Subgroups — C₂³×#14555518 #31797773

Order 17536.

1976 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14555518	C₂	1	28	yes	maximal, direct product factor
2	C₂³×#11764047	C₂	1	2	yes	maximal, complemented
3	C₂³×C₄×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#14555518	C₂²	1	112	yes	direct product factor
5	C₂²×#11764047	C₂²	1	28	yes	complemented
6	C₂⁴×D₁₃₇	C₄	1	1	yes	characteristic
7	C₂³×Dic₁₃₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄×D₁₃₇	C₄	1	14	yes	—
9	C₂²×C₄×D₁₃₇	C₂²	1	14	yes	—
10	C₂³×C₄×C₁₃₇	C₄	1	1	yes	characteristic
11	#14555518	C₂³	1	64	yes	direct product factor
12	C₂×#11764047	C₂³	1	56	yes	complemented
13	C₂³×D₁₃₇	C₈	1	16	yes	complemented
14	C₂³×D₁₃₇	C₂×C₄	1	14	yes	—
15	C₂²×Dic₁₃₇	C₂×C₄	1	7	yes	—
16	C₂²×Dic₁₃₇	C₂³	1	7	yes	—
17	C₂×C₄×D₁₃₇	C₂×C₄	1	84	yes	—
18	C₂×C₄×D₁₃₇	C₂³	1	28	yes	—
19	C₂⁴×C₁₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₃₇	C₂×C₄	1	14	yes	—
21	#11764047	C₂⁴	1	16	yes	complemented
22	C₂²×D₁₃₇	C₂×C₈	1	112	yes	complemented
23	C₂²×D₁₃₇	C₂²×C₄	1	28	yes	—
24	C₂×Dic₁₃₇	C₂²×C₄	1	21	yes	—
25	C₂×Dic₁₃₇	C₂⁴	1	7	yes	—
26	C₄×D₁₃₇	C₂²×C₄	1	56	yes	—
27	C₄×D₁₃₇	C₂⁴	1	8	yes	—
28	C₂³×C₁₃₇	C₂×C₈	1	8	yes	complemented
29	C₂³×C₁₃₇	C₂²×C₄	1	7	yes	—
30	C₂×C₄×C₁₃₇	C₂²×C₄	1	28	yes	—
31	C₂²×C₁₃₇	C₂²×C₈	1	28	yes	complemented
32	C₂²×C₁₃₇	C₂³×C₄	1	7	yes	—
33	C₂×D₁₃₇	C₂²×C₈	1	112	yes	complemented
34	C₂×D₁₃₇	C₂³×C₄	1	8	yes	—
35	C₄×C₁₃₇	C₂³×C₄	1	8	yes	—
36	Dic₁₃₇	C₂³×C₄	1	7	yes	—
37	Dic₁₃₇	C₂⁵	1	1	yes	characteristic
38	C₂×C₁₃₇	C₂³×C₈	1	14	yes	complemented
39	C₂×C₁₃₇	C₂⁴×C₄	1	1	yes	characteristic
40	D₁₃₇	C₂³×C₈	1	16	yes	complemented
41	C₁₃₇	C₂⁴×C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂⁴×C₈	—	137	1	no	maximal, Sylow
43	C₂⁴×C₄	—	137	1	no	—
44	C₂³×C₈	—	137	2	no	—
45	C₂³×C₈	—	137	28	no	—
46	C₂⁵	—	137	1	no	—
47	C₂³×C₄	—	137	1	no	—
48	C₂³×C₄	—	137	14	no	—
49	C₂³×C₄	—	137	14	no	—
50	C₂³×C₄	#10587739	1	1	yes	characteristic
51	C₂²×C₈	—	137	28	no	—
52	C₂²×C₈	—	137	112	no	—
53	C₂⁴	—	137	16	no	—
54	C₂⁴	—	137	14	no	—
55	C₂⁴	C₂×#10587739	1	1	yes	characteristic
56	C₂²×C₄	—	137	7	no	—
57	C₂²×C₄	—	137	7	no	—
58	C₂²×C₄	—	137	84	no	—
59	C₂²×C₄	—	137	28	no	—
60	C₂²×C₄	C₂×#10587739	1	14	yes	—
61	C₂×C₈	—	137	56	no	—
62	C₂×C₈	—	137	64	no	—
63	C₂³	—	137	112	no	—
64	C₂³	—	137	28	no	—
65	C₂³	#14555518	1	8	yes	direct product factor
66	C₂³	C₂²×#10587739	1	7	yes	—
67	C₂×C₄	—	137	7	no	—
68	C₂×C₄	—	137	21	no	—
69	C₂×C₄	—	137	8	no	—
70	C₂×C₄	—	137	56	no	—
71	C₂×C₄	C₂²×#10587739	1	28	yes	—
72	C₈	—	137	16	no	—
73	C₂²	—	137	112	no	—
74	C₂²	—	137	8	no	—
75	C₂²	C₂×#14555518	1	28	yes	direct product factor
76	C₂²	C₂³×#10587739	1	7	yes	—
77	C₄	—	137	1	no	—
78	C₄	—	137	7	no	—
79	C₄	C₂³×#10587739	1	8	yes	—
80	C₂	—	137	16	no	—
81	C₂	C₂²×#14555518	1	14	yes	direct product factor
82	C₂	C₂⁴×#10587739	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	#10587739
commutator, hypocenter	C₁₃₇	C₂⁴×C₈
Frattini	C₂	C₂⁴×#10587739
Fitting	C₂³×C₄×C₁₃₇	C₄
socle	C₂⁴×C₁₃₇	C₂×C₄
prime radical	C₂⁴×D₁₃₇	C₄
cosocle	Dic₁₃₇	C₂⁵
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14555518	C₂	direct (N × H)
2. C₂³×#11764047	C₂	semidirect
4. C₂×#14555518	C₂²	direct (N × H)
5. C₂²×#11764047	C₂²	semidirect
11. #14555518	C₂³	direct (N × H)
12. C₂×#11764047	C₂³	semidirect
13. C₂³×D₁₃₇	C₈	semidirect
21. #11764047	C₂⁴	semidirect
22. C₂²×D₁₃₇	C₂×C₈	semidirect
28. C₂³×C₁₃₇	C₂×C₈	semidirect
31. C₂²×C₁₃₇	C₂²×C₈	semidirect
33. C₂×D₁₃₇	C₂²×C₈	semidirect
38. C₂×C₁₃₇	C₂³×C₈	semidirect
40. D₁₃₇	C₂³×C₈	semidirect
41. C₁₃₇	C₂⁴×C₈	semidirect
65. C₂³	#14555518	direct (N × H)
75. C₂²	C₂×#14555518	direct (N × H)
81. C₂	C₂²×#14555518	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×#14555518	2
C₂³×#11764047	2
C₂³×C₄×D₁₃₇	2
C₂⁴×C₈	137

Subgroups — C23×#14555518 #31797773

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14555518 #31797773