Subgroups — #27163781

Order 12736.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₁₉₉	C₂	1	2	yes	maximal
2	C₂²×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₁₉₉	—	2	2	no	—
4	C₄⋊Dic₁₉₉	—	2	2	no	—
5	C₄⋊Dic₁₉₉	C₄	1	4	yes	complemented
6	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₁₉₉	—	2	2	no	—
8	C₂×C₈×C₁₉₉	C₄	1	2	yes	complemented
9	C₂×Dic₁₉₉	—	2	4	no	—
10	C₂×Dic₁₉₉	—	4	2	no	—
11	C₂×Dic₁₉₉	—	4	2	no	—
12	C₂³×C₁₉₉	D₄	1	1	yes	characteristic
13	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
16	C₂×C₄×C₁₉₉	Q₈	1	1	yes	characteristic
17	C₈×C₁₉₉	—	2	2	no	—
18	C₂²×C₁₉₉	C₂²⋊C₄	1	2	yes	—
19	C₂²×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₁₉₉	D₈	1	1	yes	characteristic
21	C₂²×C₁₉₉	SD₁₆	1	2	yes	—
22	C₂²×C₁₉₉	Q₁₆	1	1	yes	characteristic
23	C₄×C₁₉₉	C₄²	1	1	yes	characteristic
24	C₄×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
26	Dic₁₉₉	—	4	4	no	—
27	C₂×C₁₉₉	#95	1	1	yes	characteristic
28	C₂×C₁₉₉	#102	1	2	yes	—
29	C₂×C₁₉₉	#103	1	2	yes	—
30	C₂×C₁₉₉	#106	1	1	yes	characteristic
31	C₂×C₁₉₉	#107	1	1	yes	characteristic
32	C₁₉₉	#340	1	1	yes	characteristic, complemented, Sylow
33	#340	—	199	1	no	maximal, Sylow
34	C₂²×C₈	D₁₉₉	1	1	yes	characteristic
35	C₂×C₄⋊C₄	—	199	2	no	—
36	C₂²×C₄	—	398	2	no	—
37	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	Dic₁₉₉	1	2	yes	complemented
40	C₄⋊C₄	—	398	2	no	—
41	C₄⋊C₄	—	199	4	no	—
42	C₂³	D₇₉₆	1	1	yes	characteristic
43	C₂×C₄	—	796	2	no	—
44	C₂×C₄	—	796	2	no	—
45	C₂×C₄	—	398	4	no	—
46	C₂×C₄	C₄×D₁₉₉	1	2	yes	—
47	C₂×C₄	C₂×Dic₁₉₉	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
49	C₂×C₄	Dic₃₉₈	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₂²	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
52	C₂²	C₄⋊Dic₁₉₉	1	1	yes	characteristic
53	C₂²	D₁₅₉₂	1	1	yes	characteristic
54	C₂²	#15691875	1	2	yes	—
55	C₂²	Dic₇₉₆	1	1	yes	characteristic
56	C₄	—	796	4	no	—
57	C₄	C₄×Dic₁₉₉	1	1	yes	characteristic
58	C₄	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
59	C₄	Dic₁₉₉⋊C₄	1	2	yes	—
60	C₂	C₁₉₉⋊#95	1	1	yes	characteristic
61	C₂	#20263877	1	2	yes	—
62	C₂	#20263880	1	2	yes	—
63	C₂	#20263887	1	1	yes	characteristic
64	C₂	#20263889	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₇₉₆
commutator	C₄×C₁₉₉	C₄²
Frattini	C₂²×C₄	C₂×D₁₉₉
Fitting	C₂²×C₈×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	D₄
cosocle	C₂²×C₄×C₁₉₉	C₂²
hypercenter	C₂²×C₈	D₁₉₉
hypocenter	C₁₉₉	#340
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
5. C₄⋊Dic₁₉₉	C₄	semidirect
8. C₂×C₈×C₁₉₉	C₄	semidirect
32. C₁₉₉	#340	semidirect
39. C₂×C₈	Dic₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊Dic₁₉₉	2
C₂²×C₈×C₁₉₉	2
#340	199