Subgroups — C₅×#76203 #14327115

Order 1920.

1408 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#893	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#893	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4989	C₂	1	2	yes	maximal, complemented
6	C₅×#4987	C₂	1	2	yes	maximal, complemented
7	C₅×#4984	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4977	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4668	C₂	1	2	yes	maximal, complemented
11	C₅×D₃×#95	C₂	1	2	yes	maximal, complemented
12	C₅×#2349	—	3	1	no	maximal, Hall
13	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
15	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
16	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
22	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
24	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
25	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
26	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
30	C₅×#893	—	2	4	no	—
31	C₅×#893	C₄	1	8	yes	complemented
32	C₄×C₅×D₁₂	—	2	4	no	—
33	C₃×C₅×#95	C₂²	1	2	yes	—
34	C₅×C₃⋊#95	C₂²	1	1	yes	characteristic
35	C₅×C₃⋊#95	C₂²	1	2	yes	—
36	C₅×C₃⋊#95	C₂²	1	2	yes	—
37	C₅×C₃⋊#95	C₂²	1	2	yes	—
38	C₅×C₃⋊#95	C₂²	1	1	yes	characteristic
39	#76203	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
40	C₂×C₄×C₅×D₄	—	3	1	no	—
41	C₂×C₄×C₅×D₄	—	3	1	no	—
42	C₂×C₄×C₅×D₄	—	3	1	no	—
43	C₅×#389	—	3	1	no	—
44	C₅×#389	D₃	1	1	yes	characteristic, complemented
45	C₅×#385	—	3	2	no	—
46	C₅×#385	—	3	2	no	—
47	C₅×#382	—	3	1	no	—
48	C₅×#380	—	3	2	no	—
49	C₄×C₅×C₄⋊C₄	—	3	1	no	—
50	C₂×C₅×#95	—	3	2	no	—
51	C₂³×C₅×D₃	C₂³	1	2	yes	—
52	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
53	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
54	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
55	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
58	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
59	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
62	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
63	C₂×C₅×D₁₂	D₄	1	4	yes	—
64	C₂×C₅×D₁₂	Q₈	1	4	yes	—
65	C₂×C₄×C₅×D₃	—	2	4	no	—
66	C₂×C₄×C₅×D₃	—	2	4	no	—
67	C₂×C₄×C₅×D₃	—	2	4	no	—
68	C₂×C₄×C₅×D₃	—	2	4	no	—
69	C₂×C₄×C₅×D₃	—	4	2	no	—
70	C₂×C₄×C₅×D₃	—	4	2	no	—
71	C₂×C₄×C₅×D₃	—	4	2	no	—
72	C₂×C₄×C₅×D₃	—	4	2	no	—
73	C₂×C₄×C₅×D₃	C₂×C₄	1	8	yes	complemented
74	C₃×C₅×C₄⋊C₄	—	2	2	no	—
75	C₃×C₅×C₄⋊C₄	—	4	2	no	—
76	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
77	C₃×C₄²×C₅	—	2	2	no	—
78	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
79	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
80	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	complemented
81	C₅×C₄⋊Dic₃	—	2	2	no	—
82	C₅×C₄⋊Dic₃	—	4	2	no	—
83	C₅×Dic₃⋊C₄	—	2	2	no	—
84	C₅×Dic₃⋊C₄	—	2	2	no	—
85	C₄×C₅×Dic₃	—	2	2	no	—
86	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	complemented
87	C₂×#893	C₂×C₅	1	1	yes	characteristic
88	C₂×#893	C₂×C₅	1	1	yes	characteristic
89	C₂×C₄×D₁₂	C₂×C₅	1	1	yes	characteristic
90	C₃×#389	C₂×C₅	1	1	yes	characteristic, complemented
