Subgroups — C₂²×#15691870 #27164778

Order 12736.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₉₉
commutator	C₂×C₁₉₉	C₂³×C₄
Frattini	C₄	C₂³×D₁₉₉
Fitting	C₂³×C₄×C₁₉₉	C₂
socle	C₂³×C₁₉₉	C₂×C₄
prime radical	C₂⁴×C₁₉₉	C₄
cosocle	C₄×C₁₉₉	C₂⁴
hypercenter	C₂³×C₄	D₁₉₉
hypocenter	C₁₉₉	C₂²×M₁₆
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#15691870	C₂	direct (N × H)
2. C₂²×C₁₉₉⋊C₈	C₂	semidirect
4. #15691870	C₂²	direct (N × H)
5. C₂×C₁₉₉⋊C₈	C₂²	semidirect
11. C₁₉₉⋊C₈	C₂³	semidirect
21. C₂²×C₁₉₉	M₁₆	semidirect
28. C₂×C₁₉₉	C₂×M₁₆	semidirect
30. C₁₉₉	C₂²×M₁₆	semidirect
52. C₂²	#15691870	direct (N × H)
59. C₂	C₂×#15691870	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#15691870	2
C₂²×C₁₉₉⋊C₈	2
C₂³×C₄×C₁₉₉	2
C₂²×M₁₆	199

Subgroups — C22×#15691870 #27164778

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15691870 #27164778