Subgroups — C₅×#70227 #14305875

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#947	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#103	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#835	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#821	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×#815	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4975	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#809	C₂	1	1	yes	maximal, characteristic
8	C₅×#1959	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	C₅×#947	—	2	2	no	—
14	C₅×#947	C₄	1	4	yes	complemented
15	C₂×C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
16	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₅×Dic₃⋊C₄	—	2	2	no	—
18	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
19	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₃×C₅×#103	—	2	2	no	—
21	C₅×#835	—	2	2	no	—
22	C₅×#821	—	2	2	no	—
23	C₅×#815	—	2	2	no	—
24	C₅×#809	—	2	2	no	—
25	#70227	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
26	C₂×C₅×#124	—	3	1	no	—
27	C₂×C₅×#103	—	3	1	no	—
28	C₂×C₅×#103	D₃	1	1	yes	characteristic, complemented
29	C₂×C₅×#102	—	3	1	no	—
30	C₂×C₅×#102	—	3	1	no	—
31	C₂×C₅×#97	—	3	1	no	—
32	C₅×#384	—	3	1	no	—
33	C₂³×C₅×D₃	—	2	1	no	—
34	C₂×C₃×C₅×Q₈	—	2	1	no	—
35	C₂×C₃×C₅×Q₈	—	2	1	no	—
36	C₂×C₃×C₅×Q₈	—	2	1	no	—
37	C₂×C₃×C₅×Q₈	—	2	1	no	—
38	C₂×C₃×C₅×Q₈	—	2	1	no	—
39	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
40	C₂²×C₃×C₄×C₅	—	2	1	no	—
41	C₂²×C₃×C₄×C₅	—	2	1	no	—
42	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂²×C₅×Dic₃	—	2	1	no	—
44	C₂²×C₅×Dic₃	D₄	1	2	yes	—
45	C₂×C₅×D₁₂	—	2	1	no	—
46	C₂×C₅×D₁₂	—	2	1	no	—
47	C₂×C₅×D₁₂	—	2	1	no	—
48	C₂×C₅×D₁₂	—	2	1	no	—
49	C₂×C₅×D₁₂	—	2	1	no	—
50	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
51	C₂×C₃×C₅×C₈	—	2	1	no	—
52	C₂×C₃×C₅×C₈	—	2	1	no	—
53	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
54	C₃×C₅×C₄⋊C₄	—	2	1	no	—
55	C₃×C₅×C₄⋊C₄	—	2	1	no	—
56	C₅×(C₂×D₃)⋊C₄	—	2	8	no	—
57	C₅×C₄⋊Dic₃	—	2	1	no	—
58	C₅×C₄⋊Dic₃	—	2	1	no	—
59	C₅×Dic₃⋊C₄	—	4	4	no	—
60	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
61	C₄×C₅×Dic₃	D₄	1	2	yes	—
62	C₂×C₅×C₃⋊C₈	—	2	1	no	—
63	C₂×C₅×C₃⋊C₈	—	2	1	no	—
64	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
65	C₂×#947	C₂×C₅	1	1	yes	characteristic
66	C₂×C₃×#103	C₂×C₅	1	1	yes	characteristic, complemented
67	C₂×#835	C₂×C₅	1	1	yes	characteristic, complemented
68	C₂×#821	C₂×C₅	1	1	yes	characteristic
69	C₂×#815	C₂×C₅	1	1	yes	characteristic, complemented
70	#4975	C₂×C₅	1	1	yes	characteristic, complemented
71	C₂×#809	C₂×C₅	1	1	yes	characteristic
72	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
73	C₂²×C₅×D₄	—	3	1	no	—
74	C₂²×C₅×C₈	—	3	1	no	—
75	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
76	C₅×#124	—	3	4	no	—
77	C₅×#124	—	6	2	no	—
78	C₂×C₅×C₄⋊C₄	—	3	1	no	—
79	C₂×C₅×C₄⋊C₄	—	6	2	no	—
80	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
81	C₂×C₅×C₂²⋊C₄	—	6	2	no	—
82	C₂×C₄²×C₅	—	3	1	no	—
83	C₅×#103	—	2	2	no	—
