Subgroups — #27163773

Order 12736.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉₉⋊C₈	C₂	1	2	yes	maximal
2	C₂×C₄²×C₁₉₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₉₉⋊C₈	—	2	2	no	—
4	C₂×C₁₉₉⋊C₈	—	2	2	no	—
5	C₂×C₁₉₉⋊C₈	C₄	1	4	yes	complemented
6	C₂²×C₄×C₁₉₉	C₄	1	2	yes	—
7	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
8	C₄²×C₁₉₉	—	2	2	no	—
9	C₁₉₉⋊C₈	—	2	4	no	—
10	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
11	C₂×C₄×C₁₉₉	—	2	2	no	—
12	C₂×C₄×C₁₉₉	—	2	2	no	—
13	C₂×C₄×C₁₉₉	C₈	1	4	yes	complemented
14	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₁₉₉	D₄	1	2	yes	—
16	C₂×C₄×C₁₉₉	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₉₉	Q₈	1	1	yes	characteristic
18	C₂²×C₁₉₉	C₄²	1	1	yes	characteristic
19	C₂²×C₁₉₉	C₂²⋊C₄	1	1	yes	characteristic
20	C₂²×C₁₉₉	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₁₉₉	C₂×C₈	1	2	yes	—
22	C₂²×C₁₉₉	M₁₆	1	2	yes	—
23	C₄×C₁₉₉	—	2	4	no	—
24	C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
25	C₄×C₁₉₉	C₄⋊C₄	1	2	yes	—
26	C₂×C₁₉₉	#95	1	1	yes	characteristic
27	C₂×C₁₉₉	C₄×C₈	1	1	yes	characteristic
28	C₂×C₁₉₉	#97	1	1	yes	characteristic
29	C₂×C₁₉₉	C₂²⋊C₈	1	2	yes	—
30	C₂×C₁₉₉	C₄⋊C₈	1	2	yes	—
31	C₁₉₉	#336	1	1	yes	characteristic, complemented, Sylow
32	#336	—	199	1	no	maximal, Sylow
33	C₂²×C₈	—	199	2	no	—
34	C₂×C₄²	D₁₉₉	1	1	yes	characteristic
35	C₂²×C₄	Dic₁₉₉	1	2	yes	—
36	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
37	C₂×C₈	—	398	2	no	—
38	C₂×C₈	—	398	2	no	—
39	C₂×C₈	—	199	4	no	—
40	C₄²	—	2	2	no	—
41	C₂³	C₂×Dic₁₉₉	1	1	yes	characteristic
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	C₁₉₉⋊C₈	1	4	yes	complemented
45	C₂×C₄	C₄×D₁₉₉	1	2	yes	—
46	C₂×C₄	C₂²⋊D₁₉₉	1	2	yes	—
47	C₂×C₄	D₇₉₆	1	1	yes	characteristic
48	C₂×C₄	Dic₃₉₈	1	1	yes	characteristic
49	C₈	—	398	4	no	—
50	C₂²	C₄×Dic₁₉₉	1	1	yes	characteristic
51	C₂²	C₂²⋊Dic₁₉₉	1	1	yes	characteristic
52	C₂²	C₄⋊Dic₁₉₉	1	1	yes	characteristic
53	C₂²	C₂×C₁₉₉⋊C₈	1	2	yes	—
54	C₂²	#15691870	1	2	yes	—
55	C₄	—	2	4	no	—
56	C₄	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
57	C₄	Dic₁₉₉⋊C₄	1	2	yes	—
58	C₂	C₁₉₉⋊#95	1	1	yes	characteristic
59	C₂	C₄×C₁₉₉⋊C₈	1	1	yes	characteristic
60	C₂	#20263866	1	1	yes	characteristic
61	C₂	C₂²⋊(C₁₉₉⋊C₈)	1	2	yes	—
62	C₂	C₄⋊(C₁₉₉⋊C₈)	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₁₉₉
commutator	C₂×C₁₉₉	C₄×C₈
Fitting	C₂×C₄²×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	C₂×C₄
cosocle	C₂²×C₄×C₁₉₉	C₂²
hypercenter	C₂×C₄²	D₁₉₉
hypocenter	C₁₉₉	#336
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
5. C₂×C₁₉₉⋊C₈	C₄	semidirect
13. C₂×C₄×C₁₉₉	C₈	semidirect
31. C₁₉₉	#336	semidirect
44. C₂×C₄	C₁₉₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₉₉⋊C₈	2
C₂×C₄²×C₁₉₉	2
#336	199