Subgroups — C₅×#78714 #14412158

Order 1920.

1512 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#1009	C₂	1	1	yes	maximal, characteristic
2	C₅×#5745	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#413	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5237	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#823	C₂	1	2	yes	maximal, complemented
6	C₅×#4751	C₂	1	4	yes	maximal, complemented
7	C₅×D₃×#100	C₂	1	4	yes	maximal, complemented
8	C₅×#2888	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
10	C₅×#1009	—	2	4	no	—
11	C₅×#1009	—	2	4	no	—
12	C₂×C₅×#239	C₂²	1	2	yes	—
13	C₂×C₅×D₃×D₄	C₄	1	4	yes	—
14	C₂×C₅×D₃×D₄	C₂²	1	2	yes	—
15	C₂×C₅×#235	C₄	1	2	yes	—
16	C₂×C₅×#235	C₂²	1	1	yes	characteristic
17	C₂²×C₅×D₁₂	C₄	1	2	yes	—
18	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	complemented
20	C₂×C₅×#208	C₂²	1	2	yes	complemented
21	C₅×#907	—	2	4	no	—
22	C₅×D₃×M₁₆	C₂²	1	8	yes	complemented
23	C₅×#901	—	2	4	no	—
24	C₃×C₅×#101	C₂²	1	2	yes	complemented
25	C₃×C₅×#100	C₂²	1	2	yes	complemented
26	C₅×#836	C₂²	1	2	yes	complemented
27	C₅×#833	C₂²	1	2	yes	complemented
28	C₅×#823	C₂²	1	8	yes	complemented
29	#78714	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₂×C₅×H(5,𝔽₂)	—	3	1	no	—
31	C₅×#568	—	3	2	no	—
32	C₅×#413	—	3	4	no	—
33	C₅×#413	—	3	1	no	—
34	C₅×#413	D₃	1	1	yes	characteristic, complemented
35	C₂×C₅×#100	—	3	4	no	—
36	C₂×C₅×#100	—	3	2	no	—
37	C₂³×C₅×D₃	C₂×C₄	1	4	yes	—
38	C₂³×C₅×D₃	C₂×C₄	1	2	yes	—
39	C₃×C₅×#41	—	2	2	no	—
40	C₃×C₅×#41	—	2	2	no	—
41	C₂×C₃×C₅×Q₈	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
43	C₂×C₃×C₅×D₄	C₂³	1	1	yes	characteristic
44	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
45	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₂×C₅×#67	C₂×C₄	1	4	yes	—
47	C₂×C₅×#67	C₂×C₄	1	2	yes	—
48	C₅×#239	—	2	8	no	—
49	C₅×D₃×D₄	—	2	16	no	—
50	C₅×D₃×D₄	—	2	8	no	—
51	C₅×#235	—	2	4	no	—
52	C₅×#235	—	2	4	no	—
53	C₅×#235	D₄	1	4	yes	—
54	C₅×#235	D₄	1	4	yes	—
55	C₂×C₅×D₁₂	—	2	2	no	—
56	C₂×C₅×D₁₂	—	2	2	no	—
57	C₂×C₅×D₁₂	—	2	4	no	—
58	C₂×C₅×D₁₂	—	2	4	no	—
59	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
60	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
61	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
62	C₂×C₅×D₁₂	C₂³	1	2	yes	—
63	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	—
64	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
65	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
66	C₃×C₅×M₁₆	—	2	2	no	—
67	C₃×C₅×M₁₆	C₂³	1	4	yes	—
68	C₂×C₃×C₅×C₈	—	2	2	no	—
69	C₅×#208	—	2	2	no	—
70	C₅×#208	C₂³	1	4	yes	—
71	C₂×C₅×C₃⋊C₈	—	2	2	no	—
72	C₅×ΓL(1,𝔽₂₅)	—	2	8	no	—
73	C₅×D₃×C₈	—	2	8	no	—
74	C₂×#1009	C₂×C₅	1	1	yes	characteristic
75	#5745	C₂×C₅	1	2	yes	complemented
76	C₃×#413	C₂×C₅	1	1	yes	characteristic, complemented
77	#5237	C₂×C₅	1	1	yes	characteristic, complemented
