Group #13056351 — subgroups

Subgroups — #13056351

Order 1344.

220 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 67 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	#10621357	C₂	1	1	yes	maximal, characteristic
2	#10621353	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	#10620660	C₂	1	4	yes	maximal, complemented
5	#89485	—	3	1	no	maximal, Hall
6	C₂×D₈₄	—	2	2	no	—
7	C₂×D₈₄	C₂²	1	2	yes	—
8	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
9	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₂₁	C₂²	1	2	yes	—
11	#66629	—	2	2	no	—
12	#6437	—	3	1	no	—
13	#6433	—	3	1	no	—
14	C₇×#97	D₃	1	1	yes	characteristic, complemented
15	#6395	—	3	4	no	—
16	#4713	—	7	1	no	maximal, Hall
17	C₂²×D₂₁	—	2	2	no	—
18	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
19	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₂₁	—	2	2	no	—
21	D₈₄	—	2	4	no	—
22	D₈₄	—	4	4	no	—
23	C₃×C₇×C₈	—	2	2	no	—
24	C₂×D₂₈	—	3	2	no	—
25	C₂×D₂₈	—	6	2	no	—
26	C₂×C₇×C₈	C₂×D₃	1	2	yes	—
27	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
28	C₄⋊Dic₇	—	3	2	no	—
29	Dic₇⋊C₄	—	6	2	no	—
30	#874	—	7	1	no	—
31	#870	—	7	1	no	—
32	C₃×#97	D₇	1	1	yes	characteristic, complemented
33	#821	—	7	4	no	—
34	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
35	C₂×D₂₁	—	4	2	no	—
36	C₂×D₂₁	—	4	4	no	—
37	C₃×C₄×C₇	—	2	2	no	—
38	C₃×C₄×C₇	#41	1	2	yes	—
39	Dic₂₁	—	4	2	no	—
40	#490	—	21	1	no	Sylow
41	C₂²×D₇	—	6	2	no	—
42	C₂×C₄×C₇	D₁₂	1	2	yes	—
43	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
44	C₂×Dic₇	—	6	2	no	—
45	D₂₈	—	6	4	no	—
46	D₂₈	—	12	4	no	—
47	C₇×C₈	—	2	2	no	—
48	C₂×D₁₂	—	7	2	no	—
49	C₂×D₁₂	—	14	2	no	—
50	C₂×C₃×C₈	C₂×D₇	1	2	yes	—
51	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
52	C₄⋊Dic₃	—	7	2	no	—
53	Dic₃⋊C₄	—	14	2	no	—
54	C₂×C₃×C₇	#124	1	1	yes	characteristic
55	C₂×C₃×C₇	AGL(1,ℤ/8)	1	2	yes	—
56	D₂₁	—	8	2	no	—
57	#127	—	21	1	no	—
58	#125	—	21	1	no	—
59	#102	—	21	4	no	—
60	#97	D₂₁	1	1	yes	characteristic, complemented
61	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
62	C₂×D₇	—	12	2	no	—
63	C₂×D₇	—	12	4	no	—
64	C₄×C₇	—	2	2	no	—
65	C₄×C₇	#235	1	2	yes	—
66	Dic₇	—	12	2	no	—
67	C₂²×D₃	—	14	2	no	—
68	C₂×C₃×C₄	D₂₈	1	2	yes	—
69	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
70	C₂×Dic₃	—	14	2	no	—
71	D₁₂	—	14	4	no	—
72	D₁₂	—	28	4	no	—
73	C₃×C₈	—	2	2	no	—
74	C₃×C₇	#490	1	1	yes	characteristic, complemented, Hall
75	C₂×D₄	—	21	2	no	—
76	C₂×D₄	—	42	2	no	—
77	C₂×C₈	C₂×D₂₁	1	2	yes	—
78	C₄⋊C₄	—	21	2	no	—
79	C₄⋊C₄	—	42	2	no	—
80	C₄²	C₂×D₂₁	1	1	yes	characteristic
81	C₂×C₇	#875	1	1	yes	characteristic
82	C₂×C₇	#908	1	2	yes	—
83	D₇	—	24	2	no	—
84	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
85	C₂×D₃	—	28	2	no	—
86	C₂×D₃	—	28	4	no	—
87	C₃×C₄	—	2	2	no	—
88	C₃×C₄	#1157	1	2	yes	—
89	Dic₃	—	28	2	no	—
90	C₂³	—	42	2	no	—
91	D₄	—	42	4	no	—
92	D₄	—	84	4	no	—
93	C₂×C₄	—	42	2	no	—
94	C₂×C₄	D₈₄	1	2	yes	—
95	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
96	C₈	—	2	2	no	—
97	C₇	#4713	1	1	yes	characteristic, complemented, Sylow
98	C₂×C₃	#6438	1	1	yes	characteristic
99	C₂×C₃	#6471	1	2	yes	—
100	D₃	—	56	2	no	—
101	C₂²	—	84	2	no	—
102	C₂²	—	84	4	no	—
103	C₂²	C₂×D₈₄	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	84	2	no	—
106	C₄	#66728	1	2	yes	—
107	C₃	#89485	1	1	yes	characteristic, complemented, Sylow
108	C₂	—	168	2	no	—
109	C₂	#10621358	1	1	yes	characteristic
110	C₂	#10621391	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10621357

