Subgroups — #36374118

Order 22336.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013598	C₂	1	2	yes	maximal, complemented
2	#26013554	C₂	1	4	yes	maximal, complemented
3	C₂×#41×C₃₄₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₃₄₉	—	2	2	no	—
5	C₂²⋊Dic₃₄₉	C₂²	1	4	yes	—
6	C₄×Dic₃₄₉	—	2	2	no	—
7	#41×C₃₄₉	—	2	2	no	—
8	#41×C₃₄₉	—	2	2	no	—
9	C₂×Q₈×C₃₄₉	C₄	1	1	yes	characteristic, complemented
10	C₂×D₄×C₃₄₉	C₄	1	1	yes	characteristic, complemented
11	C₂×D₄×C₃₄₉	C₂²	1	2	yes	—
12	C₂²×C₄×C₃₄₉	C₄	1	2	yes	complemented
13	C₂²×C₄×C₃₄₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₄₉	—	2	4	no	—
15	C₂³×C₃₄₉	C₂×C₄	1	2	yes	—
16	C₂³×C₃₄₉	C₂³	1	1	yes	characteristic
17	Q₈×C₃₄₉	—	2	2	no	—
18	D₄×C₃₄₉	—	2	4	no	—
19	D₄×C₃₄₉	—	2	2	no	—
20	C₂×C₄×C₃₄₉	—	2	2	no	—
21	C₂×C₄×C₃₄₉	—	2	2	no	—
22	C₂×C₄×C₃₄₉	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₄₉	D₄	1	2	yes	—
26	C₂×C₄×C₃₄₉	D₄	1	2	yes	—
27	C₂²×C₃₄₉	—	2	2	no	—
28	C₂²×C₃₄₉	—	4	1	no	—
29	C₂²×C₃₄₉	—	4	2	no	—
30	C₂²×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₃₄₉	C₂×D₄	1	2	yes	—
32	C₄×C₃₄₉	—	2	2	no	—
33	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₄×C₃₄₉	C₂²⋊C₄	1	2	yes	—
35	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₃₄₉	—	4	4	no	—
37	C₂×C₃₄₉	—	2	2	no	—
38	C₂×C₃₄₉	—	2	1	no	—
39	C₂×C₃₄₉	—	4	2	no	—
40	C₂×C₃₄₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₃₄₉	#410	1	1	yes	characteristic, complemented, Sylow
42	#410	—	349	1	no	maximal, Sylow
43	C₂×#41	D₃₄₉	1	1	yes	characteristic
44	#117	—	349	2	no	—
45	#99	—	349	4	no	—
46	#41	—	2	2	no	—
47	#41	—	2	2	no	—
48	C₂×Q₈	Dic₃₄₉	1	1	yes	characteristic, complemented
49	C₂×D₄	Dic₃₄₉	1	1	yes	characteristic, complemented
50	C₂×D₄	C₂×D₃₄₉	1	2	yes	—
51	C₂²×C₄	Dic₃₄₉	1	2	yes	complemented
52	C₂²×C₄	C₂×D₃₄₉	1	1	yes	characteristic
53	C₄⋊C₄	—	698	2	no	—
54	C₂²⋊C₄	—	349	4	no	—
55	C₄²	—	698	2	no	—
56	C₂³	C₂×Dic₃₄₉	1	2	yes	—
57	C₂³	C₂²×D₃₄₉	1	1	yes	characteristic
58	Q₈	—	2	2	no	—
59	D₄	—	2	4	no	—
60	D₄	—	2	2	no	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	698	4	no	—
64	C₂×C₄	C₂×Dic₃₄₉	1	2	yes	—
65	C₂×C₄	C₂×Dic₃₄₉	1	1	yes	characteristic
66	C₂×C₄	C₂×Dic₃₄₉	1	1	yes	characteristic
67	C₂×C₄	C₂²⋊D₃₄₉	1	2	yes	—
68	C₂×C₄	C₂²⋊D₃₄₉	1	2	yes	—
69	C₂²	—	2	2	no	—
70	C₂²	—	4	1	no	—
71	C₂²	—	4	2	no	—
72	C₂²	C₂²×Dic₃₄₉	1	1	yes	characteristic
73	C₂²	C₂×C₂²⋊D₃₄₉	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	1396	4	no	—
76	C₄	C₂²⋊Dic₃₄₉	1	1	yes	characteristic
77	C₄	C₂²⋊Dic₃₄₉	1	2	yes	—
78	C₄	C₂²⋊Dic₃₄₉	1	1	yes	characteristic
79	C₂	—	2	2	no	—
80	C₂	—	2	1	no	—
81	C₂	—	4	2	no	—
82	C₂	C₂×C₂²⋊Dic₃₄₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²⋊Dic₃₄₉
commutator	C₂²×C₃₄₉	C₂²×C₄
Frattini	C₂³	C₂²×D₃₄₉
Fitting, prime radical	C₂×#41×C₃₄₉	C₂
socle	C₂×C₃₄₉	C₂×C₂²⋊C₄
cosocle	C₂³×C₃₄₉	C₂³
hypercenter	C₂×#41	D₃₄₉
hypocenter	C₃₄₉	#410
LCS	G ▷ C₂²×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #26013598	C₂	semidirect
2. #26013554	C₂	semidirect
9. C₂×Q₈×C₃₄₉	C₄	semidirect
10. C₂×D₄×C₃₄₉	C₄	semidirect
12. C₂²×C₄×C₃₄₉	C₄	semidirect
41. C₃₄₉	#410	semidirect
48. C₂×Q₈	Dic₃₄₉	semidirect
49. C₂×D₄	Dic₃₄₉	semidirect
51. C₂²×C₄	Dic₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013598	2
#26013554	2
C₂×#41×C₃₄₉	2
#410	349