Subgroups — C₅×#74230 #14331127

Order 1920.

1040 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#875	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#382	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4990	C₂	1	2	yes	maximal, complemented
4	C₅×#4989	C₂	1	2	yes	maximal, complemented
5	C₅×#4936	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4670	C₂	1	2	yes	maximal, complemented
7	C₅×#4668	C₂	1	2	yes	maximal, complemented
8	C₅×#4665	C₂	1	2	yes	maximal
9	C₅×#4646	C₂	1	2	yes	maximal, complemented
10	C₅×#2379	—	3	1	no	maximal, Hall
11	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
13	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
17	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
18	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
19	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
20	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
21	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
22	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
23	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
24	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
25	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
26	C₅×#875	C₄	1	8	yes	complemented
27	C₃×C₅×#95	C₂²	1	2	yes	—
28	C₃×C₅×#95	C₂²	1	2	yes	—
29	C₅×C₃⋊#95	C₂²	1	2	yes	—
30	C₅×C₃⋊#95	C₂²	1	2	yes	—
31	C₅×C₃⋊#95	C₂²	1	2	yes	—
32	C₅×C₃⋊#95	C₂²	1	2	yes	—
33	#74230	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
34	C₂×C₅×#124	—	3	1	no	—
35	C₅×#391	—	3	2	no	—
36	C₅×#388	—	3	2	no	—
37	C₅×#388	—	3	2	no	—
38	C₅×#386	—	3	2	no	—
39	C₅×#385	—	3	2	no	—
40	C₅×#385	—	3	2	no	—
41	C₅×#382	—	3	1	no	—
42	C₅×#382	D₃	1	1	yes	characteristic, complemented
43	C₂³×C₅×D₃	C₂³	1	2	yes	—
44	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
45	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
46	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
48	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
49	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
50	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
51	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
52	C₂×C₅×D₁₂	—	4	1	no	—
53	C₂×C₅×D₁₂	—	4	1	no	—
54	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
55	C₂×C₄×C₅×D₃	—	4	2	no	—
56	C₂×C₄×C₅×D₃	—	4	2	no	—
57	C₂×C₄×C₅×D₃	—	4	2	no	—
58	C₂×C₅×Dic₆	—	4	1	no	—
59	C₂×C₅×Dic₆	—	4	1	no	—
60	C₂×C₅×Dic₆	C₂×C₄	1	4	yes	complemented
61	C₃×C₅×C₄⋊C₄	—	4	1	no	—
62	C₃×C₅×C₄⋊C₄	—	4	1	no	—
63	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
64	C₅×(C₂×D₃)⋊C₄	—	4	2	no	—
65	C₅×(C₂×D₃)⋊C₄	—	4	2	no	—
66	C₅×(C₂×D₃)⋊C₄	—	4	2	no	—
67	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	complemented
68	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	complemented
69	C₅×C₄⋊Dic₃	—	4	1	no	—
70	C₅×C₄⋊Dic₃	—	4	1	no	—
71	C₅×Dic₃⋊C₄	—	4	2	no	—
72	C₄×C₅×Dic₃	—	4	2	no	—
73	C₂×#875	C₂×C₅	1	1	yes	characteristic
74	C₃×#382	C₂×C₅	1	1	yes	characteristic, complemented
75	#4990	C₂×C₅	1	2	yes	complemented
76	#4989	C₂×C₅	1	2	yes	complemented
