Subgroups — C₂×C₄²×#10587739 #31797214

Order 17536.

1272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#10587739	C₂	1	6	yes	maximal
2	C₄²×#10587739	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄²×D₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂³×#10587739	C₂²	1	4	yes	—
5	C₂×C₄×#10587739	C₄	1	48	yes	direct product factor
6	C₂×C₄×#10587739	C₂²	1	24	yes	—
7	C₂²×C₄×D₁₃₇	C₂²	1	3	yes	—
8	C₂×C₄×Dic₁₃₇	C₄	1	3	yes	complemented
9	C₄²×D₁₃₇	C₄	1	4	yes	complemented
10	C₄²×D₁₃₇	C₂²	1	4	yes	—
11	C₂×C₄²×C₁₃₇	C₄	1	1	yes	characteristic, complemented
12	C₂²×#10587739	C₂×C₄	1	48	yes	—
13	C₂²×#10587739	C₂³	1	8	yes	—
14	C₄×#10587739	C₂×C₄	1	96	yes	direct product factor
15	C₂³×D₁₃₇	C₂³	1	1	yes	characteristic
16	C₂²×Dic₁₃₇	C₂×C₄	1	3	yes	—
17	C₂×C₄×D₁₃₇	C₂×C₄	1	24	yes	—
18	C₂×C₄×D₁₃₇	C₂×C₄	1	6	yes	—
19	C₂×C₄×D₁₃₇	C₂³	1	6	yes	—
20	C₄×Dic₁₃₇	C₂×C₄	1	12	yes	complemented
21	C₂²×C₄×C₁₃₇	C₂×C₄	1	3	yes	—
22	C₄²×C₁₃₇	C₂×C₄	1	4	yes	complemented
23	C₂×#10587739	C₄²	1	64	yes	direct product factor
24	C₂×#10587739	C₂²×C₄	1	48	yes	—
25	C₂²×D₁₃₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×D₁₃₇	C₂²×C₄	1	12	yes	—
27	C₂²×D₁₃₇	C₂⁴	1	1	yes	characteristic
28	C₂×Dic₁₃₇	C₄²	1	12	yes	complemented
29	C₂×Dic₁₃₇	C₂²×C₄	1	6	yes	—
30	C₄×D₁₃₇	C₄²	1	24	yes	complemented
31	C₄×D₁₃₇	C₂²×C₄	1	24	yes	—
32	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃₇	C₄²	1	12	yes	complemented
34	C₂×C₄×C₁₃₇	C₂²×C₄	1	6	yes	—
35	C₂²×C₁₃₇	C₂×C₄²	1	6	yes	—
36	C₂²×C₁₃₇	C₂³×C₄	1	1	yes	characteristic
37	C₂×D₁₃₇	C₂×C₄²	1	6	yes	—
38	C₂×D₁₃₇	C₂×C₄²	1	16	yes	—
39	C₂×D₁₃₇	C₂³×C₄	1	6	yes	—
40	#10587739	C₂×C₄²	1	32	yes	direct product factor
41	C₄×C₁₃₇	C₂×C₄²	1	12	yes	complemented
42	Dic₁₃₇	C₂×C₄²	1	12	yes	complemented
43	C₂×C₁₃₇	C₄³	1	4	yes	complemented
44	C₂×C₁₃₇	C₂²×C₄²	1	3	yes	—
45	D₁₃₇	C₄³	1	4	yes	complemented
46	D₁₃₇	C₂²×C₄²	1	4	yes	—
47	C₁₃₇	C₂×C₄³	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₄³	—	137	1	no	maximal, Sylow
49	C₂²×C₄²	—	137	6	no	—
50	C₂²×C₄²	—	137	1	no	—
51	C₄³	—	137	8	no	—
52	C₂³×C₄	—	137	4	no	—
53	C₂³×C₄	—	137	3	no	—
54	C₂×C₄²	—	137	24	no	—
55	C₂×C₄²	—	137	48	no	—
56	C₂×C₄²	—	137	3	no	—
57	C₂×C₄²	—	137	4	no	—
58	C₂×C₄²	—	137	4	no	—
59	C₂×C₄²	#10587739	1	1	yes	characteristic, direct product factor
60	C₂⁴	—	137	1	no	—
61	C₂²×C₄	—	137	8	no	—
62	C₂²×C₄	—	137	48	no	—
63	C₂²×C₄	—	137	3	no	—
64	C₂²×C₄	—	137	24	no	—
65	C₂²×C₄	—	137	6	no	—
66	C₂²×C₄	—	137	6	no	—
67	C₂²×C₄	C₂×#10587739	1	3	yes	—
68	C₄²	—	137	96	no	—
69	C₄²	—	137	12	no	—
70	C₄²	C₂×#10587739	1	4	yes	direct product factor
71	C₂³	—	137	12	no	—
72	C₂³	—	137	1	no	—
73	C₂³	—	137	1	no	—
74	C₂³	C₂²×#10587739	1	1	yes	characteristic
75	C₂×C₄	—	137	48	no	—
76	C₂×C₄	—	137	64	no	—
77	C₂×C₄	—	137	12	no	—
78	C₂×C₄	—	137	6	no	—
79	C₂×C₄	—	137	24	no	—
80	C₂×C₄	—	137	24	no	—
81	C₂×C₄	C₄×#10587739	1	12	yes	direct product factor
82	C₂×C₄	C₂²×#10587739	1	6	yes	—
83	C₂²	—	137	6	no	—
84	C₂²	—	137	6	no	—
85	C₂²	—	137	16	no	—
86	C₂²	C₂×C₄×#10587739	1	6	yes	—
87	C₂²	C₂³×#10587739	1	1	yes	characteristic
88	C₄	—	137	32	no	—
89	C₄	—	137	12	no	—
90	C₄	C₂×C₄×#10587739	1	12	yes	direct product factor
91	C₂	—	137	4	no	—
92	C₂	—	137	4	no	—
93	C₂	C₄²×#10587739	1	4	yes	direct product factor
94	C₂	C₂²×C₄×#10587739	1	3	yes	—

