Subgroups — C₂×#26013631 #36374666

Order 22336.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013631	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊Dic₃₄₉	C₂	1	2	yes	maximal
3	C₂×#19150212	C₂	1	4	yes	maximal
4	C₂×C₄²×C₃₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₄₉	C₂²	1	4	yes	—
6	C₄⋊Dic₃₄₉	C₂²	1	8	yes	—
7	#19150212	C₂²	1	16	yes	—
8	C₂²×C₄×C₃₄₉	C₂²	1	2	yes	—
9	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
10	C₄²×C₃₄₉	C₂²	1	4	yes	—
11	C₂×Dic₃₄₉	—	2	8	no	—
12	C₂×Dic₃₄₉	C₂³	1	8	yes	—
13	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₃₄₉	—	2	4	no	—
15	C₂×C₄×C₃₄₉	Q₈	1	4	yes	—
16	C₂×C₄×C₃₄₉	C₂³	1	4	yes	—
17	C₂×C₄×C₃₄₉	C₂³	1	2	yes	—
18	C₂²×C₃₄₉	C₂×Q₈	1	2	yes	—
19	C₂²×C₃₄₉	#41	1	4	yes	—
20	C₂²×C₃₄₉	C₂⁴	1	1	yes	characteristic
21	C₄×C₃₄₉	—	2	4	no	—
22	C₄×C₃₄₉	C₂×Q₈	1	4	yes	—
23	Dic₃₄₉	—	2	8	no	—
24	C₂×C₃₄₉	#125	1	4	yes	complemented
25	C₂×C₃₄₉	C₂²×Q₈	1	1	yes	characteristic
26	C₂×C₃₄₉	C₂×#41	1	2	yes	—
27	C₃₄₉	C₂×#125	1	1	yes	characteristic, complemented, Sylow
28	C₂×#125	—	349	1	no	maximal, Sylow
29	#125	—	349	8	no	—
30	C₂×C₄⋊C₄	—	349	4	no	—
31	C₂×C₄⋊C₄	—	349	2	no	—
32	C₂×C₄²	D₃₄₉	1	1	yes	characteristic
33	C₂²×C₄	—	349	4	no	—
34	C₂²×C₄	C₂×D₃₄₉	1	2	yes	—
35	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
36	C₄⋊C₄	—	349	16	no	—
37	C₄⋊C₄	—	349	8	no	—
38	C₄²	C₂×D₃₄₉	1	4	yes	—
39	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
40	C₂×C₄	—	2	4	no	—
41	C₂×C₄	—	349	8	no	—
42	C₂×C₄	—	698	8	no	—
43	C₂×C₄	Dic₆₉₈	1	4	yes	—
44	C₂×C₄	C₂²×D₃₄₉	1	4	yes	—
45	C₂×C₄	C₂²×D₃₄₉	1	2	yes	—
46	C₂²	C₂×Dic₆₉₈	1	2	yes	—
47	C₂²	#19150233	1	4	yes	—
48	C₂²	C₂³×D₃₄₉	1	1	yes	characteristic
49	C₄	—	2	4	no	—
50	C₄	—	698	8	no	—
51	C₄	C₂×Dic₆₉₈	1	4	yes	—
52	C₂	#26013631	1	4	yes	direct product factor
53	C₂	C₂²×Dic₆₉₈	1	1	yes	characteristic
54	C₂	C₂×#19150233	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₄₉
commutator, cosocle	C₂²×C₃₄₉	C₂⁴
Frattini	C₂²	C₂³×D₃₄₉
Fitting	C₂×C₄²×C₃₄₉	C₂
socle, prime radical	C₂³×C₃₄₉	C₂³
hypercenter	C₂×C₄²	D₃₄₉
hypocenter	C₃₄₉	C₂×#125
LCS	G ▷ C₂²×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #26013631	C₂	direct (N × H)
24. C₂×C₃₄₉	#125	semidirect
27. C₃₄₉	C₂×#125	semidirect
52. C₂	#26013631	direct (N × H)

Maximal subgroups

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

subgroup	index
#26013631	2
C₂×C₄⋊Dic₃₄₉	2
C₂×#19150212	2
C₂×C₄²×C₃₄₉	2
C₂×#125	349

Subgroups — C2×#26013631 #36374666

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013631 #36374666