84	C₅×#103	—	6	2	no	—
85	C₅×#102	—	6	2	no	—
86	C₅×#102	—	6	2	no	—
87	C₅×#97	—	6	2	no	—
88	#1959	—	3	1	no	Sylow
89	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₅×D₃	—	2	2	no	—
91	C₂²×C₅×D₃	—	4	1	no	—
92	C₂²×C₅×D₃	—	4	1	no	—
93	C₂²×C₅×D₃	—	4	2	no	—
94	C₂²×C₅×D₃	—	4	2	no	—
95	C₃×C₅×Q₈	—	2	2	no	—
96	C₃×C₅×Q₈	—	2	2	no	—
97	C₃×C₅×Q₈	—	4	1	no	—
98	C₃×C₅×Q₈	—	4	1	no	—
99	C₂×C₃×C₄×C₅	—	2	2	no	—
100	C₂×C₃×C₄×C₅	—	4	1	no	—
101	C₂×C₃×C₄×C₅	—	4	1	no	—
102	C₂×C₃×C₄×C₅	—	4	1	no	—
103	C₂×C₃×C₄×C₅	—	4	2	no	—
104	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
106	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
107	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
108	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
109	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
110	C₂×C₅×Dic₃	—	2	2	no	—
111	C₂×C₅×Dic₃	—	2	2	no	—
112	C₂×C₅×Dic₃	—	4	1	no	—
113	C₂×C₅×Dic₃	—	4	2	no	—
114	C₂×C₅×Dic₃	C₂²⋊C₄	1	4	yes	—
115	C₅×D₁₂	—	2	2	no	—
116	C₅×D₁₂	—	2	2	no	—
117	C₅×D₁₂	—	4	1	no	—
118	C₅×D₁₂	—	4	1	no	—
119	C₃×C₅×C₈	—	2	2	no	—
120	C₅×C₃⋊C₈	—	2	2	no	—
121	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
122	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
123	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
124	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
125	#947	—	2	2	no	—
126	#947	C₄×C₅	1	4	yes	complemented
127	C₂×(C₂×D₃)⋊C₄	—	2	2	no	—
128	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
129	C₂×Dic₃⋊C₄	—	2	2	no	—
130	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
131	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
132	C₃×#103	—	2	2	no	—
133	#835	—	2	2	no	—
134	#821	—	2	2	no	—
135	#815	—	2	2	no	—
136	#809	—	2	2	no	—
137	C₂⁴×C₅	—	6	1	no	—
138	C₂×C₅×Q₈	—	2	1	no	—
139	C₂×C₅×Q₈	—	2	1	no	—
140	C₂×C₅×Q₈	—	2	1	no	—
141	C₂×C₅×Q₈	—	2	1	no	—
142	C₂×C₅×Q₈	—	2	1	no	—
143	C₂×C₅×Q₈	C₄×D₃	1	2	yes	—
144	C₂×C₅×D₄	—	3	2	no	—
145	C₂×C₅×D₄	—	6	1	no	—
146	C₂×C₅×D₄	—	6	1	no	—
147	C₂×C₅×D₄	—	6	1	no	—
148	C₂×C₅×D₄	—	6	1	no	—
149	C₂×C₅×D₄	—	6	1	no	—
150	C₂²×C₄×C₅	—	2	1	no	—
151	C₂²×C₄×C₅	—	2	1	no	—
152	C₂²×C₄×C₅	—	3	2	no	—
153	C₂²×C₄×C₅	—	6	1	no	—
154	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
155	C₂×C₅×C₈	—	2	1	no	—
156	C₂×C₅×C₈	—	2	1	no	—
157	C₂×C₅×C₈	—	3	2	no	—
158	C₂×C₅×C₈	—	6	1	no	—
159	C₂×C₅×C₈	—	6	1	no	—
160	C₂×C₅×C₈	#67	1	2	yes	—
161	C₅×C₄⋊C₄	—	2	1	no	—
162	C₅×C₄⋊C₄	—	2	1	no	—
163	C₅×C₄⋊C₄	—	6	1	no	—
164	C₅×C₄⋊C₄	—	6	1	no	—
165	C₅×C₄⋊C₄	—	12	4	no	—
166	C₅×C₂²⋊C₄	—	6	8	no	—
167	C₄²×C₅	—	3	2	no	—
168	C₄²×C₅	—	3	2	no	—
169	C₂×#124	—	3	1	no	—
170	C₂×#103	—	3	1	no	—
171	C₂×#103	C₅×D₃	1	1	yes	characteristic, complemented
172	C₂×#102	—	3	1	no	—
173	C₂×#102	—	3	1	no	—
174	C₂×#97	—	3	1	no	—
175	#384	—	3	1	no	—
176	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₂²×C₃×C₅	#124	1	1	yes	characteristic
178	C₂²×C₃×C₅	#127	1	1	yes	characteristic
179	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
180	C₂²×C₃×C₅	#137	1	2	yes	—
181	C₂×C₅×D₃	—	4	2	no	—
