Subgroups — #38767038

Order 25472.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27164450	C₂	1	1	yes	maximal, characteristic
2	#27163846	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#20263977	—	2	1	no	—
5	Dic₁₅₉₂	C₄	1	1	yes	characteristic, complemented
6	#20263900	—	2	1	no	—
7	D₃₁₈₄	C₄	1	1	yes	characteristic, complemented
8	#20263889	—	2	1	no	—
9	C₂×C₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₉₉	—	2	1	no	—
11	#15691886	—	4	1	no	—
12	Dic₇₉₆	—	2	1	no	—
13	#15691875	—	4	1	no	—
14	D₁₅₉₂	—	2	1	no	—
15	C₄⋊Dic₁₉₉	—	4	1	no	—
16	C₂×C₈×C₁₉₉	D₄	1	1	yes	characteristic
17	C₁₆×C₁₉₉	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₉₉	D₄	1	1	yes	characteristic
19	Dic₃₉₈	—	4	1	no	—
20	D₇₉₆	—	4	1	no	—
21	C₂²⋊D₁₉₉	—	8	1	no	—
22	C₂×Dic₁₉₉	—	8	1	no	—
23	C₄×D₁₉₉	—	8	1	no	—
24	C₂×C₄×C₁₉₉	D₈	1	1	yes	characteristic
25	C₈×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₉₉	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₉₉	D₁₆	1	1	yes	characteristic
28	C₂×D₁₉₉	—	8	1	no	—
29	C₄×C₁₉₉	#102	1	1	yes	characteristic
30	C₄×C₁₉₉	SD₃₂	1	1	yes	characteristic
31	Dic₁₉₉	—	8	1	no	—
32	Dic₁₉₉	—	16	1	no	—
33	C₂×C₁₉₉	—	2	1	no	—
34	C₂×C₁₉₉	#357	1	1	yes	characteristic
35	D₁₉₉	—	16	1	no	—
36	C₁₉₉	#1418	1	1	yes	characteristic, complemented, Sylow
37	#1418	—	199	1	no	maximal, Sylow
38	#508	—	199	1	no	—
39	M₆₄	D₁₉₉	1	1	yes	characteristic, complemented
40	#367	—	199	1	no	—
41	#135	—	398	1	no	—
42	Q₃₂	—	199	1	no	—
43	SD₃₂	—	398	1	no	—
44	D₁₆	—	199	1	no	—
45	C₂×C₁₆	C₂×D₁₉₉	1	1	yes	characteristic
46	#107	—	398	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	796	1	no	—
49	Q₁₆	—	398	1	no	—
50	SD₁₆	—	796	1	no	—
51	D₈	—	398	1	no	—
52	C₂×C₈	D₇₉₆	1	1	yes	characteristic
53	C₄⋊C₄	—	796	1	no	—
54	C₁₆	C₄×D₁₉₉	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₁₉₉	1	1	yes	characteristic
56	Q₈	—	796	1	no	—
57	D₄	—	796	1	no	—
58	D₄	—	1592	1	no	—
59	C₂×C₄	—	1592	1	no	—
60	C₂×C₄	—	1592	1	no	—
61	C₂×C₄	D₁₅₉₂	1	1	yes	characteristic
62	C₈	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
63	C₈	#15691875	1	1	yes	characteristic
64	C₂²	—	1592	1	no	—
65	C₂²	D₃₁₈₄	1	1	yes	characteristic
66	C₄	—	1592	1	no	—
67	C₄	—	3184	1	no	—
68	C₄	#20263877	1	1	yes	characteristic
69	C₄	#20263900	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	3184	1	no	—
72	C₂	#27163820	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#27163820
commutator	C₁₆×C₁₉₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₉₉
Fitting	M₆₄×C₁₉₉	C₂
socle	C₂×C₁₉₉	#357
prime radical	#27164450	C₂
cosocle	C₂×C₁₆×C₁₉₉	C₂²
hypercenter	M₆₄	D₁₉₉
hypocenter	C₁₉₉	#1418
LCS	G ▷ C₁₆×C₁₉₉ ▷ C₈×C₁₉₉ ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #27163846	C₂	semidirect
3. M₆₄×C₁₉₉	C₂	semidirect
5. Dic₁₅₉₂	C₄	semidirect
7. D₃₁₈₄	C₄	semidirect
36. C₁₉₉	#1418	semidirect
39. M₆₄	D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
#27164450	2
#27163846	2
M₆₄×C₁₉₉	2
#1418	199