Subgroups — C₅×#78925 #14412626

Order 1920.

1556 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#1004	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#907	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5523	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#418	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#832	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#5133	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#5002	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4772	C₂	1	2	yes	maximal, complemented
9	C₅×#4771	C₂	1	2	yes	maximal, complemented
10	C₅×#4770	C₂	1	2	yes	maximal, complemented
11	C₅×D₃×#102	C₂	1	2	yes	maximal, complemented
12	C₅×#2897	—	3	1	no	maximal, Hall
13	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
14	C₂²×C₅×#67	—	2	2	no	—
15	C₅×#1004	—	2	4	no	—
16	C₅×#1004	C₄	1	8	yes	complemented
17	C₂×C₅×#237	—	2	1	no	—
18	C₂×C₅×#237	C₂²	1	2	yes	—
19	C₂×C₅×D₃×D₄	—	2	1	no	—
20	C₂×C₅×D₃×D₄	C₂²	1	2	yes	—
21	C₂×C₅×#235	C₂²	1	1	yes	characteristic
22	C₂×C₅×#235	C₂²	1	1	yes	characteristic
23	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic, complemented
24	C₃×C₅×#117	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
26	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic, complemented
27	C₅×#907	—	2	4	no	—
28	C₂×C₅×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	complemented
29	C₂×C₅×D₃×C₈	C₂²	1	2	yes	complemented
30	C₅×#892	C₂²	1	2	yes	complemented
31	C₅×D₃×C₄⋊C₄	C₂²	1	2	yes	complemented
32	C₅×#67⋊C₄	—	2	2	no	—
33	C₄×C₅×D₁₂	—	2	1	no	—
34	C₄×C₅×Dic₆	—	2	1	no	—
35	C₃×C₅×#102	C₂²	1	2	yes	complemented
36	C₃×C₅×#102	C₂²	1	2	yes	complemented
37	C₅×#832	C₂²	1	2	yes	complemented
38	C₅×#832	C₂²	1	2	yes	complemented
39	C₅×#821	C₂²	1	2	yes	complemented
40	C₅×#816	C₂²	1	2	yes	complemented
41	C₅×#810	C₂²	1	1	yes	characteristic, complemented
42	C₅×#810	C₂²	1	1	yes	characteristic, complemented
43	C₅×#809	C₂²	1	1	yes	characteristic, complemented
44	C₅×#809	C₂²	1	1	yes	characteristic, complemented
45	#78925	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
46	C₂×C₅×H(5,𝔽₂)	—	3	1	no	—
47	C₂×C₅×#131	—	3	1	no	—
48	C₅×#520	—	3	1	no	—
49	C₅×#418	—	3	2	no	—
50	C₅×#418	D₃	1	1	yes	characteristic, complemented
51	C₅×#417	—	3	2	no	—
52	C₅×#417	—	3	1	no	—
53	C₅×#416	—	3	2	no	—
54	C₅×#416	—	3	1	no	—
55	C₂×C₅×#102	—	3	2	no	—
56	C₂×C₅×#102	—	3	1	no	—
57	C₂⁴×C₃×C₅	—	2	1	no	—
58	C₂³×C₅×D₃	—	2	2	no	—
59	C₂×C₃×C₅×D₄	—	2	1	no	—
60	C₂×C₃×C₅×D₄	—	2	2	no	—
61	C₂×C₃×C₅×D₄	C₂×C₄	1	4	yes	complemented
62	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
63	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
64	C₂×C₅×#67	—	2	1	no	—
65	C₂×C₅×#67	—	2	2	no	—
66	C₂×C₅×#67	—	2	16	no	—
67	C₂×C₅×#67	—	2	2	no	—
68	C₂×C₅×#67	D₄	1	2	yes	—
69	C₂²×C₅×Dic₃	—	2	2	no	—
70	C₂²×C₅×Dic₃	—	2	1	no	—
71	C₅×#237	—	2	4	no	—
72	C₅×#237	—	2	8	no	—
73	C₅×#237	C₂×C₄	1	8	yes	complemented
74	C₅×D₃×D₄	—	2	8	no	—
