Subgroups — C₅×#79081 #14413167

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#907	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#901	C₂	1	1	yes	maximal, characteristic
3	C₅×#5522	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#424	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#5118	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4999	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4839	C₂	1	2	yes	maximal, complemented
9	C₅×#4839	C₂	1	2	yes	maximal, complemented
10	C₅×#4837	C₂	1	2	yes	maximal, complemented
11	C₅×D₃×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
12	C₅×#2906	—	3	1	no	maximal, Hall
13	C₂×C₅×#235	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×#117	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
18	C₂×C₅×#208	C₂²	1	1	yes	characteristic
19	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₅×#907	—	2	4	no	—
21	C₅×#901	—	2	4	no	—
22	C₂×C₅×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
23	C₂×C₅×ΓL(1,𝔽₂₅)	C₂²	1	2	yes	—
24	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
25	C₂×C₅×D₃×C₈	C₂²	1	2	yes	—
26	C₅×#893	C₄	1	1	yes	characteristic
27	C₅×#893	C₄	1	1	yes	characteristic
28	C₅×#887	C₄	1	1	yes	characteristic
29	C₅×#887	C₄	1	1	yes	characteristic
30	C₅×#67⋊C₄	C₄	1	2	yes	—
31	C₅×#67⋊C₄	C₄	1	2	yes	—
32	C₅×#872	C₂²	1	2	yes	—
33	C₄²×C₅×D₃	C₂²	1	2	yes	—
34	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
35	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
36	C₅×Dic₃⋊C₈	C₂²	1	1	yes	characteristic
37	C₅×Dic₃⋊C₈	C₂²	1	2	yes	—
38	C₅×Dic₃⋊C₈	C₂²	1	1	yes	characteristic
39	C₅×Dic₃⋊C₈	C₂²	1	1	yes	characteristic
40	C₅×Dic₃⋊C₈	C₂²	1	1	yes	characteristic
41	C₅×Dic₃⋊C₈	C₂²	1	2	yes	—
42	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
43	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
44	#79081	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
45	C₂×C₅×#131	—	3	1	no	—
46	C₂×C₅×#131	—	3	1	no	—
47	C₄×C₅×#41	—	3	1	no	—
48	C₅×#424	—	3	2	no	—
49	C₅×#424	—	3	2	no	—
50	C₅×#424	—	3	1	no	—
51	C₅×#424	D₃	1	1	yes	characteristic, complemented
52	C₅×#423	—	3	2	no	—
53	C₅×#423	—	3	1	no	—
54	C₂×C₅×C₄⋊C₈	—	3	2	no	—
55	C₂×C₅×C₄⋊C₈	—	3	1	no	—
56	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₅×#67	C₂×C₄	1	2	yes	—
58	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
59	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
60	C₅×#235	D₄	1	4	yes	—
61	C₅×#235	Q₈	1	4	yes	—
62	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
64	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
65	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
66	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
67	C₃×C₅×M₁₆	—	2	2	no	—
68	C₂×C₃×C₅×C₈	—	2	2	no	—
69	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
70	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
71	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
72	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
73	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
74	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
75	C₃×C₄²×C₅	C₂³	1	2	yes	—
76	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
77	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	2	yes	—
78	C₅×Dic₃⋊C₄	C₂×C₄	1	2	yes	—
79	C₅×Dic₃⋊C₄	C₂×C₄	1	2	yes	—
80	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
