Subgroups — #534×C₁₃₉ #22580619

Order 8896.

346 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄×C₁₃₉	C₂	1	2	yes	maximal, complemented
2	#123×C₁₃₉	C₂	1	4	yes	maximal, complemented
3	#121×C₁₃₉	C₂	1	4	yes	maximal, complemented
4	#120×C₁₃₉	C₂	1	4	yes	maximal, complemented
5	C₂×C₂²⋊C₄×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₁₃₉	C₂²	1	1	yes	characteristic
7	C₂⁴×C₁₃₉	C₂²	1	2	yes	complemented
8	C₂×D₄×C₁₃₉	—	2	8	no	—
9	C₂×D₄×C₁₃₉	C₂²	1	4	yes	—
10	C₂×D₄×C₁₃₉	C₂²	1	4	yes	complemented
11	C₂×D₄×C₁₃₉	C₂²	1	4	yes	complemented
12	C₂²×C₄×C₁₃₉	C₂²	1	2	yes	complemented
13	C₂²×C₄×C₁₃₉	C₂²	1	2	yes	complemented
14	C₄⋊C₄×C₁₃₉	C₂²	1	4	yes	complemented
15	C₂²⋊C₄×C₁₃₉	C₂²	1	4	yes	complemented
16	C₂²⋊C₄×C₁₃₉	C₂²	1	8	yes	complemented
17	C₂³×C₁₃₉	—	2	4	no	—
18	C₂³×C₁₃₉	—	4	2	no	—
19	C₂³×C₁₃₉	—	4	4	no	—
20	C₂³×C₁₃₉	D₄	1	4	yes	complemented
21	C₂³×C₁₃₉	C₂³	1	4	yes	—
22	C₂³×C₁₃₉	C₂³	1	2	yes	—
23	C₂³×C₁₃₉	C₂³	1	1	yes	characteristic
24	D₄×C₁₃₉	—	2	8	no	—
25	D₄×C₁₃₉	—	2	8	no	—
26	D₄×C₁₃₉	—	4	4	no	—
27	C₂×C₄×C₁₃₉	—	2	4	no	—
28	C₂×C₄×C₁₃₉	—	4	2	no	—
29	C₂×C₄×C₁₃₉	C₂³	1	4	yes	—
30	C₂×C₄×C₁₃₉	C₂³	1	4	yes	—
31	C₂²×C₁₃₉	—	2	2	no	—
32	C₂²×C₁₃₉	—	2	4	no	—
33	C₂²×C₁₃₉	—	2	4	no	—
34	C₂²×C₁₃₉	—	2	8	no	—
35	C₂²×C₁₃₉	—	4	4	no	—
36	C₂²×C₁₃₉	—	4	8	no	—
37	C₂²×C₁₃₉	C₂×D₄	1	4	yes	—
38	C₂²×C₁₃₉	C₂×D₄	1	2	yes	—
39	C₂²×C₁₃₉	C₂⁴	1	1	yes	characteristic
40	C₄×C₁₃₉	—	2	4	no	—
41	C₄×C₁₃₉	—	2	4	no	—
42	C₂×C₁₃₉	—	2	4	no	—
43	C₂×C₁₃₉	—	2	2	no	—
44	C₂×C₁₃₉	—	4	4	no	—
45	C₂×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
46	C₂×C₁₃₉	H(5,𝔽₂)	1	2	yes	—
47	C₁₃₉	#534	1	1	yes	characteristic, direct product factor, Sylow
48	#534	C₁₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂²×D₄	C₂×C₁₃₉	1	2	yes	complemented
50	#123	C₂×C₁₃₉	1	4	yes	complemented
51	#121	C₂×C₁₃₉	1	4	yes	complemented
52	#120	C₂×C₁₃₉	1	4	yes	complemented
53	C₂×C₂²⋊C₄	C₂×C₁₃₉	1	1	yes	characteristic, complemented
54	C₂⁴	C₂²×C₁₃₉	1	1	yes	characteristic
55	C₂⁴	C₂²×C₁₃₉	1	2	yes	complemented
56	C₂×D₄	—	2	8	no	—
57	C₂×D₄	C₂²×C₁₃₉	1	4	yes	—
58	C₂×D₄	C₂²×C₁₃₉	1	4	yes	complemented
59	C₂×D₄	C₂²×C₁₃₉	1	4	yes	complemented
60	C₂²×C₄	C₂²×C₁₃₉	1	2	yes	complemented
