Subgroups — #45666078

Order 32896.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22551532	C₂	1	1	yes	maximal, characteristic
2	#30656692	C₂	1	1	yes	maximal, characteristic
3	#30656555	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#16861602	C₄	1	1	yes	characteristic
6	#22551551	C₄	1	1	yes	characteristic
7	#22551532	—	2	2	no	—
8	C₂×#16861593	C₂²	1	2	yes	—
9	#22551502	C₄	1	2	yes	—
10	C₈×Dic₂₅₇	C₂²	1	2	yes	—
11	C₂²×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₂₅₇	—	2	2	no	—
13	C₂×C₁₆×C₂₅₇	C₂²	1	2	yes	—
14	#16861602	—	2	2	no	—
15	C₂×#14523339	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₂₅₇	C₈	1	2	yes	—
17	C₂²⋊Dic₂₅₇	C₈	1	2	yes	—
18	C₄×Dic₂₅₇	C₂×C₄	1	2	yes	—
19	#16861593	—	2	2	no	—
20	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₂₅₇	D₄	1	2	yes	—
23	C₂×C₈×C₂₅₇	Q₈	1	2	yes	—
24	C₂×C₈×C₂₅₇	C₂³	1	1	yes	characteristic
25	C₁₆×C₂₅₇	—	2	2	no	—
26	C₂×Dic₂₅₇	C₂×C₈	1	4	yes	—
27	#14523339	—	2	2	no	—
28	C₂³×C₂₅₇	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₂₅₇	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₂₅₇	M₁₆	1	2	yes	—
32	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₂₅₇	C₄⋊C₄	1	2	yes	—
34	C₈×C₂₅₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₂₅₇	—	2	2	no	—
37	C₂²×C₂₅₇	C₄⋊C₈	1	2	yes	—
38	C₂²×C₂₅₇	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₂₅₇	C₄⋊C₈	1	2	yes	—
40	C₄×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₂₅₇	C₂×M₁₆	1	1	yes	characteristic
42	Dic₂₅₇	—	2	4	no	—
43	C₂×C₂₅₇	—	2	2	no	—
44	C₂×C₂₅₇	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₂₅₇	#504	1	2	yes	—
46	C₂₅₇	#2151	1	1	yes	characteristic, complemented, Sylow
47	#2151	—	257	1	no	maximal, Sylow
48	C₂×M₃₂	—	257	1	no	—
49	C₂²×C₁₆	D₂₅₇	1	1	yes	characteristic
50	#405	—	257	1	no	—
51	C₄⋊C₁₆	—	257	4	no	—
52	C₂×M₁₆	—	257	1	no	—
53	C₂²×C₈	C₂×D₂₅₇	1	1	yes	characteristic
54	#117	—	257	1	no	—
55	M₃₂	—	514	2	no	—
56	C₂×C₁₆	—	257	2	no	—
57	C₂×C₁₆	—	2	2	no	—
58	C₂×C₁₆	C₂×D₂₅₇	1	2	yes	—
59	#97	—	257	2	no	—
60	C₄×C₈	—	257	2	no	—
61	C₂²×C₄	C₄×D₂₅₇	1	1	yes	characteristic
62	M₁₆	—	514	2	no	—
63	C₂×C₈	—	257	2	no	—
64	C₂×C₈	C₄×D₂₅₇	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊D₂₅₇	1	2	yes	—
66	C₂×C₈	Dic₅₁₄	1	2	yes	—
67	C₂×C₈	C₂²×D₂₅₇	1	1	yes	characteristic
68	C₄⋊C₄	—	257	2	no	—
69	C₂²⋊C₄	—	257	2	no	—
70	C₄²	—	257	2	no	—
71	C₁₆	—	2	2	no	—
72	C₁₆	—	514	2	no	—
73	C₂³	C₈×D₂₅₇	1	1	yes	characteristic
74	C₂×C₄	—	257	4	no	—
75	C₂×C₄	#16861597	1	2	yes	—
76	C₂×C₄	C₈×D₂₅₇	1	1	yes	characteristic
77	C₂×C₄	#16861601	1	2	yes	—
78	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
79	C₈	—	514	2	no	—
80	C₈	#16861597	1	2	yes	—
81	C₈	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
82	C₈	C₂×Dic₅₁₄	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	#22551521	1	2	yes	—
85	C₂²	C₂×C₈×D₂₅₇	1	1	yes	characteristic
86	C₄	—	514	4	no	—
87	C₄	#22551521	1	2	yes	—
88	C₄	C₂×#16861597	1	1	yes	characteristic
89	C₄	C₂×#16861601	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×#22551521	1	1	yes	characteristic
92	C₂	#30657153	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₅₇
commutator, socle	C₂²×C₂₅₇	C₂²×C₈
Fitting	C₂²×C₁₆×C₂₅₇	C₂
prime radical	C₂³×C₂₅₇	C₂×C₈
cosocle	C₂×C₈×C₂₅₇	C₂³
hypercenter	C₂²×C₁₆	D₂₅₇
hypocenter	C₂₅₇	#2151
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #30656555	C₂	semidirect
46. C₂₅₇	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22551532	2
#30656692	2
#30656555	2
C₂²×C₁₆×C₂₅₇	2
#2151	257