91	#4989	C₂×C₅	1	2	yes	complemented
92	#4987	C₂×C₅	1	2	yes	complemented
93	#4984	C₂×C₅	1	1	yes	characteristic, complemented
94	#4977	C₂×C₅	1	1	yes	characteristic, complemented
95	C₄×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
96	#4668	C₂×C₅	1	2	yes	complemented
97	D₃×#95	C₂×C₅	1	2	yes	complemented
98	C₂²×C₅×D₄	—	3	1	no	—
99	C₂³×C₄×C₅	—	3	2	no	—
100	C₂³×C₄×C₅	—	3	2	no	—
101	C₂³×C₄×C₅	—	3	2	no	—
102	C₄×C₅×D₄	—	3	8	no	—
103	C₄×C₅×D₄	—	6	4	no	—
104	C₄×C₅×D₄	—	6	4	no	—
105	C₂×C₅×C₄⋊C₄	—	3	1	no	—
106	C₂×C₅×C₄⋊C₄	—	3	1	no	—
107	C₂×C₅×C₄⋊C₄	—	3	1	no	—
108	C₂×C₅×C₄⋊C₄	—	3	2	no	—
109	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
110	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
111	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
112	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
113	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
114	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
115	C₂×C₄²×C₅	—	3	1	no	—
116	C₂×C₄²×C₅	—	3	1	no	—
117	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
118	C₅×#95	—	3	1	no	—
119	C₅×#95	—	3	2	no	—
120	C₅×#95	—	3	2	no	—
121	C₅×#95	—	3	1	no	—
122	C₅×#95	—	3	2	no	—
123	C₅×#95	C₂×D₃	1	2	yes	—
124	#2349	—	3	1	no	Sylow
125	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
126	C₂²×C₅×D₃	—	2	2	no	—
127	C₂²×C₅×D₃	—	2	2	no	—
128	C₂²×C₅×D₃	—	2	2	no	—
129	C₂²×C₅×D₃	—	2	2	no	—
130	C₂²×C₅×D₃	C₂²×C₄	1	4	yes	—
131	C₂²×C₅×D₃	C₂×D₄	1	4	yes	—
132	C₂²×C₅×D₃	C₂×Q₈	1	4	yes	—
133	C₂×C₃×C₄×C₅	—	2	1	no	—
134	C₂×C₃×C₄×C₅	—	2	2	no	—
135	C₂×C₃×C₄×C₅	—	2	2	no	—
136	C₂×C₃×C₄×C₅	—	2	1	no	—
137	C₂×C₃×C₄×C₅	—	2	1	no	—
138	C₂×C₃×C₄×C₅	—	2	2	no	—
139	C₂×C₃×C₄×C₅	—	2	1	no	—
140	C₂×C₃×C₄×C₅	—	2	2	no	—
141	C₂×C₃×C₄×C₅	—	2	1	no	—
142	C₂×C₃×C₄×C₅	—	2	2	no	—
143	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
144	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
145	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
146	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
147	C₂×C₃×C₄×C₅	#41	1	2	yes	—
148	C₂×C₅×Dic₃	—	2	1	no	—
149	C₂×C₅×Dic₃	—	2	1	no	—
150	C₂×C₅×Dic₃	—	2	1	no	—
151	C₂×C₅×Dic₃	—	2	1	no	—
152	C₂×C₅×Dic₃	—	2	1	no	—
153	C₂×C₅×Dic₃	—	2	1	no	—
154	C₂×C₅×Dic₃	—	2	1	no	—
155	C₂×C₅×Dic₃	—	2	2	no	—
156	C₂×C₅×Dic₃	—	2	1	no	—
157	C₂×C₅×Dic₃	—	2	2	no	—
158	C₂×C₅×Dic₃	—	2	1	no	—
159	C₂×C₅×Dic₃	—	2	2	no	—
160	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
161	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
162	C₂×C₅×Dic₃	#41	1	2	yes	—
163	C₅×D₁₂	C₄⋊C₄	1	16	yes	complemented
164	C₄×C₅×D₃	—	4	8	no	—
165	C₄×C₅×D₃	—	4	8	no	—
166	C₄×C₅×D₃	—	4	8	no	—
167	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
168	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
169	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
170	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
171	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
172	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
