Subgroups — #1716×C₂₃₉ #44480486

Order 30592.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#498×C₂₃₉	C₂	1	2	yes	maximal
2	#498×C₂₃₉	C₂	1	1	yes	maximal, characteristic
3	#494×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	#487×C₂₃₉	C₂	1	2	yes	maximal
5	#322×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₂₃₉	—	2	2	no	—
7	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
8	C₄⋊Q₈×C₂₃₉	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
10	#106×C₂₃₉	—	2	4	no	—
11	#106×C₂₃₉	—	2	2	no	—
12	#103×C₂₃₉	—	2	4	no	—
13	C₄×C₈×C₂₃₉	C₂²	1	2	yes	—
14	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₂₃₉	—	2	2	no	—
16	C₂×Q₈×C₂₃₉	—	2	1	no	—
17	C₂×Q₈×C₂₃₉	—	2	1	no	—
18	Q₁₆×C₂₃₉	—	4	4	no	—
19	C₂×C₈×C₂₃₉	D₄	1	4	yes	—
20	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
21	C₄⋊C₄×C₂₃₉	—	2	2	no	—
22	C₄⋊C₄×C₂₃₉	—	2	2	no	—
23	C₄⋊C₄×C₂₃₉	—	2	2	no	—
24	C₄⋊C₄×C₂₃₉	—	2	2	no	—
25	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
26	Q₈×C₂₃₉	—	4	2	no	—
27	Q₈×C₂₃₉	—	4	2	no	—
28	Q₈×C₂₃₉	—	4	4	no	—
29	C₂×C₄×C₂₃₉	—	4	1	no	—
30	C₂×C₄×C₂₃₉	—	4	1	no	—
31	C₂×C₄×C₂₃₉	—	4	2	no	—
32	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
34	C₈×C₂₃₉	—	2	4	no	—
35	C₈×C₂₃₉	SD₁₆	1	4	yes	—
36	C₂²×C₂₃₉	#127	1	1	yes	characteristic
37	C₄×C₂₃₉	—	8	1	no	—
38	C₄×C₂₃₉	—	8	1	no	—
39	C₄×C₂₃₉	—	8	2	no	—
40	C₄×C₂₃₉	C₂×SD₁₆	1	2	yes	—
41	C₄×C₂₃₉	#137	1	4	yes	—
42	C₂×C₂₃₉	#492	1	1	yes	characteristic
43	C₂×C₂₃₉	#497	1	2	yes	—
44	C₂₃₉	#1716	1	1	yes	characteristic, direct product factor, Sylow
45	#1716	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	#498	C₂×C₂₃₉	1	2	yes	—
47	#498	C₂×C₂₃₉	1	1	yes	characteristic
48	#494	C₂×C₂₃₉	1	1	yes	characteristic
49	#487	C₂×C₂₃₉	1	2	yes	—
50	#322	C₂×C₂₃₉	1	1	yes	characteristic
51	C₂×Q₁₆	—	2	2	no	—
52	C₄⋊Q₈	C₂²×C₂₃₉	1	1	yes	characteristic
53	C₄⋊Q₈	C₂²×C₂₃₉	1	2	yes	—
54	C₄⋊Q₈	C₂²×C₂₃₉	1	1	yes	characteristic
55	#106	—	2	4	no	—
56	#106	—	2	2	no	—
57	#103	—	2	4	no	—
58	C₄×C₈	C₂²×C₂₃₉	1	2	yes	—
59	C₄×C₈	C₂²×C₂₃₉	1	1	yes	characteristic
60	C₂×Q₈	—	2	2	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×Q₈	—	2	1	no	—
63	Q₁₆	—	4	4	no	—
64	C₂×C₈	D₄×C₂₃₉	1	4	yes	—
65	C₂×C₈	D₄×C₂₃₉	1	2	yes	—
66	C₄⋊C₄	—	2	2	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄²	C₂³×C₂₃₉	1	1	yes	characteristic
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	Q₈	—	4	4	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	C₂×D₄×C₂₃₉	1	1	yes	characteristic
78	C₂×C₄	C₂×D₄×C₂₃₉	1	2	yes	—
79	C₈	—	2	4	no	—
80	C₈	SD₁₆×C₂₃₉	1	4	yes	—
81	C₂²	#127×C₂₃₉	1	1	yes	characteristic
82	C₄	—	8	1	no	—
83	C₄	—	8	1	no	—
84	C₄	—	8	2	no	—
85	C₄	C₂×SD₁₆×C₂₃₉	1	2	yes	—
86	C₄	#137×C₂₃₉	1	4	yes	—
87	C₂	#492×C₂₃₉	1	1	yes	characteristic
88	C₂	#497×C₂₃₉	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₃₉	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₂₃₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃₉ ◁ C₄²×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
44. C₂₃₉	#1716	direct (N × H)
45. #1716	C₂₃₉	direct (N × H)

Maximal subgroups

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

subgroup	index
#498×C₂₃₉	2
#494×C₂₃₉	2
#487×C₂₃₉	2
#322×C₂₃₉	2
#1716	239

Subgroups — #1716×C239 #44480486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1716×C₂₃₉ #44480486