61	C₂²×C₄	C₂²×C₁₃₉	1	2	yes	complemented
62	C₄⋊C₄	C₂²×C₁₃₉	1	4	yes	complemented
63	C₂²⋊C₄	C₂²×C₁₃₉	1	4	yes	complemented
64	C₂²⋊C₄	C₂²×C₁₃₉	1	8	yes	complemented
65	C₂³	—	2	4	no	—
66	C₂³	—	4	2	no	—
67	C₂³	—	4	4	no	—
68	C₂³	D₄×C₁₃₉	1	4	yes	complemented
69	C₂³	C₂³×C₁₃₉	1	4	yes	—
70	C₂³	C₂³×C₁₃₉	1	2	yes	—
71	C₂³	C₂³×C₁₃₉	1	1	yes	characteristic
72	D₄	—	2	8	no	—
73	D₄	—	2	8	no	—
74	D₄	—	4	4	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	C₂³×C₁₃₉	1	4	yes	—
78	C₂×C₄	C₂³×C₁₃₉	1	4	yes	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	4	no	—
81	C₂²	—	2	4	no	—
82	C₂²	—	2	8	no	—
83	C₂²	—	4	4	no	—
84	C₂²	—	4	8	no	—
85	C₂²	C₂×D₄×C₁₃₉	1	4	yes	—
86	C₂²	C₂×D₄×C₁₃₉	1	2	yes	—
87	C₂²	C₂⁴×C₁₃₉	1	1	yes	characteristic
88	C₄	—	2	4	no	—
89	C₄	—	2	4	no	—
90	C₂	—	2	4	no	—
91	C₂	—	2	2	no	—
92	C₂	—	4	4	no	—
93	C₂	C₂²×D₄×C₁₃₉	1	1	yes	characteristic
94	C₂	H(5,𝔽₂)×C₁₃₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₉	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₃₉
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×D₄×C₁₃₉	C₂	semidirect
2. #123×C₁₃₉	C₂	semidirect
3. #121×C₁₃₉	C₂	semidirect
4. #120×C₁₃₉	C₂	semidirect
5. C₂×C₂²⋊C₄×C₁₃₉	C₂	semidirect
7. C₂⁴×C₁₃₉	C₂²	semidirect
10. C₂×D₄×C₁₃₉	C₂²	semidirect
11. C₂×D₄×C₁₃₉	C₂²	semidirect
12. C₂²×C₄×C₁₃₉	C₂²	semidirect
13. C₂²×C₄×C₁₃₉	C₂²	semidirect
14. C₄⋊C₄×C₁₃₉	C₂²	semidirect
15. C₂²⋊C₄×C₁₃₉	C₂²	semidirect
16. C₂²⋊C₄×C₁₃₉	C₂²	semidirect
20. C₂³×C₁₃₉	D₄	semidirect
47. C₁₃₉	#534	direct (N × H)
48. #534	C₁₃₉	direct (N × H)
49. C₂²×D₄	C₂×C₁₃₉	semidirect
50. #123	C₂×C₁₃₉	semidirect
51. #121	C₂×C₁₃₉	semidirect
52. #120	C₂×C₁₃₉	semidirect
53. C₂×C₂²⋊C₄	C₂×C₁₃₉	semidirect
55. C₂⁴	C₂²×C₁₃₉	semidirect
58. C₂×D₄	C₂²×C₁₃₉	semidirect
59. C₂×D₄	C₂²×C₁₃₉	semidirect
60. C₂²×C₄	C₂²×C₁₃₉	semidirect
61. C₂²×C₄	C₂²×C₁₃₉	semidirect
62. C₄⋊C₄	C₂²×C₁₃₉	semidirect
63. C₂²⋊C₄	C₂²×C₁₃₉	semidirect
64. C₂²⋊C₄	C₂²×C₁₃₉	semidirect
68. C₂³	D₄×C₁₃₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₄×C₁₃₉	2
#123×C₁₃₉	2
#121×C₁₃₉	2
#120×C₁₃₉	2
C₂×C₂²⋊C₄×C₁₃₉	2
#534	139

Subgroups — #534×C139 #22580619

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #534×C₁₃₉ #22580619