173	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
174	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
175	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
176	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
177	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
178	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
179	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
180	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
181	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
182	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
183	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
184	#893	—	2	4	no	—
185	#893	C₄×C₅	1	8	yes	complemented
186	C₄×D₁₂	—	2	4	no	—
187	C₃×#95	C₂²×C₅	1	2	yes	—
188	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
189	C₃⋊#95	C₂²×C₅	1	2	yes	—
190	C₃⋊#95	C₂²×C₅	1	2	yes	—
191	C₃⋊#95	C₂²×C₅	1	2	yes	—
192	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
193	C₂⁴×C₅	—	3	2	no	—
194	C₂×C₅×D₄	—	3	4	no	—
195	C₂×C₅×D₄	—	3	4	no	—
196	C₂×C₅×D₄	—	3	4	no	—
197	C₂²×C₄×C₅	—	3	1	no	—
198	C₂²×C₄×C₅	—	3	1	no	—
199	C₂²×C₄×C₅	—	3	1	no	—
200	C₂²×C₄×C₅	—	3	8	no	—
201	C₂²×C₄×C₅	—	3	1	no	—
202	C₂²×C₄×C₅	—	3	2	no	—
203	C₂²×C₄×C₅	—	6	4	no	—
204	C₂²×C₄×C₅	—	6	4	no	—
205	C₂²×C₄×C₅	—	6	4	no	—
206	C₂²×C₄×C₅	—	6	4	no	—
207	C₂²×C₄×C₅	—	12	2	no	—
208	C₂²×C₄×C₅	—	12	2	no	—
209	C₂²×C₄×C₅	—	12	2	no	—
210	C₂²×C₄×C₅	—	12	2	no	—
211	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
212	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
213	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
214	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
215	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
216	C₅×C₄⋊C₄	—	2	2	no	—
217	C₅×C₄⋊C₄	—	4	2	no	—
218	C₅×C₄⋊C₄	—	6	2	no	—
219	C₅×C₄⋊C₄	—	6	2	no	—
220	C₅×C₄⋊C₄	—	6	2	no	—
221	C₅×C₄⋊C₄	—	12	2	no	—
222	C₅×C₄⋊C₄	C₄×D₃	1	4	yes	complemented
223	C₅×C₂²⋊C₄	—	3	8	no	—
224	C₅×C₂²⋊C₄	—	6	4	no	—
225	C₅×C₂²⋊C₄	—	6	4	no	—
226	C₄²×C₅	—	2	2	no	—
227	C₄²×C₅	—	3	4	no	—
228	C₄²×C₅	—	6	2	no	—
229	C₂×C₄×D₄	—	3	1	no	—
230	C₂×C₄×D₄	—	3	1	no	—
231	C₂×C₄×D₄	—	3	1	no	—
232	#389	—	3	1	no	—
233	#389	C₅×D₃	1	1	yes	characteristic, complemented
234	#385	—	3	2	no	—
235	#385	—	3	2	no	—
236	#382	—	3	1	no	—
237	#380	—	3	2	no	—
238	C₄×C₄⋊C₄	—	3	1	no	—
239	C₂×#95	—	3	2	no	—
240	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
241	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
242	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
243	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
244	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
245	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
246	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
247	C₂×C₅×D₃	—	2	4	no	—
248	C₂×C₅×D₃	—	2	4	no	—
249	C₂×C₅×D₃	—	2	4	no	—
250	C₂×C₅×D₃	—	2	4	no	—
251	C₂×C₅×D₃	—	2	4	no	—
252	C₂×C₅×D₃	—	2	4	no	—
253	C₂×C₅×D₃	C₂×C₄⋊C₄	1	8	yes	—
254	C₃×C₄×C₅	—	2	4	no	—
255	C₃×C₄×C₅	—	2	2	no	—
256	C₃×C₄×C₅	—	4	2	no	—
257	C₃×C₄×C₅	—	4	1	no	—