77	#4936	C₂×C₅	1	1	yes	characteristic, complemented
78	#4670	C₂×C₅	1	2	yes	complemented
79	#4668	C₂×C₅	1	2	yes	complemented
80	#4665	C₂×C₅	1	2	yes	—
81	#4646	C₂×C₅	1	2	yes	complemented
82	C₂²×C₅×Q₈	—	3	1	no	—
83	C₂²×C₅×D₄	—	3	1	no	—
84	C₂³×C₄×C₅	—	3	2	no	—
85	C₅×#124	—	3	8	no	—
86	C₂×C₅×C₄⋊C₄	—	3	1	no	—
87	C₂×C₅×C₄⋊C₄	—	3	1	no	—
88	C₂×C₅×C₄⋊C₄	—	3	2	no	—
89	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
90	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
91	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
92	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
93	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
94	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
95	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
96	C₂×C₄²×C₅	—	3	2	no	—
97	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
98	C₅×#95	—	3	2	no	—
99	C₅×#95	—	3	2	no	—
100	C₅×#95	—	3	2	no	—
101	C₅×#95	—	3	2	no	—
102	C₅×#95	C₂×D₃	1	2	yes	—
103	C₅×#95	C₂×D₃	1	2	yes	—
104	#2379	—	3	1	no	Sylow
105	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
106	C₂²×C₅×D₃	—	2	2	no	—
107	C₂²×C₅×D₃	—	2	2	no	—
108	C₂²×C₅×D₃	—	2	2	no	—
109	C₂²×C₅×D₃	—	2	2	no	—
110	C₂²×C₅×D₃	—	2	2	no	—
111	C₂²×C₅×D₃	—	2	2	no	—
112	C₂²×C₅×D₃	C₂²×C₄	1	4	yes	—
113	C₂×C₃×C₄×C₅	—	2	1	no	—
114	C₂×C₃×C₄×C₅	—	2	1	no	—
115	C₂×C₃×C₄×C₅	—	2	1	no	—
116	C₂×C₃×C₄×C₅	—	2	1	no	—
117	C₂×C₃×C₄×C₅	—	2	1	no	—
118	C₂×C₃×C₄×C₅	—	2	1	no	—
119	C₂×C₃×C₄×C₅	—	2	2	no	—
120	C₂×C₃×C₄×C₅	—	2	2	no	—
121	C₂×C₃×C₄×C₅	—	2	2	no	—
122	C₂×C₃×C₄×C₅	—	2	2	no	—
123	C₂×C₃×C₄×C₅	—	2	2	no	—
124	C₂×C₃×C₄×C₅	—	2	2	no	—
125	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
126	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
127	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
128	C₂×C₅×Dic₃	—	2	2	no	—
129	C₂×C₅×Dic₃	—	2	2	no	—
130	C₂×C₅×Dic₃	—	2	2	no	—
131	C₂×C₅×Dic₃	—	2	2	no	—
132	C₂×C₅×Dic₃	—	2	2	no	—
133	C₂×C₅×Dic₃	—	2	2	no	—
134	C₂×C₅×Dic₃	—	2	2	no	—
135	C₂×C₅×Dic₃	—	2	2	no	—
136	C₂×C₅×Dic₃	C₂²×C₄	1	4	yes	—
137	C₅×D₁₂	—	4	4	no	—
138	C₄×C₅×D₃	—	4	8	no	—
139	C₅×Dic₆	—	4	4	no	—
140	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
141	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
142	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic
143	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
144	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
145	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
146	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
147	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
148	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
149	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
150	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
151	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
152	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
153	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
154	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
155	#875	C₄×C₅	1	8	yes	complemented
156	C₃×#95	C₂²×C₅	1	2	yes	—
157	C₃×#95	C₂²×C₅	1	2	yes	—
