Subgroups — #2254×C₁₃₇ #31777265

Order 17536.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#500×C₁₃₇	C₂	1	2	yes	maximal
2	#367×C₁₃₇	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
5	C₂×C₁₆×C₁₃₇	C₂²	1	2	yes	—
6	#107×C₁₃₇	—	2	2	no	—
7	#107×C₁₃₇	C₂²	1	4	yes	—
8	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	C₂×Q₈×C₁₃₇	—	4	2	no	—
10	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
11	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
12	C₄⋊C₄×C₁₃₇	—	2	4	no	—
13	C₄⋊C₄×C₁₃₇	—	4	2	no	—
14	C₄²×C₁₃₇	D₄	1	1	yes	characteristic
15	C₁₆×C₁₃₇	Q₈	1	4	yes	complemented
16	Q₈×C₁₃₇	—	8	4	no	—
17	C₂×C₄×C₁₃₇	—	4	4	no	—
18	C₂×C₄×C₁₃₇	D₈	1	2	yes	—
19	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
20	C₈×C₁₃₇	Q₁₆	1	2	yes	—
21	C₈×C₁₃₇	C₂×Q₈	1	2	yes	—
22	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
23	C₄×C₁₃₇	—	8	4	no	—
24	C₄×C₁₃₇	SD₃₂	1	4	yes	—
25	C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
26	C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
27	C₂×C₁₃₇	#500	1	1	yes	characteristic
28	C₂×C₁₃₇	C₂×SD₃₂	1	2	yes	—
29	C₁₃₇	#2254	1	1	yes	characteristic, direct product factor, Sylow
30	#2254	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	#500	C₂×C₁₃₇	1	2	yes	—
32	#367	C₂×C₁₃₇	1	4	yes	—
33	C₄×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic
34	C₄⋊Q₈	—	2	2	no	—
35	C₂×C₁₆	C₂²×C₁₃₇	1	2	yes	—
36	#107	—	2	2	no	—
37	#107	C₂²×C₁₃₇	1	4	yes	—
38	C₄×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
39	C₂×Q₈	—	4	2	no	—
40	C₂×C₈	D₄×C₁₃₇	1	1	yes	characteristic
41	C₂×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
42	C₄⋊C₄	—	2	4	no	—
43	C₄⋊C₄	—	4	2	no	—
44	C₄²	D₄×C₁₃₇	1	1	yes	characteristic
45	C₁₆	Q₈×C₁₃₇	1	4	yes	complemented
46	Q₈	—	8	4	no	—
47	C₂×C₄	—	4	4	no	—
48	C₂×C₄	D₈×C₁₃₇	1	2	yes	—
49	C₂×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
50	C₈	Q₁₆×C₁₃₇	1	2	yes	—
51	C₈	C₂×Q₈×C₁₃₇	1	2	yes	—
52	C₂²	C₂×D₈×C₁₃₇	1	1	yes	characteristic
53	C₄	—	8	4	no	—
54	C₄	SD₃₂×C₁₃₇	1	4	yes	—
55	C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
56	C₄	C₂×Q₁₆×C₁₃₇	1	1	yes	characteristic
57	C₂	#500×C₁₃₇	1	1	yes	characteristic
58	C₂	C₂×SD₃₂×C₁₃₇	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₁₃₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ C₄×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₁₃₇	Q₈	semidirect
29. C₁₃₇	#2254	direct (N × H)
30. #2254	C₁₃₇	direct (N × H)
45. C₁₆	Q₈×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#500×C₁₃₇	2
#367×C₁₃₇	2
C₄×C₁₆×C₁₃₇	2
#2254	137

Subgroups — #2254×C137 #31777265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2254×C₁₃₇ #31777265