	no	—
92	D₄	—	8	2	no	—
93	D₄	—	8	2	no	—
94	D₄	—	104	2	no	—
95	D₄	—	104	2	no	—
96	D₄	—	104	2	no	—
97	C₂×C₄	#88115	1	2	yes	—
98	C₂×C₄	#88121	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	13	2	no	—
101	C₈	—	13	2	no	—
102	C₈	—	26	2	no	—
103	C₂²	—	16	1	no	—
104	C₂²	—	16	2	no	—
105	C₂²	—	208	1	no	—
106	C₂²	—	208	2	no	—
107	C₂²	#11718297	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	#11718276	1	2	yes	—
110	C₄	#11718278	1	2	yes	—
111	C₂	—	32	1	no	—
112	C₂	—	416	1	no	—
113	C₂	#13129844	1	1	yes	characteristic
114	C₂	#13129850	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13129866

C₂

yes

maximal, complemented

#13114450

C₂

yes

maximal, characteristic

C₁₃×#493

C₄

yes

characteristic

#11718734

—

#11718356

C₄

yes

characteristic

C₄⋊#6092

—

C₈×#1055

C₂²

yes

characteristic

C₂×C₁₃×D₈

—

C₁₃×#127

—

C₂×#6104

—

C₂×#6104

—

C₂×D₁₀₄

—

#88128

—

C₂×#6092

—

C₄×C₈×C₁₃

C₂×C₄

yes

characteristic

C₄×#1055

D₄

yes

—

#7096

—

maximal, Sylow

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₅₂

—

C₂×D₅₂

—

C₁₃×D₈

—

C₂×C₈×C₁₃

—

C₄²×C₁₃

C₂²⋊C₄

yes

—

—

—

D₈

yes

—

C₂×#1055

SD₁₆

yes

—

D₁₀₄

—

#6092

—

#1712

—

#1370

—

C₂³×C₁₃

—

C₂²×D₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

D₄×C₁₃

—

C₂×C₄×C₁₃

#100

yes

characteristic

C₂×C₄×C₁₃

#102

yes

—

D₅₂

—

D₅₂

—

D₅₂

—

C₈×C₁₃

—

#1055

—

#1055

D₁₆

yes

—

#1055

SD₃₂

yes

—

—

—

yes

C₄⋊C₁₆

—

C₈²

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#331

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#357

yes

—

C₄×C₁₃

#361

yes

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

#127

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

C₂×#261

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

#1357

yes

characteristic

C₂×C₁₃

#1361

yes

—

D₁₃

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

104

—

D₈

—

104

—

D₈

—

104

—

D₈

—

104

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₂²⋊#261

yes

characteristic

C₁₆

—

104

—

C₁₃

#7096

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

104

—

D₄

—

D₄

—

D₄

—

D₄

—

104

—

D₄

—

104

—

D₄

—

104

—

C₂×C₄

#88115

yes

—

C₂×C₄

#88121

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

102

C₈

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

208

—

106

C₂²

—

208

—

107

C₂²

#11718297

yes

characteristic

108

C₄

—

109

C₄

#11718276

yes

—

110

C₄

#11718278

yes

—

111

C₂

—

112

C₂

—

416

—

113

C₂

#13129844

yes

characteristic

114

C₂

#13129850

yes

—

	subgroup	G/H
center	C₂²	#11718297
commutator	C₄×C₈×C₁₃	C₂×C₄
Frattini	C₄×C₈	C₂×#261
Fitting	C₁₃×#493	C₄
socle	C₂²×C₁₃	#331
prime radical	#13114450	C₂
cosocle	C₈×#1055	C₂²
hypercenter	#493	#261
hypocenter	C₁₃	#7096
LCS	G ▷ C₄×C₈×C₁₃ ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

subgroup

G/H

center

C₂²

#11718297

commutator

C₄×C₈×C₁₃

C₂×C₄

Frattini

C₄×C₈

C₂×#261

Fitting

C₁₃×#493

C₄

C₂²×C₁₃

C₂

C₂²

C₁₃

G ▷ C₄×C₈×C₁₃ ▷ C₄²×C₁₃ ▷ C₂²×C₁₃ ▷ C₁₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

N	H ≅ G/N	type
1. #13129866	C₂	semidirect
88. C₁₃	#7096	semidirect

H ≅ G/N

type

1. #13129866

C₂

semidirect

88. C₁₃

#7096

semidirect

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

subgroup	index
#13129866	2
#13114450	2
#7096	13

subgroup

index

#13129866

#13114450

#7096

Subgroups — #16777002

Conjugacy classes of subgroups

Complements

Maximal subgroups