Subgroups — #44486831

Order 30592.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29505174	C₂	1	1	yes	maximal, characteristic
2	#29504599	C₂	1	2	yes	maximal
3	#29504597	C₂	1	2	yes	maximal
4	#29504548	C₂	1	1	yes	maximal, characteristic
5	#500×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	#21439952	—	2	1	no	—
7	C₂×C₂₃₉⋊C₁₆	C₂²	1	2	yes	—
8	#21439894	C₂²	1	1	yes	characteristic
9	#21439894	C₂²	1	1	yes	characteristic
10	#21439892	—	2	1	no	—
11	C₄⋊Q₈×C₂₃₉	—	2	1	no	—
12	#107×C₂₃₉	—	2	1	no	—
13	#107×C₂₃₉	C₂²	1	2	yes	—
14	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₂₃₉	—	2	1	no	—
16	C₄⋊Dic₂₃₉	—	2	1	no	—
17	Dic₂₃₉⋊C₄	—	4	1	no	—
18	Dic₂₃₉⋊C₄	—	4	1	no	—
19	C₂₃₉⋊C₁₆	Q₈	1	4	yes	complemented
20	C₂×Q₈×C₂₃₉	—	4	1	no	—
21	C₂×C₈×C₂₃₉	D₄	1	1	yes	characteristic
22	C₂×C₈×C₂₃₉	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₂₃₉	—	2	2	no	—
24	C₄⋊C₄×C₂₃₉	—	4	1	no	—
25	C₄²×C₂₃₉	D₄	1	1	yes	characteristic
26	C₂×Dic₂₃₉	—	4	1	no	—
27	C₂×Dic₂₃₉	—	4	1	no	—
28	Q₈×C₂₃₉	—	8	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	Q₁₆	1	2	yes	—
34	C₈×C₂₃₉	C₂×Q₈	1	2	yes	—
35	C₂²×C₂₃₉	C₂×D₈	1	1	yes	characteristic
36	C₄×C₂₃₉	—	2	1	no	—
37	C₄×C₂₃₉	—	2	1	no	—
38	C₄×C₂₃₉	—	8	2	no	—
39	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
40	C₄×C₂₃₉	C₂×Q₁₆	1	1	yes	characteristic
41	Dic₂₃₉	—	8	1	no	—
42	Dic₂₃₉	—	8	1	no	—
43	C₂×C₂₃₉	#500	1	1	yes	characteristic
44	C₂×C₂₃₉	#509	1	1	yes	characteristic
45	C₂×C₂₃₉	#510	1	1	yes	characteristic
46	C₂₃₉	#2255	1	1	yes	characteristic, complemented, Sylow
47	#2255	—	239	1	no	maximal, Sylow
48	#500	D₂₃₉	1	1	yes	characteristic
49	#499	—	239	1	no	—
50	#367	—	239	2	no	—
51	#366	—	239	2	no	—
52	#346	—	239	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	#125	—	478	1	no	—
55	C₂×C₁₆	—	239	2	no	—
56	#107	—	2	1	no	—
57	#107	—	239	1	no	—
58	#107	—	239	1	no	—
59	#107	C₂×D₂₃₉	1	2	yes	—
60	#106	—	478	1	no	—
61	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
62	C₂×Q₈	—	4	1	no	—
63	C₂×C₈	C₂²⋊D₂₃₉	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₂₃₉	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	956	1	no	—
68	C₄⋊C₄	—	956	1	no	—
69	C₄⋊C₄	—	478	1	no	—
70	C₄⋊C₄	—	478	1	no	—
71	C₄²	C₂²⋊D₂₃₉	1	1	yes	characteristic
72	C₁₆	—	239	4	no	—
73	Q₈	—	8	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	956	1	no	—
76	C₂×C₄	—	956	1	no	—
77	C₂×C₄	#16838533	1	1	yes	characteristic
78	C₂×C₄	#16838533	1	1	yes	characteristic
79	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
80	C₈	#16838538	1	2	yes	—
81	C₈	Q₈×D₂₃₉	1	2	yes	—
82	C₂²	C₂×#16838533	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	8	2	no	—
86	C₄	—	1912	1	no	—
87	C₄	—	1912	1	no	—
88	C₄	#21439958	1	1	yes	characteristic
89	C₄	C₂×#16838538	1	1	yes	characteristic
90	C₂	#29505175	1	1	yes	characteristic
91	C₂	#29505208	1	1	yes	characteristic
92	C₂	#29505213	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#16838533
commutator, cosocle	C₂×C₈×C₂₃₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₃₉
Fitting	#500×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂×D₈
hypercenter	#500	D₂₃₉
hypocenter	C₂₃₉	#2255
LCS	G ▷ C₂×C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
19. C₂₃₉⋊C₁₆	Q₈	semidirect
46. C₂₃₉	#2255	semidirect

Maximal subgroups

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

subgroup	index
#29505174	2
#29504599	2
#29504597	2
#29504548	2
#500×C₂₃₉	2
#2255	239