Group #27170761 — subgroups

Subgroups — #27170761

Order 12928.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20265554	C₂	1	1	yes	maximal, characteristic
2	#20265551	C₂	1	2	yes	maximal, complemented
3	#20265525	C₂	1	2	yes	maximal, complemented
4	#20264898	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13108061	—	2	4	no	—
7	C₂×D₈₀₈	—	2	2	no	—
8	#15692252	C₂²	1	1	yes	characteristic
9	#15692252	C₂²	1	1	yes	characteristic
10	#15692175	—	2	4	no	—
11	C₄×#11717579	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₀₁	C₂²	1	2	yes	—
13	#106×C₁₀₁	—	2	2	no	—
14	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×D₄₀₄	—	2	2	no	—
16	C₂×D₄₀₄	—	2	1	no	—
17	C₂×D₄₀₄	—	2	1	no	—
18	C₂×D₄₀₄	—	2	2	no	—
19	#13108061	—	4	8	no	—
20	D₈₀₈	—	4	4	no	—
21	C₂×#11717579	D₄	1	4	yes	—
22	C₂×Q₈×C₁₀₁	—	2	2	no	—
23	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
24	C₄⋊C₄×C₁₀₁	—	2	2	no	—
25	C₄⋊C₄×C₁₀₁	—	2	2	no	—
26	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
27	C₂²×D₁₀₁	—	4	1	no	—
28	C₂²×D₁₀₁	—	4	1	no	—
29	Q₈×C₁₀₁	—	4	4	no	—
30	C₂×C₄×C₁₀₁	—	4	2	no	—
31	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
33	D₄₀₄	—	4	2	no	—
34	D₄₀₄	—	4	2	no	—
35	D₄₀₄	—	4	4	no	—
36	D₄₀₄	—	4	4	no	—
37	C₈×C₁₀₁	SD₁₆	1	4	yes	complemented
38	#11717579	—	2	4	no	—
39	C₂²×C₁₀₁	#127	1	1	yes	characteristic
40	C₂×D₁₀₁	—	8	1	no	—
41	C₂×D₁₀₁	—	8	1	no	—
42	C₂×D₁₀₁	—	8	2	no	—
43	C₂×D₁₀₁	—	8	2	no	—
44	C₄×C₁₀₁	—	8	2	no	—
45	C₄×C₁₀₁	C₂×SD₁₆	1	2	yes	—
46	C₄×C₁₀₁	AGL(1,ℤ/8)	1	4	yes	—
47	C₂×C₁₀₁	#492	1	1	yes	characteristic
48	C₂×C₁₀₁	#496	1	2	yes	—
49	D₁₀₁	—	16	1	no	—
50	D₁₀₁	—	16	1	no	—
51	#1715	—	101	1	no	maximal, Sylow
52	C₁₀₁	#1715	1	1	yes	characteristic, complemented, Sylow
53	#498	D₁₀₁	1	1	yes	characteristic, complemented
54	#493	—	101	1	no	—
55	#492	—	101	2	no	—
56	#486	—	101	2	no	—
57	#322	—	101	1	no	—
58	C₂×SD₁₆	—	202	4	no	—
59	C₂×D₈	—	202	2	no	—
60	C₄⋊Q₈	C₂×D₁₀₁	1	2	yes	—
61	#127	—	101	1	no	—
62	#127	—	101	1	no	—
63	#106	—	2	2	no	—
64	#102	—	202	4	no	—
65	C₄×C₈	—	101	2	no	—
66	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	202	2	no	—
69	C₂×D₄	—	202	2	no	—
70	C₂×D₄	—	202	1	no	—
71	C₂×D₄	—	202	1	no	—
72	SD₁₆	—	404	8	no	—
73	D₈	—	404	4	no	—
74	C₂×C₈	—	101	4	no	—
75	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
79	C₂³	—	404	1	no	—
80	C₂³	—	404	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	404	2	no	—
83	D₄	—	404	2	no	—
84	D₄	—	404	4	no	—
85	D₄	—	404	4	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
89	C₈	—	202	4	no	—
90	C₈	#13108061	1	4	yes	complemented
91	C₂²	—	808	1	no	—
92	C₂²	—	808	1	no	—
93	C₂²	—	808	2	no	—
94	C₂²	—	808	2	no	—
95	C₂²	#15692251	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×#13108061	1	2	yes	—
98	C₄	#15692282	1	4	yes	—
99	C₂	—	1616	1	no	—
100	C₂	—	1616	1	no	—
101	C₂	#20265551	1	1	yes	characteristic
102	C₂	#20265563	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265554