182	C₂×C₅×D₃	—	4	4	no	—
183	C₂×C₅×D₃	—	8	2	no	—
184	C₂×C₅×D₃	—	8	1	no	—
185	C₂×C₅×D₃	—	8	1	no	—
186	C₃×C₄×C₅	—	4	2	no	—
187	C₃×C₄×C₅	—	8	1	no	—
188	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
189	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
190	C₃×C₄×C₅	#121	1	1	yes	characteristic
191	C₃×C₄×C₅	#121	1	1	yes	characteristic
192	C₅×Dic₃	—	2	4	no	—
193	C₅×Dic₃	—	8	1	no	—
194	C₂³×D₃	—	2	1	no	—
195	C₂×C₃×Q₈	—	2	1	no	—
196	C₂×C₃×Q₈	—	2	1	no	—
197	C₂×C₃×Q₈	—	2	1	no	—
198	C₂×C₃×Q₈	—	2	1	no	—
199	C₂×C₃×Q₈	—	2	1	no	—
200	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
201	C₂²×C₃×C₄	—	2	1	no	—
202	C₂²×C₃×C₄	—	2	1	no	—
203	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
204	C₂²×Dic₃	—	2	1	no	—
205	C₂²×Dic₃	C₅×D₄	1	2	yes	—
206	C₂×D₁₂	—	2	1	no	—
207	C₂×D₁₂	—	2	1	no	—
208	C₂×D₁₂	—	2	1	no	—
209	C₂×D₁₂	—	2	1	no	—
210	C₂×D₁₂	—	2	1	no	—
211	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
212	C₂×C₃×C₈	—	2	1	no	—
213	C₂×C₃×C₈	—	2	1	no	—
214	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
215	C₃×C₄⋊C₄	—	2	1	no	—
216	C₃×C₄⋊C₄	—	2	1	no	—
217	(C₂×D₃)⋊C₄	—	2	8	no	—
218	C₄⋊Dic₃	—	2	1	no	—
219	C₄⋊Dic₃	—	2	1	no	—
220	Dic₃⋊C₄	—	4	4	no	—
221	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
222	C₄×Dic₃	C₅×D₄	1	2	yes	—
223	C₂×C₃⋊C₈	—	2	1	no	—
224	C₂×C₃⋊C₈	—	2	1	no	—
225	C₂×C₃⋊C₈	C₅×D₄	1	2	yes	—
226	C₂³×C₅	—	6	2	no	—
227	C₂³×C₅	—	12	1	no	—
228	C₂³×C₅	—	12	2	no	—
229	C₂³×C₅	—	12	2	no	—
230	C₂³×C₅	—	12	1	no	—
231	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
232	C₅×Q₈	—	2	2	no	—
233	C₅×Q₈	—	2	2	no	—
234	C₅×Q₈	—	4	1	no	—
235	C₅×Q₈	—	4	1	no	—
236	C₅×D₄	—	6	2	no	—
237	C₅×D₄	—	6	2	no	—
238	C₅×D₄	—	12	1	no	—
239	C₅×D₄	—	12	1	no	—
240	C₂×C₄×C₅	—	2	2	no	—
241	C₂×C₄×C₅	—	3	4	no	—
242	C₂×C₄×C₅	—	4	1	no	—
243	C₂×C₄×C₅	—	4	1	no	—
244	C₂×C₄×C₅	—	4	2	no	—
245	C₂×C₄×C₅	—	4	1	no	—
246	C₂×C₄×C₅	—	6	2	no	—
247	C₂×C₄×C₅	—	6	2	no	—
248	C₂×C₄×C₅	—	12	2	no	—
249	C₂×C₄×C₅	—	12	1	no	—
250	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
251	C₂×C₄×C₅	#235	1	1	yes	characteristic
252	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
253	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
254	C₂×C₄×C₅	#237	1	1	yes	characteristic
255	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
256	C₅×C₈	—	2	2	no	—
257	C₅×C₈	—	6	2	no	—
258	C₂²×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
259	C₂²×D₄	—	3	1	no	—
260	C₂²×C₈	—	3	1	no	—
261	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
262	#124	—	3	4	no	—
263	#124	—	6	2	no	—
264	C₂×C₄⋊C₄	—	3	1	no	—
265	C₂×C₄⋊C₄	—	6	2	no	—
266	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
267	C₂×C₂²⋊C₄	—	6	2	no	—
268	C₂×C₄²	—	3	1	no	—
269	#103	—	2	2	no	—
270	#103	—	6	2	no	—
271	#102	—	6	2	no	—
272	#102	—	6	2	no	—
273	#97	—	6	2	no	—
274	C₂×C₃×C₅	#390	1	1	yes	characteristic
275	C₂×C₃×C₅	#417	1	2	yes	—
276	C₂×C₃×C₅	#489	1	2	yes	—
277	C₂×C₃×C₅	#496	1	1	yes	characteristic
278	C₂×C₃×C₅	#497	1	1	yes	characteristic
279	C₅×D₃	—	8	2	no	—