78	C₂×#823	C₂×C₅	1	2	yes	complemented
79	#4751	C₂×C₅	1	4	yes	complemented
80	D₃×#100	C₂×C₅	1	4	yes	complemented
81	C₅×H(5,𝔽₂)	—	6	4	no	—
82	C₅×H(5,𝔽₂)	—	6	4	no	—
83	C₂×C₅×#41	—	3	2	no	—
84	C₂×C₅×#41	—	3	2	no	—
85	C₂×C₅×#41	—	3	1	no	—
86	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
87	C₂²×C₅×D₄	—	3	4	no	—
88	C₂²×C₅×D₄	—	3	2	no	—
89	C₂²×C₅×D₄	—	3	2	no	—
90	C₂²×C₅×D₄	—	3	1	no	—
91	C₅×#131	—	6	4	no	—
92	C₅×#131	—	6	4	no	—
93	C₂×C₅×M₁₆	—	3	2	no	—
94	C₂×C₅×M₁₆	—	3	8	no	—
95	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	complemented
96	C₅×#101	—	3	2	no	—
97	C₅×#101	C₂×D₃	1	2	yes	complemented
98	C₅×#100	—	3	2	no	—
99	C₅×#100	—	3	8	no	—
100	C₅×#100	C₂×D₃	1	2	yes	complemented
101	#2888	—	3	1	no	Sylow
102	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
103	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₅×D₃	—	2	4	no	—
105	C₂²×C₅×D₃	—	2	8	no	—
106	C₂²×C₅×D₃	—	8	4	no	—
107	C₂²×C₅×D₃	—	8	2	no	—
108	C₂²×C₅×D₃	C₂²×C₄	1	4	yes	—
109	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
110	C₃×C₅×Q₈	—	2	2	no	—
111	C₃×C₅×D₄	—	2	4	no	—
112	C₃×C₅×D₄	—	2	2	no	—
113	C₂×C₃×C₄×C₅	—	2	2	no	—
114	C₂×C₃×C₄×C₅	—	2	2	no	—
115	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
116	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
118	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
119	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
120	C₅×#67	—	2	8	no	—
121	C₅×#67	C₂²⋊C₄	1	8	yes	—
122	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
123	C₅×D₁₂	—	2	8	no	—
124	C₅×D₁₂	—	2	2	no	—
125	C₅×D₁₂	—	2	2	no	—
126	C₅×D₁₂	—	2	4	no	—
127	C₅×D₁₂	C₂²⋊C₄	1	2	yes	—
128	C₅×D₁₂	C₂²⋊C₄	1	2	yes	—
129	C₄×C₅×D₃	—	2	8	no	—
130	C₄×C₅×D₃	C₂×D₄	1	8	yes	—
131	C₅×Dic₆	C₂²⋊C₄	1	2	yes	—
132	C₅×Dic₆	C₂²⋊C₄	1	2	yes	—
133	C₃×C₅×C₈	—	2	4	no	—
134	C₅×C₃⋊C₈	—	2	4	no	—
135	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
136	#1009	—	2	4	no	—
137	#1009	—	2	4	no	—
138	C₂×#239	C₂²×C₅	1	2	yes	—
139	C₂×D₃×D₄	C₄×C₅	1	4	yes	—
140	C₂×D₃×D₄	C₂²×C₅	1	2	yes	—
141	C₂×#235	C₄×C₅	1	2	yes	—
142	C₂×#235	C₂²×C₅	1	1	yes	characteristic
143	C₂²×D₁₂	C₄×C₅	1	2	yes	—
144	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
145	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	complemented
146	C₂×#208	C₂²×C₅	1	2	yes	complemented
147	#907	—	2	4	no	—
148	D₃×M₁₆	C₂²×C₅	1	8	yes	complemented
149	#901	—	2	4	no	—
150	C₃×#101	C₂²×C₅	1	2	yes	complemented
151	C₃×#100	C₂²×C₅	1	2	yes	complemented
152	#836	C₂²×C₅	1	2	yes	complemented
153	#833	C₂²×C₅	1	2	yes	complemented
154	#823	C₂²×C₅	1	8	yes	complemented
155	C₂⁴×C₅	—	3	4	no	—
156	C₂⁴×C₅	—	3	2	no	—
157	C₅×#41	—	2	2	no	—
158	C₅×#41	—	2	2	no	—
159	C₅×#41	—	3	4	no	—
160	C₅×#41	—	3	4	no	—
161	C₅×#41	—	6	4	no	—
162	C₅×#41	—	6	4	no	—
163	C₅×#41	—	6	8	no	—
164	C₂×C₅×Q₈	—	3	1	no	—
