Subgroups — #31795859

Order 17536.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₃₇⋊#108	C₂	1	2	yes	maximal
2	C₂×#17984027	C₂	1	1	yes	maximal, characteristic
3	D₁₃₇⋊M₁₆	—	2	2	no	—
4	#17984159	—	2	2	no	—
5	#17984158	C₄	1	4	yes	—
6	C₂×C₈×D₁₃₇	C₂²	1	1	yes	characteristic
7	#17984027	—	2	2	no	—
8	C₂×#14555485	C₄	1	1	yes	characteristic
9	C₂×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
10	#14555520	—	2	4	no	—
11	#14555519	—	4	2	no	—
12	#14555518	—	4	2	no	—
13	C₂×C₄×D₁₃₇	D₄	1	1	yes	characteristic
14	C₂×#11764040	Q₈	1	1	yes	characteristic
15	C₈×D₁₃₇	C₂×C₄	1	2	yes	—
16	C₈×D₁₃₇	D₄	1	2	yes	—
17	#14555485	—	2	1	no	—
18	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
19	C₁₆×C₁₃₇	—	2	1	no	—
20	#11764047	—	4	4	no	—
21	C₂²×D₁₃₇	Q₁₆	1	1	yes	characteristic
22	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
23	C₄×D₁₃₇	C₂²⋊C₄	1	2	yes	—
24	C₄×D₁₃₇	SD₁₆	1	2	yes	—
25	#11764040	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
27	C₈×C₁₃₇	C₄²	1	1	yes	characteristic
28	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₁₃₇	#107	1	1	yes	characteristic
30	C₂×D₁₃₇	—	4	1	no	—
31	C₂×D₁₃₇	#103	1	2	yes	—
32	C₄×C₁₃₇	#95	1	1	yes	characteristic
33	C₄×C₁₃₇	#106	1	1	yes	characteristic
34	Dic₁₃₇	#102	1	2	yes	—
35	C₂×C₁₃₇	—	2	1	no	—
36	C₂×C₁₃₇	#340	1	1	yes	characteristic
37	D₁₃₇	—	2	2	no	—
38	C₁₃₇	#1385	1	1	yes	characteristic, complemented, Sylow
39	#1385	—	137	1	no	maximal, Sylow
40	C₂×M₃₂	—	137	1	no	—
41	C₂×#108	—	137	2	no	—
42	C₂×M₁₆	—	274	2	no	—
43	C₂²×C₈	—	137	1	no	—
44	M₃₂	—	274	2	no	—
45	C₂×C₁₆	—	137	1	no	—
46	C₂×C₁₆	#10587739	1	1	yes	characteristic
47	#108	—	274	2	no	—
48	#108	—	137	4	no	—
49	C₂²×C₄	—	137	1	no	—
50	M₁₆	—	274	4	no	—
51	M₁₆	—	548	2	no	—
52	C₂×C₈	—	548	2	no	—
53	C₂×C₈	—	137	1	no	—
54	C₂×C₈	—	137	2	no	—
55	C₂×C₈	—	137	2	no	—
56	C₂×C₈	C₂×#10587739	1	1	yes	characteristic
57	C₁₆	—	2	1	no	—
58	C₁₆	—	274	1	no	—
59	C₂³	—	137	1	no	—
60	C₂×C₄	—	137	1	no	—
61	C₂×C₄	—	137	2	no	—
62	C₂×C₄	—	137	2	no	—
63	C₂×C₄	C₄⋊#10587739	1	1	yes	characteristic
64	C₈	—	548	4	no	—
65	C₈	—	137	2	no	—
66	C₈	C₄×#10587739	1	1	yes	characteristic
67	C₈	C₂²⋊#10587739	1	1	yes	characteristic
68	C₂²	—	548	1	no	—
69	C₂²	—	137	2	no	—
70	C₂²	#17984157	1	1	yes	characteristic
71	C₄	—	137	2	no	—
72	C₄	(C₂×#10587739)⋊C₄	1	1	yes	characteristic
73	C₄	#17984156	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	274	2	no	—
76	C₂	#22578277	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#17984156
commutator	C₈×C₁₃₇	C₄²
Frattini	C₂×C₈	C₂×#10587739
Fitting	C₂×C₁₆×C₁₃₇	C₄
socle	C₂×C₁₃₇	#340
prime radical	C₂²×D₁₃₇	Q₁₆
cosocle	C₂×C₈×D₁₃₇	C₂²
hypercenter	C₂×C₁₆	#10587739
hypocenter	C₁₃₇	#1385
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
38. C₁₃₇	#1385	semidirect

Maximal subgroups

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

subgroup	index
D₁₃₇⋊#108	2
C₂×#17984027	2
#1385	137