258	C₃×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
259	C₅×Dic₃	—	2	2	no	—
260	C₅×Dic₃	—	2	2	no	—
261	C₅×Dic₃	—	2	2	no	—
262	C₅×Dic₃	—	4	2	no	—
263	C₅×Dic₃	—	4	1	no	—
264	C₂³×D₃	C₂³×C₅	1	2	yes	—
265	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
266	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
267	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
268	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
269	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
270	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
271	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
272	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
273	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
274	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
275	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	—
276	C₂×D₁₂	C₅×D₄	1	4	yes	—
277	C₂×D₁₂	C₅×Q₈	1	4	yes	—
278	C₂×C₄×D₃	—	2	4	no	—
279	C₂×C₄×D₃	—	2	4	no	—
280	C₂×C₄×D₃	—	2	4	no	—
281	C₂×C₄×D₃	—	2	4	no	—
282	C₂×C₄×D₃	—	4	2	no	—
283	C₂×C₄×D₃	—	4	2	no	—
284	C₂×C₄×D₃	—	4	2	no	—
285	C₂×C₄×D₃	—	4	2	no	—
286	C₂×C₄×D₃	C₂×C₄×C₅	1	8	yes	complemented
287	C₃×C₄⋊C₄	—	2	2	no	—
288	C₃×C₄⋊C₄	—	4	2	no	—
289	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	complemented
290	C₃×C₄²	—	2	2	no	—
291	(C₂×D₃)⋊C₄	—	2	4	no	—
292	(C₂×D₃)⋊C₄	—	2	4	no	—
293	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	8	yes	complemented
294	C₄⋊Dic₃	—	2	2	no	—
295	C₄⋊Dic₃	—	4	2	no	—
296	Dic₃⋊C₄	—	2	2	no	—
297	Dic₃⋊C₄	—	2	2	no	—
298	C₄×Dic₃	—	2	2	no	—
299	C₄×Dic₃	C₂×C₄×C₅	1	4	yes	complemented
300	C₂³×C₅	—	3	4	no	—
301	C₂³×C₅	—	3	4	no	—
302	C₂³×C₅	—	3	4	no	—
303	C₂³×C₅	—	6	2	no	—
304	C₂³×C₅	—	6	2	no	—
305	C₂³×C₅	—	6	2	no	—
306	C₂³×C₅	—	6	2	no	—
307	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
308	C₅×D₄	—	3	16	no	—
309	C₂×C₄×C₅	—	2	1	no	—
310	C₂×C₄×C₅	—	2	2	no	—
311	C₂×C₄×C₅	—	2	2	no	—
312	C₂×C₄×C₅	—	2	1	no	—
313	C₂×C₄×C₅	—	2	1	no	—
314	C₂×C₄×C₅	—	2	2	no	—
315	C₂×C₄×C₅	—	2	1	no	—
316	C₂×C₄×C₅	—	2	2	no	—
317	C₂×C₄×C₅	—	2	1	no	—
318	C₂×C₄×C₅	—	2	2	no	—
319	C₂×C₄×C₅	—	3	2	no	—
320	C₂×C₄×C₅	—	3	2	no	—
321	C₂×C₄×C₅	—	3	2	no	—
322	C₂×C₄×C₅	—	6	1	no	—
323	C₂×C₄×C₅	—	6	1	no	—
324	C₂×C₄×C₅	—	6	1	no	—
325	C₂×C₄×C₅	—	6	1	no	—
326	C₂×C₄×C₅	—	6	1	no	—
327	C₂×C₄×C₅	—	6	1	no	—
328	C₂×C₄×C₅	—	6	1	no	—
329	C₂×C₄×C₅	—	6	2	no	—
330	C₂×C₄×C₅	—	6	1	no	—
331	C₂×C₄×C₅	—	6	2	no	—
332	C₂×C₄×C₅	—	6	1	no	—
333	C₂×C₄×C₅	—	6	2	no	—
334	C₂×C₄×C₅	—	12	8	no	—
335	C₂×C₄×C₅	—	12	8	no	—
336	C₂×C₄×C₅	—	12	8	no	—
337	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
338	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
339	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
340	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
341	C₂×C₄×C₅	#239	1	2	yes	—
342	C₂²×D₄	—	3	1	no	—
343	C₂³×C₄	—	3	2	no	—
344	C₂³×C₄	—	3	2	no	—
345	C₂³×C₄	—	3	2	no	—
346	C₄×D₄	—	3	8	no	—
347	C₄×D₄	—	6	4	no	—
348	C₄×D₄	—	6	4	no	—
349	C₂×C₄⋊C₄	—	3	1	no	—
350	C₂×C₄⋊C₄	—	3	1	no	—
351	C₂×C₄⋊C₄	—	3	1	no	—
352	C₂×C₄⋊C₄	—	3	2	no	—
