Subgroups — C₅×#76198 #14323873

Order 1920.

1336 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×#895	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#893	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#872	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#389	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4989	C₂	1	2	yes	maximal, complemented
6	C₅×#4988	C₂	1	2	yes	maximal, complemented
7	C₅×#4987	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×Dic₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#4939	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4669	C₂	1	2	yes	maximal, complemented
11	C₅×#4665	C₂	1	2	yes	maximal, complemented
12	C₅×#2324	—	3	1	no	maximal, Hall
13	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
14	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
15	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
21	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
25	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
26	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
27	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
28	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
32	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
33	C₅×#895	C₄	1	8	yes	complemented
34	C₅×#893	—	2	4	no	—
35	C₅×#872	—	2	4	no	—
36	C₃×C₅×#95	C₂²	1	2	yes	—
37	C₅×C₃⋊#95	C₂²	1	2	yes	—
38	C₅×C₃⋊#95	C₂²	1	2	yes	—
39	#76198	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
40	C₂×C₅×#121	—	3	1	no	—
41	C₂×C₄×C₅×D₄	—	3	1	no	—
42	C₂×C₅×#117	—	3	1	no	—
43	C₅×#390	—	3	2	no	—
44	C₅×#390	—	3	1	no	—
45	C₅×#389	D₃	1	1	yes	characteristic, complemented
46	C₅×#388	—	3	2	no	—
47	C₅×#386	—	3	2	no	—
48	C₅×#385	—	3	2	no	—
49	C₅×#380	—	3	1	no	—
50	C₄×C₅×C₄⋊C₄	—	3	1	no	—
51	C₂³×C₅×D₃	C₂³	1	1	yes	characteristic
52	C₂³×C₅×D₃	C₂³	1	2	yes	—
53	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
56	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
57	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
59	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
60	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
62	C₂×C₅×D₁₂	—	2	2	no	—
63	C₂×C₅×D₁₂	—	2	2	no	—
64	C₂×C₅×D₁₂	—	4	2	no	—
65	C₂×C₅×D₁₂	—	4	2	no	—
66	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
67	C₂×C₅×D₁₂	C₂×C₄	1	8	yes	complemented
68	C₂×C₄×C₅×D₃	—	2	2	no	—
69	C₂×C₄×C₅×D₃	—	2	2	no	—
70	C₂×C₄×C₅×D₃	—	2	4	no	—
71	C₂×C₄×C₅×D₃	—	4	2	no	—
72	C₂×C₄×C₅×D₃	—	4	2	no	—
73	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	complemented
74	C₃×C₅×C₄⋊C₄	—	2	2	no	—
75	C₃×C₅×C₄⋊C₄	—	4	2	no	—
76	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
77	C₃×C₄²×C₅	—	2	2	no	—
78	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
79	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
80	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