75	C₅×D₃×D₄	—	2	4	no	—
76	C₅×D₃×D₄	C₂×C₄	1	8	yes	complemented
77	C₅×#235	—	2	2	no	—
78	C₅×#235	C₂×C₄	1	4	yes	—
79	C₅×#235	C₂×C₄	1	4	yes	complemented
80	C₅×#235	D₄	1	4	yes	—
81	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
82	C₂×C₅×D₁₂	C₂³	1	1	yes	characteristic
83	C₂×C₄×C₅×D₃	—	2	1	no	—
84	C₂×C₄×C₅×D₃	—	2	2	no	—
85	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
86	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
87	C₂×C₅×Dic₆	C₂³	1	1	yes	characteristic
88	C₃×C₅×M₁₆	—	2	2	no	—
89	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
90	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
91	C₃×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
92	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
93	C₃×C₄²×C₅	—	2	1	no	—
94	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
95	C₅×C₄⋊Dic₃	—	2	1	no	—
96	C₅×C₄⋊Dic₃	C₂³	1	2	yes	—
97	C₅×Dic₃⋊C₄	—	2	2	no	—
98	C₄×C₅×Dic₃	—	2	2	no	—
99	C₂×C₅×C₃⋊C₈	—	2	2	no	—
100	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
102	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
103	C₅×D₃×C₈	—	2	4	no	—
104	C₂×#1004	C₂×C₅	1	1	yes	characteristic
105	C₂×#907	C₂×C₅	1	1	yes	characteristic, complemented
106	#5523	C₂×C₅	1	1	yes	characteristic, complemented
107	C₃×#418	C₂×C₅	1	1	yes	characteristic, complemented
108	C₂×#832	C₂×C₅	1	1	yes	characteristic, complemented
109	#5133	C₂×C₅	1	1	yes	characteristic, complemented
110	#5002	C₂×C₅	1	1	yes	characteristic, complemented
111	#4772	C₂×C₅	1	2	yes	complemented
112	#4771	C₂×C₅	1	2	yes	complemented
113	#4770	C₂×C₅	1	2	yes	complemented
114	D₃×#102	C₂×C₅	1	2	yes	complemented
115	C₅×H(5,𝔽₂)	—	3	8	no	—
116	C₅×H(5,𝔽₂)	—	6	4	no	—
117	C₂×C₅×#41	—	3	1	no	—
118	C₂×C₅×#41	—	3	2	no	—
119	C₂×C₅×#41	—	3	1	no	—
120	C₂×C₅×#41	—	6	1	no	—
121	C₂²×C₅×D₄	—	3	2	no	—
122	C₂²×C₅×D₄	—	6	1	no	—
123	C₂²×C₅×D₄	—	6	2	no	—
124	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
125	C₅×#131	—	6	4	no	—
126	C₂×C₅×M₁₆	—	3	2	no	—
127	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic, complemented
128	C₂²×C₅×C₈	—	3	1	no	—
129	C₂²×C₅×C₈	—	3	2	no	—
130	C₄×C₅×Q₈	—	6	1	no	—
131	C₄×C₅×D₄	—	6	1	no	—
132	C₄×C₅×D₄	—	6	2	no	—
133	C₅×#117	—	3	2	no	—
134	C₅×#117	C₂×D₃	1	1	yes	characteristic, complemented
135	C₂×C₅×C₄⋊C₄	—	3	1	no	—
136	C₂×C₅×C₄⋊C₄	—	3	2	no	—
137	C₅×#103	—	3	2	no	—
138	C₅×#103	—	3	1	no	—
139	C₅×#103	—	3	1	no	—
140	C₅×#102	—	3	2	no	—
141	C₅×#102	—	3	2	no	—
142	C₅×#102	—	3	2	no	—
143	C₅×#102	—	3	1	no	—
144	C₅×#102	—	3	1	no	—
145	C₅×#102	C₂×D₃	1	2	yes	complemented
146	C₅×#102	C₂×D₃	1	2	yes	complemented
147	#2897	—	3	1	no	Sylow
148	C₂³×C₃×C₅	—	2	2	no	—
149	C₂³×C₃×C₅	—	2	4	no	—
150	C₂³×C₃×C₅	—	8	1	no	—
151	C₂³×C₃×C₅	—	8	1	no	—
152	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₅×D₃	—	2	1	no	—
154	C₂²×C₅×D₃	—	2	8	no	—
155	C₂²×C₅×D₃	—	8	2	no	—
156	C₂²×C₅×D₃	—	8	2	no	—
157	C₂²×C₅×D₃	C₂×D₄	1	2	yes	—
158	C₃×C₅×D₄	—	2	4	no	—
159	C₃×C₅×D₄	—	2	2	no	—
