Subgroups — #36375018

Order 22336.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#13063703	C₂	1	2	yes	maximal
2	C₂×C₈×D₃₄₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#13063703	—	2	2	no	—
4	C₄⋊#13063703	—	2	2	no	—
5	C₄⋊#13063703	C₄	1	4	yes	complemented
6	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
7	C₂×#14584095	C₄	1	1	yes	characteristic, complemented
8	C₈×D₃₄₉	—	2	2	no	—
9	C₂×C₈×C₃₄₉	C₄	1	1	yes	characteristic, complemented
10	C₂×#13063703	—	2	4	no	—
11	C₂×#13063703	—	4	2	no	—
12	C₂×#13063703	—	4	2	no	—
13	C₂²×D₃₄₉	D₄	1	1	yes	characteristic
14	C₂×Dic₃₄₉	Q₈	1	1	yes	characteristic
15	C₄×D₃₄₉	C₂×C₄	1	2	yes	—
16	C₄×D₃₄₉	D₄	1	2	yes	—
17	#14584095	—	2	1	no	—
18	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
19	C₈×C₃₄₉	—	2	1	no	—
20	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
21	C₂×D₃₄₉	C₂²⋊C₄	1	2	yes	—
22	C₂×D₃₄₉	D₈	1	1	yes	characteristic
23	C₂×D₃₄₉	SD₁₆	1	2	yes	—
24	C₂×D₃₄₉	Q₁₆	1	1	yes	characteristic
25	#13063703	—	4	4	no	—
26	C₄×C₃₄₉	C₄²	1	1	yes	characteristic
27	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
28	Dic₃₄₉	C₄⋊C₄	1	2	yes	—
29	C₂×C₃₄₉	#95	1	1	yes	characteristic
30	C₂×C₃₄₉	#106	1	1	yes	characteristic
31	C₂×C₃₄₉	#107	1	1	yes	characteristic
32	D₃₄₉	#102	1	2	yes	—
33	D₃₄₉	#103	1	2	yes	—
34	C₃₄₉	#340	1	1	yes	characteristic, complemented, Sylow
35	#340	—	349	1	no	maximal, Sylow
36	C₂²×C₈	—	349	1	no	—
37	C₂×C₄⋊C₄	—	349	2	no	—
38	C₂²×C₄	—	349	1	no	—
39	C₂²×C₄	—	698	2	no	—
40	C₂×C₈	—	349	1	no	—
41	C₂×C₈	—	698	2	no	—
42	C₂×C₈	#13063703	1	1	yes	characteristic, complemented
43	C₄⋊C₄	—	349	4	no	—
44	C₄⋊C₄	—	698	2	no	—
45	C₂³	—	349	1	no	—
46	C₂×C₄	—	349	1	no	—
47	C₂×C₄	—	349	2	no	—
48	C₂×C₄	—	349	2	no	—
49	C₂×C₄	—	1396	2	no	—
50	C₂×C₄	—	1396	2	no	—
51	C₂×C₄	—	698	4	no	—
52	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	698	1	no	—
55	C₂²	—	349	2	no	—
56	C₂²	—	349	1	no	—
57	C₂²	—	349	1	no	—
58	C₂²	—	349	2	no	—
59	C₂²	C₄⋊#13063703	1	1	yes	characteristic
60	C₄	—	349	2	no	—
61	C₄	—	1396	4	no	—
62	C₄	C₄×#13063703	1	1	yes	characteristic
63	C₄	C₂²⋊#13063703	1	1	yes	characteristic
64	C₂	—	349	2	no	—
65	C₂	—	349	2	no	—
66	C₂	(C₂×#13063703)⋊C₄	1	1	yes	characteristic
67	C₂	#26013707	1	1	yes	characteristic
68	C₂	#26013708	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#13063703
commutator	C₄×C₃₄₉	C₄²
Frattini	C₂×C₄	C₂×#13063703
Fitting	C₂×C₈×C₃₄₉	C₄
socle	C₂²×C₃₄₉	C₄⋊C₄
prime radical	C₂²×D₃₄₉	D₄
cosocle	C₂×C₄×D₃₄₉	C₂²
hypercenter	C₂×C₈	#13063703
hypocenter	C₃₄₉	#340
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. C₄⋊#13063703	C₄	semidirect
7. C₂×#14584095	C₄	semidirect
9. C₂×C₈×C₃₄₉	C₄	semidirect
34. C₃₄₉	#340	semidirect
42. C₂×C₈	#13063703	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#13063703	2
C₂×C₈×D₃₄₉	2
#340	349