Subgroups — #36374268

Order 22336.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₆₉₈	C₂	1	1	yes	maximal, characteristic
2	#26013568	C₂	1	2	yes	maximal
3	C₄×#14584095	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₃₄₉	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₃₄₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₉₈	C₄	1	1	yes	characteristic
7	C₂×#14584095	C₂²	1	1	yes	characteristic
8	C₂×#14584095	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₃₄₉	C₄	1	1	yes	characteristic
10	#19150212	C₄	1	2	yes	—
11	C₄×Dic₃₄₉	C₂²	1	2	yes	—
12	C₂×C₈×C₃₄₉	C₂²	1	2	yes	—
13	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
14	Dic₆₉₈	C₈	1	4	yes	complemented
15	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
16	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
17	#14584095	—	2	1	no	—
18	#14584095	Q₈	1	2	yes	complemented
19	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
22	C₈×C₃₄₉	—	2	2	no	—
23	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₃₄₉	—	2	1	no	—
25	C₄×C₃₄₉	C₂×C₈	1	2	yes	—
26	C₄×C₃₄₉	C₂×Q₈	1	1	yes	characteristic
27	C₄×C₃₄₉	#41	1	1	yes	characteristic
28	Dic₃₄₉	—	2	2	no	—
29	Dic₃₄₉	C₂×C₈	1	4	yes	—
30	C₂×C₃₄₉	C₄×Q₈	1	1	yes	characteristic
31	C₂×C₃₄₉	C₂²×C₈	1	1	yes	characteristic
32	C₂×C₃₄₉	#131	1	1	yes	characteristic
33	C₃₄₉	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
34	C₈×Q₈	—	349	1	no	maximal, Sylow
35	C₄×Q₈	—	349	1	no	—
36	C₄⋊C₈	—	349	2	no	—
37	C₄⋊C₈	D₃₄₉	1	1	yes	characteristic
38	C₄×C₈	—	349	2	no	—
39	C₄×C₈	—	349	1	no	—
40	C₂×Q₈	—	349	1	no	—
41	C₂×C₈	—	349	1	no	—
42	C₂×C₈	—	349	1	no	—
43	C₂×C₈	C₂×D₃₄₉	1	2	yes	—
44	C₄⋊C₄	—	349	2	no	—
45	C₄⋊C₄	—	349	1	no	—
46	C₄²	—	349	2	no	—
47	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
48	Q₈	—	349	4	no	—
49	C₂×C₄	—	349	2	no	—
50	C₂×C₄	—	349	2	no	—
51	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
52	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
54	C₈	—	2	2	no	—
55	C₈	—	349	2	no	—
56	C₈	—	698	1	no	—
57	C₂²	C₂×C₄×D₃₄₉	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	—	349	4	no	—
60	C₄	—	698	2	no	—
61	C₄	C₈×D₃₄₉	1	2	yes	—
62	C₄	Q₈×D₃₄₉	1	1	yes	characteristic
63	C₄	#19150232	1	1	yes	characteristic
64	C₂	#26013604	1	1	yes	characteristic
65	C₂	C₂×C₈×D₃₄₉	1	1	yes	characteristic
66	C₂	#26013647	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₄₉
commutator	C₂×C₃₄₉	C₂²×C₈
Fitting	C₄⋊C₈×C₃₄₉	C₂
socle, prime radical	C₂²×C₃₄₉	C₂²×C₄
cosocle	C₂×C₄×C₃₄₉	C₂³
hypercenter	C₄⋊C₈	D₃₄₉
hypocenter	C₃₄₉	C₈×Q₈
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
14. Dic₆₉₈	C₈	semidirect
18. #14584095	Q₈	semidirect
33. C₃₄₉	C₈×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₆₉₈	2
#26013568	2
C₄×#14584095	2
C₈×Dic₃₄₉	2
C₄⋊C₈×C₃₄₉	2
C₈×Q₈	349