Group #28412556 — subgroups

Subgroups — C₂×#20301075 #28412556

Order 14656.

266 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 140 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	#20301075	C₂	1	8	yes	maximal, direct product factor
2	C₂×#16814676	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16814675	C₂	1	2	yes	maximal, complemented
4	C₂×D₂₂₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#14270087	C₂	1	2	yes	maximal, complemented
6	C₂×#16814665	C₂	1	1	yes	maximal, characteristic
7	#16814676	C₂²	1	4	yes	complemented
8	#16814675	C₂²	1	8	yes	complemented
9	D₂₂₉⋊C₈	C₂²	1	4	yes	complemented
10	C₂×#14270087	C₂²	1	1	yes	characteristic
11	C₂×#14270087	C₂²	1	2	yes	—
12	C₂×#14270087	C₂²	1	1	yes	characteristic
13	C₂×#14270087	C₂²	1	8	yes	complemented
14	#16814665	C₄	1	4	yes	—
15	#16814665	C₂²	1	4	yes	—
16	C₂×Dic₄₅₈	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊D₂₂₉	C₄	1	2	yes	—
18	C₂²×Dic₂₂₉	C₂²	1	2	yes	—
19	C₂×C₄×D₂₂₉	C₂²	1	1	yes	characteristic
20	C₂×D₄×C₂₂₉	C₄	1	1	yes	characteristic
21	#14270087	C₂³	1	2	yes	—
22	#14270087	C₂³	1	4	yes	—
23	#14270087	C₂³	1	2	yes	—
24	C₂²×D₂₂₉	C₂×C₄	1	1	yes	characteristic
25	Dic₄₅₈	C₂×C₄	1	4	yes	—
26	C₂²⋊D₂₂₉	C₂×C₄	1	8	yes	—
27	C₂×Dic₂₂₉	C₂×C₄	1	2	yes	—
28	C₂×Dic₂₂₉	C₂×C₄	1	4	yes	—
29	C₂×Dic₂₂₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₂₂₉	C₂³	1	4	yes	—
31	C₄×D₂₂₉	C₂×C₄	1	2	yes	—
32	C₄×D₂₂₉	C₂³	1	2	yes	—
33	C₂³×C₂₂₉	C₂×C₄	1	2	yes	—
34	D₄×C₂₂₉	C₂×C₄	1	4	yes	—
35	C₂×C₄×C₂₂₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₂₂₉	—	2	4	no	—
37	C₂²×C₂₂₉	C₂²×C₄	1	4	yes	—
38	C₂²×C₂₂₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₂₂₉	—	2	2	no	—
40	C₂×D₂₂₉	C₂²×C₄	1	2	yes	—
41	C₄×C₂₂₉	C₂²×C₄	1	2	yes	—
42	Dic₂₂₉	C₂²×C₄	1	1	yes	characteristic
43	Dic₂₂₉	C₂²×C₄	1	4	yes	—
44	Dic₂₂₉	C₂⁴	1	1	yes	characteristic
45	C₂×C₂₂₉	—	2	4	no	—
46	C₂×C₂₂₉	#131	1	2	yes	complemented
47	C₂×C₂₂₉	C₂³×C₄	1	1	yes	characteristic
48	D₂₂₉	—	2	2	no	—
49	C₂₂₉	C₂×#131	1	1	yes	characteristic, complemented, Sylow
50	C₂×#131	—	229	1	no	maximal, Sylow
51	C₂×#41	—	229	1	no	—
52	#131	—	229	8	no	—
53	C₂×M₁₆	—	229	2	no	—
54	C₂×M₁₆	—	229	1	no	—
55	C₂²×C₈	—	229	2	no	—
56	C₂²×C₈	—	229	1	no	—
57	#41	—	229	4	no	—
58	#41	—	229	4	no	—
59	C₂×Q₈	—	229	1	no	—
60	C₂×D₄	—	229	2	no	—
61	C₂×D₄	#11745021	1	1	yes	characteristic
62	C₂²×C₄	—	229	2	no	—
63	C₂²×C₄	—	229	1	no	—
64	M₁₆	—	229	8	no	—
65	M₁₆	—	229	4	no	—
66	C₂×C₈	—	229	1	no	—
67	C₂×C₈	—	229	2	no	—
68	C₂×C₈	—	229	1	no	—
69	C₂×C₈	—	229	8	no	—
70	C₂×C₈	—	229	4	no	—
71	C₂³	—	229	1	no	—
72	C₂³	C₂×#11745021	1	2	yes	—
73	Q₈	—	229	4	no	—
74	D₄	—	229	8	no	—
75	D₄	C₂×#11745021	1	4	yes	—
76	C₂×C₄	—	229	4	no	—
77	C₂×C₄	—	229	4	no	—
78	C₂×C₄	—	229	1	no	—
79	C₂×C₄	—	229	2	no	—
80	C₂×C₄	—	229	2	no	—
81	C₂×C₄	—	229	2	no	—
82	C₂×C₄	C₂×#11745021	1	1	yes	characteristic
83	C₈	—	229	2	no	—
84	C₈	—	229	4	no	—
85	C₈	—	229	2	no	—
86	C₂²	—	2	4	no	—
87	C₂²	—	458	2	no	—
88	C₂²	—	229	2	no	—
89	C₂²	C₂²×#11745021	1	4	yes	—
90	C₂²	C₂²×#11745021	1	1	yes	characteristic
91	C₄	—	229	1	no	—
92	C₄	—	229	1	no	—
93	C₄	—	229	4	no	—
94	C₄	C₂²×#11745021	1	2	yes	—
95	C₂	—	2	4	no	—
96	C₂	—	458	2	no	—
97	C₂	#20301075	1	2	yes	direct product factor
98	C₂	C₂³×#11745021	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20301075

