Subgroups — C₅×#74011 #14327100

Order 1920.

1408 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×#67	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#67⋊C₄	C₂	1	2	yes	maximal
3	C₃×C₅×#380	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×C₃⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4989	C₂	1	2	yes	maximal, complemented
6	C₅×#4976	C₂	1	2	yes	maximal, complemented
7	C₅×#4946	C₂	1	2	yes	maximal, complemented
8	C₅×#4667	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×D₃×#95	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4650	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#4649	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#2349	—	3	1	no	maximal, Hall
13	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂²×C₅×#67	C₂²	1	1	yes	characteristic
15	C₂³×C₅×Dic₃	C₂²	1	2	yes	—
16	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
17	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₅×#217	C₂²	1	1	yes	characteristic
21	C₄×C₅×#67	C₄	1	8	yes	complemented
22	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
25	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
26	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
27	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
29	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
30	C₅×#67⋊C₄	—	2	8	no	—
31	C₃×C₅×#95	C₂²	1	1	yes	characteristic
32	C₃×C₅×#95	C₂²	1	1	yes	characteristic
33	C₅×C₃⋊#95	C₂²	1	1	yes	characteristic
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	#74011	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
39	C₂×C₄×C₅×D₄	—	3	1	no	—
40	C₂×C₄×C₅×D₄	—	3	2	no	—
41	C₅×#389	—	3	2	no	—
42	C₅×#385	—	3	2	no	—
43	C₅×#385	—	3	2	no	—
44	C₅×#382	—	3	1	no	—
45	C₅×#380	—	3	1	no	—
46	C₅×#380	D₃	1	1	yes	characteristic, complemented
47	C₄×C₅×C₄⋊C₄	—	3	1	no	—
48	C₂×C₅×#95	—	3	1	no	—
49	C₂×C₅×#95	—	3	1	no	—
50	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
51	C₂³×C₅×D₃	C₂³	1	1	yes	characteristic
52	C₂²×C₃×C₄×C₅	—	4	2	no	—
53	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	complemented
54	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
55	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
56	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂×C₅×#67	C₂×C₄	1	4	yes	—
59	C₂×C₅×#67	D₄	1	4	yes	—
60	C₂×C₅×#67	Q₈	1	4	yes	—
61	C₂²×C₅×Dic₃	—	2	4	no	—
62	C₂²×C₅×Dic₃	—	2	4	no	—
63	C₂²×C₅×Dic₃	—	4	2	no	—
64	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
65	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
66	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
67	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
68	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
69	C₂×C₄×C₅×D₃	—	2	4	no	—
70	C₂×C₄×C₅×D₃	—	2	4	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	4	yes	complemented
74	C₃×C₅×C₄⋊C₄	—	4	2	no	—
75	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
76	C₅×#217	C₂×C₄	1	4	yes	complemented