81	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
82	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
83	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
84	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
85	C₄×C₅×Dic₃	C₂³	1	1	yes	characteristic
86	C₅×#208	—	2	2	no	—
87	C₂×C₅×C₃⋊C₈	—	2	2	no	—
88	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
89	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
90	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
91	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
92	C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
93	C₅×D₃×C₈	—	2	4	no	—
94	C₅×D₃×C₈	—	2	4	no	—
95	C₂×#907	C₂×C₅	1	1	yes	characteristic
96	C₂×#901	C₂×C₅	1	1	yes	characteristic
97	#5522	C₂×C₅	1	1	yes	characteristic
98	C₃×#424	C₂×C₅	1	1	yes	characteristic, complemented
99	#5118	C₂×C₅	1	1	yes	characteristic, complemented
100	C₂×Dic₃⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
101	#4999	C₂×C₅	1	1	yes	characteristic, complemented
102	#4839	C₂×C₅	1	2	yes	complemented
103	#4839	C₂×C₅	1	2	yes	complemented
104	#4837	C₂×C₅	1	2	yes	complemented
105	D₃×C₄⋊C₈	C₂×C₅	1	2	yes	complemented
106	C₂×C₅×#41	—	3	1	no	—
107	C₅×#131	—	6	4	no	—
108	C₅×#131	—	6	4	no	—
109	C₂×C₅×M₁₆	—	3	1	no	—
110	C₂×C₅×M₁₆	—	3	2	no	—
111	C₂×C₅×M₁₆	—	3	2	no	—
112	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
113	C₂²×C₅×C₈	—	3	1	no	—
114	C₂²×C₅×C₈	—	3	2	no	—
115	C₂²×C₅×C₈	—	3	2	no	—
116	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
117	C₄×C₅×Q₈	—	3	1	no	—
118	C₄×C₅×Q₈	—	3	1	no	—
119	C₄×C₅×D₄	—	3	1	no	—
120	C₄×C₅×D₄	—	3	1	no	—
121	C₄×C₅×D₄	—	3	2	no	—
122	C₄×C₅×D₄	—	3	2	no	—
123	C₅×#117	—	3	2	no	—
124	C₅×#117	C₂×D₃	1	1	yes	characteristic
125	C₂×C₄²×C₅	—	3	1	no	—
126	C₂×C₄²×C₅	—	3	2	no	—
127	C₅×C₄⋊C₈	—	3	1	no	—
128	C₅×C₄⋊C₈	—	3	2	no	—
129	C₅×C₄⋊C₈	—	3	1	no	—
130	C₅×C₄⋊C₈	—	3	1	no	—
131	C₅×C₄⋊C₈	—	3	1	no	—
132	C₅×C₄⋊C₈	—	3	2	no	—
133	C₅×C₄⋊C₈	—	3	2	no	—
134	C₅×C₄⋊C₈	—	3	2	no	—
135	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	—
136	C₅×C₄⋊C₈	C₂×D₃	1	2	yes	—
137	#2906	—	3	1	no	Sylow
138	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
139	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
140	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
141	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
142	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
143	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
144	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
145	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
146	C₅×#67	C₄⋊C₄	1	8	yes	—
147	C₂×C₅×Dic₃	—	2	2	no	—
148	C₂×C₅×Dic₃	—	4	1	no	—
149	C₂×C₅×Dic₃	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₅×Dic₃	C₂²×C₄	1	1	yes	characteristic
151	C₂×C₅×Dic₃	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
153	C₂×C₅×Dic₃	C₂²×C₄	1	1	yes	characteristic
154	C₅×D₁₂	C₄⋊C₄	1	4	yes	—
155	C₄×C₅×D₃	—	2	4	no	—
156	C₄×C₅×D₃	—	4	2	no	—
157	C₄×C₅×D₃	C₂×D₄	1	4	yes	—
158	C₄×C₅×D₃	C₂×Q₈	1	4	yes	—
159	C₅×Dic₆	C₄⋊C₄	1	4	yes	—
160	C₃×C₅×C₈	—	2	1	no	—
161	C₃×C₅×C₈	—	2	2	no	—
162	C₃×C₅×C₈	—	2	1	no	—
163	C₅×C₃⋊C₈	—	2	1	no	—
164	C₅×C₃⋊C₈	—	2	2	no	—
165	C₅×C₃⋊C₈	—	2	1	no	—
166	C₂×#235	C₂²×C₅	1	1	yes	characteristic
167	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
168	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
