Subgroups — #44482623

Order 30592.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29504593	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂₃₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#364×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	#21439900	—	2	1	no	—
5	C₂×C₂₃₉⋊C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₂₃₉⋊C₁₆	C₄	1	1	yes	characteristic
7	M₃₂×C₂₃₉	—	2	1	no	—
8	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
9	C₂₃₉⋊C₁₆	—	2	1	no	—
10	C₂₃₉⋊C₁₆	—	4	1	no	—
11	C₂₃₉⋊C₁₆	C₈	1	2	yes	—
12	C₂×C₈×C₂₃₉	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₂₃₉	D₄	1	1	yes	characteristic
14	C₄²×C₂₃₉	Q₈	1	1	yes	characteristic
15	C₁₆×C₂₃₉	—	4	1	no	—
16	C₂×C₄×C₂₃₉	—	2	1	no	—
17	C₂×C₄×C₂₃₉	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₂₃₉	C₂×C₈	1	1	yes	characteristic
19	C₈×C₂₃₉	M₁₆	1	1	yes	characteristic
20	C₈×C₂₃₉	D₈	1	1	yes	characteristic
21	C₈×C₂₃₉	Q₁₆	1	1	yes	characteristic
22	C₂²×C₂₃₉	#108	1	1	yes	characteristic
23	C₄×C₂₃₉	—	2	2	no	—
24	C₄×C₂₃₉	C₄⋊C₈	1	1	yes	characteristic
25	C₄×C₂₃₉	#107	1	1	yes	characteristic
26	C₂×C₂₃₉	—	2	1	no	—
27	C₂×C₂₃₉	#335	1	1	yes	characteristic
28	C₂₃₉	#1373	1	1	yes	characteristic, complemented, Sylow
29	#1373	—	239	1	no	maximal, Sylow
30	#364	—	239	1	no	—
31	#364	D₂₃₉	1	1	yes	characteristic
32	C₄×C₁₆	—	239	1	no	—
33	M₃₂	—	2	1	no	—
34	M₃₂	—	478	1	no	—
35	C₂×C₁₆	—	239	1	no	—
36	C₂×C₁₆	—	239	1	no	—
37	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
38	C₂×C₈	C₄×D₂₃₉	1	1	yes	characteristic
39	C₂×C₈	C₂²⋊D₂₃₉	1	1	yes	characteristic
40	C₄²	Dic₄₇₈	1	1	yes	characteristic
41	C₁₆	—	4	1	no	—
42	C₁₆	—	956	1	no	—
43	C₁₆	—	478	1	no	—
44	C₁₆	—	239	2	no	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	Dic₂₃₉⋊C₄	1	1	yes	characteristic
47	C₈	C₈×D₂₃₉	1	1	yes	characteristic
48	C₈	#16838531	1	1	yes	characteristic
49	C₈	#16838533	1	1	yes	characteristic
50	C₈	#16838538	1	1	yes	characteristic
51	C₂²	#21439895	1	1	yes	characteristic
52	C₄	—	2	2	no	—
53	C₄	Dic₂₃₉⋊C₈	1	1	yes	characteristic
54	C₄	#21439893	1	1	yes	characteristic
55	C₂	—	2	1	no	—
56	C₂	#29504524	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#16838533
commutator	C₈×C₂₃₉	C₂×C₈
Frattini	C₄×C₈	C₂×D₂₃₉
Fitting	#364×C₂₃₉	C₂
socle	C₂×C₂₃₉	#335
prime radical	C₂²×C₂₃₉	#108
cosocle	C₄×C₈×C₂₃₉	C₂²
hypercenter	#364	D₂₃₉
hypocenter	C₂₃₉	#1373
LCS	G ▷ C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
28. C₂₃₉	#1373	semidirect

Maximal subgroups

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

subgroup	index
#29504593	2
C₄×C₂₃₉⋊C₁₆	2
#364×C₂₃₉	2
#1373	239