Subgroups — #45664069

Order 32896.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22551503	C₂	1	2	yes	maximal
2	#30656492	C₂	1	4	yes	maximal
3	C₄³×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#14523339	C₂²	1	4	yes	—
5	#22551503	C₄	1	8	yes	complemented
6	C₂×C₄²×C₂₅₇	C₄	1	4	yes	—
7	C₂×C₄²×C₂₅₇	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
9	C₂×#14523339	—	2	8	no	—
10	C₂×#14523339	C₂×C₄	1	8	yes	—
11	C₂²×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
12	C₂²×C₄×C₂₅₇	C₂×C₄	1	4	yes	—
13	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
14	C₄²×C₂₅₇	—	2	8	no	—
15	C₄²×C₂₅₇	—	2	4	no	—
16	C₄²×C₂₅₇	C₂×C₄	1	4	yes	—
17	#14523339	—	2	8	no	—
18	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
19	C₂×C₄×C₂₅₇	—	2	4	no	—
20	C₂×C₄×C₂₅₇	—	2	4	no	—
21	C₂×C₄×C₂₅₇	—	2	4	no	—
22	C₂×C₄×C₂₅₇	C₄²	1	4	yes	—
23	C₂×C₄×C₂₅₇	M₁₆	1	8	yes	—
24	C₂×C₄×C₂₅₇	C₂²×C₄	1	2	yes	—
25	C₂×C₄×C₂₅₇	#41	1	4	yes	—
26	C₂²×C₂₅₇	C₂×C₄²	1	1	yes	characteristic
27	C₂²×C₂₅₇	#117	1	2	yes	—
28	C₂²×C₂₅₇	C₂×M₁₆	1	4	yes	—
29	C₄×C₂₅₇	—	2	4	no	—
30	C₄×C₂₅₇	—	2	8	no	—
31	C₄×C₂₅₇	#117	1	4	yes	—
32	C₂×C₂₅₇	#376	1	1	yes	characteristic
33	C₂×C₂₅₇	C₄×M₁₆	1	2	yes	—
34	C₂×C₂₅₇	#432	1	4	yes	—
35	C₂₅₇	#1745	1	1	yes	characteristic, complemented, Sylow
36	#1745	—	257	1	no	maximal, Sylow
37	C₂×#97	—	257	2	no	—
38	C₄³	D₂₅₇	1	1	yes	characteristic
39	#336	—	257	4	no	—
40	C₂²×C₈	—	257	4	no	—
41	C₂×C₄²	Dic₂₅₇	1	4	yes	—
42	C₂×C₄²	C₂×D₂₅₇	1	2	yes	—
43	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
44	#97	—	257	8	no	—
45	C₂²×C₄	C₂×Dic₂₅₇	1	2	yes	—
46	C₂²×C₄	C₂×Dic₂₅₇	1	4	yes	—
47	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
48	C₂×C₈	—	257	8	no	—
49	C₂×C₈	—	514	8	no	—
50	C₄²	—	2	8	no	—
51	C₄²	—	2	4	no	—
52	C₄²	C₄×D₂₅₇	1	4	yes	—
53	C₂³	C₂²×Dic₂₅₇	1	1	yes	characteristic
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	4	no	—
57	C₂×C₄	C₄×Dic₂₅₇	1	4	yes	—
58	C₂×C₄	#16861602	1	8	yes	—
59	C₂×C₄	C₂×C₄×D₂₅₇	1	2	yes	—
60	C₂×C₄	#16861618	1	4	yes	—
61	C₈	—	514	8	no	—
62	C₂²	C₂×C₄×Dic₂₅₇	1	1	yes	characteristic
63	C₂²	#22551551	1	2	yes	—
64	C₂²	C₂×#16861602	1	4	yes	—
65	C₄	—	2	8	no	—
66	C₄	—	2	4	no	—
67	C₄	#22551549	1	4	yes	—
68	C₂	#30656583	1	1	yes	characteristic
69	C₂	C₄×#16861602	1	2	yes	—
70	C₂	#30656794	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₅₇
commutator	C₂²×C₂₅₇	C₂×C₄²
Fitting	C₄³×C₂₅₇	C₂
socle, prime radical	C₂³×C₂₅₇	C₂²×C₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	C₄³	D₂₅₇
hypocenter	C₂₅₇	#1745
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

Complements

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

N	H ≅ G/N	type
5. #22551503	C₄	semidirect
35. C₂₅₇	#1745	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22551503	2
#30656492	2
C₄³×C₂₅₇	2
#1745	257