Subgroups — #45662605

Order 32896.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30656788	C₂	1	2	yes	maximal
2	C₄×C₈×D₂₅₇	C₂	1	1	yes	maximal, characteristic
3	#30656458	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×#14523339	C₂	1	2	yes	maximal, complemented
5	#322×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₂₅₇	C₄	1	2	yes	—
7	C₄²×D₂₅₇	C₂²	1	1	yes	characteristic
8	#22551521	C₄	1	4	yes	—
9	#22551504	C₄	1	4	yes	—
10	C₄×#14523339	C₂²	1	2	yes	—
11	C₄×#14523339	C₂²	1	1	yes	characteristic
12	C₈×Dic₂₅₇	C₄	1	2	yes	—
13	C₄×C₈×C₂₅₇	C₂²	1	2	yes	—
14	C₄×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₂₅₇	C₂×C₄	1	2	yes	—
16	C₂×C₄×D₂₅₇	C₂×C₄	1	1	yes	characteristic
17	C₂×#14523339	C₂×C₄	1	4	yes	—
18	C₂×#14523339	C₂×C₄	1	2	yes	—
19	C₈×D₂₅₇	C₈	1	8	yes	complemented
20	C₄×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₂₅₇	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₂₅₇	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₂₅₇	C₂×C₄	1	2	yes	—
24	C₄²×C₂₅₇	C₂³	1	1	yes	characteristic
25	C₂²×D₂₅₇	C₄²	1	1	yes	characteristic
26	C₂×Dic₂₅₇	C₄²	1	2	yes	—
27	C₂×Dic₂₅₇	C₄²	1	1	yes	characteristic
28	C₄×D₂₅₇	—	2	4	no	—
29	C₄×D₂₅₇	C₂×C₈	1	4	yes	—
30	#14523339	—	2	4	no	—
31	#14523339	C₂×C₈	1	4	yes	—
32	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₅₇	C₂²×C₄	1	2	yes	—
34	C₈×C₂₅₇	—	2	4	no	—
35	C₈×C₂₅₇	C₂×C₈	1	4	yes	—
36	C₂²×C₂₅₇	C₂×C₄²	1	1	yes	characteristic
37	C₂×D₂₅₇	—	2	2	no	—
38	C₂×D₂₅₇	C₄×C₈	1	2	yes	—
39	C₄×C₂₅₇	C₂²×C₈	1	2	yes	—
40	C₄×C₂₅₇	#131	1	4	yes	—
41	Dic₂₅₇	—	2	2	no	—
42	Dic₂₅₇	C₄×C₈	1	2	yes	—
43	C₂×C₂₅₇	C₂×C₄×C₈	1	1	yes	characteristic
44	C₂×C₂₅₇	#405	1	2	yes	—
45	D₂₅₇	—	2	2	no	—
46	C₂₅₇	#1450	1	1	yes	characteristic, complemented, Sylow
47	#1450	—	257	1	no	maximal, Sylow
48	#431	—	257	2	no	—
49	C₂×C₄×C₈	—	257	1	no	—
50	#322	—	257	1	no	—
51	#322	D₂₅₇	1	1	yes	characteristic, complemented
52	C₈²	—	257	2	no	—
53	C₂²×C₈	—	257	2	no	—
54	C₂×C₄²	—	257	1	no	—
55	C₄⋊C₈	—	257	4	no	—
56	C₂²⋊C₈	—	257	4	no	—
57	C₄×C₈	—	257	2	no	—
58	C₄×C₈	—	257	1	no	—
59	C₄×C₈	—	257	2	no	—
60	C₄×C₈	C₂×D₂₅₇	1	2	yes	—
61	C₄×C₈	C₂×D₂₅₇	1	1	yes	characteristic
62	C₂²×C₄	—	257	1	no	—
63	C₂²×C₄	—	257	2	no	—
64	C₂×C₈	—	257	2	no	—
65	C₂×C₈	—	257	8	no	—
66	C₂×C₈	—	257	4	no	—
67	C₂×C₈	C₄×D₂₅₇	1	4	yes	—
68	C₂×C₈	C₄×D₂₅₇	1	2	yes	—
69	C₄²	—	257	1	no	—
70	C₄²	—	257	2	no	—
71	C₄²	C₂²×D₂₅₇	1	1	yes	characteristic
72	C₂³	—	257	1	no	—
73	C₂×C₄	—	257	1	no	—
74	C₂×C₄	—	257	4	no	—
75	C₂×C₄	—	257	2	no	—
76	C₂×C₄	—	514	4	no	—
77	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
78	C₂×C₄	C₂×C₄×D₂₅₇	1	2	yes	—
79	C₈	—	257	4	no	—
80	C₈	—	2	4	no	—
81	C₈	—	514	4	no	—
82	C₈	C₈×D₂₅₇	1	4	yes	—
83	C₂²	—	257	2	no	—
84	C₂²	—	514	2	no	—
85	C₂²	C₄²×D₂₅₇	1	1	yes	characteristic
86	C₄	—	257	2	no	—
87	C₄	—	514	2	no	—
88	C₄	C₂×C₈×D₂₅₇	1	2	yes	—
89	C₄	#22551600	1	4	yes	—
90	C₂	—	514	2	no	—
91	C₂	C₄×C₈×D₂₅₇	1	1	yes	characteristic
92	C₂	#30656694	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₂₅₇
commutator	C₂×C₂₅₇	C₂×C₄×C₈
Fitting	#322×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₄²
prime radical	C₂²×D₂₅₇	C₄²
cosocle	C₄²×C₂₅₇	C₂³
hypercenter	#322	D₂₅₇
hypocenter	C₂₅₇	#1450
LCS	G ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
3. #30656458	C₂	semidirect
4. C₈×#14523339	C₂	semidirect
5. #322×C₂₅₇	C₂	semidirect
19. C₈×D₂₅₇	C₈	semidirect
46. C₂₅₇	#1450	semidirect
51. #322	D₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#30656788	2
C₄×C₈×D₂₅₇	2
#30656458	2
C₈×#14523339	2
#322×C₂₅₇	2
#1450	257