Group #27195278 — subgroups

Subgroups — #27195278

Order 13184.

224 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 109 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	#20267661	C₂	1	1	yes	maximal, characteristic
2	#20267657	C₂	1	1	yes	maximal, characteristic
3	#20267554	C₂	1	4	yes	maximal
4	C₂×#97×C₁₀₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₁₀₃	—	2	2	no	—
6	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₃	C₂²	1	2	yes	—
8	C₁₀₃⋊#95	—	2	2	no	—
9	C₂²×C₈×C₁₀₃	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
11	#97×C₁₀₃	C₄	1	4	yes	complemented
12	C₂²×Dic₁₀₃	—	2	2	no	—
13	C₂²×Dic₁₀₃	—	2	2	no	—
14	C₄⋊Dic₁₀₃	—	2	4	no	—
15	C₄⋊Dic₁₀₃	—	2	4	no	—
16	C₄⋊Dic₁₀₃	—	4	4	no	—
17	C₂²×C₄×C₁₀₃	D₄	1	2	yes	—
18	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₁₀₃	—	2	2	no	—
20	C₂×C₈×C₁₀₃	—	2	2	no	—
21	C₂×C₈×C₁₀₃	C₂×C₄	1	4	yes	—
22	C₄²×C₁₀₃	C₂×C₄	1	2	yes	—
23	C₄²×C₁₀₃	Q₈	1	2	yes	—
24	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₃	—	2	2	no	—
26	C₂×C₄×C₁₀₃	—	2	2	no	—
27	C₂×C₄×C₁₀₃	C₄⋊C₄	1	4	yes	—
28	C₂×C₄×C₁₀₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₃	#41	1	2	yes	—
31	C₂×C₄×C₁₀₃	#41	1	2	yes	—
32	C₂×Dic₁₀₃	—	4	2	no	—
33	C₂×Dic₁₀₃	—	4	2	no	—
34	C₂×Dic₁₀₃	—	4	4	no	—
35	C₂×Dic₁₀₃	—	4	2	no	—
36	C₂×Dic₁₀₃	—	4	2	no	—
37	C₈×C₁₀₃	—	2	4	no	—
38	C₂²×C₁₀₃	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₀₃	#124	1	1	yes	characteristic
40	C₂²×C₁₀₃	#124	1	1	yes	characteristic
41	C₂²×C₁₀₃	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₁₀₃	#137	1	2	yes	—
43	C₄×C₁₀₃	—	2	4	no	—
44	C₄×C₁₀₃	#117	1	2	yes	—
45	C₄×C₁₀₃	#125	1	2	yes	—
46	Dic₁₀₃	—	8	2	no	—
47	Dic₁₀₃	—	8	2	no	—
48	C₂×C₁₀₃	#382	1	1	yes	characteristic
49	C₂×C₁₀₃	#428	1	2	yes	—
50	C₂×C₁₀₃	#489	1	2	yes	—
51	C₂×C₁₀₃	#490	1	1	yes	characteristic
52	C₂×C₁₀₃	#491	1	1	yes	characteristic
53	#1836	—	103	1	no	maximal, Sylow
54	C₁₀₃	#1836	1	1	yes	characteristic, complemented, Sylow
55	C₂×#97	D₁₀₃	1	1	yes	characteristic
56	#384	—	103	1	no	—
57	#382	—	103	1	no	—
58	#340	—	103	4	no	—
59	C₂²×C₈	C₂×D₁₀₃	1	2	yes	—
60	C₂×C₄⋊C₄	—	103	2	no	—
61	C₂×C₄⋊C₄	—	103	2	no	—
62	C₂×C₄⋊C₄	—	206	2	no	—
63	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
64	#97	Dic₁₀₃	1	4	yes	complemented
65	#95	—	206	2	no	—
66	C₂²×C₄	—	206	2	no	—
67	C₂²×C₄	—	206	2	no	—
68	C₂²×C₄	D₄₁₂	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×Dic₁₀₃	1	4	yes	—
73	C₄⋊C₄	—	412	4	no	—
74	C₄⋊C₄	—	206	4	no	—
75	C₄⋊C₄	—	206	4	no	—
76	C₄²	Dic₂₀₆	1	2	yes	—
77	C₄²	C₂×Dic₁₀₃	1	2	yes	—
78	C₂³	C₂×D₄₁₂	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	412	2	no	—
82	C₂×C₄	—	412	2	no	—
83	C₂×C₄	—	412	4	no	—
84	C₂×C₄	—	412	2	no	—
85	C₂×C₄	—	412	2	no	—
86	C₂×C₄	C₄⋊Dic₁₀₃	1	4	yes	—
87	C₂×C₄	C₂²×Dic₁₀₃	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₂₀₆	1	1	yes	characteristic
89	C₂×C₄	#13110105	1	2	yes	—
90	C₂×C₄	#13110105	1	2	yes	—
91	C₈	—	2	4	no	—
92	C₂²	C₂×C₄⋊Dic₁₀₃	1	1	yes	characteristic
93	C₂²	#15693009	1	1	yes	characteristic
94	C₂²	#15693009	1	1	yes	characteristic
95	C₂²	#15693049	1	2	yes	—
96	C₂²	#15693054	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	824	2	no	—
99	C₄	—	824	2	no	—
100	C₄	#15692980	1	2	yes	—
101	C₄	#15693013	1	2	yes	—
102	C₂	#20267657	1	1	yes	characteristic
103	C₂	#20267833	1	2	yes	—
104	C₂	#20268151	1	2	yes	—
105	C₂	#20268156	1	1	yes	characteristic
106	C₂	#20268160	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20267661