353	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
355	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
356	C₂×C₂²⋊C₄	—	3	2	no	—
357	C₂×C₂²⋊C₄	—	3	2	no	—
358	C₂×C₂²⋊C₄	—	3	2	no	—
359	C₂×C₄²	—	3	1	no	—
360	C₂×C₄²	—	3	1	no	—
361	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
362	#95	—	3	1	no	—
363	#95	—	3	2	no	—
364	#95	—	3	2	no	—
365	#95	—	3	1	no	—
366	#95	—	3	2	no	—
367	#95	C₂×C₅×D₃	1	2	yes	—
368	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
369	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
370	C₂×C₃×C₅	#518	1	1	yes	characteristic
371	C₂×C₃×C₅	#546	1	1	yes	characteristic
372	C₂×C₃×C₅	#546	1	1	yes	characteristic
373	C₂×C₃×C₅	#554	1	1	yes	characteristic
374	C₂×C₃×C₅	#554	1	1	yes	characteristic
375	C₅×D₃	—	2	8	no	—
376	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
377	C₂²×D₃	—	2	2	no	—
378	C₂²×D₃	—	2	2	no	—
379	C₂²×D₃	—	2	2	no	—
380	C₂²×D₃	—	2	2	no	—
381	C₂²×D₃	C₂²×C₄×C₅	1	4	yes	—
382	C₂²×D₃	C₂×C₅×D₄	1	4	yes	—
383	C₂²×D₃	C₂×C₅×Q₈	1	4	yes	—
384	C₂×C₃×C₄	—	2	1	no	—
385	C₂×C₃×C₄	—	2	2	no	—
386	C₂×C₃×C₄	—	2	2	no	—
387	C₂×C₃×C₄	—	2	1	no	—
388	C₂×C₃×C₄	—	2	1	no	—
389	C₂×C₃×C₄	—	2	2	no	—
390	C₂×C₃×C₄	—	2	1	no	—
391	C₂×C₃×C₄	—	2	2	no	—
392	C₂×C₃×C₄	—	2	1	no	—
393	C₂×C₃×C₄	—	2	2	no	—
394	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
395	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
396	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
397	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
398	C₂×C₃×C₄	C₅×#41	1	2	yes	—
399	C₂×Dic₃	—	2	1	no	—
400	C₂×Dic₃	—	2	1	no	—
401	C₂×Dic₃	—	2	1	no	—
402	C₂×Dic₃	—	2	1	no	—
403	C₂×Dic₃	—	2	1	no	—
404	C₂×Dic₃	—	2	1	no	—
405	C₂×Dic₃	—	2	1	no	—
406	C₂×Dic₃	—	2	2	no	—
407	C₂×Dic₃	—	2	1	no	—
408	C₂×Dic₃	—	2	2	no	—
409	C₂×Dic₃	—	2	1	no	—
410	C₂×Dic₃	—	2	2	no	—
411	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
412	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
413	C₂×Dic₃	C₅×#41	1	2	yes	—
414	D₁₂	C₅×C₄⋊C₄	1	16	yes	complemented
415	C₄×D₃	—	4	8	no	—
416	C₄×D₃	—	4	8	no	—
417	C₄×D₃	—	4	8	no	—
418	C₂²×C₅	—	3	8	no	—
419	C₂²×C₅	—	6	4	no	—
420	C₂²×C₅	—	6	4	no	—
421	C₂²×C₅	—	6	4	no	—
422	C₂²×C₅	—	6	4	no	—
423	C₂²×C₅	—	6	4	no	—
424	C₂²×C₅	—	6	4	no	—
425	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
426	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
427	C₂²×C₅	#1004	1	1	yes	characteristic
428	C₂²×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
429	C₂²×C₅	C₂×#239	1	1	yes	characteristic
430	C₂²×C₅	D₃×#41	1	1	yes	characteristic
431	C₂²×C₅	#1009	1	1	yes	characteristic
432	C₄×C₅	—	2	4	no	—
433	C₄×C₅	—	2	2	no	—
434	C₄×C₅	—	4	1	no	—
435	C₄×C₅	—	4	2	no	—
436	C₄×C₅	—	6	2	no	—
437	C₄×C₅	—	6	2	no	—
438	C₄×C₅	—	6	2	no	—
439	C₄×C₅	—	12	1	no	—
440	C₄×C₅	—	12	2	no	—
441	C₄×C₅	D₃×C₄⋊C₄	1	4	yes	—
442	C₂⁴	—	3	2	no	—
443	C₂×D₄	—	3	4	no	—
444	C₂×D₄	—	3	4	no	—
445	C₂×D₄	—	3	4	no	—
446	C₂²×C₄	—	3	1	no	—
447	C₂²×C₄	—	3	1	no	—
448	C₂²×C₄	—	3	1	no	—
449	C₂²×C₄	—	3	8	no	—
450	C₂²×C₄	—	3	2	no	—
451	C₂²×C₄	—	3	1	no	—
452	C₂²×C₄	—	6	4	no	—