169	C₃×#117	C₂²×C₅	1	1	yes	characteristic
170	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
171	C₂×#208	C₂²×C₅	1	1	yes	characteristic
172	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
173	#907	—	2	4	no	—
174	#901	—	2	4	no	—
175	C₂×ΓL(1,𝔽₂₅)	C₂²×C₅	1	2	yes	—
176	C₂×ΓL(1,𝔽₂₅)	C₂²×C₅	1	2	yes	—
177	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
178	C₂×D₃×C₈	C₂²×C₅	1	2	yes	—
179	#893	C₄×C₅	1	1	yes	characteristic
180	#893	C₄×C₅	1	1	yes	characteristic
181	#887	C₄×C₅	1	1	yes	characteristic
182	#887	C₄×C₅	1	1	yes	characteristic
183	#67⋊C₄	C₄×C₅	1	2	yes	—
184	#67⋊C₄	C₄×C₅	1	2	yes	—
185	#872	C₂²×C₅	1	2	yes	—
186	C₄²×D₃	C₂²×C₅	1	2	yes	—
187	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
188	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
189	Dic₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
190	Dic₃⋊C₈	C₂²×C₅	1	2	yes	—
191	Dic₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
192	Dic₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
193	Dic₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
194	Dic₃⋊C₈	C₂²×C₅	1	2	yes	—
195	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
196	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	2	yes	—
197	C₅×#41	—	3	4	no	—
198	C₅×#41	—	3	4	no	—
199	C₂×C₅×Q₈	—	3	1	no	—
200	C₂×C₅×D₄	—	3	2	no	—
201	C₂×C₅×D₄	—	3	1	no	—
202	C₂²×C₄×C₅	—	3	2	no	—
203	C₂²×C₄×C₅	—	3	1	no	—
204	C₂²×C₄×C₅	—	3	1	no	—
205	C₂²×C₄×C₅	—	3	2	no	—
206	C₂²×C₄×C₅	—	3	2	no	—
207	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
208	C₅×M₁₆	—	2	2	no	—
209	C₅×M₁₆	—	6	4	no	—
210	C₅×M₁₆	—	6	4	no	—
211	C₅×M₁₆	—	6	2	no	—
212	C₂×C₅×C₈	—	2	2	no	—
213	C₂×C₅×C₈	—	3	1	no	—
214	C₂×C₅×C₈	—	3	1	no	—
215	C₂×C₅×C₈	—	3	2	no	—
216	C₂×C₅×C₈	—	6	4	no	—
217	C₂×C₅×C₈	—	6	4	no	—
218	C₂×C₅×C₈	—	6	2	no	—
219	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
220	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
221	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
222	C₅×C₄⋊C₄	—	3	2	no	—
223	C₅×C₄⋊C₄	—	3	2	no	—
224	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
225	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
226	C₅×C₂²⋊C₄	—	3	2	no	—
227	C₅×C₂²⋊C₄	—	3	2	no	—
228	C₅×C₂²⋊C₄	C₄×D₃	1	2	yes	—
229	C₄²×C₅	—	3	2	no	—
230	C₄²×C₅	—	3	1	no	—
231	C₄²×C₅	—	3	2	no	—
232	C₄²×C₅	—	3	1	no	—
233	C₄²×C₅	—	3	1	no	—
234	C₄²×C₅	—	3	1	no	—
235	C₄²×C₅	C₂²×D₃	1	2	yes	—
236	C₂×#131	—	3	1	no	—
237	C₂×#131	—	3	1	no	—
238	C₄×#41	—	3	1	no	—
239	#424	—	3	1	no	—
240	#424	—	3	2	no	—
241	#424	—	3	2	no	—
242	#424	C₅×D₃	1	1	yes	characteristic, complemented
243	#423	—	3	1	no	—
244	#423	—	3	2	no	—
245	C₂×C₄⋊C₈	—	3	1	no	—
246	C₂×C₄⋊C₈	—	3	2	no	—
247	C₂²×C₃×C₅	—	2	1	no	—
248	C₂²×C₃×C₅	—	2	1	no	—
249	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
250	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
251	C₂×C₅×D₃	—	2	2	no	—
252	C₂×C₅×D₃	—	2	2	no	—
253	C₂×C₅×D₃	C₂×C₄⋊C₄	1	4	yes	—
254	C₃×C₄×C₅	—	2	2	no	—
255	C₃×C₄×C₅	—	2	2	no	—
256	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
257	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
258	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
259	C₅×Dic₃	—	2	1	no	—
260	C₅×Dic₃	—	2	1	no	—
261	C₅×Dic₃	—	2	1	no	—
262	C₅×Dic₃	C₂×C₄⋊C₄	1	4	yes	—
263	C₅×Dic₃	#131	1	2	yes	complemented
264	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
