Subgroups — #2069×C₁₃₇ #31777080

Order 17536.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#430×C₁₃₇	C₂	1	3	yes	maximal, complemented
2	C₂×#100×C₁₃₇	C₂	1	3	yes	maximal
3	#338×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₄×C₁₃₇	C₂²	1	1	yes	characteristic
5	C₂×M₁₆×C₁₃₇	C₂²	1	3	yes	—
6	C₂×M₁₆×C₁₃₇	C₂²	1	3	yes	—
7	#108×C₁₃₇	—	2	6	no	—
8	#100×C₁₃₇	—	2	6	no	—
9	C₂⁴×C₁₃₇	Q₈	1	2	yes	—
10	C₂×D₄×C₁₃₇	—	2	3	no	—
11	C₂×D₄×C₁₃₇	—	2	3	no	—
12	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
13	M₁₆×C₁₃₇	—	2	3	no	—
14	M₁₆×C₁₃₇	—	2	3	no	—
15	M₁₆×C₁₃₇	—	2	3	no	—
16	M₁₆×C₁₃₇	D₄	1	6	yes	—
17	C₂×C₈×C₁₃₇	—	2	3	no	—
18	C₂×C₈×C₁₃₇	—	2	3	no	—
19	C₂³×C₁₃₇	—	2	6	no	—
20	C₂³×C₁₃₇	—	8	2	no	—
21	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
22	D₄×C₁₃₇	—	4	3	no	—
23	D₄×C₁₃₇	—	4	1	no	—
24	C₂×C₄×C₁₃₇	C₂×D₄	1	3	yes	—
25	C₂×C₄×C₁₃₇	#41	1	3	yes	—
26	C₈×C₁₃₇	—	2	6	no	—
27	C₈×C₁₃₇	—	4	3	no	—
28	C₂²×C₁₃₇	—	4	1	no	—
29	C₂²×C₁₃₇	—	4	2	no	—
30	C₂²×C₁₃₇	—	8	6	no	—
31	C₂²×C₁₃₇	C₂²⋊Q₈	1	3	yes	—
32	C₄×C₁₃₇	#121	1	3	yes	—
33	C₄×C₁₃₇	#126	1	1	yes	characteristic
34	C₂×C₁₃₇	—	2	3	no	—
35	C₂×C₁₃₇	—	8	2	no	—
36	C₂×C₁₃₇	#396	1	1	yes	characteristic
37	C₁₃₇	#2069	1	1	yes	characteristic, direct product factor, Sylow
38	#2069	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	#430	C₂×C₁₃₇	1	3	yes	complemented
40	C₂×#100	C₂×C₁₃₇	1	3	yes	—
41	#338	C₂×C₁₃₇	1	1	yes	characteristic, complemented
42	C₂²×D₄	C₂²×C₁₃₇	1	1	yes	characteristic
43	C₂×M₁₆	C₂²×C₁₃₇	1	3	yes	—
44	C₂×M₁₆	C₂²×C₁₃₇	1	3	yes	—
45	#108	—	2	6	no	—
46	#100	—	2	6	no	—
47	C₂⁴	Q₈×C₁₃₇	1	2	yes	—
48	C₂×D₄	—	2	3	no	—
49	C₂×D₄	—	2	3	no	—
50	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
51	M₁₆	—	2	3	no	—
52	M₁₆	—	2	3	no	—
53	M₁₆	—	2	3	no	—
54	M₁₆	D₄×C₁₃₇	1	6	yes	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	—	2	3	no	—
57	C₂³	—	2	6	no	—
58	C₂³	—	8	2	no	—
59	C₂³	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
60	D₄	—	4	3	no	—
61	D₄	—	4	1	no	—
62	C₂×C₄	C₂×D₄×C₁₃₇	1	3	yes	—
63	C₂×C₄	#41×C₁₃₇	1	3	yes	—
64	C₈	—	2	6	no	—
65	C₈	—	4	3	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	2	no	—
68	C₂²	—	8	6	no	—
69	C₂²	C₂²⋊Q₈×C₁₃₇	1	3	yes	—
70	C₄	#121×C₁₃₇	1	3	yes	—
71	C₄	#126×C₁₃₇	1	1	yes	characteristic
72	C₂	—	2	3	no	—
73	C₂	—	8	2	no	—
74	C₂	#396×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃₇	#396
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₃₇
prime radical	C₂²×D₄×C₁₃₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #430×C₁₃₇	C₂	semidirect
3. #338×C₁₃₇	C₂	semidirect
37. C₁₃₇	#2069	direct (N × H)
38. #2069	C₁₃₇	direct (N × H)
39. #430	C₂×C₁₃₇	semidirect
41. #338	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#430×C₁₃₇	2
C₂×#100×C₁₃₇	2
#338×C₁₃₇	2
#2069	137

Subgroups — #2069×C137 #31777080

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2069×C₁₃₇ #31777080