Subgroups — C₅×#78689 #14412009

Order 1920.

1432 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#1010	C₂	1	1	yes	maximal, characteristic
2	C₅×#5516	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#410	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5240	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5006	C₂	1	2	yes	maximal, complemented
6	C₅×#4740	C₂	1	8	yes	maximal, complemented
7	C₅×#2884	—	3	1	no	maximal, Hall
8	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
9	C₅×#1010	—	2	4	no	—
10	C₅×#1010	—	2	4	no	—
11	C₂×C₅×D₃×Q₈	C₄	1	2	yes	complemented
12	C₂×C₅×#237	C₄	1	2	yes	complemented
13	C₂×C₅×#237	C₂²	1	4	yes	—
14	C₂×C₅×#235	C₄	1	2	yes	complemented
15	C₂×C₅×#235	C₂²	1	1	yes	characteristic
16	C₂²×C₅×Dic₆	C₄	1	2	yes	complemented
17	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
18	C₃×C₅×#117	C₂²	1	2	yes	—
19	C₅×#926	C₂²	1	2	yes	—
20	C₅×#887	—	2	4	no	—
21	C₅×#877	C₂²	1	8	yes	—
22	C₄×C₅×Dic₆	—	2	4	no	—
23	C₃×C₅×#99	C₂²	1	4	yes	—
24	C₅×#834	C₂²	1	4	yes	—
25	C₅×#806	C₂²	1	8	yes	—
26	#78689	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₂×C₅×2₋¹⁺⁴	—	3	1	no	—
28	C₅×#519	—	3	2	no	—
29	C₅×#410	—	3	8	no	—
30	C₅×#410	—	3	1	no	—
31	C₅×#410	—	3	2	no	—
32	C₅×#410	D₃	1	1	yes	characteristic, complemented
33	C₃×C₅×#41	—	2	2	no	—
34	C₃×C₅×#41	—	2	2	no	—
35	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
38	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
39	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂×C₅×#67	C₂×C₄	1	4	yes	—
41	C₂×C₅×#67	C₂³	1	2	yes	—
42	C₂²×C₅×Dic₃	C₂×C₄	1	4	yes	—
43	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
44	C₅×D₃×Q₈	—	2	8	no	—
45	C₅×#237	—	2	16	no	—
46	C₅×#237	—	2	8	no	—
47	C₅×#235	—	2	2	no	—
48	C₅×#235	—	2	2	no	—
49	C₅×#235	—	2	4	no	—
50	C₅×#235	—	2	4	no	—
51	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	—
53	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
54	C₂×C₅×Dic₆	—	2	4	no	—
55	C₂×C₅×Dic₆	—	2	4	no	—
56	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
57	C₂×C₅×Dic₆	C₂×C₄	1	4	yes	—
58	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
59	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	—
60	C₂×C₅×Dic₆	D₄	1	4	yes	—
61	C₂×C₅×Dic₆	D₄	1	4	yes	—
62	C₃×C₅×C₄⋊C₄	—	2	2	no	—
63	C₃×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
64	C₃×C₄²×C₅	—	2	2	no	—
65	C₅×#217	C₂³	1	4	yes	—
66	C₅×C₄⋊Dic₃	—	2	2	no	—
67	C₅×Dic₃⋊C₄	—	2	8	no	—
68	C₄×C₅×Dic₃	—	2	2	no	—
69	C₄×C₅×Dic₃	—	2	8	no	—
70	C₂×#1010	C₂×C₅	1	1	yes	characteristic
71	#5516	C₂×C₅	1	2	yes	complemented
