Subgroups — #44486791

Order 30592.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29505149	C₂	1	1	yes	maximal, characteristic
2	#29504572	C₂	1	4	yes	maximal, complemented
3	#29504548	C₂	1	1	yes	maximal, characteristic, complemented
4	#499×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₉₁₂	—	2	1	no	—
6	C₂×D₁₉₁₂	C₂²	1	2	yes	—
7	#21439957	—	2	1	no	—
8	C₂×C₂₃₉⋊C₁₆	C₂²	1	2	yes	—
9	#125×C₂₃₉	—	2	1	no	—
10	#107×C₂₃₉	C₂²	1	2	yes	—
11	#106×C₂₃₉	—	2	1	no	—
12	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
13	C₂×D₉₅₆	—	2	2	no	—
14	C₂×D₉₅₆	—	4	2	no	—
15	D₁₉₁₂	—	2	4	no	—
16	D₁₉₁₂	—	4	2	no	—
17	C₂₃₉⋊C₁₆	—	2	2	no	—
18	C₂×C₈×C₂₃₉	D₄	1	1	yes	characteristic
19	C₂×C₈×C₂₃₉	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₂₃₉	—	2	2	no	—
21	C₄⋊C₄×C₂₃₉	—	4	2	no	—
22	C₄²×C₂₃₉	D₄	1	1	yes	characteristic
23	C₂²×D₂₃₉	—	4	2	no	—
24	D₉₅₆	—	4	2	no	—
25	D₉₅₆	—	4	2	no	—
26	D₉₅₆	—	8	4	no	—
27	C₂×C₄×C₂₃₉	—	4	2	no	—
28	C₂×C₄×C₂₃₉	D₈	1	2	yes	—
29	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
30	C₈×C₂₃₉	SD₁₆	1	2	yes	—
31	C₈×C₂₃₉	#41	1	2	yes	—
32	C₂²×C₂₃₉	C₂×D₈	1	1	yes	characteristic
33	C₂×D₂₃₉	—	8	2	no	—
34	C₂×D₂₃₉	—	8	4	no	—
35	C₄×C₂₃₉	—	2	2	no	—
36	C₄×C₂₃₉	—	8	2	no	—
37	C₄×C₂₃₉	#124	1	1	yes	characteristic
38	C₄×C₂₃₉	C₂×SD₁₆	1	1	yes	characteristic
39	C₂×C₂₃₉	#486	1	1	yes	characteristic
40	C₂×C₂₃₉	#509	1	2	yes	—
41	D₂₃₉	—	16	2	no	—
42	C₂₃₉	#2244	1	1	yes	characteristic, complemented, Sylow
43	#2244	—	239	1	no	maximal, Sylow
44	#499	D₂₃₉	1	1	yes	characteristic, complemented
45	#493	—	239	1	no	—
46	#357	—	239	4	no	—
47	#346	—	239	1	no	—
48	C₂×D₈	—	478	1	no	—
49	C₂×D₈	—	239	2	no	—
50	#127	—	478	1	no	—
51	#125	—	2	1	no	—
52	C₂×C₁₆	—	239	2	no	—
53	#107	C₂×D₂₃₉	1	2	yes	—
54	#106	—	2	1	no	—
55	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
56	C₂×D₄	—	956	2	no	—
57	C₂×D₄	—	478	2	no	—
58	D₈	—	956	2	no	—
59	D₈	—	478	4	no	—
60	C₂×C₈	C₂²⋊D₂₃₉	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₂₃₉	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄²	C₂²⋊D₂₃₉	1	1	yes	characteristic
65	C₁₆	—	478	2	no	—
66	C₂³	—	956	2	no	—
67	D₄	—	1912	4	no	—
68	D₄	—	956	2	no	—
69	D₄	—	956	2	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	#16838533	1	2	yes	—
72	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
73	C₈	#16838535	1	2	yes	—
74	C₈	#16838549	1	2	yes	—
75	C₂²	—	1912	2	no	—
76	C₂²	—	1912	4	no	—
77	C₂²	C₂×#16838533	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	8	2	no	—
80	C₄	#21439949	1	1	yes	characteristic
81	C₄	C₂×#16838535	1	1	yes	characteristic
82	C₂	—	3824	2	no	—
83	C₂	#29505120	1	1	yes	characteristic
84	C₂	#29505207	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#16838533
commutator, cosocle	C₂×C₈×C₂₃₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₃₉
Fitting	#499×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×D₈
prime radical	#29505149	C₂
hypercenter	#499	D₂₃₉
hypocenter	C₂₃₉	#2244
LCS	G ▷ C₂×C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #499

Complements

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

N	H ≅ G/N	type
2. #29504572	C₂	semidirect
3. #29504548	C₂	semidirect
4. #499×C₂₃₉	C₂	semidirect
42. C₂₃₉	#2244	semidirect
44. #499	D₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#29505149	2
#29504572	2
#29504548	2
#499×C₂₃₉	2
#2244	239