Group #27205648 — subgroups

Subgroups — C₂×#20268193 #27205648

Order 13184.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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×#15693013	C₂	1	2	yes	maximal
2	#20268193	C₂	1	8	yes	maximal, direct product factor
3	C₂×#15692955	C₂	1	2	yes	maximal
4	C₂×#15692953	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	#15693013	—	2	4	no	—
7	#15693013	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₁₀₃	C₂²	1	4	yes	—
9	C₂×Dic₁₀₃⋊C₄	—	2	4	no	—
10	#15692955	C₂²	1	8	yes	—
11	#15692953	C₂²	1	8	yes	—
12	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₀₃	C₂²	1	4	yes	—
16	C₂²×Dic₁₀₃	—	2	4	no	—
17	C₄⋊Dic₁₀₃	—	2	4	no	—
18	C₄⋊Dic₁₀₃	C₂³	1	8	yes	—
19	Dic₁₀₃⋊C₄	—	2	16	no	—
20	C₂²×C₄×C₁₀₃	D₄	1	2	yes	—
21	C₂²×C₄×C₁₀₃	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₀₃	Q₈	1	4	yes	—
23	C₂×C₈×C₁₀₃	Q₈	1	4	yes	—
24	C₂×C₈×C₁₀₃	C₂³	1	2	yes	—
25	C₂×C₈×C₁₀₃	C₂³	1	2	yes	—
26	C₄²×C₁₀₃	D₄	1	2	yes	—
27	C₄²×C₁₀₃	C₂³	1	2	yes	—
28	C₂³×C₁₀₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₃	—	2	4	no	—
30	C₂×C₄×C₁₀₃	C₂×D₄	1	4	yes	—
31	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₃	C₂×Q₈	1	2	yes	—
33	C₂×C₄×C₁₀₃	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₁₀₃	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₁₀₃	—	2	8	no	—
36	C₂×Dic₁₀₃	—	4	8	no	—
37	C₈×C₁₀₃	C₂×Q₈	1	4	yes	—
38	C₈×C₁₀₃	C₂×Q₈	1	4	yes	—
39	C₂²×C₁₀₃	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₁₀₃	#135	1	4	yes	—
41	C₂²×C₁₀₃	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₁₀₃	—	2	4	no	—
43	C₄×C₁₀₃	C₄⋊Q₈	1	2	yes	—
44	C₄×C₁₀₃	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₁₀₃	C₂²×Q₈	1	1	yes	characteristic
46	Dic₁₀₃	—	4	8	no	—
47	C₂×C₁₀₃	#499	1	4	yes	complemented
48	C₂×C₁₀₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₁₀₃	C₂×#135	1	2	yes	—
50	C₂×#499	—	103	1	no	maximal, Sylow
51	C₁₀₃	C₂×#499	1	1	yes	characteristic, complemented, Sylow
52	C₂×#125	—	103	2	no	—
53	#499	—	103	8	no	—
54	C₂×#107	—	103	2	no	—
55	C₂×#106	—	103	2	no	—
56	C₂×C₄×C₈	D₁₀₃	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₁₀₃	1	1	yes	characteristic
59	#125	—	103	8	no	—
60	#125	—	206	4	no	—
61	C₂×C₄⋊C₄	—	103	4	no	—
62	C₂×C₄⋊C₄	—	206	4	no	—
63	C₂×C₄²	C₂×D₁₀₃	1	1	yes	characteristic
64	#107	—	103	8	no	—
65	#106	—	103	8	no	—
66	C₄×C₈	C₂×D₁₀₃	1	4	yes	—
67	C₂²×C₄	—	206	4	no	—
68	C₂²×C₄	D₄₁₂	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₀₃	1	1	yes	characteristic
70	C₂×C₈	Dic₂₀₆	1	4	yes	—
71	C₂×C₈	Dic₂₀₆	1	4	yes	—
72	C₂×C₈	C₂²×D₁₀₃	1	2	yes	—
73	C₂×C₈	C₂²×D₁₀₃	1	2	yes	—
74	C₄⋊C₄	—	103	8	no	—
75	C₄⋊C₄	—	206	16	no	—
76	C₄⋊C₄	—	206	4	no	—
77	C₄²	D₄₁₂	1	2	yes	—
78	C₄²	C₂²×D₁₀₃	1	2	yes	—
79	C₂³	C₂×D₄₁₂	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	412	8	no	—
82	C₂×C₄	—	206	8	no	—
83	C₂×C₄	C₂×D₄₁₂	1	4	yes	—
84	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₂₀₆	1	2	yes	—
86	C₂×C₄	C₂×Dic₂₀₆	1	2	yes	—
87	C₂×C₄	C₂³×D₁₀₃	1	1	yes	characteristic
88	C₈	C₂×Dic₂₀₆	1	4	yes	—
89	C₈	C₂×Dic₂₀₆	1	4	yes	—
90	C₂²	#15693021	1	2	yes	—
91	C₂²	#15693043	1	4	yes	—
92	C₂²	C₂²×D₄₁₂	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	412	8	no	—
95	C₄	#15693021	1	2	yes	—
96	C₄	C₂²×Dic₂₀₆	1	1	yes	characteristic
97	C₄	C₂²×Dic₂₀₆	1	1	yes	characteristic
98	C₂	#20268193	1	4	yes	direct product factor
99	C₂	C₂×#15693021	1	1	yes	characteristic
100	C₂	C₂×#15693043	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15693013

