Subgroups — C₄×#17984183 #31797942

Order 17536.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17984183	C₂	1	2	yes	maximal
2	C₄×#14555522	C₂	1	4	yes	maximal, complemented
3	C₄×#14555518	C₂	1	1	yes	maximal, characteristic
4	#17984183	C₄	1	8	yes	direct product factor
5	C₂×#14555522	C₂²	1	4	yes	—
6	C₂×#14555518	C₄	1	1	yes	characteristic
7	C₂×#14555518	C₂²	1	1	yes	characteristic
8	C₄×#11764047	C₄	1	2	yes	—
9	C₄×#11764047	C₂²	1	2	yes	—
10	C₄²×D₁₃₇	C₄	1	1	yes	characteristic
11	#14555522	C₂×C₄	1	8	yes	complemented
12	#14555518	C₈	1	4	yes	—
13	#14555518	C₂×C₄	1	4	yes	—
14	C₂×#11764047	C₂×C₄	1	1	yes	characteristic
15	C₂×#11764047	C₂×C₄	1	2	yes	—
16	C₂×#11764047	C₂³	1	1	yes	characteristic
17	C₂×C₄×D₁₃₇	C₈	1	2	yes	—
18	C₂×C₄×D₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₁₃₇	C₈	1	1	yes	characteristic
20	C₄×Dic₁₃₇	C₂×C₄	1	2	yes	—
21	C₄²×C₁₃₇	C₈	1	1	yes	characteristic
22	#11764047	C₄²	1	2	yes	—
23	#11764047	C₂×C₈	1	4	yes	—
24	#11764047	C₂²×C₄	1	2	yes	—
25	C₂²×D₁₃₇	C₂×C₈	1	1	yes	characteristic
26	C₂×Dic₁₃₇	C₂×C₈	1	2	yes	—
27	C₂×Dic₁₃₇	C₂²×C₄	1	1	yes	characteristic
28	C₄×D₁₃₇	C₁₆	1	8	yes	complemented
29	C₄×D₁₃₇	C₄²	1	2	yes	—
30	C₄×D₁₃₇	C₂×C₈	1	2	yes	—
31	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₁₃₇	C₂×C₈	1	2	yes	—
33	C₂²×C₁₃₇	C₂²×C₈	1	1	yes	characteristic
34	C₂×D₁₃₇	C₄×C₈	1	2	yes	—
35	C₂×D₁₃₇	C₂×C₁₆	1	4	yes	—
36	C₄×C₁₃₇	C₄×C₈	1	2	yes	—
37	C₄×C₁₃₇	C₂×C₁₆	1	4	yes	complemented
38	Dic₁₃₇	C₂×C₁₆	1	4	yes	complemented
39	Dic₁₃₇	C₂×C₄²	1	1	yes	characteristic
40	Dic₁₃₇	C₂²×C₈	1	1	yes	characteristic
41	C₂×C₁₃₇	C₂×C₄×C₈	1	1	yes	characteristic
42	C₂×C₁₃₇	C₂²×C₁₆	1	2	yes	—
43	D₁₃₇	C₄×C₁₆	1	4	yes	complemented
44	C₁₃₇	C₂×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄×C₁₆	—	137	1	no	maximal, Sylow
46	C₂²×C₁₆	—	137	2	no	—
47	C₂×C₄×C₈	—	137	1	no	—
48	C₄×C₁₆	—	137	4	no	—
49	C₂²×C₈	—	137	1	no	—
50	C₂²×C₈	—	137	1	no	—
51	C₂×C₄²	—	137	1	no	—
52	C₂×C₁₆	—	137	4	no	—
53	C₂×C₁₆	—	137	8	no	—
54	C₄×C₈	—	137	2	no	—
55	C₄×C₈	—	137	2	no	—
56	C₂²×C₄	—	137	2	no	—
57	C₂²×C₄	—	137	1	no	—
58	C₂×C₈	—	137	1	no	—
59	C₂×C₈	—	137	1	no	—
60	C₂×C₈	—	137	2	no	—
61	C₂×C₈	—	137	4	no	—
62	C₂×C₈	—	137	4	no	—
63	C₄²	—	137	1	no	—
64	C₄²	—	137	2	no	—
65	C₄²	#11764049	1	1	yes	characteristic
66	C₁₆	—	137	8	no	—
67	C₂³	—	137	1	no	—
68	C₂×C₄	—	137	1	no	—
69	C₂×C₄	—	137	2	no	—
70	C₂×C₄	—	137	2	no	—
71	C₂×C₄	—	137	2	no	—
72	C₂×C₄	—	137	8	no	—
73	C₂×C₄	C₂×#11764049	1	1	yes	characteristic
74	C₂×C₄	C₂×#11764049	1	2	yes	—
75	C₈	—	137	2	no	—
76	C₈	—	137	2	no	—
77	C₈	—	137	4	no	—
78	C₂²	—	137	2	no	—
79	C₂²	—	137	4	no	—
80	C₂²	C₂²×#11764049	1	1	yes	characteristic
81	C₄	—	137	1	no	—
82	C₄	—	137	1	no	—
83	C₄	—	137	4	no	—
84	C₄	C₄×#11764049	1	2	yes	—
85	C₄	#17984183	1	4	yes	direct product factor
86	C₂	—	137	4	no	—
87	C₂	C₂×C₄×#11764049	1	1	yes	characteristic
88	C₂	C₂×#17984183	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₄²	#11764049
commutator, hypocenter	C₁₃₇	C₂×C₄×C₁₆
Frattini	C₂²	C₂²×#11764049
Fitting	C₄²×C₁₃₇	C₈
socle	C₂²×C₁₃₇	C₂²×C₈
prime radical	C₂²×D₁₃₇	C₂×C₈
cosocle	C₂×#11764047	C₂³
LCS	G ▷ C₁₃₇
UCS	C₁ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×#14555522	C₂	semidirect
4. #17984183	C₄	direct (N × H)
11. #14555522	C₂×C₄	semidirect
28. C₄×D₁₃₇	C₁₆	semidirect
37. C₄×C₁₃₇	C₂×C₁₆	semidirect
38. Dic₁₃₇	C₂×C₁₆	semidirect
43. D₁₃₇	C₄×C₁₆	semidirect
44. C₁₃₇	C₂×C₄×C₁₆	semidirect
85. C₄	#17984183	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×#17984183	2
C₄×#14555522	2
C₄×#14555518	2
C₂×C₄×C₁₆	137

Subgroups — C4×#17984183 #31797942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#17984183 #31797942