Subgroups — #1711×C₁₃₇ #31776722

Order 17536.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#487×C₁₃₇	C₂	1	3	yes	maximal, complemented
2	#486×C₁₃₇	C₂	1	3	yes	maximal
3	C₈²×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₁₃₇	C₂²	1	3	yes	—
5	#127×C₁₃₇	C₂²	1	1	yes	characteristic
6	#103×C₁₃₇	—	2	6	no	—
7	#102×C₁₃₇	—	2	6	no	—
8	C₄×C₈×C₁₃₇	C₂²	1	3	yes	—
9	C₂×Q₈×C₁₃₇	—	2	3	no	—
10	C₂×D₄×C₁₃₇	—	2	3	no	—
11	C₂×C₈×C₁₃₇	D₄	1	6	yes	—
12	C₄⋊C₄×C₁₃₇	—	2	3	no	—
13	C₄⋊C₄×C₁₃₇	—	2	3	no	—
14	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
15	C₂³×C₁₃₇	—	4	1	no	—
16	Q₈×C₁₃₇	—	4	6	no	—
17	D₄×C₁₃₇	—	4	6	no	—
18	C₂×C₄×C₁₃₇	—	4	3	no	—
19	C₂×C₄×C₁₃₇	C₂×D₄	1	3	yes	—
20	C₈×C₁₃₇	—	2	6	no	—
21	C₂²×C₁₃₇	—	8	3	no	—
22	C₂²×C₁₃₇	#127	1	1	yes	characteristic
23	C₄×C₁₃₇	—	8	3	no	—
24	C₄×C₁₃₇	#135	1	6	yes	—
25	C₂×C₁₃₇	—	16	1	no	—
26	C₂×C₁₃₇	#495	1	3	yes	—
27	C₁₃₇	#1711	1	1	yes	characteristic, direct product factor, Sylow
28	#1711	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	#487	C₂×C₁₃₇	1	3	yes	complemented
30	#486	C₂×C₁₃₇	1	3	yes	—
31	C₈²	C₂×C₁₃₇	1	1	yes	characteristic, complemented
32	C₄⋊Q₈	C₂²×C₁₃₇	1	3	yes	—
33	#127	C₂²×C₁₃₇	1	1	yes	characteristic
34	#103	—	2	6	no	—
35	#102	—	2	6	no	—
36	C₄×C₈	C₂²×C₁₃₇	1	3	yes	—
37	C₂×Q₈	—	2	3	no	—
38	C₂×D₄	—	2	3	no	—
39	C₂×C₈	D₄×C₁₃₇	1	6	yes	—
40	C₄⋊C₄	—	2	3	no	—
41	C₄⋊C₄	—	2	3	no	—
42	C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
43	C₂³	—	4	1	no	—
44	Q₈	—	4	6	no	—
45	D₄	—	4	6	no	—
46	C₂×C₄	—	4	3	no	—
47	C₂×C₄	C₂×D₄×C₁₃₇	1	3	yes	—
48	C₈	—	2	6	no	—
49	C₂²	—	8	3	no	—
50	C₂²	#127×C₁₃₇	1	1	yes	characteristic
51	C₄	—	8	3	no	—
52	C₄	#135×C₁₃₇	1	6	yes	—
53	C₂	—	16	1	no	—
54	C₂	#495×C₁₃₇	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₁₃₇
prime radical	#127×C₁₃₇	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ 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. #487×C₁₃₇	C₂	semidirect
3. C₈²×C₁₃₇	C₂	semidirect
27. C₁₃₇	#1711	direct (N × H)
28. #1711	C₁₃₇	direct (N × H)
29. #487	C₂×C₁₃₇	semidirect
31. C₈²	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#487×C₁₃₇	2
#486×C₁₃₇	2
C₈²×C₁₃₇	2
#1711	137

Subgroups — #1711×C137 #31776722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1711×C₁₃₇ #31776722