280	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
281	C₂²×D₃	—	2	2	no	—
282	C₂²×D₃	—	4	1	no	—
283	C₂²×D₃	—	4	1	no	—
284	C₂²×D₃	—	4	2	no	—
285	C₂²×D₃	—	4	2	no	—
286	C₃×Q₈	—	2	2	no	—
287	C₃×Q₈	—	2	2	no	—
288	C₃×Q₈	—	4	1	no	—
289	C₃×Q₈	—	4	1	no	—
290	C₂×C₃×C₄	—	2	2	no	—
291	C₂×C₃×C₄	—	4	1	no	—
292	C₂×C₃×C₄	—	4	1	no	—
293	C₂×C₃×C₄	—	4	1	no	—
294	C₂×C₃×C₄	—	4	2	no	—
295	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
296	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
297	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
298	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
299	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
300	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
301	C₂×Dic₃	—	2	2	no	—
302	C₂×Dic₃	—	2	2	no	—
303	C₂×Dic₃	—	4	1	no	—
304	C₂×Dic₃	—	4	2	no	—
305	C₂×Dic₃	C₅×C₂²⋊C₄	1	4	yes	—
306	D₁₂	—	2	2	no	—
307	D₁₂	—	2	2	no	—
308	D₁₂	—	4	1	no	—
309	D₁₂	—	4	1	no	—
310	C₃×C₈	—	2	2	no	—
311	C₃⋊C₈	—	2	2	no	—
312	C₂²×C₅	—	12	2	no	—
313	C₂²×C₅	—	12	4	no	—
314	C₂²×C₅	—	24	1	no	—
315	C₂²×C₅	—	24	1	no	—
316	C₂²×C₅	—	24	2	no	—
317	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
318	C₂²×C₅	#885	1	1	yes	characteristic
319	C₂²×C₅	#914	1	2	yes	—
320	C₂²×C₅	#918	1	2	yes	—
321	C₂²×C₅	#940	1	1	yes	characteristic
322	C₄×C₅	—	4	2	no	—
323	C₄×C₅	—	6	4	no	—
324	C₄×C₅	—	8	1	no	—
325	C₄×C₅	—	24	1	no	—
326	C₄×C₅	#67⋊C₄	1	1	yes	characteristic
327	C₄×C₅	#884	1	1	yes	characteristic
328	C₄×C₅	C₄×#67	1	1	yes	characteristic
329	C₄×C₅	#939	1	1	yes	characteristic
330	C₂⁴	—	6	1	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	—	2	1	no	—
333	C₂×Q₈	—	2	1	no	—
334	C₂×Q₈	—	2	1	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	C₄×C₅×D₃	1	2	yes	—
337	C₂×D₄	—	3	2	no	—
338	C₂×D₄	—	6	1	no	—
339	C₂×D₄	—	6	1	no	—
340	C₂×D₄	—	6	1	no	—
341	C₂×D₄	—	6	1	no	—
342	C₂×D₄	—	6	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	3	2	no	—
346	C₂²×C₄	—	6	1	no	—
347	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	3	2	no	—
351	C₂×C₈	—	6	1	no	—
352	C₂×C₈	—	6	1	no	—
353	C₂×C₈	C₅×#67	1	2	yes	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	6	1	no	—
357	C₄⋊C₄	—	6	1	no	—
358	C₄⋊C₄	—	12	4	no	—
359	C₂²⋊C₄	—	6	8	no	—
360	C₄²	—	3	2	no	—
361	C₄²	—	3	2	no	—
362	C₃×C₅	#1959	1	1	yes	characteristic, complemented, Hall
363	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
364	C₂²×C₃	C₅×#124	1	1	yes	characteristic
365	C₂²×C₃	C₅×#127	1	1	yes	characteristic
366	C₂²×C₃	C₅×AGL(1,ℤ/8)	1	2	yes	—
367	C₂²×C₃	C₅×#137	1	2	yes	—
368	C₂×D₃	—	4	2	no	—
369	C₂×D₃	—	4	4	no	—
370	C₂×D₃	—	8	2	no	—
371	C₂×D₃	—	8	1	no	—
372	C₂×D₃	—	8	1	no	—
373	C₃×C₄	—	4	2	no	—
374	C₃×C₄	—	8	1	no	—
375	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
376	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
377	C₃×C₄	C₅×#121	1	1	yes	characteristic
378	C₃×C₄	C₅×#121	1	1	yes	characteristic
379	Dic₃	—	2	4	no	—
380	Dic₃	—	8	1	no	—
381	C₂×C₅	—	24	2	no	—
382	C₂×C₅	#4794	1	2	yes	—
383	C₂×C₅	#4803	1	2	yes	—
384	C₂×C₅	#4957	1	1	yes	characteristic
