Subgroups — C₅×#69978 #14283265

Order 1920.

652 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#930	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#407	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5211	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4470	C₂	1	4	yes	maximal, complemented
5	C₅×#1510	—	3	1	no	maximal, Hall
6	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₅×#67	—	2	1	no	—
8	C₂²×C₅×D₁₂	C₄	1	1	yes	characteristic, complemented
9	C₂×C₃×C₅×M₁₆	—	2	1	no	—
10	C₂×C₅×#217	—	2	1	no	—
11	C₅×#930	—	2	1	no	—
12	C₅×#930	—	2	1	no	—
13	C₅×#930	C₄	1	2	yes	complemented
14	C₅×#930	C₂²	1	2	yes	—
15	C₂×C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
16	C₂×C₅×C₄⋊Dic₃	C₄	1	1	yes	characteristic, complemented
17	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
18	C₃×C₅×C₂²⋊C₈	C₂²	1	2	yes	complemented
19	C₅×C₃⋊#95	—	2	1	no	—
20	C₅×C₃⋊#95	C₂²	1	2	yes	complemented
21	#69978	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×#121	—	3	1	no	—
23	C₅×#407	D₃	1	1	yes	characteristic, complemented
24	C₅×#381	—	3	1	no	—
25	C₅×#327	—	3	4	no	—
26	C₂⁴×C₃×C₅	D₄	1	1	yes	characteristic
27	C₂³×C₅×D₃	—	2	1	no	—
28	C₂²×C₃×C₄×C₅	—	4	2	no	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₅×#67	—	2	2	no	—
36	C₂×C₅×#67	—	2	2	no	—
37	C₂×C₅×#67	—	4	4	no	—
38	C₂²×C₅×Dic₃	—	2	1	no	—
39	C₂²×C₅×Dic₃	—	2	2	no	—
40	C₂×C₅×D₁₂	—	2	1	no	—
41	C₂×C₅×D₁₂	—	4	2	no	—
42	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
43	C₃×C₅×M₁₆	—	4	2	no	—
44	C₂×C₃×C₅×C₈	—	2	2	no	—
45	C₅×#217	—	4	2	no	—
46	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
47	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
48	C₅×C₄⋊Dic₃	—	2	1	no	—
49	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
50	C₂×#930	C₂×C₅	1	1	yes	characteristic
51	C₃×#407	C₂×C₅	1	1	yes	characteristic, complemented
52	#5211	C₂×C₅	1	1	yes	characteristic, complemented
53	#4470	C₂×C₅	1	4	yes	complemented
54	C₂²×C₅×D₄	—	3	1	no	—
55	C₂²×C₅×D₄	—	6	1	no	—
56	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
57	C₂×C₅×M₁₆	—	2	1	no	—
58	C₅×#121	—	3	2	no	—
59	C₅×#121	—	3	2	no	—
60	C₅×#121	—	6	1	no	—
61	C₅×#121	—	6	1	no	—
62	C₂×C₅×C₄⋊C₄	—	3	1	no	—
63	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
64	C₂×C₅×C₂²⋊C₄	—	6	1	no	—
65	C₅×C₂²⋊C₈	—	2	1	no	—
66	C₅×C₂²⋊C₈	C₂×D₃	1	2	yes	complemented
67	C₅×#95	—	3	2	no	—
68	C₅×#95	—	6	1	no	—
69	#1510	—	3	1	no	Sylow
70	C₂³×C₃×C₅	—	2	2	no	—
71	C₂³×C₃×C₅	—	2	2	no	—
72	C₂³×C₃×C₅	—	4	1	no	—
73	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₅×D₃	—	2	2	no	—
77	C₂²×C₅×D₃	—	4	2	no	—
78	C₂²×C₅×D₃	—	4	2	no	—
79	C₂²×C₅×D₃	—	4	2	no	—
80	C₂×C₃×C₄×C₅	—	2	2	no	—