C₂

yes

maximal, characteristic

#20267657

C₂

yes

maximal, characteristic

#20267554

C₂

yes

maximal

C₂×#97×C₁₀₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₀₃

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

C₁₀₃⋊#95

—

C₂²×C₈×C₁₀₃

C₂²

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

#97×C₁₀₃

C₄

yes

complemented

C₂²×Dic₁₀₃

—

C₂²×Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₂²×C₄×C₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

C₂×C₄

yes

—

C₄²×C₁₀₃

C₂×C₄

yes

—

C₄²×C₁₀₃

Q₈

yes

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₀₃

#41

yes

—

C₂×C₄×C₁₀₃

#41

yes

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₈×C₁₀₃

—

C₂²×C₁₀₃

C₂×C₄⋊C₄

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

—

C₂²×C₁₀₃

#137

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

#117

yes

—

C₄×C₁₀₃

#125

yes

—

Dic₁₀₃

—

Dic₁₀₃

—

C₂×C₁₀₃

#382

yes

characteristic

C₂×C₁₀₃

#428

yes

—

C₂×C₁₀₃

#489

yes

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

103

maximal, Sylow

C₁₀₃

#1836

yes

characteristic, complemented, Sylow

C₂×#97

D₁₀₃

yes

characteristic

#384

—

103

—

#382

—

103

—

#340

—

103

—

C₂²×C₈

C₂×D₁₀₃

yes

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

206

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#97

Dic₁₀₃

yes

complemented

#95

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

—

206

—

C₂²×C₄

D₄₁₂

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₀₃

yes

—

C₄⋊C₄

—

412

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄²

Dic₂₀₆

yes

—

C₄²

C₂×Dic₁₀₃

yes

—

C₂³

C₂×D₄₁₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

412

—

C₂×C₄

—

412

—

C₂×C₄

—

412

—

C₂×C₄

—

412

—

C₂×C₄

—

412

—

C₂×C₄

C₄⋊Dic₁₀₃

yes

—

C₂×C₄

C₂²×Dic₁₀₃

yes

characteristic

C₂×C₄

C₂×Dic₂₀₆

yes

characteristic

C₂×C₄

#13110105

yes

—

C₂×C₄

#13110105

yes

—

C₈

—

C₂²

C₂×C₄⋊Dic₁₀₃

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#15693054

yes

—

C₄

—

C₄

—

824

—

C₄

—

824

—

100

C₄

#15692980

yes

—

101

C₄

#15693013

yes

—

102

C₂

#20267657

yes

characteristic

103

C₂

#20267833

yes

—

104

C₂

#20268151

yes

—

105

C₂

#20268156

yes

characteristic

106

C₂

#20268160

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₄₁₂
commutator	C₂×C₄×C₁₀₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₃
Fitting	C₂×#97×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂×D₄
cosocle	C₂²×C₄×C₁₀₃	C₂³
hypercenter	C₂×#97	D₁₀₃
hypocenter	C₁₀₃	#1836
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₄₁₂

commutator

C₂×C₄×C₁₀₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₃

Fitting

C₂×#97×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂×D₄

cosocle

C₂²×C₄×C₁₀₃

C₂³

hypercenter

C₂×#97

D₁₀₃

hypocenter

C₁₀₃

#1836

LCS

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
11. #97×C₁₀₃	C₄	semidirect
54. C₁₀₃	#1836	semidirect
64. #97	Dic₁₀₃	semidirect

H ≅ G/N

type

11. #97×C₁₀₃

C₄

semidirect

54. C₁₀₃

#1836

semidirect

64. #97

Dic₁₀₃

semidirect

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

subgroup	index
#20267661	2
#20267657	2
#20267554	2
C₂×#97×C₁₀₃	2
#1836	103

subgroup

index

C₂×#97×C₁₀₃

103

Subgroups — #27195278

Conjugacy classes of subgroups

Complements

Maximal subgroups