Subgroups — #45664403

Order 32896.

272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30656701	C₂	1	2	yes	maximal
2	#30656506	C₂	1	4	yes	maximal, complemented
3	C₂×#117×C₂₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#16861602	C₂²	1	4	yes	—
5	C₂²⋊#14523339	—	2	4	no	—
6	C₂³×C₄×C₂₅₇	C₂²	1	1	yes	characteristic
7	#117×C₂₅₇	—	2	4	no	—
8	C₂×C₄⋊C₄×C₂₅₇	C₂²	1	2	yes	—
9	C₂×C₂²⋊C₄×C₂₅₇	C₄	1	2	yes	—
10	C₂×C₄²×C₂₅₇	C₄	1	2	yes	—
11	#16861602	—	2	8	no	—
12	C₂×#14523339	—	2	4	no	—
13	C₂⁴×C₂₅₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₅₇	—	2	4	no	—
15	C₂²×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₅₇	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₂₅₇	D₄	1	4	yes	—
18	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₂₅₇	—	2	4	no	—
20	C₂²⋊C₄×C₂₅₇	—	2	4	no	—
21	C₄²×C₂₅₇	—	2	4	no	—
22	#14523339	—	4	4	no	—
23	C₂³×C₂₅₇	—	2	2	no	—
24	C₂³×C₂₅₇	—	4	2	no	—
25	C₂³×C₂₅₇	C₂²⋊C₄	1	2	yes	—
26	C₂³×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₅₇	—	2	2	no	—
28	C₂×C₄×C₂₅₇	—	2	8	no	—
29	C₂×C₄×C₂₅₇	—	2	4	no	—
30	C₂×C₄×C₂₅₇	—	4	4	no	—
31	C₂×C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₂₅₇	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₂₅₇	#41	1	4	yes	—
34	C₂²×C₂₅₇	—	2	2	no	—
35	C₂²×C₂₅₇	—	2	2	no	—
36	C₂²×C₂₅₇	—	2	4	no	—
37	C₂²×C₂₅₇	—	4	4	no	—
38	C₂²×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₂₅₇	#117	1	2	yes	—
40	C₄×C₂₅₇	—	2	2	no	—
41	C₄×C₂₅₇	—	4	4	no	—
42	C₄×C₂₅₇	#123	1	4	yes	—
43	C₂×C₂₅₇	—	2	2	no	—
44	C₂×C₂₅₇	—	4	2	no	—
45	C₂×C₂₅₇	#381	1	1	yes	characteristic
46	C₂×C₂₅₇	#413	1	2	yes	—
47	C₂₅₇	#1822	1	1	yes	characteristic, complemented, Sylow
48	#1822	—	257	1	no	maximal, Sylow
49	C₂×#117	D₂₅₇	1	1	yes	characteristic
50	#407	—	257	2	no	—
51	#343	—	257	4	no	—
52	C₂³×C₄	C₂×D₂₅₇	1	1	yes	characteristic
53	C₂×M₁₆	—	257	4	no	—
54	#117	—	2	4	no	—
55	C₂×C₄⋊C₄	C₂×D₂₅₇	1	2	yes	—
56	C₂×C₂²⋊C₄	Dic₂₅₇	1	2	yes	—
57	C₂×C₄²	Dic₂₅₇	1	2	yes	—
58	C₂²⋊C₈	—	514	4	no	—
59	C₂⁴	C₂×Dic₂₅₇	1	1	yes	characteristic
60	C₂²×C₄	—	2	4	no	—
61	C₂²×C₄	C₂×Dic₂₅₇	1	1	yes	characteristic
62	C₂²×C₄	C₂×Dic₂₅₇	1	4	yes	—
63	C₂²×C₄	C₂²⋊D₂₅₇	1	4	yes	—
64	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
65	M₁₆	—	514	8	no	—
66	C₂×C₈	—	514	4	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₂²⋊C₄	—	2	4	no	—
69	C₄²	—	2	4	no	—
70	C₂³	—	2	2	no	—
71	C₂³	—	4	2	no	—
72	C₂³	C₂²⋊Dic₂₅₇	1	2	yes	—
73	C₂³	C₂²×Dic₂₅₇	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	8	no	—
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	4	4	no	—
78	C₂×C₄	C₂²⋊Dic₂₅₇	1	2	yes	—
79	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	2	yes	—
80	C₂×C₄	#16861618	1	4	yes	—
81	C₈	—	1028	4	no	—
82	C₂²	—	2	2	no	—
83	C₂²	—	2	2	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	4	4	no	—
86	C₂²	C₂×C₂²⋊Dic₂₅₇	1	1	yes	characteristic
87	C₂²	#22551551	1	2	yes	—
88	C₄	—	2	2	no	—
89	C₄	—	4	4	no	—
90	C₄	#22551575	1	4	yes	—
91	C₂	—	2	2	no	—
92	C₂	—	4	2	no	—
93	C₂	#30656600	1	1	yes	characteristic
94	C₂	#30656721	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₅₇
commutator	C₂³×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	C₂×#117×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₅₇	C₂×C₄
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	C₂×#117	D₂₅₇
hypocenter	C₂₅₇	#1822
LCS	G ▷ C₂³×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #30656506	C₂	semidirect
47. C₂₅₇	#1822	semidirect

Maximal subgroups

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

subgroup	index
#30656701	2
#30656506	2
C₂×#117×C₂₅₇	2
#1822	257