81	C₅×(C₂×D₃)⋊C₄	—	4	2	no	—
82	C₅×(C₂×D₃)⋊C₄	—	4	2	no	—
83	C₅×(C₂×D₃)⋊C₄	—	4	2	no	—
84	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	complemented
85	C₅×C₄⋊Dic₃	—	2	2	no	—
86	C₅×Dic₃⋊C₄	—	2	2	no	—
87	C₅×Dic₃⋊C₄	—	2	2	no	—
88	C₄×C₅×Dic₃	—	4	2	no	—
89	C₄×C₅×Dic₃	D₄	1	4	yes	complemented
90	C₂×#895	C₂×C₅	1	1	yes	characteristic
91	C₂×#893	C₂×C₅	1	1	yes	characteristic, complemented
92	C₂×#872	C₂×C₅	1	1	yes	characteristic, complemented
93	C₃×#389	C₂×C₅	1	1	yes	characteristic, complemented
94	#4989	C₂×C₅	1	2	yes	complemented
95	#4988	C₂×C₅	1	2	yes	complemented
96	#4987	C₂×C₅	1	1	yes	characteristic, complemented
97	Dic₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
98	#4939	C₂×C₅	1	1	yes	characteristic, complemented
99	#4669	C₂×C₅	1	2	yes	complemented
100	#4665	C₂×C₅	1	2	yes	complemented
101	C₂²×C₅×D₄	—	3	1	no	—
102	C₂²×C₅×D₄	—	3	2	no	—
103	C₂³×C₄×C₅	—	3	1	no	—
104	C₂³×C₄×C₅	—	3	2	no	—
105	C₅×#121	—	3	8	no	—
106	C₄×C₅×D₄	—	6	4	no	—
107	C₅×#117	—	6	4	no	—
108	C₂×C₅×C₄⋊C₄	—	3	1	no	—
109	C₂×C₅×C₄⋊C₄	—	3	1	no	—
110	C₂×C₅×C₄⋊C₄	—	3	1	no	—
111	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
112	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
113	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
114	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
115	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
116	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
117	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
118	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
119	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
120	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
121	C₂×C₄²×C₅	—	3	1	no	—
122	C₂×C₄²×C₅	—	3	2	no	—
123	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
124	C₅×#95	—	3	2	no	—
125	C₅×#95	—	3	2	no	—
126	C₅×#95	C₂×D₃	1	2	yes	—
127	#2324	—	3	1	no	Sylow
128	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
129	C₂²×C₅×D₃	—	2	1	no	—
130	C₂²×C₅×D₃	—	2	2	no	—
131	C₂²×C₅×D₃	—	2	1	no	—
132	C₂²×C₅×D₃	—	2	2	no	—
133	C₂²×C₅×D₃	—	2	1	no	—
134	C₂²×C₅×D₃	—	2	2	no	—
135	C₂²×C₅×D₃	—	2	2	no	—
136	C₂²×C₅×D₃	—	2	2	no	—
137	C₂²×C₅×D₃	—	2	1	no	—
138	C₂²×C₅×D₃	—	2	1	no	—
139	C₂²×C₅×D₃	—	2	1	no	—
140	C₂²×C₅×D₃	—	2	2	no	—
141	C₂²×C₅×D₃	C₂²×C₄	1	4	yes	—
142	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
143	C₂×C₃×C₄×C₅	—	2	2	no	—
144	C₂×C₃×C₄×C₅	—	2	2	no	—
145	C₂×C₃×C₄×C₅	—	2	1	no	—
146	C₂×C₃×C₄×C₅	—	2	1	no	—
147	C₂×C₃×C₄×C₅	—	2	2	no	—
148	C₂×C₃×C₄×C₅	—	2	1	no	—
149	C₂×C₃×C₄×C₅	—	2	1	no	—
150	C₂×C₃×C₄×C₅	—	2	2	no	—
151	C₂×C₃×C₄×C₅	—	2	1	no	—
152	C₂×C₃×C₄×C₅	—	2	2	no	—
153	C₂×C₃×C₄×C₅	—	2	1	no	—
154	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
155	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
156	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
