Subgroups — C₅×#76202 #14327007

Order 1920.

1552 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#893	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#893	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×D₃×C₄⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×#389	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4989	C₂	1	2	yes	maximal, complemented
7	C₅×#4984	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#4975	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₅×Dic₃⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4668	C₂	1	2	yes	maximal, complemented
11	C₅×#4667	C₂	1	2	yes	maximal, complemented
12	C₅×D₄×C₄⋊C₄	—	3	1	no	maximal, Hall
13	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
15	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
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	2	yes	—
22	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
23	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
24	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
25	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
26	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
28	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
29	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
32	C₅×#893	—	2	4	no	—
33	C₅×#893	C₄	1	8	yes	complemented
34	C₅×D₃×C₄⋊C₄	—	2	8	no	—
35	C₄×C₅×D₁₂	—	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	1	yes	characteristic
39	C₅×C₃⋊#95	C₂²	1	1	yes	characteristic
40	#76202	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
41	C₂×C₄×C₅×D₄	—	3	1	no	—
42	C₂×C₄×C₅×D₄	—	3	1	no	—
43	C₂×C₄×C₅×D₄	—	3	1	no	—
44	C₂²×C₅×C₄⋊C₄	—	3	2	no	—
45	C₅×#389	—	3	1	no	—
46	C₅×#389	D₃	1	1	yes	characteristic, complemented
47	C₅×#385	—	3	2	no	—
48	C₅×#385	—	3	2	no	—
49	C₅×#384	—	3	1	no	—
50	C₅×#380	—	3	2	no	—
51	C₄×C₅×C₄⋊C₄	—	3	1	no	—
52	C₂³×C₅×D₃	C₂³	1	2	yes	—
53	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
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	1	yes	characteristic
57	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
59	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
60	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
61	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
62	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
63	C₂×C₅×D₁₂	C₂×C₄	1	4	yes	—
64	C₂×C₅×D₁₂	D₄	1	4	yes	—
65	C₂×C₅×D₁₂	Q₈	1	4	yes	—
66	C₂×C₄×C₅×D₃	—	2	4	no	—
67	C₂×C₄×C₅×D₃	—	2	4	no	—
68	C₂×C₄×C₅×D₃	—	2	4	no	—
69	C₂×C₄×C₅×D₃	—	2	4	no	—
70	C₂×C₄×C₅×D₃	—	2	4	no	—
71	C₂×C₄×C₅×D₃	—	2	4	no	—
72	C₂×C₄×C₅×D₃	—	4	2	no	—
73	C₂×C₄×C₅×D₃	—	4	2	no	—
74	C₂×C₄×C₅×D₃	C₂×C₄	1	8	yes	complemented
75	C₃×C₅×C₄⋊C₄	—	2	4	no	—
76	C₃×C₅×C₄⋊C₄	—	2	2	no	—
77	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	complemented
78	C₃×C₄²×C₅	—	2	2	no	—
79	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
80	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
81	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	8	yes	complemented
82	C₅×C₄⋊Dic₃	—	2	2	no	—
83	C₅×C₄⋊Dic₃	—	2	4	no	—
84	C₅×Dic₃⋊C₄	—	2	4	no	—
85	C₅×Dic₃⋊C₄	—	2	2	no	—
86	C₅×Dic₃⋊C₄	—	4	2	no	—
87	C₅×Dic₃⋊C₄	D₄	1	4	yes	complemented
88	C₄×C₅×Dic₃	—	2	2	no	—
89	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	complemented
90	C₂×#893	C₂×C₅	1	1	yes	characteristic
91	C₂×#893	C₂×C₅	1	1	yes	characteristic
92	C₂×D₃×C₄⋊C₄	C₂×C₅	1	2	yes	complemented
93	C₂×C₄×D₁₂	C₂×C₅	1	1	yes	characteristic
94	C₃×#389	C₂×C₅	1	1	yes	characteristic, complemented
95	#4989	C₂×C₅	1	2	yes	complemented
96	#4984	C₂×C₅	1	1	yes	characteristic, complemented
97	#4975	C₂×C₅	1	1	yes	characteristic, complemented
98	C₄×Dic₃⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
99	#4668	C₂×C₅	1	2	yes	complemented
100	#4667	C₂×C₅	1	2	yes	complemented
101	C₂²×C₅×D₄	—	3	1	no	—
102	C₂³×C₄×C₅	—	3	2	no	—