160	C₃×C₅×D₄	C₂²×C₄	1	4	yes	—
161	C₂×C₃×C₄×C₅	—	2	1	no	—
162	C₂×C₃×C₄×C₅	—	2	1	no	—
163	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
164	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
165	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
166	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
167	C₅×#67	—	2	4	no	—
168	C₅×#67	—	2	8	no	—
169	C₅×#67	—	2	8	no	—
170	C₅×#67	C₂²⋊C₄	1	8	yes	—
171	C₂×C₅×Dic₃	—	2	1	no	—
172	C₂×C₅×Dic₃	—	2	8	no	—
173	C₂×C₅×Dic₃	—	2	2	no	—
174	C₂×C₅×Dic₃	—	8	1	no	—
175	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
176	C₅×D₁₂	—	2	1	no	—
177	C₅×D₁₂	C₂²⋊C₄	1	4	yes	—
178	C₅×D₁₂	C₂²×C₄	1	2	yes	—
179	C₄×C₅×D₃	—	2	4	no	—
180	C₄×C₅×D₃	—	8	2	no	—
181	C₄×C₅×D₃	C₂²×C₄	1	4	yes	—
182	C₄×C₅×D₃	C₂×D₄	1	4	yes	—
183	C₅×Dic₆	—	2	1	no	—
184	C₅×Dic₆	C₂²⋊C₄	1	4	yes	—
185	C₅×Dic₆	C₂²×C₄	1	2	yes	—
186	C₃×C₅×C₈	—	2	2	no	—
187	C₅×C₃⋊C₈	—	2	1	no	—
188	C₅×C₃⋊C₈	—	2	1	no	—
189	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
190	C₂²×#67	—	2	2	no	—
191	#1004	—	2	4	no	—
192	#1004	C₄×C₅	1	8	yes	complemented
193	C₂×#237	—	2	1	no	—
194	C₂×#237	C₂²×C₅	1	2	yes	—
195	C₂×D₃×D₄	—	2	1	no	—
196	C₂×D₃×D₄	C₂²×C₅	1	2	yes	—
197	C₂×#235	C₂²×C₅	1	1	yes	characteristic
198	C₂×#235	C₂²×C₅	1	1	yes	characteristic
199	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
200	C₃×#117	C₂²×C₅	1	1	yes	characteristic, complemented
201	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
202	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic, complemented
203	#907	—	2	4	no	—
204	C₂×ΓL(1,𝔽₂₅)	C₂²×C₅	1	2	yes	complemented
205	C₂×D₃×C₈	C₂²×C₅	1	2	yes	complemented
206	#892	C₂²×C₅	1	2	yes	complemented
207	D₃×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
208	#67⋊C₄	—	2	2	no	—
209	C₄×D₁₂	—	2	1	no	—
210	C₄×Dic₆	—	2	1	no	—
211	C₃×#102	C₂²×C₅	1	2	yes	complemented
212	C₃×#102	C₂²×C₅	1	2	yes	complemented
213	#832	C₂²×C₅	1	2	yes	complemented
214	#832	C₂²×C₅	1	2	yes	complemented
215	#821	C₂²×C₅	1	2	yes	complemented
216	#816	C₂²×C₅	1	2	yes	complemented
217	#810	C₂²×C₅	1	1	yes	characteristic, complemented
218	#810	C₂²×C₅	1	1	yes	characteristic, complemented
219	#809	C₂²×C₅	1	1	yes	characteristic, complemented
220	#809	C₂²×C₅	1	1	yes	characteristic, complemented
221	C₂⁴×C₅	—	2	1	no	—
222	C₂⁴×C₅	—	6	2	no	—
223	C₅×#41	—	3	4	no	—
224	C₅×#41	—	3	4	no	—
225	C₅×#41	—	3	8	no	—
226	C₅×#41	—	3	4	no	—
227	C₅×#41	—	6	4	no	—
228	C₅×#41	—	6	8	no	—
229	C₅×#41	—	6	2	no	—
230	C₂×C₅×Q₈	—	3	1	no	—
231	C₂×C₅×Q₈	—	3	1	no	—
232	C₂×C₅×D₄	—	2	1	no	—
233	C₂×C₅×D₄	—	2	2	no	—
234	C₂×C₅×D₄	—	3	2	no	—
235	C₂×C₅×D₄	—	3	8	no	—
236	C₂×C₅×D₄	—	3	1	no	—
237	C₂×C₅×D₄	—	3	1	no	—
238	C₂×C₅×D₄	—	6	4	no	—
239	C₂×C₅×D₄	—	6	2	no	—
240	C₂×C₅×D₄	—	6	8	no	—
241	C₂×C₅×D₄	—	6	16	no	—
242	C₂×C₅×D₄	—	6	2	no	—
243	C₂×C₅×D₄	—	6	1	no	—
244	C₂×C₅×D₄	C₄×D₃	1	4	yes	complemented
245	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
246	C₂²×C₄×C₅	—	3	2	no	—
247	C₂²×C₄×C₅	—	6	2	no	—