157	C₂×C₃×C₄×C₅	#41	1	2	yes	—
158	C₂×C₅×Dic₃	—	2	1	no	—
159	C₂×C₅×Dic₃	—	2	1	no	—
160	C₂×C₅×Dic₃	—	2	1	no	—
161	C₂×C₅×Dic₃	—	2	1	no	—
162	C₂×C₅×Dic₃	—	2	1	no	—
163	C₂×C₅×Dic₃	—	2	1	no	—
164	C₂×C₅×Dic₃	—	2	2	no	—
165	C₂×C₅×Dic₃	—	2	2	no	—
166	C₂×C₅×Dic₃	—	2	2	no	—
167	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
168	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
169	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
170	C₅×D₁₂	—	2	8	no	—
171	C₅×D₁₂	—	4	8	no	—
172	C₄×C₅×D₃	—	2	8	no	—
173	C₄×C₅×D₃	—	4	8	no	—
174	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
175	C₂²×D₁₂	C₂²×C₅	1	2	yes	—
176	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
177	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
178	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
179	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
180	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
181	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
182	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
183	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
184	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
185	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
186	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
187	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
188	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
189	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
190	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
191	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
192	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
193	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
194	#895	C₄×C₅	1	8	yes	complemented
195	#893	—	2	4	no	—
196	#872	—	2	4	no	—
197	C₃×#95	C₂²×C₅	1	2	yes	—
198	C₃⋊#95	C₂²×C₅	1	2	yes	—
199	C₃⋊#95	C₂²×C₅	1	2	yes	—
200	C₂⁴×C₅	—	3	2	no	—
201	C₂⁴×C₅	—	3	1	no	—
202	C₂×C₅×D₄	—	3	4	no	—
203	C₂×C₅×D₄	—	3	8	no	—
204	C₂×C₅×D₄	—	6	2	no	—
205	C₂×C₅×D₄	—	6	2	no	—
206	C₂×C₅×D₄	—	12	2	no	—
207	C₂×C₅×D₄	—	12	2	no	—
208	C₂²×C₄×C₅	—	3	1	no	—
209	C₂²×C₄×C₅	—	3	1	no	—
210	C₂²×C₄×C₅	—	3	1	no	—
211	C₂²×C₄×C₅	—	3	4	no	—
212	C₂²×C₄×C₅	—	3	2	no	—
213	C₂²×C₄×C₅	—	6	4	no	—
214	C₂²×C₄×C₅	—	6	2	no	—
215	C₂²×C₄×C₅	—	6	2	no	—
216	C₂²×C₄×C₅	—	12	2	no	—
217	C₂²×C₄×C₅	—	12	2	no	—
218	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
219	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
220	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
221	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
222	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
223	C₅×C₄⋊C₄	—	2	2	no	—
224	C₅×C₄⋊C₄	—	4	2	no	—
225	C₅×C₄⋊C₄	—	6	2	no	—
226	C₅×C₄⋊C₄	—	6	2	no	—
227	C₅×C₄⋊C₄	—	6	2	no	—
228	C₅×C₄⋊C₄	C₄×D₃	1	4	yes	complemented
229	C₅×C₂²⋊C₄	—	3	8	no	—
