Group #27170760 — subgroups

Subgroups — #27170760

Order 12928.

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	#20265575	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265554	C₂	1	1	yes	maximal, characteristic
3	#20265550	C₂	1	2	yes	maximal, complemented
4	#20265524	C₂	1	2	yes	maximal, complemented
5	#322×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13108063	—	2	4	no	—
7	C₂×D₈₀₈	—	2	2	no	—
8	#15692255	C₂²	1	2	yes	—
9	#15692252	C₂²	1	1	yes	characteristic
10	#15692252	C₂²	1	1	yes	characteristic
11	#15692187	—	2	2	no	—
12	#15692177	—	2	4	no	—
13	C₄×C₈×C₁₀₁	C₂²	1	2	yes	—
14	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₀₂	—	2	2	no	—
16	C₂×D₄₀₄	—	2	2	no	—
17	C₂×D₄₀₄	—	2	1	no	—
18	C₂×D₄₀₄	—	2	1	no	—
19	C₂×D₄₀₄	—	2	2	no	—
20	#13108063	—	4	8	no	—
21	D₈₀₈	—	4	4	no	—
22	C₄⋊Dic₁₀₁	—	2	2	no	—
23	C₄⋊Dic₁₀₁	—	2	2	no	—
24	C₂×C₈×C₁₀₁	D₄	1	4	yes	—
25	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
26	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
27	C₂²×D₁₀₁	—	4	1	no	—
28	C₂²×D₁₀₁	—	4	1	no	—
29	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
31	C₂×Dic₁₀₁	—	4	2	no	—
32	D₄₀₄	—	4	2	no	—
33	D₄₀₄	—	4	2	no	—
34	D₄₀₄	—	4	4	no	—
35	D₄₀₄	—	4	4	no	—
36	Dic₂₀₂	—	4	4	no	—
37	C₈×C₁₀₁	—	2	4	no	—
38	C₈×C₁₀₁	SD₁₆	1	4	yes	complemented
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	C₂×SD₁₆	1	2	yes	—
45	C₄×C₁₀₁	AGL(1,ℤ/8)	1	4	yes	—
46	Dic₁₀₁	—	8	2	no	—
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	—	101	1	no	—
54	#493	—	101	1	no	—
55	#492	—	101	2	no	—
56	#486	—	101	2	no	—
57	#322	D₁₀₁	1	1	yes	characteristic, complemented
58	C₂×SD₁₆	—	202	4	no	—
59	C₂×D₈	—	202	2	no	—
60	C₄⋊Q₈	—	101	2	no	—
61	#127	—	101	1	no	—
62	#127	—	101	1	no	—
63	#106	—	202	2	no	—
64	#102	—	202	4	no	—
65	C₄×C₈	C₂×D₁₀₁	1	2	yes	—
66	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
67	C₂×Q₈	—	202	2	no	—
68	C₂×D₄	—	202	2	no	—
69	C₂×D₄	—	202	1	no	—
70	C₂×D₄	—	202	1	no	—
71	C₂×D₄	—	202	2	no	—
72	SD₁₆	—	404	8	no	—
73	D₈	—	404	4	no	—
74	C₂×C₈	D₄₀₄	1	4	yes	—
75	C₂×C₈	D₄₀₄	1	2	yes	—
76	C₄⋊C₄	—	202	2	no	—
77	C₄⋊C₄	—	202	2	no	—
78	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
79	C₂³	—	404	1	no	—
80	C₂³	—	404	1	no	—
81	Q₈	—	404	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	—	404	2	no	—
87	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
88	C₂×C₄	C₂×D₄₀₄	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	#13108063	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	#15692252	1	1	yes	characteristic
96	C₄	—	808	2	no	—
97	C₄	C₂×#13108063	1	2	yes	—
98	C₄	#15692283	1	4	yes	—
99	C₂	—	1616	1	no	—
100	C₂	—	1616	1	no	—
101	C₂	#20265550	1	1	yes	characteristic
102	C₂	#20265566	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265575