81	C₂×C₃×C₄×C₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	4	1	no	—
83	C₂×C₃×C₄×C₅	—	4	1	no	—
84	C₂×C₃×C₄×C₅	—	4	1	no	—
85	C₂×C₃×C₄×C₅	—	4	1	no	—
86	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
90	C₅×#67	—	4	4	no	—
91	C₅×#67	—	4	4	no	—
92	C₂×C₅×Dic₃	—	2	2	no	—
93	C₂×C₅×Dic₃	—	4	2	no	—
94	C₂×C₅×Dic₃	—	4	2	no	—
95	C₂×C₅×Dic₃	—	4	2	no	—
96	C₂×C₅×Dic₃	—	4	2	no	—
97	C₅×D₁₂	—	4	2	no	—
98	C₅×D₁₂	—	4	2	no	—
99	C₃×C₅×C₈	—	4	2	no	—
100	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
101	C₂²×#67	—	2	1	no	—
102	C₂²×D₁₂	C₄×C₅	1	1	yes	characteristic, complemented
103	C₂×C₃×M₁₆	—	2	1	no	—
104	C₂×#217	—	2	1	no	—
105	#930	—	2	1	no	—
106	#930	—	2	1	no	—
107	#930	C₄×C₅	1	2	yes	complemented
108	#930	C₂²×C₅	1	2	yes	—
109	C₂×(C₂×D₃)⋊C₄	—	2	1	no	—
110	C₂×C₄⋊Dic₃	C₄×C₅	1	1	yes	characteristic, complemented
111	C₃×C₂²⋊C₈	—	2	1	no	—
112	C₃×C₂²⋊C₈	C₂²×C₅	1	2	yes	complemented
113	C₃⋊#95	—	2	1	no	—
114	C₃⋊#95	C₂²×C₅	1	2	yes	complemented
115	C₂⁴×C₅	—	6	1	no	—
116	C₂⁴×C₅	D₁₂	1	1	yes	characteristic
117	C₂×C₅×D₄	—	3	2	no	—
118	C₂×C₅×D₄	—	6	2	no	—
119	C₂×C₅×D₄	—	6	2	no	—
120	C₂×C₅×D₄	—	6	1	no	—
121	C₂×C₅×D₄	—	12	2	no	—
122	C₂×C₅×D₄	—	12	4	no	—
123	C₂²×C₄×C₅	—	4	2	no	—
124	C₂²×C₄×C₅	—	6	1	no	—
125	C₂²×C₄×C₅	—	6	2	no	—
126	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
127	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
128	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
129	C₂²×C₄×C₅	#67	1	1	yes	characteristic
130	C₂²×C₄×C₅	#67	1	1	yes	characteristic
131	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
132	C₅×M₁₆	—	4	2	no	—
133	C₂×C₅×C₈	—	2	2	no	—
134	C₅×C₄⋊C₄	—	3	2	no	—
135	C₅×C₄⋊C₄	—	6	1	no	—
136	C₅×C₂²⋊C₄	—	6	2	no	—
137	C₅×C₂²⋊C₄	—	6	2	no	—
138	C₅×C₂²⋊C₄	—	12	2	no	—
139	C₂×#121	—	3	1	no	—
140	#407	C₅×D₃	1	1	yes	characteristic, complemented
141	#381	—	3	1	no	—
142	#327	—	3	4	no	—
143	C₂²×C₃×C₅	—	2	1	no	—
144	C₂²×C₃×C₅	—	2	1	no	—
145	C₂²×C₃×C₅	—	2	1	no	—
146	C₂²×C₃×C₅	—	2	1	no	—
147	C₂²×C₃×C₅	—	2	1	no	—
148	C₂²×C₃×C₅	—	2	1	no	—
149	C₂²×C₃×C₅	—	2	1	no	—
150	C₂²×C₃×C₅	—	4	4	no	—
151	C₂²×C₃×C₅	#99	1	2	yes	—
152	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
154	C₂×C₅×D₃	—	4	2	no	—
155	C₂×C₅×D₃	—	4	2	no	—
156	C₂×C₅×D₃	—	4	2	no	—
157	C₂×C₅×D₃	—	8	4	no	—
158	C₃×C₄×C₅	—	2	1	no	—
159	C₃×C₄×C₅	—	2	1	no	—
160	C₃×C₄×C₅	—	2	1	no	—
161	C₃×C₄×C₅	—	2	1	no	—
162	C₅×Dic₃	—	4	2	no	—
163	C₅×Dic₃	—	8	2	no	—
164	C₂⁴×C₃	C₅×D₄	1	1	yes	characteristic
165	C₂³×D₃	—	2	1	no	—
166	C₂²×C₃×C₄	—	4	2	no	—
167	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
168	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
169	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