265	C₂×#67	C₂×C₄×C₅	1	2	yes	—
266	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
267	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
268	#235	C₅×D₄	1	4	yes	—
269	#235	C₅×Q₈	1	4	yes	—
270	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
271	C₂×C₄×D₃	C₂×C₄×C₅	1	2	yes	—
272	C₂×C₄×D₃	C₂×C₄×C₅	1	2	yes	—
273	C₂×C₄×D₃	C₂³×C₅	1	2	yes	—
274	C₂×Dic₆	C₂×C₄×C₅	1	1	yes	characteristic
275	C₃×M₁₆	—	2	2	no	—
276	C₂×C₃×C₈	—	2	2	no	—
277	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
278	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
279	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
280	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
281	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
282	C₃×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
283	C₃×C₄²	C₂³×C₅	1	2	yes	—
284	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	2	yes	—
285	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	2	yes	—
286	Dic₃⋊C₄	C₂×C₄×C₅	1	2	yes	—
287	Dic₃⋊C₄	C₂×C₄×C₅	1	2	yes	—
288	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
289	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
290	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
291	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
292	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
293	C₄×Dic₃	C₂³×C₅	1	1	yes	characteristic
294	#208	—	2	2	no	—
295	C₂×C₃⋊C₈	—	2	2	no	—
296	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
297	C₂×C₃⋊C₈	C₂³×C₅	1	2	yes	—
298	C₂×C₃⋊C₈	C₂³×C₅	1	1	yes	characteristic
299	ΓL(1,𝔽₂₅)	—	2	4	no	—
300	ΓL(1,𝔽₂₅)	—	2	4	no	—
301	D₃×C₈	—	2	4	no	—
302	D₃×C₈	—	2	4	no	—
303	C₂³×C₅	—	3	2	no	—
304	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
305	C₅×Q₈	—	3	4	no	—
306	C₅×D₄	—	3	8	no	—
307	C₅×D₄	—	3	4	no	—
308	C₂×C₄×C₅	—	3	1	no	—
309	C₂×C₄×C₅	—	3	2	no	—
310	C₂×C₄×C₅	—	3	4	no	—
311	C₂×C₄×C₅	—	3	4	no	—
312	C₂×C₄×C₅	—	3	1	no	—
313	C₂×C₄×C₅	—	3	1	no	—
314	C₂×C₄×C₅	—	3	1	no	—
315	C₂×C₄×C₅	—	6	4	no	—
316	C₂×C₄×C₅	—	6	2	no	—
317	C₂×C₄×C₅	—	12	2	no	—
318	C₂×C₄×C₅	—	12	1	no	—
319	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
320	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
321	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
322	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
323	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
324	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
325	C₅×C₈	—	2	1	no	—
326	C₅×C₈	—	2	1	no	—
327	C₅×C₈	—	2	2	no	—
328	C₅×C₈	—	6	1	no	—
329	C₅×C₈	—	6	2	no	—
330	C₅×C₈	—	6	1	no	—
331	C₂×#41	—	3	1	no	—
332	#131	—	6	4	no	—
333	#131	—	6	4	no	—
334	C₂×M₁₆	—	3	1	no	—
335	C₂×M₁₆	—	3	2	no	—
336	C₂×M₁₆	—	3	2	no	—
337	C₂×M₁₆	C₂×C₅×D₃	1	1	yes	characteristic
338	C₂²×C₈	—	3	1	no	—
339	C₂²×C₈	—	3	2	no	—
340	C₂²×C₈	—	3	2	no	—
341	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
342	C₄×Q₈	—	3	1	no	—
343	C₄×Q₈	—	3	1	no	—
344	C₄×D₄	—	3	1	no	—
345	C₄×D₄	—	3	1	no	—
346	C₄×D₄	—	3	2	no	—
347	C₄×D₄	—	3	2	no	—
348	#117	—	3	2	no	—
349	#117	C₂×C₅×D₃	1	1	yes	characteristic
350	C₂×C₄²	—	3	1	no	—
351	C₂×C₄²	—	3	2	no	—
352	C₄⋊C₈	—	3	1	no	—
353	C₄⋊C₈	—	3	2	no	—
354	C₄⋊C₈	—	3	1	no	—
355	C₄⋊C₈	—	3	1	no	—
356	C₄⋊C₈	—	3	1	no	—
357	C₄⋊C₈	—	3	2	no	—
358	C₄⋊C₈	—	3	2	no	—
359	C₄⋊C₈	—	3	2	no	—
360	C₄⋊C₈	C₂×C₅×D₃	1	2	yes	—
361	C₄⋊C₈	C₂×C₅×D₃	1	2	yes	—
362	C₂×C₃×C₅	—	2	2	no	—
