Subgroups — #31782660

Order 17536.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×Dic₁₃₇	C₂	1	2	yes	maximal
2	#22577231	C₂	1	1	yes	maximal, characteristic
3	#22577207	C₂	1	2	yes	maximal
4	#22577178	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₃₇	—	2	2	no	—
7	C₂×#14555485	C₂²	1	1	yes	characteristic
8	#17984022	C₂²	1	2	yes	—
9	#17984012	—	2	2	no	—
10	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
11	C₂×Q₁₆×C₁₃₇	C₂²	1	2	yes	—
12	Q₃₂×C₁₃₇	C₄	1	4	yes	complemented
13	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	C₂×#11764040	D₄	1	1	yes	characteristic
15	C₄⋊Dic₁₃₇	—	2	2	no	—
16	C₄⋊Dic₁₃₇	—	4	2	no	—
17	C₄×Dic₁₃₇	—	4	2	no	—
18	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
19	#14555485	—	2	1	no	—
20	C₂×Q₈×C₁₃₇	—	2	2	no	—
21	Q₁₆×C₁₃₇	—	2	2	no	—
22	Q₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
24	C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
25	C₂×Dic₁₃₇	—	4	2	no	—
26	C₂×Dic₁₃₇	D₈	1	2	yes	—
27	#11764040	—	2	1	no	—
28	Q₈×C₁₃₇	—	2	4	no	—
29	Q₈×C₁₃₇	—	4	2	no	—
30	C₂×C₄×C₁₃₇	—	4	2	no	—
31	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
32	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₃₇	#41	1	1	yes	characteristic
34	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
35	C₄×C₁₃₇	—	4	4	no	—
36	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
37	C₄×C₁₃₇	#135	1	1	yes	characteristic
38	Dic₁₃₇	—	2	2	no	—
39	Dic₁₃₇	—	8	2	no	—
40	C₂×C₁₃₇	C₄×D₈	1	1	yes	characteristic
41	C₂×C₁₃₇	#510	1	2	yes	—
42	C₁₃₇	#2176	1	1	yes	characteristic, complemented, Sylow
43	#2176	—	137	1	no	maximal, Sylow
44	C₂×Q₃₂	D₁₃₇	1	1	yes	characteristic
45	C₄×Q₁₆	—	137	2	no	—
46	#367	—	137	1	no	—
47	#358	—	137	2	no	—
48	#346	—	137	1	no	—
49	C₂×Q₁₆	C₂×D₁₃₇	1	2	yes	—
50	C₄×Q₈	—	274	2	no	—
51	Q₃₂	Dic₁₃₇	1	4	yes	complemented
52	C₂×C₁₆	—	137	1	no	—
53	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
54	#107	—	137	2	no	—
55	#103	—	274	2	no	—
56	C₄×C₈	—	137	1	no	—
57	C₂×Q₈	—	2	2	no	—
58	Q₁₆	—	2	2	no	—
59	Q₁₆	C₂×Dic₁₃₇	1	4	yes	—
60	C₂×C₈	—	137	1	no	—
61	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
62	C₄⋊C₄	—	274	2	no	—
63	C₄⋊C₄	—	548	2	no	—
64	C₄²	—	548	2	no	—
65	C₄²	—	137	1	no	—
66	C₁₆	—	274	1	no	—
67	C₁₆	C₂×Dic₁₃₇	1	2	yes	—
68	Q₈	—	2	4	no	—
69	Q₈	—	4	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	548	2	no	—
72	C₂×C₄	—	137	2	no	—
73	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
74	C₈	—	274	1	no	—
75	C₈	C₂²×Dic₁₃₇	1	1	yes	characteristic
76	C₈	#14555509	1	1	yes	characteristic
77	C₂²	D₈×D₁₃₇	1	1	yes	characteristic
78	C₄	—	4	4	no	—
79	C₄	—	274	2	no	—
80	C₄	—	1096	2	no	—
81	C₄	D₄×Dic₁₃₇	1	1	yes	characteristic
82	C₄	#17984107	1	1	yes	characteristic
83	C₂	D₈×Dic₁₃₇	1	1	yes	characteristic
84	C₂	#22577845	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₃₇
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	C₂×Q₃₂×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	C₂×D₈
cosocle	C₂×C₈×C₁₃₇	C₂³
hypercenter	C₂×Q₃₂	D₁₃₇
hypocenter	C₁₃₇	#2176
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
12. Q₃₂×C₁₃₇	C₄	semidirect
42. C₁₃₇	#2176	semidirect
51. Q₃₂	Dic₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×Dic₁₃₇	2
#22577231	2
#22577207	2
#22577178	2
C₂×Q₃₂×C₁₃₇	2
#2176	137