170	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
171	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
172	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
173	C₂×#67	—	2	2	no	—
174	C₂×#67	—	2	2	no	—
175	C₂×#67	—	4	4	no	—
176	C₂²×Dic₃	—	2	1	no	—
177	C₂²×Dic₃	—	2	2	no	—
178	C₂×D₁₂	—	2	1	no	—
179	C₂×D₁₂	—	4	2	no	—
180	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
181	C₃×M₁₆	—	4	2	no	—
182	C₂×C₃×C₈	—	2	2	no	—
183	#217	—	4	2	no	—
184	(C₂×D₃)⋊C₄	—	2	2	no	—
185	(C₂×D₃)⋊C₄	—	2	2	no	—
186	C₄⋊Dic₃	—	2	1	no	—
187	C₄⋊Dic₃	C₂×C₄×C₅	1	2	yes	—
188	C₂³×C₅	—	2	2	no	—
189	C₂³×C₅	—	2	2	no	—
190	C₂³×C₅	—	4	1	no	—
191	C₂³×C₅	—	6	2	no	—
192	C₂³×C₅	—	12	2	no	—
193	C₂³×C₅	—	12	2	no	—
194	C₂³×C₅	—	12	2	no	—
195	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
196	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
197	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
198	C₅×D₄	—	12	2	no	—
199	C₅×D₄	—	12	2	no	—
200	C₅×D₄	—	12	4	no	—
201	C₅×D₄	—	12	4	no	—
202	C₂×C₄×C₅	—	2	2	no	—
203	C₂×C₄×C₅	—	2	2	no	—
204	C₂×C₄×C₅	—	4	1	no	—
205	C₂×C₄×C₅	—	4	1	no	—
206	C₂×C₄×C₅	—	4	1	no	—
207	C₂×C₄×C₅	—	4	1	no	—
208	C₂×C₄×C₅	—	6	2	no	—
209	C₂×C₄×C₅	—	12	2	no	—
210	C₂×C₄×C₅	—	12	2	no	—
211	C₂×C₄×C₅	—	12	2	no	—
212	C₂×C₄×C₅	—	12	2	no	—
213	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
214	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
215	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
216	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
217	C₅×C₈	—	4	2	no	—
218	C₂²×D₄	—	3	1	no	—
219	C₂²×D₄	—	6	1	no	—
220	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
221	C₂×M₁₆	—	2	1	no	—
222	#121	—	3	2	no	—
223	#121	—	3	2	no	—
224	#121	—	6	1	no	—
225	#121	—	6	1	no	—
226	C₂×C₄⋊C₄	—	3	1	no	—
227	C₂×C₂²⋊C₄	—	6	1	no	—
228	C₂×C₂²⋊C₄	—	6	1	no	—
229	C₂²⋊C₈	—	2	1	no	—
230	C₂²⋊C₈	C₂×C₅×D₃	1	2	yes	complemented
231	#95	—	3	2	no	—
232	#95	—	6	1	no	—
233	C₂×C₃×C₅	—	2	2	no	—
234	C₂×C₃×C₅	—	2	2	no	—
235	C₂×C₃×C₅	—	4	1	no	—
236	C₂×C₃×C₅	C₂×#99	1	1	yes	characteristic
237	C₂×C₃×C₅	#418	1	1	yes	characteristic
238	C₂×C₃×C₅	#421	1	1	yes	characteristic
239	C₅×D₃	—	8	2	no	—
240	C₂³×C₃	—	2	2	no	—
241	C₂³×C₃	—	2	2	no	—
242	C₂³×C₃	—	4	1	no	—
243	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
244	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
245	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
246	C₂²×D₃	—	2	2	no	—
247	C₂²×D₃	—	4	2	no	—
248	C₂²×D₃	—	4	2	no	—
249	C₂²×D₃	—	4	2	no	—
250	C₂×C₃×C₄	—	2	2	no	—
251	C₂×C₃×C₄	—	2	2	no	—
252	C₂×C₃×C₄	—	4	1	no	—
253	C₂×C₃×C₄	—	4	1	no	—
254	C₂×C₃×C₄	—	4	1	no	—
255	C₂×C₃×C₄	—	4	1	no	—
256	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