77	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
78	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	complemented
79	C₅×Dic₃⋊C₄	—	2	4	no	—
80	C₅×Dic₃⋊C₄	—	2	4	no	—
81	C₅×Dic₃⋊C₄	—	4	2	no	—
82	C₅×Dic₃⋊C₄	C₂×C₄	1	4	yes	complemented
83	C₄×C₅×Dic₃	—	2	4	no	—
84	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	complemented
85	C₂×C₄×#67	C₂×C₅	1	1	yes	characteristic
86	C₂×#67⋊C₄	C₂×C₅	1	2	yes	—
87	C₃×#380	C₂×C₅	1	1	yes	characteristic, complemented
88	C₂×C₃⋊#95	C₂×C₅	1	1	yes	characteristic, complemented
89	#4989	C₂×C₅	1	2	yes	complemented
90	#4976	C₂×C₅	1	2	yes	complemented
91	#4946	C₂×C₅	1	2	yes	complemented
92	#4667	C₂×C₅	1	1	yes	characteristic, complemented
93	D₃×#95	C₂×C₅	1	1	yes	characteristic, complemented
94	#4650	C₂×C₅	1	1	yes	characteristic, complemented
95	#4649	C₂×C₅	1	1	yes	characteristic, complemented
96	C₂²×C₅×D₄	—	3	1	no	—
97	C₂³×C₄×C₅	—	3	1	no	—
98	C₂³×C₄×C₅	—	3	2	no	—
99	C₂³×C₄×C₅	—	3	2	no	—
100	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
101	C₄×C₅×D₄	—	3	8	no	—
102	C₄×C₅×D₄	—	6	8	no	—
103	C₂×C₅×C₄⋊C₄	—	3	1	no	—
104	C₂×C₅×C₄⋊C₄	—	3	2	no	—
105	C₂×C₅×C₄⋊C₄	—	3	2	no	—
106	C₂×C₅×C₄⋊C₄	—	3	2	no	—
107	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
108	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
109	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
110	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
111	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
112	C₂×C₄²×C₅	—	3	1	no	—
113	C₂×C₄²×C₅	—	3	2	no	—
114	C₅×#95	—	3	1	no	—
115	C₅×#95	—	3	1	no	—
116	C₅×#95	—	3	2	no	—
117	C₅×#95	—	3	2	no	—
118	C₅×#95	—	3	2	no	—
119	C₅×#95	C₂×D₃	1	1	yes	characteristic
120	C₅×#95	C₂×D₃	1	1	yes	characteristic
121	#2349	—	3	1	no	Sylow
122	C₂³×C₃×C₅	—	2	2	no	—
123	C₂³×C₃×C₅	—	2	2	no	—
124	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
125	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
126	C₂³×C₃×C₅	C₂×Q₈	1	2	yes	—
127	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
128	C₂²×C₅×D₃	—	2	2	no	—
129	C₂²×C₅×D₃	—	2	2	no	—
130	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
131	C₂²×C₅×D₃	C₂×D₄	1	2	yes	—
132	C₂²×C₅×D₃	C₂×Q₈	1	2	yes	—
133	C₂×C₃×C₄×C₅	—	2	2	no	—
134	C₂×C₃×C₄×C₅	—	2	2	no	—
135	C₂×C₃×C₄×C₅	—	2	2	no	—
136	C₂×C₃×C₄×C₅	—	2	2	no	—
137	C₂×C₃×C₄×C₅	—	2	2	no	—
138	C₂×C₃×C₄×C₅	—	2	2	no	—
139	C₂×C₃×C₄×C₅	—	2	2	no	—
140	C₂×C₃×C₄×C₅	—	2	2	no	—
141	C₂×C₃×C₄×C₅	—	4	4	no	—
142	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
143	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
144	C₅×#67	C₄⋊C₄	1	16	yes	complemented
145	C₂×C₅×Dic₃	—	2	2	no	—
146	C₂×C₅×Dic₃	—	2	2	no	—
147	C₂×C₅×Dic₃	—	2	2	no	—
148	C₂×C₅×Dic₃	—	2	2	no	—
149	C₂×C₅×Dic₃	—	2	2	no	—
150	C₂×C₅×Dic₃	—	2	2	no	—
151	C₂×C₅×Dic₃	—	2	2	no	—
152	C₂×C₅×Dic₃	—	4	8	no	—
153	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
154	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
155	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
