Subgroups — #44486406

Order 30592.

150 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₉₁₂	C₂	1	1	yes	maximal, characteristic
2	#29504866	C₂	1	2	yes	maximal
3	#29504575	C₂	1	2	yes	maximal
4	C₁₆×Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
5	#366×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₉₅₆	C₂²	1	2	yes	—
7	#21439925	—	2	2	no	—
8	Dic₁₉₁₂	C₄	1	4	yes	complemented
9	C₂×C₂₃₉⋊C₁₆	C₂²	1	1	yes	characteristic
10	#21439883	—	2	2	no	—
11	C₈×Dic₂₃₉	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₂₃₉	C₂²	1	1	yes	characteristic
13	#107×C₂₃₉	C₂²	1	2	yes	—
14	C₂×Dic₄₇₈	—	2	2	no	—
15	Dic₉₅₆	—	2	2	no	—
16	Dic₉₅₆	C₂×C₄	1	4	yes	—
17	C₂×C₂₃₉⋊C₈	D₄	1	1	yes	characteristic
18	Dic₂₃₉⋊C₄	—	4	2	no	—
19	C₄×Dic₂₃₉	—	4	2	no	—
20	C₄×Dic₂₃₉	D₄	1	1	yes	characteristic
21	C₂₃₉⋊C₁₆	—	2	1	no	—
22	C₂×C₈×C₂₃₉	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₂₃₉	—	2	2	no	—
24	C₁₆×C₂₃₉	C₂×C₄	1	2	yes	—
25	Dic₄₇₈	—	2	4	no	—
26	Dic₄₇₈	—	4	2	no	—
27	C₂×Dic₂₃₉	—	4	2	no	—
28	C₂×Dic₂₃₉	D₈	1	1	yes	characteristic
29	C₂×Dic₂₃₉	D₈	1	1	yes	characteristic
30	C₂₃₉⋊C₈	—	2	1	no	—
31	C₂×C₄×C₂₃₉	—	4	2	no	—
32	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
33	C₈×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₂₃₉	#41	1	1	yes	characteristic
35	C₂²×C₂₃₉	C₂×D₈	1	1	yes	characteristic
36	C₄×C₂₃₉	—	8	2	no	—
37	C₄×C₂₃₉	C₄×D₄	1	1	yes	characteristic
38	C₄×C₂₃₉	#135	1	1	yes	characteristic
39	Dic₂₃₉	—	2	1	no	—
40	Dic₂₃₉	—	4	4	no	—
41	Dic₂₃₉	Q₃₂	1	2	yes	complemented
42	C₂×C₂₃₉	C₄×D₈	1	1	yes	characteristic
43	C₂×C₂₃₉	C₂×Q₃₂	1	1	yes	characteristic
44	C₂×C₂₃₉	#508	1	1	yes	characteristic
45	C₂₃₉	C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
46	C₄×Q₃₂	—	239	1	no	maximal, Sylow
47	C₂×Q₃₂	—	239	1	no	—
48	C₄×Q₁₆	—	239	2	no	—
49	#366	D₂₃₉	1	1	yes	characteristic
50	#358	—	239	2	no	—
51	C₄×C₁₆	—	239	1	no	—
52	C₂×Q₁₆	—	239	2	no	—
53	C₄×Q₈	—	478	2	no	—
54	Q₃₂	—	239	4	no	—
55	C₂×C₁₆	—	239	1	no	—
56	C₂×C₁₆	C₂×D₂₃₉	1	1	yes	characteristic
57	#107	C₂×D₂₃₉	1	2	yes	—
58	#103	—	478	2	no	—
59	C₄×C₈	—	239	1	no	—
60	C₂×Q₈	—	478	2	no	—
61	Q₁₆	—	478	2	no	—
62	Q₁₆	—	239	4	no	—
63	C₂×C₈	—	239	1	no	—
64	C₂×C₈	C₂²×D₂₃₉	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	956	2	no	—
67	C₄²	—	956	2	no	—
68	C₄²	—	239	1	no	—
69	C₁₆	—	478	1	no	—
70	C₁₆	C₄×D₂₃₉	1	2	yes	—
71	Q₈	—	956	2	no	—
72	Q₈	—	478	4	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	956	2	no	—
75	C₂×C₄	—	239	1	no	—
76	C₂×C₄	—	239	1	no	—
77	C₂×C₄	D₄×D₂₃₉	1	1	yes	characteristic
78	C₈	—	478	1	no	—
79	C₈	C₂×C₄×D₂₃₉	1	1	yes	characteristic
80	C₈	#16838549	1	1	yes	characteristic
81	C₂²	D₈×D₂₃₉	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	956	4	no	—
84	C₄	—	478	1	no	—
85	C₄	—	239	2	no	—
86	C₄	#21439922	1	1	yes	characteristic
87	C₄	#21439983	1	1	yes	characteristic
88	C₂	#29504854	1	1	yes	characteristic
89	C₂	Q₃₂×D₂₃₉	1	1	yes	characteristic
90	C₂	#29505201	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₂₃₉
commutator	C₈×C₂₃₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₃₉
Fitting	#366×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂×D₈
cosocle	C₂×C₈×C₂₃₉	C₂³
hypercenter	#366	D₂₃₉
hypocenter	C₂₃₉	C₄×Q₃₂
LCS	G ▷ C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
8. Dic₁₉₁₂	C₄	semidirect
41. Dic₂₃₉	Q₃₂	semidirect
45. C₂₃₉	C₄×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₉₁₂	2
#29504866	2
#29504575	2
C₁₆×Dic₂₃₉	2
#366×C₂₃₉	2
C₄×Q₃₂	239