258	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
259	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
260	#67	—	4	4	no	—
261	#67	—	4	4	no	—
262	C₂×Dic₃	—	2	2	no	—
263	C₂×Dic₃	—	4	2	no	—
264	C₂×Dic₃	—	4	2	no	—
265	C₂×Dic₃	—	4	2	no	—
266	C₂×Dic₃	—	4	2	no	—
267	D₁₂	—	4	2	no	—
268	D₁₂	—	4	2	no	—
269	C₃×C₈	—	4	2	no	—
270	C₂²×C₅	—	2	1	no	—
271	C₂²×C₅	—	2	1	no	—
272	C₂²×C₅	—	2	1	no	—
273	C₂²×C₅	—	2	1	no	—
274	C₂²×C₅	—	2	1	no	—
275	C₂²×C₅	—	2	1	no	—
276	C₂²×C₅	—	2	1	no	—
277	C₂²×C₅	—	4	4	no	—
278	C₂²×C₅	—	12	2	no	—
279	C₂²×C₅	—	12	2	no	—
280	C₂²×C₅	—	12	2	no	—
281	C₂²×C₅	—	24	4	no	—
282	C₂²×C₅	#806	1	2	yes	—
283	C₂²×C₅	#908	1	2	yes	—
284	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
285	C₄×C₅	—	2	1	no	—
286	C₄×C₅	—	2	1	no	—
287	C₄×C₅	—	2	1	no	—
288	C₄×C₅	—	2	1	no	—
289	C₄×C₅	—	12	2	no	—
290	C₄×C₅	—	24	2	no	—
291	C₂⁴	—	6	1	no	—
292	C₂⁴	C₅×D₁₂	1	1	yes	characteristic
293	C₂×D₄	—	3	2	no	—
294	C₂×D₄	—	6	2	no	—
295	C₂×D₄	—	6	2	no	—
296	C₂×D₄	—	6	1	no	—
297	C₂×D₄	—	12	2	no	—
298	C₂×D₄	—	12	4	no	—
299	C₂²×C₄	—	4	2	no	—
300	C₂²×C₄	—	6	1	no	—
301	C₂²×C₄	—	6	2	no	—
302	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
303	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
304	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
305	C₂²×C₄	C₅×#67	1	1	yes	characteristic
306	C₂²×C₄	C₅×#67	1	1	yes	characteristic
307	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
308	M₁₆	—	4	2	no	—
309	C₂×C₈	—	2	2	no	—
310	C₄⋊C₄	—	3	2	no	—
311	C₄⋊C₄	—	6	1	no	—
312	C₂²⋊C₄	—	6	2	no	—
313	C₂²⋊C₄	—	6	2	no	—
314	C₂²⋊C₄	—	12	2	no	—
315	C₃×C₅	#1510	1	1	yes	characteristic, complemented, Hall
316	C₂²×C₃	—	2	1	no	—
317	C₂²×C₃	—	2	1	no	—
318	C₂²×C₃	—	2	1	no	—
319	C₂²×C₃	—	2	1	no	—
320	C₂²×C₃	—	2	1	no	—
321	C₂²×C₃	—	2	1	no	—
322	C₂²×C₃	—	2	1	no	—
323	C₂²×C₃	—	4	4	no	—
324	C₂²×C₃	C₅×#99	1	2	yes	—
325	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
326	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
327	C₂×D₃	—	4	2	no	—
328	C₂×D₃	—	4	2	no	—
329	C₂×D₃	—	4	2	no	—
330	C₂×D₃	—	8	4	no	—
331	C₃×C₄	—	2	1	no	—
332	C₃×C₄	—	2	1	no	—
333	C₃×C₄	—	2	1	no	—
334	C₃×C₄	—	2	1	no	—
335	Dic₃	—	4	2	no	—
336	Dic₃	—	8	2	no	—
337	C₂×C₅	—	2	2	no	—
338	C₂×C₅	—	2	2	no	—
339	C₂×C₅	—	4	1	no	—
340	C₂×C₅	—	24	2	no	—
341	C₂×C₅	C₂×#806	1	1	yes	characteristic
342	C₂×C₅	#5131	1	1	yes	characteristic
343	C₂×C₅	#5139	1	1	yes	characteristic
344	C₂³	—	2	2	no	—
345	C₂³	—	2	2	no	—
346	C₂³	—	4	1	no	—
347	C₂³	—	6	2	no	—
348	C₂³	—	12	2	no	—
349	C₂³	—	12	2	no	—
350	C₂³	—	12	2	no	—
351	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
352	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