385	C₂×C₅	#5176	1	1	yes	characteristic
386	C₂×C₅	#5190	1	1	yes	characteristic
387	C₂³	—	6	2	no	—
388	C₂³	—	12	1	no	—
389	C₂³	—	12	2	no	—
390	C₂³	—	12	2	no	—
391	C₂³	—	12	1	no	—
392	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
393	Q₈	—	2	2	no	—
394	Q₈	—	2	2	no	—
395	Q₈	—	4	1	no	—
396	Q₈	—	4	1	no	—
397	D₄	—	6	2	no	—
398	D₄	—	6	2	no	—
399	D₄	—	12	1	no	—
400	D₄	—	12	1	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	3	4	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	6	2	no	—
408	C₂×C₄	—	6	2	no	—
409	C₂×C₄	—	12	2	no	—
410	C₂×C₄	—	12	1	no	—
411	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
412	C₂×C₄	C₅×#235	1	1	yes	characteristic
413	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
414	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
415	C₂×C₄	C₅×#237	1	1	yes	characteristic
416	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
417	C₈	—	2	2	no	—
418	C₈	—	6	2	no	—
419	C₂×C₃	C₅×#390	1	1	yes	characteristic
420	C₂×C₃	C₅×#417	1	2	yes	—
421	C₂×C₃	C₅×#489	1	2	yes	—
422	C₂×C₃	C₅×#496	1	1	yes	characteristic
423	C₂×C₃	C₅×#497	1	1	yes	characteristic
424	D₃	—	8	2	no	—
425	C₅	#70227	1	1	yes	characteristic, direct product factor, Sylow
426	C₂²	—	12	2	no	—
427	C₂²	—	12	4	no	—
428	C₂²	—	24	1	no	—
429	C₂²	—	24	1	no	—
430	C₂²	—	24	2	no	—
431	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
432	C₂²	C₅×#885	1	1	yes	characteristic
433	C₂²	C₅×#914	1	2	yes	—
434	C₂²	C₅×#918	1	2	yes	—
435	C₂²	C₅×#940	1	1	yes	characteristic
436	C₄	—	4	2	no	—
437	C₄	—	6	4	no	—
438	C₄	—	8	1	no	—
439	C₄	—	24	1	no	—
440	C₄	C₅×#67⋊C₄	1	1	yes	characteristic
441	C₄	C₅×#884	1	1	yes	characteristic
442	C₄	C₄×C₅×#67	1	1	yes	characteristic
443	C₄	C₅×#939	1	1	yes	characteristic
444	C₃	C₅×#1959	1	1	yes	characteristic, complemented, Sylow
445	C₂	—	24	2	no	—
446	C₂	C₅×#4794	1	2	yes	—
447	C₂	C₅×#4803	1	2	yes	—
448	C₂	C₅×#4957	1	1	yes	characteristic
449	C₂	C₅×#5176	1	1	yes	characteristic
450	C₂	C₅×#5190	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#103	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂²×C₅×D₁₂	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#103	D₃
hypocenter	C₃	C₅×#1959
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#103

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#103	C₂	semidirect
3. C₂×C₅×#835	C₂	semidirect
5. C₂×C₅×#815	C₂	semidirect
6. C₅×#4975	C₂	semidirect
14. C₅×#947	C₄	semidirect
25. #70227	C₅	direct (N × H)
28. C₂×C₅×#103	D₃	semidirect
66. C₂×C₃×#103	C₂×C₅	semidirect
67. C₂×#835	C₂×C₅	semidirect
69. C₂×#815	C₂×C₅	semidirect
70. #4975	C₂×C₅	semidirect
126. #947	C₄×C₅	semidirect
171. C₂×#103	C₅×D₃	semidirect
362. C₃×C₅	#1959	semidirect
425. C₅	#70227	direct (N × H)
444. C₃	C₅×#1959	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#947	2
C₂×C₃×C₅×#103	2
C₂×C₅×#835	2
C₂×C₅×#821	2
C₂×C₅×#815	2
C₅×#4975	2
C₂×C₅×#809	2
C₅×#1959	3
#70227	5

Subgroups — C5×#70227 #14305875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70227 #14305875