Subgroups — #28385322

Order 14368.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₄₄₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄₄₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊Dic₄₄₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297715	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊Dic₄₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#20297713	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₄×C₄₄₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₄₉	C₂²	1	1	yes	characteristic
9	C₂²⋊D₄₄₉	—	2	2	no	—
10	C₂×Dic₄₄₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₄₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₄₄₉	C₂²	1	1	yes	characteristic
13	C₄×D₄₄₉	—	2	2	no	—
14	C₂³×C₄₄₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₄₄₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₄₄₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄₄₉	—	2	1	no	—
18	C₂²×C₄₄₉	—	2	1	no	—
19	C₂²×C₄₄₉	—	2	1	no	—
20	C₂²×C₄₄₉	C₂³	1	1	yes	characteristic
21	C₂×D₄₄₉	—	2	1	no	—
22	C₂×D₄₄₉	—	2	1	no	—
23	C₂×D₄₄₉	D₄	1	2	yes	—
24	C₄×C₄₄₉	—	2	1	no	—
25	C₄×C₄₄₉	—	2	1	no	—
26	Dic₄₄₉	—	2	1	no	—
27	Dic₄₄₉	—	2	1	no	—
28	Dic₄₄₉	—	2	1	no	—
29	C₂×C₄₄₉	—	4	1	no	—
30	C₂×C₄₄₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄₄₉	#41	1	1	yes	characteristic
32	C₂×C₄₄₉	#41	1	1	yes	characteristic
33	D₄₄₉	—	2	2	no	—
34	C₄₄₉	#123	1	1	yes	characteristic, complemented, Sylow
35	#123	—	449	1	no	maximal, Sylow
36	C₂×D₄	—	449	1	no	—
37	C₂²×C₄	—	449	1	no	—
38	C₄⋊C₄	—	449	1	no	—
39	C₄⋊C₄	—	449	1	no	—
40	C₂²⋊C₄	—	449	1	no	—
41	C₂²⋊C₄	—	449	1	no	—
42	C₂²⋊C₄	D₄₄₉	1	1	yes	characteristic, complemented
43	C₂³	—	449	1	no	—
44	C₂³	C₂×D₄₄₉	1	1	yes	characteristic
45	D₄	—	898	2	no	—
46	C₂×C₄	—	898	2	no	—
47	C₂×C₄	—	449	1	no	—
48	C₂×C₄	—	449	1	no	—
49	C₂×C₄	—	449	1	no	—
50	C₂×C₄	C₂×D₄₄₉	1	1	yes	characteristic
51	C₂×C₄	C₂×D₄₄₉	1	1	yes	characteristic
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	898	1	no	—
56	C₂²	—	898	1	no	—
57	C₂²	—	449	2	no	—
58	C₂²	C₂²×D₄₄₉	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	898	1	no	—
62	C₄	—	898	1	no	—
63	C₄	—	898	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	898	2	no	—
66	C₂	D₄×D₄₄₉	1	1	yes	characteristic
67	C₂	#20297735	1	1	yes	characteristic
68	C₂	#20297736	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₄₄₉
commutator, socle, cosocle	C₂²×C₄₄₉	C₂³
Fitting	C₂²⋊C₄×C₄₄₉	C₂
prime radical	C₂×C₂²⋊D₄₄₉	C₂
hypercenter	C₂²⋊C₄	D₄₄₉
hypocenter	C₄₄₉	#123
LCS	G ▷ C₂²×C₄₄₉ ▷ C₄₄₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₄×D₄₄₉	C₂	semidirect
3. C₄⋊Dic₄₄₉	C₂	semidirect
4. #20297715	C₂	semidirect
5. C₂²⋊Dic₄₄₉	C₂	semidirect
6. #20297713	C₂	semidirect
7. C₂²⋊C₄×C₄₄₉	C₂	semidirect
34. C₄₄₉	#123	semidirect
42. C₂²⋊C₄	D₄₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊D₄₄₉	2
C₂×C₄×D₄₄₉	2
C₄⋊Dic₄₄₉	2
#20297715	2
C₂²⋊Dic₄₄₉	2
#20297713	2
C₂²⋊C₄×C₄₄₉	2
#123	449