Subgroups — #30657409

Order 16448.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16861619	C₂	1	2	yes	maximal, complemented
2	#22551589	C₂	1	3	yes	maximal, complemented
3	#22551567	C₂	1	6	yes	maximal, complemented
4	C₄×D₁₀₂₈	C₂	1	3	yes	maximal, complemented
5	C₄×Q₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16861619	—	2	8	no	—
7	C₂×D₁₀₂₈	C₂²	1	3	yes	—
8	C₂×D₁₀₂₈	C₂²	1	6	yes	—
9	C₂×D₁₀₂₈	C₂²	1	6	yes	—
10	C₂×C₄×D₂₅₇	C₂²	1	6	yes	—
11	C₄⋊Dic₂₅₇	C₂²	1	1	yes	characteristic
12	#16861595	C₂²	1	6	yes	—
13	C₂×Q₈×C₂₅₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₂₅₇	C₂²	1	3	yes	—
15	C₄²×C₂₅₇	C₂²	1	3	yes	—
16	C₂²×D₂₅₇	C₂³	1	6	yes	—
17	D₁₀₂₈	—	2	12	no	—
18	D₁₀₂₈	—	2	6	no	—
19	D₁₀₂₈	—	4	6	no	—
20	C₂×Dic₂₅₇	C₂³	1	2	yes	—
21	C₄×D₂₅₇	—	2	12	no	—
22	Q₈×C₂₅₇	D₄	1	4	yes	complemented
23	C₂×C₄×C₂₅₇	C₂³	1	3	yes	—
24	C₂×C₄×C₂₅₇	C₂³	1	3	yes	—
25	C₂×C₄×C₂₅₇	C₂³	1	1	yes	characteristic
26	C₂²×C₂₅₇	C₂⁴	1	1	yes	characteristic
27	C₂×D₂₅₇	—	2	6	no	—
28	C₂×D₂₅₇	—	2	6	no	—
29	C₂×D₂₅₇	—	2	6	no	—
30	C₄×C₂₅₇	—	2	3	no	—
31	C₄×C₂₅₇	C₂×D₄	1	6	yes	—
32	C₄×C₂₅₇	#41	1	2	yes	complemented
33	Dic₂₅₇	—	2	2	no	—
34	C₂×C₂₅₇	C₂²×D₄	1	1	yes	characteristic
35	C₂×C₂₅₇	C₂×#41	1	1	yes	characteristic
36	C₂×C₂₅₇	H(5,𝔽₂)	1	1	yes	characteristic
37	D₂₅₇	—	4	6	no	—
38	C₂₅₇	#550	1	1	yes	characteristic, complemented, Sylow
39	#550	—	257	1	no	maximal, Sylow
40	C₂×#41	—	257	2	no	—
41	#127	—	257	3	no	—
42	#121	—	257	6	no	—
43	C₄×Q₈	D₂₅₇	1	1	yes	characteristic, complemented
44	C₄×D₄	—	257	3	no	—
45	#41	—	514	8	no	—
46	C₂×Q₈	C₂×D₂₅₇	1	1	yes	characteristic
47	C₂×D₄	—	257	3	no	—
48	C₂×D₄	—	257	6	no	—
49	C₂×D₄	—	257	6	no	—
50	C₂²×C₄	—	257	6	no	—
51	C₄⋊C₄	—	257	1	no	—
52	C₄⋊C₄	C₂×D₂₅₇	1	3	yes	—
53	C₂²⋊C₄	—	257	6	no	—
54	C₄²	C₂×D₂₅₇	1	3	yes	—
55	C₂³	—	257	6	no	—
56	Q₈	D₁₀₂₈	1	4	yes	complemented
57	D₄	—	514	12	no	—
58	D₄	—	514	6	no	—
59	D₄	—	1028	6	no	—
60	C₂×C₄	—	257	2	no	—
61	C₂×C₄	—	514	12	no	—
62	C₂×C₄	C₂²×D₂₅₇	1	3	yes	—
63	C₂×C₄	C₂²×D₂₅₇	1	3	yes	—
64	C₂×C₄	C₂²×D₂₅₇	1	1	yes	characteristic
65	C₂²	—	514	6	no	—
66	C₂²	—	514	6	no	—
67	C₂²	—	514	6	no	—
68	C₂²	C₂³×D₂₅₇	1	1	yes	characteristic
69	C₄	—	2	3	no	—
70	C₄	—	514	2	no	—
71	C₄	C₂×D₁₀₂₈	1	6	yes	—
72	C₄	#16861619	1	2	yes	complemented
73	C₂	—	1028	6	no	—
74	C₂	C₂²×D₁₀₂₈	1	1	yes	characteristic
75	C₂	C₂×#16861619	1	1	yes	characteristic
76	C₂	#22551638	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₅₇
commutator, socle, cosocle	C₂²×C₂₅₇	C₂⁴
Fitting	C₄×Q₈×C₂₅₇	C₂
hypercenter	C₄×Q₈	D₂₅₇
hypocenter	C₂₅₇	#550
LCS	G ▷ C₂²×C₂₅₇ ▷ C₂₅₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#16861619	C₂	semidirect
2. #22551589	C₂	semidirect
3. #22551567	C₂	semidirect
4. C₄×D₁₀₂₈	C₂	semidirect
5. C₄×Q₈×C₂₅₇	C₂	semidirect
22. Q₈×C₂₅₇	D₄	semidirect
32. C₄×C₂₅₇	#41	semidirect
38. C₂₅₇	#550	semidirect
43. C₄×Q₈	D₂₅₇	semidirect
56. Q₈	D₁₀₂₈	semidirect
72. C₄	#16861619	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16861619	2
#22551589	2
#22551567	2
C₄×D₁₀₂₈	2
C₄×Q₈×C₂₅₇	2
#550	257