Group #41013065 — subgroups

Subgroups — #41013065

Order 27008.

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	#27213522	C₂	1	2	yes	maximal, complemented
2	#27213519	C₂	1	1	yes	maximal, characteristic, complemented
3	#27213492	C₂	1	2	yes	maximal, complemented
4	#27212867	C₂	1	1	yes	maximal, characteristic, complemented
5	#493×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16809951	—	2	2	no	—
7	C₂×#16809947	—	2	4	no	—
8	#20269977	C₂²	1	1	yes	characteristic
9	#20269974	C₂²	1	1	yes	characteristic
10	#20269898	—	2	4	no	—
11	C₄×C₂₁₁⋊C₈	C₂²	1	2	yes	—
12	C₂×D₈×C₂₁₁	—	2	2	no	—
13	#127×C₂₁₁	C₂²	1	2	yes	—
14	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₂₂	—	2	1	no	—
16	C₂×D₈₄₄	—	2	2	no	—
17	C₂×D₈₄₄	—	2	1	no	—
18	#16809951	—	4	4	no	—
19	#16809947	—	4	8	no	—
20	C₂×C₂₁₁⋊C₈	D₄	1	4	yes	—
21	C₄⋊Dic₂₁₁	—	2	2	no	—
22	C₂×D₄×C₂₁₁	—	2	2	no	—
23	C₂×D₄×C₂₁₁	—	2	2	no	—
24	C₂×D₄×C₂₁₁	—	2	2	no	—
25	D₈×C₂₁₁	—	4	4	no	—
26	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
27	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₂²×D₂₁₁	—	4	1	no	—
29	Dic₄₂₂	—	4	2	no	—
30	D₈₄₄	—	4	2	no	—
31	D₈₄₄	—	4	4	no	—
32	C₂×Dic₂₁₁	—	4	1	no	—
33	C₂₁₁⋊C₈	—	2	4	no	—
34	C₂³×C₂₁₁	—	4	2	no	—
35	D₄×C₂₁₁	—	4	4	no	—
36	D₄×C₂₁₁	—	4	4	no	—
37	D₄×C₂₁₁	—	4	4	no	—
38	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
40	C₈×C₂₁₁	D₈	1	4	yes	complemented
41	C₂²×C₂₁₁	—	8	2	no	—
42	C₂²×C₂₁₁	—	8	2	no	—
43	C₂²×C₂₁₁	—	8	2	no	—
44	C₂²×C₂₁₁	#127	1	1	yes	characteristic
45	C₂×D₂₁₁	—	8	1	no	—
46	C₂×D₂₁₁	—	8	2	no	—
47	C₄×C₂₁₁	C₂×D₈	1	2	yes	—
48	C₄×C₂₁₁	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₂₁₁	—	8	1	no	—
50	C₂×C₂₁₁	—	16	2	no	—
51	C₂×C₂₁₁	#493	1	1	yes	characteristic
52	C₂×C₂₁₁	#496	1	2	yes	—
53	D₂₁₁	—	16	1	no	—
54	C₂₁₁	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	211	1	no	maximal, Sylow
56	#493	—	211	2	no	—
57	#493	D₂₁₁	1	1	yes	characteristic, complemented
58	#492	—	211	1	no	—
59	#486	—	211	2	no	—
60	#322	—	211	1	no	—
61	C₂×SD₁₆	—	422	2	no	—
62	C₂×D₈	—	2	2	no	—
63	C₂×D₈	—	422	4	no	—
64	C₄⋊Q₈	—	211	1	no	—
65	#127	—	211	1	no	—
66	#127	C₂×D₂₁₁	1	2	yes	—
67	#102	—	422	4	no	—
68	C₄×C₈	—	211	2	no	—
69	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
70	C₂×Q₈	—	422	1	no	—
71	C₂×D₄	—	2	2	no	—
72	C₂×D₄	—	2	2	no	—
73	C₂×D₄	—	2	2	no	—
74	C₂×D₄	—	422	2	no	—
75	C₂×D₄	—	422	1	no	—
76	SD₁₆	—	844	4	no	—
77	D₈	—	4	4	no	—
78	D₈	—	844	8	no	—
79	C₂×C₈	—	211	4	no	—
80	C₂×C₈	C₂²⋊D₂₁₁	1	2	yes	—
81	C₄⋊C₄	—	422	2	no	—
82	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
83	C₂³	—	4	2	no	—
84	C₂³	—	844	1	no	—
85	Q₈	—	844	2	no	—
86	D₄	—	4	4	no	—
87	D₄	—	4	4	no	—
88	D₄	—	4	4	no	—
89	D₄	—	844	2	no	—
90	D₄	—	844	4	no	—
91	C₂×C₄	—	844	1	no	—
92	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
94	C₈	—	422	4	no	—
95	C₈	#16809947	1	4	yes	complemented
96	C₂²	—	8	2	no	—
97	C₂²	—	8	2	no	—
98	C₂²	—	8	2	no	—
99	C₂²	—	1688	1	no	—
100	C₂²	—	1688	2	no	—
101	C₂²	#20269973	1	1	yes	characteristic
102	C₄	—	1688	1	no	—
103	C₄	C₂×#16809947	1	2	yes	—
104	C₄	#20270001	1	4	yes	—
105	C₂	—	16	2	no	—
106	C₂	—	3376	1	no	—
107	C₂	#27213522	1	1	yes	characteristic
108	C₂	#27213534	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27213522