353	C₂³	C₂×C₅×D₁₂	1	1	yes	characteristic
354	D₄	—	12	2	no	—
355	D₄	—	12	2	no	—
356	D₄	—	12	4	no	—
357	D₄	—	12	4	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	6	2	no	—
365	C₂×C₄	—	12	2	no	—
366	C₂×C₄	—	12	2	no	—
367	C₂×C₄	—	12	2	no	—
368	C₂×C₄	—	12	2	no	—
369	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
370	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
371	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
372	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
373	C₈	—	4	2	no	—
374	C₂×C₃	—	2	2	no	—
375	C₂×C₃	—	2	2	no	—
376	C₂×C₃	—	4	1	no	—
377	C₂×C₃	C₂×C₅×#99	1	1	yes	characteristic
378	C₂×C₃	C₅×#418	1	1	yes	characteristic
379	C₂×C₃	C₅×#421	1	1	yes	characteristic
380	D₃	—	8	2	no	—
381	C₅	#69978	1	1	yes	characteristic, direct product factor, Sylow
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	2	1	no	—
389	C₂²	—	4	4	no	—
390	C₂²	—	12	2	no	—
391	C₂²	—	12	2	no	—
392	C₂²	—	12	2	no	—
393	C₂²	—	24	4	no	—
394	C₂²	C₅×#806	1	2	yes	—
395	C₂²	C₅×#908	1	2	yes	—
396	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
397	C₄	—	2	1	no	—
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	2	1	no	—
401	C₄	—	12	2	no	—
402	C₄	—	24	2	no	—
403	C₃	C₅×#1510	1	1	yes	characteristic, complemented, Sylow
404	C₂	—	2	2	no	—
405	C₂	—	2	2	no	—
406	C₂	—	4	1	no	—
407	C₂	—	24	2	no	—
408	C₂	C₂×C₅×#806	1	1	yes	characteristic
409	C₂	C₅×#5131	1	1	yes	characteristic
410	C₂	C₅×#5139	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#407	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₅×#930	C₂
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#407	D₃
hypocenter	C₃	C₅×#1510
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂³×C₄×C₅ ◁ C₅×#407

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#407	C₂	semidirect
3. C₅×#5211	C₂	semidirect
4. C₅×#4470	C₂	semidirect
8. C₂²×C₅×D₁₂	C₄	semidirect
13. C₅×#930	C₄	semidirect
16. C₂×C₅×C₄⋊Dic₃	C₄	semidirect
18. C₃×C₅×C₂²⋊C₈	C₂²	semidirect
20. C₅×C₃⋊#95	C₂²	semidirect
21. #69978	C₅	direct (N × H)
23. C₅×#407	D₃	semidirect
51. C₃×#407	C₂×C₅	semidirect
52. #5211	C₂×C₅	semidirect
53. #4470	C₂×C₅	semidirect
66. C₅×C₂²⋊C₈	C₂×D₃	semidirect
102. C₂²×D₁₂	C₄×C₅	semidirect
107. #930	C₄×C₅	semidirect
110. C₂×C₄⋊Dic₃	C₄×C₅	semidirect
112. C₃×C₂²⋊C₈	C₂²×C₅	semidirect
114. C₃⋊#95	C₂²×C₅	semidirect
140. #407	C₅×D₃	semidirect
230. C₂²⋊C₈	C₂×C₅×D₃	semidirect
315. C₃×C₅	#1510	semidirect
381. C₅	#69978	direct (N × H)
403. C₃	C₅×#1510	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#930	2
C₃×C₅×#407	2
C₅×#5211	2
C₅×#4470	2
C₅×#1510	3
#69978	5

Subgroups — C5×#69978 #14283265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69978 #14283265