156	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
157	C₂×C₅×Dic₃	C₂×Q₈	1	2	yes	—
158	C₂×C₅×Dic₃	#41	1	4	yes	—
159	C₄×C₅×D₃	—	4	4	no	—
160	C₄×C₅×D₃	—	4	8	no	—
161	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
162	C₂²×#67	C₂²×C₅	1	1	yes	characteristic
163	C₂³×Dic₃	C₂²×C₅	1	2	yes	—
164	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
165	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
166	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
167	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
168	C₂×#217	C₂²×C₅	1	1	yes	characteristic
169	C₄×#67	C₄×C₅	1	8	yes	complemented
170	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
171	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
172	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
173	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
174	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
175	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
176	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
177	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
178	#67⋊C₄	—	2	8	no	—
179	C₃×#95	C₂²×C₅	1	1	yes	characteristic
180	C₃×#95	C₂²×C₅	1	1	yes	characteristic
181	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
182	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
183	C₃⋊#95	C₂²×C₅	1	2	yes	—
184	C₃⋊#95	C₂²×C₅	1	2	yes	—
185	C₃⋊#95	C₂²×C₅	1	2	yes	—
186	C₂⁴×C₅	—	3	1	no	—
187	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
188	C₂×C₅×D₄	—	3	4	no	—
189	C₂×C₅×D₄	—	3	4	no	—
190	C₂×C₅×D₄	—	3	4	no	—
191	C₂²×C₄×C₅	—	3	1	no	—
192	C₂²×C₄×C₅	—	3	1	no	—
193	C₂²×C₄×C₅	—	3	2	no	—
194	C₂²×C₄×C₅	—	3	1	no	—
195	C₂²×C₄×C₅	—	3	4	no	—
196	C₂²×C₄×C₅	—	3	2	no	—
197	C₂²×C₄×C₅	—	4	2	no	—
198	C₂²×C₄×C₅	—	6	4	no	—
199	C₂²×C₄×C₅	—	6	4	no	—
200	C₂²×C₄×C₅	—	6	4	no	—
201	C₂²×C₄×C₅	—	6	4	no	—
202	C₂²×C₄×C₅	—	12	2	no	—
203	C₂²×C₄×C₅	—	12	2	no	—
204	C₂²×C₄×C₅	—	12	2	no	—
205	C₂²×C₄×C₅	C₄×D₃	1	4	yes	complemented
206	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
207	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
208	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
209	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
210	C₅×C₄⋊C₄	—	3	4	no	—
211	C₅×C₄⋊C₄	—	4	2	no	—
212	C₅×C₄⋊C₄	—	6	4	no	—
213	C₅×C₄⋊C₄	—	6	4	no	—
214	C₅×C₄⋊C₄	—	12	2	no	—
215	C₅×C₂²⋊C₄	—	2	4	no	—
216	C₅×C₂²⋊C₄	—	3	4	no	—
217	C₅×C₂²⋊C₄	—	3	4	no	—
218	C₅×C₂²⋊C₄	—	6	4	no	—
219	C₄²×C₅	—	3	4	no	—
220	C₄²×C₅	—	6	4	no	—
221	C₂×C₄×D₄	—	3	1	no	—
222	C₂×C₄×D₄	—	3	2	no	—
223	#389	—	3	2	no	—
224	#385	—	3	2	no	—
225	#385	—	3	2	no	—
226	#382	—	3	1	no	—
227	#380	—	3	1	no	—
228	#380	C₅×D₃	1	1	yes	characteristic, complemented
229	C₄×C₄⋊C₄	—	3	1	no	—
230	C₂×#95	—	3	1	no	—
231	C₂×#95	—	3	1	no	—
232	C₂²×C₃×C₅	—	2	2	no	—
233	C₂²×C₃×C₅	—	2	2	no	—
234	C₂²×C₃×C₅	—	2	4	no	—
235	C₂²×C₃×C₅	—	2	4	no	—
236	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	4	yes	—
237	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