363	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
364	C₂×C₃×C₅	C₂×#131	1	1	yes	characteristic
365	C₂×C₃×C₅	#568	1	1	yes	characteristic
366	C₅×D₃	—	2	4	no	—
367	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
368	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
369	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
370	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
371	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
372	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
373	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
374	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
375	#67	C₅×C₄⋊C₄	1	8	yes	—
376	C₂×Dic₃	—	2	2	no	—
377	C₂×Dic₃	—	4	1	no	—
378	C₂×Dic₃	C₂²×C₄×C₅	1	1	yes	characteristic
379	C₂×Dic₃	C₂²×C₄×C₅	1	1	yes	characteristic
380	C₂×Dic₃	C₂²×C₄×C₅	1	1	yes	characteristic
381	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
382	C₂×Dic₃	C₂²×C₄×C₅	1	1	yes	characteristic
383	D₁₂	C₅×C₄⋊C₄	1	4	yes	—
384	C₄×D₃	—	2	4	no	—
385	C₄×D₃	—	4	2	no	—
386	C₄×D₃	C₂×C₅×D₄	1	4	yes	—
387	C₄×D₃	C₂×C₅×Q₈	1	4	yes	—
388	Dic₆	C₅×C₄⋊C₄	1	4	yes	—
389	C₃×C₈	—	2	1	no	—
390	C₃×C₈	—	2	2	no	—
391	C₃×C₈	—	2	1	no	—
392	C₃⋊C₈	—	2	1	no	—
393	C₃⋊C₈	—	2	2	no	—
394	C₃⋊C₈	—	2	1	no	—
395	C₂²×C₅	—	2	1	no	—
396	C₂²×C₅	—	2	1	no	—
397	C₂²×C₅	—	3	4	no	—
398	C₂²×C₅	—	6	2	no	—
399	C₂²×C₅	—	6	2	no	—
400	C₂²×C₅	D₃×C₄⋊C₄	1	2	yes	—
401	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
402	C₄×C₅	—	2	2	no	—
403	C₄×C₅	—	2	2	no	—
404	C₄×C₅	—	3	2	no	—
405	C₄×C₅	—	3	4	no	—
406	C₄×C₅	—	6	1	no	—
407	C₄×C₅	—	6	1	no	—
408	C₄×C₅	—	6	1	no	—
409	C₄×C₅	D₃×C₄⋊C₄	1	2	yes	—
410	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
411	C₄×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
412	#41	—	3	4	no	—
413	#41	—	3	4	no	—
414	C₂×Q₈	—	3	1	no	—
415	C₂×D₄	—	3	2	no	—
416	C₂×D₄	—	3	1	no	—
417	C₂²×C₄	—	3	2	no	—
418	C₂²×C₄	—	3	1	no	—
419	C₂²×C₄	—	3	1	no	—
420	C₂²×C₄	—	3	2	no	—
421	C₂²×C₄	—	3	2	no	—
422	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
423	M₁₆	—	2	2	no	—
424	M₁₆	—	6	4	no	—
425	M₁₆	—	6	4	no	—
426	M₁₆	—	6	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	3	1	no	—
429	C₂×C₈	—	3	1	no	—
430	C₂×C₈	—	3	2	no	—
431	C₂×C₈	—	6	4	no	—
432	C₂×C₈	—	6	4	no	—
433	C₂×C₈	—	6	2	no	—
434	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
435	C₂×C₈	C₂²×C₅×D₃	1	2	yes	—
436	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
437	C₄⋊C₄	—	3	2	no	—
438	C₄⋊C₄	—	3	2	no	—
439	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
440	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
441	C₂²⋊C₄	—	3	2	no	—
442	C₂²⋊C₄	—	3	2	no	—
443	C₂²⋊C₄	C₄×C₅×D₃	1	2	yes	—
444	C₄²	—	3	2	no	—
445	C₄²	—	3	1	no	—
446	C₄²	—	3	2	no	—
447	C₄²	—	3	1	no	—
448	C₄²	—	3	1	no	—
449	C₄²	—	3	1	no	—
450	C₄²	C₂²×C₅×D₃	1	2	yes	—
451	C₃×C₅	#2906	1	1	yes	characteristic, complemented, Hall
452	C₂²×C₃	—	2	1	no	—
453	C₂²×C₃	—	2	1	no	—
454	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	2	yes	—
455	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
456	C₂×D₃	—	2	2	no	—
457	C₂×D₃	—	2	2	no	—
458	C₂×D₃	C₂×C₅×C₄⋊C₄	1	4	yes	—
459	C₃×C₄	—	2	2	no	—
460	C₃×C₄	—	2	2	no	—
461	C₃×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
462	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
463	C₃×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
464	Dic₃	—	2	1	no	—
