Group #27194733 — subgroups

Subgroups — #27194733

Order 13184.

224 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 69 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	#20268196	C₂	1	1	yes	maximal, characteristic, complemented
2	#20268163	C₂	1	2	yes	maximal, complemented
3	#20268137	C₂	1	2	yes	maximal, complemented
4	C₈×C₁₀₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₁₀₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13110093	—	2	4	no	—
7	#15693021	C₂²	1	2	yes	—
8	#15692955	—	2	2	no	—
9	#15692942	—	2	4	no	—
10	C₄×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₁₀₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×D₈×C₁₀₃	—	2	2	no	—
13	#127×C₁₀₃	C₂²	1	2	yes	—
14	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₀₆	—	2	2	no	—
16	#13110093	—	4	8	no	—
17	C₂×C₁₀₃⋊C₈	D₄	1	2	yes	—
18	C₂×C₁₀₃⋊C₈	D₄	1	2	yes	—
19	C₄⋊Dic₁₀₃	—	2	2	no	—
20	C₄⋊Dic₁₀₃	—	2	2	no	—
21	C₂×D₄×C₁₀₃	—	2	2	no	—
22	C₂×D₄×C₁₀₃	—	2	2	no	—
23	C₂×D₄×C₁₀₃	—	2	2	no	—
24	D₈×C₁₀₃	—	4	4	no	—
25	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
26	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
27	C₂³×C₁₀₃	—	4	2	no	—
28	D₄×C₁₀₃	—	4	4	no	—
29	D₄×C₁₀₃	—	4	4	no	—
30	D₄×C₁₀₃	—	4	4	no	—
31	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₁₀₃	—	4	2	no	—
35	Dic₂₀₆	—	4	4	no	—
36	C₈×C₁₀₃	SD₁₆	1	4	yes	complemented
37	C₁₀₃⋊C₈	—	2	2	no	—
38	C₁₀₃⋊C₈	D₈	1	4	yes	complemented
39	C₂²×C₁₀₃	—	8	2	no	—
40	C₂²×C₁₀₃	—	8	2	no	—
41	C₂²×C₁₀₃	—	8	2	no	—
42	C₂²×C₁₀₃	#127	1	1	yes	characteristic
43	C₄×C₁₀₃	C₂×D₈	1	2	yes	—
44	C₄×C₁₀₃	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₀₃	#135	1	2	yes	—
46	Dic₁₀₃	—	8	2	no	—
47	C₂×C₁₀₃	—	16	2	no	—
48	C₂×C₁₀₃	#492	1	1	yes	characteristic
49	C₂×C₁₀₃	#493	1	1	yes	characteristic
50	C₂×C₁₀₃	#495	1	1	yes	characteristic
51	#1706	—	103	1	no	maximal, Sylow
52	C₁₀₃	#1706	1	1	yes	characteristic, complemented, Sylow
53	#500	—	103	1	no	—
54	#493	D₁₀₃	1	1	yes	characteristic
55	#492	—	103	2	no	—
56	#486	—	103	2	no	—
57	C₈²	—	103	1	no	—
58	C₂×SD₁₆	—	206	4	no	—
59	C₂×D₈	—	2	2	no	—
60	C₄⋊Q₈	—	103	2	no	—
61	#127	C₂×D₁₀₃	1	2	yes	—
62	#107	—	206	2	no	—
63	#102	—	206	4	no	—
64	C₄×C₈	—	103	1	no	—
65	C₄×C₈	—	103	1	no	—
66	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
67	C₂×Q₈	—	206	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	2	no	—
70	C₂×D₄	—	2	2	no	—
71	SD₁₆	—	412	8	no	—
72	D₈	—	4	4	no	—
73	C₂×C₈	—	103	2	no	—
74	C₂×C₈	—	103	2	no	—
75	C₂×C₈	C₂²⋊D₁₀₃	1	2	yes	—
76	C₄⋊C₄	—	206	2	no	—
77	C₄⋊C₄	—	206	2	no	—
78	C₄²	C₂²×D₁₀₃	1	1	yes	characteristic
79	C₂³	—	4	2	no	—
80	Q₈	—	412	4	no	—
81	D₄	—	4	4	no	—
82	D₄	—	4	4	no	—
83	D₄	—	4	4	no	—
84	C₂×C₄	—	412	2	no	—
85	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
86	C₂×C₄	D₄×D₁₀₃	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₁₀₃	1	1	yes	characteristic
88	C₈	—	103	4	no	—
89	C₈	—	206	2	no	—
90	C₈	#13110093	1	4	yes	complemented
91	C₂²	—	8	2	no	—
92	C₂²	—	8	2	no	—
93	C₂²	—	8	2	no	—
94	C₂²	#15693017	1	1	yes	characteristic
95	C₄	—	824	2	no	—
96	C₄	D₈×D₁₀₃	1	2	yes	—
97	C₄	C₂×#13110093	1	2	yes	—
98	C₄	#15693040	1	2	yes	—
99	C₂	—	16	2	no	—
100	C₂	#20268163	1	1	yes	characteristic
101	C₂	#20268166	1	1	yes	characteristic
102	C₂	#20268174	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20268196