238	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
239	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
240	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
241	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
242	C₂×C₅×D₃	—	2	2	no	—
243	C₂×C₅×D₃	—	2	2	no	—
244	C₂×C₅×D₃	—	2	4	no	—
245	C₂×C₅×D₃	—	2	4	no	—
246	C₂×C₅×D₃	C₂×C₄⋊C₄	1	4	yes	—
247	C₃×C₄×C₅	—	2	2	no	—
248	C₃×C₄×C₅	—	2	2	no	—
249	C₃×C₄×C₅	—	4	2	no	—
250	C₃×C₄×C₅	—	4	2	no	—
251	C₅×Dic₃	—	2	2	no	—
252	C₅×Dic₃	—	2	2	no	—
253	C₅×Dic₃	—	2	4	no	—
254	C₅×Dic₃	—	4	2	no	—
255	C₅×Dic₃	C₂×C₄⋊C₄	1	4	yes	—
256	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
257	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
258	C₂²×C₃×C₄	—	4	2	no	—
259	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	complemented
260	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
261	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
262	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
263	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
264	C₂×#67	C₂×C₄×C₅	1	4	yes	—
265	C₂×#67	C₅×D₄	1	4	yes	—
266	C₂×#67	C₅×Q₈	1	4	yes	—
267	C₂²×Dic₃	—	2	4	no	—
268	C₂²×Dic₃	—	2	4	no	—
269	C₂²×Dic₃	—	4	2	no	—
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	2	yes	—
274	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
275	C₂×C₄×D₃	—	2	4	no	—
276	C₂×C₄×D₃	—	2	4	no	—
277	C₂×C₄×D₃	—	4	2	no	—
278	C₂×C₄×D₃	—	4	2	no	—
279	C₂×C₄×D₃	C₂×C₄×C₅	1	4	yes	complemented
280	C₃×C₄⋊C₄	—	4	2	no	—
281	C₃×C₂²⋊C₄	—	2	4	no	—
282	#217	C₂×C₄×C₅	1	4	yes	complemented
283	(C₂×D₃)⋊C₄	—	2	4	no	—
284	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	4	yes	complemented
285	Dic₃⋊C₄	—	2	4	no	—
286	Dic₃⋊C₄	—	2	4	no	—
287	Dic₃⋊C₄	—	4	2	no	—
288	Dic₃⋊C₄	C₂×C₄×C₅	1	4	yes	complemented
289	C₄×Dic₃	—	2	4	no	—
290	C₄×Dic₃	C₂×C₄×C₅	1	4	yes	complemented
291	C₂³×C₅	—	2	2	no	—
292	C₂³×C₅	—	2	2	no	—
293	C₂³×C₅	—	3	2	no	—
294	C₂³×C₅	—	3	2	no	—
295	C₂³×C₅	—	3	2	no	—
296	C₂³×C₅	—	6	2	no	—
297	C₂³×C₅	—	6	2	no	—
298	C₂³×C₅	C₂×C₄×D₃	1	2	yes	—
299	C₂³×C₅	D₃×D₄	1	2	yes	—
300	C₂³×C₅	D₃×Q₈	1	2	yes	—
301	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
302	C₅×D₄	—	3	16	no	—
303	C₂×C₄×C₅	—	2	2	no	—
304	C₂×C₄×C₅	—	2	2	no	—
305	C₂×C₄×C₅	—	2	2	no	—
306	C₂×C₄×C₅	—	2	2	no	—
307	C₂×C₄×C₅	—	2	2	no	—
308	C₂×C₄×C₅	—	2	2	no	—
309	C₂×C₄×C₅	—	2	2	no	—
310	C₂×C₄×C₅	—	2	2	no	—
311	C₂×C₄×C₅	—	3	2	no	—
312	C₂×C₄×C₅	—	3	2	no	—
313	C₂×C₄×C₅	—	3	2	no	—
314	C₂×C₄×C₅	—	3	2	no	—
315	C₂×C₄×C₅	—	3	2	no	—
316	C₂×C₄×C₅	—	3	4	no	—
317	C₂×C₄×C₅	—	4	4	no	—
318	C₂×C₄×C₅	—	6	2	no	—
319	C₂×C₄×C₅	—	6	2	no	—
320	C₂×C₄×C₅	—	6	2	no	—
321	C₂×C₄×C₅	—	6	2	no	—
322	C₂×C₄×C₅	—	6	2	no	—
323	C₂×C₄×C₅	—	6	2	no	—
324	C₂×C₄×C₅	—	6	2	no	—
325	C₂×C₄×C₅	—	12	4	no	—
326	C₂×C₄×C₅	—	12	8	no	—
327	C₂×C₄×C₅	—	12	8	no	—
328	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
329	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
330	C₂²×D₄	—	3	1	no	—
331	C₂³×C₄	—	3	1	no	—
332	C₂³×C₄	—	3	2	no	—
