Subgroups — #31780423

Order 17536.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22577799	C₂	1	2	yes	maximal
2	#22577783	C₂	1	1	yes	maximal, characteristic
3	#22577753	C₂	1	2	yes	maximal
4	#22577124	C₂	1	1	yes	maximal, characteristic
5	#498×C₁₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₄₈	—	2	2	no	—
7	#17984088	C₂²	1	1	yes	characteristic
8	#17984088	C₂²	1	1	yes	characteristic
9	#17984019	—	2	4	no	—
10	#17984012	—	2	4	no	—
11	C₄×#11764040	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₃₇	C₂²	1	2	yes	—
13	#106×C₁₃₇	—	2	2	no	—
14	C₄×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₇₄	—	2	1	no	—
16	C₂×Dic₂₇₄	—	2	1	no	—
17	Dic₅₄₈	—	4	4	no	—
18	C₂×#11764040	D₄	1	4	yes	—
19	C₄⋊Dic₁₃₇	—	2	2	no	—
20	C₄⋊Dic₁₃₇	—	2	2	no	—
21	C₂×Q₈×C₁₃₇	—	2	2	no	—
22	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₃₇	—	2	2	no	—
24	C₄⋊C₄×C₁₃₇	—	2	2	no	—
25	C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
26	Dic₂₇₄	—	4	2	no	—
27	Dic₂₇₄	—	4	2	no	—
28	C₂×Dic₁₃₇	—	4	1	no	—
29	C₂×Dic₁₃₇	—	4	1	no	—
30	#11764040	—	2	4	no	—
31	Q₈×C₁₃₇	—	4	4	no	—
32	C₂×C₄×C₁₃₇	—	4	2	no	—
33	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₃₇	SD₁₆	1	4	yes	—
36	C₂²×C₁₃₇	#127	1	1	yes	characteristic
37	C₄×C₁₃₇	—	8	2	no	—
38	C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
39	C₄×C₁₃₇	#137	1	4	yes	—
40	Dic₁₃₇	—	8	1	no	—
41	Dic₁₃₇	—	8	1	no	—
42	C₂×C₁₃₇	#492	1	1	yes	characteristic
43	C₂×C₁₃₇	#497	1	2	yes	—
44	C₁₃₇	#1716	1	1	yes	characteristic, complemented, Sylow
45	#1716	—	137	1	no	maximal, Sylow
46	#498	—	137	2	no	—
47	#498	D₁₃₇	1	1	yes	characteristic
48	#494	—	137	1	no	—
49	#487	—	137	2	no	—
50	#322	—	137	1	no	—
51	C₂×Q₁₆	—	274	2	no	—
52	C₄⋊Q₈	—	137	1	no	—
53	C₄⋊Q₈	—	137	1	no	—
54	C₄⋊Q₈	C₂×D₁₃₇	1	2	yes	—
55	#106	—	2	2	no	—
56	#106	—	274	4	no	—
57	#103	—	274	4	no	—
58	C₄×C₈	—	137	2	no	—
59	C₄×C₈	C₂×D₁₃₇	1	1	yes	characteristic
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	274	1	no	—
62	C₂×Q₈	—	274	1	no	—
63	Q₁₆	—	548	4	no	—
64	C₂×C₈	—	137	4	no	—
65	C₂×C₈	C₂²⋊D₁₃₇	1	2	yes	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	274	2	no	—
69	C₄⋊C₄	—	274	2	no	—
70	C₄²	C₂²×D₁₃₇	1	1	yes	characteristic
71	Q₈	—	4	4	no	—
72	Q₈	—	548	2	no	—
73	Q₈	—	548	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	548	1	no	—
76	C₂×C₄	—	548	1	no	—
77	C₂×C₄	D₄×D₁₃₇	1	2	yes	—
78	C₂×C₄	C₂×C₂²⋊D₁₃₇	1	1	yes	characteristic
79	C₈	—	274	4	no	—
80	C₈	#14555497	1	4	yes	—
81	C₂²	#17984084	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	1096	1	no	—
84	C₄	—	1096	1	no	—
85	C₄	C₂×#14555497	1	2	yes	—
86	C₄	#17984117	1	4	yes	—
87	C₂	#22577777	1	1	yes	characteristic
88	C₂	#22577794	1	2	yes	—

	subgroup	G/H
center	C₂²	#17984084
commutator, cosocle	C₄²×C₁₃₇	C₂³
Frattini	C₄²	C₂²×D₁₃₇
Fitting	#498×C₁₃₇	C₂
socle, prime radical	C₂²×C₁₃₇	#127
hypercenter	#498	D₁₃₇
hypocenter	C₁₃₇	#1716
LCS	G ▷ C₄²×C₁₃₇ ▷ C₂²×C₁₃₇ ▷ C₁₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

Complements

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

N	H ≅ G/N	type
44. C₁₃₇	#1716	semidirect

Maximal subgroups

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

subgroup	index
#22577799	2
#22577783	2
#22577753	2
#22577124	2
#498×C₁₃₇	2
#1716	137