Subgroups — #486×C₁₃₇ #22577021

Order 8768.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#127×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#102×C₁₃₇	C₂	1	4	yes	maximal, complemented
4	C₄×C₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×C₁₃₇	—	2	1	no	—
6	C₂×D₄×C₁₃₇	—	2	1	no	—
7	C₂×D₄×C₁₃₇	—	2	1	no	—
8	C₂×D₄×C₁₃₇	C₂²	1	2	yes	—
9	C₂×C₈×C₁₃₇	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₃₇	—	2	1	no	—
11	C₄⋊C₄×C₁₃₇	C₂²	1	2	yes	—
12	C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂³×C₁₃₇	—	2	2	no	—
14	Q₈×C₁₃₇	—	4	2	no	—
15	D₄×C₁₃₇	—	2	4	no	—
16	D₄×C₁₃₇	—	4	2	no	—
17	D₄×C₁₃₇	—	4	2	no	—
18	C₂×C₄×C₁₃₇	—	2	2	no	—
19	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₃₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
22	C₈×C₁₃₇	—	2	2	no	—
23	C₂²×C₁₃₇	—	4	2	no	—
24	C₂²×C₁₃₇	—	4	2	no	—
25	C₂²×C₁₃₇	—	4	2	no	—
26	C₂²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
27	C₄×C₁₃₇	—	4	2	no	—
28	C₄×C₁₃₇	D₈	1	2	yes	—
29	C₄×C₁₃₇	SD₁₆	1	2	yes	—
30	C₄×C₁₃₇	#41	1	2	yes	—
31	C₂×C₁₃₇	—	8	2	no	—
32	C₂×C₁₃₇	#124	1	1	yes	characteristic
33	C₂×C₁₃₇	C₂×D₈	1	1	yes	characteristic
34	C₂×C₁₃₇	C₂×SD₁₆	1	1	yes	characteristic
35	C₁₃₇	#486	1	1	yes	characteristic, direct product factor, Sylow
36	#486	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₄⋊Q₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
38	#127	C₂×C₁₃₇	1	1	yes	characteristic
39	#102	C₂×C₁₃₇	1	4	yes	complemented
40	C₄×C₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
41	C₂×Q₈	—	2	1	no	—
42	C₂×D₄	—	2	1	no	—
43	C₂×D₄	—	2	1	no	—
44	C₂×D₄	C₂²×C₁₃₇	1	2	yes	—
45	C₂×C₈	C₂²×C₁₃₇	1	2	yes	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂²×C₁₃₇	1	2	yes	—
48	C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
49	C₂³	—	2	2	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	2	4	no	—
52	D₄	—	4	2	no	—
53	D₄	—	4	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	D₄×C₁₃₇	1	1	yes	characteristic
56	C₂×C₄	D₄×C₁₃₇	1	1	yes	characteristic
57	C₂×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
58	C₈	—	2	2	no	—
59	C₂²	—	4	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	D₈×C₁₃₇	1	2	yes	—
65	C₄	SD₁₆×C₁₃₇	1	2	yes	—
66	C₄	#41×C₁₃₇	1	2	yes	—
67	C₂	—	8	2	no	—
68	C₂	#124×C₁₃₇	1	1	yes	characteristic
69	C₂	C₂×D₈×C₁₃₇	1	1	yes	characteristic
70	C₂	C₂×SD₁₆×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₁₃₇
prime radical	#127×C₁₃₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄⋊Q₈×C₁₃₇	C₂	semidirect
3. #102×C₁₃₇	C₂	semidirect
4. C₄×C₈×C₁₃₇	C₂	semidirect
35. C₁₃₇	#486	direct (N × H)
36. #486	C₁₃₇	direct (N × H)
37. C₄⋊Q₈	C₂×C₁₃₇	semidirect
39. #102	C₂×C₁₃₇	semidirect
40. C₄×C₈	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×C₁₃₇	2
#127×C₁₃₇	2
#102×C₁₃₇	2
C₄×C₈×C₁₃₇	2
#486	137

Subgroups — #486×C137 #22577021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #486×C₁₃₇ #22577021