333	C₂³×C₄	—	3	2	no	—
334	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
335	C₄×D₄	—	3	8	no	—
336	C₄×D₄	—	6	8	no	—
337	C₂×C₄⋊C₄	—	3	1	no	—
338	C₂×C₄⋊C₄	—	3	2	no	—
339	C₂×C₄⋊C₄	—	3	2	no	—
340	C₂×C₄⋊C₄	—	3	2	no	—
341	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
342	C₂×C₂²⋊C₄	—	3	1	no	—
343	C₂×C₂²⋊C₄	—	3	1	no	—
344	C₂×C₂²⋊C₄	—	3	2	no	—
345	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
346	C₂×C₄²	—	3	1	no	—
347	C₂×C₄²	—	3	2	no	—
348	#95	—	3	1	no	—
349	#95	—	3	1	no	—
350	#95	—	3	2	no	—
351	#95	—	3	2	no	—
352	#95	—	3	2	no	—
353	#95	C₂×C₅×D₃	1	1	yes	characteristic
354	#95	C₂×C₅×D₃	1	1	yes	characteristic
355	C₂×C₃×C₅	—	2	4	no	—
356	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
357	C₂×C₃×C₅	C₄×#41	1	1	yes	characteristic
358	C₂×C₃×C₅	#518	1	1	yes	characteristic
359	C₂×C₃×C₅	#546	1	2	yes	—
360	C₂×C₃×C₅	#554	1	2	yes	—
361	C₅×D₃	—	2	4	no	—
362	C₂³×C₃	—	2	2	no	—
363	C₂³×C₃	—	2	2	no	—
364	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
365	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
366	C₂³×C₃	C₂×C₅×Q₈	1	2	yes	—
367	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
368	C₂²×D₃	—	2	2	no	—
369	C₂²×D₃	—	2	2	no	—
370	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
371	C₂²×D₃	C₂×C₅×D₄	1	2	yes	—
372	C₂²×D₃	C₂×C₅×Q₈	1	2	yes	—
373	C₂×C₃×C₄	—	2	2	no	—
374	C₂×C₃×C₄	—	2	2	no	—
375	C₂×C₃×C₄	—	2	2	no	—
376	C₂×C₃×C₄	—	2	2	no	—
377	C₂×C₃×C₄	—	2	2	no	—
378	C₂×C₃×C₄	—	2	2	no	—
379	C₂×C₃×C₄	—	2	2	no	—
380	C₂×C₃×C₄	—	2	2	no	—
381	C₂×C₃×C₄	—	4	4	no	—
382	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
383	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
384	#67	C₅×C₄⋊C₄	1	16	yes	complemented
385	C₂×Dic₃	—	2	2	no	—
386	C₂×Dic₃	—	2	2	no	—
387	C₂×Dic₃	—	2	2	no	—
388	C₂×Dic₃	—	2	2	no	—
389	C₂×Dic₃	—	2	2	no	—
390	C₂×Dic₃	—	2	2	no	—
391	C₂×Dic₃	—	2	2	no	—
392	C₂×Dic₃	—	4	8	no	—
393	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
394	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
395	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
396	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
397	C₂×Dic₃	C₂×C₅×Q₈	1	2	yes	—
398	C₂×Dic₃	C₅×#41	1	4	yes	—
399	C₄×D₃	—	4	4	no	—
400	C₄×D₃	—	4	8	no	—
401	C₂²×C₅	—	2	4	no	—
402	C₂²×C₅	—	2	2	no	—
403	C₂²×C₅	—	2	2	no	—
404	C₂²×C₅	—	2	4	no	—
405	C₂²×C₅	—	3	4	no	—
406	C₂²×C₅	—	6	4	no	—
407	C₂²×C₅	—	6	2	no	—
408	C₂²×C₅	—	6	2	no	—
409	C₂²×C₅	—	6	4	no	—
410	C₂²×C₅	D₃×C₄⋊C₄	1	4	yes	—
411	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
412	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
413	C₂²×C₅	#1004	1	2	yes	—
414	C₂²×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
415	C₂²×C₅	D₃×#41	1	2	yes	—
416	C₄×C₅	—	2	2	no	—
417	C₄×C₅	—	2	2	no	—
418	C₄×C₅	—	3	4	no	—
419	C₄×C₅	—	4	2	no	—
420	C₄×C₅	—	4	2	no	—
421	C₄×C₅	—	6	2	no	—
422	C₄×C₅	—	6	2	no	—
423	C₄×C₅	—	6	4	no	—
424	C₄×C₅	—	12	2	no	—
425	C₂⁴	—	3	1	no	—
426	C₂⁴	C₂²×C₅×D₃	1	1	yes	characteristic
