Subgroups — C₂×C₃×C₅×#500 #14276193

Order 1920.

848 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₃×C₅×#500	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×C₅×#107	C₂	1	4	yes	maximal
4	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#500	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	—	2	2	no	—
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₃×C₅×C₄⋊Q₈	—	2	4	no	—
9	C₃×C₅×C₄⋊Q₈	C₂²	1	8	yes	—
10	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
11	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₃×C₅×#107	C₂²	1	16	yes	—
14	C₃×C₄×C₅×C₈	C₂²	1	4	yes	—
15	C₂×C₃×#500	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	2	yes	—
17	C₅×#500	C₂×C₃	1	8	yes	direct product factor
18	C₂×C₅×#107	C₂×C₃	1	4	yes	—
19	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×C₃×C₅×Q₈	—	2	8	no	—
21	C₂×C₃×C₅×Q₈	—	4	8	no	—
22	C₂²×C₃×C₄×C₅	—	2	4	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	Q₈	1	8	yes	complemented
27	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
28	C₃×C₅×C₄⋊C₄	—	2	8	no	—
29	C₃×C₅×C₄⋊C₄	—	2	4	no	—
30	C₃×C₅×C₄⋊C₄	C₂³	1	8	yes	—
31	C₃×C₄²×C₅	D₄	1	2	yes	—
32	C₃×C₄²×C₅	C₂³	1	2	yes	—
33	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	2	yes	—
34	C₃×#500	C₂×C₅	1	8	yes	direct product factor
35	C₂×C₃×#107	C₂×C₅	1	4	yes	—
36	C₂×C₃×C₄×C₈	C₂×C₅	1	1	yes	characteristic
37	C₂²×C₅×Q₈	—	2	2	no	—
38	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
39	C₅×C₄⋊Q₈	—	2	4	no	—
40	C₅×C₄⋊Q₈	C₂²×C₃	1	8	yes	—
41	C₂×C₅×C₄⋊C₄	—	2	2	no	—
42	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	4	yes	—
43	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
44	C₅×#107	C₂²×C₃	1	16	yes	—
45	C₄×C₅×C₈	C₂²×C₃	1	4	yes	—
46	C₂×#500	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
48	C₃×C₅×Q₈	—	4	16	no	—
49	C₂×C₃×C₄×C₅	—	2	8	no	—
50	C₂×C₃×C₄×C₅	—	4	8	no	—
51	C₂×C₃×C₄×C₅	D₈	1	4	yes	—
52	C₂×C₃×C₄×C₅	Q₁₆	1	4	yes	—
53	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
57	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
58	C₃×C₅×C₈	C₂×Q₈	1	8	yes	complemented
59	C₂²×C₃×Q₈	—	2	2	no	—
60	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
61	C₃×C₄⋊Q₈	—	2	4	no	—
62	C₃×C₄⋊Q₈	C₂²×C₅	1	8	yes	—
63	C₂×C₃×C₄⋊C₄	—	2	2	no	—
64	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
65	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
66	C₃×#107	C₂²×C₅	1	16	yes	—
67	C₃×C₄×C₈	C₂²×C₅	1	4	yes	—
68	C₂×C₅×Q₈	—	2	8	no	—
69	C₂×C₅×Q₈	—	4	8	no	—
70	C₂²×C₄×C₅	—	2	4	no	—
71	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
74	C₂×C₅×C₈	C₃×Q₈	1	8	yes	complemented
75	C₂×C₅×C₈	C₂³×C₃	1	4	yes	—
76	C₅×C₄⋊C₄	—	2	8	no	—
77	C₅×C₄⋊C₄	—	2	4	no	—
78	C₅×C₄⋊C₄	C₂³×C₃	1	8	yes	—
79	C₄²×C₅	C₃×D₄	1	2	yes	—
80	C₄²×C₅	C₂³×C₃	1	2	yes	—
81	C₂×C₄⋊Q₈	C₂×C₃×C₅	1	2	yes	—
82	#500	C₂×C₃×C₅	1	8	yes	direct product factor
83	C₂×#107	C₂×C₃×C₅	1	4	yes	—
84	C₂×C₄×C₈	C₂×C₃×C₅	1	1	yes	characteristic