C₂

yes

maximal, characteristic, complemented

#20268163

C₂

yes

maximal, complemented

#20268137

C₂

yes

maximal, complemented

C₈×C₁₀₃⋊C₈

C₂

yes

maximal, characteristic, complemented

#493×C₁₀₃

C₂

yes

maximal, characteristic

C₂×#13110093

—

#15693021

C₂²

yes

—

#15692955

—

#15692942

—

C₄×C₁₀₃⋊C₈

C₂²

yes

characteristic

C₄×C₁₀₃⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₁₀₃

—

#127×C₁₀₃

C₂²

yes

—

C₄×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×Dic₂₀₆

—

#13110093

—

C₂×C₁₀₃⋊C₈

D₄

yes

—

C₂×C₁₀₃⋊C₈

D₄

yes

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₂×D₄×C₁₀₃

—

C₂×D₄×C₁₀₃

—

C₂×D₄×C₁₀₃

—

D₈×C₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₄²×C₁₀₃

C₂³

yes

characteristic

C₂³×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

D₄×C₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×Dic₁₀₃

—

Dic₂₀₆

—

C₈×C₁₀₃

SD₁₆

yes

complemented

C₁₀₃⋊C₈

—

C₁₀₃⋊C₈

D₈

yes

complemented

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

—

C₂²×C₁₀₃

#127

yes

characteristic

C₄×C₁₀₃

C₂×D₈

yes

—

C₄×C₁₀₃

C₂×SD₁₆

yes

—

C₄×C₁₀₃

#135

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

—

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

C₂×C₁₀₃

yes

—

103

maximal, Sylow

C₁₀₃

#1706

yes

characteristic, complemented, Sylow

#500

—

103

—

#493

D₁₀₃

yes

characteristic

#492

—

103

—

#486

—

103

—

C₈²

—

103

—

C₂×SD₁₆

—

206

—

C₂×D₈

—

C₄⋊Q₈

—

103

—

#127

C₂×D₁₀₃

yes

—

#107

—

206

—

#102

—

206

—

C₄×C₈

—

103

—

C₄×C₈

—

103

—

C₄×C₈

C₂×D₁₀₃

yes

characteristic

C₂×Q₈

—

206

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

412

—

D₈

—

C₂×C₈

—

103

—

C₂×C₈

—

103

—

C₂×C₈

C₂²⋊D₁₀₃

yes

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄²

C₂²×D₁₀₃

yes

characteristic

C₂³

—

Q₈

—

412

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

412

—

C₂×C₄

D₄×D₁₀₃

yes

characteristic

C₂×C₄

D₄×D₁₀₃

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₃

yes

characteristic

C₈

—

103

—

C₈

—

206

—

C₈

#13110093

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

#15693017

yes

characteristic

C₄

—

824

—

C₄

D₈×D₁₀₃

yes

—

C₄

C₂×#13110093

yes

—

C₄

#15693040

yes

—

C₂

—

100

C₂

#20268163

yes

characteristic

101

C₂

#20268166

yes

characteristic

102

C₂

#20268174

yes

characteristic

	subgroup	G/H
center	C₂²	#15693017
commutator, cosocle	C₄²×C₁₀₃	C₂³
Frattini	C₄²	C₂²×D₁₀₃
Fitting, prime radical	#493×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#127
hypercenter	#493	D₁₀₃
hypocenter	C₁₀₃	#1706
LCS	G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#15693017

commutator, cosocle

C₄²×C₁₀₃

C₂³

Frattini

C₄²

C₂²×D₁₀₃

Fitting, prime radical

#493×C₁₀₃

C₂

C₂²×C₁₀₃

D₁₀₃

C₁₀₃

G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #20268196	C₂	semidirect
2. #20268163	C₂	semidirect
3. #20268137	C₂	semidirect
4. C₈×C₁₀₃⋊C₈	C₂	semidirect
36. C₈×C₁₀₃	SD₁₆	semidirect
38. C₁₀₃⋊C₈	D₈	semidirect
52. C₁₀₃	#1706	semidirect
90. C₈	#13110093	semidirect

H ≅ G/N

type

1. #20268196

C₂

semidirect

2. #20268163

C₂

semidirect

3. #20268137

C₂

semidirect

4. C₈×C₁₀₃⋊C₈

C₂

semidirect

36. C₈×C₁₀₃

SD₁₆

semidirect

38. C₁₀₃⋊C₈

D₈

semidirect

52. C₁₀₃

#1706

semidirect

90. C₈

#13110093

semidirect

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

subgroup	index
#20268196	2
#20268163	2
#20268137	2
C₈×C₁₀₃⋊C₈	2
#493×C₁₀₃	2
#1706	103

subgroup

index

#20268196

#20268163

#20268137

C₈×C₁₀₃⋊C₈

#493×C₁₀₃

#1706

103

Subgroups — #27194733

Conjugacy classes of subgroups

Complements

Maximal subgroups