Subgroups — #30657607

Order 16448.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861628	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11759967	C₂	1	1	yes	maximal, characteristic
3	#22551550	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11759967	—	2	1	no	—
5	#16861628	—	2	1	no	—
6	#16861628	C₄	1	2	yes	complemented
7	C₂×#14523346	—	2	1	no	—
8	C₄×#11759967	—	2	1	no	—
9	C₄×#11759967	C₄	1	2	yes	complemented
10	C₂×C₄×D₂₅₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₂₅₇	C₄	1	1	yes	characteristic, complemented
12	#16861595	—	2	1	no	—
13	C₄×Dic₂₅₇	—	2	1	no	—
14	C₄⋊C₄×C₂₅₇	C₄	1	1	yes	characteristic, complemented
15	C₂×#11759967	—	2	2	no	—
16	C₂×#11759967	—	2	2	no	—
17	C₂×#11759967	—	2	2	no	—
18	#14523346	—	2	2	no	—
19	C₂²×D₂₅₇	D₄	1	1	yes	characteristic
20	C₂×Dic₂₅₇	—	2	1	no	—
21	C₂×Dic₂₅₇	D₄	1	1	yes	characteristic
22	C₄×D₂₅₇	C₂×C₄	1	1	yes	characteristic
23	C₄×D₂₅₇	C₂×C₄	1	1	yes	characteristic
24	C₄×D₂₅₇	D₄	1	1	yes	characteristic
25	C₄×D₂₅₇	Q₈	1	1	yes	characteristic
26	C₂×C₄×C₂₅₇	—	2	1	no	—
27	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₂₅₇	—	2	1	no	—
30	C₂×D₂₅₇	—	2	1	no	—
31	C₂×D₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×D₂₅₇	C₄⋊C₄	1	1	yes	characteristic
33	#11759967	—	2	4	no	—
34	C₄×C₂₅₇	—	4	1	no	—
35	C₄×C₂₅₇	C₄²	1	1	yes	characteristic
36	C₄×C₂₅₇	C₄⋊C₄	1	1	yes	characteristic
37	Dic₂₅₇	—	4	1	no	—
38	Dic₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
39	Dic₂₅₇	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₂₅₇	#95	1	1	yes	characteristic
41	C₂×C₂₅₇	#104	1	1	yes	characteristic
42	C₂×C₂₅₇	#104	1	1	yes	characteristic
43	D₂₅₇	—	2	1	no	—
44	D₂₅₇	—	2	1	no	—
45	C₂₅₇	#339	1	1	yes	characteristic, complemented, Sylow
46	#339	—	257	1	no	maximal, Sylow
47	C₂×M₁₆	—	257	1	no	—
48	#117	—	257	1	no	—
49	C₂×C₄²	—	257	1	no	—
50	C₂²×C₄	—	257	1	no	—
51	C₂²×C₄	—	514	1	no	—
52	M₁₆	—	257	2	no	—
53	M₁₆	—	514	1	no	—
54	C₂×C₈	—	514	1	no	—
55	C₄⋊C₄	—	257	1	no	—
56	C₄⋊C₄	#11759967	1	1	yes	characteristic, complemented
57	C₂²⋊C₄	—	514	1	no	—
58	C₄²	—	257	2	no	—
59	C₄²	—	514	1	no	—
60	C₄²	—	514	1	no	—
61	C₂³	—	257	1	no	—
62	C₂×C₄	—	257	1	no	—
63	C₂×C₄	—	257	1	no	—
64	C₂×C₄	—	257	1	no	—
65	C₂×C₄	—	257	1	no	—
66	C₂×C₄	—	257	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	514	2	no	—
69	C₂×C₄	—	514	2	no	—
70	C₂×C₄	—	514	2	no	—
71	C₂×C₄	—	514	1	no	—
72	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
73	C₈	—	514	2	no	—
74	C₂²	—	257	1	no	—
75	C₂²	—	257	1	no	—
76	C₂²	—	514	1	no	—
77	C₂²	—	514	1	no	—
78	C₂²	C₂²⋊#11759967	1	1	yes	characteristic
79	C₄	—	257	1	no	—
80	C₄	—	257	1	no	—
81	C₄	—	514	4	no	—
82	C₄	—	4	1	no	—
83	C₄	—	1028	1	no	—
84	C₄	C₄×#11759967	1	1	yes	characteristic
85	C₄	C₄⋊#11759967	1	1	yes	characteristic
86	C₂	—	514	1	no	—
87	C₂	—	514	1	no	—
88	C₂	(C₂×#11759967)⋊C₄	1	1	yes	characteristic
89	C₂	#22551656	1	1	yes	characteristic
90	C₂	#22551657	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#11759967
commutator	C₄×C₂₅₇	C₄²
Frattini	C₂×C₄	C₂×#11759967
Fitting	C₄⋊C₄×C₂₅₇	C₄
socle	C₂²×C₂₅₇	C₂²⋊C₄
prime radical	C₂²×D₂₅₇	D₄
cosocle	C₂×C₄×D₂₅₇	C₂²
hypercenter	C₄⋊C₄	#11759967
hypocenter	C₂₅₇	#339
LCS	G ▷ C₄×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. #16861628	C₄	semidirect
9. C₄×#11759967	C₄	semidirect
11. C₄⋊Dic₂₅₇	C₄	semidirect
14. C₄⋊C₄×C₂₅₇	C₄	semidirect
45. C₂₅₇	#339	semidirect
56. C₄⋊C₄	#11759967	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861628	2
C₂×C₄×#11759967	2
#22551550	2
#339	257