Group #14580910 — subgroups

Subgroups — #14580910

Order 2624.

270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 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	C₂×#10620306	C₂	1	1	yes	maximal, characteristic
2	#13050162	C₂	1	2	yes	maximal, complemented
3	#13050153	C₂	1	2	yes	maximal, complemented
4	#13050147	C₂	1	4	yes	maximal, complemented
5	#13050140	C₂	1	4	yes	maximal, complemented
6	#13050123	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×C₄₁	—	2	4	no	—
9	C₂×Q₈×C₄₁	C₂²	1	2	yes	—
10	C₂×Q₈×C₄₁	C₂²	1	2	yes	—
11	C₂²×C₄×C₄₁	C₂²	1	2	yes	—
12	C₂²×C₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₄₁	C₂²	1	2	yes	—
14	#10620306	—	2	4	no	—
15	C₂×D₁₆₄	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄₁	C₂²	1	2	yes	—
17	C₂×Dic₈₂	C₂²	1	1	yes	characteristic
18	C₂²⋊Dic₄₁	C₂²	1	2	yes	—
19	#10620281	C₂²	1	8	yes	—
20	C₄⋊Dic₄₁	C₂²	1	2	yes	—
21	#10620279	C₂²	1	8	yes	—
22	C₄×Dic₄₁	C₂²	1	2	yes	—
23	C₂³×C₄₁	C₂³	1	1	yes	characteristic
24	C₂²×D₄₁	C₂³	1	2	yes	—
25	Q₈×C₄₁	—	2	4	no	—
26	Q₈×C₄₁	—	2	4	no	—
27	C₂×C₄×C₄₁	—	2	4	no	—
28	C₂×C₄×C₄₁	D₄	1	4	yes	—
29	C₂×C₄×C₄₁	C₂³	1	4	yes	—
30	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
32	C₂²⋊D₄₁	—	2	4	no	—
33	C₂×Dic₄₁	C₂³	1	4	yes	—
34	C₂×Dic₄₁	C₂³	1	2	yes	—
35	D₁₆₄	—	2	2	no	—
36	C₄×D₄₁	—	2	4	no	—
37	Dic₈₂	—	2	2	no	—
38	C₂²×C₄₁	—	2	2	no	—
39	C₂²×C₄₁	C₂×D₄	1	2	yes	—
40	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
41	C₂×D₄₁	—	2	2	no	—
42	C₂×D₄₁	—	2	4	no	—
43	C₄×C₄₁	—	2	4	no	—
44	C₄×C₄₁	C₂×D₄	1	2	yes	—
45	C₄×C₄₁	C₂×D₄	1	2	yes	—
46	Dic₄₁	—	2	2	no	—
47	Dic₄₁	—	2	4	no	—
48	C₂×C₄₁	—	2	2	no	—
49	C₂×C₄₁	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₄₁	2₋¹⁺⁴	1	2	yes	—
51	D₄₁	—	4	2	no	—
52	#536	—	41	1	no	maximal, Sylow
53	C₄₁	#536	1	1	yes	characteristic, complemented, Sylow
54	C₂×#41	—	41	1	no	—
55	C₂²×Q₈	D₄₁	1	1	yes	characteristic, complemented
56	C₄⋊Q₈	—	41	2	no	—
57	#124	—	41	2	no	—
58	#123	—	41	4	no	—
59	C₂²⋊Q₈	—	41	4	no	—
60	#117	—	41	1	no	—
61	#41	—	82	4	no	—
62	C₂×Q₈	—	2	4	no	—
63	C₂×Q₈	—	41	1	no	—
64	C₂×Q₈	C₂×D₄₁	1	2	yes	—
65	C₂×Q₈	C₂×D₄₁	1	2	yes	—
66	C₂×D₄	—	41	2	no	—
67	C₂×D₄	—	41	1	no	—
68	C₂²×C₄	—	41	2	no	—
69	C₂²×C₄	C₂×D₄₁	1	2	yes	—
70	C₂²×C₄	C₂×D₄₁	1	1	yes	characteristic
71	C₄⋊C₄	—	41	2	no	—
72	C₄⋊C₄	—	41	8	no	—
73	C₂²⋊C₄	—	41	2	no	—
74	C₂²⋊C₄	—	41	8	no	—
75	C₄²	—	41	2	no	—
76	C₂³	—	41	2	no	—
77	C₂³	C₂²×D₄₁	1	1	yes	characteristic
78	Q₈	—	2	4	no	—
79	Q₈	—	2	4	no	—
80	Q₈	—	82	2	no	—
81	D₄	—	82	4	no	—
82	D₄	—	82	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	41	4	no	—
85	C₂×C₄	—	41	2	no	—
86	C₂×C₄	—	82	4	no	—
87	C₂×C₄	C₂²⋊D₄₁	1	4	yes	—
88	C₂×C₄	C₂²×D₄₁	1	4	yes	—
89	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	82	2	no	—
93	C₂²	—	82	4	no	—
94	C₂²	C₂×C₂²⋊D₄₁	1	2	yes	—
95	C₂²	C₂³×D₄₁	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	82	2	no	—
98	C₄	—	82	4	no	—
99	C₄	C₂×C₂²⋊D₄₁	1	2	yes	—
100	C₄	C₂×C₂²⋊D₄₁	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	164	2	no	—
103	C₂	C₂²×C₂²⋊D₄₁	1	1	yes	characteristic
104	C₂	#13050212	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10620306

