Subgroups — #30665595

Order 16768.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553708	C₂	1	2	yes	maximal
2	#22553692	C₂	1	1	yes	maximal, characteristic
3	#22553662	C₂	1	2	yes	maximal
4	#22553033	C₂	1	1	yes	maximal, characteristic
5	#498×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₂₄	—	2	2	no	—
7	#16862246	C₂²	1	1	yes	characteristic
8	#16862246	C₂²	1	1	yes	characteristic
9	#16862177	—	2	4	no	—
10	#16862170	—	2	4	no	—
11	C₄×C₁₃₁⋊C₈	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
13	#106×C₁₃₁	—	2	2	no	—
14	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₆₂	—	2	1	no	—
16	C₂×Dic₂₆₂	—	2	1	no	—
17	Dic₅₂₄	—	4	4	no	—
18	C₂×C₁₃₁⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₁₃₁	—	2	2	no	—
20	C₄⋊Dic₁₃₁	—	2	2	no	—
21	C₂×Q₈×C₁₃₁	—	2	2	no	—
22	C₂×C₈×C₁₃₁	D₄	1	2	yes	—
23	C₄⋊C₄×C₁₃₁	—	2	2	no	—
24	C₄⋊C₄×C₁₃₁	—	2	2	no	—
25	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
26	Dic₂₆₂	—	4	2	no	—
27	Dic₂₆₂	—	4	2	no	—
28	C₂×Dic₁₃₁	—	4	1	no	—
29	C₂×Dic₁₃₁	—	4	1	no	—
30	C₁₃₁⋊C₈	—	2	4	no	—
31	Q₈×C₁₃₁	—	4	4	no	—
32	C₂×C₄×C₁₃₁	—	4	2	no	—
33	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₃₁	SD₁₆	1	4	yes	—
36	C₂²×C₁₃₁	#127	1	1	yes	characteristic
37	C₄×C₁₃₁	—	8	2	no	—
38	C₄×C₁₃₁	C₂×SD₁₆	1	2	yes	—
39	C₄×C₁₃₁	#137	1	4	yes	—
40	Dic₁₃₁	—	8	1	no	—
41	Dic₁₃₁	—	8	1	no	—
42	C₂×C₁₃₁	#492	1	1	yes	characteristic
43	C₂×C₁₃₁	#497	1	2	yes	—
44	C₁₃₁	#1716	1	1	yes	characteristic, complemented, Sylow
45	#1716	—	131	1	no	maximal, Sylow
46	#498	—	131	2	no	—
47	#498	D₁₃₁	1	1	yes	characteristic
48	#494	—	131	1	no	—
49	#487	—	131	2	no	—
50	#322	—	131	1	no	—
51	C₂×Q₁₆	—	262	2	no	—
52	C₄⋊Q₈	—	131	1	no	—
53	C₄⋊Q₈	—	131	1	no	—
54	C₄⋊Q₈	C₂×D₁₃₁	1	2	yes	—
55	#106	—	2	2	no	—
56	#106	—	262	4	no	—
57	#103	—	262	4	no	—
58	C₄×C₈	—	131	2	no	—
59	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	262	1	no	—
62	C₂×Q₈	—	262	1	no	—
63	Q₁₆	—	524	4	no	—
64	C₂×C₈	—	131	4	no	—
65	C₂×C₈	C₂²⋊D₁₃₁	1	2	yes	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	262	2	no	—
69	C₄⋊C₄	—	262	2	no	—
70	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
71	Q₈	—	4	4	no	—
72	Q₈	—	524	2	no	—
73	Q₈	—	524	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	524	1	no	—
76	C₂×C₄	—	524	1	no	—
77	C₂×C₄	D₄×D₁₃₁	1	2	yes	—
78	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
79	C₈	—	262	4	no	—
80	C₈	#14523500	1	4	yes	—
81	C₂²	#16862242	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	1048	1	no	—
84	C₄	—	1048	1	no	—
85	C₄	C₂×#14523500	1	2	yes	—
86	C₄	#16862275	1	4	yes	—
87	C₂	#22553686	1	1	yes	characteristic
88	C₂	#22553703	1	2	yes	—

	subgroup	G/H
center	C₂²	#16862242
commutator, cosocle	C₄²×C₁₃₁	C₂³
Frattini	C₄²	C₂²×D₁₃₁
Fitting	#498×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	#127
hypercenter	#498	D₁₃₁
hypocenter	C₁₃₁	#1716
LCS	G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

Complements

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

N	H ≅ G/N	type
44. C₁₃₁	#1716	semidirect

Maximal subgroups

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

subgroup	index
#22553708	2
#22553692	2
#22553662	2
#22553033	2
#498×C₁₃₁	2
#1716	131