85	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
86	C₂²×C₃×C₅	C₂×D₈	1	2	yes	—
87	C₂²×C₃×C₅	C₂×Q₁₆	1	2	yes	—
88	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
89	C₃×C₄×C₅	—	4	8	no	—
90	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
91	C₃×C₄×C₅	C₂×D₈	1	4	yes	—
92	C₃×C₄×C₅	C₂×Q₁₆	1	4	yes	—
93	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
94	C₂×C₃×Q₈	—	2	8	no	—
95	C₂×C₃×Q₈	—	4	8	no	—
96	C₂²×C₃×C₄	—	2	4	no	—
97	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
98	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
100	C₂×C₃×C₈	C₅×Q₈	1	8	yes	complemented
101	C₂×C₃×C₈	C₂³×C₅	1	4	yes	—
102	C₃×C₄⋊C₄	—	2	8	no	—
103	C₃×C₄⋊C₄	—	2	4	no	—
104	C₃×C₄⋊C₄	C₂³×C₅	1	8	yes	—
105	C₃×C₄²	C₅×D₄	1	2	yes	—
106	C₃×C₄²	C₂³×C₅	1	2	yes	—
107	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
108	C₅×Q₈	—	4	16	no	—
109	C₂×C₄×C₅	—	2	8	no	—
110	C₂×C₄×C₅	—	4	8	no	—
111	C₂×C₄×C₅	C₃×D₈	1	4	yes	—
112	C₂×C₄×C₅	C₃×Q₁₆	1	4	yes	—
113	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
115	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
116	C₂×C₄×C₅	C₂×C₃×Q₈	1	4	yes	—
117	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
118	C₅×C₈	C₂×C₃×Q₈	1	8	yes	complemented
119	C₂²×Q₈	—	2	2	no	—
120	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	C₂²×C₃×C₅	1	8	yes	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	C₂²×C₃×C₅	1	4	yes	—
125	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
126	#107	C₂²×C₃×C₅	1	16	yes	—
127	C₄×C₈	C₂²×C₃×C₅	1	4	yes	—
128	C₂×C₃×C₅	#500	1	4	yes	direct product factor
129	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₂×C₃×C₅	C₂²×D₈	1	1	yes	characteristic
131	C₂×C₃×C₅	C₂²×Q₁₆	1	1	yes	characteristic
132	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
133	C₃×Q₈	—	4	16	no	—
134	C₂×C₃×C₄	—	2	8	no	—
135	C₂×C₃×C₄	—	4	8	no	—
136	C₂×C₃×C₄	C₅×D₈	1	4	yes	—
137	C₂×C₃×C₄	C₅×Q₁₆	1	4	yes	—
138	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
139	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
140	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
141	C₂×C₃×C₄	C₂×C₅×Q₈	1	4	yes	—
142	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
143	C₃×C₈	C₂×C₅×Q₈	1	8	yes	complemented
144	C₂²×C₅	C₃×C₄⋊Q₈	1	2	yes	—
145	C₂²×C₅	C₂×C₃×D₈	1	2	yes	—
146	C₂²×C₅	C₂×C₃×Q₁₆	1	2	yes	—
147	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
148	C₄×C₅	—	4	8	no	—
149	C₄×C₅	C₃×C₄⋊Q₈	1	2	yes	—
150	C₄×C₅	C₂×C₃×D₈	1	4	yes	—
151	C₄×C₅	C₂×C₃×Q₁₆	1	4	yes	—
152	C₄×C₅	C₂²×C₃×Q₈	1	2	yes	—
153	C₂×Q₈	—	2	8	no	—
154	C₂×Q₈	—	4	8	no	—
155	C₂²×C₄	—	2	4	no	—
156	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
157	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
159	C₂×C₈	C₃×C₅×Q₈	1	8	yes	complemented
160	C₂×C₈	C₂³×C₃×C₅	1	4	yes	—
161	C₄⋊C₄	—	2	8	no	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	C₂³×C₃×C₅	1	8	yes	—
164	C₄²	C₃×C₅×D₄	1	2	yes	—