465	Dic₃	—	2	1	no	—
466	Dic₃	—	2	1	no	—
467	Dic₃	C₂×C₅×C₄⋊C₄	1	4	yes	—
468	Dic₃	C₅×#131	1	2	yes	complemented
469	C₂×C₅	—	2	2	no	—
470	C₂×C₅	—	6	4	no	—
471	C₂×C₅	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
472	C₂×C₅	D₃×#131	1	1	yes	characteristic
473	C₂×C₅	#5750	1	1	yes	characteristic
474	C₂³	—	3	2	no	—
475	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
476	Q₈	—	3	4	no	—
477	D₄	—	3	8	no	—
478	D₄	—	3	4	no	—
479	C₂×C₄	—	3	1	no	—
480	C₂×C₄	—	3	2	no	—
481	C₂×C₄	—	3	4	no	—
482	C₂×C₄	—	3	4	no	—
483	C₂×C₄	—	3	1	no	—
484	C₂×C₄	—	3	1	no	—
485	C₂×C₄	—	3	1	no	—
486	C₂×C₄	—	6	4	no	—
487	C₂×C₄	—	6	2	no	—
488	C₂×C₄	—	12	2	no	—
489	C₂×C₄	—	12	1	no	—
490	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
491	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
492	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
493	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
494	C₂×C₄	C₅×D₃×Q₈	1	2	yes	—
495	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
496	C₈	—	2	1	no	—
497	C₈	—	2	1	no	—
498	C₈	—	2	2	no	—
499	C₈	—	6	1	no	—
500	C₈	—	6	2	no	—
501	C₈	—	6	1	no	—
502	C₂×C₃	—	2	2	no	—
503	C₂×C₃	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
504	C₂×C₃	C₂×C₅×#131	1	1	yes	characteristic
505	C₂×C₃	C₅×#568	1	1	yes	characteristic
506	D₃	—	2	4	no	—
507	C₅	#79081	1	1	yes	characteristic, direct product factor, Sylow
508	C₂²	—	2	1	no	—
509	C₂²	—	2	1	no	—
510	C₂²	—	3	4	no	—
511	C₂²	—	6	2	no	—
512	C₂²	—	6	2	no	—
513	C₂²	C₅×D₃×C₄⋊C₄	1	2	yes	—
514	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
515	C₄	—	2	2	no	—
516	C₄	—	2	2	no	—
517	C₄	—	3	2	no	—
518	C₄	—	3	4	no	—
519	C₄	—	6	1	no	—
520	C₄	—	6	1	no	—
521	C₄	—	6	1	no	—
522	C₄	C₅×D₃×C₄⋊C₄	1	2	yes	—
523	C₄	C₂×C₅×D₃×D₄	1	1	yes	characteristic
524	C₄	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
525	C₃	C₅×#2906	1	1	yes	characteristic, complemented, Sylow
526	C₂	—	2	2	no	—
527	C₂	—	6	4	no	—
528	C₂	C₂×C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
529	C₂	C₅×D₃×#131	1	1	yes	characteristic
530	C₂	C₅×#5750	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₅×#424	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂×C₅×#235	C₂²
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₅×#424	D₃
hypocenter	C₃	C₅×#2906
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₅×#424

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#424	C₂	semidirect
5. C₅×#5118	C₂	semidirect
6. C₂×C₅×Dic₃⋊C₈	C₂	semidirect
7. C₅×#4999	C₂	semidirect
8. C₅×#4839	C₂	semidirect
9. C₅×#4839	C₂	semidirect
10. C₅×#4837	C₂	semidirect
11. C₅×D₃×C₄⋊C₈	C₂	semidirect
44. #79081	C₅	direct (N × H)
51. C₅×#424	D₃	semidirect
98. C₃×#424	C₂×C₅	semidirect
99. #5118	C₂×C₅	semidirect
100. C₂×Dic₃⋊C₈	C₂×C₅	semidirect
101. #4999	C₂×C₅	semidirect
102. #4839	C₂×C₅	semidirect
103. #4839	C₂×C₅	semidirect
104. #4837	C₂×C₅	semidirect
105. D₃×C₄⋊C₈	C₂×C₅	semidirect
242. #424	C₅×D₃	semidirect
263. C₅×Dic₃	#131	semidirect
451. C₃×C₅	#2906	semidirect
468. Dic₃	C₅×#131	semidirect
507. C₅	#79081	direct (N × H)
525. C₃	C₅×#2906	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#907	2
C₂×C₅×#901	2
C₅×#5522	2
C₃×C₅×#424	2
C₅×#5118	2
C₂×C₅×Dic₃⋊C₈	2
C₅×#4999	2
C₅×#4839	2
C₅×#4837	2
C₅×D₃×C₄⋊C₈	2
C₅×#2906	3
#79081	5

Subgroups — C5×#79081 #14413167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#79081 #14413167