Subgroups — #44486823

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	#29504598	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	1	yes	characteristic
19	C₂×C₄×C₂₃₉	D₈	1	1	yes	characteristic
20	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
21	C₈×C₂₃₉	Q₁₆	1	2	yes	—
22	C₈×C₂₃₉	C₂×Q₈	1	2	yes	—
23	C₂²×C₂₃₉	C₂×D₈	1	1	yes	characteristic
24	C₄×C₂₃₉	D₁₆	1	2	yes	—
25	C₄×C₂₃₉	Q₃₂	1	2	yes	—
26	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
27	C₄×C₂₃₉	C₂×Q₁₆	1	1	yes	characteristic
28	Dic₂₃₉	—	8	4	no	—
29	C₂×C₂₃₉	#500	1	1	yes	characteristic
30	C₂×C₂₃₉	C₂×D₁₆	1	1	yes	characteristic
31	C₂×C₂₃₉	C₂×Q₃₂	1	1	yes	characteristic
32	C₂₃₉	#2252	1	1	yes	characteristic, complemented, Sylow
33	#2252	—	239	1	no	maximal, Sylow
34	#500	—	239	2	no	—
35	#366	—	239	4	no	—
36	C₄×C₁₆	D₂₃₉	1	1	yes	characteristic
37	C₄⋊Q₈	—	478	2	no	—
38	C₂×C₁₆	C₂×D₂₃₉	1	2	yes	—
39	#107	—	478	2	no	—
40	#107	—	239	4	no	—
41	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
42	C₂×Q₈	—	956	2	no	—
43	C₂×C₈	D₉₅₆	1	1	yes	characteristic
44	C₂×C₈	C₂²×D₂₃₉	1	1	yes	characteristic
45	C₄⋊C₄	—	956	2	no	—
46	C₄⋊C₄	—	478	4	no	—
47	C₄²	D₉₅₆	1	1	yes	characteristic
48	C₁₆	Dic₄₇₈	1	4	yes	complemented
49	Q₈	—	1912	4	no	—
50	C₂×C₄	—	956	4	no	—
51	C₂×C₄	D₁₉₁₂	1	1	yes	characteristic
52	C₂×C₄	D₁₉₁₂	1	1	yes	characteristic
53	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
54	C₈	Dic₉₅₆	1	2	yes	—
55	C₈	C₂×Dic₄₇₈	1	2	yes	—
56	C₂²	C₂×D₁₉₁₂	1	1	yes	characteristic
57	C₄	—	1912	4	no	—
58	C₄	D₃₈₂₄	1	2	yes	—
59	C₄	Dic₁₉₁₂	1	2	yes	—
60	C₄	#21439960	1	1	yes	characteristic
61	C₄	C₂×Dic₉₅₆	1	1	yes	characteristic
62	C₂	#29505177	1	1	yes	characteristic
63	C₂	C₂×D₃₈₂₄	1	1	yes	characteristic
64	C₂	C₂×Dic₁₉₁₂	1	1	yes	characteristic

	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	#2252
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
32. C₂₃₉	#2252	semidirect
48. 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
#29504598	2
C₄×C₁₆×C₂₃₉	2
#2252	239