Subgroups — C₂×C₃×C₅×#499 #14276192

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#125	C₂	1	2	yes	maximal
2	C₃×C₅×#499	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×C₅×#107	C₂	1	2	yes	maximal
4	C₂×C₃×C₅×#106	C₂	1	2	yes	maximal
5	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#499	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#125	—	2	4	no	—
10	C₃×C₅×#125	C₂²	1	8	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	—	2	4	no	—
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₃×C₅×#107	C₂²	1	8	yes	—
15	C₃×C₅×#106	C₂²	1	8	yes	—
16	C₃×C₄×C₅×C₈	C₂²	1	4	yes	—
17	C₂×C₃×#499	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#125	C₂×C₃	1	2	yes	—
19	C₅×#499	C₂×C₃	1	8	yes	direct product factor
20	C₂×C₅×#107	C₂×C₃	1	2	yes	—
21	C₂×C₅×#106	C₂×C₃	1	2	yes	—
22	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	—	2	4	no	—
24	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	Q₈	1	4	yes	—
27	C₂×C₃×C₅×C₈	Q₈	1	4	yes	—
28	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
29	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
30	C₃×C₅×C₄⋊C₄	—	2	16	no	—
31	C₃×C₅×C₄⋊C₄	—	2	4	no	—
32	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
33	C₃×C₄²×C₅	D₄	1	2	yes	—
34	C₃×C₄²×C₅	C₂³	1	2	yes	—
35	C₂×C₃×#125	C₂×C₅	1	2	yes	—
36	C₃×#499	C₂×C₅	1	8	yes	direct product factor
37	C₂×C₃×#107	C₂×C₅	1	2	yes	—
38	C₂×C₃×#106	C₂×C₅	1	2	yes	—
39	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
40	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₅×#125	—	2	4	no	—
43	C₅×#125	C₂²×C₃	1	8	yes	—
44	C₂×C₅×C₄⋊C₄	—	2	4	no	—
45	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
46	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
47	C₅×#107	C₂²×C₃	1	8	yes	—
48	C₅×#106	C₂²×C₃	1	8	yes	—
49	C₄×C₅×C₈	C₂²×C₃	1	4	yes	—
50	C₂×#499	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
51	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	—	2	4	no	—
53	C₂×C₃×C₄×C₅	—	2	8	no	—
54	C₂×C₃×C₄×C₅	—	4	8	no	—
55	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
56	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
58	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
59	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
60	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
61	C₃×C₅×C₈	C₂×Q₈	1	4	yes	—
62	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
63	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
64	C₃×#125	—	2	4	no	—
65	C₃×#125	C₂²×C₅	1	8	yes	—
66	C₂×C₃×C₄⋊C₄	—	2	4	no	—
67	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
68	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
69	C₃×#107	C₂²×C₅	1	8	yes	—
70	C₃×#106	C₂²×C₅	1	8	yes	—
71	C₃×C₄×C₈	C₂²×C₅	1	4	yes	—
72	C₂²×C₄×C₅	—	2	4	no	—
73	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
75	C₂×C₅×C₈	C₃×Q₈	1	4	yes	—
76	C₂×C₅×C₈	C₃×Q₈	1	4	yes	—
77	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
78	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
79	C₅×C₄⋊C₄	—	2	16	no	—
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	C₂³×C₃	1	8	yes	—
82	C₄²×C₅	C₃×D₄	1	2	yes	—
83	C₄²×C₅	C₂³×C₃	1	2	yes	—
84	C₂×#125	C₂×C₃×C₅	1	2	yes	—
85	#499	C₂×C₃×C₅	1	8	yes	direct product factor
86	C₂×#107	C₂×C₃×C₅	1	2	yes	—
87	C₂×#106	C₂×C₃×C₅	1	2	yes	—
88	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
89	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
90	C₂²×C₃×C₅	#135	1	4	yes	—
91	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
92	C₃×C₄×C₅	—	2	4	no	—
93	C₃×C₄×C₅	—	4	8	no	—
94	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
95	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
96	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
97	C₂²×C₃×C₄	—	2	4	no	—
98	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
99	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
100	C₂×C₃×C₈	C₅×Q₈	1	4	yes	—
101	C₂×C₃×C₈	C₅×Q₈	1	4	yes	—
102	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
103	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
104	C₃×C₄⋊C₄	—	2	16	no	—
105	C₃×C₄⋊C₄	—	2	4	no	—
106	C₃×C₄⋊C₄	C₂³×C₅	1	8	yes	—
107	C₃×C₄²	C₅×D₄	1	2	yes	—
108	C₃×C₄²	C₂³×C₅	1	2	yes	—
109	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	—	2	8	no	—
112	C₂×C₄×C₅	—	4	8	no	—
113	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
114	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
