Group #27194772 — subgroups

Subgroups — #27194772

Order 13184.

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

normal subgroups only

This group has 220 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	#20268189	C₂	1	1	yes	maximal, characteristic, complemented
2	#20268168	C₂	1	1	yes	maximal, characteristic
3	#20268164	C₂	1	2	yes	maximal, complemented
4	#20268138	C₂	1	2	yes	maximal, complemented
5	#322×C₁₀₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13110094	—	2	4	no	—
7	C₂×D₈₂₄	—	2	2	no	—
8	#15693021	C₂²	1	2	yes	—
9	#15693018	C₂²	1	1	yes	characteristic
10	#15693018	C₂²	1	1	yes	characteristic
11	#15692953	—	2	2	no	—
12	#15692943	—	2	4	no	—
13	C₄×C₈×C₁₀₃	C₂²	1	2	yes	—
14	C₄×C₈×C₁₀₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₀₆	—	2	2	no	—
16	C₂×D₄₁₂	—	2	2	no	—
17	C₂×D₄₁₂	—	2	1	no	—
18	C₂×D₄₁₂	—	2	1	no	—
19	C₂×D₄₁₂	—	2	2	no	—
20	#13110094	—	4	8	no	—
21	D₈₂₄	—	4	4	no	—
22	C₄⋊Dic₁₀₃	—	2	2	no	—
23	C₄⋊Dic₁₀₃	—	2	2	no	—
24	C₂×C₈×C₁₀₃	D₄	1	4	yes	—
25	C₂×C₈×C₁₀₃	D₄	1	2	yes	—
26	C₄²×C₁₀₃	C₂³	1	1	yes	characteristic
27	C₂²×D₁₀₃	—	4	1	no	—
28	C₂²×D₁₀₃	—	4	1	no	—
29	C₂×C₄×C₁₀₃	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₃	C₂×D₄	1	2	yes	—
31	C₂×Dic₁₀₃	—	4	2	no	—
32	D₄₁₂	—	4	2	no	—
33	D₄₁₂	—	4	2	no	—
34	D₄₁₂	—	4	4	no	—
35	D₄₁₂	—	4	4	no	—
36	Dic₂₀₆	—	4	4	no	—
37	C₈×C₁₀₃	—	2	4	no	—
38	C₈×C₁₀₃	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₀₃	#127	1	1	yes	characteristic
40	C₂×D₁₀₃	—	8	1	no	—
41	C₂×D₁₀₃	—	8	1	no	—
42	C₂×D₁₀₃	—	8	2	no	—
43	C₂×D₁₀₃	—	8	2	no	—
44	C₄×C₁₀₃	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₀₃	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₁₀₃	—	8	2	no	—
47	C₂×C₁₀₃	#492	1	1	yes	characteristic
48	C₂×C₁₀₃	#496	1	2	yes	—
49	D₁₀₃	—	16	1	no	—
50	D₁₀₃	—	16	1	no	—
51	#1715	—	103	1	no	maximal, Sylow
52	C₁₀₃	#1715	1	1	yes	characteristic, complemented, Sylow
53	#498	—	103	1	no	—
54	#493	—	103	1	no	—
55	#492	—	103	2	no	—
56	#486	—	103	2	no	—
57	#322	D₁₀₃	1	1	yes	characteristic, complemented
58	C₂×SD₁₆	—	206	4	no	—
59	C₂×D₈	—	206	2	no	—
60	C₄⋊Q₈	—	103	2	no	—
61	#127	—	103	1	no	—
62	#127	—	103	1	no	—
63	#106	—	206	2	no	—
64	#102	—	206	4	no	—
65	C₄×C₈	C₂×D₁₀₃	1	2	yes	—
66	C₄×C₈	C₂×D₁₀₃	1	1	yes	characteristic
67	C₂×Q₈	—	206	2	no	—
68	C₂×D₄	—	206	2	no	—
69	C₂×D₄	—	206	1	no	—
70	C₂×D₄	—	206	1	no	—
71	C₂×D₄	—	206	2	no	—
72	SD₁₆	—	412	8	no	—
73	D₈	—	412	4	no	—
74	C₂×C₈	D₄₁₂	1	4	yes	—
75	C₂×C₈	D₄₁₂	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	—	412	1	no	—
80	C₂³	—	412	1	no	—
81	Q₈	—	412	4	no	—
82	D₄	—	412	2	no	—
83	D₄	—	412	2	no	—
84	D₄	—	412	4	no	—
85	D₄	—	412	4	no	—
86	C₂×C₄	—	412	2	no	—
87	C₂×C₄	C₂×D₄₁₂	1	1	yes	characteristic
88	C₂×C₄	C₂×D₄₁₂	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	#13110094	1	4	yes	complemented
91	C₂²	—	824	1	no	—
92	C₂²	—	824	1	no	—
93	C₂²	—	824	2	no	—
94	C₂²	—	824	2	no	—
95	C₂²	#15693018	1	1	yes	characteristic
96	C₄	—	824	2	no	—
97	C₄	C₂×#13110094	1	2	yes	—
98	C₄	#15693049	1	4	yes	—
99	C₂	—	1648	1	no	—
100	C₂	—	1648	1	no	—
101	C₂	#20268164	1	1	yes	characteristic
102	C₂	#20268180	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20268189

