Subgroups — #36375154

Order 22336.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013718	C₂	1	2	yes	maximal
2	#26013708	C₂	1	2	yes	maximal, complemented
3	#26013707	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₃₄₉	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#13063703	C₂²	1	4	yes	—
6	C₂²⋊#13063703	—	2	2	no	—
7	C₄×#13063703	—	2	2	no	—
8	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
9	C₂×#14584095	C₄	1	1	yes	characteristic, complemented
10	C₈×D₃₄₉	C₄	1	1	yes	characteristic, complemented
11	C₈×D₃₄₉	C₄	1	1	yes	characteristic, complemented
12	C₈×D₃₄₉	C₂²	1	1	yes	characteristic
13	C₈×D₃₄₉	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₃₄₉	C₄	1	1	yes	characteristic, complemented
15	C₂×#13063703	—	2	4	no	—
16	C₂²×D₃₄₉	D₄	1	1	yes	characteristic
17	C₂×Dic₃₄₉	Q₈	1	1	yes	characteristic
18	C₄×D₃₄₉	C₂×C₄	1	1	yes	characteristic
19	C₄×D₃₄₉	D₄	1	1	yes	characteristic
20	C₄×D₃₄₉	Q₈	1	1	yes	characteristic
21	C₄×D₃₄₉	C₂³	1	1	yes	characteristic
22	#14584095	C₂×C₄	1	1	yes	characteristic
23	#14584095	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
25	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
26	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×D₃₄₉	—	2	2	no	—
29	C₂×D₃₄₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×D₃₄₉	C₂×D₄	1	1	yes	characteristic
31	#13063703	—	4	4	no	—
32	C₄×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
34	Dic₃₄₉	C₄⋊C₄	1	1	yes	characteristic
35	Dic₃₄₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₃₄₉	—	2	1	no	—
37	C₂×C₃₄₉	C₂×C₄⋊C₄	1	1	yes	characteristic
38	D₃₄₉	—	2	1	no	—
39	D₃₄₉	#135	1	2	yes	—
40	C₃₄₉	#427	1	1	yes	characteristic, complemented, Sylow
41	#427	—	349	1	no	maximal, Sylow
42	C₂²×C₈	—	349	1	no	—
43	#117	—	349	2	no	—
44	#107	—	349	2	no	—
45	#106	—	349	2	no	—
46	C₂²×C₄	—	349	1	no	—
47	C₂×C₈	—	349	1	no	—
48	C₂×C₈	—	349	1	no	—
49	C₂×C₈	—	349	1	no	—
50	C₂×C₈	—	349	1	no	—
51	C₂×C₈	—	349	1	no	—
52	C₂×C₈	#13063703	1	1	yes	characteristic, complemented
53	C₄⋊C₄	—	349	4	no	—
54	C₂²⋊C₄	—	698	2	no	—
55	C₄²	—	698	2	no	—
56	C₂³	—	349	1	no	—
57	C₂×C₄	—	349	1	no	—
58	C₂×C₄	—	349	1	no	—
59	C₂×C₄	—	349	1	no	—
60	C₂×C₄	—	349	1	no	—
61	C₂×C₄	—	349	1	no	—
62	C₂×C₄	—	698	4	no	—
63	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
64	C₈	—	349	1	no	—
65	C₈	—	349	1	no	—
66	C₈	C₂×#13063703	1	1	yes	characteristic
67	C₈	C₂×#13063703	1	1	yes	characteristic
68	C₂²	—	349	1	no	—
69	C₂²	—	349	1	no	—
70	C₂²	—	698	2	no	—
71	C₂²	C₄⋊#13063703	1	1	yes	characteristic
72	C₄	—	349	1	no	—
73	C₄	—	349	1	no	—
74	C₄	—	1396	4	no	—
75	C₄	C₄⋊#13063703	1	1	yes	characteristic
76	C₄	C₂²×#13063703	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	349	2	no	—
79	C₂	—	698	1	no	—
80	C₂	C₂×C₄⋊#13063703	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊#13063703
commutator	C₄×C₃₄₉	C₂²×C₄
Frattini	C₄	C₂²×#13063703
Fitting	C₂×C₈×C₃₄₉	C₄
socle	C₂×C₃₄₉	C₂×C₄⋊C₄
prime radical	C₂²×D₃₄₉	D₄
cosocle	C₄×D₃₄₉	C₂³
hypercenter	C₂×C₈	#13063703
hypocenter	C₃₄₉	#427
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. #26013708	C₂	semidirect
3. #26013707	C₂	semidirect
9. C₂×#14584095	C₄	semidirect
10. C₈×D₃₄₉	C₄	semidirect
11. C₈×D₃₄₉	C₄	semidirect
14. C₂×C₈×C₃₄₉	C₄	semidirect
40. C₃₄₉	#427	semidirect
52. 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
#26013718	2
#26013708	2
#26013707	2
C₂×C₈×D₃₄₉	2
#427	349