Subgroups — #36375193

Order 22336.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013706	C₂	1	2	yes	maximal
2	C₄⋊#14584102	C₂	1	1	yes	maximal, characteristic
3	C₄×#14584102	C₂	1	1	yes	maximal, characteristic
4	C₄×#14584102	C₂	1	2	yes	maximal
5	#26013604	C₂	1	1	yes	maximal, characteristic
6	C₂×#14584102	C₂²	1	2	yes	—
7	C₂×#14584102	C₂²	1	1	yes	characteristic
8	C₂×#14584102	C₂²	1	1	yes	characteristic
9	C₂×Dic₆₉₈	C₄	1	1	yes	characteristic
10	#19150212	C₄	1	2	yes	—
11	C₄×Dic₃₄₉	C₂²	1	1	yes	characteristic
12	C₄×Dic₃₄₉	C₂²	1	2	yes	—
13	C₄⋊C₄×C₃₄₉	C₄	1	1	yes	characteristic
14	#14584102	—	2	2	no	—
15	#14584102	—	2	1	no	—
16	#14584102	Q₈	1	2	yes	complemented
17	Dic₆₉₈	C₈	1	4	yes	complemented
18	C₂×Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₃₄₉	C₂×C₄	1	2	yes	—
20	C₂×Dic₃₄₉	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₃₄₉	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₃₄₉	—	2	2	no	—
25	C₄×C₃₄₉	C₂×C₈	1	2	yes	—
26	Dic₃₄₉	—	2	1	no	—
27	Dic₃₄₉	C₂×C₈	1	4	yes	—
28	Dic₃₄₉	C₂×Q₈	1	1	yes	characteristic
29	Dic₃₄₉	#41	1	1	yes	characteristic
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	1	no	—
37	C₄⋊C₈	—	349	2	no	—
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	2	no	—
43	C₂×C₈	—	349	1	no	—
44	C₄⋊C₄	—	349	2	no	—
45	C₄⋊C₄	#13063703	1	1	yes	characteristic
46	C₄²	—	349	2	no	—
47	C₄²	—	349	1	no	—
48	Q₈	—	349	4	no	—
49	C₂×C₄	—	349	1	no	—
50	C₂×C₄	—	349	2	no	—
51	C₂×C₄	—	349	1	no	—
52	C₂×C₄	C₂×#13063703	1	2	yes	—
53	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
54	C₈	—	349	2	no	—
55	C₈	—	698	2	no	—
56	C₈	—	698	1	no	—
57	C₂²	C₂²×#13063703	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	349	1	no	—
60	C₄	—	349	1	no	—
61	C₄	—	349	4	no	—
62	C₄	—	698	1	no	—
63	C₄	D₃₄₉⋊C₈	1	2	yes	—
64	C₂	Q₈×#13063703	1	1	yes	characteristic
65	C₂	C₂×D₃₄₉⋊C₈	1	1	yes	characteristic
66	C₂	#26013726	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#13063703
commutator	C₂×C₃₄₉	C₂²×C₈
Fitting	C₄⋊C₄×C₃₄₉	C₄
socle, prime radical	C₂²×C₃₄₉	C₂²×C₄
cosocle	C₂×Dic₃₄₉	C₂³
hypercenter	C₄⋊C₄	#13063703
hypocenter	C₃₄₉	C₈×Q₈
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
16. #14584102	Q₈	semidirect
17. Dic₆₉₈	C₈	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
#26013706	2
C₄⋊#14584102	2
C₄×#14584102	2
#26013604	2
C₈×Q₈	349