72	C₃×#410	C₂×C₅	1	1	yes	characteristic, complemented
73	#5240	C₂×C₅	1	1	yes	characteristic, complemented
74	#5006	C₂×C₅	1	2	yes	complemented
75	#4740	C₂×C₅	1	8	yes	complemented
76	C₅×2₋¹⁺⁴	—	6	4	no	—
77	C₅×2₋¹⁺⁴	—	6	4	no	—
78	C₂×C₅×#41	—	3	2	no	—
79	C₂×C₅×#41	—	3	4	no	—
80	C₂×C₅×#41	—	3	2	no	—
81	C₂×C₅×#41	—	3	1	no	—
82	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
83	C₂²×C₅×Q₈	—	3	2	no	—
84	C₂²×C₅×Q₈	—	3	2	no	—
85	C₂²×C₅×Q₈	—	3	1	no	—
86	C₄×C₅×Q₈	—	6	4	no	—
87	C₄×C₅×Q₈	—	6	4	no	—
88	C₅×#117	—	3	2	no	—
89	C₅×#117	—	3	8	no	—
90	C₅×#117	C₂×D₃	1	2	yes	—
91	C₅×#99	—	3	4	no	—
92	C₅×#99	—	3	8	no	—
93	C₅×#99	C₂×D₃	1	4	yes	—
94	#2884	—	3	1	no	Sylow
95	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
96	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
97	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
98	C₃×C₅×Q₈	—	2	2	no	—
99	C₃×C₅×D₄	—	2	4	no	—
100	C₃×C₅×D₄	—	2	2	no	—
101	C₂×C₃×C₄×C₅	—	2	2	no	—
102	C₂×C₃×C₄×C₅	—	2	2	no	—
103	C₂×C₃×C₄×C₅	—	2	4	no	—
104	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
105	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
108	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
109	C₅×#67	—	2	4	no	—
110	C₅×#67	—	2	8	no	—
111	C₂×C₅×Dic₃	—	2	8	no	—
112	C₂×C₅×Dic₃	—	2	4	no	—
113	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
114	C₂×C₅×Dic₃	C₂²×C₄	1	4	yes	—
115	C₂×C₅×Dic₃	C₂×D₄	1	8	yes	—
116	C₅×D₁₂	—	2	1	no	—
117	C₅×D₁₂	—	2	1	no	—
118	C₄×C₅×D₃	—	2	4	no	—
119	C₄×C₅×D₃	—	2	8	no	—
120	C₅×Dic₆	—	2	1	no	—
121	C₅×Dic₆	—	2	1	no	—
122	C₅×Dic₆	—	2	8	no	—
123	C₅×Dic₆	C₂²⋊C₄	1	4	yes	—
124	C₅×Dic₆	C₂²⋊C₄	1	4	yes	—
125	C₅×Dic₆	C₂²⋊C₄	1	8	yes	—
126	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
127	#1010	—	2	4	no	—
128	#1010	—	2	4	no	—
129	C₂×D₃×Q₈	C₄×C₅	1	2	yes	complemented
130	C₂×#237	C₄×C₅	1	2	yes	complemented
131	C₂×#237	C₂²×C₅	1	4	yes	—
132	C₂×#235	C₄×C₅	1	2	yes	complemented
133	C₂×#235	C₂²×C₅	1	1	yes	characteristic
134	C₂²×Dic₆	C₄×C₅	1	2	yes	complemented
135	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic
136	C₃×#117	C₂²×C₅	1	2	yes	—
137	#926	C₂²×C₅	1	2	yes	—
138	#887	—	2	4	no	—
139	#877	C₂²×C₅	1	8	yes	—
140	C₄×Dic₆	—	2	4	no	—
141	C₃×#99	C₂²×C₅	1	4	yes	—
142	#834	C₂²×C₅	1	4	yes	—
143	#806	C₂²×C₅	1	8	yes	—
144	C₅×#41	—	2	2	no	—
145	C₅×#41	—	2	2	no	—
146	C₅×#41	—	6	4	no	—
147	C₅×#41	—	6	4	no	—
148	C₅×#41	—	6	16	no	—
149	C₅×#41	—	6	8	no	—
150	C₅×#41	—	6	2	no	—
151	C₅×#41	—	6	2	no	—
152	C₂×C₅×Q₈	—	3	2	no	—
153	C₂×C₅×Q₈	—	3	2	no	—
154	C₂×C₅×Q₈	—	3	4	no	—
155	C₂×C₅×Q₈	—	3	1	no	—
156	C₂×C₅×Q₈	—	3	4	no	—
157	C₂×C₅×Q₈	—	3	4	no	—
158	C₂×C₅×Q₈	—	6	4	no	—
