Subgroups — #30656866

Order 16448.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₅₁₄	C₂	1	1	yes	maximal, characteristic
2	#22551521	C₂	1	1	yes	maximal, characteristic
3	#22551521	C₂	1	1	yes	maximal, characteristic
4	#22551503	C₂	1	1	yes	maximal, characteristic
5	#22551502	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₁₄	C₄	1	1	yes	characteristic
9	C₂×#14523339	C₂²	1	1	yes	characteristic
10	C₂×#14523339	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₅₇	C₄	1	1	yes	characteristic
12	#16861597	C₄	1	1	yes	characteristic
13	#16861597	C₄	1	1	yes	characteristic
14	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
15	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
18	C₄²×C₂₅₇	C₂²	1	1	yes	characteristic
19	Dic₅₁₄	—	2	2	no	—
20	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
24	#14523339	—	2	1	no	—
25	#14523339	Q₈	1	2	yes	complemented
26	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
29	C₈×C₂₅₇	—	2	1	no	—
30	C₈×C₂₅₇	—	2	1	no	—
31	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
32	C₄×C₂₅₇	—	2	1	no	—
33	C₄×C₂₅₇	—	2	1	no	—
34	C₄×C₂₅₇	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₂₅₇	#41	1	1	yes	characteristic
36	Dic₂₅₇	—	2	1	no	—
37	Dic₂₅₇	—	2	1	no	—
38	Dic₂₅₇	—	2	1	no	—
39	Dic₂₅₇	M₁₆	1	2	yes	—
40	C₂×C₂₅₇	C₄×Q₈	1	1	yes	characteristic
41	C₂×C₂₅₇	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₂₅₇	#131	1	1	yes	characteristic
43	C₂₅₇	#446	1	1	yes	characteristic, complemented, Sylow
44	#446	—	257	1	no	maximal, Sylow
45	C₄×Q₈	—	257	1	no	—
46	C₄⋊C₈	—	257	1	no	—
47	C₄⋊C₈	—	257	1	no	—
48	C₄⋊C₈	D₂₅₇	1	1	yes	characteristic
49	#97	—	257	1	no	—
50	#97	—	257	1	no	—
51	C₄×C₈	—	257	1	no	—
52	C₂×Q₈	—	257	1	no	—
53	C₂×C₈	—	257	1	no	—
54	C₂×C₈	—	257	1	no	—
55	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
56	C₂×C₈	C₂×D₂₅₇	1	1	yes	characteristic
57	C₄⋊C₄	—	257	1	no	—
58	C₄⋊C₄	—	257	1	no	—
59	C₄⋊C₄	—	257	1	no	—
60	C₄²	—	257	1	no	—
61	C₄²	—	257	1	no	—
62	C₄²	C₂×D₂₅₇	1	1	yes	characteristic
63	Q₈	—	514	2	no	—
64	C₂×C₄	—	257	1	no	—
65	C₂×C₄	—	257	1	no	—
66	C₂×C₄	—	257	1	no	—
67	C₂×C₄	—	257	1	no	—
68	C₂×C₄	C₄×D₂₅₇	1	1	yes	characteristic
69	C₂×C₄	C₄×D₂₅₇	1	1	yes	characteristic
70	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
71	C₈	—	257	2	no	—
72	C₈	—	2	1	no	—
73	C₈	—	2	1	no	—
74	C₈	—	514	1	no	—
75	C₂²	C₂×C₄×D₂₅₇	1	1	yes	characteristic
76	C₄	—	257	2	no	—
77	C₄	—	2	1	no	—
78	C₄	—	2	1	no	—
79	C₄	—	514	1	no	—
80	C₄	—	514	1	no	—
81	C₄	—	514	1	no	—
82	C₄	Q₈×D₂₅₇	1	1	yes	characteristic
83	C₄	#16861617	1	1	yes	characteristic
84	C₂	#22551557	1	1	yes	characteristic
85	C₂	M₁₆×D₂₅₇	1	1	yes	characteristic
86	C₂	#22551598	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₅₇
commutator, socle, prime radical	C₂²×C₂₅₇	C₂²×C₄
Fitting	C₄⋊C₈×C₂₅₇	C₂
cosocle	C₂×C₄×C₂₅₇	C₂³
hypercenter	C₄⋊C₈	D₂₅₇
hypocenter	C₂₅₇	#446
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
25. #14523339	Q₈	semidirect
43. C₂₅₇	#446	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₅₁₄	2
#22551521	2
#22551503	2
#22551502	2
C₈×Dic₂₅₇	2
C₄⋊C₈×C₂₅₇	2
#446	257