165	C₄²	C₂³×C₃×C₅	1	2	yes	—
166	C₃×C₅	C₂×#500	1	1	yes	characteristic, direct product factor, Hall
167	C₂²×C₃	C₅×C₄⋊Q₈	1	2	yes	—
168	C₂²×C₃	C₂×C₅×D₈	1	2	yes	—
169	C₂²×C₃	C₂×C₅×Q₁₆	1	2	yes	—
170	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
171	C₃×C₄	—	4	8	no	—
172	C₃×C₄	C₅×C₄⋊Q₈	1	2	yes	—
173	C₃×C₄	C₂×C₅×D₈	1	4	yes	—
174	C₃×C₄	C₂×C₅×Q₁₆	1	4	yes	—
175	C₃×C₄	C₂²×C₅×Q₈	1	2	yes	—
176	C₂×C₅	C₃×#500	1	4	yes	direct product factor
177	C₂×C₅	C₂×C₃×C₄⋊Q₈	1	1	yes	characteristic
178	C₂×C₅	C₂²×C₃×D₈	1	1	yes	characteristic
179	C₂×C₅	C₂²×C₃×Q₁₆	1	1	yes	characteristic
180	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
181	Q₈	—	4	16	no	—
182	C₂×C₄	—	2	8	no	—
183	C₂×C₄	—	4	8	no	—
184	C₂×C₄	C₃×C₅×D₈	1	4	yes	—
185	C₂×C₄	C₃×C₅×Q₁₆	1	4	yes	—
186	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
187	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
188	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
189	C₂×C₄	C₂×C₃×C₅×Q₈	1	4	yes	—
190	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
191	C₈	C₂×C₃×C₅×Q₈	1	8	yes	complemented
192	C₂×C₃	C₅×#500	1	4	yes	direct product factor
193	C₂×C₃	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
194	C₂×C₃	C₂²×C₅×D₈	1	1	yes	characteristic
195	C₂×C₃	C₂²×C₅×Q₁₆	1	1	yes	characteristic
196	C₅	C₂×C₃×#500	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²	C₃×C₅×C₄⋊Q₈	1	2	yes	—
198	C₂²	C₂×C₃×C₅×D₈	1	2	yes	—
199	C₂²	C₂×C₃×C₅×Q₁₆	1	2	yes	—
200	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
201	C₄	—	4	8	no	—
202	C₄	C₃×C₅×C₄⋊Q₈	1	2	yes	—
203	C₄	C₂×C₃×C₅×D₈	1	4	yes	—
204	C₄	C₂×C₃×C₅×Q₁₆	1	4	yes	—
205	C₄	C₂²×C₃×C₅×Q₈	1	2	yes	—
206	C₃	C₂×C₅×#500	1	1	yes	characteristic, direct product factor, Sylow
207	C₂	C₃×C₅×#500	1	4	yes	direct product factor
208	C₂	C₂×C₃×C₅×C₄⋊Q₈	1	1	yes	characteristic
209	C₂	C₂²×C₃×C₅×D₈	1	1	yes	characteristic
210	C₂	C₂²×C₃×C₅×Q₁₆	1	1	yes	characteristic

	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×#500	C₂	direct (N × H)
5. C₂×C₅×#500	C₃	direct (N × H)
15. C₂×C₃×#500	C₅	direct (N × H)
17. C₅×#500	C₂×C₃	direct (N × H)
26. C₂×C₃×C₅×C₈	Q₈	semidirect
34. C₃×#500	C₂×C₅	direct (N × H)
46. C₂×#500	C₃×C₅	direct (N × H)
58. C₃×C₅×C₈	C₂×Q₈	semidirect
74. C₂×C₅×C₈	C₃×Q₈	semidirect
82. #500	C₂×C₃×C₅	direct (N × H)
100. C₂×C₃×C₈	C₅×Q₈	semidirect
118. C₅×C₈	C₂×C₃×Q₈	semidirect
128. C₂×C₃×C₅	#500	direct (N × H)
143. C₃×C₈	C₂×C₅×Q₈	semidirect
159. C₂×C₈	C₃×C₅×Q₈	semidirect
166. C₃×C₅	C₂×#500	direct (N × H)
176. C₂×C₅	C₃×#500	direct (N × H)
191. C₈	C₂×C₃×C₅×Q₈	semidirect
192. C₂×C₃	C₅×#500	direct (N × H)
196. C₅	C₂×C₃×#500	direct (N × H)
206. C₃	C₂×C₅×#500	direct (N × H)
207. C₂	C₃×C₅×#500	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×C₄⋊Q₈	2
C₃×C₅×#500	2
C₂×C₃×C₅×#107	2
C₂×C₃×C₄×C₅×C₈	2
C₂×C₅×#500	3
C₂×C₃×#500	5

Subgroups — C2×C3×C5×#500 #14276193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₅×#500 #14276193