453	C₂²×C₄	—	6	4	no	—
454	C₂²×C₄	—	6	4	no	—
455	C₂²×C₄	—	6	4	no	—
456	C₂²×C₄	—	12	2	no	—
457	C₂²×C₄	—	12	2	no	—
458	C₂²×C₄	—	12	2	no	—
459	C₂²×C₄	—	12	2	no	—
460	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
461	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
462	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
463	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
464	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
465	C₄⋊C₄	—	2	2	no	—
466	C₄⋊C₄	—	4	2	no	—
467	C₄⋊C₄	—	6	2	no	—
468	C₄⋊C₄	—	6	2	no	—
469	C₄⋊C₄	—	6	2	no	—
470	C₄⋊C₄	—	12	2	no	—
471	C₄⋊C₄	C₄×C₅×D₃	1	4	yes	complemented
472	C₂²⋊C₄	—	3	8	no	—
473	C₂²⋊C₄	—	6	4	no	—
474	C₂²⋊C₄	—	6	4	no	—
475	C₄²	—	2	2	no	—
476	C₄²	—	3	4	no	—
477	C₄²	—	6	2	no	—
478	C₃×C₅	#2349	1	1	yes	characteristic, complemented, Hall
479	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
480	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
481	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
482	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
483	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
484	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
485	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
486	C₂×D₃	—	2	4	no	—
487	C₂×D₃	—	2	4	no	—
488	C₂×D₃	—	2	4	no	—
489	C₂×D₃	—	2	4	no	—
490	C₂×D₃	—	2	4	no	—
491	C₂×D₃	—	2	4	no	—
492	C₂×D₃	C₂×C₅×C₄⋊C₄	1	8	yes	—
493	C₃×C₄	—	2	4	no	—
494	C₃×C₄	—	2	2	no	—
495	C₃×C₄	—	4	2	no	—
496	C₃×C₄	—	4	1	no	—
497	C₃×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
498	Dic₃	—	2	2	no	—
499	Dic₃	—	2	2	no	—
500	Dic₃	—	2	2	no	—
501	Dic₃	—	4	2	no	—
502	Dic₃	—	4	1	no	—
503	C₂×C₅	—	6	8	no	—
504	C₂×C₅	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
505	C₂×C₅	#5545	1	1	yes	characteristic
506	C₂×C₅	C₄×#239	1	1	yes	characteristic
507	C₂×C₅	#5612	1	1	yes	characteristic
508	C₂×C₅	#5630	1	1	yes	characteristic
509	C₂×C₅	#5690	1	1	yes	characteristic
510	C₂×C₅	#5730	1	1	yes	characteristic
511	C₂³	—	3	4	no	—
512	C₂³	—	3	4	no	—
513	C₂³	—	3	4	no	—
514	C₂³	—	6	2	no	—
515	C₂³	—	6	2	no	—
516	C₂³	—	6	2	no	—
517	C₂³	—	6	2	no	—
518	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
519	D₄	—	3	16	no	—
520	C₂×C₄	—	2	1	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	1	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	3	2	no	—
531	C₂×C₄	—	3	2	no	—
532	C₂×C₄	—	3	2	no	—
533	C₂×C₄	—	6	1	no	—
534	C₂×C₄	—	6	1	no	—
535	C₂×C₄	—	6	1	no	—
536	C₂×C₄	—	6	1	no	—
537	C₂×C₄	—	6	1	no	—
538	C₂×C₄	—	6	1	no	—
539	C₂×C₄	—	6	1	no	—
540	C₂×C₄	—	6	2	no	—
541	C₂×C₄	—	6	1	no	—
542	C₂×C₄	—	6	2	no	—
543	C₂×C₄	—	6	1	no	—
544	C₂×C₄	—	6	2	no	—
545	C₂×C₄	—	12	8	no	—
546	C₂×C₄	—	12	8	no	—
547	C₂×C₄	—	12	8	no	—
548	C₂×C₄	C₂×C₄×C₅×D₃	1	4	yes	—
549	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
550	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
551	C₂×C₄	C₅×D₃×Q₈	1	2	yes	—
552	C₂×C₄	C₅×#239	1	2	yes	—
553	C₂×C₃	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
554	C₂×C₃	C₄×C₅×#41	1	1	yes	characteristic
