Subgroups — C₂²×#19150221 #36374917

Order 22336.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₉₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19150221	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#14584095	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₄×C₃₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₉₈	—	2	3	no	—
6	C₂×Dic₆₉₈	C₂²	1	6	yes	complemented
7	C₂²×Dic₃₄₉	—	2	1	no	—
8	#19150221	C₂²	1	16	yes	direct product factor
9	C₂×#14584095	C₂²	1	6	yes	complemented
10	C₂⁴×C₃₄₉	—	2	1	no	—
11	C₂×D₄×C₃₄₉	—	2	3	no	—
12	C₂×D₄×C₃₄₉	C₂²	1	6	yes	—
13	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
14	Dic₆₉₈	—	2	6	no	—
15	Dic₆₉₈	C₂³	1	4	yes	complemented
16	C₂×Dic₃₄₉	—	2	6	no	—
17	#14584095	C₂³	1	4	yes	complemented
18	C₂³×C₃₄₉	—	2	6	no	—
19	C₂³×C₃₄₉	—	4	4	no	—
20	C₂³×C₃₄₉	D₄	1	1	yes	characteristic
21	D₄×C₃₄₉	—	2	6	no	—
22	D₄×C₃₄₉	C₂³	1	4	yes	—
23	C₂×C₄×C₃₄₉	D₄	1	3	yes	—
24	C₂×C₄×C₃₄₉	C₂³	1	3	yes	—
25	C₂²×C₃₄₉	—	2	4	no	—
26	C₂²×C₃₄₉	—	4	12	no	—
27	C₂²×C₃₄₉	SD₁₆	1	4	yes	complemented
28	C₂²×C₃₄₉	C₂×D₄	1	3	yes	—
29	C₄×C₃₄₉	C₂×D₄	1	3	yes	—
30	C₄×C₃₄₉	C₂⁴	1	1	yes	characteristic
31	Dic₃₄₉	—	2	4	no	—
32	C₂×C₃₄₉	—	4	4	no	—
33	C₂×C₃₄₉	C₂×SD₁₆	1	6	yes	complemented
34	C₂×C₃₄₉	C₂²×D₄	1	1	yes	characteristic
35	C₃₄₉	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
36	C₂²×SD₁₆	—	349	1	no	maximal, Sylow
37	C₂²×Q₈	—	349	1	no	—
38	C₂²×D₄	D₃₄₉	1	1	yes	characteristic
39	C₂×SD₁₆	—	349	12	no	—
40	C₂²×C₈	—	349	1	no	—
41	C₂⁴	—	2	1	no	—
42	C₂×Q₈	—	349	6	no	—
43	C₂×Q₈	—	698	3	no	—
44	C₂×D₄	—	2	3	no	—
45	C₂×D₄	C₂×D₃₄₉	1	6	yes	—
46	C₂²×C₄	—	698	1	no	—
47	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
48	SD₁₆	—	349	16	no	—
49	C₂×C₈	—	349	6	no	—
50	C₂³	—	2	6	no	—
51	C₂³	—	4	4	no	—
52	C₂³	C₂²⋊D₃₄₉	1	1	yes	characteristic
53	Q₈	—	349	4	no	—
54	Q₈	—	698	6	no	—
55	D₄	—	2	6	no	—
56	D₄	C₂²×D₃₄₉	1	4	yes	—
57	C₂×C₄	—	698	6	no	—
58	C₂×C₄	C₂²⋊D₃₄₉	1	3	yes	—
59	C₂×C₄	C₂²×D₃₄₉	1	3	yes	—
60	C₈	—	349	4	no	—
61	C₂²	—	2	4	no	—
62	C₂²	—	4	12	no	—
63	C₂²	#19150221	1	4	yes	direct product factor
64	C₂²	C₂×C₂²⋊D₃₄₉	1	3	yes	—
65	C₄	—	698	4	no	—
66	C₄	C₂×C₂²⋊D₃₄₉	1	3	yes	—
67	C₄	C₂³×D₃₄₉	1	1	yes	characteristic
68	C₂	—	4	4	no	—
69	C₂	C₂×#19150221	1	6	yes	direct product factor
70	C₂	C₂²×C₂²⋊D₃₄₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₃₄₉
commutator, cosocle	C₄×C₃₄₉	C₂⁴
Frattini	C₄	C₂³×D₃₄₉
Fitting, prime radical	C₂²×D₄×C₃₄₉	C₂
socle	C₂³×C₃₄₉	D₄
hypercenter	C₂²×D₄	D₃₄₉
hypocenter	C₃₄₉	C₂²×SD₁₆
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×Dic₆₉₈	C₂	semidirect
2. C₂×#19150221	C₂	direct (N × H)
3. C₂²×#14584095	C₂	semidirect
6. C₂×Dic₆₉₈	C₂²	semidirect
8. #19150221	C₂²	direct (N × H)
9. C₂×#14584095	C₂²	semidirect
15. Dic₆₉₈	C₂³	semidirect
17. #14584095	C₂³	semidirect
27. C₂²×C₃₄₉	SD₁₆	semidirect
33. C₂×C₃₄₉	C₂×SD₁₆	semidirect
35. C₃₄₉	C₂²×SD₁₆	semidirect
63. C₂²	#19150221	direct (N × H)
69. C₂	C₂×#19150221	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₆₉₈	2
C₂×#19150221	2
C₂²×#14584095	2
C₂²×D₄×C₃₄₉	2
C₂²×SD₁₆	349

Subgroups — C22×#19150221 #36374917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19150221 #36374917