159	C₂×C₅×Q₈	—	6	4	no	—
160	C₂×C₅×Q₈	—	6	8	no	—
161	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
162	C₂×C₅×D₄	—	3	4	no	—
163	C₂×C₅×D₄	—	3	2	no	—
164	C₂×C₅×D₄	—	3	1	no	—
165	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
166	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
167	C₂²×C₄×C₅	—	3	2	no	—
168	C₂²×C₄×C₅	—	3	4	no	—
169	C₂²×C₄×C₅	—	3	4	no	—
170	C₂²×C₄×C₅	—	3	2	no	—
171	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
172	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
173	C₅×C₄⋊C₄	—	2	2	no	—
174	C₅×C₄⋊C₄	—	6	8	no	—
175	C₅×C₄⋊C₄	—	6	2	no	—
176	C₅×C₂²⋊C₄	—	3	4	no	—
177	C₅×C₂²⋊C₄	C₂²×D₃	1	4	yes	—
178	C₄²×C₅	—	2	2	no	—
179	C₄²×C₅	—	6	8	no	—
180	C₄²×C₅	—	6	2	no	—
181	C₂×2₋¹⁺⁴	—	3	1	no	—
182	#519	—	3	2	no	—
183	#410	—	3	1	no	—
184	#410	—	3	2	no	—
185	#410	—	3	8	no	—
186	#410	C₅×D₃	1	1	yes	characteristic, complemented
187	C₂²×C₃×C₅	—	2	2	no	—
188	C₂²×C₃×C₅	—	4	1	no	—
189	C₂²×C₃×C₅	—	4	2	no	—
190	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
191	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
192	C₂×C₅×D₃	—	2	2	no	—
193	C₂×C₅×D₃	—	4	2	no	—
194	C₃×C₄×C₅	—	2	2	no	—
195	C₃×C₄×C₅	—	4	4	no	—
196	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
197	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
198	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
199	C₅×Dic₃	—	2	2	no	—
200	C₅×Dic₃	—	4	4	no	—
201	C₅×Dic₃	C₂×C₂²⋊C₄	1	8	yes	—
202	C₃×#41	—	2	2	no	—
203	C₃×#41	—	2	2	no	—
204	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
205	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
206	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
207	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
208	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
209	C₂×#67	C₂×C₄×C₅	1	4	yes	—
210	C₂×#67	C₂³×C₅	1	2	yes	—
211	C₂²×Dic₃	C₂×C₄×C₅	1	4	yes	—
212	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
213	D₃×Q₈	—	2	8	no	—
214	#237	—	2	16	no	—
215	#237	—	2	8	no	—
216	#235	—	2	2	no	—
217	#235	—	2	2	no	—
218	#235	—	2	4	no	—
219	#235	—	2	4	no	—
220	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
221	C₂×C₄×D₃	C₂×C₄×C₅	1	4	yes	—
222	C₂×C₄×D₃	C₂×C₄×C₅	1	2	yes	—
223	C₂×Dic₆	—	2	4	no	—
224	C₂×Dic₆	—	2	4	no	—
225	C₂×Dic₆	C₂×C₄×C₅	1	1	yes	characteristic
226	C₂×Dic₆	C₂×C₄×C₅	1	4	yes	—
227	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
228	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	—
229	C₂×Dic₆	C₅×D₄	1	4	yes	—
230	C₂×Dic₆	C₅×D₄	1	4	yes	—
231	C₃×C₄⋊C₄	—	2	2	no	—
232	C₃×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
