Subgroups — #30657618

Order 16448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861627	C₂	1	1	yes	maximal, characteristic
2	C₂×#16861627	C₂	1	1	yes	maximal, characteristic
3	#22551548	C₂	1	1	yes	maximal, characteristic
4	#16861627	—	2	1	no	—
5	#16861627	—	2	1	no	—
6	#16861627	—	2	1	no	—
7	#16861627	C₄	1	2	yes	complemented
8	C₂×#14523346	—	2	1	no	—
9	C₂×#14523346	C₄	1	2	yes	complemented
10	C₂²×Dic₂₅₇	C₂²	1	1	yes	characteristic
11	#16861597	—	2	1	no	—
12	C₂²⋊Dic₂₅₇	C₄	1	1	yes	characteristic
13	C₄×Dic₂₅₇	—	2	1	no	—
14	C₂²⋊C₄×C₂₅₇	C₄	1	1	yes	characteristic
15	#14523346	—	2	2	no	—
16	#14523346	—	2	2	no	—
17	C₂×Dic₂₅₇	—	2	1	no	—
18	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₂₅₇	D₄	1	1	yes	characteristic
21	C₂×Dic₂₅₇	D₄	1	1	yes	characteristic
22	C₂×Dic₂₅₇	D₄	1	1	yes	characteristic
23	C₂×Dic₂₅₇	Q₈	1	1	yes	characteristic
24	C₂³×C₂₅₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₅₇	—	2	1	no	—
26	C₂²×C₂₅₇	—	4	1	no	—
27	C₂²×C₂₅₇	C₄²	1	1	yes	characteristic
28	C₂²×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₂₅₇	—	4	1	no	—
31	Dic₂₅₇	—	4	1	no	—
32	Dic₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
33	Dic₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₂₅₇	C₄⋊C₄	1	1	yes	characteristic
35	Dic₂₅₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₂₅₇	—	2	1	no	—
37	C₂×C₂₅₇	—	2	1	no	—
38	C₂×C₂₅₇	—	2	1	no	—
39	C₂×C₂₅₇	#95	1	1	yes	characteristic
40	C₂₅₇	#344	1	1	yes	characteristic, complemented, Sylow
41	#344	—	257	1	no	maximal, Sylow
42	C₂×M₁₆	—	257	1	no	—
43	C₂×M₁₆	—	257	1	no	—
44	#117	—	257	1	no	—
45	C₂²×C₄	—	257	1	no	—
46	M₁₆	—	257	2	no	—
47	M₁₆	—	514	1	no	—
48	M₁₆	—	514	1	no	—
49	M₁₆	—	514	1	no	—
50	C₂×C₈	—	257	2	no	—
51	C₂×C₈	—	514	1	no	—
52	C₄⋊C₄	—	514	1	no	—
53	C₂²⋊C₄	—	257	1	no	—
54	C₂²⋊C₄	#11759967	1	1	yes	characteristic
55	C₄²	—	514	1	no	—
56	C₂³	C₂×#11759967	1	1	yes	characteristic
57	C₂×C₄	—	257	1	no	—
58	C₂×C₄	—	257	1	no	—
59	C₂×C₄	—	257	1	no	—
60	C₂×C₄	—	257	1	no	—
61	C₂×C₄	—	257	1	no	—
62	C₂×C₄	—	257	1	no	—
63	C₂×C₄	—	2	1	no	—
64	C₂×C₄	—	514	1	no	—
65	C₈	—	514	2	no	—
66	C₈	—	514	2	no	—
67	C₂²	—	4	1	no	—
68	C₂²	C₄×#11759967	1	1	yes	characteristic
69	C₂²	C₂²⋊#11759967	1	1	yes	characteristic
70	C₂²	C₄⋊#11759967	1	1	yes	characteristic
71	C₄	—	257	1	no	—
72	C₄	—	257	1	no	—
73	C₄	—	257	1	no	—
74	C₄	—	257	1	no	—
75	C₄	—	4	1	no	—
76	C₄	—	1028	1	no	—
77	C₂	—	2	1	no	—
78	C₂	—	2	1	no	—
79	C₂	—	2	1	no	—
80	C₂	(C₂×#11759967)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#11759967)⋊C₄
commutator	C₂²×C₂₅₇	C₄²
Frattini	C₂³	C₂×#11759967
Fitting	C₂²⋊C₄×C₂₅₇	C₄
socle	C₂×C₂₅₇	#95
prime radical	C₂³×C₂₅₇	C₂×C₄
cosocle	C₂²×Dic₂₅₇	C₂²
hypercenter	C₂²⋊C₄	#11759967
hypocenter	C₂₅₇	#344
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
7. #16861627	C₄	semidirect
9. C₂×#14523346	C₄	semidirect
40. C₂₅₇	#344	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861627	2
#22551548	2
#344	257