Subgroups — C₄×#17983998 #31780473

Order 17536.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17983998	C₂	1	4	yes	maximal
2	C₂×C₄×#11764040	C₂	1	1	yes	maximal, characteristic
3	C₄²×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11764040	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₃₇	C₄	1	4	yes	—
7	C₂×C₄×Dic₁₃₇	C₂²	1	2	yes	—
8	#17983998	C₄	1	16	yes	direct product factor
9	C₄×#11764040	C₄	1	4	yes	complemented
10	C₂²×C₈×C₁₃₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₃₇	C₄	1	4	yes	complemented
13	C₂²×Dic₁₃₇	C₂×C₄	1	4	yes	—
14	C₂×#11764040	C₂×C₄	1	4	yes	—
15	C₂×#11764040	C₂×C₄	1	8	yes	—
16	C₄×Dic₁₃₇	—	2	8	no	—
17	C₄×Dic₁₃₇	C₂×C₄	1	8	yes	—
18	C₂²×C₄×C₁₃₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₁₃₇	C₂×C₄	1	8	yes	—
21	C₂×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
22	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
23	C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
24	C₂×Dic₁₃₇	—	2	8	no	—
25	C₂×Dic₁₃₇	C₄²	1	8	yes	—
26	#11764040	C₄²	1	8	yes	complemented
27	C₂³×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₃₇	C₄²	1	4	yes	—
29	C₂×C₄×C₁₃₇	M₁₆	1	8	yes	—
30	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₃₇	C₂²×C₄	1	4	yes	—
32	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₃₇	C₄²	1	8	yes	complemented
34	C₂²×C₁₃₇	C₂×C₄²	1	2	yes	—
35	C₂²×C₁₃₇	C₂×C₄²	1	1	yes	characteristic
36	C₂²×C₁₃₇	C₂×M₁₆	1	4	yes	—
37	C₄×C₁₃₇	#97	1	8	yes	complemented
38	C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
39	C₄×C₁₃₇	C₂×C₄²	1	2	yes	—
40	Dic₁₃₇	—	2	8	no	—
41	C₂×C₁₃₇	C₄³	1	1	yes	characteristic
42	C₂×C₁₃₇	C₂×#97	1	2	yes	—
43	C₂×C₁₃₇	C₄×M₁₆	1	4	yes	—
44	C₁₃₇	C₄×#97	1	1	yes	characteristic, complemented, Sylow
45	C₄×#97	—	137	1	no	maximal, Sylow
46	C₂×#97	—	137	4	no	—
47	C₂×C₄×C₈	—	137	1	no	—
48	C₂×C₄×C₈	D₁₃₇	1	1	yes	characteristic
49	C₄³	—	137	1	no	—
50	C₂²×C₈	—	137	2	no	—
51	C₂²×C₈	C₂×D₁₃₇	1	2	yes	—
52	C₂×C₄²	—	137	4	no	—
53	C₂×C₄²	—	137	2	no	—
54	C₂×C₄²	C₂×D₁₃₇	1	1	yes	characteristic
55	#97	—	137	16	no	—
56	C₄×C₈	—	137	4	no	—
57	C₄×C₈	Dic₁₃₇	1	4	yes	complemented
58	C₂²×C₄	—	137	4	no	—
59	C₂²×C₄	C₄×D₁₃₇	1	2	yes	—
60	C₂²×C₄	C₂²×D₁₃₇	1	1	yes	characteristic
61	C₂×C₈	—	137	4	no	—
62	C₂×C₈	—	137	8	no	—
63	C₂×C₈	C₄×D₁₃₇	1	8	yes	—
64	C₂×C₈	C₂×Dic₁₃₇	1	4	yes	—
65	C₄²	—	274	8	no	—
66	C₄²	—	137	8	no	—
67	C₄²	C₄×D₁₃₇	1	2	yes	—
68	C₄²	C₂×Dic₁₃₇	1	2	yes	—
69	C₂³	C₂×C₄×D₁₃₇	1	1	yes	characteristic
70	C₂×C₄	—	274	8	no	—
71	C₂×C₄	—	137	8	no	—
72	C₂×C₄	C₄×Dic₁₃₇	1	4	yes	—
73	C₂×C₄	#14555493	1	8	yes	—
74	C₂×C₄	C₂×C₄×D₁₃₇	1	1	yes	characteristic
75	C₂×C₄	C₂×C₄×D₁₃₇	1	4	yes	—
76	C₂×C₄	C₂²×Dic₁₃₇	1	1	yes	characteristic
77	C₈	—	137	8	no	—
78	C₈	C₄×Dic₁₃₇	1	8	yes	complemented
79	C₂²	C₄²×D₁₃₇	1	2	yes	—
80	C₂²	C₂×C₄×Dic₁₃₇	1	1	yes	characteristic
81	C₂²	C₂×#14555493	1	4	yes	—
82	C₄	—	274	8	no	—
83	C₄	#17983998	1	8	yes	direct product factor
84	C₄	C₄²×D₁₃₇	1	2	yes	—
85	C₄	C₂×C₄×Dic₁₃₇	1	2	yes	—
86	C₂	C₄²×Dic₁₃₇	1	1	yes	characteristic
87	C₂	C₂×#17983998	1	2	yes	—
88	C₂	C₄×#14555493	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₃₇
commutator	C₂×C₁₃₇	C₄³
Frattini	C₂²×C₄	C₂²×D₁₃₇
Fitting	C₂×C₄×C₈×C₁₃₇	C₂
socle, prime radical	C₂³×C₁₃₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₃₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₃₇
hypocenter	C₁₃₇	C₄×#97
LCS	G ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. #17983998	C₄	direct (N × H)
9. C₄×#11764040	C₄	semidirect
12. C₄×C₈×C₁₃₇	C₄	semidirect
26. #11764040	C₄²	semidirect
33. C₈×C₁₃₇	C₄²	semidirect
37. C₄×C₁₃₇	#97	semidirect
44. C₁₃₇	C₄×#97	semidirect
57. C₄×C₈	Dic₁₃₇	semidirect
78. C₈	C₄×Dic₁₃₇	semidirect
83. C₄	#17983998	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×#17983998	2
C₂×C₄×#11764040	2
C₄²×Dic₁₃₇	2
C₂×C₄×C₈×C₁₃₇	2
C₄×#97	137

Subgroups — C4×#17983998 #31780473

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17983998 #31780473