Subgroups — #30657717

Order 16448.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22551671	C₂	1	2	yes	maximal, complemented
2	#22551650	C₂	1	4	yes	maximal, complemented
3	C₂×#16861617	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11759967	—	2	2	no	—
5	C₂²⋊#11759967	C₂²	1	4	yes	—
6	C₄×#11759967	—	2	2	no	—
7	#16861617	—	2	2	no	—
8	#16861617	—	2	2	no	—
9	C₂×Dic₅₁₄	C₄	1	1	yes	characteristic, complemented
10	C₂×C₂²⋊D₂₅₇	C₂²	1	2	yes	—
11	C₂²×Dic₂₅₇	C₄	1	2	yes	complemented
12	C₂×C₄×D₂₅₇	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₂₅₇	C₄	1	1	yes	characteristic, complemented
14	C₂×#11759967	—	2	4	no	—
15	C₂²×D₂₅₇	C₂³	1	1	yes	characteristic
16	Dic₅₁₄	—	2	2	no	—
17	C₂²⋊D₂₅₇	—	2	4	no	—
18	C₂×Dic₂₅₇	—	2	2	no	—
19	C₂×Dic₂₅₇	—	2	2	no	—
20	C₂×Dic₂₅₇	C₂×C₄	1	2	yes	—
21	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
22	C₄×D₂₅₇	D₄	1	2	yes	—
23	C₄×D₂₅₇	D₄	1	2	yes	—
24	C₂³×C₂₅₇	C₂×C₄	1	2	yes	—
25	D₄×C₂₅₇	—	2	2	no	—
26	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₂₅₇	—	2	2	no	—
28	C₂²×C₂₅₇	—	4	2	no	—
29	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×D₂₅₇	—	4	1	no	—
31	C₂×D₂₅₇	C₂×D₄	1	2	yes	—
32	#11759967	—	4	4	no	—
33	C₄×C₂₅₇	C₂²⋊C₄	1	2	yes	—
34	Dic₂₅₇	—	2	2	no	—
35	Dic₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
36	Dic₂₅₇	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₂₅₇	—	2	1	no	—
38	C₂×C₂₅₇	—	4	2	no	—
39	C₂×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	D₂₅₇	—	2	2	no	—
41	C₂₅₇	#410	1	1	yes	characteristic, complemented, Sylow
42	#410	—	257	1	no	maximal, Sylow
43	C₂×#41	—	257	1	no	—
44	#117	—	257	2	no	—
45	#99	—	257	4	no	—
46	#41	—	514	2	no	—
47	#41	—	514	2	no	—
48	C₂×Q₈	—	257	1	no	—
49	C₂×D₄	—	257	2	no	—
50	C₂×D₄	#11759967	1	1	yes	characteristic, complemented
51	C₂²×C₄	—	257	1	no	—
52	C₂²×C₄	—	257	2	no	—
53	C₄⋊C₄	—	514	2	no	—
54	C₂²⋊C₄	—	257	4	no	—
55	C₄²	—	514	2	no	—
56	C₂³	—	257	1	no	—
57	C₂³	C₂×#11759967	1	2	yes	—
58	Q₈	—	514	2	no	—
59	D₄	—	2	2	no	—
60	D₄	—	514	4	no	—
61	C₂×C₄	—	257	1	no	—
62	C₂×C₄	—	257	2	no	—
63	C₂×C₄	—	257	2	no	—
64	C₂×C₄	—	257	2	no	—
65	C₂×C₄	—	514	2	no	—
66	C₂×C₄	—	514	2	no	—
67	C₂×C₄	—	514	4	no	—
68	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
69	C₂²	—	257	2	no	—
70	C₂²	—	2	2	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	1028	1	no	—
73	C₂²	C₂²×#11759967	1	1	yes	characteristic
74	C₄	—	257	1	no	—
75	C₄	—	257	1	no	—
76	C₄	—	514	2	no	—
77	C₄	—	1028	4	no	—
78	C₄	C₂²⋊#11759967	1	2	yes	—
79	C₂	—	2	1	no	—
80	C₂	—	514	2	no	—
81	C₂	—	4	2	no	—
82	C₂	C₂×C₂²⋊#11759967	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#11759967
commutator	C₂²×C₂₅₇	C₂²×C₄
Frattini	C₂²	C₂²×#11759967
Fitting	C₂×D₄×C₂₅₇	C₄
socle	C₂×C₂₅₇	C₂×C₂²⋊C₄
prime radical	C₂×#16861617	C₂
cosocle	C₂²×D₂₅₇	C₂³
hypercenter	C₂×D₄	#11759967
hypocenter	C₂₅₇	#410
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. #22551671	C₂	semidirect
2. #22551650	C₂	semidirect
9. C₂×Dic₅₁₄	C₄	semidirect
11. C₂²×Dic₂₅₇	C₄	semidirect
13. C₂×D₄×C₂₅₇	C₄	semidirect
41. C₂₅₇	#410	semidirect
50. C₂×D₄	#11759967	semidirect

Maximal subgroups

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

subgroup	index
#22551671	2
#22551650	2
C₂×#16861617	2
#410	257