248	C₂²×C₄×C₅	—	6	2	no	—
249	C₂²×C₄×C₅	—	6	1	no	—
250	C₂²×C₄×C₅	—	6	1	no	—
251	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
252	C₅×M₁₆	—	2	2	no	—
253	C₅×M₁₆	—	6	4	no	—
254	C₂×C₅×C₈	—	3	1	no	—
255	C₂×C₅×C₈	—	3	1	no	—
256	C₂×C₅×C₈	—	6	4	no	—
257	C₂×C₅×C₈	—	6	2	no	—
258	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
259	C₅×C₄⋊C₄	—	3	2	no	—
260	C₅×C₄⋊C₄	—	6	2	no	—
261	C₅×C₄⋊C₄	—	6	1	no	—
262	C₅×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
263	C₅×C₄⋊C₄	C₂²×D₃	1	1	yes	characteristic
264	C₅×C₂²⋊C₄	—	2	1	no	—
265	C₅×C₂²⋊C₄	—	6	2	no	—
266	C₄²×C₅	—	2	1	no	—
267	C₄²×C₅	—	6	2	no	—
268	C₂×H(5,𝔽₂)	—	3	1	no	—
269	C₂×#131	—	3	1	no	—
270	#520	—	3	1	no	—
271	#418	—	3	2	no	—
272	#418	C₅×D₃	1	1	yes	characteristic, complemented
273	#417	—	3	1	no	—
274	#417	—	3	2	no	—
275	#416	—	3	1	no	—
276	#416	—	3	2	no	—
277	C₂×#102	—	3	1	no	—
278	C₂×#102	—	3	2	no	—
279	C₂²×C₃×C₅	—	2	1	no	—
280	C₂²×C₃×C₅	—	2	1	no	—
281	C₂²×C₃×C₅	—	2	4	no	—
282	C₂²×C₃×C₅	—	4	2	no	—
283	C₂²×C₃×C₅	—	4	2	no	—
284	C₂²×C₃×C₅	—	8	4	no	—
285	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	2	yes	—
286	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₅×D₃	—	2	2	no	—
288	C₂×C₅×D₃	—	2	2	no	—
289	C₂×C₅×D₃	—	2	2	no	—
290	C₂×C₅×D₃	—	4	1	no	—
291	C₂×C₅×D₃	—	4	1	no	—
292	C₂×C₅×D₃	—	8	8	no	—
293	C₂×C₅×D₃	C₂×C₂²⋊C₄	1	4	yes	—
294	C₃×C₄×C₅	—	4	1	no	—
295	C₃×C₄×C₅	—	4	1	no	—
296	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
297	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
298	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
299	C₅×Dic₃	—	2	2	no	—
300	C₅×Dic₃	—	4	2	no	—
301	C₅×Dic₃	C₂×C₂²⋊C₄	1	4	yes	—
302	C₂⁴×C₃	—	2	1	no	—
303	C₂³×D₃	—	2	2	no	—
304	C₂×C₃×D₄	—	2	1	no	—
305	C₂×C₃×D₄	—	2	2	no	—
306	C₂×C₃×D₄	C₂×C₄×C₅	1	4	yes	complemented
307	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
308	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
309	C₂×#67	—	2	1	no	—
310	C₂×#67	—	2	2	no	—
311	C₂×#67	—	2	16	no	—
312	C₂×#67	—	2	2	no	—
313	C₂×#67	C₅×D₄	1	2	yes	—
314	C₂²×Dic₃	—	2	2	no	—
315	C₂²×Dic₃	—	2	1	no	—
316	#237	—	2	4	no	—
317	#237	—	2	8	no	—
318	#237	C₂×C₄×C₅	1	8	yes	complemented
319	D₃×D₄	—	2	8	no	—
320	D₃×D₄	—	2	4	no	—
321	D₃×D₄	C₂×C₄×C₅	1	8	yes	complemented
322	#235	—	2	2	no	—
323	#235	C₂×C₄×C₅	1	4	yes	—
324	#235	C₂×C₄×C₅	1	4	yes	complemented
325	#235	C₅×D₄	1	4	yes	—
326	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
327	C₂×D₁₂	C₂³×C₅	1	1	yes	characteristic
328	C₂×C₄×D₃	—	2	1	no	—
329	C₂×C₄×D₃	—	2	2	no	—
330	C₂×C₄×D₃	C₂³×C₅	1	2	yes	—
331	C₂×Dic₆	C₅×D₄	1	1	yes	characteristic
332	C₂×Dic₆	C₂³×C₅	1	1	yes	characteristic
333	C₃×M₁₆	—	2	2	no	—
334	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
335	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
336	C₃×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
337	C₃×C₂²⋊C₄	—	2	1	no	—
338	C₃×C₄²	—	2	1	no	—
339	(C₂×D₃)⋊C₄	—	2	2	no	—
340	C₄⋊Dic₃	—	2	1	no	—
341	C₄⋊Dic₃	C₂³×C₅	1	2	yes	—