233	C₃×C₄²	—	2	2	no	—
234	#217	C₂³×C₅	1	4	yes	—
235	C₄⋊Dic₃	—	2	2	no	—
236	Dic₃⋊C₄	—	2	8	no	—
237	C₄×Dic₃	—	2	2	no	—
238	C₄×Dic₃	—	2	8	no	—
239	C₂³×C₅	—	3	2	no	—
240	C₂³×C₅	C₂×C₄×D₃	1	2	yes	—
241	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
242	C₅×Q₈	—	2	2	no	—
243	C₅×Q₈	—	3	4	no	—
244	C₅×Q₈	—	3	4	no	—
245	C₅×Q₈	—	3	8	no	—
246	C₅×Q₈	—	6	8	no	—
247	C₅×Q₈	—	6	1	no	—
248	C₅×Q₈	—	6	1	no	—
249	C₅×D₄	—	2	4	no	—
250	C₅×D₄	—	2	2	no	—
251	C₅×D₄	—	6	8	no	—
252	C₅×D₄	—	6	4	no	—
253	C₅×D₄	—	6	1	no	—
254	C₅×D₄	—	6	1	no	—
255	C₂×C₄×C₅	—	2	2	no	—
256	C₂×C₄×C₅	—	2	2	no	—
257	C₂×C₄×C₅	—	2	4	no	—
258	C₂×C₄×C₅	—	3	4	no	—
259	C₂×C₄×C₅	—	3	8	no	—
260	C₂×C₄×C₅	—	3	2	no	—
261	C₂×C₄×C₅	—	6	8	no	—
262	C₂×C₄×C₅	—	6	8	no	—
263	C₂×C₄×C₅	—	6	4	no	—
264	C₂×C₄×C₅	—	6	4	no	—
265	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
266	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
267	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
268	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
269	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
270	2₋¹⁺⁴	—	6	4	no	—
271	2₋¹⁺⁴	—	6	4	no	—
272	C₂×#41	—	3	2	no	—
273	C₂×#41	—	3	4	no	—
274	C₂×#41	—	3	2	no	—
275	C₂×#41	—	3	1	no	—
276	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
277	C₂²×Q₈	—	3	2	no	—
278	C₂²×Q₈	—	3	2	no	—
279	C₂²×Q₈	—	3	1	no	—
280	C₄×Q₈	—	6	4	no	—
281	C₄×Q₈	—	6	4	no	—
282	#117	—	3	2	no	—
283	#117	—	3	8	no	—
284	#117	C₂×C₅×D₃	1	2	yes	—
285	#99	—	3	4	no	—
286	#99	—	3	8	no	—
287	#99	C₂×C₅×D₃	1	4	yes	—
288	C₂×C₃×C₅	—	2	2	no	—
289	C₂×C₃×C₅	—	2	1	no	—
290	C₂×C₃×C₅	—	4	2	no	—
291	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
292	C₅×D₃	—	4	2	no	—
293	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
294	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
295	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
296	C₃×Q₈	—	2	2	no	—
297	C₃×D₄	—	2	4	no	—
298	C₃×D₄	—	2	2	no	—
299	C₂×C₃×C₄	—	2	2	no	—
300	C₂×C₃×C₄	—	2	2	no	—
301	C₂×C₃×C₄	—	2	4	no	—
302	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
303	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
304	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
305	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
306	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
307	#67	—	2	4	no	—
308	#67	—	2	8	no	—
309	C₂×Dic₃	—	2	8	no	—
310	C₂×Dic₃	—	2	4	no	—
311	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
312	C₂×Dic₃	C₂²×C₄×C₅	1	4	yes	—
313	C₂×Dic₃	C₂×C₅×D₄	1	8	yes	—
314	D₁₂	—	2	1	no	—
315	D₁₂	—	2	1	no	—
316	C₄×D₃	—	2	4	no	—
317	C₄×D₃	—	2	8	no	—