427	C₂×D₄	—	3	4	no	—
428	C₂×D₄	—	3	4	no	—
429	C₂×D₄	—	3	4	no	—
430	C₂²×C₄	—	3	1	no	—
431	C₂²×C₄	—	3	2	no	—
432	C₂²×C₄	—	3	1	no	—
433	C₂²×C₄	—	3	1	no	—
434	C₂²×C₄	—	3	4	no	—
435	C₂²×C₄	—	3	2	no	—
436	C₂²×C₄	—	4	2	no	—
437	C₂²×C₄	—	6	4	no	—
438	C₂²×C₄	—	6	4	no	—
439	C₂²×C₄	—	6	4	no	—
440	C₂²×C₄	—	6	4	no	—
441	C₂²×C₄	—	12	2	no	—
442	C₂²×C₄	—	12	2	no	—
443	C₂²×C₄	—	12	2	no	—
444	C₂²×C₄	C₄×C₅×D₃	1	4	yes	complemented
445	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
446	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
447	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
448	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
449	C₄⋊C₄	—	3	4	no	—
450	C₄⋊C₄	—	4	2	no	—
451	C₄⋊C₄	—	6	4	no	—
452	C₄⋊C₄	—	6	4	no	—
453	C₄⋊C₄	—	12	2	no	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	3	4	no	—
456	C₂²⋊C₄	—	3	4	no	—
457	C₂²⋊C₄	—	6	4	no	—
458	C₄²	—	3	4	no	—
459	C₄²	—	6	4	no	—
460	C₃×C₅	#2349	1	1	yes	characteristic, complemented, Hall
461	C₂²×C₃	—	2	2	no	—
462	C₂²×C₃	—	2	2	no	—
463	C₂²×C₃	—	2	4	no	—
464	C₂²×C₃	—	2	4	no	—
465	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	4	yes	—
466	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
467	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
468	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
469	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
470	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
471	C₂×D₃	—	2	2	no	—
472	C₂×D₃	—	2	2	no	—
473	C₂×D₃	—	2	4	no	—
474	C₂×D₃	—	2	4	no	—
475	C₂×D₃	C₂×C₅×C₄⋊C₄	1	4	yes	—
476	C₃×C₄	—	2	2	no	—
477	C₃×C₄	—	2	2	no	—
478	C₃×C₄	—	4	2	no	—
479	C₃×C₄	—	4	2	no	—
480	Dic₃	—	2	2	no	—
481	Dic₃	—	2	2	no	—
482	Dic₃	—	2	4	no	—
483	Dic₃	—	4	2	no	—
484	Dic₃	C₂×C₅×C₄⋊C₄	1	4	yes	—
485	C₂×C₅	—	2	4	no	—
486	C₂×C₅	—	6	4	no	—
487	C₂×C₅	#5486	1	1	yes	characteristic
488	C₂×C₅	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
489	C₂×C₅	#5522	1	1	yes	characteristic
490	C₂×C₅	#5610	1	2	yes	—
491	C₂×C₅	#5636	1	2	yes	—
492	C₂³	—	2	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	3	2	no	—
495	C₂³	—	3	2	no	—
496	C₂³	—	3	2	no	—
497	C₂³	—	6	2	no	—
498	C₂³	—	6	2	no	—
499	C₂³	C₂×C₄×C₅×D₃	1	2	yes	—
500	C₂³	C₅×D₃×D₄	1	2	yes	—
501	C₂³	C₅×D₃×Q₈	1	2	yes	—
502	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
503	D₄	—	3	16	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	3	2	no	—
513	C₂×C₄	—	3	2	no	—
514	C₂×C₄	—	3	2	no	—
515	C₂×C₄	—	3	2	no	—
516	C₂×C₄	—	3	2	no	—
517	C₂×C₄	—	3	4	no	—
518	C₂×C₄	—	4	4	no	—
519	C₂×C₄	—	6	2	no	—
520	C₂×C₄	—	6	2	no	—
521	C₂×C₄	—	6	2	no	—
522	C₂×C₄	—	6	2	no	—
523	C₂×C₄	—	6	2	no	—
524	C₂×C₄	—	6	2	no	—
525	C₂×C₄	—	6	2	no	—
526	C₂×C₄	—	12	4	no	—
527	C₂×C₄	—	12	8	no	—
528	C₂×C₄	—	12	8	no	—
529	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
530	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
531	C₂×C₃	—	2	4	no	—
532	C₂×C₃	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
533	C₂×C₃	C₄×C₅×#41	1	1	yes	characteristic
