Subgroups — #27164125

Order 12736.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₁₉₉	C₂	1	2	yes	maximal
2	#20263887	C₂	1	1	yes	maximal, characteristic
3	#20263879	C₂	1	2	yes	maximal
4	#20263865	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₁₆×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₁₉₉	—	2	2	no	—
8	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
9	C₄×Dic₁₉₉	—	2	2	no	—
10	C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₁₉₉	C₂²	1	2	yes	—
12	Q₁₆×C₁₉₉	C₄	1	4	yes	complemented
13	C₂×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₉₉	—	2	2	no	—
15	C₂×Dic₁₉₉	D₄	1	2	yes	—
16	C₁₉₉⋊C₈	—	2	1	no	—
17	Q₈×C₁₉₉	—	2	2	no	—
18	Q₈×C₁₉₉	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₁₉₉	—	2	2	no	—
20	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
21	C₈×C₁₉₉	C₂×C₄	1	2	yes	—
22	C₂²×C₁₉₉	C₂×D₄	1	1	yes	characteristic
23	C₄×C₁₉₉	—	2	4	no	—
24	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₁₉₉	#41	1	1	yes	characteristic
26	Dic₁₉₉	—	2	2	no	—
27	Dic₁₉₉	—	4	2	no	—
28	C₂×C₁₉₉	C₄×D₄	1	1	yes	characteristic
29	C₂×C₁₉₉	#137	1	2	yes	—
30	C₁₉₉	#441	1	1	yes	characteristic, complemented, Sylow
31	#441	—	199	1	no	maximal, Sylow
32	C₂×Q₁₆	D₁₉₉	1	1	yes	characteristic
33	C₄×Q₈	—	199	2	no	—
34	#106	—	199	1	no	—
35	#103	—	199	2	no	—
36	#97	—	199	1	no	—
37	C₂×Q₈	C₂×D₁₉₉	1	2	yes	—
38	Q₁₆	Dic₁₉₉	1	4	yes	complemented
39	C₂×C₈	—	199	1	no	—
40	C₂×C₈	C₂×D₁₉₉	1	1	yes	characteristic
41	C₄⋊C₄	—	398	2	no	—
42	C₄⋊C₄	—	199	2	no	—
43	C₄²	—	398	2	no	—
44	C₄²	—	199	1	no	—
45	Q₈	—	2	2	no	—
46	Q₈	C₂×Dic₁₉₉	1	4	yes	—
47	C₂×C₄	—	2	2	no	—
48	C₂×C₄	—	398	2	no	—
49	C₂×C₄	—	199	2	no	—
50	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
51	C₈	—	398	1	no	—
52	C₈	C₂×Dic₁₉₉	1	2	yes	—
53	C₂²	D₄×D₁₉₉	1	1	yes	characteristic
54	C₄	—	2	4	no	—
55	C₄	—	796	2	no	—
56	C₄	—	398	2	no	—
57	C₄	C₂²×Dic₁₉₉	1	1	yes	characteristic
58	C₄	#15691885	1	1	yes	characteristic
59	C₂	D₄×Dic₁₉₉	1	1	yes	characteristic
60	C₂	#20263987	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₉₉
commutator	C₄×C₁₉₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₂×Q₁₆×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂×D₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	C₂×Q₁₆	D₁₉₉
hypocenter	C₁₉₉	#441
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
12. Q₁₆×C₁₉₉	C₄	semidirect
30. C₁₉₉	#441	semidirect
38. Q₁₆	Dic₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₁₉₉	2
#20263887	2
#20263879	2
#20263865	2
C₂×Q₁₆×C₁₉₉	2
#441	199