Subgroups — #27164142

Order 12736.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₃₉₈	C₂	1	1	yes	maximal, characteristic
2	Dic₁₉₉⋊C₈	C₂	1	2	yes	maximal
3	C₄×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₉₉	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₉₈	C₄	1	1	yes	characteristic
7	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₉₉	C₄	1	1	yes	characteristic
10	Dic₁₉₉⋊C₄	C₄	1	2	yes	—
11	C₄×Dic₁₉₉	C₂²	1	2	yes	—
12	C₂×C₈×C₁₉₉	C₂²	1	2	yes	—
13	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
14	Dic₃₉₈	C₈	1	4	yes	complemented
15	C₂×Dic₁₉₉	C₂×C₄	1	2	yes	—
16	C₂×Dic₁₉₉	C₂×C₄	1	2	yes	—
17	C₁₉₉⋊C₈	—	2	1	no	—
18	C₁₉₉⋊C₈	Q₈	1	2	yes	complemented
19	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
22	C₈×C₁₉₉	—	2	2	no	—
23	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₁₉₉	—	2	1	no	—
25	C₄×C₁₉₉	C₂×C₈	1	2	yes	—
26	C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
27	C₄×C₁₉₉	#41	1	1	yes	characteristic
28	Dic₁₉₉	—	2	2	no	—
29	Dic₁₉₉	C₂×C₈	1	4	yes	—
30	C₂×C₁₉₉	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₁₉₉	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₁₉₉	#131	1	1	yes	characteristic
33	C₁₉₉	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₈×Q₈	—	199	1	no	maximal, Sylow
35	C₄×Q₈	—	199	1	no	—
36	C₄⋊C₈	—	199	2	no	—
37	C₄⋊C₈	D₁₉₉	1	1	yes	characteristic
38	C₄×C₈	—	199	2	no	—
39	C₄×C₈	—	199	1	no	—
40	C₂×Q₈	—	199	1	no	—
41	C₂×C₈	—	199	1	no	—
42	C₂×C₈	—	199	1	no	—
43	C₂×C₈	C₂×D₁₉₉	1	2	yes	—
44	C₄⋊C₄	—	199	2	no	—
45	C₄⋊C₄	—	199	1	no	—
46	C₄²	—	199	2	no	—
47	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
48	Q₈	—	199	4	no	—
49	C₂×C₄	—	199	2	no	—
50	C₂×C₄	—	199	2	no	—
51	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
52	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	398	1	no	—
56	C₈	—	199	2	no	—
57	C₂²	C₂×C₄×D₁₉₉	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	398	2	no	—
60	C₄	—	199	4	no	—
61	C₄	C₈×D₁₉₉	1	2	yes	—
62	C₄	Q₈×D₁₉₉	1	1	yes	characteristic
63	C₄	#15691885	1	1	yes	characteristic
64	C₂	#20263920	1	1	yes	characteristic
65	C₂	C₂×C₈×D₁₉₉	1	1	yes	characteristic
66	C₂	#20263963	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₉₉
commutator	C₂×C₁₉₉	C₂²×C₈
Fitting	C₄⋊C₈×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	C₂²×C₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	C₄⋊C₈	D₁₉₉
hypocenter	C₁₉₉	C₈×Q₈
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
14. Dic₃₉₈	C₈	semidirect
18. C₁₉₉⋊C₈	Q₈	semidirect
33. C₁₉₉	C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₃₉₈	2
Dic₁₉₉⋊C₈	2
C₄×C₁₉₉⋊C₈	2
C₈×Dic₁₉₉	2
C₄⋊C₈×C₁₉₉	2
C₈×Q₈	199