C₂

yes

maximal, direct product factor

C₂×#16814676

C₂

yes

maximal, characteristic, complemented

C₂×#16814675

C₂

yes

maximal, complemented

C₂×D₂₂₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂²×#14270087

C₂

yes

maximal, complemented

C₂×#16814665

C₂

yes

maximal, characteristic

#16814676

C₂²

yes

complemented

#16814675

C₂²

yes

complemented

D₂₂₉⋊C₈

C₂²

yes

complemented

C₂×#14270087

C₂²

yes

characteristic

C₂×#14270087

C₂²

yes

—

C₂×#14270087

C₂²

yes

characteristic

C₂×#14270087

C₂²

yes

complemented

#16814665

C₄

yes

—

#16814665

C₂²

yes

—

C₂×Dic₄₅₈

C₄

yes

characteristic

C₂×C₂²⋊D₂₂₉

C₄

yes

—

C₂²×Dic₂₂₉

C₂²

yes

—

C₂×C₄×D₂₂₉

C₂²

yes

characteristic

C₂×D₄×C₂₂₉

C₄

yes

characteristic

#14270087

C₂³

yes

—

#14270087

C₂³

yes

—

#14270087

C₂³

yes

—

C₂²×D₂₂₉

C₂×C₄

yes

characteristic

Dic₄₅₈

C₂×C₄

yes

—

C₂²⋊D₂₂₉

C₂×C₄

yes

—

C₂×Dic₂₂₉

C₂×C₄

yes

—

C₂×Dic₂₂₉

C₂×C₄

yes

—

C₂×Dic₂₂₉

C₂³

yes

characteristic

C₂×Dic₂₂₉

C₂³

yes

—

C₄×D₂₂₉

C₂×C₄

yes

—

C₄×D₂₂₉

C₂³

yes

—

C₂³×C₂₂₉

C₂×C₄

yes

—

D₄×C₂₂₉

C₂×C₄

yes

—

C₂×C₄×C₂₂₉

C₂×C₄

yes

characteristic

C₂²×C₂₂₉

—

C₂²×C₂₂₉

C₂²×C₄

yes

—

C₂²×C₂₂₉

C₂²×C₄

yes

characteristic

C₂×D₂₂₉

—

C₂×D₂₂₉

C₂²×C₄

yes

—

C₄×C₂₂₉

C₂²×C₄

yes

—

Dic₂₂₉

C₂²×C₄

yes

characteristic

Dic₂₂₉

C₂²×C₄

yes

—

Dic₂₂₉

C₂⁴

yes

characteristic

C₂×C₂₂₉

—

C₂×C₂₂₉

#131

yes

complemented

C₂×C₂₂₉

C₂³×C₄

yes

characteristic

D₂₂₉

—

C₂₂₉

C₂×#131

yes

characteristic, complemented, Sylow

C₂×#131

—

229

maximal, Sylow

C₂×#41

—

229

—

#131

—

229

—

C₂×M₁₆

—

229

—

C₂×M₁₆

—

229

—

C₂²×C₈

—

229

—

C₂²×C₈

—

229

—

#41

—

229

—

#41

—

229

—

C₂×Q₈

—

229

—

C₂×D₄

—

229

—

C₂×D₄

#11745021

yes

characteristic

C₂²×C₄

—

229

—

C₂²×C₄

—

229

—

M₁₆

—

229

—

M₁₆

—

229

—

C₂×C₈

—

229

—

C₂×C₈

—

229

—

C₂×C₈

—

229

—

C₂×C₈

—

229

—

C₂×C₈

—

229

—

C₂³

—

229

—

C₂³

C₂×#11745021

yes

—

Q₈

—

229

—

D₄

—

229

—

D₄

C₂×#11745021

yes

—

C₂×C₄

—

229

—

C₂×C₄

—

229

—

C₂×C₄

—

229

—

