Group #27192979 — subgroups

Subgroups — #27192979

Order 13184.

180 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₄⋊(C₁₀₃⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₀₃⋊C₈	—	2	2	no	—
4	C₄⋊(C₁₀₃⋊C₈)	—	2	2	no	—
5	C₄⋊(C₁₀₃⋊C₈)	C₄	1	4	yes	—
6	C₂²×C₈×C₁₀₃	C₄	1	2	yes	—
7	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₁₀₃	—	2	2	no	—
9	C₂×C₁₀₃⋊C₈	—	2	4	no	—
10	C₂×C₁₀₃⋊C₈	—	4	2	no	—
11	C₂×C₁₀₃⋊C₈	—	4	2	no	—
12	C₂²×C₄×C₁₀₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₁₀₃	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₀₃	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₁₀₃	—	2	2	no	—
16	C₂×C₈×C₁₀₃	—	2	2	no	—
17	C₂×C₈×C₁₀₃	C₈	1	4	yes	complemented
18	C₄²×C₁₀₃	C₂×C₄	1	2	yes	—
19	C₄²×C₁₀₃	D₄	1	2	yes	—
20	C₂³×C₁₀₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₀₃	—	2	2	no	—
22	C₂×C₄×C₁₀₃	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄×C₁₀₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₁₀₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₁₀₃	M₁₆	1	2	yes	—
28	C₂×C₄×C₁₀₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₁₀₃	SD₁₆	1	2	yes	—
30	C₂×C₄×C₁₀₃	Q₁₆	1	1	yes	characteristic
31	C₈×C₁₀₃	—	2	4	no	—
32	C₁₀₃⋊C₈	—	4	4	no	—
33	C₂²×C₁₀₃	#95	1	1	yes	characteristic
34	C₂²×C₁₀₃	C₄⋊C₈	1	2	yes	—
35	C₂²×C₁₀₃	#106	1	1	yes	characteristic
36	C₂²×C₁₀₃	#107	1	1	yes	characteristic
37	C₂²×C₁₀₃	#108	1	2	yes	—
38	C₄×C₁₀₃	—	2	2	no	—
39	C₄×C₁₀₃	C₄×C₈	1	1	yes	characteristic
40	C₄×C₁₀₃	#97	1	1	yes	characteristic
41	C₄×C₁₀₃	C₂²⋊C₈	1	2	yes	—
42	C₄×C₁₀₃	#102	1	2	yes	—
43	C₄×C₁₀₃	#103	1	2	yes	—
44	C₂×C₁₀₃	#334	1	2	yes	—
45	C₂×C₁₀₃	#335	1	2	yes	—
46	C₂×C₁₀₃	#336	1	1	yes	characteristic
47	C₂×C₁₀₃	#340	1	1	yes	characteristic
48	C₂×C₁₀₃	#341	1	1	yes	characteristic
49	#1277	—	103	1	no	maximal, Sylow
50	C₁₀₃	#1277	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₄⋊C₈	—	103	2	no	—
52	C₂×C₄×C₈	D₁₀₃	1	1	yes	characteristic
53	C₂²×C₈	—	206	2	no	—
54	C₂²×C₈	Dic₁₀₃	1	2	yes	—
55	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
56	C₄⋊C₈	—	103	4	no	—
57	C₄⋊C₈	—	206	2	no	—
58	C₄×C₈	—	2	2	no	—
59	C₂²×C₄	Dic₂₀₆	1	1	yes	characteristic
60	C₂²×C₄	D₄₁₂	1	1	yes	characteristic
61	C₂²×C₄	C₂×Dic₁₀₃	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	412	2	no	—
65	C₂×C₈	—	412	2	no	—
66	C₂×C₈	—	206	4	no	—
67	C₂×C₈	C₁₀₃⋊C₈	1	4	yes	complemented
68	C₄²	C₄×D₁₀₃	1	2	yes	—
69	C₄²	C₂²⋊D₁₀₃	1	2	yes	—
70	C₂³	C₄⋊Dic₁₀₃	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	C₄×Dic₁₀₃	1	1	yes	characteristic
73	C₂×C₄	(C₂×D₁₀₃)⋊C₄	1	2	yes	—
74	C₂×C₄	C₂²⋊Dic₁₀₃	1	1	yes	characteristic
75	C₂×C₄	Dic₁₀₃⋊C₄	1	2	yes	—
76	C₂×C₄	C₂×C₁₀₃⋊C₈	1	2	yes	—
77	C₂×C₄	#13110089	1	2	yes	—
78	C₂×C₄	D₈₂₄	1	1	yes	characteristic
79	C₂×C₄	#13110094	1	2	yes	—
80	C₂×C₄	Dic₄₁₂	1	1	yes	characteristic
81	C₈	—	2	4	no	—
82	C₈	—	412	4	no	—
83	C₂²	C₁₀₃⋊#95	1	1	yes	characteristic
84	C₂²	C₄⋊(C₁₀₃⋊C₈)	1	2	yes	—
85	C₂²	#15692953	1	1	yes	characteristic
86	C₂²	#15692955	1	1	yes	characteristic
87	C₂²	#15692957	1	2	yes	—
88	C₄	—	2	2	no	—
89	C₄	C₄×C₁₀₃⋊C₈	1	1	yes	characteristic
90	C₄	#15692932	1	1	yes	characteristic
91	C₄	C₂²⋊(C₁₀₃⋊C₈)	1	2	yes	—
92	C₄	#15692943	1	2	yes	—
93	C₄	#15692946	1	2	yes	—
94	C₂	#20267542	1	2	yes	—
95	C₂	#20267544	1	2	yes	—
96	C₂	#20267546	1	1	yes	characteristic
97	C₂	#20267554	1	1	yes	characteristic
98	C₂	#20267556	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₀₃⋊C₈)