534	C₂×C₃	C₅×#518	1	1	yes	characteristic
535	C₂×C₃	C₅×#546	1	2	yes	—
536	C₂×C₃	C₅×#554	1	2	yes	—
537	D₃	—	2	4	no	—
538	C₅	#74011	1	1	yes	characteristic, direct product factor, Sylow
539	C₂²	—	2	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	2	4	no	—
542	C₂²	—	2	4	no	—
543	C₂²	—	3	4	no	—
544	C₂²	—	6	2	no	—
545	C₂²	—	6	2	no	—
546	C₂²	—	6	4	no	—
547	C₂²	—	6	4	no	—
548	C₂²	C₅×D₃×C₄⋊C₄	1	4	yes	—
549	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
550	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
551	C₂²	C₅×#1004	1	2	yes	—
552	C₂²	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
553	C₂²	C₅×D₃×#41	1	2	yes	—
554	C₄	—	2	2	no	—
555	C₄	—	2	2	no	—
556	C₄	—	3	4	no	—
557	C₄	—	4	2	no	—
558	C₄	—	4	2	no	—
559	C₄	—	6	2	no	—
560	C₄	—	6	2	no	—
561	C₄	—	6	4	no	—
562	C₄	—	12	2	no	—
563	C₃	C₅×#2349	1	1	yes	characteristic, complemented, Sylow
564	C₂	—	2	4	no	—
565	C₂	—	6	4	no	—
566	C₂	C₅×#5486	1	1	yes	characteristic
567	C₂	C₂×C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
568	C₂	C₅×#5522	1	1	yes	characteristic
569	C₂	C₅×#5610	1	2	yes	—
570	C₂	C₅×#5636	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#380	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂²×C₅×#67	C₂²
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#380	D₃
hypocenter	C₃	C₅×#2349
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#380

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#380	C₂	semidirect
4. C₂×C₅×C₃⋊#95	C₂	semidirect
5. C₅×#4989	C₂	semidirect
6. C₅×#4976	C₂	semidirect
7. C₅×#4946	C₂	semidirect
8. C₅×#4667	C₂	semidirect
9. C₅×D₃×#95	C₂	semidirect
10. C₅×#4650	C₂	semidirect
11. C₅×#4649	C₂	semidirect
21. C₄×C₅×#67	C₄	semidirect
38. #74011	C₅	direct (N × H)
46. C₅×#380	D₃	semidirect
53. C₂²×C₃×C₄×C₅	C₂×C₄	semidirect
73. C₂×C₄×C₅×D₃	C₂×C₄	semidirect
76. C₅×#217	C₂×C₄	semidirect
78. C₅×(C₂×D₃)⋊C₄	C₂×C₄	semidirect
82. C₅×Dic₃⋊C₄	C₂×C₄	semidirect
84. C₄×C₅×Dic₃	C₂×C₄	semidirect
87. C₃×#380	C₂×C₅	semidirect
88. C₂×C₃⋊#95	C₂×C₅	semidirect
89. #4989	C₂×C₅	semidirect
90. #4976	C₂×C₅	semidirect
91. #4946	C₂×C₅	semidirect
92. #4667	C₂×C₅	semidirect
93. D₃×#95	C₂×C₅	semidirect
94. #4650	C₂×C₅	semidirect
95. #4649	C₂×C₅	semidirect
144. C₅×#67	C₄⋊C₄	semidirect
169. C₄×#67	C₄×C₅	semidirect
205. C₂²×C₄×C₅	C₄×D₃	semidirect
228. #380	C₅×D₃	semidirect
259. C₂²×C₃×C₄	C₂×C₄×C₅	semidirect
279. C₂×C₄×D₃	C₂×C₄×C₅	semidirect
282. #217	C₂×C₄×C₅	semidirect
284. (C₂×D₃)⋊C₄	C₂×C₄×C₅	semidirect
288. Dic₃⋊C₄	C₂×C₄×C₅	semidirect
290. C₄×Dic₃	C₂×C₄×C₅	semidirect
384. #67	C₅×C₄⋊C₄	semidirect
444. C₂²×C₄	C₄×C₅×D₃	semidirect
460. C₃×C₅	#2349	semidirect
538. C₅	#74011	direct (N × H)
563. 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×C₄×C₅×#67	2
C₂×C₅×#67⋊C₄	2
C₃×C₅×#380	2
C₂×C₅×C₃⋊#95	2
C₅×#4989	2
C₅×#4976	2
C₅×#4946	2
C₅×#4667	2
C₅×D₃×#95	2
C₅×#4650	2
C₅×#4649	2
C₅×#2349	3
#74011	5

Subgroups — C5×#74011 #14327100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#74011 #14327100