Subgroups — #36374989

Order 22336.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#14584102	C₂	1	2	yes	maximal
2	C₂×C₄⋊Dic₃₄₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#14584102	—	2	2	no	—
4	C₂²×Dic₃₄₉	C₄	1	1	yes	characteristic
5	C₂²×Dic₃₄₉	C₂²	1	1	yes	characteristic
6	C₄⋊Dic₃₄₉	—	2	2	no	—
7	C₂²×C₄×C₃₄₉	C₄	1	1	yes	characteristic
8	#14584102	—	4	2	no	—
9	C₂×Dic₃₄₉	—	2	1	no	—
10	C₂×Dic₃₄₉	—	4	1	no	—
11	C₂×Dic₃₄₉	—	4	1	no	—
12	C₂×Dic₃₄₉	D₄	1	2	yes	—
13	C₂³×C₃₄₉	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₃₄₉	—	2	1	no	—
15	C₂×C₄×C₃₄₉	—	2	1	no	—
16	C₂×C₄×C₃₄₉	C₈	1	2	yes	complemented
17	C₂²×C₃₄₉	—	2	1	no	—
18	C₂²×C₃₄₉	—	2	1	no	—
19	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
20	C₂²×C₃₄₉	C₂×C₈	1	1	yes	characteristic
21	C₂²×C₃₄₉	M₁₆	1	1	yes	characteristic
22	C₄×C₃₄₉	—	2	2	no	—
23	Dic₃₄₉	—	2	2	no	—
24	Dic₃₄₉	—	4	1	no	—
25	C₂×C₃₄₉	—	2	1	no	—
26	C₂×C₃₄₉	—	2	1	no	—
27	C₂×C₃₄₉	C₂²⋊C₈	1	1	yes	characteristic
28	C₂×C₃₄₉	#99	1	1	yes	characteristic
29	C₂×C₃₄₉	#101	1	1	yes	characteristic
30	C₃₄₉	#324	1	1	yes	characteristic, complemented, Sylow
31	#324	—	349	1	no	maximal, Sylow
32	C₂×C₄⋊C₄	—	349	1	no	—
33	C₂²⋊C₈	—	349	2	no	—
34	C₂²×C₄	—	349	1	no	—
35	C₂²×C₄	—	349	1	no	—
36	C₂²×C₄	#13063703	1	1	yes	characteristic
37	C₂×C₈	—	698	2	no	—
38	C₄⋊C₄	—	698	2	no	—
39	C₂³	C₂×#13063703	1	1	yes	characteristic
40	C₂×C₄	—	2	1	no	—
41	C₂×C₄	—	2	1	no	—
42	C₂×C₄	—	349	2	no	—
43	C₂×C₄	—	1396	1	no	—
44	C₂×C₄	—	1396	1	no	—
45	C₂×C₄	—	698	1	no	—
46	C₂×C₄	#14584102	1	2	yes	complemented
47	C₈	—	1396	2	no	—
48	C₂²	—	2	1	no	—
49	C₂²	—	2	1	no	—
50	C₂²	C₂²⋊#13063703	1	1	yes	characteristic
51	C₂²	C₂×#14584102	1	1	yes	characteristic
52	C₂²	#19150242	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	—	1396	1	no	—
55	C₄	—	698	2	no	—
56	C₂	—	2	1	no	—
57	C₂	—	2	1	no	—
58	C₂	C₂²⋊#14584102	1	1	yes	characteristic
59	C₂	#26013698	1	1	yes	characteristic
60	C₂	#26013700	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#13063703
commutator	C₂²×C₃₄₉	C₂×C₈
Frattini	C₂³	C₂×#13063703
Fitting	C₂²×C₄×C₃₄₉	C₄
socle	C₂²×C₃₄₉	C₂²⋊C₄
prime radical	C₂³×C₃₄₉	C₂×C₄
cosocle	C₂²×Dic₃₄₉	C₂²
hypercenter	C₂²×C₄	#13063703
hypocenter	C₃₄₉	#324
LCS	G ▷ C₂²×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
16. C₂×C₄×C₃₄₉	C₈	semidirect
30. C₃₄₉	#324	semidirect
46. C₂×C₄	#14584102	semidirect

Maximal subgroups

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

subgroup	index
C₂²⋊#14584102	2
C₂×C₄⋊Dic₃₄₉	2
#324	349