Subgroups — #1418×C₁₃₇ #31776429

Order 17536.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#508×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	M₆₄×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#367×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#135×C₁₃₇	—	2	1	no	—
5	Q₃₂×C₁₃₇	C₄	1	1	yes	characteristic, complemented
6	SD₃₂×C₁₃₇	—	2	1	no	—
7	D₁₆×C₁₃₇	C₄	1	1	yes	characteristic, complemented
8	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
9	#107×C₁₃₇	—	2	1	no	—
10	C₃₂×C₁₃₇	—	2	1	no	—
11	#41×C₁₃₇	—	4	1	no	—
12	Q₁₆×C₁₃₇	—	2	1	no	—
13	SD₁₆×C₁₃₇	—	4	1	no	—
14	D₈×C₁₃₇	—	2	1	no	—
15	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
16	C₄⋊C₄×C₁₃₇	—	4	1	no	—
17	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
19	Q₈×C₁₃₇	—	4	1	no	—
20	D₄×C₁₃₇	—	4	1	no	—
21	D₄×C₁₃₇	—	8	1	no	—
22	C₂×C₄×C₁₃₇	—	8	1	no	—
23	C₂×C₄×C₁₃₇	—	8	1	no	—
24	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
25	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₃₇	—	8	1	no	—
28	C₂²×C₁₃₇	D₁₆	1	1	yes	characteristic
29	C₄×C₁₃₇	—	8	1	no	—
30	C₄×C₁₃₇	—	16	1	no	—
31	C₄×C₁₃₇	#102	1	1	yes	characteristic
32	C₄×C₁₃₇	SD₃₂	1	1	yes	characteristic
33	C₂×C₁₃₇	—	2	1	no	—
34	C₂×C₁₃₇	—	16	1	no	—
35	C₂×C₁₃₇	#357	1	1	yes	characteristic
36	C₁₃₇	#1418	1	1	yes	characteristic, direct product factor, Sylow
37	#1418	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	#508	C₂×C₁₃₇	1	1	yes	characteristic
39	M₆₄	C₂×C₁₃₇	1	1	yes	characteristic, complemented
40	#367	C₂×C₁₃₇	1	1	yes	characteristic, complemented
41	#135	—	2	1	no	—
42	Q₃₂	C₄×C₁₃₇	1	1	yes	characteristic, complemented
43	SD₃₂	—	2	1	no	—
44	D₁₆	C₄×C₁₃₇	1	1	yes	characteristic, complemented
45	C₂×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
46	#107	—	2	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	4	1	no	—
49	Q₁₆	—	2	1	no	—
50	SD₁₆	—	4	1	no	—
51	D₈	—	2	1	no	—
52	C₂×C₈	D₄×C₁₃₇	1	1	yes	characteristic
53	C₄⋊C₄	—	4	1	no	—
54	C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
55	C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
56	Q₈	—	4	1	no	—
57	D₄	—	4	1	no	—
58	D₄	—	8	1	no	—
59	C₂×C₄	—	8	1	no	—
60	C₂×C₄	—	8	1	no	—
61	C₂×C₄	D₈×C₁₃₇	1	1	yes	characteristic
62	C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
63	C₈	SD₁₆×C₁₃₇	1	1	yes	characteristic
64	C₂²	—	8	1	no	—
65	C₂²	D₁₆×C₁₃₇	1	1	yes	characteristic
66	C₄	—	8	1	no	—
67	C₄	—	16	1	no	—
68	C₄	#102×C₁₃₇	1	1	yes	characteristic
69	C₄	SD₃₂×C₁₃₇	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	16	1	no	—
72	C₂	#357×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃₇	#357
commutator	C₁₆	C₂×C₄×C₁₃₇
Frattini, cosocle	C₂×C₁₆	C₂²×C₁₃₇
prime radical	#508×C₁₃₇	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₇ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₈×C₁₃₇ ◁ C₂×C₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. M₆₄×C₁₃₇	C₂	semidirect
3. #367×C₁₃₇	C₂	semidirect
5. Q₃₂×C₁₃₇	C₄	semidirect
7. D₁₆×C₁₃₇	C₄	semidirect
36. C₁₃₇	#1418	direct (N × H)
37. #1418	C₁₃₇	direct (N × H)
39. M₆₄	C₂×C₁₃₇	semidirect
40. #367	C₂×C₁₃₇	semidirect
42. Q₃₂	C₄×C₁₃₇	semidirect
44. D₁₆	C₄×C₁₃₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#508×C₁₃₇	2
M₆₄×C₁₃₇	2
#367×C₁₃₇	2
#1418	137

Subgroups — #1418×C137 #31776429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1418×C₁₃₇ #31776429