Subgroups — #30657771

Order 16448.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861628	C₂	1	2	yes	maximal, complemented
2	C₄⋊#14523346	C₂	1	1	yes	maximal, characteristic, complemented
3	#22551649	C₂	1	2	yes	maximal, complemented
4	C₄×#14523346	C₂	1	1	yes	maximal, characteristic, complemented
5	#22551554	C₂	1	1	yes	maximal, characteristic
6	#16861628	—	2	4	no	—
7	C₂×#14523346	C₂²	1	1	yes	characteristic
8	C₂×#14523346	C₂²	1	2	yes	—
9	C₂×#14523346	C₂²	1	1	yes	characteristic
10	C₂×D₁₀₂₈	C₄	1	1	yes	characteristic
11	C₂×C₄×D₂₅₇	C₂²	1	2	yes	—
12	#16861595	C₄	1	2	yes	—
13	C₄×Dic₂₅₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₂₅₇	C₄	1	1	yes	characteristic
15	#14523346	—	2	2	no	—
16	#14523346	—	2	1	no	—
17	#14523346	D₄	1	2	yes	complemented
18	C₂²×D₂₅₇	C₂×C₄	1	2	yes	—
19	D₁₀₂₈	—	2	2	no	—
20	C₂×Dic₂₅₇	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₂₅₇	C₂³	1	1	yes	characteristic
22	C₄×D₂₅₇	—	2	2	no	—
23	C₄×D₂₅₇	—	2	2	no	—
24	C₂×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₂₅₇	—	2	2	no	—
28	C₂×D₂₅₇	—	2	2	no	—
29	C₂×D₂₅₇	—	2	2	no	—
30	C₄×C₂₅₇	—	2	2	no	—
31	C₄×C₂₅₇	M₁₆	1	2	yes	complemented
32	Dic₂₅₇	—	2	1	no	—
33	Dic₂₅₇	C₂×D₄	1	1	yes	characteristic
34	Dic₂₅₇	#41	1	1	yes	characteristic
35	C₂×C₂₅₇	C₄×D₄	1	1	yes	characteristic
36	C₂×C₂₅₇	C₂×M₁₆	1	1	yes	characteristic
37	C₂×C₂₅₇	#131	1	1	yes	characteristic
38	D₂₅₇	—	4	2	no	—
39	C₂₅₇	#436	1	1	yes	characteristic, complemented, Sylow
40	#436	—	257	1	no	maximal, Sylow
41	C₂×M₁₆	—	257	2	no	—
42	C₄×D₄	—	257	1	no	—
43	C₄⋊C₈	—	257	1	no	—
44	C₂²⋊C₈	—	257	2	no	—
45	C₄×C₈	—	257	1	no	—
46	C₂×D₄	—	257	1	no	—
47	C₂²×C₄	—	257	2	no	—
48	M₁₆	—	514	4	no	—
49	C₂×C₈	—	257	1	no	—
50	C₂×C₈	—	257	1	no	—
51	C₂×C₈	—	257	2	no	—
52	C₄⋊C₄	#11759967	1	1	yes	characteristic
53	C₂²⋊C₄	—	257	2	no	—
54	C₄²	—	257	1	no	—
55	C₂³	—	257	2	no	—
56	D₄	—	514	2	no	—
57	C₂×C₄	—	257	1	no	—
58	C₂×C₄	—	257	1	no	—
59	C₂×C₄	—	514	2	no	—
60	C₂×C₄	—	514	2	no	—
61	C₂×C₄	C₂×#11759967	1	2	yes	—
62	C₂×C₄	C₂×#11759967	1	1	yes	characteristic
63	C₈	—	257	2	no	—
64	C₈	—	514	2	no	—
65	C₈	—	514	1	no	—
66	C₂²	—	514	2	no	—
67	C₂²	—	514	2	no	—
68	C₂²	—	514	2	no	—
69	C₂²	C₂²×#11759967	1	1	yes	characteristic
70	C₄	—	257	1	no	—
71	C₄	—	257	1	no	—
72	C₄	—	2	2	no	—
73	C₄	—	514	1	no	—
74	C₄	#16861628	1	2	yes	complemented
75	C₂	—	1028	2	no	—
76	C₂	D₄×#11759967	1	1	yes	characteristic
77	C₂	C₂×#16861628	1	1	yes	characteristic
78	C₂	#22551680	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#11759967
commutator, socle	C₂²×C₂₅₇	C₂²×C₄
Fitting	C₄⋊C₄×C₂₅₇	C₄
prime radical	C₂×D₁₀₂₈	C₄
cosocle	C₂×Dic₂₅₇	C₂³
hypercenter	C₄⋊C₄	#11759967
hypocenter	C₂₅₇	#436
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#16861628	C₂	semidirect
2. C₄⋊#14523346	C₂	semidirect
3. #22551649	C₂	semidirect
4. C₄×#14523346	C₂	semidirect
17. #14523346	D₄	semidirect
31. C₄×C₂₅₇	M₁₆	semidirect
39. C₂₅₇	#436	semidirect
74. C₄	#16861628	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
C₄⋊#14523346	2
#22551649	2
C₄×#14523346	2
#22551554	2
#436	257