Group #27170796 — subgroups

Subgroups — #27170796

Order 12928.

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

normal subgroups only

This group has 242 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, complemented
2	#20265553	C₂	1	2	yes	maximal, complemented
3	#20265525	C₂	1	2	yes	maximal, complemented
4	#20264898	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₀₈	—	2	2	no	—
7	C₂×#13108059	—	2	4	no	—
8	#15692252	C₂²	1	2	yes	—
9	#15692175	—	2	4	no	—
10	C₄×#11717579	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₁₀₁	—	2	2	no	—
12	C₄⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
13	#127×C₁₀₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂×D₄₀₄	—	2	2	no	—
16	C₂×D₄₀₄	—	2	2	no	—
17	C₂×D₄₀₄	—	2	2	no	—
18	D₈₀₈	—	4	4	no	—
19	#13108059	—	4	8	no	—
20	C₂×#11717579	D₄	1	4	yes	—
21	C₂×Q₈×C₁₀₁	—	2	1	no	—
22	C₂×D₄×C₁₀₁	—	2	1	no	—
23	C₂×D₄×C₁₀₁	—	2	2	no	—
24	SD₁₆×C₁₀₁	—	4	4	no	—
25	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₀₁	—	2	2	no	—
27	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
28	C₂³×C₁₀₁	—	4	1	no	—
29	C₂²×D₁₀₁	—	4	2	no	—
30	Q₈×C₁₀₁	—	4	2	no	—
31	D₄×C₁₀₁	—	4	2	no	—
32	D₄×C₁₀₁	—	4	4	no	—
33	C₂×C₄×C₁₀₁	—	4	1	no	—
34	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
36	D₄₀₄	—	4	4	no	—
37	D₄₀₄	—	4	4	no	—
38	D₄₀₄	—	4	4	no	—
39	C₈×C₁₀₁	D₈	1	4	yes	complemented
40	#11717579	—	2	4	no	—
41	C₂²×C₁₀₁	—	8	1	no	—
42	C₂²×C₁₀₁	—	8	2	no	—
43	C₂²×C₁₀₁	#127	1	1	yes	characteristic
44	C₂×D₁₀₁	—	8	2	no	—
45	C₂×D₁₀₁	—	8	2	no	—
46	C₂×D₁₀₁	—	8	2	no	—
47	C₄×C₁₀₁	—	8	1	no	—
48	C₄×C₁₀₁	C₂×D₈	1	2	yes	—
49	C₄×C₁₀₁	AGL(1,ℤ/8)	1	4	yes	—
50	C₂×C₁₀₁	—	16	1	no	—
51	C₂×C₁₀₁	#493	1	1	yes	characteristic
52	C₂×C₁₀₁	#496	1	2	yes	—
53	D₁₀₁	—	16	2	no	—
54	#1721	—	101	1	no	maximal, Sylow
55	C₁₀₁	#1721	1	1	yes	characteristic, complemented, Sylow
56	#493	—	101	2	no	—
57	#493	—	101	1	no	—
58	#492	D₁₀₁	1	1	yes	characteristic, complemented
59	#486	—	101	2	no	—
60	#322	—	101	1	no	—
61	C₂×SD₁₆	—	2	2	no	—
62	C₂×D₈	—	202	4	no	—
63	C₂×D₈	—	202	2	no	—
64	C₄⋊Q₈	C₂×D₁₀₁	1	1	yes	characteristic
65	#127	—	101	2	no	—
66	#127	C₂×D₁₀₁	1	1	yes	characteristic
67	#102	—	202	4	no	—
68	C₄×C₈	—	101	2	no	—
69	C₄×C₈	C₂×D₁₀₁	1	1	yes	characteristic
70	C₂×Q₈	—	2	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	1	no	—
73	C₂×D₄	—	202	2	no	—
74	C₂×D₄	—	202	2	no	—
75	C₂×D₄	—	202	2	no	—
76	SD₁₆	—	4	4	no	—
77	D₈	—	404	8	no	—
78	D₈	—	404	4	no	—
79	C₂×C₈	—	101	4	no	—
80	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
83	C₂³	—	4	1	no	—
84	C₂³	—	404	2	no	—
85	Q₈	—	4	2	no	—
86	D₄	—	4	2	no	—
87	D₄	—	4	4	no	—
88	D₄	—	404	4	no	—
89	D₄	—	404	4	no	—
90	D₄	—	404	4	no	—
91	C₂×C₄	—	4	1	no	—
92	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
94	C₈	—	202	4	no	—
95	C₈	#13108059	1	4	yes	complemented
96	C₂²	—	8	1	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	808	2	no	—
99	C₂²	—	808	2	no	—
100	C₂²	—	808	2	no	—
101	C₂²	#15692251	1	1	yes	characteristic
102	C₄	—	8	1	no	—
103	C₄	C₂×#13108059	1	2	yes	—
104	C₄	#15692282	1	4	yes	—
105	C₂	—	16	1	no	—
106	C₂	—	1616	2	no	—
107	C₂	#20265553	1	1	yes	characteristic
108	C₂	#20265563	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265554