158	C₃⋊#95	C₂²×C₅	1	2	yes	—
159	C₃⋊#95	C₂²×C₅	1	2	yes	—
160	C₃⋊#95	C₂²×C₅	1	2	yes	—
161	C₃⋊#95	C₂²×C₅	1	2	yes	—
162	C₂⁴×C₅	—	3	2	no	—
163	C₂×C₅×Q₈	—	3	4	no	—
164	C₂×C₅×Q₈	—	12	1	no	—
165	C₂×C₅×Q₈	—	12	1	no	—
166	C₂×C₅×D₄	—	3	4	no	—
167	C₂×C₅×D₄	—	12	1	no	—
168	C₂×C₅×D₄	—	12	1	no	—
169	C₂²×C₄×C₅	—	3	2	no	—
170	C₂²×C₄×C₅	—	3	2	no	—
171	C₂²×C₄×C₅	—	3	2	no	—
172	C₂²×C₄×C₅	—	12	2	no	—
173	C₂²×C₄×C₅	—	12	2	no	—
174	C₂²×C₄×C₅	—	12	2	no	—
175	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
176	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
177	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
178	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
179	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
180	C₅×C₄⋊C₄	—	4	1	no	—
181	C₅×C₄⋊C₄	—	4	1	no	—
182	C₅×C₄⋊C₄	—	12	2	no	—
183	C₅×C₄⋊C₄	—	12	1	no	—
184	C₅×C₄⋊C₄	—	12	1	no	—
185	C₅×C₂²⋊C₄	—	3	8	no	—
186	C₅×C₂²⋊C₄	—	3	8	no	—
187	C₅×C₂²⋊C₄	—	12	2	no	—
188	C₅×C₂²⋊C₄	—	12	2	no	—
189	C₅×C₂²⋊C₄	—	12	2	no	—
190	C₄²×C₅	—	12	2	no	—
191	C₄²×C₅	C₄×D₃	1	4	yes	complemented
192	C₂×#124	—	3	1	no	—
193	#391	—	3	2	no	—
194	#388	—	3	2	no	—
195	#388	—	3	2	no	—
196	#386	—	3	2	no	—
197	#385	—	3	2	no	—
198	#385	—	3	2	no	—
199	#382	—	3	1	no	—
200	#382	C₅×D₃	1	1	yes	characteristic, complemented
201	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
202	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
203	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
204	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
205	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
206	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
207	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
208	C₂×C₅×D₃	—	2	4	no	—
209	C₂×C₅×D₃	—	2	4	no	—
210	C₂×C₅×D₃	—	2	4	no	—
211	C₂×C₅×D₃	—	4	2	no	—
212	C₂×C₅×D₃	—	4	2	no	—
213	C₂×C₅×D₃	—	4	2	no	—
214	C₂×C₅×D₃	—	4	2	no	—
215	C₃×C₄×C₅	—	2	2	no	—
216	C₃×C₄×C₅	—	2	2	no	—
217	C₃×C₄×C₅	—	2	2	no	—
218	C₃×C₄×C₅	—	4	2	no	—
219	C₃×C₄×C₅	—	4	2	no	—
220	C₅×Dic₃	—	2	4	no	—
221	C₅×Dic₃	—	4	2	no	—
222	C₅×Dic₃	—	4	2	no	—
223	C₂³×D₃	C₂³×C₅	1	2	yes	—
224	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
225	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
226	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
227	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
228	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
229	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
230	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
231	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
232	C₂×D₁₂	—	4	1	no	—
233	C₂×D₁₂	—	4	1	no	—
234	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	—
235	C₂×C₄×D₃	—	4	2	no	—
236	C₂×C₄×D₃	—	4	2	no	—
237	C₂×C₄×D₃	—	4	2	no	—
238	C₂×Dic₆	—	4	1	no	—
239	C₂×Dic₆	—	4	1	no	—
240	C₂×Dic₆	C₂×C₄×C₅	1	4	yes	complemented
241	C₃×C₄⋊C₄	—	4	1	no	—
242	C₃×C₄⋊C₄	—	4	1	no	—
243	C₃×C₄²	C₂×C₄×C₅	1	4	yes	complemented
244	(C₂×D₃)⋊C₄	—	4	2	no	—
245	(C₂×D₃)⋊C₄	—	4	2	no	—
246	(C₂×D₃)⋊C₄	—	4	2	no	—
247	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	8	yes	complemented