230	C₅×C₂²⋊C₄	—	6	2	no	—
231	C₅×C₂²⋊C₄	—	6	2	no	—
232	C₅×C₂²⋊C₄	—	6	4	no	—
233	C₅×C₂²⋊C₄	—	12	2	no	—
234	C₅×C₂²⋊C₄	—	12	2	no	—
235	C₅×C₂²⋊C₄	—	12	2	no	—
236	C₄²×C₅	—	2	2	no	—
237	C₄²×C₅	—	3	4	no	—
238	C₄²×C₅	—	12	2	no	—
239	C₂×#121	—	3	1	no	—
240	C₂×C₄×D₄	—	3	1	no	—
241	C₂×#117	—	3	1	no	—
242	#390	—	3	2	no	—
243	#390	—	3	1	no	—
244	#389	C₅×D₃	1	1	yes	characteristic, complemented
245	#388	—	3	2	no	—
246	#386	—	3	2	no	—
247	#385	—	3	2	no	—
248	#380	—	3	1	no	—
249	C₄×C₄⋊C₄	—	3	1	no	—
250	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
251	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
252	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
253	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
254	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
255	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
256	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
257	C₂×C₅×D₃	—	2	2	no	—
258	C₂×C₅×D₃	—	2	4	no	—
259	C₂×C₅×D₃	—	2	2	no	—
260	C₂×C₅×D₃	—	2	4	no	—
261	C₂×C₅×D₃	—	2	2	no	—
262	C₂×C₅×D₃	—	2	4	no	—
263	C₂×C₅×D₃	—	4	2	no	—
264	C₂×C₅×D₃	—	4	1	no	—
265	C₂×C₅×D₃	—	4	1	no	—
266	C₂×C₅×D₃	—	4	2	no	—
267	C₂×C₅×D₃	—	4	1	no	—
268	C₂×C₅×D₃	—	4	2	no	—
269	C₂×C₅×D₃	—	4	1	no	—
270	C₂×C₅×D₃	—	4	2	no	—
271	C₃×C₄×C₅	—	2	4	no	—
272	C₃×C₄×C₅	—	4	2	no	—
273	C₃×C₄×C₅	—	4	1	no	—
274	C₃×C₄×C₅	—	4	1	no	—
275	C₃×C₄×C₅	C₄×D₄	1	4	yes	—
276	C₅×Dic₃	—	2	2	no	—
277	C₅×Dic₃	—	2	2	no	—
278	C₅×Dic₃	—	2	2	no	—
279	C₅×Dic₃	—	4	2	no	—
280	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
281	C₂³×D₃	C₂³×C₅	1	2	yes	—
282	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
283	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
284	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
285	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
286	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
287	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
288	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
289	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
290	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
291	C₂×D₁₂	—	2	2	no	—
292	C₂×D₁₂	—	2	2	no	—
293	C₂×D₁₂	—	4	2	no	—
294	C₂×D₁₂	—	4	2	no	—
295	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	—
296	C₂×D₁₂	C₂×C₄×C₅	1	8	yes	complemented
297	C₂×C₄×D₃	—	2	2	no	—
298	C₂×C₄×D₃	—	2	2	no	—
299	C₂×C₄×D₃	—	2	4	no	—
300	C₂×C₄×D₃	—	4	2	no	—
301	C₂×C₄×D₃	—	4	2	no	—
302	C₂×C₄×D₃	C₂×C₄×C₅	1	4	yes	complemented
303	C₃×C₄⋊C₄	—	2	2	no	—
304	C₃×C₄⋊C₄	—	4	2	no	—
305	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	complemented
306	C₃×C₄²	—	2	2	no	—
307	(C₂×D₃)⋊C₄	—	2	2	no	—
308	(C₂×D₃)⋊C₄	—	2	2	no	—
309	(C₂×D₃)⋊C₄	—	2	4	no	—
310	(C₂×D₃)⋊C₄	—	4	2	no	—
311	(C₂×D₃)⋊C₄	—	4	2	no	—
312	(C₂×D₃)⋊C₄	—	4	2	no	—
313	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	8	yes	complemented
