Subgroups — #2030×C₁₃₇ #31777041

Order 17536.

290 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#543×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#381×C₁₃₇	C₂	1	3	yes	maximal
3	#342×C₁₃₇	C₂	1	3	yes	maximal
4	C₂³×C₄×C₁₃₇	C₂²	1	1	yes	characteristic
5	C₂²⋊Q₈×C₁₃₇	—	2	6	no	—
6	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	3	yes	—
7	C₂×C₂²⋊C₄×C₁₃₇	C₂²	1	3	yes	—
8	#95×C₁₃₇	—	2	6	no	—
9	C₂⁴×C₁₃₇	C₂³	1	1	yes	characteristic
10	C₂×Q₈×C₁₃₇	—	2	2	no	—
11	C₂²×C₄×C₁₃₇	—	2	3	no	—
12	C₂²×C₄×C₁₃₇	—	2	3	no	—
13	C₂²×C₄×C₁₃₇	—	2	3	no	—
14	C₂²×C₄×C₁₃₇	D₄	1	6	yes	—
15	C₂²×C₄×C₁₃₇	Q₈	1	2	yes	complemented
16	C₄⋊C₄×C₁₃₇	—	2	6	no	—
17	C₂²⋊C₄×C₁₃₇	—	2	6	no	—
18	C₂²⋊C₄×C₁₃₇	—	2	6	no	—
19	C₂³×C₁₃₇	—	4	2	no	—
20	C₂³×C₁₃₇	C₂×D₄	1	3	yes	—
21	C₂³×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
22	C₂³×C₁₃₇	#41	1	3	yes	—
23	Q₈×C₁₃₇	—	8	2	no	—
24	C₂×C₄×C₁₃₇	—	2	6	no	—
25	C₂×C₄×C₁₃₇	—	2	6	no	—
26	C₂×C₄×C₁₃₇	—	4	1	no	—
27	C₂×C₄×C₁₃₇	—	4	3	no	—
28	C₂×C₄×C₁₃₇	—	4	6	no	—
29	C₂×C₄×C₁₃₇	—	4	6	no	—
30	C₂×C₄×C₁₃₇	—	4	3	no	—
31	C₂²×C₁₃₇	—	2	6	no	—
32	C₂²×C₁₃₇	—	4	1	no	—
33	C₂²×C₁₃₇	—	4	3	no	—
34	C₂²×C₁₃₇	#120	1	1	yes	characteristic
35	C₂²×C₁₃₇	C₂²⋊Q₈	1	3	yes	—
36	C₂²×C₁₃₇	#124	1	3	yes	—
37	C₄×C₁₃₇	—	4	2	no	—
38	C₄×C₁₃₇	—	4	6	no	—
39	C₄×C₁₃₇	—	8	3	no	—
40	C₂×C₁₃₇	—	2	3	no	—
41	C₂×C₁₃₇	—	2	3	no	—
42	C₂×C₁₃₇	#393	1	1	yes	characteristic
43	C₂×C₁₃₇	#458	1	2	yes	—
44	C₁₃₇	#2030	1	1	yes	characteristic, direct product factor, Sylow
45	#2030	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	#543	C₂×C₁₃₇	1	1	yes	characteristic
47	#381	C₂×C₁₃₇	1	3	yes	—
48	#342	C₂×C₁₃₇	1	3	yes	—
49	C₂³×C₄	C₂²×C₁₃₇	1	1	yes	characteristic
50	C₂²⋊Q₈	—	2	6	no	—
51	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	3	yes	—
52	C₂×C₂²⋊C₄	C₂²×C₁₃₇	1	3	yes	—
53	#95	—	2	6	no	—
54	C₂⁴	C₂³×C₁₃₇	1	1	yes	characteristic
55	C₂×Q₈	—	2	2	no	—
56	C₂²×C₄	—	2	3	no	—
57	C₂²×C₄	—	2	3	no	—
58	C₂²×C₄	—	2	3	no	—
59	C₂²×C₄	D₄×C₁₃₇	1	6	yes	—
60	C₂²×C₄	Q₈×C₁₃₇	1	2	yes	complemented
61	C₄⋊C₄	—	2	6	no	—
62	C₂²⋊C₄	—	2	6	no	—
63	C₂²⋊C₄	—	2	6	no	—
64	C₂³	—	4	2	no	—
65	C₂³	C₂×D₄×C₁₃₇	1	3	yes	—
66	C₂³	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
67	C₂³	#41×C₁₃₇	1	3	yes	—
68	Q₈	—	8	2	no	—
69	C₂×C₄	—	2	6	no	—
70	C₂×C₄	—	2	6	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	—	4	3	no	—
73	C₂×C₄	—	4	6	no	—
74	C₂×C₄	—	4	6	no	—
75	C₂×C₄	—	4	3	no	—
76	C₂²	—	2	6	no	—
77	C₂²	—	4	1	no	—
78	C₂²	—	4	3	no	—
79	C₂²	#120×C₁₃₇	1	1	yes	characteristic
80	C₂²	C₂²⋊Q₈×C₁₃₇	1	3	yes	—
81	C₂²	#124×C₁₃₇	1	3	yes	—
82	C₄	—	4	2	no	—
83	C₄	—	4	6	no	—
84	C₄	—	8	3	no	—
85	C₂	—	2	3	no	—
86	C₂	—	2	3	no	—
87	C₂	#393×C₁₃₇	1	1	yes	characteristic
88	C₂	#458×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#120
commutator, Frattini, cosocle	C₂⁴	C₂³×C₁₃₇
prime radical	C₂⁴×C₁₃₇	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂⁴×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₂²×C₄×C₁₃₇	Q₈	semidirect
44. C₁₃₇	#2030	direct (N × H)
45. #2030	C₁₃₇	direct (N × H)
60. C₂²×C₄	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#543×C₁₃₇	2
#381×C₁₃₇	2
#342×C₁₃₇	2
#2030	137

Subgroups — #2030×C137 #31777041

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2030×C₁₃₇ #31777041