248	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	8	yes	complemented
249	C₄⋊Dic₃	—	4	1	no	—
250	C₄⋊Dic₃	—	4	1	no	—
251	Dic₃⋊C₄	—	4	2	no	—
252	C₄×Dic₃	—	4	2	no	—
253	C₂³×C₅	—	3	4	no	—
254	C₂³×C₅	—	6	2	no	—
255	C₂³×C₅	—	6	2	no	—
256	C₂³×C₅	—	6	2	no	—
257	C₂³×C₅	—	6	2	no	—
258	C₂³×C₅	—	6	2	no	—
259	C₂³×C₅	—	6	2	no	—
260	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
261	C₅×Q₈	—	12	4	no	—
262	C₅×D₄	—	12	4	no	—
263	C₂×C₄×C₅	—	2	1	no	—
264	C₂×C₄×C₅	—	2	1	no	—
265	C₂×C₄×C₅	—	2	1	no	—
266	C₂×C₄×C₅	—	2	1	no	—
267	C₂×C₄×C₅	—	2	1	no	—
268	C₂×C₄×C₅	—	2	1	no	—
269	C₂×C₄×C₅	—	2	2	no	—
270	C₂×C₄×C₅	—	2	2	no	—
271	C₂×C₄×C₅	—	2	2	no	—
272	C₂×C₄×C₅	—	2	2	no	—
273	C₂×C₄×C₅	—	2	2	no	—
274	C₂×C₄×C₅	—	2	2	no	—
275	C₂×C₄×C₅	—	3	4	no	—
276	C₂×C₄×C₅	—	6	2	no	—
277	C₂×C₄×C₅	—	6	2	no	—
278	C₂×C₄×C₅	—	6	2	no	—
279	C₂×C₄×C₅	—	6	2	no	—
280	C₂×C₄×C₅	—	6	2	no	—
281	C₂×C₄×C₅	—	6	2	no	—
282	C₂×C₄×C₅	—	6	2	no	—
283	C₂×C₄×C₅	—	6	2	no	—
284	C₂×C₄×C₅	—	12	8	no	—
285	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
286	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
287	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
288	C₂²×Q₈	—	3	1	no	—
289	C₂²×D₄	—	3	1	no	—
290	C₂³×C₄	—	3	2	no	—
291	#124	—	3	8	no	—
292	C₂×C₄⋊C₄	—	3	1	no	—
293	C₂×C₄⋊C₄	—	3	1	no	—
294	C₂×C₄⋊C₄	—	3	2	no	—
295	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
297	C₂×C₂²⋊C₄	—	3	2	no	—
298	C₂×C₂²⋊C₄	—	3	2	no	—
299	C₂×C₂²⋊C₄	—	3	2	no	—
300	C₂×C₂²⋊C₄	—	3	2	no	—
301	C₂×C₂²⋊C₄	—	3	2	no	—
302	C₂×C₄²	—	3	2	no	—
303	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
304	#95	—	3	2	no	—
305	#95	—	3	2	no	—
306	#95	—	3	2	no	—
307	#95	—	3	2	no	—
308	#95	C₂×C₅×D₃	1	2	yes	—
309	#95	C₂×C₅×D₃	1	2	yes	—
310	C₂×C₃×C₅	#518	1	1	yes	characteristic
311	C₂×C₃×C₅	#520	1	1	yes	characteristic
312	C₂×C₃×C₅	#520	1	1	yes	characteristic
313	C₂×C₃×C₅	H(3,𝔽₄)	1	1	yes	characteristic
314	C₂×C₃×C₅	#562	1	1	yes	characteristic
315	C₂×C₃×C₅	#562	1	1	yes	characteristic
316	C₂×C₃×C₅	#563	1	1	yes	characteristic
317	C₅×D₃	—	4	4	no	—
318	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
319	C₂²×D₃	—	2	2	no	—
320	C₂²×D₃	—	2	2	no	—
321	C₂²×D₃	—	2	2	no	—
322	C₂²×D₃	—	2	2	no	—
323	C₂²×D₃	—	2	2	no	—
324	C₂²×D₃	—	2	2	no	—
325	C₂²×D₃	C₂²×C₄×C₅	1	4	yes	—
326	C₂×C₃×C₄	—	2	1	no	—
327	C₂×C₃×C₄	—	2	1	no	—
328	C₂×C₃×C₄	—	2	1	no	—
329	C₂×C₃×C₄	—	2	1	no	—
330	C₂×C₃×C₄	—	2	1	no	—
331	C₂×C₃×C₄	—	2	1	no	—
332	C₂×C₃×C₄	—	2	2	no	—
333	C₂×C₃×C₄	—	2	2	no	—
334	C₂×C₃×C₄	—	2	2	no	—
335	C₂×C₃×C₄	—	2	2	no	—
336	C₂×C₃×C₄	—	2	2	no	—
337	C₂×C₃×C₄	—	2	2	no	—
338	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
339	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
340	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
341	C₂×Dic₃	—	2	2	no	—
342	C₂×Dic₃	—	2	2	no	—
343	C₂×Dic₃	—	2	2	no	—
344	C₂×Dic₃	—	2	2	no	—
345	C₂×Dic₃	—	2	2	no	—
346	C₂×Dic₃	—	2	2	no	—
347	C₂×Dic₃	—	2	2	no	—
348	C₂×Dic₃	—	2	2	no	—
349	C₂×Dic₃	C₂²×C₄×C₅	1	4	yes	—
350	D₁₂	—	4	4	no	—
351	C₄×D₃	—	4	8	no	—
352	Dic₆	—	4	4	no	—
353	C₂²×C₅	—	6	4	no	—
