Subgroups — C₄²×#11764047 #31796441

Order 17536.

408 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11764047	C₂	1	6	yes	maximal
2	C₄²×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11764047	C₂²	1	4	yes	—
4	C₄×#11764047	C₄	1	24	yes	direct product factor
5	C₂×C₄×Dic₁₃₇	C₄	1	3	yes	—
6	C₂×C₄×Dic₁₃₇	C₂²	1	3	yes	—
7	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic
8	C₂×#11764047	C₂×C₄	1	24	yes	—
9	C₂²×Dic₁₃₇	C₂×C₄	1	3	yes	—
10	C₂²×Dic₁₃₇	C₂³	1	1	yes	characteristic
11	C₄×Dic₁₃₇	C₈	1	12	yes	complemented
12	C₄×Dic₁₃₇	C₂×C₄	1	12	yes	—
13	C₂²×C₄×C₁₃₇	C₂×C₄	1	3	yes	—
14	C₄²×C₁₃₇	C₈	1	4	yes	complemented
15	#11764047	C₄²	1	16	yes	direct product factor
16	C₂×Dic₁₃₇	C₄²	1	6	yes	—
17	C₂×Dic₁₃₇	C₂×C₈	1	12	yes	—
18	C₂×Dic₁₃₇	C₂²×C₄	1	6	yes	—
19	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₃₇	C₄²	1	6	yes	—
21	C₂×C₄×C₁₃₇	C₂×C₈	1	6	yes	—
22	C₂×C₄×C₁₃₇	C₂×C₈	1	6	yes	—
23	C₂²×C₁₃₇	C₂×C₄²	1	3	yes	—
24	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
25	C₂²×C₁₃₇	C₂²×C₈	1	3	yes	—
26	C₄×C₁₃₇	C₄×C₈	1	12	yes	complemented
27	Dic₁₃₇	C₄×C₈	1	12	yes	complemented
28	Dic₁₃₇	C₂×C₄²	1	4	yes	—
29	C₂×C₁₃₇	C₄³	1	1	yes	characteristic
30	C₂×C₁₃₇	C₂×C₄×C₈	1	3	yes	—
31	C₂×C₁₃₇	C₂×C₄×C₈	1	3	yes	—
32	C₁₃₇	C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
33	C₄²×C₈	—	137	1	no	maximal, Sylow
34	C₂×C₄×C₈	—	137	6	no	—
35	C₄³	—	137	1	no	—
36	C₂²×C₈	—	137	4	no	—
37	C₂×C₄²	—	137	3	no	—
38	C₂×C₄²	—	137	3	no	—
39	C₂×C₄²	#10587739	1	1	yes	characteristic
40	C₄×C₈	—	137	24	no	—
41	C₂²×C₄	—	137	3	no	—
42	C₂²×C₄	—	137	1	no	—
43	C₂²×C₄	C₂×#10587739	1	3	yes	—
44	C₂×C₈	—	137	24	no	—
45	C₄²	—	137	12	no	—
46	C₄²	—	137	12	no	—
47	C₄²	#11764047	1	4	yes	direct product factor
48	C₂³	C₂²×#10587739	1	1	yes	characteristic
49	C₂×C₄	—	137	6	no	—
50	C₂×C₄	—	137	6	no	—
51	C₂×C₄	—	137	12	no	—
52	C₂×C₄	C₄×#10587739	1	6	yes	—
53	C₂×C₄	C₂×#11764047	1	6	yes	—
54	C₂×C₄	#14555518	1	6	yes	—
55	C₈	—	137	16	no	—
56	C₂²	C₂×C₄×#10587739	1	3	yes	—
57	C₂²	C₂²×#11764047	1	1	yes	characteristic
58	C₂²	C₂×#14555518	1	3	yes	—
59	C₄	—	137	4	no	—
60	C₄	—	137	12	no	—
61	C₄	C₄×#11764047	1	12	yes	direct product factor
62	C₂	C₄²×#10587739	1	1	yes	characteristic
63	C₂	C₂×C₄×#11764047	1	3	yes	—
64	C₂	C₄×#14555518	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄²	#10587739
commutator, hypocenter	C₁₃₇	C₄²×C₈
Frattini	C₂³	C₂²×#10587739
Fitting	C₂×C₄²×C₁₃₇	C₄
socle, prime radical	C₂³×C₁₃₇	C₂²×C₄
cosocle	C₂²×Dic₁₃₇	C₂³
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. C₄×#11764047	C₄	direct (N × H)
11. C₄×Dic₁₃₇	C₈	semidirect
14. C₄²×C₁₃₇	C₈	semidirect
15. #11764047	C₄²	direct (N × H)
26. C₄×C₁₃₇	C₄×C₈	semidirect
27. Dic₁₃₇	C₄×C₈	semidirect
32. C₁₃₇	C₄²×C₈	semidirect
47. C₄²	#11764047	direct (N × H)
61. C₄	C₄×#11764047	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×#11764047	2
C₄²×Dic₁₃₇	2
C₄²×C₈	137

Subgroups — C42×#11764047 #31796441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×#11764047 #31796441