Subgroups — #45664457

Order 32896.

188 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30656761	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#14523339	C₂	1	1	yes	maximal, characteristic
3	#30656501	C₂	1	4	yes	maximal
4	#423×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#16861602	C₂²	1	2	yes	—
6	C₂²×#14523339	C₂²	1	2	yes	—
7	C₄⋊#14523339	—	2	2	no	—
8	C₄×#14523339	C₄	1	2	yes	—
9	C₄×#14523339	C₄	1	2	yes	—
10	C₂×M₁₆×C₂₅₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₂₅₇	—	2	2	no	—
13	#16861602	—	2	4	no	—
14	C₂×#14523339	—	2	2	no	—
15	C₂×#14523339	—	2	2	no	—
16	C₂×#14523339	—	2	2	no	—
17	C₂×#14523339	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₂₅₇	Q₈	1	2	yes	—
19	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
20	M₁₆×C₂₅₇	—	2	4	no	—
21	C₂×C₈×C₂₅₇	—	2	2	no	—
22	C₄²×C₂₅₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₂₅₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₅₇	D₄	1	2	yes	—
25	#14523339	—	2	4	no	—
26	#14523339	—	4	2	no	—
27	C₂³×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₂₅₇	—	2	4	no	—
29	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₂₅₇	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₅₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₅₇	#41	1	2	yes	—
34	C₂×C₄×C₂₅₇	#41	1	2	yes	—
35	C₈×C₂₅₇	—	4	2	no	—
36	C₂²×C₂₅₇	—	2	2	no	—
37	C₂²×C₂₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₂₅₇	#125	1	2	yes	—
39	C₄×C₂₅₇	—	2	2	no	—
40	C₄×C₂₅₇	#108	1	4	yes	—
41	C₄×C₂₅₇	#117	1	2	yes	—
42	C₄×C₂₅₇	#124	1	1	yes	characteristic
43	C₄×C₂₅₇	#124	1	1	yes	characteristic
44	C₂×C₂₅₇	—	2	2	no	—
45	C₂×C₂₅₇	#382	1	1	yes	characteristic
46	C₂×C₂₅₇	C₂×#108	1	2	yes	—
47	C₂₅₇	#1837	1	1	yes	characteristic, complemented, Sylow
48	#1837	—	257	1	no	maximal, Sylow
49	#423	—	257	1	no	—
50	#423	D₂₅₇	1	1	yes	characteristic
51	C₂×C₄×C₈	—	257	1	no	—
52	#341	—	257	4	no	—
53	C₂×M₁₆	—	257	2	no	—
54	C₂×M₁₆	C₂×D₂₅₇	1	2	yes	—
55	C₂²×C₈	—	257	2	no	—
56	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	514	2	no	—
59	C₄×C₈	—	257	2	no	—
60	C₄×C₈	—	257	2	no	—
61	C₂²×C₄	Dic₅₁₄	1	2	yes	—
62	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
63	M₁₆	—	2	4	no	—
64	M₁₆	—	514	4	no	—
65	C₂×C₈	—	257	4	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	514	2	no	—
68	C₂×C₈	—	514	2	no	—
69	C₂×C₈	—	514	2	no	—
70	C₄²	C₄×D₂₅₇	1	1	yes	characteristic
71	C₄²	C₄×D₂₅₇	1	1	yes	characteristic
72	C₄²	C₂²⋊D₂₅₇	1	2	yes	—
73	C₂³	C₂×Dic₅₁₄	1	1	yes	characteristic
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	#16861597	1	2	yes	—
76	C₂×C₄	#16861597	1	2	yes	—
77	C₂×C₄	C₂×C₄×D₂₅₇	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₂₅₇	1	1	yes	characteristic
79	C₂×C₄	#16861617	1	2	yes	—
80	C₂×C₄	#16861619	1	2	yes	—
81	C₈	—	514	4	no	—
82	C₈	—	4	2	no	—
83	C₈	—	1028	2	no	—
84	C₂²	—	2	2	no	—
85	C₂²	C₂×#16861597	1	1	yes	characteristic
86	C₂²	#22551583	1	2	yes	—
87	C₄	—	2	2	no	—
88	C₄	#22551527	1	4	yes	—
89	C₄	#22551550	1	2	yes	—
90	C₄	#22551579	1	1	yes	characteristic
91	C₄	#22551581	1	1	yes	characteristic
92	C₂	—	2	2	no	—
93	C₂	#30656602	1	1	yes	characteristic
94	C₂	C₂×#22551527	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₅₇
commutator	C₂×C₄×C₂₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₅₇
Fitting	#423×C₂₅₇	C₂
socle	C₂²×C₂₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₂₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₂₅₇	C₂³
hypercenter	#423	D₂₅₇
hypocenter	C₂₅₇	#1837
LCS	G ▷ C₂×C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
47. C₂₅₇	#1837	semidirect

Maximal subgroups

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

subgroup	index
#30656761	2
C₂×C₄×#14523339	2
#30656501	2
#423×C₂₅₇	2
#1837	257