314	C₄⋊Dic₃	—	2	2	no	—
315	Dic₃⋊C₄	—	2	2	no	—
316	Dic₃⋊C₄	—	2	2	no	—
317	C₄×Dic₃	—	4	2	no	—
318	C₄×Dic₃	C₅×D₄	1	4	yes	complemented
319	C₂³×C₅	—	3	4	no	—
320	C₂³×C₅	—	3	2	no	—
321	C₂³×C₅	—	6	1	no	—
322	C₂³×C₅	—	6	2	no	—
323	C₂³×C₅	—	6	2	no	—
324	C₂³×C₅	—	6	1	no	—
325	C₂³×C₅	—	6	1	no	—
326	C₂³×C₅	—	6	2	no	—
327	C₂³×C₅	—	6	2	no	—
328	C₂³×C₅	—	6	1	no	—
329	C₂³×C₅	—	6	1	no	—
330	C₂³×C₅	—	6	2	no	—
331	C₂³×C₅	—	6	2	no	—
332	C₂³×C₅	—	6	1	no	—
333	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
334	C₅×D₄	—	6	8	no	—
335	C₅×D₄	—	12	8	no	—
336	C₂×C₄×C₅	—	2	2	no	—
337	C₂×C₄×C₅	—	2	2	no	—
338	C₂×C₄×C₅	—	2	1	no	—
339	C₂×C₄×C₅	—	2	1	no	—
340	C₂×C₄×C₅	—	2	2	no	—
341	C₂×C₄×C₅	—	2	1	no	—
342	C₂×C₄×C₅	—	2	2	no	—
343	C₂×C₄×C₅	—	2	1	no	—
344	C₂×C₄×C₅	—	2	1	no	—
345	C₂×C₄×C₅	—	2	1	no	—
346	C₂×C₄×C₅	—	2	2	no	—
347	C₂×C₄×C₅	—	3	2	no	—
348	C₂×C₄×C₅	—	3	2	no	—
349	C₂×C₄×C₅	—	3	2	no	—
350	C₂×C₄×C₅	—	6	1	no	—
351	C₂×C₄×C₅	—	6	1	no	—
352	C₂×C₄×C₅	—	6	1	no	—
353	C₂×C₄×C₅	—	6	1	no	—
354	C₂×C₄×C₅	—	6	1	no	—
355	C₂×C₄×C₅	—	6	1	no	—
356	C₂×C₄×C₅	—	6	8	no	—
357	C₂×C₄×C₅	—	6	2	no	—
358	C₂×C₄×C₅	—	6	2	no	—
359	C₂×C₄×C₅	—	6	2	no	—
360	C₂×C₄×C₅	—	12	8	no	—
361	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
362	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
363	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
364	C₂×C₄×C₅	#239	1	2	yes	—
365	C₂²×D₄	—	3	1	no	—
366	C₂²×D₄	—	3	2	no	—
367	C₂³×C₄	—	3	1	no	—
368	C₂³×C₄	—	3	2	no	—
369	#121	—	3	8	no	—
370	C₄×D₄	—	6	4	no	—
371	#117	—	6	4	no	—
372	C₂×C₄⋊C₄	—	3	1	no	—
373	C₂×C₄⋊C₄	—	3	1	no	—
374	C₂×C₄⋊C₄	—	3	1	no	—
375	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
377	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
378	C₂×C₂²⋊C₄	—	3	2	no	—
379	C₂×C₂²⋊C₄	—	3	1	no	—
380	C₂×C₂²⋊C₄	—	3	1	no	—
381	C₂×C₂²⋊C₄	—	3	2	no	—
382	C₂×C₂²⋊C₄	—	3	2	no	—
383	C₂×C₂²⋊C₄	—	3	2	no	—
384	C₂×C₂²⋊C₄	—	3	2	no	—
385	C₂×C₄²	—	3	1	no	—
386	C₂×C₄²	—	3	2	no	—
387	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
388	#95	—	3	2	no	—
389	#95	—	3	2	no	—
390	#95	C₂×C₅×D₃	1	2	yes	—
391	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
392	C₂×C₃×C₅	#518	1	1	yes	characteristic
393	C₂×C₃×C₅	#520	1	1	yes	characteristic
394	C₂×C₃×C₅	#535	1	1	yes	characteristic
395	C₂×C₃×C₅	#537	1	1	yes	characteristic
396	C₂×C₃×C₅	#540	1	1	yes	characteristic
397	C₂×C₃×C₅	#542	1	1	yes	characteristic
398	C₅×D₃	—	4	2	no	—
399	C₅×D₃	—	4	4	no	—
400	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
401	C₂²×D₃	—	2	1	no	—
402	C₂²×D₃	—	2	2	no	—
403	C₂²×D₃	—	2	1	no	—
404	C₂²×D₃	—	2	2	no	—
405	C₂²×D₃	—	2	1	no	—
406	C₂²×D₃	—	2	2	no	—
407	C₂²×D₃	—	2	2	no	—
408	C₂²×D₃	—	2	2	no	—
409	C₂²×D₃	—	2	1	no	—
410	C₂²×D₃	—	2	1	no	—
411	C₂²×D₃	—	2	1	no	—
412	C₂²×D₃	—	2	2	no	—
413	C₂²×D₃	C₂²×C₄×C₅	1	4	yes	—