C₂

yes

maximal, characteristic, complemented

#20268168

C₂

yes

maximal, characteristic

#20268164

C₂

yes

maximal, complemented

#20268138

C₂

yes

maximal, complemented

#322×C₁₀₃

C₂

yes

maximal, characteristic, complemented

C₂×#13110094

—

C₂×D₈₂₄

—

#15693021

C₂²

yes

—

#15693018

C₂²

yes

characteristic

#15693018

C₂²

yes

characteristic

#15692953

—

#15692943

—

C₄×C₈×C₁₀₃

C₂²

yes

—

C₄×C₈×C₁₀₃

C₂²

yes

characteristic

C₂×Dic₂₀₆

—

C₂×D₄₁₂

—

C₂×D₄₁₂

—

C₂×D₄₁₂

—

C₂×D₄₁₂

—

#13110094

—

D₈₂₄

—

C₄⋊Dic₁₀₃

—

C₄⋊Dic₁₀₃

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₂×C₈×C₁₀₃

D₄

yes

—

C₄²×C₁₀₃

C₂³

yes

characteristic

C₂²×D₁₀₃

—

C₂²×D₁₀₃

—

C₂×C₄×C₁₀₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₃

C₂×D₄

yes

—

C₂×Dic₁₀₃

—

D₄₁₂

—

D₄₁₂

—

D₄₁₂

—

D₄₁₂

—

Dic₂₀₆

—

C₈×C₁₀₃

—

C₈×C₁₀₃

SD₁₆

yes

complemented

C₂²×C₁₀₃

#127

yes

characteristic

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₂×D₁₀₃

—

C₄×C₁₀₃

C₂×SD₁₆

yes

—

C₄×C₁₀₃

AGL(1,ℤ/8)

yes

—

Dic₁₀₃

—

C₂×C₁₀₃

#492

yes

characteristic

C₂×C₁₀₃

#496

yes

—

D₁₀₃

—

D₁₀₃

—

#1715

—

103

maximal, Sylow

C₁₀₃

#1715

yes

characteristic, complemented, Sylow

#498

—

103

—

#493

—

103

—

#492

—

103

—

#486

—

103

—

#322

D₁₀₃

yes

characteristic, complemented

C₂×SD₁₆

—

206

—

C₂×D₈

—

206

—

C₄⋊Q₈

—

103

—

#127

—

103

—

#127

—

103

—

#106

—

206

—

#102

—

206

—

C₄×C₈

C₂×D₁₀₃

yes

—

C₄×C₈

C₂×D₁₀₃

yes

characteristic

C₂×Q₈

—

206

—

C₂×D₄

—

206

—

C₂×D₄

—

206

—

C₂×D₄

—

206

—

C₂×D₄

—

206

—

SD₁₆

—

412

—

D₈

—

412

—

C₂×C₈

D₄₁₂

yes

—

C₂×C₈

D₄₁₂

yes

—

C₄⋊C₄

—

206

—

C₄⋊C₄

—

206

—

C₄²

C₂²×D₁₀₃

yes

characteristic

C₂³

—

412

—

C₂³

—

412

—

Q₈

—

412

—

D₄

—

412

—

D₄

—

412

—

D₄

—

412

—

D₄

—

412

—

C₂×C₄

—

412

—

C₂×C₄

C₂×D₄₁₂

yes

characteristic

C₂×C₄

C₂×D₄₁₂

yes

—

C₈

—

C₈

#13110094

yes

complemented

C₂²

—

824

—

C₂²

—

824

—

C₂²

—

824

—

C₂²

—

824

—

C₂²

#15693018

yes

characteristic

C₄

—

824

—

C₄

C₂×#13110094

yes

—

C₄

#15693049

yes

—

C₂

—

1648

—

100

C₂

—

1648

—

101

C₂

#20268164

yes

characteristic

102

C₂

#20268180

yes

—

	subgroup	G/H
center	C₂²	#15693018
commutator, cosocle	C₄²×C₁₀₃	C₂³
Frattini	C₄²	C₂²×D₁₀₃
Fitting	#322×C₁₀₃	C₂
socle	C₂²×C₁₀₃	#127
prime radical	#20268168	C₂
hypercenter	#322	D₁₀₃
hypocenter	C₁₀₃	#1715
LCS	G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#15693018

commutator, cosocle

C₄²×C₁₀₃

C₂³

Frattini

C₄²

C₂²×D₁₀₃

Fitting

#322×C₁₀₃

C₂

C₂²×C₁₀₃

C₂

D₁₀₃

C₁₀₃

G ▷ C₄²×C₁₀₃ ▷ C₂²×C₁₀₃ ▷ C₁₀₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #20268189	C₂	semidirect
3. #20268164	C₂	semidirect
4. #20268138	C₂	semidirect
5. #322×C₁₀₃	C₂	semidirect
38. C₈×C₁₀₃	SD₁₆	semidirect
52. C₁₀₃	#1715	semidirect
57. #322	D₁₀₃	semidirect
90. C₈	#13110094	semidirect

H ≅ G/N

type

1. #20268189

C₂

semidirect

3. #20268164

C₂

semidirect

4. #20268138

C₂

semidirect

5. #322×C₁₀₃

C₂

semidirect

38. C₈×C₁₀₃

SD₁₆

semidirect

52. C₁₀₃

#1715

semidirect

57. #322

D₁₀₃

semidirect

90. C₈

#13110094

semidirect

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

subgroup	index
#20268189	2
#20268168	2
#20268164	2
#20268138	2
#322×C₁₀₃	2
#1715	103

subgroup

index

#322×C₁₀₃

103

Subgroups — #27194772

Conjugacy classes of subgroups

Complements

Maximal subgroups