C₂

yes

maximal, characteristic

#13050162

C₂

yes

maximal, complemented

#13050153

C₂

yes

maximal, complemented

#13050147

C₂

yes

maximal, complemented

#13050140

C₂

yes

maximal, complemented

#13050123

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₄₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₄₁

—

C₂×Q₈×C₄₁

C₂²

yes

—

C₂×Q₈×C₄₁

C₂²

yes

—

C₂²×C₄×C₄₁

C₂²

yes

—

C₂²×C₄×C₄₁

C₂²

yes

characteristic

C₂×C₂²⋊D₄₁

C₂²

yes

—

#10620306

—

C₂×D₁₆₄

C₂²

yes

characteristic

C₂×C₄×D₄₁

C₂²

yes

—

C₂×Dic₈₂

C₂²

yes

characteristic

C₂²⋊Dic₄₁

C₂²

yes

—

#10620281

C₂²

yes

—

C₄⋊Dic₄₁

C₂²

yes

—

#10620279

C₂²

yes

—

C₄×Dic₄₁

C₂²

yes

—

C₂³×C₄₁

C₂³

yes

characteristic

C₂²×D₄₁

C₂³

yes

—

Q₈×C₄₁

—

Q₈×C₄₁

—

C₂×C₄×C₄₁

—

C₂×C₄×C₄₁

D₄

yes

—

C₂×C₄×C₄₁

C₂³

yes

—

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂×C₄×C₄₁

C₂³

yes

characteristic

C₂²⋊D₄₁

—

C₂×Dic₄₁

C₂³

yes

—

C₂×Dic₄₁

C₂³

yes

—

D₁₆₄

—

C₄×D₄₁

—

Dic₈₂

—

C₂²×C₄₁

—

C₂²×C₄₁

C₂×D₄

yes

—

C₂²×C₄₁

C₂⁴

yes

characteristic

C₂×D₄₁

—

C₂×D₄₁

—

C₄×C₄₁

—

C₄×C₄₁

C₂×D₄

yes

—

C₄×C₄₁

C₂×D₄

yes

—

Dic₄₁

—

Dic₄₁

—

C₂×C₄₁

—

C₂×C₄₁

C₂²×D₄

yes

characteristic

C₂×C₄₁

2₋¹⁺⁴

yes

—

D₄₁

—

#536

—

maximal, Sylow

C₄₁

#536

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

D₄₁

yes

characteristic, complemented

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₁

yes

—

C₂×Q₈

C₂×D₄₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄₁

yes

—

C₂²×C₄

C₂×D₄₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

C₂³

C₂²×D₄₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊D₄₁

yes

—

C₂×C₄

C₂²×D₄₁

yes

—

C₂×C₄

C₂²×D₄₁

yes

characteristic

C₂×C₄

C₂²×D₄₁

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊D₄₁

yes

—

C₂²

C₂³×D₄₁

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂×C₂²⋊D₄₁

yes

—

100

C₄

C₂×C₂²⋊D₄₁

yes

—

101

C₂

—

102

C₂

—

164

—

103

C₂

C₂²×C₂²⋊D₄₁

yes

characteristic

104

C₂

#13050212

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator, socle, cosocle	C₂²×C₄₁	C₂⁴
Fitting	C₂²×Q₈×C₄₁	C₂
prime radical	C₂×#10620306	C₂
hypercenter	C₂²×Q₈	D₄₁
hypocenter	C₄₁	#536
LCS	G ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₁

commutator, socle, cosocle

C₂²×C₄₁

C₂⁴

Fitting

C₂²×Q₈×C₄₁

C₂

prime radical

C₂×#10620306

C₂

hypercenter

C₂²×Q₈

D₄₁

C₄₁

G ▷ C₂²×C₄₁ ▷ C₄₁

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #13050162	C₂	semidirect
3. #13050153	C₂	semidirect
4. #13050147	C₂	semidirect
5. #13050140	C₂	semidirect
6. #13050123	C₂	semidirect
7. C₂²×Q₈×C₄₁	C₂	semidirect
53. C₄₁	#536	semidirect
55. C₂²×Q₈	D₄₁	semidirect

H ≅ G/N

type

2. #13050162

C₂

semidirect

3. #13050153

C₂

semidirect

4. #13050147

C₂

semidirect

5. #13050140

C₂

semidirect

6. #13050123

C₂

semidirect

7. C₂²×Q₈×C₄₁

C₂

semidirect

53. C₄₁

#536

semidirect

55. C₂²×Q₈

D₄₁

semidirect

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

subgroup	index
C₂×#10620306	2
#13050162	2
#13050153	2
#13050147	2
#13050140	2
#13050123	2
C₂²×Q₈×C₄₁	2
#536	41

subgroup

index

C₂²×Q₈×C₄₁

Subgroups — #14580910

Conjugacy classes of subgroups

Complements

Maximal subgroups