Subgroups — C₂×#20263962 #27164781

Order 12736.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20263962	C₂	1	8	yes	maximal, direct product factor
2	C₂×#15691870	C₂	1	3	yes	maximal, complemented
3	C₂²×C₁₉₉⋊C₈	C₂	1	3	yes	maximal, complemented
4	C₂×#41×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	#15691870	C₂²	1	12	yes	complemented
6	C₂×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₉₉⋊C₈	C₂²	1	3	yes	—
8	C₂×C₁₉₉⋊C₈	C₂²	1	12	yes	complemented
9	#41×C₁₉₉	C₄	1	4	yes	—
10	#41×C₁₉₉	C₂²	1	4	yes	—
11	C₂×Q₈×C₁₉₉	C₄	1	1	yes	characteristic
12	C₂×D₄×C₁₉₉	C₄	1	3	yes	—
13	C₂²×C₄×C₁₉₉	C₂²	1	3	yes	—
14	C₁₉₉⋊C₈	C₂³	1	2	yes	—
15	C₁₉₉⋊C₈	C₂³	1	6	yes	—
16	C₂³×C₁₉₉	C₂×C₄	1	3	yes	—
17	Q₈×C₁₉₉	C₂×C₄	1	4	yes	—
18	D₄×C₁₉₉	C₂×C₄	1	12	yes	—
19	C₂×C₄×C₁₉₉	C₂×C₄	1	3	yes	—
20	C₂×C₄×C₁₉₉	C₂×C₄	1	6	yes	—
21	C₂×C₄×C₁₉₉	C₂³	1	6	yes	—
22	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
23	C₂²×C₁₉₉	—	2	6	no	—
24	C₂²×C₁₉₉	C₂²×C₄	1	6	yes	—
25	C₂²×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₁₉₉	C₂²×C₄	1	6	yes	—
28	C₄×C₁₉₉	C₂⁴	1	1	yes	characteristic
29	C₂×C₁₉₉	—	2	6	no	—
30	C₂×C₁₉₉	#131	1	2	yes	complemented
31	C₂×C₁₉₉	C₂³×C₄	1	1	yes	characteristic
32	C₁₉₉	C₂×#131	1	1	yes	characteristic, complemented, Sylow
33	C₂×#131	—	199	1	no	maximal, Sylow
34	C₂×#41	D₁₉₉	1	1	yes	characteristic
35	#131	—	199	8	no	—
36	C₂×M₁₆	—	199	3	no	—
37	C₂²×C₈	—	199	3	no	—
38	#41	Dic₁₉₉	1	4	yes	—
39	#41	C₂×D₁₉₉	1	4	yes	—
40	C₂×Q₈	Dic₁₉₉	1	1	yes	characteristic
41	C₂×D₄	Dic₁₉₉	1	3	yes	—
42	C₂²×C₄	C₂×D₁₉₉	1	3	yes	—
43	M₁₆	—	199	12	no	—
44	C₂×C₈	—	199	12	no	—
45	C₂×C₈	—	199	1	no	—
46	C₂×C₈	—	199	3	no	—
47	C₂³	C₂×Dic₁₉₉	1	3	yes	—
48	Q₈	C₂×Dic₁₉₉	1	4	yes	—
49	D₄	C₂×Dic₁₉₉	1	12	yes	—
50	C₂×C₄	C₂×Dic₁₉₉	1	3	yes	—
51	C₂×C₄	C₂×Dic₁₉₉	1	6	yes	—
52	C₂×C₄	C₂²×D₁₉₉	1	6	yes	—
53	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
54	C₈	—	199	2	no	—
55	C₈	—	199	6	no	—
56	C₂²	—	2	6	no	—
57	C₂²	C₂²×Dic₁₉₉	1	6	yes	—
58	C₂²	C₂²×Dic₁₉₉	1	1	yes	characteristic
59	C₄	C₂²×Dic₁₉₉	1	1	yes	characteristic
60	C₄	C₂²×Dic₁₉₉	1	6	yes	—
61	C₄	C₂³×D₁₉₉	1	1	yes	characteristic
62	C₂	—	2	6	no	—
63	C₂	#20263962	1	2	yes	direct product factor
64	C₂	C₂³×Dic₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₉₉
commutator	C₂×C₁₉₉	C₂³×C₄
Frattini	C₄	C₂³×D₁₉₉
Fitting, prime radical	C₂×#41×C₁₉₉	C₂
socle	C₂²×C₁₉₉	C₂²×C₄
cosocle	C₄×C₁₉₉	C₂⁴
hypercenter	C₂×#41	D₁₉₉
hypocenter	C₁₉₉	C₂×#131
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #20263962	C₂	direct (N × H)
2. C₂×#15691870	C₂	semidirect
3. C₂²×C₁₉₉⋊C₈	C₂	semidirect
5. #15691870	C₂²	semidirect
8. C₂×C₁₉₉⋊C₈	C₂²	semidirect
30. C₂×C₁₉₉	#131	semidirect
32. C₁₉₉	C₂×#131	semidirect
63. C₂	#20263962	direct (N × H)

Maximal subgroups

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

subgroup	index
#20263962	2
C₂×#15691870	2
C₂²×C₁₉₉⋊C₈	2
C₂×#41×C₁₉₉	2
C₂×#131	199

Subgroups — C2×#20263962 #27164781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20263962 #27164781