555	C₂×C₃	C₅×#518	1	1	yes	characteristic
556	C₂×C₃	C₅×#546	1	1	yes	characteristic
557	C₂×C₃	C₅×#546	1	1	yes	characteristic
558	C₂×C₃	C₅×#554	1	1	yes	characteristic
559	C₂×C₃	C₅×#554	1	1	yes	characteristic
560	D₃	—	2	8	no	—
561	C₅	#76203	1	1	yes	characteristic, direct product factor, Sylow
562	C₂²	—	3	8	no	—
563	C₂²	—	6	4	no	—
564	C₂²	—	6	4	no	—
565	C₂²	—	6	4	no	—
566	C₂²	—	6	4	no	—
567	C₂²	—	6	4	no	—
568	C₂²	—	6	4	no	—
569	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
570	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
571	C₂²	C₅×#1004	1	1	yes	characteristic
572	C₂²	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
573	C₂²	C₂×C₅×#239	1	1	yes	characteristic
574	C₂²	C₅×D₃×#41	1	1	yes	characteristic
575	C₂²	C₅×#1009	1	1	yes	characteristic
576	C₄	—	2	4	no	—
577	C₄	—	2	2	no	—
578	C₄	—	4	2	no	—
579	C₄	—	4	1	no	—
580	C₄	—	6	2	no	—
581	C₄	—	6	2	no	—
582	C₄	—	6	2	no	—
583	C₄	—	12	2	no	—
584	C₄	—	12	1	no	—
585	C₄	C₅×D₃×C₄⋊C₄	1	4	yes	—
586	C₃	C₅×#2349	1	1	yes	characteristic, complemented, Sylow
587	C₂	—	6	8	no	—
588	C₂	C₂×C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
589	C₂	C₅×#5545	1	1	yes	characteristic
590	C₂	C₄×C₅×#239	1	1	yes	characteristic
591	C₂	C₅×#5612	1	1	yes	characteristic
592	C₂	C₅×#5630	1	1	yes	characteristic
593	C₂	C₅×#5690	1	1	yes	characteristic
594	C₂	C₅×#5730	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#389	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂²×C₅×D₁₂	C₂²
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#389	D₃
hypocenter	C₃	C₅×#2349
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#389

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#389	C₂	semidirect
5. C₅×#4989	C₂	semidirect
6. C₅×#4987	C₂	semidirect
7. C₅×#4984	C₂	semidirect
8. C₅×#4977	C₂	semidirect
9. C₄×C₅×Dic₃⋊C₄	C₂	semidirect
10. C₅×#4668	C₂	semidirect
11. C₅×D₃×#95	C₂	semidirect
31. C₅×#893	C₄	semidirect
39. #76203	C₅	direct (N × H)
44. C₅×#389	D₃	semidirect
73. C₂×C₄×C₅×D₃	C₂×C₄	semidirect
76. C₃×C₅×C₄⋊C₄	C₂×C₄	semidirect
80. C₅×(C₂×D₃)⋊C₄	C₂×C₄	semidirect
86. C₄×C₅×Dic₃	C₂×C₄	semidirect
90. C₃×#389	C₂×C₅	semidirect
91. #4989	C₂×C₅	semidirect
92. #4987	C₂×C₅	semidirect
93. #4984	C₂×C₅	semidirect
94. #4977	C₂×C₅	semidirect
95. C₄×Dic₃⋊C₄	C₂×C₅	semidirect
96. #4668	C₂×C₅	semidirect
97. D₃×#95	C₂×C₅	semidirect
163. C₅×D₁₂	C₄⋊C₄	semidirect
185. #893	C₄×C₅	semidirect
222. C₅×C₄⋊C₄	C₄×D₃	semidirect
233. #389	C₅×D₃	semidirect
286. C₂×C₄×D₃	C₂×C₄×C₅	semidirect
289. C₃×C₄⋊C₄	C₂×C₄×C₅	semidirect
293. (C₂×D₃)⋊C₄	C₂×C₄×C₅	semidirect
299. C₄×Dic₃	C₂×C₄×C₅	semidirect
414. D₁₂	C₅×C₄⋊C₄	semidirect
471. C₄⋊C₄	C₄×C₅×D₃	semidirect
478. C₃×C₅	#2349	semidirect
561. C₅	#76203	direct (N × H)
586. C₃	C₅×#2349	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#893	2
C₂×C₄×C₅×D₁₂	2
C₃×C₅×#389	2
C₅×#4989	2
C₅×#4987	2
C₅×#4984	2
C₅×#4977	2
C₄×C₅×Dic₃⋊C₄	2
C₅×#4668	2
C₅×D₃×#95	2
C₅×#2349	3
#76203	5

Subgroups — C5×#76203 #14327115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#76203 #14327115