C₂

yes

maximal, characteristic, complemented

#20265554

C₂

yes

maximal, characteristic

#20265550

C₂

yes

maximal, complemented

#20265524

C₂

yes

maximal, complemented

#322×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#13108063

—

C₂×D₈₀₈

—

#15692255

C₂²

yes

—

#15692252

C₂²

yes

characteristic

#15692252

C₂²

yes

characteristic

#15692187

—

#15692177

—

C₄×C₈×C₁₀₁

C₂²

yes

—

C₄×C₈×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

#13108063

—

D₈₀₈

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×Dic₁₀₁

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

Dic₂₀₂

—

C₈×C₁₀₁

—

C₈×C₁₀₁

SD₁₆

yes

complemented

C₂²×C₁₀₁

#127

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

C₂×SD₁₆

yes

—

C₄×C₁₀₁

AGL(1,ℤ/8)

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

#492

yes

characteristic

C₂×C₁₀₁

#496

yes

—

D₁₀₁

—

D₁₀₁

—

#1715

—

101

maximal, Sylow

C₁₀₁

#1715

yes

characteristic, complemented, Sylow

#498

—

101

—

#493

—

101

—

#492

—

101

—

#486

—

101

—

#322

D₁₀₁

yes

characteristic, complemented

C₂×SD₁₆

—

202

—

C₂×D₈

—

202

—

C₄⋊Q₈

—

101

—

#127

—

101

—

#127

—

101

—

#106

—

202

—

#102

—

202

—

C₄×C₈

C₂×D₁₀₁

yes

—

C₄×C₈

C₂×D₁₀₁

yes

characteristic

C₂×Q₈

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

SD₁₆

—

404

—

D₈

—

404

—

C₂×C₈

D₄₀₄

yes

—

C₂×C₈

D₄₀₄

yes

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

C₄²

C₂²×D₁₀₁

yes

characteristic

C₂³

—

404

—

C₂³

—

404

—

Q₈

—

404

—

D₄

—

404

—

D₄

—

404

—

D₄

—

404

—

D₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

C₂×D₄₀₄

yes

characteristic

C₂×C₄

C₂×D₄₀₄

yes

—

C₈

—

C₈

#13108063

yes

complemented

C₂²

—

808

—

C₂²

—

808

—

C₂²

—

808

—

C₂²

—

808

—

C₂²

#15692252

yes

characteristic

C₄

—

808

—

C₄

C₂×#13108063

yes

—

C₄

#15692283

yes

—

C₂

—

1616

—

100

C₂

—

1616

—

101

C₂

#20265550

yes

characteristic

102

C₂

#20265566

yes

—

	subgroup	G/H
center	C₂²	#15692252
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#322×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#127
prime radical	#20265554	C₂
hypercenter	#322	D₁₀₁
hypocenter	C₁₀₁	#1715
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#15692252

commutator, cosocle

C₄²×C₁₀₁

C₂³

Frattini

C₄²

C₂²×D₁₀₁

Fitting

#322×C₁₀₁

C₂

C₂²×C₁₀₁

C₂

D₁₀₁

C₁₀₁

G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #20265575	C₂	semidirect
3. #20265550	C₂	semidirect
4. #20265524	C₂	semidirect
5. #322×C₁₀₁	C₂	semidirect
38. C₈×C₁₀₁	SD₁₆	semidirect
52. C₁₀₁	#1715	semidirect
57. #322	D₁₀₁	semidirect
90. C₈	#13108063	semidirect

H ≅ G/N

type

1. #20265575

C₂

semidirect

3. #20265550

C₂

semidirect

4. #20265524

C₂

semidirect

5. #322×C₁₀₁

C₂

semidirect

38. C₈×C₁₀₁

SD₁₆

semidirect

52. C₁₀₁

#1715

semidirect

57. #322

D₁₀₁

semidirect

90. C₈

#13108063

semidirect

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

subgroup	index
#20265575	2
#20265554	2
#20265550	2
#20265524	2
#322×C₁₀₁	2
#1715	101

subgroup

index

#322×C₁₀₁

101

Subgroups — #27170760

Conjugacy classes of subgroups

Complements

Maximal subgroups