Subgroups — C₂×#26013708 #36375153

Order 22336.

130 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	#26013708	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×D₃₄₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×#13063703	—	2	2	no	—
5	C₄⋊#13063703	—	2	2	no	—
6	C₄⋊#13063703	C₂²	1	4	yes	—
7	C₂×C₄×D₃₄₉	C₂²	1	1	yes	characteristic
8	C₂×#14584095	C₄	1	1	yes	characteristic, complemented
9	C₈×D₃₄₉	C₄	1	2	yes	complemented
10	C₈×D₃₄₉	C₂²	1	2	yes	—
11	C₂×C₈×C₃₄₉	C₄	1	1	yes	characteristic, complemented
12	C₂×#13063703	—	2	4	no	—
13	C₂×#13063703	—	4	4	no	—
14	C₂²×D₃₄₉	D₄	1	1	yes	characteristic
15	C₂×Dic₃₄₉	Q₈	1	1	yes	characteristic
16	C₄×D₃₄₉	C₂×C₄	1	1	yes	characteristic
17	C₄×D₃₄₉	D₄	1	1	yes	characteristic
18	C₄×D₃₄₉	Q₈	1	1	yes	characteristic
19	C₄×D₃₄₉	C₂³	1	1	yes	characteristic
20	#14584095	C₂×C₄	1	2	yes	complemented
21	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
22	C₈×C₃₄₉	C₂×C₄	1	2	yes	complemented
23	C₂²×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
24	C₂×D₃₄₉	C₄⋊C₄	1	1	yes	characteristic
25	C₂×D₃₄₉	D₈	1	2	yes	—
26	C₂×D₃₄₉	Q₁₆	1	2	yes	—
27	C₂×D₃₄₉	C₂×D₄	1	1	yes	characteristic
28	#13063703	—	4	4	no	—
29	C₄×C₃₄₉	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
31	Dic₃₄₉	C₄⋊C₄	1	1	yes	characteristic
32	Dic₃₄₉	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₃₄₉	#107	1	2	yes	complemented
34	C₂×C₃₄₉	C₂×C₄⋊C₄	1	1	yes	characteristic
35	D₃₄₉	#107	1	2	yes	complemented
36	D₃₄₉	C₂×D₈	1	1	yes	characteristic
37	D₃₄₉	C₂×Q₁₆	1	1	yes	characteristic
38	C₃₄₉	C₂×#107	1	1	yes	characteristic, complemented, Sylow
39	C₂×#107	—	349	1	no	maximal, Sylow
40	C₂²×C₈	—	349	1	no	—
41	C₂×C₄⋊C₄	—	349	2	no	—
42	#107	—	349	4	no	—
43	C₂²×C₄	—	349	1	no	—
44	C₂²×C₄	—	698	2	no	—
45	C₂×C₈	—	349	1	no	—
46	C₂×C₈	—	349	2	no	—
47	C₂×C₈	—	349	2	no	—
48	C₂×C₈	#13063703	1	1	yes	characteristic, complemented
49	C₄⋊C₄	—	349	4	no	—
50	C₄⋊C₄	—	698	2	no	—
51	C₂³	—	349	1	no	—
52	C₂×C₄	—	349	1	no	—
53	C₂×C₄	—	349	1	no	—
54	C₂×C₄	—	349	1	no	—
55	C₂×C₄	—	349	1	no	—
56	C₂×C₄	—	349	1	no	—
57	C₂×C₄	—	1396	4	no	—
58	C₂×C₄	—	698	4	no	—
59	C₂×C₄	C₂×#13063703	1	1	yes	characteristic
60	C₈	—	349	2	no	—
61	C₈	C₂×#13063703	1	2	yes	complemented
62	C₂²	—	349	2	no	—
63	C₂²	—	349	2	no	—
64	C₂²	—	349	1	no	—
65	C₂²	—	349	1	no	—
66	C₂²	C₄⋊#13063703	1	1	yes	characteristic
67	C₄	—	349	1	no	—
68	C₄	—	349	1	no	—
69	C₄	—	1396	4	no	—
70	C₄	C₄⋊#13063703	1	1	yes	characteristic
71	C₄	C₂²×#13063703	1	1	yes	characteristic
72	C₂	—	349	1	no	—
73	C₂	—	349	2	no	—
74	C₂	—	349	1	no	—
75	C₂	#26013708	1	2	yes	direct product factor
76	C₂	C₂×C₄⋊#13063703	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#13063703
commutator	C₄×C₃₄₉	C₂²×C₄
Frattini	C₄	C₂²×#13063703
Fitting	C₂×C₈×C₃₄₉	C₄
socle	C₂²×C₃₄₉	C₄⋊C₄
prime radical	C₂²×D₃₄₉	D₄
cosocle	C₄×D₃₄₉	C₂³
hypercenter	C₂×C₈	#13063703
hypocenter	C₃₄₉	C₂×#107
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
2. #26013708	C₂	direct (N × H)
8. C₂×#14584095	C₄	semidirect
9. C₈×D₃₄₉	C₄	semidirect
11. C₂×C₈×C₃₄₉	C₄	semidirect
20. #14584095	C₂×C₄	semidirect
22. C₈×C₃₄₉	C₂×C₄	semidirect
33. C₂×C₃₄₉	#107	semidirect
35. D₃₄₉	#107	semidirect
38. C₃₄₉	C₂×#107	semidirect
48. C₂×C₈	#13063703	semidirect
61. C₈	C₂×#13063703	semidirect
75. C₂	#26013708	direct (N × H)

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
#26013708	2
C₂×C₈×D₃₄₉	2
C₂×#107	349

Subgroups — C2×#26013708 #36375153

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26013708 #36375153