Subgroups — C₂×#20263865 #27163966

Order 12736.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₉₉	C₂	1	1	yes	maximal, characteristic
3	#20263865	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₉₉	C₄	1	2	yes	—
6	C₂×C₁₉₉⋊C₈	C₄	1	4	yes	complemented
7	C₂×C₁₉₉⋊C₈	C₂²	1	2	yes	—
8	C₄×Dic₁₉₉	C₄	1	2	yes	—
9	C₄×Dic₁₉₉	C₂²	1	2	yes	—
10	C₂²×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₉₉	C₄	1	4	yes	complemented
12	C₂×C₈×C₁₉₉	C₂²	1	2	yes	—
13	C₂×Dic₁₉₉	—	2	4	no	—
14	C₂×Dic₁₉₉	C₂×C₄	1	4	yes	—
15	C₁₉₉⋊C₈	C₂×C₄	1	4	yes	complemented
16	C₂³×C₁₉₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
19	C₂×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
21	C₈×C₁₉₉	C₂×C₄	1	4	yes	complemented
22	C₂²×C₁₉₉	C₄²	1	2	yes	—
23	C₂²×C₁₉₉	M₁₆	1	4	yes	—
24	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
25	C₄×C₁₉₉	C₄²	1	2	yes	—
26	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
28	Dic₁₉₉	—	2	4	no	—
29	C₂×C₁₉₉	#97	1	4	yes	complemented
30	C₂×C₁₉₉	C₂×C₄²	1	1	yes	characteristic
31	C₂×C₁₉₉	C₂×M₁₆	1	2	yes	—
32	C₁₉₉	C₂×#97	1	1	yes	characteristic, complemented, Sylow
33	C₂×#97	—	199	1	no	maximal, Sylow
34	C₂²×C₈	—	199	1	no	—
35	C₂²×C₈	D₁₉₉	1	1	yes	characteristic
36	C₂×C₄²	—	199	1	no	—
37	#97	—	199	4	no	—
38	C₂²×C₄	—	199	2	no	—
39	C₂²×C₄	C₂×D₁₉₉	1	1	yes	characteristic
40	C₂×C₈	—	199	4	no	—
41	C₂×C₈	—	199	2	no	—
42	C₂×C₈	Dic₁₉₉	1	4	yes	complemented
43	C₂×C₈	C₂×D₁₉₉	1	2	yes	—
44	C₄²	—	199	2	no	—
45	C₄²	—	199	2	no	—
46	C₂³	C₄×D₁₉₉	1	1	yes	characteristic
47	C₂×C₄	—	398	4	no	—
48	C₂×C₄	—	199	4	no	—
49	C₂×C₄	C₄×D₁₉₉	1	1	yes	characteristic
50	C₂×C₄	C₄×D₁₉₉	1	2	yes	—
51	C₂×C₄	C₂×Dic₁₉₉	1	2	yes	—
52	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
53	C₈	—	199	4	no	—
54	C₈	C₂×Dic₁₉₉	1	4	yes	complemented
55	C₂²	C₄×Dic₁₉₉	1	2	yes	—
56	C₂²	#15691869	1	4	yes	—
57	C₂²	C₂×C₄×D₁₉₉	1	1	yes	characteristic
58	C₄	—	398	4	no	—
59	C₄	C₄×Dic₁₉₉	1	2	yes	—
60	C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
61	C₄	C₂²×Dic₁₉₉	1	1	yes	characteristic
62	C₂	#20263865	1	4	yes	direct product factor
63	C₂	C₂×C₄×Dic₁₉₉	1	1	yes	characteristic
64	C₂	C₂×#15691869	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₉₉
commutator	C₂×C₁₉₉	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₁₉₉
Fitting	C₂²×C₈×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	C₂×C₄
cosocle	C₂×C₄×C₁₉₉	C₂³
hypercenter	C₂²×C₈	D₁₉₉
hypocenter	C₁₉₉	C₂×#97
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #20263865	C₂	direct (N × H)
6. C₂×C₁₉₉⋊C₈	C₄	semidirect
11. C₂×C₈×C₁₉₉	C₄	semidirect
15. C₁₉₉⋊C₈	C₂×C₄	semidirect
21. C₈×C₁₉₉	C₂×C₄	semidirect
29. C₂×C₁₉₉	#97	semidirect
32. C₁₉₉	C₂×#97	semidirect
42. C₂×C₈	Dic₁₉₉	semidirect
54. C₈	C₂×Dic₁₉₉	semidirect
62. C₂	#20263865	direct (N × H)

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×C₄×Dic₁₉₉	2
#20263865	2
C₂²×C₈×C₁₉₉	2
C₂×#97	199

Subgroups — C2×#20263865 #27163966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20263865 #27163966