Subgroups — #29505509

Order 15296.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21439954	C₂	1	2	yes	maximal, complemented
2	#21439941	C₂	1	3	yes	maximal, complemented
3	#21439932	C₂	1	6	yes	maximal, complemented
4	#21439928	C₂	1	3	yes	maximal, complemented
5	#127×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₂₃₉	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₂₃₉	C₂²	1	6	yes	—
8	C₂²×Dic₂₃₉	C₂²	1	3	yes	complemented
9	Dic₂₃₉⋊C₄	C₂²	1	6	yes	complemented
10	C₂²⋊Dic₂₃₉	C₂²	1	3	yes	—
11	C₂²⋊Dic₂₃₉	C₂²	1	3	yes	complemented
12	(C₂×D₂₃₉)⋊C₄	C₂²	1	6	yes	—
13	C₂×D₄×C₂₃₉	C₂²	1	3	yes	—
14	C₂×D₄×C₂₃₉	C₂²	1	3	yes	complemented
15	C₄²×C₂₃₉	C₂²	1	1	yes	characteristic, complemented
16	C₂²×D₂₃₉	—	4	3	no	—
17	C₂²×D₂₃₉	C₂³	1	2	yes	—
18	C₂²⋊D₂₃₉	—	4	6	no	—
19	C₂×Dic₂₃₉	—	4	3	no	—
20	C₂×Dic₂₃₉	C₂³	1	6	yes	—
21	C₂³×C₂₃₉	C₂³	1	1	yes	characteristic
22	C₂³×C₂₃₉	C₂³	1	3	yes	—
23	D₄×C₂₃₉	—	4	3	no	—
24	D₄×C₂₃₉	—	4	3	no	—
25	C₂×C₄×C₂₃₉	C₂³	1	3	yes	—
26	C₂²×C₂₃₉	—	2	3	no	—
27	C₂²×C₂₃₉	—	2	6	no	—
28	C₂²×C₂₃₉	—	2	3	no	—
29	C₂²×C₂₃₉	C₂⁴	1	1	yes	characteristic
30	C₂×D₂₃₉	—	2	6	no	—
31	C₂×D₂₃₉	—	4	4	no	—
32	C₄×C₂₃₉	—	2	3	no	—
33	Dic₂₃₉	—	2	6	no	—
34	C₂×C₂₃₉	—	4	1	no	—
35	C₂×C₂₃₉	—	4	3	no	—
36	C₂×C₂₃₉	H(5,𝔽₂)	1	3	yes	—
37	D₂₃₉	—	4	2	no	—
38	C₂₃₉	#560	1	1	yes	characteristic, complemented, Sylow
39	#560	—	239	1	no	maximal, Sylow
40	#127	D₂₃₉	1	1	yes	characteristic, complemented
41	#126	—	239	2	no	—
42	#123	—	239	3	no	—
43	#121	—	239	6	no	—
44	#120	—	239	3	no	—
45	C₂⁴	—	239	1	no	—
46	C₂×D₄	—	239	6	no	—
47	C₂×D₄	C₂×D₂₃₉	1	3	yes	—
48	C₂×D₄	C₂×D₂₃₉	1	3	yes	complemented
49	C₂²×C₄	—	239	3	no	—
50	C₄⋊C₄	—	239	6	no	—
51	C₂²⋊C₄	—	239	3	no	—
52	C₂²⋊C₄	—	239	3	no	—
53	C₂²⋊C₄	—	239	6	no	—
54	C₄²	C₂×D₂₃₉	1	1	yes	characteristic, complemented
55	C₂³	—	956	3	no	—
56	C₂³	—	239	2	no	—
57	C₂³	C₂²×D₂₃₉	1	1	yes	characteristic
58	C₂³	C₂²×D₂₃₉	1	3	yes	—
59	D₄	—	4	3	no	—
60	D₄	—	4	3	no	—
61	D₄	—	956	6	no	—
62	C₂×C₄	—	956	3	no	—
63	C₂×C₄	—	239	6	no	—
64	C₂×C₄	C₂²×D₂₃₉	1	3	yes	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	956	4	no	—
69	C₂²	—	478	6	no	—
70	C₂²	C₂³×D₂₃₉	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	478	6	no	—
73	C₂	—	4	1	no	—
74	C₂	—	4	3	no	—
75	C₂	—	956	2	no	—
76	C₂	#21440005	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃₉
commutator, socle, cosocle	C₂²×C₂₃₉	C₂⁴
Fitting	#127×C₂₃₉	C₂
hypercenter	#127	D₂₃₉
hypocenter	C₂₃₉	#560
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #21439954	C₂	semidirect
2. #21439941	C₂	semidirect
3. #21439932	C₂	semidirect
4. #21439928	C₂	semidirect
5. #127×C₂₃₉	C₂	semidirect
8. C₂²×Dic₂₃₉	C₂²	semidirect
9. Dic₂₃₉⋊C₄	C₂²	semidirect
11. C₂²⋊Dic₂₃₉	C₂²	semidirect
14. C₂×D₄×C₂₃₉	C₂²	semidirect
15. C₄²×C₂₃₉	C₂²	semidirect
38. C₂₃₉	#560	semidirect
40. #127	D₂₃₉	semidirect
48. C₂×D₄	C₂×D₂₃₉	semidirect
54. C₄²	C₂×D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#21439954	2
#21439941	2
#21439932	2
#21439928	2
#127×C₂₃₉	2
#560	239