165	C₂×C₅×Q₈	C₂²×D₃	1	1	yes	characteristic
166	C₂×C₅×D₄	—	3	2	no	—
167	C₂×C₅×D₄	—	3	4	no	—
168	C₂×C₅×D₄	—	3	1	no	—
169	C₂×C₅×D₄	—	3	4	no	—
170	C₂×C₅×D₄	—	3	2	no	—
171	C₂×C₅×D₄	—	3	2	no	—
172	C₂×C₅×D₄	—	6	4	no	—
173	C₂×C₅×D₄	—	6	4	no	—
174	C₂×C₅×D₄	—	6	8	no	—
175	C₂×C₅×D₄	—	6	16	no	—
176	C₂×C₅×D₄	—	6	2	no	—
177	C₂×C₅×D₄	—	6	2	no	—
178	C₂×C₅×D₄	C₄×D₃	1	2	yes	—
179	C₂×C₅×D₄	C₂²×D₃	1	1	yes	characteristic
180	C₂²×C₄×C₅	—	3	2	no	—
181	C₂²×C₄×C₅	—	3	4	no	—
182	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
183	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
184	C₅×M₁₆	—	2	2	no	—
185	C₅×M₁₆	—	3	4	no	—
186	C₅×M₁₆	—	6	8	no	—
187	C₅×M₁₆	—	6	2	no	—
188	C₅×M₁₆	C₂²×D₃	1	4	yes	—
189	C₂×C₅×C₈	—	2	2	no	—
190	C₂×C₅×C₈	—	6	8	no	—
191	C₂×C₅×C₈	—	6	2	no	—
192	C₂×H(5,𝔽₂)	—	3	1	no	—
193	#568	—	3	2	no	—
194	#413	—	3	1	no	—
195	#413	—	3	4	no	—
196	#413	C₅×D₃	1	1	yes	characteristic, complemented
197	C₂×#100	—	3	2	no	—
198	C₂×#100	—	3	4	no	—
199	C₂²×C₃×C₅	—	2	2	no	—
200	C₂²×C₃×C₅	—	4	1	no	—
201	C₂²×C₃×C₅	—	4	2	no	—
202	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	2	yes	—
203	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
204	C₂×C₅×D₃	—	2	4	no	—
205	C₂×C₅×D₃	—	4	2	no	—
206	C₂×C₅×D₃	—	4	4	no	—
207	C₂×C₅×D₃	—	8	8	no	—
208	C₂×C₅×D₃	—	8	4	no	—
209	C₂×C₅×D₃	C₂×C₂²⋊C₄	1	4	yes	—
210	C₃×C₄×C₅	—	2	2	no	—
211	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
212	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
213	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
214	C₅×Dic₃	C₂×C₂²⋊C₄	1	4	yes	—
215	C₂³×D₃	C₂×C₄×C₅	1	4	yes	—
216	C₂³×D₃	C₂×C₄×C₅	1	2	yes	—
217	C₃×#41	—	2	2	no	—
218	C₃×#41	—	2	2	no	—
219	C₂×C₃×Q₈	C₂³×C₅	1	1	yes	characteristic
220	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
221	C₂×C₃×D₄	C₂³×C₅	1	1	yes	characteristic
222	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
223	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
224	C₂×#67	C₂×C₄×C₅	1	4	yes	—
225	C₂×#67	C₂×C₄×C₅	1	2	yes	—
226	#239	—	2	8	no	—
227	D₃×D₄	—	2	16	no	—
228	D₃×D₄	—	2	8	no	—
229	#235	—	2	4	no	—
230	#235	—	2	4	no	—
231	#235	C₅×D₄	1	4	yes	—
232	#235	C₅×D₄	1	4	yes	—
233	C₂×D₁₂	—	2	2	no	—
234	C₂×D₁₂	—	2	2	no	—
235	C₂×D₁₂	—	2	4	no	—
236	C₂×D₁₂	—	2	4	no	—
237	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	—
238	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	—
239	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
240	C₂×D₁₂	C₂³×C₅	1	2	yes	—
241	C₂×C₄×D₃	C₂×C₄×C₅	1	4	yes	—
242	C₂×C₄×D₃	C₂³×C₅	1	2	yes	—
243	C₂×Dic₆	C₂×C₄×C₅	1	1	yes	characteristic
244	C₃×M₁₆	—	2	2	no	—
245	C₃×M₁₆	C₂³×C₅	1	4	yes	—
246	C₂×C₃×C₈	—	2	2	no	—
247	#208	—	2	2	no	—
248	#208	C₂³×C₅	1	4	yes	—
249	C₂×C₃⋊C₈	—	2	2	no	—
250	ΓL(1,𝔽₂₅)	—	2	8	no	—
251	D₃×C₈	—	2	8	no	—
252	C₂³×C₅	—	3	2	no	—
253	C₂³×C₅	—	3	4	no	—
254	C₂³×C₅	—	6	8	no	—
255	C₂³×C₅	—	6	4	no	—