	subgroup	G/H
center, hypercenter	C₂×C₄²	#10587739
commutator, hypocenter	C₁₃₇	C₂×C₄³
Frattini	C₂²	C₂³×#10587739
Fitting	C₂×C₄²×C₁₃₇	C₄
socle	C₂³×C₁₃₇	C₂²×C₄
prime radical	C₂³×D₁₃₇	C₂³
cosocle	C₂²×D₁₃₇	C₂⁴
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₄²×#10587739	C₂	direct (N × H)
5. C₂×C₄×#10587739	C₄	direct (N × H)
8. C₂×C₄×Dic₁₃₇	C₄	semidirect
9. C₄²×D₁₃₇	C₄	semidirect
11. C₂×C₄²×C₁₃₇	C₄	semidirect
14. C₄×#10587739	C₂×C₄	direct (N × H)
20. C₄×Dic₁₃₇	C₂×C₄	semidirect
22. C₄²×C₁₃₇	C₂×C₄	semidirect
23. C₂×#10587739	C₄²	direct (N × H)
28. C₂×Dic₁₃₇	C₄²	semidirect
30. C₄×D₁₃₇	C₄²	semidirect
33. C₂×C₄×C₁₃₇	C₄²	semidirect
40. #10587739	C₂×C₄²	direct (N × H)
41. C₄×C₁₃₇	C₂×C₄²	semidirect
42. Dic₁₃₇	C₂×C₄²	semidirect
43. C₂×C₁₃₇	C₄³	semidirect
45. D₁₃₇	C₄³	semidirect
47. C₁₃₇	C₂×C₄³	semidirect
59. C₂×C₄²	#10587739	direct (N × H)
70. C₄²	C₂×#10587739	direct (N × H)
81. C₂×C₄	C₄×#10587739	direct (N × H)
90. C₄	C₂×C₄×#10587739	direct (N × H)
93. C₂	C₄²×#10587739	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×C₄×#10587739	2
C₄²×#10587739	2
C₂×C₄²×D₁₃₇	2
C₂×C₄³	137

Subgroups — C2×C42×#10587739 #31797214

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×#10587739 #31797214