318	Dic₆	—	2	1	no	—
319	Dic₆	—	2	1	no	—
320	Dic₆	—	2	8	no	—
321	Dic₆	C₅×C₂²⋊C₄	1	4	yes	—
322	Dic₆	C₅×C₂²⋊C₄	1	4	yes	—
323	Dic₆	C₅×C₂²⋊C₄	1	8	yes	—
324	C₂²×C₅	—	2	2	no	—
325	C₂²×C₅	—	4	1	no	—
326	C₂²×C₅	—	4	2	no	—
327	C₂²×C₅	—	6	2	no	—
328	C₂²×C₅	—	12	2	no	—
329	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
330	C₂²×C₅	C₂×D₃×D₄	1	2	yes	—
331	C₄×C₅	—	2	2	no	—
332	C₄×C₅	—	3	8	no	—
333	C₄×C₅	—	4	4	no	—
334	C₄×C₅	—	6	2	no	—
335	C₄×C₅	—	12	4	no	—
336	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
337	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
338	C₄×C₅	D₃×C₂²⋊C₄	1	2	yes	—
339	#41	—	2	2	no	—
340	#41	—	2	2	no	—
341	#41	—	6	4	no	—
342	#41	—	6	4	no	—
343	#41	—	6	16	no	—
344	#41	—	6	8	no	—
345	#41	—	6	2	no	—
346	#41	—	6	2	no	—
347	C₂×Q₈	—	3	2	no	—
348	C₂×Q₈	—	3	2	no	—
349	C₂×Q₈	—	3	4	no	—
350	C₂×Q₈	—	3	1	no	—
351	C₂×Q₈	—	3	4	no	—
352	C₂×Q₈	—	3	4	no	—
353	C₂×Q₈	—	6	4	no	—
354	C₂×Q₈	—	6	4	no	—
355	C₂×Q₈	—	6	8	no	—
356	C₂×Q₈	C₄×C₅×D₃	1	1	yes	characteristic
357	C₂×D₄	—	3	4	no	—
358	C₂×D₄	—	3	2	no	—
359	C₂×D₄	—	3	1	no	—
360	C₂×D₄	C₄×C₅×D₃	1	1	yes	characteristic
361	C₂×D₄	C₂²×C₅×D₃	1	2	yes	—
362	C₂²×C₄	—	3	2	no	—
363	C₂²×C₄	—	3	4	no	—
364	C₂²×C₄	—	3	4	no	—
365	C₂²×C₄	—	3	2	no	—
366	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
367	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	6	8	no	—
370	C₄⋊C₄	—	6	2	no	—
371	C₂²⋊C₄	—	3	4	no	—
372	C₂²⋊C₄	C₂²×C₅×D₃	1	4	yes	—
373	C₄²	—	2	2	no	—
374	C₄²	—	6	8	no	—
375	C₄²	—	6	2	no	—
376	C₃×C₅	#2884	1	1	yes	characteristic, complemented, Hall
377	C₂²×C₃	—	2	2	no	—
378	C₂²×C₃	—	4	1	no	—
379	C₂²×C₃	—	4	2	no	—
380	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
381	C₂²×C₃	C₂²×C₅×D₄	1	2	yes	—
382	C₂×D₃	—	2	2	no	—
383	C₂×D₃	—	4	2	no	—
384	C₃×C₄	—	2	2	no	—
385	C₃×C₄	—	4	4	no	—
386	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
387	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
388	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
389	Dic₃	—	2	2	no	—
390	Dic₃	—	4	4	no	—
391	Dic₃	C₂×C₅×C₂²⋊C₄	1	8	yes	—
392	C₂×C₅	—	2	2	no	—
393	C₂×C₅	—	2	1	no	—
394	C₂×C₅	—	4	2	no	—
395	C₂×C₅	—	12	2	no	—
396	C₂×C₅	C₂×D₃×C₂²⋊C₄	1	1	yes	characteristic
397	C₂³	—	3	2	no	—
398	C₂³	C₂×C₄×C₅×D₃	1	2	yes	—
399	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
400	Q₈	—	2	2	no	—
401	Q₈	—	3	4	no	—
402	Q₈	—	3	4	no	—
403	Q₈	—	3	8	no	—
404	Q₈	—	6	8	no	—
405	Q₈	—	6	1	no	—
406	Q₈	—	6	1	no	—
407	D₄	—	2	4	no	—
408	D₄	—	2	2	no	—
409	D₄	—	6	8	no	—
410	D₄	—	6	4	no	—
411	D₄	—	6	1	no	—
412	D₄	—	6	1	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	3	4	no	—