414	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
415	C₂×C₃×C₄	—	2	2	no	—
416	C₂×C₃×C₄	—	2	2	no	—
417	C₂×C₃×C₄	—	2	1	no	—
418	C₂×C₃×C₄	—	2	1	no	—
419	C₂×C₃×C₄	—	2	2	no	—
420	C₂×C₃×C₄	—	2	1	no	—
421	C₂×C₃×C₄	—	2	1	no	—
422	C₂×C₃×C₄	—	2	2	no	—
423	C₂×C₃×C₄	—	2	1	no	—
424	C₂×C₃×C₄	—	2	2	no	—
425	C₂×C₃×C₄	—	2	1	no	—
426	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
427	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
428	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
429	C₂×C₃×C₄	C₅×#41	1	2	yes	—
430	C₂×Dic₃	—	2	1	no	—
431	C₂×Dic₃	—	2	1	no	—
432	C₂×Dic₃	—	2	1	no	—
433	C₂×Dic₃	—	2	1	no	—
434	C₂×Dic₃	—	2	1	no	—
435	C₂×Dic₃	—	2	1	no	—
436	C₂×Dic₃	—	2	2	no	—
437	C₂×Dic₃	—	2	2	no	—
438	C₂×Dic₃	—	2	2	no	—
439	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
440	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
441	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
442	D₁₂	—	2	8	no	—
443	D₁₂	—	4	8	no	—
444	C₄×D₃	—	2	8	no	—
445	C₄×D₃	—	4	8	no	—
446	C₂²×C₅	—	6	2	no	—
447	C₂²×C₅	—	6	4	no	—
448	C₂²×C₅	—	6	2	no	—
449	C₂²×C₅	—	6	4	no	—
450	C₂²×C₅	—	6	4	no	—
451	C₂²×C₅	—	6	2	no	—
452	C₂²×C₅	—	12	1	no	—
453	C₂²×C₅	—	12	2	no	—
454	C₂²×C₅	—	12	2	no	—
455	C₂²×C₅	—	12	1	no	—
456	C₂²×C₅	—	12	2	no	—
457	C₂²×C₅	—	12	1	no	—
458	C₂²×C₅	—	12	1	no	—
459	C₂²×C₅	—	12	2	no	—
460	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
461	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
462	C₂²×C₅	#1004	1	1	yes	characteristic
463	C₂²×C₅	C₂×#239	1	1	yes	characteristic
464	C₂²×C₅	#1007	1	1	yes	characteristic
465	C₂²×C₅	#1009	1	1	yes	characteristic
466	C₂²×C₅	#1009	1	1	yes	characteristic
467	C₄×C₅	—	2	4	no	—
468	C₄×C₅	—	4	1	no	—
469	C₄×C₅	—	4	2	no	—
470	C₄×C₅	—	4	1	no	—
471	C₄×C₅	—	6	2	no	—
472	C₄×C₅	—	6	2	no	—
473	C₄×C₅	—	6	2	no	—
474	C₄×C₅	—	12	2	no	—
475	C₄×C₅	#893	1	4	yes	—
476	C₂⁴	—	3	2	no	—
477	C₂⁴	—	3	1	no	—
478	C₂×D₄	—	3	4	no	—
479	C₂×D₄	—	3	8	no	—
480	C₂×D₄	—	6	2	no	—
481	C₂×D₄	—	6	2	no	—
482	C₂×D₄	—	12	2	no	—
483	C₂×D₄	—	12	2	no	—
484	C₂²×C₄	—	3	1	no	—
485	C₂²×C₄	—	3	1	no	—
486	C₂²×C₄	—	3	1	no	—
487	C₂²×C₄	—	3	4	no	—
488	C₂²×C₄	—	3	2	no	—
489	C₂²×C₄	—	6	4	no	—
490	C₂²×C₄	—	6	2	no	—
491	C₂²×C₄	—	6	2	no	—
492	C₂²×C₄	—	12	2	no	—
493	C₂²×C₄	—	12	2	no	—
494	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
495	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
496	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
497	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
498	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	4	2	no	—
501	C₄⋊C₄	—	6	2	no	—
502	C₄⋊C₄	—	6	2	no	—
503	C₄⋊C₄	—	6	2	no	—
504	C₄⋊C₄	C₄×C₅×D₃	1	4	yes	complemented
505	C₂²⋊C₄	—	3	8	no	—
506	C₂²⋊C₄	—	6	2	no	—
507	C₂²⋊C₄	—	6	2	no	—
508	C₂²⋊C₄	—	6	4	no	—
509	C₂²⋊C₄	—	12	2	no	—
510	C₂²⋊C₄	—	12	2	no	—
511	C₂²⋊C₄	—	12	2	no	—
512	C₄²	—	2	2	no	—
