Subgroups — #43243772

Order 28096.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28380768	C₂	1	2	yes	maximal, complemented
2	#28380739	C₂	1	1	yes	maximal, characteristic, complemented
3	#28380670	C₂	1	2	yes	maximal, complemented
4	#28380660	C₂	1	1	yes	maximal, characteristic, complemented
5	H(5,𝔽₂)×C₄₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₇₅₆	C₂²	1	1	yes	characteristic, complemented
7	C₂×C₂²⋊D₄₃₉	—	2	2	no	—
8	#20296316	—	2	2	no	—
9	#20296314	—	2	2	no	—
10	#20296313	C₂²	1	2	yes	complemented
11	C₄×Dic₄₃₉	C₂²	1	1	yes	characteristic, complemented
12	#41×C₄₃₉	—	2	2	no	—
13	#41×C₄₃₉	C₂²	1	2	yes	—
14	C₂×D₄×C₄₃₉	—	2	2	no	—
15	C₂×D₄×C₄₃₉	—	2	2	no	—
16	C₂×D₄×C₄₃₉	C₂²	1	1	yes	characteristic
17	C₂²×D₄₃₉	—	2	1	no	—
18	D₁₇₅₆	—	2	2	no	—
19	C₂²⋊D₄₃₉	—	4	4	no	—
20	C₂×Dic₄₃₉	—	2	1	no	—
21	C₄₃₉⋊C₈	—	2	2	no	—
22	C₂³×C₄₃₉	—	2	2	no	—
23	C₂³×C₄₃₉	D₄	1	2	yes	—
24	Q₈×C₄₃₉	D₄	1	2	yes	complemented
25	D₄×C₄₃₉	—	2	4	no	—
26	D₄×C₄₃₉	—	2	2	no	—
27	D₄×C₄₃₉	—	2	2	no	—
28	D₄×C₄₃₉	D₄	1	2	yes	complemented
29	C₂×C₄×C₄₃₉	—	2	2	no	—
30	C₂×C₄×C₄₃₉	—	2	2	no	—
31	C₂×C₄×C₄₃₉	C₂³	1	1	yes	characteristic
32	C₂²×C₄₃₉	—	2	2	no	—
33	C₂²×C₄₃₉	—	2	2	no	—
34	C₂²×C₄₃₉	—	4	2	no	—
35	C₂²×C₄₃₉	—	8	2	no	—
36	C₂²×C₄₃₉	C₂×D₄	1	1	yes	characteristic
37	C₂×D₄₃₉	—	4	2	no	—
38	C₂×D₄₃₉	—	4	1	no	—
39	C₄×C₄₃₉	—	2	2	no	—
40	C₄×C₄₃₉	C₂×D₄	1	2	yes	—
41	Dic₄₃₉	—	2	2	no	—
42	C₂×C₄₃₉	—	2	1	no	—
43	C₂×C₄₃₉	—	4	2	no	—
44	C₂×C₄₃₉	—	4	2	no	—
45	C₂×C₄₃₉	#120	1	1	yes	characteristic
46	D₄₃₉	—	8	1	no	—
47	C₄₃₉	#453	1	1	yes	characteristic, complemented, Sylow
48	#453	—	439	1	no	maximal, Sylow
49	H(5,𝔽₂)	D₄₃₉	1	1	yes	characteristic, complemented
50	AGL(1,ℤ/8)	—	439	2	no	—
51	#127	—	439	1	no	—
52	#104	—	439	2	no	—
53	#100	—	439	1	no	—
54	#41	—	2	2	no	—
55	#41	C₂×D₄₃₉	1	2	yes	—
56	C₂×D₄	—	2	2	no	—
57	C₂×D₄	—	2	2	no	—
58	C₂×D₄	—	878	2	no	—
59	C₂×D₄	—	439	1	no	—
60	C₂×D₄	C₂×D₄₃₉	1	1	yes	characteristic
61	SD₁₆	—	878	2	no	—
62	D₈	—	878	2	no	—
63	M₁₆	—	439	2	no	—
64	C₄²	—	439	1	no	—
65	C₂³	—	2	2	no	—
66	C₂³	—	878	1	no	—
67	C₂³	C₂²⋊D₄₃₉	1	2	yes	—
68	Q₈	C₂²⋊D₄₃₉	1	2	yes	complemented
69	D₄	—	2	4	no	—
70	D₄	—	2	2	no	—
71	D₄	—	2	2	no	—
72	D₄	—	878	2	no	—
73	D₄	—	1756	4	no	—
74	D₄	C₂²⋊D₄₃₉	1	2	yes	complemented
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	878	1	no	—
78	C₂×C₄	C₂²×D₄₃₉	1	1	yes	characteristic
79	C₈	—	878	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	1756	2	no	—
85	C₂²	—	1756	1	no	—
86	C₂²	C₂×C₂²⋊D₄₃₉	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	878	2	no	—
89	C₄	C₂×C₂²⋊D₄₃₉	1	2	yes	—
90	C₂	—	2	1	no	—
91	C₂	—	4	2	no	—
92	C₂	—	4	2	no	—
93	C₂	—	3512	1	no	—
94	C₂	#28380712	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28380712
commutator, cosocle	C₂×C₄×C₄₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₉
Fitting	H(5,𝔽₂)×C₄₃₉	C₂
socle	C₂×C₄₃₉	#120
hypercenter	H(5,𝔽₂)	D₄₃₉
hypocenter	C₄₃₉	#453
LCS	G ▷ C₂×C₄×C₄₃₉ ▷ C₂×C₄₃₉ ▷ C₄₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #28380768	C₂	semidirect
2. #28380739	C₂	semidirect
3. #28380670	C₂	semidirect
4. #28380660	C₂	semidirect
5. H(5,𝔽₂)×C₄₃₉	C₂	semidirect
6. C₂×D₁₇₅₆	C₂²	semidirect
10. #20296313	C₂²	semidirect
11. C₄×Dic₄₃₉	C₂²	semidirect
24. Q₈×C₄₃₉	D₄	semidirect
28. D₄×C₄₃₉	D₄	semidirect
47. C₄₃₉	#453	semidirect
49. H(5,𝔽₂)	D₄₃₉	semidirect
68. Q₈	C₂²⋊D₄₃₉	semidirect
74. D₄	C₂²⋊D₄₃₉	semidirect

Maximal subgroups

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

subgroup	index
#28380768	2
#28380739	2
#28380670	2
#28380660	2
H(5,𝔽₂)×C₄₃₉	2
#453	439