Subgroups — #50306252

Order 38336.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#34062703	C₂	1	1	yes	maximal, characteristic
2	#34062638	C₂	1	2	yes	maximal, complemented
3	#34062635	C₂	1	2	yes	maximal
4	#34062626	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₅₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₁₁₉₈	C₂²	1	1	yes	characteristic
7	C₂×D₂₃₉₆	C₂²	1	1	yes	characteristic
8	C₂×C₅₉₉⋊C₈	C₂²	1	2	yes	—
9	(C₂×D₅₉₉)⋊C₄	—	2	1	no	—
10	(C₂×D₅₉₉)⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₅₉₉	—	2	1	no	—
12	C₄⋊C₄×C₅₉₉	—	2	1	no	—
13	C₄⋊C₄×C₅₉₉	C₂²	1	2	yes	—
14	C₄²×C₅₉₉	C₂²	1	1	yes	characteristic
15	C₂²×D₅₉₉	—	2	1	no	—
16	Dic₁₁₉₈	—	2	2	no	—
17	D₂₃₉₆	—	2	2	no	—
18	C₂×Dic₅₉₉	—	2	1	no	—
19	C₅₉₉⋊C₈	—	2	2	no	—
20	Q₈×C₅₉₉	—	4	2	no	—
21	C₂×C₄×C₅₉₉	—	2	2	no	—
22	C₂×C₄×C₅₉₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₅₉₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₅₉₉	C₂³	1	1	yes	characteristic
25	C₂²×C₅₉₉	C₂×D₄	1	1	yes	characteristic
26	C₂×D₅₉₉	—	4	1	no	—
27	C₂×D₅₉₉	—	4	1	no	—
28	C₂×D₅₉₉	—	4	1	no	—
29	C₄×C₅₉₉	—	2	1	no	—
30	C₄×C₅₉₉	—	2	1	no	—
31	C₄×C₅₉₉	—	4	2	no	—
32	C₄×C₅₉₉	#41	1	2	yes	—
33	Dic₅₉₉	—	4	1	no	—
34	C₂×C₅₉₉	#124	1	1	yes	characteristic
35	C₂×C₅₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₅₉₉	#137	1	1	yes	characteristic
37	D₅₉₉	—	8	1	no	—
38	C₅₉₉	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	599	1	no	maximal, Sylow
40	C₄⋊Q₈	D₅₉₉	1	1	yes	characteristic, complemented
41	#124	—	599	1	no	—
42	#103	—	599	2	no	—
43	#102	—	599	2	no	—
44	#97	—	599	1	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	599	1	no	—
47	C₂×D₄	—	599	1	no	—
48	C₂×C₈	—	599	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	C₂×D₅₉₉	1	2	yes	—
51	C₂²⋊C₄	—	1198	1	no	—
52	C₂²⋊C₄	—	1198	1	no	—
53	C₄²	C₂×D₅₉₉	1	1	yes	characteristic
54	C₂³	—	1198	1	no	—
55	Q₈	—	4	2	no	—
56	Q₈	—	1198	2	no	—
57	D₄	—	1198	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	1198	1	no	—
60	C₂×C₄	C₂²⋊D₅₉₉	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₅₉₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₅₉₉	1	1	yes	characteristic
63	C₈	—	1198	2	no	—
64	C₂²	—	2396	1	no	—
65	C₂²	—	2396	1	no	—
66	C₂²	—	2396	1	no	—
67	C₂²	C₂×C₂²⋊D₅₉₉	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	2396	1	no	—
72	C₄	#23751079	1	2	yes	—
73	C₂	—	4792	1	no	—
74	C₂	#34062704	1	1	yes	characteristic
75	C₂	#34062741	1	1	yes	characteristic
76	C₂	#34062746	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₅₉₉
commutator, cosocle	C₂×C₄×C₅₉₉	C₂³
Frattini	C₂×C₄	C₂²×D₅₉₉
Fitting	C₄⋊Q₈×C₅₉₉	C₂
socle	C₂²×C₅₉₉	C₂×D₄
prime radical	C₂×D₂₃₉₆	C₂²
hypercenter	C₄⋊Q₈	D₅₉₉
hypocenter	C₅₉₉	#489
LCS	G ▷ C₂×C₄×C₅₉₉ ▷ C₂×C₅₉₉ ▷ C₅₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #34062638	C₂	semidirect
4. #34062626	C₂	semidirect
5. C₄⋊Q₈×C₅₉₉	C₂	semidirect
38. C₅₉₉	#489	semidirect
40. C₄⋊Q₈	D₅₉₉	semidirect

Maximal subgroups

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

subgroup	index
#34062703	2
#34062638	2
#34062635	2
#34062626	2
C₄⋊Q₈×C₅₉₉	2
#489	599