342	Dic₃⋊C₄	—	2	2	no	—
343	C₄×Dic₃	—	2	2	no	—
344	C₂×C₃⋊C₈	—	2	2	no	—
345	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
346	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
347	ΓL(1,𝔽₂₅)	—	2	4	no	—
348	D₃×C₈	—	2	4	no	—
349	C₂³×C₅	—	2	2	no	—
350	C₂³×C₅	—	2	4	no	—
351	C₂³×C₅	—	3	2	no	—
352	C₂³×C₅	—	6	8	no	—
353	C₂³×C₅	—	6	1	no	—
354	C₂³×C₅	—	8	1	no	—
355	C₂³×C₅	—	8	1	no	—
356	C₂³×C₅	—	24	2	no	—
357	C₂³×C₅	—	24	2	no	—
358	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
359	C₅×Q₈	—	3	4	no	—
360	C₅×Q₈	—	3	2	no	—
361	C₅×Q₈	—	6	1	no	—
362	C₅×D₄	—	2	4	no	—
363	C₅×D₄	—	2	2	no	—
364	C₅×D₄	—	3	8	no	—
365	C₅×D₄	—	3	4	no	—
366	C₅×D₄	—	3	2	no	—
367	C₅×D₄	—	6	8	no	—
368	C₅×D₄	—	6	8	no	—
369	C₅×D₄	—	6	4	no	—
370	C₅×D₄	—	6	1	no	—
371	C₅×D₄	C₂×C₄×D₃	1	4	yes	—
372	C₂×C₄×C₅	—	2	1	no	—
373	C₂×C₄×C₅	—	2	1	no	—
374	C₂×C₄×C₅	—	3	2	no	—
375	C₂×C₄×C₅	—	3	4	no	—
376	C₂×C₄×C₅	—	3	4	no	—
377	C₂×C₄×C₅	—	6	8	no	—
378	C₂×C₄×C₅	—	6	1	no	—
379	C₂×C₄×C₅	—	6	4	no	—
380	C₂×C₄×C₅	—	6	2	no	—
381	C₂×C₄×C₅	—	24	2	no	—
382	C₂×C₄×C₅	—	24	1	no	—
383	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
384	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
385	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
386	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
387	C₅×C₈	—	2	2	no	—
388	C₅×C₈	—	6	1	no	—
389	C₅×C₈	—	6	1	no	—
390	H(5,𝔽₂)	—	3	8	no	—
391	H(5,𝔽₂)	—	6	4	no	—
392	C₂×#41	—	3	1	no	—
393	C₂×#41	—	3	2	no	—
394	C₂×#41	—	3	1	no	—
395	C₂×#41	—	6	1	no	—
396	C₂²×D₄	—	3	2	no	—
397	C₂²×D₄	—	6	1	no	—
398	C₂²×D₄	—	6	2	no	—
399	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
400	#131	—	6	4	no	—
401	C₂×M₁₆	—	3	2	no	—
402	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic, complemented
403	C₂²×C₈	—	3	1	no	—
404	C₂²×C₈	—	3	2	no	—
405	C₄×Q₈	—	6	1	no	—
406	C₄×D₄	—	6	1	no	—
407	C₄×D₄	—	6	2	no	—
408	#117	—	3	2	no	—
409	#117	C₂×C₅×D₃	1	1	yes	characteristic, complemented
410	C₂×C₄⋊C₄	—	3	1	no	—
411	C₂×C₄⋊C₄	—	3	2	no	—
412	#103	—	3	2	no	—
413	#103	—	3	1	no	—
414	#103	—	3	1	no	—
415	#102	—	3	2	no	—
416	#102	—	3	2	no	—
417	#102	—	3	2	no	—
418	#102	—	3	1	no	—
419	#102	—	3	1	no	—
420	#102	C₂×C₅×D₃	1	2	yes	complemented
421	#102	C₂×C₅×D₃	1	2	yes	complemented
422	C₂×C₃×C₅	—	2	2	no	—
423	C₂×C₃×C₅	—	4	4	no	—
424	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
425	C₂×C₃×C₅	#576	1	1	yes	characteristic
426	C₂×C₃×C₅	#577	1	1	yes	characteristic
427	C₅×D₃	—	2	4	no	—
428	C₅×D₃	—	4	2	no	—
429	C₂³×C₃	—	2	2	no	—
430	C₂³×C₃	—	2	4	no	—
431	C₂³×C₃	—	8	1	no	—
432	C₂³×C₃	—	8	1	no	—
433	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
434	C₂²×D₃	—	2	1	no	—
435	C₂²×D₃	—	2	8	no	—
436	C₂²×D₃	—	8	2	no	—
437	C₂²×D₃	—	8	2	no	—
438	C₂²×D₃	C₂×C₅×D₄	1	2	yes	—
439	C₃×D₄	—	2	4	no	—
440	C₃×D₄	—	2	2	no	—
441	C₃×D₄	C₂²×C₄×C₅	1	4	yes	—
442	C₂×C₃×C₄	—	2	1	no	—
443	C₂×C₃×C₄	—	2	1	no	—
444	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
445	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
