Subgroups — #45667501

Order 32896.

424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30656644	C₂	1	4	yes	maximal
2	#30656623	C₂	1	8	yes	maximal
3	#30656605	C₂	1	2	yes	maximal
4	C₂×C₄⋊Q₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₂₅₇	C₂²	1	4	yes	—
6	C₂×C₄×Dic₂₅₇	C₂²	1	4	yes	—
7	#22551499	C₂²	1	8	yes	—
8	#22551499	C₂²	1	4	yes	—
9	#22551499	C₂²	1	8	yes	—
10	C₂²×Q₈×C₂₅₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₂₅₇	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₄×C₂₅₇	C₂²	1	4	yes	—
13	C₂×C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₅₇	C₂³	1	8	yes	—
15	C₄⋊Dic₂₅₇	—	4	4	no	—
16	C₄×Dic₂₅₇	—	4	4	no	—
17	C₂×Q₈×C₂₅₇	—	4	2	no	—
18	C₂×Q₈×C₂₅₇	—	4	2	no	—
19	C₂×Q₈×C₂₅₇	C₂×C₄	1	8	yes	—
20	C₂²×C₄×C₂₅₇	C₂³	1	2	yes	—
21	C₂²×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₂₅₇	C₂³	1	4	yes	—
23	C₄⋊C₄×C₂₅₇	C₂×C₄	1	16	yes	—
24	C₄²×C₂₅₇	C₂×C₄	1	4	yes	—
25	C₂×Dic₂₅₇	—	2	8	no	—
26	C₂×Dic₂₅₇	—	2	16	no	—
27	C₂³×C₂₅₇	C₂⁴	1	1	yes	characteristic
28	Q₈×C₂₅₇	—	4	8	no	—
29	C₂×C₄×C₂₅₇	—	2	2	no	—
30	C₂×C₄×C₂₅₇	—	2	4	no	—
31	C₂×C₄×C₂₅₇	—	2	4	no	—
32	C₂×C₄×C₂₅₇	—	2	4	no	—
33	C₂×C₄×C₂₅₇	C₂²×C₄	1	4	yes	—
34	C₂×C₄×C₂₅₇	C₂²×C₄	1	2	yes	—
35	C₂×C₄×C₂₅₇	C₂²×C₄	1	8	yes	—
36	C₂²×C₂₅₇	C₂³×C₄	1	1	yes	characteristic
37	C₂²×C₂₅₇	H(5,𝔽₂)	1	2	yes	—
38	C₂²×C₂₅₇	2₋¹⁺⁴	1	4	yes	—
39	C₄×C₂₅₇	—	2	8	no	—
40	C₄×C₂₅₇	—	2	2	no	—
41	C₄×C₂₅₇	—	2	4	no	—
42	Dic₂₅₇	—	4	8	no	—
43	C₂×C₂₅₇	#518	1	1	yes	characteristic
44	C₂×C₂₅₇	#519	1	2	yes	—
45	C₂×C₂₅₇	#563	1	4	yes	—
46	C₂₅₇	#2381	1	1	yes	characteristic, complemented, Sylow
47	#2381	—	257	1	no	maximal, Sylow
48	C₂×C₄⋊Q₈	D₂₅₇	1	1	yes	characteristic
49	#391	—	257	4	no	—
50	#387	—	257	8	no	—
51	#383	—	257	2	no	—
52	C₂²×Q₈	C₂×D₂₅₇	1	2	yes	—
53	C₄⋊Q₈	Dic₂₅₇	1	8	yes	complemented
54	C₂×C₄⋊C₄	—	257	4	no	—
55	C₂×C₄⋊C₄	C₂×D₂₅₇	1	4	yes	—
56	C₂×C₄²	—	257	4	no	—
57	C₂×C₄²	C₂×D₂₅₇	1	1	yes	characteristic
58	#95	—	257	8	no	—
59	#95	—	257	4	no	—
60	#95	—	257	8	no	—
61	C₂×Q₈	—	4	2	no	—
62	C₂×Q₈	—	4	2	no	—
63	C₂×Q₈	C₂×Dic₂₅₇	1	8	yes	—
64	C₂²×C₄	—	257	8	no	—
65	C₂²×C₄	C₂²×D₂₅₇	1	2	yes	—
66	C₂²×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
67	C₂²×C₄	C₂²×D₂₅₇	1	4	yes	—
68	C₄⋊C₄	—	1028	4	no	—
69	C₄⋊C₄	C₂×Dic₂₅₇	1	16	yes	—
70	C₄²	—	1028	4	no	—
71	C₄²	C₂×Dic₂₅₇	1	4	yes	—
72	C₂³	C₂³×D₂₅₇	1	1	yes	characteristic
73	Q₈	—	4	8	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	514	8	no	—
79	C₂×C₄	—	514	16	no	—
80	C₂×C₄	C₂²×Dic₂₅₇	1	4	yes	—
81	C₂×C₄	C₂²×Dic₂₅₇	1	2	yes	—
82	C₂×C₄	C₂²×Dic₂₅₇	1	8	yes	—
83	C₂²	C₂³×Dic₂₅₇	1	1	yes	characteristic
84	C₂²	#22551637	1	2	yes	—
85	C₂²	#22551640	1	4	yes	—
86	C₄	—	2	2	no	—
87	C₄	—	2	8	no	—
88	C₄	—	2	4	no	—
89	C₄	—	1028	8	no	—
90	C₂	#30657212	1	1	yes	characteristic
91	C₂	#30657215	1	2	yes	—
92	C₂	#30657486	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₇
commutator	C₂²×C₂₅₇	C₂³×C₄
Fitting	C₂×C₄⋊Q₈×C₂₅₇	C₂
socle, prime radical, cosocle	C₂³×C₂₅₇	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₂₅₇
hypocenter	C₂₅₇	#2381
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
11. C₄⋊Q₈×C₂₅₇	C₄	semidirect
46. C₂₅₇	#2381	semidirect
53. C₄⋊Q₈	Dic₂₅₇	semidirect

Maximal subgroups

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

subgroup	index
#30656644	2
#30656623	2
#30656605	2
C₂×C₄⋊Q₈×C₂₅₇	2
#2381	257