Subgroups — #44486830

Order 30592.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29505177	C₂	1	2	yes	maximal
2	#29504600	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	#21439960	—	2	2	no	—
5	#21439894	—	2	2	no	—
6	#21439894	C₂²	1	4	yes	—
7	C₂×C₁₆×C₂₃₉	C₂²	1	2	yes	—
8	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
9	C₂×Dic₄₇₈	—	4	2	no	—
10	C₄⋊Dic₂₃₉	—	2	4	no	—
11	C₄⋊Dic₂₃₉	—	4	2	no	—
12	C₂×C₈×C₂₃₉	D₄	1	1	yes	characteristic
13	C₂×C₈×C₂₃₉	C₂³	1	1	yes	characteristic
14	C₄²×C₂₃₉	D₄	1	1	yes	characteristic
15	C₁₆×C₂₃₉	Q₈	1	4	yes	complemented
16	Dic₄₇₈	—	8	4	no	—
17	C₂×Dic₂₃₉	—	4	4	no	—
18	C₂×C₄×C₂₃₉	D₈	1	2	yes	—
19	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
20	C₈×C₂₃₉	Q₁₆	1	2	yes	—
21	C₈×C₂₃₉	C₂×Q₈	1	2	yes	—
22	C₂²×C₂₃₉	C₂×D₈	1	1	yes	characteristic
23	C₄×C₂₃₉	SD₃₂	1	4	yes	—
24	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
25	C₄×C₂₃₉	C₂×Q₁₆	1	1	yes	characteristic
26	Dic₂₃₉	—	8	4	no	—
27	C₂×C₂₃₉	#500	1	1	yes	characteristic
28	C₂×C₂₃₉	C₂×SD₃₂	1	2	yes	—
29	C₂₃₉	#2254	1	1	yes	characteristic, complemented, Sylow
30	#2254	—	239	1	no	maximal, Sylow
31	#500	—	239	2	no	—
32	#367	—	239	4	no	—
33	C₄×C₁₆	D₂₃₉	1	1	yes	characteristic
34	C₄⋊Q₈	—	478	2	no	—
35	C₂×C₁₆	C₂×D₂₃₉	1	2	yes	—
36	#107	—	478	2	no	—
37	#107	—	239	4	no	—
38	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
39	C₂×Q₈	—	956	2	no	—
40	C₂×C₈	D₉₅₆	1	1	yes	characteristic
41	C₂×C₈	C₂²×D₂₃₉	1	1	yes	characteristic
42	C₄⋊C₄	—	956	2	no	—
43	C₄⋊C₄	—	478	4	no	—
44	C₄²	D₉₅₆	1	1	yes	characteristic
45	C₁₆	Dic₄₇₈	1	4	yes	complemented
46	Q₈	—	1912	4	no	—
47	C₂×C₄	—	956	4	no	—
48	C₂×C₄	D₁₉₁₂	1	2	yes	—
49	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
50	C₈	Dic₉₅₆	1	2	yes	—
51	C₈	C₂×Dic₄₇₈	1	2	yes	—
52	C₂²	C₂×D₁₉₁₂	1	1	yes	characteristic
53	C₄	—	1912	4	no	—
54	C₄	#21439905	1	4	yes	—
55	C₄	#21439960	1	1	yes	characteristic
56	C₄	C₂×Dic₉₅₆	1	1	yes	characteristic
57	C₂	#29505177	1	1	yes	characteristic
58	C₂	C₂×#21439905	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₉₁₂
commutator, cosocle	C₂×C₈×C₂₃₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₃₉
Fitting	C₄×C₁₆×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂×D₈
hypercenter	C₄×C₁₆	D₂₃₉
hypocenter	C₂₃₉	#2254
LCS	G ▷ C₂×C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₂₃₉	Q₈	semidirect
29. C₂₃₉	#2254	semidirect
45. C₁₆	Dic₄₇₈	semidirect

Maximal subgroups

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

subgroup	index
#29505177	2
#29504600	2
C₄×C₁₆×C₂₃₉	2
#2254	239