256	C₂³×C₅	—	24	2	no	—
257	C₂³×C₅	—	24	4	no	—
258	C₂³×C₅	C₂×C₄×D₃	1	2	yes	—
259	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
260	C₅×Q₈	—	2	2	no	—
261	C₅×Q₈	—	3	2	no	—
262	C₅×Q₈	—	3	2	no	—
263	C₅×D₄	—	2	4	no	—
264	C₅×D₄	—	2	2	no	—
265	C₅×D₄	—	3	8	no	—
266	C₅×D₄	—	3	2	no	—
267	C₅×D₄	—	3	2	no	—
268	C₅×D₄	—	6	8	no	—
269	C₅×D₄	—	6	8	no	—
270	C₅×D₄	—	6	2	no	—
271	C₅×D₄	—	6	2	no	—
272	C₅×D₄	—	6	4	no	—
273	C₂×C₄×C₅	—	2	2	no	—
274	C₂×C₄×C₅	—	2	2	no	—
275	C₂×C₄×C₅	—	3	2	no	—
276	C₂×C₄×C₅	—	3	8	no	—
277	C₂×C₄×C₅	—	6	8	no	—
278	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
279	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
280	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
281	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
282	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
283	C₅×C₈	—	2	4	no	—
284	C₅×C₈	—	6	4	no	—
285	H(5,𝔽₂)	—	6	4	no	—
286	H(5,𝔽₂)	—	6	4	no	—
287	C₂×#41	—	3	2	no	—
288	C₂×#41	—	3	2	no	—
289	C₂×#41	—	3	1	no	—
290	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
291	C₂²×D₄	—	3	4	no	—
292	C₂²×D₄	—	3	2	no	—
293	C₂²×D₄	—	3	2	no	—
294	C₂²×D₄	—	3	1	no	—
295	#131	—	6	4	no	—
296	#131	—	6	4	no	—
297	C₂×M₁₆	—	3	2	no	—
298	C₂×M₁₆	—	3	8	no	—
299	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	complemented
300	#101	—	3	2	no	—
301	#101	C₂×C₅×D₃	1	2	yes	complemented
302	#100	—	3	2	no	—
303	#100	—	3	8	no	—
304	#100	C₂×C₅×D₃	1	2	yes	complemented
305	C₂×C₃×C₅	—	2	1	no	—
306	C₂×C₃×C₅	—	2	2	no	—
307	C₂×C₃×C₅	—	4	2	no	—
308	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
309	C₅×D₃	—	2	4	no	—
310	C₅×D₃	—	4	4	no	—
311	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
312	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
313	C₂²×D₃	—	2	4	no	—
314	C₂²×D₃	—	2	8	no	—
315	C₂²×D₃	—	8	4	no	—
316	C₂²×D₃	—	8	2	no	—
317	C₂²×D₃	C₂²×C₄×C₅	1	4	yes	—
318	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
319	C₃×Q₈	—	2	2	no	—
320	C₃×D₄	—	2	4	no	—
321	C₃×D₄	—	2	2	no	—
322	C₂×C₃×C₄	—	2	2	no	—
323	C₂×C₃×C₄	—	2	2	no	—
324	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
325	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
326	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
327	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
328	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
329	#67	—	2	8	no	—
330	#67	C₅×C₂²⋊C₄	1	8	yes	—
331	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
332	D₁₂	—	2	8	no	—
333	D₁₂	—	2	2	no	—
334	D₁₂	—	2	2	no	—
335	D₁₂	—	2	4	no	—
336	D₁₂	C₅×C₂²⋊C₄	1	2	yes	—
337	D₁₂	C₅×C₂²⋊C₄	1	2	yes	—
338	C₄×D₃	—	2	8	no	—
339	C₄×D₃	C₂×C₅×D₄	1	8	yes	—
340	Dic₆	C₅×C₂²⋊C₄	1	2	yes	—
341	Dic₆	C₅×C₂²⋊C₄	1	2	yes	—
342	C₃×C₈	—	2	4	no	—
343	C₃⋊C₈	—	2	4	no	—
344	C₂²×C₅	—	2	2	no	—
345	C₂²×C₅	—	3	4	no	—
346	C₂²×C₅	—	4	1	no	—
347	C₂²×C₅	—	4	2	no	—
