Subgroups — C₂×C₄×#14582115 #36342415

Order 21568.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14582115	C₂	1	2	yes	maximal
2	C₄×#14582115	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×#13062438	C₂	1	1	yes	maximal, characteristic
4	C₂×#14582115	C₄	1	8	yes	direct product factor
5	C₂×#14582115	C₂²	1	4	yes	—
6	C₂²×#13062438	C₄	1	1	yes	characteristic
7	C₂²×#13062438	C₂²	1	1	yes	characteristic
8	C₄×#13062438	C₄	1	2	yes	—
9	C₄×#13062438	C₂²	1	2	yes	—
10	C₂×C₄×D₃₃₇	C₄	1	1	yes	characteristic
11	#14582115	C₂×C₄	1	8	yes	direct product factor
12	C₂×#13062438	C₈	1	4	yes	complemented
13	C₂×#13062438	C₂×C₄	1	1	yes	characteristic
14	C₂×#13062438	C₂×C₄	1	4	yes	—
15	C₂×#13062438	C₂×C₄	1	2	yes	—
16	C₂×#13062438	C₂³	1	1	yes	characteristic
17	C₂²×D₃₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₃₃₇	C₈	1	1	yes	characteristic, complemented
19	C₄×D₃₃₇	C₈	1	2	yes	complemented
20	C₄×D₃₃₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₃₃₇	C₈	1	1	yes	characteristic, complemented
22	C₂²×C₃₃₇	C₂×C₈	1	1	yes	characteristic
23	C₂×D₃₃₇	C₄²	1	2	yes	—
24	C₂×D₃₃₇	C₂×C₈	1	1	yes	characteristic
25	C₂×D₃₃₇	C₂×C₈	1	2	yes	—
26	C₂×D₃₃₇	C₂²×C₄	1	1	yes	characteristic
27	#13062438	C₄²	1	2	yes	—
28	#13062438	C₂×C₈	1	4	yes	complemented
29	#13062438	C₂²×C₄	1	2	yes	—
30	C₄×C₃₃₇	C₂×C₈	1	2	yes	complemented
31	Dic₃₃₇	C₂×C₈	1	2	yes	complemented
32	C₂×C₃₃₇	C₄×C₈	1	2	yes	complemented
33	C₂×C₃₃₇	C₂²×C₈	1	1	yes	characteristic
34	D₃₃₇	C₄×C₈	1	2	yes	complemented
35	D₃₃₇	C₂×C₄²	1	1	yes	characteristic
36	D₃₃₇	C₂²×C₈	1	1	yes	characteristic
37	C₃₃₇	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
38	C₂×C₄×C₈	—	337	1	no	maximal, Sylow
39	C₂²×C₈	—	337	2	no	—
40	C₂×C₄²	—	337	1	no	—
41	C₄×C₈	—	337	4	no	—
42	C₂²×C₄	—	337	1	no	—
43	C₂²×C₄	—	337	1	no	—
44	C₂²×C₄	—	337	1	no	—
45	C₂×C₈	—	337	4	no	—
46	C₂×C₈	—	337	8	no	—
47	C₄²	—	337	2	no	—
48	C₄²	—	337	2	no	—
49	C₂³	—	337	1	no	—
50	C₂×C₄	—	337	1	no	—
51	C₂×C₄	—	337	1	no	—
52	C₂×C₄	—	337	2	no	—
53	C₂×C₄	—	337	4	no	—
54	C₂×C₄	—	337	4	no	—
55	C₂×C₄	—	337	1	no	—
56	C₂×C₄	—	337	2	no	—
57	C₂×C₄	—	337	2	no	—
58	C₂×C₄	#14582115	1	1	yes	characteristic, direct product factor
59	C₈	—	337	8	no	—
60	C₂²	—	337	2	no	—
61	C₂²	—	337	2	no	—
62	C₂²	—	337	1	no	—
63	C₂²	—	337	1	no	—
64	C₂²	C₂×#14582115	1	1	yes	characteristic
65	C₄	—	337	2	no	—
66	C₄	—	337	2	no	—
67	C₄	—	337	4	no	—
68	C₄	—	337	2	no	—
69	C₄	C₂×#14582115	1	2	yes	direct product factor
70	C₂	—	337	1	no	—
71	C₂	—	337	1	no	—
72	C₂	—	337	2	no	—
73	C₂	C₄×#14582115	1	2	yes	direct product factor
74	C₂	C₂²×#14582115	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄	#14582115
commutator, hypocenter	C₃₃₇	C₂×C₄×C₈
Frattini	C₂	C₂²×#14582115
Fitting	C₂×C₄×C₃₃₇	C₈
socle	C₂²×C₃₃₇	C₂×C₈
prime radical	C₂²×D₃₃₇	C₂×C₄
cosocle	C₂×#13062438	C₂³
LCS	G ▷ C₃₃₇
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₄×#14582115	C₂	direct (N × H)
4. C₂×#14582115	C₄	direct (N × H)
11. #14582115	C₂×C₄	direct (N × H)
12. C₂×#13062438	C₈	semidirect
18. C₂×Dic₃₃₇	C₈	semidirect
19. C₄×D₃₃₇	C₈	semidirect
21. C₂×C₄×C₃₃₇	C₈	semidirect
28. #13062438	C₂×C₈	semidirect
30. C₄×C₃₃₇	C₂×C₈	semidirect
31. Dic₃₃₇	C₂×C₈	semidirect
32. C₂×C₃₃₇	C₄×C₈	semidirect
34. D₃₃₇	C₄×C₈	semidirect
37. C₃₃₇	C₂×C₄×C₈	semidirect
58. C₂×C₄	#14582115	direct (N × H)
69. C₄	C₂×#14582115	direct (N × H)
73. C₂	C₄×#14582115	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14582115	2
C₄×#14582115	2
C₂×C₄×#13062438	2
C₂×C₄×C₈	337

Subgroups — C2×C4×#14582115 #36342415

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#14582115 #36342415