513	C₄²	—	3	4	no	—
514	C₄²	—	12	2	no	—
515	C₃×C₅	#2324	1	1	yes	characteristic, complemented, Hall
516	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
517	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
518	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
519	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
520	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
521	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
522	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
523	C₂×D₃	—	2	2	no	—
524	C₂×D₃	—	2	4	no	—
525	C₂×D₃	—	2	2	no	—
526	C₂×D₃	—	2	4	no	—
527	C₂×D₃	—	2	2	no	—
528	C₂×D₃	—	2	4	no	—
529	C₂×D₃	—	4	2	no	—
530	C₂×D₃	—	4	1	no	—
531	C₂×D₃	—	4	1	no	—
532	C₂×D₃	—	4	2	no	—
533	C₂×D₃	—	4	1	no	—
534	C₂×D₃	—	4	2	no	—
535	C₂×D₃	—	4	1	no	—
536	C₂×D₃	—	4	2	no	—
537	C₃×C₄	—	2	4	no	—
538	C₃×C₄	—	4	2	no	—
539	C₃×C₄	—	4	1	no	—
540	C₃×C₄	—	4	1	no	—
541	C₃×C₄	C₄×C₅×D₄	1	4	yes	—
542	Dic₃	—	2	2	no	—
543	Dic₃	—	2	2	no	—
544	Dic₃	—	2	2	no	—
545	Dic₃	—	4	2	no	—
546	C₂×C₅	—	12	2	no	—
547	C₂×C₅	—	12	4	no	—
548	C₂×C₅	C₂×#893	1	1	yes	characteristic
549	C₂×C₅	#5545	1	1	yes	characteristic
550	C₂×C₅	#5574	1	1	yes	characteristic
551	C₂×C₅	#5607	1	1	yes	characteristic
552	C₂×C₅	#5629	1	1	yes	characteristic
553	C₂×C₅	#5697	1	1	yes	characteristic
554	C₂×C₅	#5733	1	1	yes	characteristic
555	C₂³	—	3	4	no	—
556	C₂³	—	3	2	no	—
557	C₂³	—	6	1	no	—
558	C₂³	—	6	2	no	—
559	C₂³	—	6	2	no	—
560	C₂³	—	6	1	no	—
561	C₂³	—	6	1	no	—
562	C₂³	—	6	2	no	—
563	C₂³	—	6	2	no	—
564	C₂³	—	6	1	no	—
565	C₂³	—	6	1	no	—
566	C₂³	—	6	2	no	—
567	C₂³	—	6	2	no	—
568	C₂³	—	6	1	no	—
569	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
570	D₄	—	6	8	no	—
571	D₄	—	12	8	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	1	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	3	2	no	—
584	C₂×C₄	—	3	2	no	—
585	C₂×C₄	—	3	2	no	—
586	C₂×C₄	—	6	1	no	—
587	C₂×C₄	—	6	1	no	—
588	C₂×C₄	—	6	1	no	—
589	C₂×C₄	—	6	1	no	—
590	C₂×C₄	—	6	1	no	—
591	C₂×C₄	—	6	1	no	—
592	C₂×C₄	—	6	8	no	—
593	C₂×C₄	—	6	2	no	—
594	C₂×C₄	—	6	2	no	—
595	C₂×C₄	—	6	2	no	—
596	C₂×C₄	—	12	8	no	—
597	C₂×C₄	C₂×C₄×C₅×D₃	1	4	yes	—
598	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
599	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
600	C₂×C₄	C₅×#239	1	2	yes	—
601	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
602	C₂×C₃	C₅×#518	1	1	yes	characteristic
603	C₂×C₃	C₅×#520	1	1	yes	characteristic
604	C₂×C₃	C₅×#535	1	1	yes	characteristic
605	C₂×C₃	C₅×#537	1	1	yes	characteristic
606	C₂×C₃	C₅×#540	1	1	yes	characteristic
607	C₂×C₃	C₅×#542	1	1	yes	characteristic
608	D₃	—	4	2	no	—
609	D₃	—	4	4	no	—
610	C₅	#76198	1	1	yes	characteristic, direct product factor, Sylow
611	C₂²	—	6	2	no	—
612	C₂²	—	6	4	no	—
613	C₂²	—	6	2	no	—
614	C₂²	—	6	4	no	—
615	C₂²	—	6	4	no	—
616	C₂²	—	6	2	no	—
617	C₂²	—	12	1	no	—
