Subgroups — #30657707

Order 16448.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861627	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#14523346	C₂	1	4	yes	maximal, complemented
3	C₂³×Dic₂₅₇	C₂	1	1	yes	maximal, characteristic
4	#16861627	—	2	4	no	—
5	C₂×#14523346	C₂²	1	4	yes	complemented
6	C₂²×Dic₂₅₇	—	2	4	no	—
7	C₂²×Dic₂₅₇	C₄	1	1	yes	characteristic
8	C₂²×Dic₂₅₇	C₄	1	2	yes	—
9	C₂²×Dic₂₅₇	C₂²	1	2	yes	—
10	C₂²×Dic₂₅₇	C₂²	1	1	yes	characteristic
11	C₂⁴×C₂₅₇	C₄	1	1	yes	characteristic
12	#14523346	—	2	4	no	—
13	C₂×Dic₂₅₇	—	2	2	no	—
14	C₂×Dic₂₅₇	—	2	4	no	—
15	C₂×Dic₂₅₇	—	2	4	no	—
16	C₂×Dic₂₅₇	C₂×C₄	1	2	yes	—
17	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₂₅₇	D₄	1	4	yes	—
19	C₂×Dic₂₅₇	C₂³	1	1	yes	characteristic
20	C₂³×C₂₅₇	—	2	2	no	—
21	C₂³×C₂₅₇	—	2	4	no	—
22	C₂³×C₂₅₇	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₂₅₇	C₂×C₄	1	2	yes	—
24	C₂²×C₂₅₇	—	2	8	no	—
25	C₂²×C₂₅₇	—	2	2	no	—
26	C₂²×C₂₅₇	—	2	2	no	—
27	C₂²×C₂₅₇	—	2	2	no	—
28	C₂²×C₂₅₇	C₂²⋊C₄	1	2	yes	—
29	C₂²×C₂₅₇	M₁₆	1	4	yes	complemented
30	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
31	Dic₂₅₇	—	2	2	no	—
32	Dic₂₅₇	C₂²⋊C₄	1	2	yes	—
33	Dic₂₅₇	C₂×D₄	1	2	yes	—
34	C₂×C₂₅₇	—	2	2	no	—
35	C₂×C₂₅₇	—	2	4	no	—
36	C₂×C₂₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₂₅₇	C₂×M₁₆	1	2	yes	—
38	C₂₅₇	#407	1	1	yes	characteristic, complemented, Sylow
39	#407	—	257	1	no	maximal, Sylow
40	C₂³×C₄	—	257	1	no	—
41	C₂×M₁₆	—	257	2	no	—
42	C₂²⋊C₈	—	257	4	no	—
43	C₂⁴	#11759967	1	1	yes	characteristic
44	C₂²×C₄	—	257	1	no	—
45	C₂²×C₄	—	257	2	no	—
46	C₂²×C₄	—	257	2	no	—
47	C₂²×C₄	—	257	1	no	—
48	C₂²×C₄	—	514	4	no	—
49	M₁₆	—	514	4	no	—
50	C₂×C₈	—	257	4	no	—
51	C₂³	—	2	2	no	—
52	C₂³	—	2	4	no	—
53	C₂³	C₂×#11759967	1	1	yes	characteristic
54	C₂³	C₂×#11759967	1	2	yes	—
55	C₂×C₄	—	257	4	no	—
56	C₂×C₄	—	257	1	no	—
57	C₂×C₄	—	257	1	no	—
58	C₂×C₄	—	257	2	no	—
59	C₂×C₄	—	514	4	no	—
60	C₂×C₄	—	514	4	no	—
61	C₂×C₄	—	514	2	no	—
62	C₈	—	514	4	no	—
63	C₂²	—	2	8	no	—
64	C₂²	—	2	2	no	—
65	C₂²	—	2	2	no	—
66	C₂²	—	2	2	no	—
67	C₂²	C₂²⋊#11759967	1	2	yes	—
68	C₂²	#16861627	1	4	yes	complemented
69	C₂²	C₂²×#11759967	1	1	yes	characteristic
70	C₄	—	257	2	no	—
71	C₄	—	257	2	no	—
72	C₄	—	514	2	no	—
73	C₂	—	2	2	no	—
74	C₂	—	2	4	no	—
75	C₂	C₂×C₂²⋊#11759967	1	1	yes	characteristic
76	C₂	C₂×#16861627	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11759967
commutator, socle	C₂²×C₂₅₇	C₂²×C₄
Fitting, prime radical	C₂⁴×C₂₅₇	C₄
cosocle	C₂×Dic₂₅₇	C₂³
hypercenter	C₂⁴	#11759967
hypocenter	C₂₅₇	#407
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂×#16861627	C₂	semidirect
2. C₂²⋊#14523346	C₂	semidirect
5. C₂×#14523346	C₂²	semidirect
29. C₂²×C₂₅₇	M₁₆	semidirect
38. C₂₅₇	#407	semidirect
68. C₂²	#16861627	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
C₂²⋊#14523346	2
C₂³×Dic₂₅₇	2
#407	257