C₂

yes

maximal, characteristic, complemented

#20265553

C₂

yes

maximal, complemented

#20265525

C₂

yes

maximal, complemented

#20264898

C₂

yes

maximal, characteristic, complemented

#492×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×D₈₀₈

—

C₂×#13108059

—

#15692252

C₂²

yes

—

#15692175

—

C₄×#11717579

C₂²

yes

—

C₂×SD₁₆×C₁₀₁

—

C₄⋊Q₈×C₁₀₁

C₂²

yes

characteristic

#127×C₁₀₁

C₂²

yes

characteristic

C₄×C₈×C₁₀₁

C₂²

yes

characteristic

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

D₈₀₈

—

#13108059

—

C₂×#11717579

D₄

yes

—

C₂×Q₈×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

SD₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

—

C₂²×D₁₀₁

—

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

C₈×C₁₀₁

D₈

yes

complemented

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#127

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×D₈

yes

—

C₄×C₁₀₁

AGL(1,ℤ/8)

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#493

yes

characteristic

C₂×C₁₀₁

#496

yes

—

D₁₀₁

—

#1721

—

101

maximal, Sylow

C₁₀₁

#1721

yes

characteristic, complemented, Sylow

#493

—

101

—

#493

—

101

—

#492

D₁₀₁

yes

characteristic, complemented

#486

—

101

—

#322

—

101

—

C₂×SD₁₆

—

C₂×D₈

—

202

—

C₂×D₈

—

202

—

C₄⋊Q₈

C₂×D₁₀₁

yes

characteristic

#127

—

101

—

#127

C₂×D₁₀₁

yes

characteristic

#102

—

202

—

C₄×C₈

—

101

—

C₄×C₈

C₂×D₁₀₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

SD₁₆

—

D₈

—

404

—

D₈

—

404

—

C₂×C₈

—

101

—

C₂×C₈

C₂²⋊D₁₀₁

yes

—

C₄⋊C₄

—

C₄²

C₂²×D₁₀₁

yes

characteristic

C₂³

—

C₂³

—

404

—

Q₈

—

D₄

—

D₄

—

D₄

—

404

—

D₄

—

404

—

D₄

—

404

—

C₂×C₄

—

C₂×C₄

D₄×D₁₀₁

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

C₈

—

202

—

C₈

#13108059

yes

complemented

C₂²

—

C₂²

—

C₂²

—

808

—

C₂²

—

808

—

100

C₂²

—

808

—

101

C₂²

#15692251

yes

characteristic

102

C₄

—

103

C₄

C₂×#13108059

yes

—

104

C₄

#15692282

yes

—

105

C₂

—

106

C₂

—

1616

—

107

C₂

#20265553

yes

characteristic

108

C₂

#20265563

yes

—

	subgroup	G/H
center	C₂²	#15692251
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#492×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#127
hypercenter	#492	D₁₀₁
hypocenter	C₁₀₁	#1721
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#15692251

commutator, cosocle

C₄²×C₁₀₁

C₂³

Frattini

C₄²

C₂²×D₁₀₁

Fitting

#492×C₁₀₁

C₂

C₂²×C₁₀₁

D₁₀₁

C₁₀₁

G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #20265554	C₂	semidirect
2. #20265553	C₂	semidirect
3. #20265525	C₂	semidirect
4. #20264898	C₂	semidirect
5. #492×C₁₀₁	C₂	semidirect
39. C₈×C₁₀₁	D₈	semidirect
55. C₁₀₁	#1721	semidirect
58. #492	D₁₀₁	semidirect
95. C₈	#13108059	semidirect

H ≅ G/N

type

1. #20265554

C₂

semidirect

2. #20265553

C₂

semidirect

3. #20265525

C₂

semidirect

4. #20264898

C₂

semidirect

5. #492×C₁₀₁

C₂

semidirect

39. C₈×C₁₀₁

D₈

semidirect

55. C₁₀₁

#1721

semidirect

58. #492

D₁₀₁

semidirect

95. C₈

#13108059

semidirect

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

subgroup	index
#20265554	2
#20265553	2
#20265525	2
#20264898	2
#492×C₁₀₁	2
#1721	101

subgroup

index

#492×C₁₀₁

101

Subgroups — #27170796

Conjugacy classes of subgroups

Complements

Maximal subgroups