Subgroups — #43243871

Order 28096.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28380769	C₂	1	2	yes	maximal, complemented
2	#28380749	C₂	1	1	yes	maximal, characteristic, complemented
3	#28380679	C₂	1	1	yes	maximal, characteristic, complemented
4	#28380660	C₂	1	2	yes	maximal, complemented
5	C₂×D₈×C₄₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#20296329	C₂²	1	1	yes	characteristic
7	#20296317	—	2	2	no	—
8	#20296314	—	2	2	no	—
9	#20296313	C₂²	1	2	yes	—
10	#20296313	C₂²	1	1	yes	characteristic
11	#20296312	—	2	1	no	—
12	C₈×D₄₃₉	—	2	1	no	—
13	C₂×D₄×C₄₃₉	C₂²	1	2	yes	—
14	D₈×C₄₃₉	—	2	1	no	—
15	D₈×C₄₃₉	—	2	1	no	—
16	C₂×C₈×C₄₃₉	C₂²	1	1	yes	characteristic
17	Dic₈₇₈	D₄	1	1	yes	characteristic
18	D₁₇₅₆	D₄	1	1	yes	characteristic
19	C₂²⋊D₄₃₉	—	2	1	no	—
20	C₄×D₄₃₉	—	2	1	no	—
21	C₄₃₉⋊C₈	—	2	1	no	—
22	C₄₃₉⋊C₈	—	2	2	no	—
23	C₂³×C₄₃₉	—	2	2	no	—
24	D₄×C₄₃₉	—	2	2	no	—
25	D₄×C₄₃₉	—	2	2	no	—
26	C₂×C₄×C₄₃₉	C₂³	1	1	yes	characteristic
27	C₈×C₄₃₉	D₄	1	1	yes	characteristic
28	C₈×C₄₃₉	D₄	1	1	yes	characteristic
29	C₂²×C₄₃₉	—	4	2	no	—
30	C₂²×C₄₃₉	—	8	2	no	—
31	C₂²×C₄₃₉	#41	1	1	yes	characteristic
32	C₂×D₄₃₉	—	2	1	no	—
33	C₄×C₄₃₉	C₂×D₄	1	1	yes	characteristic
34	C₄×C₄₃₉	C₂×D₄	1	1	yes	characteristic
35	Dic₄₃₉	—	2	1	no	—
36	C₂×C₄₃₉	—	2	1	no	—
37	C₂×C₄₃₉	—	8	2	no	—
38	C₂×C₄₃₉	#121	1	1	yes	characteristic
39	D₄₃₉	—	4	1	no	—
40	C₄₃₉	#472	1	1	yes	characteristic, complemented, Sylow
41	#472	—	439	1	no	maximal, Sylow
42	AGL(1,ℤ/8)	—	439	2	no	—
43	C₂×D₈	D₄₃₉	1	1	yes	characteristic, complemented
44	#131	—	439	1	no	—
45	#108	—	439	1	no	—
46	#100	—	439	2	no	—
47	#41	—	439	1	no	—
48	C₂×D₄	C₂×D₄₃₉	1	2	yes	—
49	SD₁₆	—	878	2	no	—
50	D₈	—	2	1	no	—
51	D₈	—	2	1	no	—
52	D₈	—	878	2	no	—
53	M₁₆	—	878	1	no	—
54	M₁₆	—	439	1	no	—
55	M₁₆	—	439	2	no	—
56	C₂×C₈	—	878	1	no	—
57	C₂×C₈	C₂×D₄₃₉	1	1	yes	characteristic
58	C₂³	—	2	2	no	—
59	Q₈	—	439	1	no	—
60	D₄	—	2	2	no	—
61	D₄	—	2	2	no	—
62	D₄	—	878	1	no	—
63	D₄	—	439	1	no	—
64	C₂×C₄	—	878	1	no	—
65	C₂×C₄	C₂²×D₄₃₉	1	1	yes	characteristic
66	C₈	—	878	2	no	—
67	C₈	—	878	1	no	—
68	C₈	C₂²⋊D₄₃₉	1	1	yes	characteristic
69	C₈	C₂²⋊D₄₃₉	1	1	yes	characteristic
70	C₂²	—	4	2	no	—
71	C₂²	—	8	2	no	—
72	C₂²	—	878	1	no	—
73	C₂²	#20296328	1	1	yes	characteristic
74	C₄	—	878	1	no	—
75	C₄	D₄×D₄₃₉	1	1	yes	characteristic
76	C₄	C₂×C₂²⋊D₄₃₉	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	8	2	no	—
79	C₂	—	1756	1	no	—
80	C₂	#28380714	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#28380714
commutator, cosocle	C₂×C₄×C₄₃₉	C₂³
Frattini	C₂×C₄	C₂²×D₄₃₉
Fitting	C₂×D₈×C₄₃₉	C₂
socle	C₂×C₄₃₉	#121
hypercenter	C₂×D₈	D₄₃₉
hypocenter	C₄₃₉	#472
LCS	G ▷ C₂×C₄×C₄₃₉ ▷ C₂×C₄₃₉ ▷ C₄₃₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₈

Complements

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

N	H ≅ G/N	type
1. #28380769	C₂	semidirect
2. #28380749	C₂	semidirect
3. #28380679	C₂	semidirect
4. #28380660	C₂	semidirect
5. C₂×D₈×C₄₃₉	C₂	semidirect
40. C₄₃₉	#472	semidirect
43. C₂×D₈	D₄₃₉	semidirect

Maximal subgroups

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

subgroup	index
#28380769	2
#28380749	2
#28380679	2
#28380660	2
C₂×D₈×C₄₃₉	2
#472	439