Subgroups — #38767027

Order 25472.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₁₈₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₉₉⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#366×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₅₉₂	—	2	1	no	—
5	D₃₁₈₄	—	2	1	no	—
6	D₃₁₈₄	C₄	1	2	yes	complemented
7	C₁₉₉⋊C₃₂	—	2	1	no	—
8	C₂×C₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
9	#107×C₁₉₉	—	2	1	no	—
10	C₂×D₇₉₆	—	4	1	no	—
11	D₁₅₉₂	—	2	2	no	—
12	D₁₅₉₂	—	4	1	no	—
13	C₂×C₈×C₁₉₉	D₄	1	1	yes	characteristic
14	C₄⋊C₄×C₁₉₉	—	4	1	no	—
15	C₁₆×C₁₉₉	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₉₉	D₄	1	1	yes	characteristic
17	C₂²×D₁₉₉	—	8	1	no	—
18	D₇₉₆	—	4	2	no	—
19	D₇₉₆	—	8	1	no	—
20	C₂×C₄×C₁₉₉	—	8	1	no	—
21	C₂×C₄×C₁₉₉	D₈	1	1	yes	characteristic
22	C₈×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₁₉₉	SD₁₆	1	1	yes	characteristic
24	C₂²×C₁₉₉	D₁₆	1	1	yes	characteristic
25	C₂×D₁₉₉	—	8	2	no	—
26	C₂×D₁₉₉	—	16	1	no	—
27	C₂×D₁₉₉	—	16	1	no	—
28	C₄×C₁₉₉	—	16	1	no	—
29	C₄×C₁₉₉	#102	1	1	yes	characteristic
30	C₄×C₁₉₉	SD₃₂	1	1	yes	characteristic
31	C₂×C₁₉₉	#357	1	1	yes	characteristic
32	C₂×C₁₉₉	D₃₂	1	1	yes	characteristic
33	C₂×C₁₉₉	SD₆₄	1	1	yes	characteristic
34	D₁₉₉	—	16	2	no	—
35	C₁₉₉	#1415	1	1	yes	characteristic, complemented, Sylow
36	#1415	—	199	1	no	maximal, Sylow
37	C₂×D₁₆	—	199	1	no	—
38	C₂×C₃₂	—	199	1	no	—
39	#366	D₁₉₉	1	1	yes	characteristic, complemented
40	C₂×D₈	—	398	1	no	—
41	D₁₆	—	398	1	no	—
42	D₁₆	—	199	2	no	—
43	C₂×C₁₆	C₂×D₁₉₉	1	1	yes	characteristic
44	#107	—	2	1	no	—
45	C₃₂	—	398	1	no	—
46	C₂×D₄	—	796	1	no	—
47	D₈	—	796	1	no	—
48	D₈	—	398	2	no	—
49	C₂×C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
50	C₄⋊C₄	—	4	1	no	—
51	C₁₆	C₄×D₁₉₉	1	1	yes	characteristic
52	C₁₆	D₇₉₆	1	1	yes	characteristic
53	C₂³	—	1592	1	no	—
54	D₄	—	796	2	no	—
55	D₄	—	1592	1	no	—
56	C₂×C₄	—	8	1	no	—
57	C₂×C₄	#15691871	1	1	yes	characteristic
58	C₈	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
59	C₈	#15691873	1	1	yes	characteristic
60	C₂²	—	1592	2	no	—
61	C₂²	—	3184	1	no	—
62	C₂²	—	3184	1	no	—
63	C₂²	#20263896	1	1	yes	characteristic
64	C₄	—	16	1	no	—
65	C₄	#20263875	1	1	yes	characteristic
66	C₄	#20263898	1	1	yes	characteristic
67	C₂	—	3184	2	no	—
68	C₂	#27163818	1	1	yes	characteristic
69	C₂	#27163853	1	1	yes	characteristic
70	C₂	#27163855	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#20263896
commutator	C₁₆×C₁₉₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₉₉
Fitting	#366×C₁₉₉	C₂
socle	C₂²×C₁₉₉	D₁₆
prime radical	C₂×D₃₁₈₄	C₂
cosocle	C₂×C₁₆×C₁₉₉	C₂²
hypercenter	#366	D₁₉₉
hypocenter	C₁₉₉	#1415
LCS	G ▷ C₁₆×C₁₉₉ ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₉₉⋊C₃₂	C₂	semidirect
3. #366×C₁₉₉	C₂	semidirect
6. D₃₁₈₄	C₄	semidirect
35. C₁₉₉	#1415	semidirect
39. #366	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₃₁₈₄	2
C₂×C₁₉₉⋊C₃₂	2
#366×C₁₉₉	2
#1415	199