354	C₂²×C₅	—	6	4	no	—
355	C₂²×C₅	—	6	4	no	—
356	C₂²×C₅	—	12	2	no	—
357	C₂²×C₅	—	12	2	no	—
358	C₂²×C₅	—	12	2	no	—
359	C₂²×C₅	—	12	2	no	—
360	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
361	C₂²×C₅	#1004	1	1	yes	characteristic
362	C₂²×C₅	#1007	1	1	yes	characteristic
363	C₂²×C₅	#1009	1	1	yes	characteristic
364	C₂²×C₅	#1009	1	1	yes	characteristic
365	C₂²×C₅	#1009	1	1	yes	characteristic
366	C₂²×C₅	#1010	1	1	yes	characteristic
367	C₄×C₅	—	2	2	no	—
368	C₄×C₅	—	2	2	no	—
369	C₄×C₅	—	2	2	no	—
370	C₄×C₅	—	4	2	no	—
371	C₄×C₅	—	4	2	no	—
372	C₄×C₅	—	6	4	no	—
373	C₄×C₅	—	12	2	no	—
374	C₄×C₅	—	12	2	no	—
375	C₂⁴	—	3	2	no	—
376	C₂×Q₈	—	3	4	no	—
377	C₂×Q₈	—	12	1	no	—
378	C₂×Q₈	—	12	1	no	—
379	C₂×D₄	—	3	4	no	—
380	C₂×D₄	—	12	1	no	—
381	C₂×D₄	—	12	1	no	—
382	C₂²×C₄	—	3	2	no	—
383	C₂²×C₄	—	3	2	no	—
384	C₂²×C₄	—	3	2	no	—
385	C₂²×C₄	—	12	2	no	—
386	C₂²×C₄	—	12	2	no	—
387	C₂²×C₄	—	12	2	no	—
388	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
389	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
390	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
391	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
392	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
393	C₄⋊C₄	—	4	1	no	—
394	C₄⋊C₄	—	4	1	no	—
395	C₄⋊C₄	—	12	2	no	—
396	C₄⋊C₄	—	12	1	no	—
397	C₄⋊C₄	—	12	1	no	—
398	C₂²⋊C₄	—	3	8	no	—
399	C₂²⋊C₄	—	3	8	no	—
400	C₂²⋊C₄	—	12	2	no	—
401	C₂²⋊C₄	—	12	2	no	—
402	C₂²⋊C₄	—	12	2	no	—
403	C₄²	—	12	2	no	—
404	C₄²	C₄×C₅×D₃	1	4	yes	complemented
405	C₃×C₅	#2379	1	1	yes	characteristic, complemented, Hall
406	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
407	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
408	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
409	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
410	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
411	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
412	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
413	C₂×D₃	—	2	4	no	—
414	C₂×D₃	—	2	4	no	—
415	C₂×D₃	—	2	4	no	—
416	C₂×D₃	—	4	2	no	—
417	C₂×D₃	—	4	2	no	—
418	C₂×D₃	—	4	2	no	—
419	C₂×D₃	—	4	2	no	—
420	C₃×C₄	—	2	2	no	—
421	C₃×C₄	—	2	2	no	—
422	C₃×C₄	—	2	2	no	—
423	C₃×C₄	—	4	2	no	—
424	C₃×C₄	—	4	2	no	—
425	Dic₃	—	2	4	no	—
426	Dic₃	—	4	2	no	—
427	Dic₃	—	4	2	no	—
428	C₂×C₅	—	12	4	no	—
429	C₂×C₅	#5504	1	1	yes	characteristic
430	C₂×C₅	#5521	1	1	yes	characteristic
431	C₂×C₅	#5523	1	1	yes	characteristic
432	C₂×C₅	#5683	1	1	yes	characteristic
433	C₂×C₅	#5684	1	1	yes	characteristic
434	C₂×C₅	#5699	1	1	yes	characteristic
435	C₂×C₅	#5700	1	1	yes	characteristic
436	C₂³	—	3	4	no	—
437	C₂³	—	6	2	no	—
438	C₂³	—	6	2	no	—
439	C₂³	—	6	2	no	—
440	C₂³	—	6	2	no	—
441	C₂³	—	6	2	no	—
442	C₂³	—	6	2	no	—
443	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
444	Q₈	—	12	4	no	—
445	D₄	—	12	4	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	3	4	no	—
459	C₂×C₄	—	6	2	no	—
460	C₂×C₄	—	6	2	no	—
461	C₂×C₄	—	6	2	no	—
462	C₂×C₄	—	6	2	no	—
463	C₂×C₄	—	6	2	no	—
464	C₂×C₄	—	6	2	no	—
465	C₂×C₄	—	6	2	no	—
466	C₂×C₄	—	6	2	no	—
