Subgroups — C₂×#27211710 #41000883

Order 26816.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27211710	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊Dic₄₁₉	C₂	1	2	yes	maximal
3	C₂×Dic₄₁₉⋊C₄	C₂	1	4	yes	maximal
4	C₂×C₄²×C₄₁₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₁₉	C₂²	1	4	yes	—
6	C₄⋊Dic₄₁₉	C₂²	1	8	yes	—
7	Dic₄₁₉⋊C₄	C₂²	1	16	yes	—
8	C₂²×C₄×C₄₁₉	C₂²	1	2	yes	—
9	C₂²×C₄×C₄₁₉	C₂²	1	1	yes	characteristic
10	C₄²×C₄₁₉	C₂²	1	4	yes	—
11	C₂×Dic₄₁₉	—	2	8	no	—
12	C₂×Dic₄₁₉	C₂³	1	8	yes	—
13	C₂³×C₄₁₉	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₄₁₉	—	2	4	no	—
15	C₂×C₄×C₄₁₉	Q₈	1	4	yes	—
16	C₂×C₄×C₄₁₉	C₂³	1	4	yes	—
17	C₂×C₄×C₄₁₉	C₂³	1	2	yes	—
18	C₂²×C₄₁₉	C₂×Q₈	1	2	yes	—
19	C₂²×C₄₁₉	#41	1	4	yes	—
20	C₂²×C₄₁₉	C₂⁴	1	1	yes	characteristic
21	C₄×C₄₁₉	—	2	4	no	—
22	C₄×C₄₁₉	C₂×Q₈	1	4	yes	—
23	Dic₄₁₉	—	2	8	no	—
24	C₂×C₄₁₉	#125	1	4	yes	complemented
25	C₂×C₄₁₉	C₂²×Q₈	1	1	yes	characteristic
26	C₂×C₄₁₉	C₂×#41	1	2	yes	—
27	C₄₁₉	C₂×#125	1	1	yes	characteristic, complemented, Sylow
28	C₂×#125	—	419	1	no	maximal, Sylow
29	#125	—	419	8	no	—
30	C₂×C₄⋊C₄	—	419	4	no	—
31	C₂×C₄⋊C₄	—	419	2	no	—
32	C₂×C₄²	D₄₁₉	1	1	yes	characteristic
33	C₂²×C₄	—	419	4	no	—
34	C₂²×C₄	C₂×D₄₁₉	1	2	yes	—
35	C₂²×C₄	C₂×D₄₁₉	1	1	yes	characteristic
36	C₄⋊C₄	—	419	16	no	—
37	C₄⋊C₄	—	419	8	no	—
38	C₄²	C₂×D₄₁₉	1	4	yes	—
39	C₂³	C₂²×D₄₁₉	1	1	yes	characteristic
40	C₂×C₄	—	2	4	no	—
41	C₂×C₄	—	838	8	no	—
42	C₂×C₄	—	419	8	no	—
43	C₂×C₄	Dic₈₃₈	1	4	yes	—
44	C₂×C₄	C₂²×D₄₁₉	1	4	yes	—
45	C₂×C₄	C₂²×D₄₁₉	1	2	yes	—
46	C₂²	C₂×Dic₈₃₈	1	2	yes	—
47	C₂²	#20269411	1	4	yes	—
48	C₂²	C₂³×D₄₁₉	1	1	yes	characteristic
49	C₄	—	2	4	no	—
50	C₄	—	838	8	no	—
51	C₄	C₂×Dic₈₃₈	1	4	yes	—
52	C₂	#27211710	1	4	yes	direct product factor
53	C₂	C₂²×Dic₈₃₈	1	1	yes	characteristic
54	C₂	C₂×#20269411	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₄₁₉
commutator, cosocle	C₂²×C₄₁₉	C₂⁴
Frattini	C₂²	C₂³×D₄₁₉
Fitting	C₂×C₄²×C₄₁₉	C₂
socle, prime radical	C₂³×C₄₁₉	C₂³
hypercenter	C₂×C₄²	D₄₁₉
hypocenter	C₄₁₉	C₂×#125
LCS	G ▷ C₂²×C₄₁₉ ▷ C₄₁₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #27211710	C₂	direct (N × H)
24. C₂×C₄₁₉	#125	semidirect
27. C₄₁₉	C₂×#125	semidirect
52. C₂	#27211710	direct (N × H)

Maximal subgroups

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

subgroup	index
#27211710	2
C₂×C₄⋊Dic₄₁₉	2
C₂×Dic₄₁₉⋊C₄	2
C₂×C₄²×C₄₁₉	2
C₂×#125	419

Subgroups — C2×#27211710 #41000883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27211710 #41000883