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₃

C₂

yes

maximal, characteristic

C₂²×C₁₀₃⋊C₈

—

C₄⋊(C₁₀₃⋊C₈)

—

C₄⋊(C₁₀₃⋊C₈)

C₄

yes

—

C₂²×C₈×C₁₀₃

C₄

yes

—

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₄×C₈×C₁₀₃

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂×C₁₀₃⋊C₈

—

C₂²×C₄×C₁₀₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₃

D₄

yes

characteristic

C₂²×C₄×C₁₀₃

Q₈

yes

characteristic

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

—

C₂×C₈×C₁₀₃

C₈

yes

complemented

C₄²×C₁₀₃

C₂×C₄

yes

—

C₄²×C₁₀₃

D₄

yes

—

C₂³×C₁₀₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₄²

yes

characteristic

C₂×C₄×C₁₀₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₃

C₂×C₈

yes

—

C₂×C₄×C₁₀₃

M₁₆

yes

—

C₂×C₄×C₁₀₃

D₈

yes

characteristic

C₂×C₄×C₁₀₃

SD₁₆

yes

—

C₂×C₄×C₁₀₃

Q₁₆

yes

characteristic

C₈×C₁₀₃

—

C₁₀₃⋊C₈

—

C₂²×C₁₀₃

#95

yes

characteristic

C₂²×C₁₀₃

C₄⋊C₈

yes

—

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

C₂²×C₁₀₃

yes

—

C₄×C₁₀₃

—

C₄×C₁₀₃

C₄×C₈

yes

characteristic

C₄×C₁₀₃

#97

yes

characteristic

C₄×C₁₀₃

C₂²⋊C₈

yes

—

C₄×C₁₀₃

#102

yes

—

C₄×C₁₀₃

#103

yes

—

C₂×C₁₀₃

#334

yes

—

C₂×C₁₀₃

#335

yes

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

103

maximal, Sylow

C₁₀₃

#1277

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

103

—

C₂×C₄×C₈

D₁₀₃

yes

characteristic

C₂²×C₈

—

206

—

C₂²×C₈

Dic₁₀₃

yes

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

C₄⋊C₈

—

103

—

C₄⋊C₈

—

206

—

C₄×C₈

—

C₂²×C₄

Dic₂₀₆

yes

characteristic

C₂²×C₄

D₄₁₂

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

412

—

C₂×C₈

—

412

—

C₂×C₈

—

206

—

C₂×C₈

C₁₀₃⋊C₈

yes

complemented

C₄²

C₄×D₁₀₃

yes

—

C₄²

C₂²⋊D₁₀₃

yes

—

C₂³

C₄⋊Dic₁₀₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×Dic₁₀₃

yes

characteristic

C₂×C₄

(C₂×D₁₀₃)⋊C₄

yes

—

C₂×C₄

C₂²⋊Dic₁₀₃

yes

characteristic

C₂×C₄

Dic₁₀₃⋊C₄

yes

—

C₂×C₄

C₂×C₁₀₃⋊C₈

yes

—

C₂×C₄

#13110089

yes

—

C₂×C₄

D₈₂₄

yes

characteristic

C₂×C₄

#13110094

yes

—

C₂×C₄

Dic₄₁₂

yes

characteristic

C₈

—

C₈

—

412

—

C₂²

C₁₀₃⋊#95

yes

characteristic

C₂²

C₄⋊(C₁₀₃⋊C₈)

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

C₄×C₁₀₃⋊C₈

yes

characteristic

C₄

#15692932

yes

characteristic

C₄

C₂²⋊(C₁₀₃⋊C₈)

yes

—

C₄

#15692943

yes

—

C₄

#15692946

yes

—

C₂

#20267542

yes

—

C₂

#20267544

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₄	D₄₁₂
commutator	C₄×C₁₀₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₀₃
Fitting	C₂×C₄×C₈×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₄⋊C₄
cosocle	C₂×C₄²×C₁₀₃	C₂²
hypercenter	C₂×C₄×C₈	D₁₀₃
hypocenter	C₁₀₃	#1277
LCS	G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₄₁₂

commutator

C₄×C₁₀₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₁₀₃

Fitting

C₂×C₄×C₈×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₄⋊C₄

cosocle

C₂×C₄²×C₁₀₃

C₂²

hypercenter

C₂×C₄×C₈

D₁₀₃

hypocenter

C₁₀₃

#1277

LCS

G ▷ C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
17. C₂×C₈×C₁₀₃	C₈	semidirect
50. C₁₀₃	#1277	semidirect
67. C₂×C₈	C₁₀₃⋊C₈	semidirect

H ≅ G/N

type

17. C₂×C₈×C₁₀₃

C₈

semidirect

50. C₁₀₃

#1277

semidirect

67. C₂×C₈

C₁₀₃⋊C₈

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₀₃⋊C₈)	2
C₂×C₄×C₈×C₁₀₃	2
#1277	103

subgroup

index

C₂×C₄⋊(C₁₀₃⋊C₈)

C₂×C₄×C₈×C₁₀₃

#1277

103

Subgroups — #27192979

Conjugacy classes of subgroups

Complements

Maximal subgroups