348	C₂²×C₅	—	6	4	no	—
349	C₂²×C₅	—	12	2	no	—
350	C₂²×C₅	—	12	4	no	—
351	C₂²×C₅	—	24	4	no	—
352	C₂²×C₅	—	24	8	no	—
353	C₂²×C₅	D₃×C₂²⋊C₄	1	2	yes	—
354	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
355	C₄×C₅	—	2	2	no	—
356	C₄×C₅	—	3	4	no	—
357	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
358	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
359	C₄×C₅	C₂×D₃×D₄	1	2	yes	—
360	C₂⁴	—	3	4	no	—
361	C₂⁴	—	3	2	no	—
362	#41	—	2	2	no	—
363	#41	—	2	2	no	—
364	#41	—	3	4	no	—
365	#41	—	3	4	no	—
366	#41	—	6	4	no	—
367	#41	—	6	4	no	—
368	#41	—	6	8	no	—
369	C₂×Q₈	—	3	1	no	—
370	C₂×Q₈	C₂²×C₅×D₃	1	1	yes	characteristic
371	C₂×D₄	—	3	2	no	—
372	C₂×D₄	—	3	4	no	—
373	C₂×D₄	—	3	1	no	—
374	C₂×D₄	—	3	4	no	—
375	C₂×D₄	—	3	2	no	—
376	C₂×D₄	—	3	2	no	—
377	C₂×D₄	—	6	4	no	—
378	C₂×D₄	—	6	4	no	—
379	C₂×D₄	—	6	8	no	—
380	C₂×D₄	—	6	16	no	—
381	C₂×D₄	—	6	2	no	—
382	C₂×D₄	—	6	2	no	—
383	C₂×D₄	C₄×C₅×D₃	1	2	yes	—
384	C₂×D₄	C₂²×C₅×D₃	1	1	yes	characteristic
385	C₂²×C₄	—	3	2	no	—
386	C₂²×C₄	—	3	4	no	—
387	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
388	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
389	M₁₆	—	2	2	no	—
390	M₁₆	—	3	4	no	—
391	M₁₆	—	6	8	no	—
392	M₁₆	—	6	2	no	—
393	M₁₆	C₂²×C₅×D₃	1	4	yes	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	6	8	no	—
396	C₂×C₈	—	6	2	no	—
397	C₃×C₅	#2888	1	1	yes	characteristic, complemented, Hall
398	C₂²×C₃	—	2	2	no	—
399	C₂²×C₃	—	4	1	no	—
400	C₂²×C₃	—	4	2	no	—
401	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	2	yes	—
402	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
403	C₂×D₃	—	2	4	no	—
404	C₂×D₃	—	4	2	no	—
405	C₂×D₃	—	4	4	no	—
406	C₂×D₃	—	8	8	no	—
407	C₂×D₃	—	8	4	no	—
408	C₂×D₃	C₂×C₅×C₂²⋊C₄	1	4	yes	—
409	C₃×C₄	—	2	2	no	—
410	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
411	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
412	C₃×C₄	C₂²×C₅×D₄	1	2	yes	—
413	Dic₃	C₂×C₅×C₂²⋊C₄	1	4	yes	—
414	C₂×C₅	—	2	1	no	—
415	C₂×C₅	—	2	2	no	—
416	C₂×C₅	—	4	2	no	—
417	C₂×C₅	—	6	4	no	—
418	C₂×C₅	—	12	4	no	—
419	C₂×C₅	C₂×D₃×C₂²⋊C₄	1	1	yes	characteristic
420	C₂³	—	3	2	no	—
421	C₂³	—	3	4	no	—
422	C₂³	—	6	8	no	—
423	C₂³	—	6	4	no	—
424	C₂³	—	24	2	no	—
425	C₂³	—	24	4	no	—
426	C₂³	C₂×C₄×C₅×D₃	1	2	yes	—
427	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
428	Q₈	—	2	2	no	—
429	Q₈	—	3	2	no	—
430	Q₈	—	3	2	no	—
431	D₄	—	2	4	no	—
432	D₄	—	2	2	no	—
433	D₄	—	3	8	no	—
434	D₄	—	3	2	no	—
435	D₄	—	3	2	no	—
436	D₄	—	6	8	no	—
437	D₄	—	6	8	no	—
438	D₄	—	6	2	no	—
439	D₄	—	6	2	no	—
440	D₄	—	6	4	no	—
441	C₂×C₄	—	2	2	no	—
442	C₂×C₄	—	2	2	no	—
443	C₂×C₄	—	3	2	no	—
444	C₂×C₄	—	3	8	no	—
445	C₂×C₄	—	6	8	no	—
446	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
447	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
448	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