C₂

yes

maximal, characteristic

#20265551

C₂

yes

maximal, complemented

#20265525

C₂

yes

maximal, complemented

#20264898

C₂

yes

maximal, characteristic, complemented

#498×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#13108061

—

C₂×D₈₀₈

—

#15692252

C₂²

yes

characteristic

#15692252

C₂²

yes

characteristic

#15692175

—

C₄×#11717579

C₂²

yes

—

C₄⋊Q₈×C₁₀₁

C₂²

yes

—

#106×C₁₀₁

—

C₄×C₈×C₁₀₁

C₂²

yes

characteristic

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

#13108061

—

D₈₀₈

—

C₂×#11717579

D₄

yes

—

C₂×Q₈×C₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

C₈×C₁₀₁

SD₁₆

yes

complemented

#11717579

—

C₂²×C₁₀₁

#127

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×SD₁₆

yes

—

C₄×C₁₀₁

AGL(1,ℤ/8)

yes

—

C₂×C₁₀₁

#492

yes

characteristic

C₂×C₁₀₁

#496

yes

—

D₁₀₁

—

D₁₀₁

—

#1715

—

101

maximal, Sylow

C₁₀₁

#1715

yes

characteristic, complemented, Sylow

#498

D₁₀₁

yes

characteristic, complemented

#493

—

101

—

#492

—

101

—

#486

—

101

—

#322

—

101

—

C₂×SD₁₆

—

202

—

C₂×D₈

—

202

—

C₄⋊Q₈

C₂×D₁₀₁

yes

—

#127

—

101

—

#127

—

101

—

#106

—

#102

—

202

—

C₄×C₈

—

101

—

C₄×C₈

C₂×D₁₀₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

SD₁₆

—

404

—

D₈

—

404

—

C₂×C₈

—

101

—

C₂×C₈

C₂²⋊D₁₀₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₁₀₁

yes

characteristic

C₂³

—

404

—

C₂³

—

404

—

Q₈

—

D₄

—

404

—

D₄

—

404

—

D₄

—

404

—

D₄

—

404

—

C₂×C₄

—

C₂×C₄

D₄×D₁₀₁

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

C₈

—

202

—

C₈

#13108061

yes

complemented

C₂²

—

808

—

C₂²

—

808

—

C₂²

—

808

—

C₂²

—

808

—

C₂²

#15692251

yes

characteristic

C₄

—

C₄

C₂×#13108061

yes

—

C₄

#15692282

yes

—

C₂

—

1616

—

100

C₂

—

1616

—

101

C₂

#20265551

yes

characteristic

102

C₂

#20265563

yes

—

	subgroup	G/H
center	C₂²	#15692251
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#498×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#127
prime radical	#20265554	C₂
hypercenter	#498	D₁₀₁
hypocenter	C₁₀₁	#1715
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#15692251

commutator, cosocle

C₄²×C₁₀₁

C₂³

Frattini

C₄²

C₂²×D₁₀₁

Fitting

#498×C₁₀₁

C₂

C₂²×C₁₀₁

C₂

D₁₀₁

C₁₀₁

G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #20265551	C₂	semidirect
3. #20265525	C₂	semidirect
4. #20264898	C₂	semidirect
5. #498×C₁₀₁	C₂	semidirect
37. C₈×C₁₀₁	SD₁₆	semidirect
52. C₁₀₁	#1715	semidirect
53. #498	D₁₀₁	semidirect
90. C₈	#13108061	semidirect

H ≅ G/N

type

2. #20265551

C₂

semidirect

3. #20265525

C₂

semidirect

4. #20264898

C₂

semidirect

5. #498×C₁₀₁

C₂

semidirect

37. C₈×C₁₀₁

SD₁₆

semidirect

52. C₁₀₁

#1715

semidirect

53. #498

D₁₀₁

semidirect

90. C₈

#13108061

semidirect

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

subgroup	index
#20265554	2
#20265551	2
#20265525	2
#20264898	2
#498×C₁₀₁	2
#1715	101

subgroup

index

#498×C₁₀₁

101

Subgroups — #27170761

Conjugacy classes of subgroups

Complements

Maximal subgroups