618	C₂²	—	12	2	no	—
619	C₂²	—	12	2	no	—
620	C₂²	—	12	1	no	—
621	C₂²	—	12	2	no	—
622	C₂²	—	12	1	no	—
623	C₂²	—	12	1	no	—
624	C₂²	—	12	2	no	—
625	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
626	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
627	C₂²	C₅×#1004	1	1	yes	characteristic
628	C₂²	C₂×C₅×#239	1	1	yes	characteristic
629	C₂²	C₅×#1007	1	1	yes	characteristic
630	C₂²	C₅×#1009	1	1	yes	characteristic
631	C₂²	C₅×#1009	1	1	yes	characteristic
632	C₄	—	2	4	no	—
633	C₄	—	4	2	no	—
634	C₄	—	4	1	no	—
635	C₄	—	4	1	no	—
636	C₄	—	6	2	no	—
637	C₄	—	6	2	no	—
638	C₄	—	6	2	no	—
639	C₄	—	12	2	no	—
640	C₄	C₅×#893	1	4	yes	—
641	C₃	C₅×#2324	1	1	yes	characteristic, complemented, Sylow
642	C₂	—	12	2	no	—
643	C₂	—	12	4	no	—
644	C₂	C₂×C₅×#893	1	1	yes	characteristic
645	C₂	C₅×#5545	1	1	yes	characteristic
646	C₂	C₅×#5574	1	1	yes	characteristic
647	C₂	C₅×#5607	1	1	yes	characteristic
648	C₂	C₅×#5629	1	1	yes	characteristic
649	C₂	C₅×#5697	1	1	yes	characteristic
650	C₂	C₅×#5733	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₃×C₅×#389	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂×C₅×#895	C₂
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#389	D₃
hypocenter	C₃	C₅×#2324
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₅×#389

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#893	C₂	semidirect
3. C₂×C₅×#872	C₂	semidirect
4. C₃×C₅×#389	C₂	semidirect
5. C₅×#4989	C₂	semidirect
6. C₅×#4988	C₂	semidirect
7. C₅×#4987	C₂	semidirect
8. C₅×Dic₃×C₄⋊C₄	C₂	semidirect
9. C₅×#4939	C₂	semidirect
10. C₅×#4669	C₂	semidirect
11. C₅×#4665	C₂	semidirect
33. C₅×#895	C₄	semidirect
39. #76198	C₅	direct (N × H)
45. C₅×#389	D₃	semidirect
67. C₂×C₅×D₁₂	C₂×C₄	semidirect
73. C₂×C₄×C₅×D₃	C₂×C₄	semidirect
76. C₃×C₅×C₄⋊C₄	C₂×C₄	semidirect
84. C₅×(C₂×D₃)⋊C₄	C₂×C₄	semidirect
89. C₄×C₅×Dic₃	D₄	semidirect
91. C₂×#893	C₂×C₅	semidirect
92. C₂×#872	C₂×C₅	semidirect
93. C₃×#389	C₂×C₅	semidirect
94. #4989	C₂×C₅	semidirect
95. #4988	C₂×C₅	semidirect
96. #4987	C₂×C₅	semidirect
97. Dic₃×C₄⋊C₄	C₂×C₅	semidirect
98. #4939	C₂×C₅	semidirect
99. #4669	C₂×C₅	semidirect
100. #4665	C₂×C₅	semidirect
194. #895	C₄×C₅	semidirect
228. C₅×C₄⋊C₄	C₄×D₃	semidirect
244. #389	C₅×D₃	semidirect
296. C₂×D₁₂	C₂×C₄×C₅	semidirect
302. C₂×C₄×D₃	C₂×C₄×C₅	semidirect
305. C₃×C₄⋊C₄	C₂×C₄×C₅	semidirect
313. (C₂×D₃)⋊C₄	C₂×C₄×C₅	semidirect
318. C₄×Dic₃	C₅×D₄	semidirect
504. C₄⋊C₄	C₄×C₅×D₃	semidirect
515. C₃×C₅	#2324	semidirect
610. C₅	#76198	direct (N × H)
641. C₃	C₅×#2324	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#895	2
C₂×C₅×#893	2
C₂×C₅×#872	2
C₃×C₅×#389	2
C₅×#4989	2
C₅×#4988	2
C₅×#4987	2
C₅×Dic₃×C₄⋊C₄	2
C₅×#4939	2
C₅×#4669	2
C₅×#4665	2
C₅×#2324	3
#76198	5

Subgroups — C5×#76198 #14323873

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#76198 #14323873