Subgroups — #44486775

Order 30592.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29505177	C₂	1	1	yes	maximal, characteristic
2	#29504576	C₂	1	4	yes	maximal
3	C₄×C₂₃₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#494×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	#21439960	—	2	1	no	—
6	C₂×C₂₃₉⋊C₁₆	C₂²	1	2	yes	—
7	#21439894	—	2	1	no	—
8	#21439894	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₂₃₉	—	2	1	no	—
10	C₂×Q₁₆×C₂₃₉	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₂₃₉	—	2	1	no	—
12	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₇₈	—	4	1	no	—
14	C₄⋊Dic₂₃₉	—	2	2	no	—
15	C₄⋊Dic₂₃₉	—	4	1	no	—
16	C₂₃₉⋊C₁₆	—	2	2	no	—
17	C₂×Q₈×C₂₃₉	—	2	2	no	—
18	Q₁₆×C₂₃₉	—	2	4	no	—
19	Q₁₆×C₂₃₉	—	4	2	no	—
20	C₂×C₈×C₂₃₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₂₃₉	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₂₃₉	—	4	1	no	—
23	C₄⋊C₄×C₂₃₉	—	4	1	no	—
24	C₄²×C₂₃₉	D₄	1	1	yes	characteristic
25	Dic₄₇₈	—	8	2	no	—
26	C₂×Dic₂₃₉	—	4	2	no	—
27	Q₈×C₂₃₉	—	4	2	no	—
28	Q₈×C₂₃₉	—	4	2	no	—
29	C₂×C₄×C₂₃₉	—	4	2	no	—
30	C₂×C₄×C₂₃₉	D₈	1	1	yes	characteristic
31	C₂×C₄×C₂₃₉	D₈	1	1	yes	characteristic
32	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
33	C₈×C₂₃₉	SD₁₆	1	2	yes	—
34	C₈×C₂₃₉	#41	1	2	yes	—
35	C₂²×C₂₃₉	C₂×D₈	1	1	yes	characteristic
36	C₄×C₂₃₉	—	8	2	no	—
37	C₄×C₂₃₉	SD₃₂	1	2	yes	—
38	C₄×C₂₃₉	Q₃₂	1	2	yes	—
39	C₄×C₂₃₉	#124	1	1	yes	characteristic
40	C₄×C₂₃₉	C₂×SD₁₆	1	1	yes	characteristic
41	Dic₂₃₉	—	8	2	no	—
42	C₂×C₂₃₉	#486	1	1	yes	characteristic
43	C₂×C₂₃₉	C₂×SD₃₂	1	1	yes	characteristic
44	C₂×C₂₃₉	C₂×Q₃₂	1	1	yes	characteristic
45	C₂₃₉	#2241	1	1	yes	characteristic, complemented, Sylow
46	#2241	—	239	1	no	maximal, Sylow
47	#500	—	239	1	no	—
48	#494	D₂₃₉	1	1	yes	characteristic
49	#358	—	239	4	no	—
50	C₄×C₁₆	—	239	1	no	—
51	C₂×Q₁₆	—	2	1	no	—
52	C₂×Q₁₆	C₂×D₂₃₉	1	2	yes	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	478	1	no	—
55	C₂×C₁₆	—	239	2	no	—
56	#107	—	478	1	no	—
57	#107	—	239	2	no	—
58	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
59	C₂×Q₈	—	2	2	no	—
60	C₂×Q₈	—	956	1	no	—
61	Q₁₆	—	2	4	no	—
62	Q₁₆	—	4	2	no	—
63	C₂×C₈	C₂²⋊D₂₃₉	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₂₃₉	1	1	yes	characteristic
65	C₄⋊C₄	—	4	1	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	956	1	no	—
68	C₄⋊C₄	—	478	2	no	—
69	C₄²	C₂²⋊D₂₃₉	1	1	yes	characteristic
70	C₁₆	—	478	2	no	—
71	Q₈	—	4	2	no	—
72	Q₈	—	4	2	no	—
73	Q₈	—	1912	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	956	2	no	—
76	C₂×C₄	#16838533	1	1	yes	characteristic
77	C₂×C₄	#16838533	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
79	C₈	#16838536	1	2	yes	—
80	C₈	#16838547	1	2	yes	—
81	C₂²	C₂×#16838533	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	1912	2	no	—
84	C₄	#21439903	1	2	yes	—
85	C₄	#21439906	1	2	yes	—
86	C₄	#21439947	1	1	yes	characteristic
87	C₄	C₂×#16838536	1	1	yes	characteristic
88	C₂	#29505118	1	1	yes	characteristic
89	C₂	C₂×#21439903	1	1	yes	characteristic
90	C₂	C₂×#21439906	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#16838533
commutator, cosocle	C₂×C₈×C₂₃₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₃₉
Fitting	#494×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂×D₈
hypercenter	#494	D₂₃₉
hypocenter	C₂₃₉	#2241
LCS	G ▷ C₂×C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
45. C₂₃₉	#2241	semidirect

Maximal subgroups

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

subgroup	index
#29505177	2
#29504576	2
C₄×C₂₃₉⋊C₁₆	2
#494×C₂₃₉	2
#2241	239