Subgroups — C₂×#22577236 #31783080

Order 17536.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×D₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#22577236	C₂	1	4	yes	maximal, direct product factor
3	C₂×#17983994	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₈×D₁₃₇	C₄	1	1	yes	characteristic
6	C₂×#14555485	C₄	1	1	yes	characteristic
7	C₁₆×D₁₃₇	C₄	1	2	yes	—
8	C₁₆×D₁₃₇	C₂²	1	2	yes	—
9	#17983994	C₂²	1	2	yes	complemented
10	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
11	C₃₂×C₁₃₇	C₂²	1	2	yes	complemented
12	C₂×C₄×D₁₃₇	C₈	1	1	yes	characteristic
13	C₂×#11764040	C₈	1	1	yes	characteristic
14	C₈×D₁₃₇	C₈	1	2	yes	—
15	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
16	#14555485	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₃₇	C₂³	1	1	yes	characteristic
20	C₂²×D₁₃₇	C₁₆	1	1	yes	characteristic
21	C₂×Dic₁₃₇	C₁₆	1	1	yes	characteristic
22	C₄×D₁₃₇	C₁₆	1	2	yes	—
23	C₄×D₁₃₇	C₂×C₈	1	2	yes	—
24	#11764040	C₂×C₈	1	2	yes	—
25	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
26	C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
27	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
29	C₂×D₁₃₇	—	2	2	no	—
30	C₂×D₁₃₇	C₂×C₁₆	1	2	yes	—
31	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
32	C₄×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
33	Dic₁₃₇	C₂×C₁₆	1	2	yes	—
34	C₂×C₁₃₇	M₆₄	1	2	yes	complemented
35	C₂×C₁₃₇	C₂²×C₁₆	1	1	yes	characteristic
36	D₁₃₇	—	2	2	no	—
37	C₁₃₇	C₂×M₆₄	1	1	yes	characteristic, complemented, Sylow
38	C₂×M₆₄	—	137	1	no	maximal, Sylow
39	C₂²×C₁₆	—	137	1	no	—
40	M₆₄	—	137	4	no	—
41	C₂×C₃₂	—	137	1	no	—
42	C₂×C₃₂	D₁₃₇	1	1	yes	characteristic, complemented
43	C₂²×C₈	—	137	1	no	—
44	C₂×C₁₆	—	137	1	no	—
45	C₂×C₁₆	—	137	2	no	—
46	C₂×C₁₆	—	137	2	no	—
47	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
48	C₃₂	—	137	2	no	—
49	C₃₂	C₂×D₁₃₇	1	2	yes	complemented
50	C₂²×C₄	—	137	1	no	—
51	C₂×C₈	—	137	1	no	—
52	C₂×C₈	—	137	2	no	—
53	C₂×C₈	—	137	2	no	—
54	C₂×C₈	C₄×D₁₃₇	1	1	yes	characteristic
55	C₁₆	—	137	2	no	—
56	C₁₆	C₄×D₁₃₇	1	1	yes	characteristic
57	C₁₆	C₂²×D₁₃₇	1	1	yes	characteristic
58	C₂³	—	137	1	no	—
59	C₂×C₄	—	137	1	no	—
60	C₂×C₄	—	137	2	no	—
61	C₂×C₄	—	137	2	no	—
62	C₂×C₄	C₈×D₁₃₇	1	1	yes	characteristic
63	C₈	—	137	2	no	—
64	C₈	C₈×D₁₃₇	1	1	yes	characteristic
65	C₈	C₂×C₄×D₁₃₇	1	1	yes	characteristic
66	C₂²	—	274	2	no	—
67	C₂²	—	137	2	no	—
68	C₂²	C₁₆×D₁₃₇	1	1	yes	characteristic
69	C₄	—	137	2	no	—
70	C₄	C₁₆×D₁₃₇	1	1	yes	characteristic
71	C₄	C₂×C₈×D₁₃₇	1	1	yes	characteristic
72	C₂	—	274	2	no	—
73	C₂	#22577236	1	2	yes	direct product factor
74	C₂	C₂×C₁₆×D₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂×D₁₃₇
commutator	C₂×C₁₃₇	C₂²×C₁₆
Frattini	C₁₆	C₂²×D₁₃₇
Fitting	C₂×C₃₂×C₁₃₇	C₂
socle	C₂²×C₁₃₇	C₂×C₁₆
prime radical	C₂²×D₁₃₇	C₁₆
cosocle	C₁₆×C₁₃₇	C₂³
hypercenter	C₂×C₃₂	D₁₃₇
hypocenter	C₁₃₇	C₂×M₆₄
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #22577236	C₂	direct (N × H)
3. C₂×#17983994	C₂	semidirect
4. C₂×C₃₂×C₁₃₇	C₂	semidirect
9. #17983994	C₂²	semidirect
11. C₃₂×C₁₃₇	C₂²	semidirect
34. C₂×C₁₃₇	M₆₄	semidirect
37. C₁₃₇	C₂×M₆₄	semidirect
42. C₂×C₃₂	D₁₃₇	semidirect
49. C₃₂	C₂×D₁₃₇	semidirect
73. C₂	#22577236	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×C₁₆×D₁₃₇	2
#22577236	2
C₂×#17983994	2
C₂×C₃₂×C₁₃₇	2
C₂×M₆₄	137

Subgroups — C2×#22577236 #31783080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22577236 #31783080