Subgroups — #30657850

Order 16448.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14523348	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#14523348	C₂	1	2	yes	maximal
3	C₄×#14523348	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×#11759967	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523348	C₂²	1	4	yes	—
6	C₂²×#11759967	C₄	1	1	yes	characteristic
7	C₂²×#11759967	C₂²	1	1	yes	characteristic
8	C₄×#11759967	C₄	1	1	yes	characteristic
9	C₄×#11759967	C₄	1	1	yes	characteristic
10	C₄×#11759967	C₂²	1	1	yes	characteristic
11	C₄×#11759967	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₅₇	C₄	1	1	yes	characteristic
13	#14523348	—	2	4	no	—
14	C₂×#11759967	—	2	2	no	—
15	C₂×#11759967	—	2	2	no	—
16	C₂×#11759967	C₂×C₄	1	1	yes	characteristic
17	C₂×#11759967	C₂×C₄	1	1	yes	characteristic
18	C₂×#11759967	C₂×C₄	1	1	yes	characteristic
19	C₂×#11759967	C₂³	1	1	yes	characteristic
20	C₂²×D₂₅₇	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₂₅₇	C₈	1	1	yes	characteristic, complemented
22	C₄×D₂₅₇	C₈	1	1	yes	characteristic, complemented
23	C₄×D₂₅₇	C₈	1	1	yes	characteristic, complemented
24	C₄×D₂₅₇	C₂×C₄	1	1	yes	characteristic
25	C₄×D₂₅₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₅₇	C₈	1	1	yes	characteristic, complemented
27	C₂²×C₂₅₇	C₂×C₈	1	1	yes	characteristic
28	C₂×D₂₅₇	—	2	1	no	—
29	C₂×D₂₅₇	—	2	1	no	—
30	C₂×D₂₅₇	C₂×C₈	1	1	yes	characteristic
31	C₂×D₂₅₇	C₂²×C₄	1	1	yes	characteristic
32	#11759967	—	2	1	no	—
33	#11759967	—	2	1	no	—
34	#11759967	M₁₆	1	2	yes	—
35	#11759967	#41	1	2	yes	—
36	C₄×C₂₅₇	C₂×C₈	1	1	yes	characteristic
37	C₄×C₂₅₇	C₂×C₈	1	1	yes	characteristic
38	Dic₂₅₇	C₂×C₈	1	1	yes	characteristic
39	Dic₂₅₇	C₂×C₈	1	1	yes	characteristic
40	C₂×C₂₅₇	—	2	1	no	—
41	C₂×C₂₅₇	C₂²×C₈	1	1	yes	characteristic
42	D₂₅₇	—	2	1	no	—
43	D₂₅₇	#117	1	1	yes	characteristic
44	D₂₅₇	C₂×M₁₆	1	1	yes	characteristic
45	C₂₅₇	#431	1	1	yes	characteristic, complemented, Sylow
46	#431	—	257	1	no	maximal, Sylow
47	C₂×C₄²	—	257	1	no	—
48	C₄⋊C₈	—	257	2	no	—
49	C₂²⋊C₈	—	257	2	no	—
50	C₄×C₈	—	257	2	no	—
51	C₂²×C₄	—	257	1	no	—
52	C₂²×C₄	—	257	1	no	—
53	C₂²×C₄	—	257	1	no	—
54	C₂×C₈	—	257	4	no	—
55	C₄²	—	257	1	no	—
56	C₄²	—	257	1	no	—
57	C₄²	—	257	1	no	—
58	C₄²	—	257	1	no	—
59	C₂³	—	257	1	no	—
60	C₂×C₄	—	257	1	no	—
61	C₂×C₄	—	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	—	257	1	no	—
68	C₂×C₄	—	257	1	no	—
69	C₂×C₄	—	514	2	no	—
70	C₂×C₄	—	514	2	no	—
71	C₂×C₄	#14523348	1	1	yes	characteristic, complemented
72	C₈	—	514	4	no	—
73	C₂²	—	257	1	no	—
74	C₂²	—	257	1	no	—
75	C₂²	—	514	1	no	—
76	C₂²	—	514	1	no	—
77	C₂²	C₂×#14523348	1	1	yes	characteristic
78	C₄	—	257	2	no	—
79	C₄	—	257	2	no	—
80	C₄	—	257	1	no	—
81	C₄	—	257	1	no	—
82	C₄	—	514	1	no	—
83	C₄	—	514	1	no	—
84	C₄	C₂×#14523348	1	1	yes	characteristic
85	C₄	C₂×#14523348	1	1	yes	characteristic
86	C₂	—	257	1	no	—
87	C₂	—	257	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	514	1	no	—
90	C₂	C₂²×#14523348	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#14523348
commutator, socle	C₂×C₂₅₇	C₂²×C₈
Frattini	C₂	C₂²×#14523348
Fitting	C₂×C₄×C₂₅₇	C₈
prime radical	C₂²×D₂₅₇	C₂×C₄
cosocle	C₂×#11759967	C₂³
hypercenter	C₂×C₄	#14523348
hypocenter	C₂₅₇	#431
LCS	G ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊#14523348	C₂	semidirect
3. C₄×#14523348	C₂	semidirect
21. C₂×Dic₂₅₇	C₈	semidirect
22. C₄×D₂₅₇	C₈	semidirect
23. C₄×D₂₅₇	C₈	semidirect
26. C₂×C₄×C₂₅₇	C₈	semidirect
45. C₂₅₇	#431	semidirect
71. C₂×C₄	#14523348	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14523348	2
C₂²⋊#14523348	2
C₄×#14523348	2
C₂×C₄×#11759967	2
#431	257