Subgroups — #38734734

Order 24896.

428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×D₃₈₉	C₂	1	4	yes	maximal, complemented
2	#27159355	C₂	1	4	yes	maximal, complemented
3	#27159352	C₂	1	4	yes	maximal, complemented
4	C₄×D₁₅₅₆	C₂	1	2	yes	maximal, complemented
5	#127×C₃₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₈₉	C₂²	1	4	yes	complemented
7	C₂×D₁₅₅₆	C₂²	1	2	yes	—
8	C₂×C₂²⋊D₃₈₉	C₂²	1	8	yes	complemented
9	D₄×D₃₈₉	—	2	16	no	—
10	C₂×C₄×D₃₈₉	C₂²	1	4	yes	complemented
11	C₄⋊Dic₃₈₉	C₂²	1	2	yes	complemented
12	C₂²⋊Dic₃₈₉	C₂²	1	4	yes	complemented
13	#20261943	C₂²	1	4	yes	complemented
14	C₂×D₄×C₃₈₉	C₂²	1	4	yes	complemented
15	C₂×D₄×C₃₈₉	C₂²	1	2	yes	complemented
16	C₄²×C₃₈₉	C₂²	1	1	yes	characteristic, complemented
17	C₂²×D₃₈₉	—	2	16	no	—
18	C₂²×D₃₈₉	—	4	4	no	—
19	C₂²×D₃₈₉	C₂³	1	4	yes	—
20	D₁₅₅₆	D₄	1	8	yes	complemented
21	C₂²⋊D₃₈₉	—	2	16	no	—
22	C₂×Dic₃₈₉	C₂³	1	4	yes	—
23	C₄×D₃₈₉	—	2	8	no	—
24	C₂³×C₃₈₉	C₂³	1	4	yes	—
25	D₄×C₃₈₉	—	2	8	no	—
26	D₄×C₃₈₉	—	4	2	no	—
27	C₂×C₄×C₃₈₉	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₃₈₉	C₂³	1	2	yes	—
29	C₂²×C₃₈₉	—	2	8	no	—
30	C₂²×C₃₈₉	—	2	4	no	—
31	C₂²×C₃₈₉	C₂⁴	1	1	yes	characteristic
32	C₂×D₃₈₉	—	2	4	no	—
33	C₂×D₃₈₉	—	2	4	no	—
34	C₂×D₃₈₉	—	4	16	no	—
35	C₂×D₃₈₉	C₂×D₄	1	8	yes	complemented
36	C₄×C₃₈₉	—	2	1	no	—
37	C₄×C₃₈₉	C₂×D₄	1	4	yes	complemented
38	Dic₃₈₉	—	2	4	no	—
39	C₂×C₃₈₉	—	4	4	no	—
40	C₂×C₃₈₉	C₂²×D₄	1	2	yes	—
41	C₂×C₃₈₉	H(5,𝔽₂)	1	1	yes	characteristic
42	D₃₈₉	—	2	8	no	—
43	C₃₈₉	D₄²	1	1	yes	characteristic, complemented, Sylow
44	D₄²	—	389	1	no	maximal, Sylow
45	C₂²×D₄	—	389	4	no	—
46	#127	D₃₈₉	1	1	yes	characteristic, complemented
47	#121	—	389	4	no	—
48	#120	—	389	4	no	—
49	C₄×D₄	—	389	2	no	—
50	C₂⁴	—	389	4	no	—
51	C₂×D₄	—	778	16	no	—
52	C₂×D₄	—	389	8	no	—
53	C₂×D₄	—	389	2	no	—
54	C₂×D₄	C₂×D₃₈₉	1	4	yes	complemented
55	C₂×D₄	C₂×D₃₈₉	1	2	yes	complemented
56	C₂²×C₄	—	389	4	no	—
57	C₄⋊C₄	—	389	2	no	—
58	C₂²⋊C₄	—	389	4	no	—
59	C₂²⋊C₄	—	389	4	no	—
60	C₄²	C₂×D₃₈₉	1	1	yes	characteristic, complemented
61	C₂³	—	778	16	no	—
62	C₂³	—	1556	4	no	—
63	C₂³	—	389	4	no	—
64	C₂³	C₂²×D₃₈₉	1	4	yes	—
65	D₄	—	2	8	no	—
66	D₄	—	4	2	no	—
67	D₄	—	778	16	no	—
68	D₄	—	389	8	no	—
69	C₂×C₄	—	778	8	no	—
70	C₂×C₄	—	389	4	no	—
71	C₂×C₄	C₂²×D₃₈₉	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₃₈₉	1	2	yes	—
73	C₂²	—	2	8	no	—
74	C₂²	—	2	4	no	—
75	C₂²	—	778	4	no	—
76	C₂²	—	778	4	no	—
77	C₂²	—	1556	16	no	—
78	C₂²	—	389	8	no	—
79	C₂²	C₂³×D₃₈₉	1	1	yes	characteristic
80	C₄	—	2	1	no	—
81	C₄	—	778	4	no	—
82	C₄	D₄×D₃₈₉	1	4	yes	complemented
83	C₂	—	4	4	no	—
84	C₂	—	778	8	no	—
85	C₂	C₂×D₄×D₃₈₉	1	2	yes	—
86	C₂	#27159429	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₈₉
commutator, socle, cosocle	C₂²×C₃₈₉	C₂⁴
Fitting	#127×C₃₈₉	C₂
hypercenter	#127	D₃₈₉
hypocenter	C₃₈₉	D₄²
LCS	G ▷ C₂²×C₃₈₉ ▷ C₃₈₉
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. C₂×D₄×D₃₈₉	C₂	semidirect
2. #27159355	C₂	semidirect
3. #27159352	C₂	semidirect
4. C₄×D₁₅₅₆	C₂	semidirect
5. #127×C₃₈₉	C₂	semidirect
6. C₂³×D₃₈₉	C₂²	semidirect
8. C₂×C₂²⋊D₃₈₉	C₂²	semidirect
10. C₂×C₄×D₃₈₉	C₂²	semidirect
11. C₄⋊Dic₃₈₉	C₂²	semidirect
12. C₂²⋊Dic₃₈₉	C₂²	semidirect
13. #20261943	C₂²	semidirect
14. C₂×D₄×C₃₈₉	C₂²	semidirect
15. C₂×D₄×C₃₈₉	C₂²	semidirect
16. C₄²×C₃₈₉	C₂²	semidirect
20. D₁₅₅₆	D₄	semidirect
35. C₂×D₃₈₉	C₂×D₄	semidirect
37. C₄×C₃₈₉	C₂×D₄	semidirect
43. C₃₈₉	D₄²	semidirect
46. #127	D₃₈₉	semidirect
54. C₂×D₄	C₂×D₃₈₉	semidirect
55. C₂×D₄	C₂×D₃₈₉	semidirect
60. C₄²	C₂×D₃₈₉	semidirect
82. C₄	D₄×D₃₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×D₃₈₉	2
#27159355	2
#27159352	2
C₄×D₁₅₅₆	2
#127×C₃₈₉	2
D₄²	389