C₂×C₄

—

229

—

C₂×C₄

—

229

—

C₂×C₄

—

229

—

C₂×C₄

C₂×#11745021

yes

characteristic

C₈

—

229

—

C₈

—

229

—

C₈

—

229

—

C₂²

—

C₂²

—

458

—

C₂²

—

229

—

C₂²

C₂²×#11745021

yes

—

C₂²

C₂²×#11745021

yes

characteristic

C₄

—

229

—

C₄

—

229

—

C₄

—

229

—

C₄

C₂²×#11745021

yes

—

C₂

—

C₂

—

458

—

C₂

#20301075

yes

direct product factor

C₂

C₂³×#11745021

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#11745021
commutator	C₂×C₂₂₉	C₂³×C₄
Frattini	C₂	C₂³×#11745021
Fitting	C₂×D₄×C₂₂₉	C₄
socle	C₂²×C₂₂₉	C₂²×C₄
prime radical	C₂×#16814665	C₂
cosocle	Dic₂₂₉	C₂⁴
hypercenter	C₂×D₄	#11745021
hypocenter	C₂₂₉	C₂×#131
LCS	G ▷ C₂×C₂₂₉ ▷ C₂₂₉
UCS	C₁ ◁ C₂² ◁ C₂×D₄

subgroup

G/H

center

C₂²

C₂²×#11745021

commutator

C₂×C₂₂₉

C₂³×C₄

Frattini

C₂

C₂³×#11745021

Fitting

C₂×D₄×C₂₂₉

C₄

socle

C₂²×C₂₂₉

C₂²×C₄

prime radical

C₂×#16814665

C₂

cosocle

Dic₂₂₉

C₂⁴

C₂×D₄

C₂₂₉

G ▷ C₂×C₂₂₉ ▷ C₂₂₉

C₁ ◁ C₂² ◁ C₂×D₄

N	H ≅ G/N	type
1. #20301075	C₂	direct (N × H)
2. C₂×#16814676	C₂	semidirect
3. C₂×#16814675	C₂	semidirect
4. C₂×D₂₂₉⋊C₈	C₂	semidirect
5. C₂²×#14270087	C₂	semidirect
7. #16814676	C₂²	semidirect
8. #16814675	C₂²	semidirect
9. D₂₂₉⋊C₈	C₂²	semidirect
13. C₂×#14270087	C₂²	semidirect
46. C₂×C₂₂₉	#131	semidirect
49. C₂₂₉	C₂×#131	semidirect
97. C₂	#20301075	direct (N × H)

H ≅ G/N

type

1. #20301075

C₂

direct (N × H)

2. C₂×#16814676

C₂

semidirect

3. C₂×#16814675

C₂

semidirect

4. C₂×D₂₂₉⋊C₈

C₂

semidirect

5. C₂²×#14270087

C₂

semidirect

7. #16814676

C₂²

semidirect

8. #16814675

C₂²

semidirect

9. D₂₂₉⋊C₈

C₂²

semidirect

13. C₂×#14270087

C₂²

semidirect

46. C₂×C₂₂₉

#131

semidirect

49. C₂₂₉

C₂×#131

semidirect

97. C₂

#20301075

direct (N × H)

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

subgroup	index
#20301075	2
C₂×#16814676	2
C₂×#16814675	2
C₂×D₂₂₉⋊C₈	2
C₂²×#14270087	2
C₂×#16814665	2
C₂×#131	229

subgroup

index

C₂×D₂₂₉⋊C₈

229

Subgroups — C₂×#20301075 #28412556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20301075 #28412556

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20301075 #28412556