116	C₂×C₄×C₅	C₂×C₃×Q₈	1	2	yes	—
117	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
118	C₅×C₈	C₂×C₃×Q₈	1	4	yes	—
119	C₅×C₈	C₂×C₃×Q₈	1	4	yes	—
120	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
121	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
122	#125	—	2	4	no	—
123	#125	C₂²×C₃×C₅	1	8	yes	—
124	C₂×C₄⋊C₄	—	2	4	no	—
125	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
126	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
127	#107	C₂²×C₃×C₅	1	8	yes	—
128	#106	C₂²×C₃×C₅	1	8	yes	—
129	C₄×C₈	C₂²×C₃×C₅	1	4	yes	—
130	C₂×C₃×C₅	#499	1	4	yes	direct product factor
131	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
132	C₂×C₃×C₅	C₂×#135	1	2	yes	—
133	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
134	C₂×C₃×C₄	—	2	4	no	—
135	C₂×C₃×C₄	—	2	8	no	—
136	C₂×C₃×C₄	—	4	8	no	—
137	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
138	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
140	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
141	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
142	C₃×C₈	C₂×C₅×Q₈	1	4	yes	—
143	C₃×C₈	C₂×C₅×Q₈	1	4	yes	—
144	C₂²×C₅	C₃×C₄⋊Q₈	1	2	yes	—
145	C₂²×C₅	C₃×#135	1	4	yes	—
146	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
147	C₄×C₅	—	2	4	no	—
148	C₄×C₅	—	4	8	no	—
149	C₄×C₅	C₃×C₄⋊Q₈	1	2	yes	—
150	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
151	C₄×C₅	C₂²×C₃×Q₈	1	1	yes	characteristic
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
154	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
155	C₂×C₈	C₃×C₅×Q₈	1	4	yes	—
156	C₂×C₈	C₃×C₅×Q₈	1	4	yes	—
157	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
158	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
159	C₄⋊C₄	—	2	16	no	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	C₂³×C₃×C₅	1	8	yes	—
162	C₄²	C₃×C₅×D₄	1	2	yes	—
163	C₄²	C₂³×C₃×C₅	1	2	yes	—
164	C₃×C₅	C₂×#499	1	1	yes	characteristic, direct product factor, Hall
165	C₂²×C₃	C₅×C₄⋊Q₈	1	2	yes	—
166	C₂²×C₃	C₅×#135	1	4	yes	—
167	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
168	C₃×C₄	—	2	4	no	—
169	C₃×C₄	—	4	8	no	—
170	C₃×C₄	C₅×C₄⋊Q₈	1	2	yes	—
171	C₃×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
172	C₃×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
173	C₂×C₅	C₃×#499	1	4	yes	direct product factor
174	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
175	C₂×C₅	C₂×C₃×#135	1	2	yes	—
176	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	4	8	no	—
180	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
181	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
182	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
183	C₂×C₄	C₂×C₃×C₅×Q₈	1	2	yes	—
184	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
185	C₈	C₂×C₃×C₅×Q₈	1	4	yes	—
186	C₈	C₂×C₃×C₅×Q₈	1	4	yes	—
187	C₂×C₃	C₅×#499	1	4	yes	direct product factor
188	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
189	C₂×C₃	C₂×C₅×#135	1	2	yes	—
190	C₅	C₂×C₃×#499	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²	C₃×C₅×C₄⋊Q₈	1	2	yes	—
192	C₂²	C₃×C₅×#135	1	4	yes	—
193	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
194	C₄	—	2	4	no	—
195	C₄	—	4	8	no	—
196	C₄	C₃×C₅×C₄⋊Q₈	1	2	yes	—
197	C₄	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
198	C₄	C₂²×C₃×C₅×Q₈	1	1	yes	characteristic
199	C₃	C₂×C₅×#499	1	1	yes	characteristic, direct product factor, Sylow
200	C₂	C₃×C₅×#499	1	4	yes	direct product factor
201	C₂	C₂×C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
202	C₂	C₂×C₃×C₅×#135	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#499	C₂	direct (N × H)
6. C₂×C₅×#499	C₃	direct (N × H)
17. C₂×C₃×#499	C₅	direct (N × H)
19. C₅×#499	C₂×C₃	direct (N × H)
36. C₃×#499	C₂×C₅	direct (N × H)
50. C₂×#499	C₃×C₅	direct (N × H)
85. #499	C₂×C₃×C₅	direct (N × H)
130. C₂×C₃×C₅	#499	direct (N × H)
164. C₃×C₅	C₂×#499	direct (N × H)
173. C₂×C₅	C₃×#499	direct (N × H)
187. C₂×C₃	C₅×#499	direct (N × H)
190. C₅	C₂×C₃×#499	direct (N × H)
199. C₃	C₂×C₅×#499	direct (N × H)
200. C₂	C₃×C₅×#499	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#125	2
C₃×C₅×#499	2
C₂×C₃×C₅×#107	2
C₂×C₃×C₅×#106	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₅×#499	3
C₂×C₃×#499	5

Subgroups — C2×C3×C5×#499 #14276192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#499 #14276192