Subgroups — #31782647

Order 17536.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₀₉₆	C₂	1	1	yes	maximal, characteristic
2	#22577497	C₂	1	2	yes	maximal
3	#22577206	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#366×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₄₈	C₂²	1	2	yes	—
7	#17984053	—	2	2	no	—
8	Dic₁₀₉₆	C₄	1	4	yes	complemented
9	C₂×#14555485	C₂²	1	1	yes	characteristic
10	#17984011	—	2	2	no	—
11	C₈×Dic₁₃₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
13	#107×C₁₃₇	C₂²	1	2	yes	—
14	C₂×Dic₂₇₄	—	2	2	no	—
15	Dic₅₄₈	—	2	2	no	—
16	Dic₅₄₈	C₂×C₄	1	4	yes	—
17	C₂×#11764040	D₄	1	1	yes	characteristic
18	#14555489	—	4	2	no	—
19	C₄×Dic₁₃₇	—	4	2	no	—
20	C₄×Dic₁₃₇	D₄	1	1	yes	characteristic
21	#14555485	—	2	1	no	—
22	C₂×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₃₇	—	2	2	no	—
24	C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
25	Dic₂₇₄	—	2	4	no	—
26	Dic₂₇₄	—	4	2	no	—
27	C₂×Dic₁₃₇	—	4	2	no	—
28	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
29	C₂×Dic₁₃₇	D₈	1	1	yes	characteristic
30	#11764040	—	2	1	no	—
31	C₂×C₄×C₁₃₇	—	4	2	no	—
32	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₃₇	#41	1	1	yes	characteristic
35	C₂²×C₁₃₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₃₇	—	8	2	no	—
37	C₄×C₁₃₇	C₄×D₄	1	1	yes	characteristic
38	C₄×C₁₃₇	#135	1	1	yes	characteristic
39	Dic₁₃₇	—	2	1	no	—
40	Dic₁₃₇	—	4	4	no	—
41	Dic₁₃₇	Q₃₂	1	2	yes	complemented
42	C₂×C₁₃₇	C₄×D₈	1	1	yes	characteristic
43	C₂×C₁₃₇	C₂×Q₃₂	1	1	yes	characteristic
44	C₂×C₁₃₇	#508	1	1	yes	characteristic
45	C₁₃₇	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₃₂	—	137	1	no	maximal, Sylow
47	C₂×Q₃₂	—	137	1	no	—
48	C₄×Q₁₆	—	137	2	no	—
49	#366	D₁₃₇	1	1	yes	characteristic
50	#358	—	137	2	no	—
51	C₄×C₁₆	—	137	1	no	—
52	C₂×Q₁₆	—	137	2	no	—
53	C₄×Q₈	—	274	2	no	—
54	Q₃₂	—	137	4	no	—
55	C₂×C₁₆	—	137	1	no	—
56	C₂×C₁₆	C₂×D₁₃₇	1	1	yes	characteristic
57	#107	C₂×D₁₃₇	1	2	yes	—
58	#103	—	274	2	no	—
59	C₄×C₈	—	137	1	no	—
60	C₂×Q₈	—	274	2	no	—
61	Q₁₆	—	274	2	no	—
62	Q₁₆	—	137	4	no	—
63	C₂×C₈	—	137	1	no	—
64	C₂×C₈	C₂²×D₁₃₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	548	2	no	—
67	C₄²	—	548	2	no	—
68	C₄²	—	137	1	no	—
69	C₁₆	—	274	1	no	—
70	C₁₆	C₄×D₁₃₇	1	2	yes	—
71	Q₈	—	274	4	no	—
72	Q₈	—	548	2	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	548	2	no	—
75	C₂×C₄	—	137	1	no	—
76	C₂×C₄	—	137	1	no	—
77	C₂×C₄	D₄×D₁₃₇	1	1	yes	characteristic
78	C₈	—	274	1	no	—
79	C₈	C₂×C₄×D₁₃₇	1	1	yes	characteristic
80	C₈	#14555511	1	1	yes	characteristic
81	C₂²	D₈×D₁₃₇	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	274	1	no	—
84	C₄	—	548	4	no	—
85	C₄	—	137	2	no	—
86	C₄	#17984050	1	1	yes	characteristic
87	C₄	#17984111	1	1	yes	characteristic
88	C₂	#22577485	1	1	yes	characteristic
89	C₂	Q₃₂×D₁₃₇	1	1	yes	characteristic
90	C₂	#22577832	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₃₇
commutator	C₈×C₁₃₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₃₇
Fitting	#366×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	C₂×D₈
cosocle	C₂×C₈×C₁₃₇	C₂³
hypercenter	#366	D₁₃₇
hypocenter	C₁₃₇	C₄×Q₃₂
LCS	G ▷ C₈×C₁₃₇ ▷ C₄×C₁₃₇ ▷ C₂×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
8. Dic₁₀₉₆	C₄	semidirect
41. Dic₁₃₇	Q₃₂	semidirect
45. C₁₃₇	C₄×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₀₉₆	2
#22577497	2
#22577206	2
C₁₆×Dic₁₃₇	2
#366×C₁₃₇	2
C₄×Q₃₂	137