C₂

yes

maximal, complemented

#27213519

C₂

yes

maximal, characteristic, complemented

#27213492

C₂

yes

maximal, complemented

#27212867

C₂

yes

maximal, characteristic, complemented

#493×C₂₁₁

C₂

yes

maximal, characteristic, complemented

C₂×#16809951

—

C₂×#16809947

—

#20269977

C₂²

yes

characteristic

#20269974

C₂²

yes

characteristic

#20269898

—

C₄×C₂₁₁⋊C₈

C₂²

yes

—

C₂×D₈×C₂₁₁

—

#127×C₂₁₁

C₂²

yes

—

C₄×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×Dic₄₂₂

—

C₂×D₈₄₄

—

C₂×D₈₄₄

—

#16809951

—

#16809947

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₄⋊Dic₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

C₂×D₄×C₂₁₁

—

D₈×C₂₁₁

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

C₂³

yes

characteristic

C₂²×D₂₁₁

—

Dic₄₂₂

—

D₈₄₄

—

D₈₄₄

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

D₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₈×C₂₁₁

D₈

yes

complemented

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

#127

yes

characteristic

C₂×D₂₁₁

—

C₂×D₂₁₁

—

C₄×C₂₁₁

C₂×D₈

yes

—

C₄×C₂₁₁

AGL(1,ℤ/8)

yes

—

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#493

yes

characteristic

C₂×C₂₁₁

#496

yes

—

D₂₁₁

—

C₂₁₁

#1721

yes

characteristic, complemented, Sylow

#1721

—

211

maximal, Sylow

#493

—

211

—

#493

D₂₁₁

yes

characteristic, complemented

#492

—

211

—

#486

—

211

—

#322

—

211

—

C₂×SD₁₆

—

422

—

C₂×D₈

—

C₂×D₈

—

422

—

C₄⋊Q₈

—

211

—

#127

—

211

—

#127

C₂×D₂₁₁

yes

—

#102

—

422

—

C₄×C₈

—

211

—

C₄×C₈

C₂×D₂₁₁

yes

characteristic

C₂×Q₈

—

422

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

422

—

C₂×D₄

—

422

—

SD₁₆

—

844

—

D₈

—

D₈

—

844

—

C₂×C₈

—

211

—

C₂×C₈

C₂²⋊D₂₁₁

yes

—

C₄⋊C₄

—

422

—

C₄²

C₂²×D₂₁₁

yes

characteristic

C₂³

—

C₂³

—

844

—

Q₈

—

844

—

D₄

—

D₄

—

D₄

—

D₄

—

844

—

D₄

—

844

—

C₂×C₄

—

844

—

C₂×C₄

D₄×D₂₁₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

C₈

—

422

—

C₈

#16809947

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

1688

—

100

C₂²

—

1688

—

101

C₂²

#20269973

yes

characteristic

102

C₄

—

1688

—

103

C₄

C₂×#16809947

yes

—

104

C₄

#20270001

yes

—

105

C₂

—

106

C₂

—

3376

—

107

C₂

#27213522

yes

characteristic

108

C₂

#27213534

yes

—

	subgroup	G/H
center	C₂²	#20269973
commutator, cosocle	C₄²×C₂₁₁	C₂³
Frattini	C₄²	C₂²×D₂₁₁
Fitting	#493×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#127
hypercenter	#493	D₂₁₁
hypocenter	C₂₁₁	#1721
LCS	G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#20269973

commutator, cosocle

C₄²×C₂₁₁

C₂³

Frattini

C₄²

C₂²×D₂₁₁

Fitting

#493×C₂₁₁

C₂

C₂²×C₂₁₁

D₂₁₁

C₂₁₁

G ▷ C₄²×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #27213522	C₂	semidirect
2. #27213519	C₂	semidirect
3. #27213492	C₂	semidirect
4. #27212867	C₂	semidirect
5. #493×C₂₁₁	C₂	semidirect
40. C₈×C₂₁₁	D₈	semidirect
54. C₂₁₁	#1721	semidirect
57. #493	D₂₁₁	semidirect
95. C₈	#16809947	semidirect

H ≅ G/N

type

1. #27213522

C₂

semidirect

2. #27213519

C₂

semidirect

3. #27213492

C₂

semidirect

4. #27212867

C₂

semidirect

5. #493×C₂₁₁

C₂

semidirect

40. C₈×C₂₁₁

D₈

semidirect

54. C₂₁₁

#1721

semidirect

57. #493

D₂₁₁

semidirect

95. C₈

#16809947

semidirect

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

subgroup	index
#27213522	2
#27213519	2
#27213492	2
#27212867	2
#493×C₂₁₁	2
#1721	211

subgroup

index

#493×C₂₁₁

211

Subgroups — #41013065

Conjugacy classes of subgroups

Complements

Maximal subgroups