446	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
447	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
448	#67	—	2	4	no	—
449	#67	—	2	8	no	—
450	#67	—	2	8	no	—
451	#67	C₅×C₂²⋊C₄	1	8	yes	—
452	C₂×Dic₃	—	2	1	no	—
453	C₂×Dic₃	—	2	8	no	—
454	C₂×Dic₃	—	2	2	no	—
455	C₂×Dic₃	—	8	1	no	—
456	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
457	D₁₂	—	2	1	no	—
458	D₁₂	C₅×C₂²⋊C₄	1	4	yes	—
459	D₁₂	C₂²×C₄×C₅	1	2	yes	—
460	C₄×D₃	—	2	4	no	—
461	C₄×D₃	—	8	2	no	—
462	C₄×D₃	C₂²×C₄×C₅	1	4	yes	—
463	C₄×D₃	C₂×C₅×D₄	1	4	yes	—
464	Dic₆	—	2	1	no	—
465	Dic₆	C₅×C₂²⋊C₄	1	4	yes	—
466	Dic₆	C₂²×C₄×C₅	1	2	yes	—
467	C₃×C₈	—	2	2	no	—
468	C₃⋊C₈	—	2	1	no	—
469	C₃⋊C₈	—	2	1	no	—
470	C₂²×C₅	—	2	1	no	—
471	C₂²×C₅	—	2	1	no	—
472	C₂²×C₅	—	2	4	no	—
473	C₂²×C₅	—	3	4	no	—
474	C₂²×C₅	—	4	2	no	—
475	C₂²×C₅	—	4	2	no	—
476	C₂²×C₅	—	6	2	no	—
477	C₂²×C₅	—	6	2	no	—
478	C₂²×C₅	—	6	2	no	—
479	C₂²×C₅	—	8	4	no	—
480	C₂²×C₅	—	12	1	no	—
481	C₂²×C₅	—	12	1	no	—
482	C₂²×C₅	—	24	8	no	—
483	C₂²×C₅	D₃×C₂²⋊C₄	1	2	yes	—
484	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
485	C₄×C₅	—	3	4	no	—
486	C₄×C₅	—	4	1	no	—
487	C₄×C₅	—	4	1	no	—
488	C₄×C₅	—	6	2	no	—
489	C₄×C₅	—	12	2	no	—
490	C₄×C₅	D₃×C₂²⋊C₄	1	2	yes	—
491	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
492	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
493	C₂⁴	—	2	1	no	—
494	C₂⁴	—	6	2	no	—
495	#41	—	3	4	no	—
496	#41	—	3	4	no	—
497	#41	—	3	8	no	—
498	#41	—	3	4	no	—
499	#41	—	6	4	no	—
500	#41	—	6	8	no	—
501	#41	—	6	2	no	—
502	C₂×Q₈	—	3	1	no	—
503	C₂×Q₈	—	3	1	no	—
504	C₂×D₄	—	2	1	no	—
505	C₂×D₄	—	2	2	no	—
506	C₂×D₄	—	3	2	no	—
507	C₂×D₄	—	3	8	no	—
508	C₂×D₄	—	3	1	no	—
509	C₂×D₄	—	3	1	no	—
510	C₂×D₄	—	6	4	no	—
511	C₂×D₄	—	6	2	no	—
512	C₂×D₄	—	6	8	no	—
513	C₂×D₄	—	6	16	no	—
514	C₂×D₄	—	6	2	no	—
515	C₂×D₄	—	6	1	no	—
516	C₂×D₄	C₄×C₅×D₃	1	4	yes	complemented
517	C₂×D₄	C₂²×C₅×D₃	1	2	yes	—
518	C₂²×C₄	—	3	2	no	—
519	C₂²×C₄	—	6	2	no	—
520	C₂²×C₄	—	6	2	no	—
521	C₂²×C₄	—	6	1	no	—
522	C₂²×C₄	—	6	1	no	—
523	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
524	M₁₆	—	2	2	no	—
525	M₁₆	—	6	4	no	—
526	C₂×C₈	—	3	1	no	—
527	C₂×C₈	—	3	1	no	—
528	C₂×C₈	—	6	4	no	—
529	C₂×C₈	—	6	2	no	—
530	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
531	C₄⋊C₄	—	3	2	no	—
532	C₄⋊C₄	—	6	2	no	—
533	C₄⋊C₄	—	6	1	no	—
534	C₄⋊C₄	C₂²×C₅×D₃	1	1	yes	characteristic
535	C₄⋊C₄	C₂²×C₅×D₃	1	1	yes	characteristic
536	C₂²⋊C₄	—	2	1	no	—
537	C₂²⋊C₄	—	6	2	no	—
538	C₄²	—	2	1	no	—
539	C₄²	—	6	2	no	—
540	C₃×C₅	#2897	1	1	yes	characteristic, complemented, Hall
541	C₂²×C₃	—	2	1	no	—
542	C₂²×C₃	—	2	1	no	—
543	C₂²×C₃	—	2	4	no	—
544	C₂²×C₃	—	4	2	no	—
545	C₂²×C₃	—	4	2	no	—
546	C₂²×C₃	—	8	4	no	—
547	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	2	yes	—
548	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
549	C₂×D₃	—	2	2	no	—
550	C₂×D₃	—	2	2	no	—
551	C₂×D₃	—	2	2	no	—
552	C₂×D₃	—	4	1	no	—
553	C₂×D₃	—	4	1	no	—
554	C₂×D₃	—	8	8	no	—
