Subgroups — C₂²×#15691887 #27164846

Order 12736.

890 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15691887	C₂	1	24	yes	maximal, direct product factor
2	C₂²×D₇₉₆	C₂	1	3	yes	maximal, complemented
3	C₂²×C₄×D₁₉₉	C₂	1	3	yes	maximal, complemented
4	C₂²×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₁₉₉	C₂²	1	3	yes	—
6	#15691887	C₂²	1	64	yes	direct product factor
7	C₂×D₇₉₆	C₂²	1	36	yes	complemented
8	C₂²×Dic₁₉₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₉₉	C₂²	1	36	yes	complemented
10	C₂×Q₈×C₁₉₉	C₂²	1	12	yes	complemented
11	C₂²×C₄×C₁₉₉	C₂²	1	3	yes	—
12	C₂²×D₁₉₉	—	2	12	no	—
13	C₂²×D₁₉₉	C₂³	1	18	yes	—
14	D₇₉₆	C₂³	1	48	yes	complemented
15	C₂×Dic₁₉₉	C₂³	1	6	yes	—
16	C₄×D₁₉₉	C₂³	1	48	yes	complemented
17	C₂³×C₁₉₉	C₂³	1	1	yes	characteristic
18	Q₈×C₁₉₉	C₂³	1	16	yes	complemented
19	C₂×C₄×C₁₉₉	C₂³	1	18	yes	—
20	C₂²×C₁₉₉	#41	1	4	yes	complemented
21	C₂²×C₁₉₉	C₂⁴	1	3	yes	—
22	C₂×D₁₉₉	—	2	36	no	—
23	C₂×D₁₉₉	C₂⁴	1	12	yes	—
24	C₄×C₁₉₉	C₂⁴	1	12	yes	—
25	Dic₁₉₉	C₂⁴	1	4	yes	—
26	C₂×C₁₉₉	C₂×#41	1	6	yes	complemented
27	C₂×C₁₉₉	C₂⁵	1	1	yes	characteristic
28	D₁₉₉	—	2	12	no	—
29	C₁₉₉	C₂²×#41	1	1	yes	characteristic, complemented, Sylow
30	C₂²×#41	—	199	1	no	maximal, Sylow
31	C₂×#41	—	199	24	no	—
32	C₂²×Q₈	D₁₉₉	1	1	yes	characteristic, complemented
33	C₂²×D₄	—	199	3	no	—
34	C₂³×C₄	—	199	3	no	—
35	C₂⁴	—	199	3	no	—
36	#41	—	199	64	no	—
37	C₂×Q₈	C₂×D₁₉₉	1	12	yes	complemented
38	C₂×D₄	—	199	36	no	—
39	C₂²×C₄	—	199	1	no	—
40	C₂²×C₄	—	199	36	no	—
41	C₂²×C₄	C₂×D₁₉₉	1	3	yes	—
42	C₂³	—	398	12	no	—
43	C₂³	—	199	18	no	—
44	C₂³	C₂²×D₁₉₉	1	1	yes	characteristic
45	Q₈	C₂²×D₁₉₉	1	16	yes	complemented
46	D₄	—	199	48	no	—
47	C₂×C₄	—	199	6	no	—
48	C₂×C₄	—	199	48	no	—
49	C₂×C₄	C₂²×D₁₉₉	1	18	yes	—
50	C₂²	—	398	36	no	—
51	C₂²	—	199	12	no	—
52	C₂²	#15691887	1	4	yes	direct product factor
53	C₂²	C₂³×D₁₉₉	1	3	yes	—
54	C₄	—	199	4	no	—
55	C₄	C₂³×D₁₉₉	1	12	yes	—
56	C₂	—	398	12	no	—
57	C₂	C₂×#15691887	1	6	yes	direct product factor
58	C₂	C₂⁴×D₁₉₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₉₉
commutator, cosocle	C₂×C₁₉₉	C₂⁵
Frattini	C₂	C₂⁴×D₁₉₉
Fitting	C₂²×Q₈×C₁₉₉	C₂
socle	C₂³×C₁₉₉	C₂³
hypercenter	C₂²×Q₈	D₁₉₉
hypocenter	C₁₉₉	C₂²×#41
LCS	G ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#15691887	C₂	direct (N × H)
2. C₂²×D₇₉₆	C₂	semidirect
3. C₂²×C₄×D₁₉₉	C₂	semidirect
4. C₂²×Q₈×C₁₉₉	C₂	semidirect
6. #15691887	C₂²	direct (N × H)
7. C₂×D₇₉₆	C₂²	semidirect
9. C₂×C₄×D₁₉₉	C₂²	semidirect
10. C₂×Q₈×C₁₉₉	C₂²	semidirect
14. D₇₉₆	C₂³	semidirect
16. C₄×D₁₉₉	C₂³	semidirect
18. Q₈×C₁₉₉	C₂³	semidirect
20. C₂²×C₁₉₉	#41	semidirect
26. C₂×C₁₉₉	C₂×#41	semidirect
29. C₁₉₉	C₂²×#41	semidirect
32. C₂²×Q₈	D₁₉₉	semidirect
37. C₂×Q₈	C₂×D₁₉₉	semidirect
45. Q₈	C₂²×D₁₉₉	semidirect
52. C₂²	#15691887	direct (N × H)
57. C₂	C₂×#15691887	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×#15691887	2
C₂²×D₇₉₆	2
C₂²×C₄×D₁₉₉	2
C₂²×Q₈×C₁₉₉	2
C₂²×#41	199

Subgroups — C22×#15691887 #27164846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15691887 #27164846