C₂

yes

maximal, characteristic

#10621353

C₂

yes

maximal, characteristic, complemented

C₃×C₇×#97

C₂

yes

maximal, characteristic, complemented

#10620660

C₂

yes

maximal, complemented

#89485

—

maximal, Hall

C₂×D₈₄

—

C₂×D₈₄

C₂²

yes

—

C₂×C₃×C₇×C₈

C₂²

yes

—

C₃×C₄²×C₇

C₂²

yes

characteristic

C₄⋊Dic₂₁

C₂²

yes

—

#66629

—

#6437

—

#6433

—

C₇×#97

D₃

yes

characteristic, complemented

#6395

—

#4713

—

maximal, Hall

C₂²×D₂₁

—

C₂×C₃×C₄×C₇

D₄

yes

—

C₂×C₃×C₄×C₇

C₂³

yes

characteristic

C₂×Dic₂₁

—

D₈₄

—

D₈₄

—

C₃×C₇×C₈

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×C₇×C₈

C₂×D₃

yes

—

C₄²×C₇

C₂×D₃

yes

characteristic

C₄⋊Dic₇

—

Dic₇⋊C₄

—

#874

—

#870

—

C₃×#97

D₇

yes

characteristic, complemented

#821

—

C₂²×C₃×C₇

C₂×D₄

yes

characteristic

C₂×D₂₁

—

C₂×D₂₁

—

C₃×C₄×C₇

—

C₃×C₄×C₇

#41

yes

—

Dic₂₁

—

#490

—

Sylow

C₂²×D₇

—

C₂×C₄×C₇

D₁₂

yes

—

C₂×C₄×C₇

C₂²×D₃

yes

characteristic

C₂×Dic₇

—

D₂₈

—

D₂₈

—

C₇×C₈

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×C₃×C₈

C₂×D₇

yes

—

C₃×C₄²

C₂×D₇

yes

characteristic

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂×C₃×C₇

#124

yes

characteristic

C₂×C₃×C₇

AGL(1,ℤ/8)

yes

—

D₂₁

—

—

—

—

D₂₁

yes

characteristic, complemented

C₂²×C₇

C₂×D₁₂

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

#235

yes

—

Dic₇

—

C₂²×D₃

—

C₂×C₃×C₄

D₂₈

yes

—

C₂×C₃×C₄

C₂²×D₇

yes

characteristic

C₂×Dic₃

—

D₁₂

—

D₁₂

—

C₃×C₈

—

C₃×C₇

#490

yes

characteristic, complemented, Hall

C₂×D₄

—

C₂×D₄

—

C₂×C₈

C₂×D₂₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₂₁

yes

C₂×C₇

yes

C₂×C₇

yes

—

D₇

—

C₂²×C₃

C₂×D₂₈

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

#1157

yes

—

Dic₃

—

C₂³

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

D₈₄

yes

—

C₂×C₄

C₂²×D₂₁

yes

characteristic

C₈

—

C₇

#4713

yes

characteristic, complemented, Sylow

C₂×C₃

#6438

yes

characteristic

C₂×C₃

#6471

yes

—

100

D₃

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₂×D₈₄

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

#66728

yes

—

107

C₃

#89485

yes

characteristic, complemented, Sylow

108

C₂

—

168

—

109

C₂

#10621358

yes

characteristic

110

C₂

#10621391

yes

—

	subgroup	G/H
center	C₂²	C₂×D₈₄
commutator, cosocle	C₂×C₃×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₁
Fitting	C₃×C₇×#97	C₂
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	#10621357	C₂
hypercenter	#97	D₂₁
hypocenter	C₃×C₇	#490
LCS	G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×D₈₄

commutator, cosocle

C₂×C₃×C₄×C₇

C₂³

Frattini

C₂×C₄

C₂²×D₂₁

Fitting

C₃×C₇×#97

C₂

socle

C₂²×C₃×C₇

C₂×D₄

C₂

D₂₁

C₃×C₇

G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
2. #10621353	C₂	semidirect
3. C₃×C₇×#97	C₂	semidirect
4. #10620660	C₂	semidirect
14. C₇×#97	D₃	semidirect
32. C₃×#97	D₇	semidirect
60. #97	D₂₁	semidirect
74. C₃×C₇	#490	semidirect
97. C₇	#4713	semidirect
107. C₃	#89485	semidirect

H ≅ G/N

type

2. #10621353

C₂

semidirect

3. C₃×C₇×#97

C₂

semidirect

4. #10620660

C₂

semidirect

14. C₇×#97

D₃

semidirect

32. C₃×#97

D₇

semidirect

60. #97

D₂₁

semidirect

74. C₃×C₇

#490

semidirect

97. C₇

#4713

semidirect

107. C₃

#89485

semidirect

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

subgroup	index
#10621357	2
#10621353	2
C₃×C₇×#97	2
#10620660	2
#89485	3
#4713	7

subgroup

index

C₃×C₇×#97

Subgroups — #13056351

Conjugacy classes of subgroups

Complements

Maximal subgroups