555	C₂×D₃	C₂×C₅×C₂²⋊C₄	1	4	yes	—
556	C₃×C₄	—	4	1	no	—
557	C₃×C₄	—	4	1	no	—
558	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
559	C₃×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
560	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
561	Dic₃	—	2	2	no	—
562	Dic₃	—	4	2	no	—
563	Dic₃	C₂×C₅×C₂²⋊C₄	1	4	yes	—
564	C₂×C₅	—	2	2	no	—
565	C₂×C₅	—	4	4	no	—
566	C₂×C₅	—	6	4	no	—
567	C₂×C₅	—	12	2	no	—
568	C₂×C₅	C₂×D₃×C₂²⋊C₄	1	1	yes	characteristic
569	C₂×C₅	#5774	1	1	yes	characteristic
570	C₂×C₅	#5775	1	1	yes	characteristic
571	C₂³	—	2	2	no	—
572	C₂³	—	2	4	no	—
573	C₂³	—	3	2	no	—
574	C₂³	—	6	8	no	—
575	C₂³	—	6	1	no	—
576	C₂³	—	8	1	no	—
577	C₂³	—	8	1	no	—
578	C₂³	—	24	2	no	—
579	C₂³	—	24	2	no	—
580	C₂³	C₅×D₃×D₄	1	1	yes	characteristic
581	Q₈	—	3	4	no	—
582	Q₈	—	3	2	no	—
583	Q₈	—	6	1	no	—
584	D₄	—	2	4	no	—
585	D₄	—	2	2	no	—
586	D₄	—	3	8	no	—
587	D₄	—	3	4	no	—
588	D₄	—	3	2	no	—
589	D₄	—	6	8	no	—
590	D₄	—	6	8	no	—
591	D₄	—	6	4	no	—
592	D₄	—	6	1	no	—
593	D₄	C₂×C₄×C₅×D₃	1	4	yes	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	3	2	no	—
597	C₂×C₄	—	3	4	no	—
598	C₂×C₄	—	3	4	no	—
599	C₂×C₄	—	6	8	no	—
600	C₂×C₄	—	6	1	no	—
601	C₂×C₄	—	6	4	no	—
602	C₂×C₄	—	6	2	no	—
603	C₂×C₄	—	24	2	no	—
604	C₂×C₄	—	24	1	no	—
605	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
606	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
607	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
608	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
609	C₈	—	2	2	no	—
610	C₈	—	6	1	no	—
611	C₈	—	6	1	no	—
612	C₂×C₃	—	2	2	no	—
613	C₂×C₃	—	4	4	no	—
614	C₂×C₃	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
615	C₂×C₃	C₅×#576	1	1	yes	characteristic
616	C₂×C₃	C₅×#577	1	1	yes	characteristic
617	D₃	—	2	4	no	—
618	D₃	—	4	2	no	—
619	C₅	#78925	1	1	yes	characteristic, direct product factor, Sylow
620	C₂²	—	2	1	no	—
621	C₂²	—	2	1	no	—
622	C₂²	—	2	4	no	—
623	C₂²	—	3	4	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	4	2	no	—
626	C₂²	—	6	2	no	—
627	C₂²	—	6	2	no	—
628	C₂²	—	6	2	no	—
629	C₂²	—	8	4	no	—
630	C₂²	—	12	1	no	—
631	C₂²	—	12	1	no	—
632	C₂²	—	24	8	no	—
633	C₂²	C₅×D₃×C₂²⋊C₄	1	2	yes	—
634	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
635	C₄	—	3	4	no	—
636	C₄	—	4	1	no	—
637	C₄	—	4	1	no	—
638	C₄	—	6	2	no	—
639	C₄	—	12	2	no	—
640	C₄	C₅×D₃×C₂²⋊C₄	1	2	yes	—
641	C₄	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
642	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
643	C₃	C₅×#2897	1	1	yes	characteristic, complemented, Sylow
644	C₂	—	2	2	no	—
645	C₂	—	4	4	no	—
646	C₂	—	6	4	no	—
647	C₂	—	12	2	no	—
648	C₂	C₂×C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
649	C₂	C₅×#5774	1	1	yes	characteristic
650	C₂	C₅×#5775	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₃×D₄
commutator	C₃×C₄	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#418	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂×C₅×#1004	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#418	D₃
