Subgroups — C₂×#27211954 #41001599

Order 26848.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211954	C₂	1	8	yes	maximal, direct product factor
2	C₂×Dic₁₆₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊D₈₃₉	C₂	1	2	yes	maximal, complemented
4	C₂²×Dic₈₃₉	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×D₈₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×C₈₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈₃₉	C₂²	1	1	yes	characteristic
8	Dic₁₆₇₈	C₂²	1	4	yes	complemented
9	C₂²⋊D₈₃₉	C₂²	1	8	yes	complemented
10	C₂×Dic₈₃₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₈₃₉	C₂²	1	2	yes	—
12	C₂×Dic₈₃₉	C₂²	1	8	yes	complemented
13	C₄×D₈₃₉	C₂²	1	4	yes	complemented
14	C₂³×C₈₃₉	C₂²	1	2	yes	—
15	D₄×C₈₃₉	C₂²	1	4	yes	complemented
16	C₂×C₄×C₈₃₉	C₂²	1	1	yes	characteristic
17	C₂²×C₈₃₉	—	2	4	no	—
18	C₂²×C₈₃₉	C₂³	1	4	yes	—
19	C₂²×C₈₃₉	C₂³	1	1	yes	characteristic
20	C₂×D₈₃₉	—	2	2	no	—
21	C₂×D₈₃₉	C₂³	1	2	yes	—
22	C₄×C₈₃₉	C₂³	1	2	yes	—
23	Dic₈₃₉	C₂³	1	2	yes	—
24	Dic₈₃₉	C₂³	1	4	yes	—
25	C₂×C₈₃₉	—	2	4	no	—
26	C₂×C₈₃₉	#41	1	2	yes	complemented
27	C₂×C₈₃₉	C₂⁴	1	1	yes	characteristic
28	D₈₃₉	—	2	2	no	—
29	C₈₃₉	C₂×#41	1	1	yes	characteristic, complemented, Sylow
30	C₂×#41	—	839	1	no	maximal, Sylow
31	#41	—	839	8	no	—
32	C₂×Q₈	—	839	1	no	—
33	C₂×D₄	—	839	2	no	—
34	C₂×D₄	D₈₃₉	1	1	yes	characteristic, complemented
35	C₂²×C₄	—	839	2	no	—
36	C₂²×C₄	—	839	1	no	—
37	C₂³	—	839	1	no	—
38	C₂³	C₂×D₈₃₉	1	2	yes	—
39	Q₈	—	839	4	no	—
40	D₄	—	839	8	no	—
41	D₄	C₂×D₈₃₉	1	4	yes	complemented
42	C₂×C₄	—	839	8	no	—
43	C₂×C₄	—	839	1	no	—
44	C₂×C₄	—	839	2	no	—
45	C₂×C₄	—	839	4	no	—
46	C₂×C₄	C₂×D₈₃₉	1	1	yes	characteristic
47	C₂²	—	2	4	no	—
48	C₂²	—	839	2	no	—
49	C₂²	—	1678	2	no	—
50	C₂²	C₂²×D₈₃₉	1	4	yes	—
51	C₂²	C₂²×D₈₃₉	1	1	yes	characteristic
52	C₄	—	839	2	no	—
53	C₄	—	839	4	no	—
54	C₄	C₂²×D₈₃₉	1	2	yes	—
55	C₂	—	2	4	no	—
56	C₂	—	1678	2	no	—
57	C₂	#27211954	1	2	yes	direct product factor
58	C₂	C₂³×D₈₃₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₈₃₉
commutator, cosocle	C₂×C₈₃₉	C₂⁴
Frattini	C₂	C₂³×D₈₃₉
Fitting	C₂×D₄×C₈₃₉	C₂
socle	C₂²×C₈₃₉	C₂³
hypercenter	C₂×D₄	D₈₃₉
hypocenter	C₈₃₉	C₂×#41
LCS	G ▷ C₂×C₈₃₉ ▷ C₈₃₉
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #27211954	C₂	direct (N × H)
2. C₂×Dic₁₆₇₈	C₂	semidirect
3. C₂×C₂²⋊D₈₃₉	C₂	semidirect
4. C₂²×Dic₈₃₉	C₂	semidirect
5. C₂×C₄×D₈₃₉	C₂	semidirect
6. C₂×D₄×C₈₃₉	C₂	semidirect
8. Dic₁₆₇₈	C₂²	semidirect
9. C₂²⋊D₈₃₉	C₂²	semidirect
12. C₂×Dic₈₃₉	C₂²	semidirect
13. C₄×D₈₃₉	C₂²	semidirect
15. D₄×C₈₃₉	C₂²	semidirect
26. C₂×C₈₃₉	#41	semidirect
29. C₈₃₉	C₂×#41	semidirect
34. C₂×D₄	D₈₃₉	semidirect
41. D₄	C₂×D₈₃₉	semidirect
57. C₂	#27211954	direct (N × H)

Maximal subgroups

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

subgroup	index
#27211954	2
C₂×Dic₁₆₇₈	2
C₂×C₂²⋊D₈₃₉	2
C₂²×Dic₈₃₉	2
C₂×C₄×D₈₃₉	2
C₂×D₄×C₈₃₉	2
C₂×#41	839

Subgroups — C2×#27211954 #41001599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27211954 #41001599