417	C₂×C₄	—	3	8	no	—
418	C₂×C₄	—	3	2	no	—
419	C₂×C₄	—	6	8	no	—
420	C₂×C₄	—	6	8	no	—
421	C₂×C₄	—	6	4	no	—
422	C₂×C₄	—	6	4	no	—
423	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
424	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
425	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
426	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
427	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
428	C₂×C₃	—	2	2	no	—
429	C₂×C₃	—	2	1	no	—
430	C₂×C₃	—	4	2	no	—
431	C₂×C₃	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
432	D₃	—	4	2	no	—
433	C₅	#78689	1	1	yes	characteristic, direct product factor, Sylow
434	C₂²	—	2	2	no	—
435	C₂²	—	4	1	no	—
436	C₂²	—	4	2	no	—
437	C₂²	—	6	2	no	—
438	C₂²	—	12	2	no	—
439	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
440	C₂²	C₂×C₅×D₃×D₄	1	2	yes	—
441	C₄	—	2	2	no	—
442	C₄	—	3	8	no	—
443	C₄	—	4	4	no	—
444	C₄	—	6	2	no	—
445	C₄	—	12	4	no	—
446	C₄	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
447	C₄	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
448	C₄	C₅×D₃×C₂²⋊C₄	1	2	yes	—
449	C₃	C₅×#2884	1	1	yes	characteristic, complemented, Sylow
450	C₂	—	2	2	no	—
451	C₂	—	2	1	no	—
452	C₂	—	4	2	no	—
453	C₂	—	12	2	no	—
454	C₂	C₂×C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂×D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#410	C₂
socle	C₂×C₃×C₅	C₂²×C₂²⋊C₄
prime radical	C₂×C₅×#1010	C₂
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#410	D₃
hypocenter	C₃	C₅×#2884
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#410

Complements

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

N	H ≅ G/N	type
2. C₅×#5516	C₂	semidirect
3. C₃×C₅×#410	C₂	semidirect
4. C₅×#5240	C₂	semidirect
5. C₅×#5006	C₂	semidirect
6. C₅×#4740	C₂	semidirect
11. C₂×C₅×D₃×Q₈	C₄	semidirect
12. C₂×C₅×#237	C₄	semidirect
14. C₂×C₅×#235	C₄	semidirect
16. C₂²×C₅×Dic₆	C₄	semidirect
26. #78689	C₅	direct (N × H)
32. C₅×#410	D₃	semidirect
71. #5516	C₂×C₅	semidirect
72. C₃×#410	C₂×C₅	semidirect
73. #5240	C₂×C₅	semidirect
74. #5006	C₂×C₅	semidirect
75. #4740	C₂×C₅	semidirect
129. C₂×D₃×Q₈	C₄×C₅	semidirect
130. C₂×#237	C₄×C₅	semidirect
132. C₂×#235	C₄×C₅	semidirect
134. C₂²×Dic₆	C₄×C₅	semidirect
186. #410	C₅×D₃	semidirect
376. C₃×C₅	#2884	semidirect
433. C₅	#78689	direct (N × H)
449. C₃	C₅×#2884	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#1010	2
C₅×#5516	2
C₃×C₅×#410	2
C₅×#5240	2
C₅×#5006	2
C₅×#4740	2
C₅×#2884	3
#78689	5

Subgroups — C5×#78689 #14412009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78689 #14412009