hypocenter	C₃	C₅×#2897
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#418

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#907	C₂	semidirect
3. C₅×#5523	C₂	semidirect
4. C₃×C₅×#418	C₂	semidirect
5. C₂×C₅×#832	C₂	semidirect
6. C₅×#5133	C₂	semidirect
7. C₅×#5002	C₂	semidirect
8. C₅×#4772	C₂	semidirect
9. C₅×#4771	C₂	semidirect
10. C₅×#4770	C₂	semidirect
11. C₅×D₃×#102	C₂	semidirect
16. C₅×#1004	C₄	semidirect
23. C₂×C₃×C₅×M₁₆	C₂²	semidirect
24. C₃×C₅×#117	C₂²	semidirect
25. C₂×C₅×C₄⋊Dic₃	C₂²	semidirect
26. C₂²×C₅×C₃⋊C₈	C₂²	semidirect
28. C₂×C₅×ΓL(1,𝔽₂₅)	C₂²	semidirect
29. C₂×C₅×D₃×C₈	C₂²	semidirect
30. C₅×#892	C₂²	semidirect
31. C₅×D₃×C₄⋊C₄	C₂²	semidirect
35. C₃×C₅×#102	C₂²	semidirect
36. C₃×C₅×#102	C₂²	semidirect
37. C₅×#832	C₂²	semidirect
38. C₅×#832	C₂²	semidirect
39. C₅×#821	C₂²	semidirect
40. C₅×#816	C₂²	semidirect
41. C₅×#810	C₂²	semidirect
42. C₅×#810	C₂²	semidirect
43. C₅×#809	C₂²	semidirect
44. C₅×#809	C₂²	semidirect
45. #78925	C₅	direct (N × H)
50. C₅×#418	D₃	semidirect
61. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
73. C₅×#237	C₂×C₄	semidirect
76. C₅×D₃×D₄	C₂×C₄	semidirect
79. C₅×#235	C₂×C₄	semidirect
105. C₂×#907	C₂×C₅	semidirect
106. #5523	C₂×C₅	semidirect
107. C₃×#418	C₂×C₅	semidirect
108. C₂×#832	C₂×C₅	semidirect
109. #5133	C₂×C₅	semidirect
110. #5002	C₂×C₅	semidirect
111. #4772	C₂×C₅	semidirect
112. #4771	C₂×C₅	semidirect
113. #4770	C₂×C₅	semidirect
114. D₃×#102	C₂×C₅	semidirect
127. C₂×C₅×M₁₆	C₂×D₃	semidirect
134. C₅×#117	C₂×D₃	semidirect
145. C₅×#102	C₂×D₃	semidirect
146. C₅×#102	C₂×D₃	semidirect
192. #1004	C₄×C₅	semidirect
199. C₂×C₃×M₁₆	C₂²×C₅	semidirect
200. C₃×#117	C₂²×C₅	semidirect
201. C₂×C₄⋊Dic₃	C₂²×C₅	semidirect
202. C₂²×C₃⋊C₈	C₂²×C₅	semidirect
204. C₂×ΓL(1,𝔽₂₅)	C₂²×C₅	semidirect
205. C₂×D₃×C₈	C₂²×C₅	semidirect
206. #892	C₂²×C₅	semidirect
207. D₃×C₄⋊C₄	C₂²×C₅	semidirect
211. C₃×#102	C₂²×C₅	semidirect
212. C₃×#102	C₂²×C₅	semidirect
213. #832	C₂²×C₅	semidirect
214. #832	C₂²×C₅	semidirect
215. #821	C₂²×C₅	semidirect
216. #816	C₂²×C₅	semidirect
217. #810	C₂²×C₅	semidirect
218. #810	C₂²×C₅	semidirect
219. #809	C₂²×C₅	semidirect
220. #809	C₂²×C₅	semidirect
244. C₂×C₅×D₄	C₄×D₃	semidirect
272. #418	C₅×D₃	semidirect
306. C₂×C₃×D₄	C₂×C₄×C₅	semidirect
318. #237	C₂×C₄×C₅	semidirect
321. D₃×D₄	C₂×C₄×C₅	semidirect
324. #235	C₂×C₄×C₅	semidirect
402. C₂×M₁₆	C₂×C₅×D₃	semidirect
409. #117	C₂×C₅×D₃	semidirect
420. #102	C₂×C₅×D₃	semidirect
421. #102	C₂×C₅×D₃	semidirect
516. C₂×D₄	C₄×C₅×D₃	semidirect
540. C₃×C₅	#2897	semidirect
619. C₅	#78925	direct (N × H)
643. C₃	C₅×#2897	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#1004	2
C₂×C₅×#907	2
C₅×#5523	2
C₃×C₅×#418	2
C₂×C₅×#832	2
C₅×#5133	2
C₅×#5002	2
C₅×#4772	2
C₅×#4771	2
C₅×#4770	2
C₅×D₃×#102	2
C₅×#2897	3
#78925	5

Subgroups — C5×#78925 #14412626

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78925 #14412626