449	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
450	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
451	C₈	—	2	4	no	—
452	C₈	—	6	4	no	—
453	C₂×C₃	—	2	1	no	—
454	C₂×C₃	—	2	2	no	—
455	C₂×C₃	—	4	2	no	—
456	C₂×C₃	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
457	D₃	—	2	4	no	—
458	D₃	—	4	4	no	—
459	C₅	#78714	1	1	yes	characteristic, direct product factor, Sylow
460	C₂²	—	2	2	no	—
461	C₂²	—	3	4	no	—
462	C₂²	—	4	1	no	—
463	C₂²	—	4	2	no	—
464	C₂²	—	6	4	no	—
465	C₂²	—	12	2	no	—
466	C₂²	—	12	4	no	—
467	C₂²	—	24	4	no	—
468	C₂²	—	24	8	no	—
469	C₂²	C₅×D₃×C₂²⋊C₄	1	2	yes	—
470	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
471	C₄	—	2	2	no	—
472	C₄	—	3	4	no	—
473	C₄	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
474	C₄	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
475	C₄	C₂×C₅×D₃×D₄	1	2	yes	—
476	C₃	C₅×#2888	1	1	yes	characteristic, complemented, Sylow
477	C₂	—	2	1	no	—
478	C₂	—	2	2	no	—
479	C₂	—	4	2	no	—
480	C₂	—	6	4	no	—
481	C₂	—	12	4	no	—
482	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	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#413	C₂
socle	C₂×C₃×C₅	C₂²×C₂²⋊C₄
prime radical	C₂×C₅×#1009	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#413	D₃
hypocenter	C₃	C₅×#2888
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#413

Complements

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

N	H ≅ G/N	type
2. C₅×#5745	C₂	semidirect
3. C₃×C₅×#413	C₂	semidirect
4. C₅×#5237	C₂	semidirect
5. C₂×C₅×#823	C₂	semidirect
6. C₅×#4751	C₂	semidirect
7. C₅×D₃×#100	C₂	semidirect
19. C₂×C₃×C₅×M₁₆	C₂²	semidirect
20. C₂×C₅×#208	C₂²	semidirect
22. C₅×D₃×M₁₆	C₂²	semidirect
24. C₃×C₅×#101	C₂²	semidirect
25. C₃×C₅×#100	C₂²	semidirect
26. C₅×#836	C₂²	semidirect
27. C₅×#833	C₂²	semidirect
28. C₅×#823	C₂²	semidirect
29. #78714	C₅	direct (N × H)
34. C₅×#413	D₃	semidirect
75. #5745	C₂×C₅	semidirect
76. C₃×#413	C₂×C₅	semidirect
77. #5237	C₂×C₅	semidirect
78. C₂×#823	C₂×C₅	semidirect
79. #4751	C₂×C₅	semidirect
80. D₃×#100	C₂×C₅	semidirect
95. C₂×C₅×M₁₆	C₂×D₃	semidirect
97. C₅×#101	C₂×D₃	semidirect
100. C₅×#100	C₂×D₃	semidirect
145. C₂×C₃×M₁₆	C₂²×C₅	semidirect
146. C₂×#208	C₂²×C₅	semidirect
148. D₃×M₁₆	C₂²×C₅	semidirect
150. C₃×#101	C₂²×C₅	semidirect
151. C₃×#100	C₂²×C₅	semidirect
152. #836	C₂²×C₅	semidirect
153. #833	C₂²×C₅	semidirect
154. #823	C₂²×C₅	semidirect
196. #413	C₅×D₃	semidirect
299. C₂×M₁₆	C₂×C₅×D₃	semidirect
301. #101	C₂×C₅×D₃	semidirect
304. #100	C₂×C₅×D₃	semidirect
397. C₃×C₅	#2888	semidirect
459. C₅	#78714	direct (N × H)
476. C₃	C₅×#2888	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#1009	2
C₅×#5745	2
C₃×C₅×#413	2
C₅×#5237	2
C₂×C₅×#823	2
C₅×#4751	2
C₅×D₃×#100	2
C₅×#2888	3
#78714	5

Subgroups — C5×#78714 #14412158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78714 #14412158