Subgroups — C₄²×#11764049 #31797921

Order 17536.

408 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11764049	C₂	1	6	yes	maximal
2	C₄²×#10587739	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11764049	C₂²	1	4	yes	—
4	C₄×#11764049	C₄	1	24	yes	direct product factor
5	C₂×C₄×#10587739	C₄	1	3	yes	—
6	C₂×C₄×#10587739	C₂²	1	3	yes	—
7	C₄²×D₁₃₇	C₄	1	1	yes	characteristic
8	C₂×#11764049	C₂×C₄	1	24	yes	—
9	C₂²×#10587739	C₂×C₄	1	3	yes	—
10	C₂²×#10587739	C₂³	1	1	yes	characteristic
11	C₄×#10587739	C₈	1	12	yes	complemented
12	C₄×#10587739	C₂×C₄	1	12	yes	—
13	C₂×C₄×D₁₃₇	C₂×C₄	1	3	yes	—
14	C₄×Dic₁₃₇	C₈	1	3	yes	complemented
15	C₄²×C₁₃₇	C₈	1	1	yes	characteristic, complemented
16	#11764049	C₄²	1	16	yes	direct product factor
17	C₂×#10587739	C₄²	1	6	yes	—
18	C₂×#10587739	C₂×C₈	1	12	yes	—
19	C₂×#10587739	C₂²×C₄	1	6	yes	—
20	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
21	C₂×Dic₁₃₇	C₂×C₈	1	3	yes	—
22	C₄×D₁₃₇	C₄²	1	6	yes	—
23	C₄×D₁₃₇	C₂×C₈	1	6	yes	—
24	C₂×C₄×C₁₃₇	C₂×C₈	1	3	yes	—
25	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
26	C₂×D₁₃₇	C₂×C₄²	1	3	yes	—
27	C₂×D₁₃₇	C₂²×C₈	1	3	yes	—
28	#10587739	C₄×C₈	1	12	yes	complemented
29	#10587739	C₂×C₄²	1	4	yes	—
30	C₄×C₁₃₇	C₄×C₈	1	6	yes	complemented
31	Dic₁₃₇	C₄×C₈	1	6	yes	complemented
32	C₂×C₁₃₇	C₂×C₄×C₈	1	3	yes	—
33	D₁₃₇	C₄³	1	1	yes	characteristic
34	D₁₃₇	C₂×C₄×C₈	1	3	yes	—
35	C₁₃₇	C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
36	C₄²×C₈	—	137	1	no	maximal, Sylow
37	C₂×C₄×C₈	—	137	6	no	—
38	C₄³	—	137	1	no	—
39	C₂²×C₈	—	137	4	no	—
40	C₂×C₄²	—	137	3	no	—
41	C₂×C₄²	—	137	3	no	—
42	C₂×C₄²	—	137	1	no	—
43	C₄×C₈	—	137	24	no	—
44	C₂²×C₄	—	137	1	no	—
45	C₂²×C₄	—	137	3	no	—
46	C₂²×C₄	—	137	3	no	—
47	C₂×C₈	—	137	24	no	—
48	C₄²	—	137	12	no	—
49	C₄²	—	137	12	no	—
50	C₄²	—	137	3	no	—
51	C₄²	#11764049	1	1	yes	characteristic, direct product factor
52	C₂³	—	137	1	no	—
53	C₂×C₄	—	137	6	no	—
54	C₂×C₄	—	137	6	no	—
55	C₂×C₄	—	137	12	no	—
56	C₂×C₄	—	137	3	no	—
57	C₂×C₄	—	137	6	no	—
58	C₂×C₄	—	137	6	no	—
59	C₂×C₄	C₂×#11764049	1	3	yes	—
60	C₈	—	137	16	no	—
61	C₂²	—	137	3	no	—
62	C₂²	—	137	3	no	—
63	C₂²	C₂²×#11764049	1	1	yes	characteristic
64	C₄	—	137	4	no	—
65	C₄	—	137	12	no	—
66	C₄	—	137	6	no	—
67	C₄	C₄×#11764049	1	6	yes	direct product factor
68	C₂	—	137	1	no	—
69	C₂	—	137	3	no	—
70	C₂	C₂×C₄×#11764049	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₄²	#11764049
commutator, hypocenter	C₁₃₇	C₄²×C₈
Frattini	C₂²	C₂²×#11764049
Fitting	C₄²×C₁₃₇	C₈
socle	C₂²×C₁₃₇	C₂²×C₈
prime radical	C₂²×D₁₃₇	C₂²×C₄
cosocle	C₂²×#10587739	C₂³
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₄²

Complements

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

N	H ≅ G/N	type
4. C₄×#11764049	C₄	direct (N × H)
11. C₄×#10587739	C₈	semidirect
14. C₄×Dic₁₃₇	C₈	semidirect
15. C₄²×C₁₃₇	C₈	semidirect
16. #11764049	C₄²	direct (N × H)
28. #10587739	C₄×C₈	semidirect
30. C₄×C₁₃₇	C₄×C₈	semidirect
31. Dic₁₃₇	C₄×C₈	semidirect
35. C₁₃₇	C₄²×C₈	semidirect
51. C₄²	#11764049	direct (N × H)
67. C₄	C₄×#11764049	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×#11764049	2
C₄²×#10587739	2
C₄²×C₈	137

Subgroups — C42×#11764049 #31797921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#11764049 #31797921