Subgroups — #38767041

Order 25472.

78 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	#27163848	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×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	—
7	#20263891	—	2	1	no	—
8	C₂×C₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
9	C₃₂×C₁₉₉	—	2	1	no	—
10	C₂×D₇₉₆	—	4	1	no	—
11	D₁₅₉₂	—	2	2	no	—
12	D₁₅₉₂	—	4	1	no	—
13	#15691870	—	4	1	no	—
14	C₂×C₈×C₁₉₉	D₄	1	1	yes	characteristic
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	C₁₉₉⋊C₈	—	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	SD₃₂	1	1	yes	characteristic
25	C₂×D₁₉₉	—	8	2	no	—
26	C₂×D₁₉₉	—	16	2	no	—
27	C₄×C₁₉₉	#102	1	1	yes	characteristic
28	C₄×C₁₉₉	D₁₆	1	1	yes	characteristic
29	C₂×C₁₉₉	—	2	1	no	—
30	C₂×C₁₉₉	#357	1	1	yes	characteristic
31	D₁₉₉	—	16	2	no	—
32	C₁₉₉	#1419	1	1	yes	characteristic, complemented, Sylow
33	#1419	—	199	1	no	maximal, Sylow
34	C₂×D₁₆	—	199	1	no	—
35	M₆₄	D₁₉₉	1	1	yes	characteristic, complemented
36	#368	—	199	1	no	—
37	C₂×D₈	—	398	1	no	—
38	D₁₆	—	398	1	no	—
39	D₁₆	—	199	2	no	—
40	C₂×C₁₆	C₂×D₁₉₉	1	1	yes	characteristic
41	#108	—	398	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	796	1	no	—
44	D₈	—	796	1	no	—
45	D₈	—	398	2	no	—
46	M₁₆	—	796	1	no	—
47	C₂×C₈	D₇₉₆	1	1	yes	characteristic
48	C₁₆	C₄×D₁₉₉	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₁₉₉	1	1	yes	characteristic
50	C₂³	—	1592	1	no	—
51	D₄	—	796	2	no	—
52	D₄	—	1592	1	no	—
53	C₂×C₄	D₁₅₉₂	1	1	yes	characteristic
54	C₈	—	1592	1	no	—
55	C₈	(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
56	C₈	#15691875	1	1	yes	characteristic
57	C₂²	—	1592	2	no	—
58	C₂²	—	3184	2	no	—
59	C₂²	#20263900	1	1	yes	characteristic
60	C₄	#20263877	1	1	yes	characteristic
61	C₄	D₃₁₈₄	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	3184	2	no	—
64	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	C₂×D₃₁₈₄	C₂
cosocle	C₂×C₁₆×C₁₉₉	C₂²
hypercenter	M₆₄	D₁₉₉
hypocenter	C₁₉₉	#1419
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. #27163848	C₂	semidirect
3. M₆₄×C₁₉₉	C₂	semidirect
32. C₁₉₉	#1419	semidirect
35. M₆₄	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
#27163848	2
M₆₄×C₁₉₉	2
#1419	199