C₂

yes

maximal

#20268193

C₂

yes

maximal, direct product factor

C₂×#15692955

C₂

yes

maximal

C₂×#15692953

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₃

C₂

yes

maximal, characteristic

#15693013

—

#15693013

C₂²

yes

—

C₂×C₄⋊Dic₁₀₃

C₂²

yes

—

C₂×Dic₁₀₃⋊C₄

—

#15692955

C₂²

yes

—

#15692953

C₂²

yes

—

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

C₂²×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×C₄²×C₁₀₃

C₂²

yes

characteristic

C₄×C₈×C₁₀₃

C₂²

yes

—

C₂²×Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

C₂³

yes

—

Dic₁₀₃⋊C₄

—

C₂²×C₄×C₁₀₃

D₄

yes

—

C₂²×C₄×C₁₀₃

C₂³

yes

characteristic

C₂×C₈×C₁₀₃

Q₈

yes

—

C₂×C₈×C₁₀₃

Q₈

yes

—

C₂×C₈×C₁₀₃

C₂³

yes

—

C₂×C₈×C₁₀₃

C₂³

yes

—

C₄²×C₁₀₃

D₄

yes

—

C₄²×C₁₀₃

C₂³

yes

—

C₂³×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×Q₈

yes

—

C₂×C₄×C₁₀₃

C₂×Q₈

yes

—

C₂×C₄×C₁₀₃

C₂⁴

yes

characteristic

C₂×Dic₁₀₃

—

C₂×Dic₁₀₃

—

C₈×C₁₀₃

C₂×Q₈

yes

—

C₈×C₁₀₃

C₂×Q₈

yes

—

C₂²×C₁₀₃

C₄⋊Q₈

yes

—

C₂²×C₁₀₃

#135

yes

—

C₂²×C₁₀₃

C₂²×D₄

yes

characteristic

C₄×C₁₀₃

—

C₄×C₁₀₃

C₄⋊Q₈

yes

—

C₄×C₁₀₃

C₂²×Q₈

yes

characteristic

C₄×C₁₀₃

C₂²×Q₈

yes

characteristic

Dic₁₀₃

—

C₂×C₁₀₃

#499

yes

complemented

C₂×C₁₀₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₃

C₂×#135

yes

—

C₂×#499

—

103

maximal, Sylow

C₁₀₃

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#125

—

103

—

#499

—

103

—

C₂×#107

—

103

—

C₂×#106

—

103

—

C₂×C₄×C₈

D₁₀₃

yes

characteristic

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

C₂²×C₈

C₂×D₁₀₃

yes

characteristic

#125

—

103

—

#125

—

206

—

C₂×C₄⋊C₄

—

103

—

C₂×C₄⋊C₄

—

206

—

C₂×C₄²

C₂×D₁₀₃

yes

characteristic

#107

—

103

—

#106

—

103

—

C₄×C₈

C₂×D₁₀₃

yes

—

C₂²×C₄

—

206

—

C₂²×C₄

D₄₁₂

yes

—

C₂²×C₄

C₂²×D₁₀₃

yes

characteristic

C₂×C₈

Dic₂₀₆

yes

—

C₂×C₈

Dic₂₀₆

yes

—

C₂×C₈

C₂²×D₁₀₃

yes

—

C₂×C₈

C₂²×D₁₀₃

yes

—

C₄⋊C₄

—

103

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄²

D₄₁₂

yes

—

C₄²

C₂²×D₁₀₃

yes

—

C₂³

C₂×D₄₁₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

412

—

C₂×C₄

—

206

—

C₂×C₄

C₂×D₄₁₂

yes

—

C₂×C₄

C₂×D₄₁₂

yes

characteristic

C₂×C₄

C₂×Dic₂₀₆

yes

—

C₂×C₄

C₂×Dic₂₀₆

yes

—

C₂×C₄

C₂³×D₁₀₃

yes

characteristic

C₈

C₂×Dic₂₀₆

yes

—

C₈

C₂×Dic₂₀₆

yes

—

C₂²

#15693021

yes

—

C₂²

#15693043

yes

—

C₂²

C₂²×D₄₁₂

yes

characteristic

C₄

—

C₄

—

412

—

C₄

#15693021

yes

—

C₄

C₂²×Dic₂₀₆

yes

characteristic

C₄

C₂²×Dic₂₀₆

yes

characteristic

C₂

#20268193

yes

direct product factor

C₂

C₂×#15693021

yes

characteristic

100

C₂

C₂×#15693043

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₁₂
commutator, cosocle	C₂×C₄×C₁₀₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₃
Fitting	C₂×C₄×C₈×C₁₀₃	C₂
socle, prime radical	C₂³×C₁₀₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₀₃
hypocenter	C₁₀₃	C₂×#499
LCS	G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₄₁₂

commutator, cosocle

C₂×C₄×C₁₀₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₃

Fitting

C₂×C₄×C₈×C₁₀₃

C₂

socle, prime radical

C₂³×C₁₀₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₀₃

hypocenter

C₁₀₃

C₂×#499

LCS

G ▷ C₂×C₄×C₁₀₃ ▷ C₂×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #20268193	C₂	direct (N × H)
47. C₂×C₁₀₃	#499	semidirect
51. C₁₀₃	C₂×#499	semidirect
98. C₂	#20268193	direct (N × H)

H ≅ G/N

type

2. #20268193

C₂

direct (N × H)

47. C₂×C₁₀₃

#499

semidirect

51. C₁₀₃

C₂×#499

semidirect

98. C₂

#20268193

direct (N × H)

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

subgroup	index
C₂×#15693013	2
#20268193	2
C₂×#15692955	2
C₂×#15692953	2
C₂×C₄×C₈×C₁₀₃	2
C₂×#499	103

subgroup

index

C₂×C₄×C₈×C₁₀₃

103

Subgroups — C₂×#20268193 #27205648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20268193 #27205648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20268193 #27205648