467	C₂×C₄	—	12	8	no	—
468	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
469	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
470	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
471	C₂×C₃	C₅×#518	1	1	yes	characteristic
472	C₂×C₃	C₅×#520	1	1	yes	characteristic
473	C₂×C₃	C₅×#520	1	1	yes	characteristic
474	C₂×C₃	C₅×H(3,𝔽₄)	1	1	yes	characteristic
475	C₂×C₃	C₅×#562	1	1	yes	characteristic
476	C₂×C₃	C₅×#562	1	1	yes	characteristic
477	C₂×C₃	C₅×#563	1	1	yes	characteristic
478	D₃	—	4	4	no	—
479	C₅	#74230	1	1	yes	characteristic, direct product factor, Sylow
480	C₂²	—	6	4	no	—
481	C₂²	—	6	4	no	—
482	C₂²	—	6	4	no	—
483	C₂²	—	12	2	no	—
484	C₂²	—	12	2	no	—
485	C₂²	—	12	2	no	—
486	C₂²	—	12	2	no	—
487	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
488	C₂²	C₅×#1004	1	1	yes	characteristic
489	C₂²	C₅×#1007	1	1	yes	characteristic
490	C₂²	C₅×#1009	1	1	yes	characteristic
491	C₂²	C₅×#1009	1	1	yes	characteristic
492	C₂²	C₅×#1009	1	1	yes	characteristic
493	C₂²	C₅×#1010	1	1	yes	characteristic
494	C₄	—	2	2	no	—
495	C₄	—	2	2	no	—
496	C₄	—	2	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	2	no	—
499	C₄	—	6	4	no	—
500	C₄	—	12	2	no	—
501	C₄	—	12	2	no	—
502	C₃	C₅×#2379	1	1	yes	characteristic, complemented, Sylow
503	C₂	—	12	4	no	—
504	C₂	C₅×#5504	1	1	yes	characteristic
505	C₂	C₅×#5521	1	1	yes	characteristic
506	C₂	C₅×#5523	1	1	yes	characteristic
507	C₂	C₅×#5683	1	1	yes	characteristic
508	C₂	C₅×#5684	1	1	yes	characteristic
509	C₂	C₅×#5699	1	1	yes	characteristic
510	C₂	C₅×#5700	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×#382	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂²×C₅×D₁₂	C₂²
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#382	D₃
hypocenter	C₃	C₅×#2379
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#382

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#382	C₂	semidirect
3. C₅×#4990	C₂	semidirect
4. C₅×#4989	C₂	semidirect
5. C₅×#4936	C₂	semidirect
6. C₅×#4670	C₂	semidirect
7. C₅×#4668	C₂	semidirect
9. C₅×#4646	C₂	semidirect
26. C₅×#875	C₄	semidirect
33. #74230	C₅	direct (N × H)
42. C₅×#382	D₃	semidirect
60. C₂×C₅×Dic₆	C₂×C₄	semidirect
63. C₃×C₄²×C₅	C₂×C₄	semidirect
67. C₅×(C₂×D₃)⋊C₄	C₂×C₄	semidirect
68. C₅×(C₂×D₃)⋊C₄	C₂×C₄	semidirect
74. C₃×#382	C₂×C₅	semidirect
75. #4990	C₂×C₅	semidirect
76. #4989	C₂×C₅	semidirect
77. #4936	C₂×C₅	semidirect
78. #4670	C₂×C₅	semidirect
79. #4668	C₂×C₅	semidirect
81. #4646	C₂×C₅	semidirect
155. #875	C₄×C₅	semidirect
191. C₄²×C₅	C₄×D₃	semidirect
200. #382	C₅×D₃	semidirect
240. C₂×Dic₆	C₂×C₄×C₅	semidirect
243. C₃×C₄²	C₂×C₄×C₅	semidirect
247. (C₂×D₃)⋊C₄	C₂×C₄×C₅	semidirect
248. (C₂×D₃)⋊C₄	C₂×C₄×C₅	semidirect
404. C₄²	C₄×C₅×D₃	semidirect
405. C₃×C₅	#2379	semidirect
479. C₅	#74230	direct (N × H)
502. C₃	C₅×#2379	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#875	2
C₃×C₅×#382	2
C₅×#4990	2
C₅×#4989	2
C₅×#4936	2
C₅×#4670	2
C₅×#4668	2
C₅×#4665	2
C₅×#4646	2
C₅×#2379	3
#74230	5

Subgroups — C5×#74230 #14331127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#74230 #14331127