103	C₂³×C₄×C₅	—	3	2	no	—
104	C₂³×C₄×C₅	—	3	2	no	—
105	C₄×C₅×D₄	—	3	8	no	—
106	C₄×C₅×D₄	—	6	4	no	—
107	C₄×C₅×D₄	—	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	2	no	—
111	C₂×C₅×C₄⋊C₄	—	3	1	no	—
112	C₂×C₅×C₄⋊C₄	—	3	2	no	—
113	C₂×C₅×C₄⋊C₄	—	3	2	no	—
114	C₂×C₅×C₄⋊C₄	—	6	8	no	—
115	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
116	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
117	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
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	1	no	—
123	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
124	C₅×#95	—	3	1	no	—
125	C₅×#95	—	3	1	no	—
126	C₅×#95	—	3	2	no	—
127	C₅×#95	C₂×D₃	1	2	yes	—
128	D₄×C₄⋊C₄	—	3	1	no	Sylow
129	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
130	C₂²×C₅×D₃	—	2	2	no	—
131	C₂²×C₅×D₃	—	2	2	no	—
132	C₂²×C₅×D₃	—	2	2	no	—
133	C₂²×C₅×D₃	—	2	2	no	—
134	C₂²×C₅×D₃	C₂²×C₄	1	4	yes	—
135	C₂²×C₅×D₃	C₂×D₄	1	4	yes	—
136	C₂²×C₅×D₃	C₂×Q₈	1	4	yes	—
137	C₂×C₃×C₄×C₅	—	2	2	no	—
138	C₂×C₃×C₄×C₅	—	2	1	no	—
139	C₂×C₃×C₄×C₅	—	2	2	no	—
140	C₂×C₃×C₄×C₅	—	2	1	no	—
141	C₂×C₃×C₄×C₅	—	2	2	no	—
142	C₂×C₃×C₄×C₅	—	2	1	no	—
143	C₂×C₃×C₄×C₅	—	2	2	no	—
144	C₂×C₃×C₄×C₅	—	2	1	no	—
145	C₂×C₃×C₄×C₅	—	2	2	no	—
146	C₂×C₃×C₄×C₅	—	2	1	no	—
147	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
148	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
149	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
150	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
151	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
152	C₂×C₅×Dic₃	—	2	1	no	—
153	C₂×C₅×Dic₃	—	2	1	no	—
154	C₂×C₅×Dic₃	—	2	1	no	—
155	C₂×C₅×Dic₃	—	2	1	no	—
156	C₂×C₅×Dic₃	—	2	1	no	—
157	C₂×C₅×Dic₃	—	2	2	no	—
158	C₂×C₅×Dic₃	—	2	2	no	—
159	C₂×C₅×Dic₃	—	2	1	no	—
160	C₂×C₅×Dic₃	—	2	2	no	—
161	C₂×C₅×Dic₃	—	2	1	no	—
162	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
163	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
164	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
165	C₂×C₅×Dic₃	C₂×D₄	1	2	yes	—
166	C₂×C₅×Dic₃	#41	1	2	yes	—
167	C₅×D₁₂	C₄⋊C₄	1	16	yes	complemented
168	C₄×C₅×D₃	—	2	16	no	—
169	C₄×C₅×D₃	—	4	8	no	—
170	C₄×C₅×D₃	—	4	8	no	—
171	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
172	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
173	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
174	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
175	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
176	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	2	yes	—
177	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
178	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
179	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
180	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
181	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
182	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
183	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
184	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
185	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
186	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
187	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
188	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
189	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
190	#893	—	2	4	no	—
191	#893	C₄×C₅	1	8	yes	complemented
192	D₃×C₄⋊C₄	—	2	8	no	—
193	C₄×D₁₂	—	2	4	no	—
194	C₃×#95	C₂²×C₅	1	2	yes	—
195	C₃⋊#95	C₂²×C₅	1	2	yes	—
196	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
197	C₃⋊#95	C₂²×C₅	1	1	yes	characteristic
198	C₂⁴×C₅	—	3	2	no	—
199	C₂×C₅×D₄	—	3	4	no	—
200	C₂×C₅×D₄	—	3	4	no	—
201	C₂×C₅×D₄	—	3	4	no	—
202	C₂²×C₄×C₅	—	3	1	no	—
203	C₂²×C₄×C₅	—	3	1	no	—
204	C₂²×C₄×C₅	—	3	1	no	—
205	C₂²×C₄×C₅	—	3	8	no	—
206	C₂²×C₄×C₅	—	3	1	no	—
207	C₂²×C₄×C₅	—	3	2	no	—
208	C₂²×C₄×C₅	—	6	4	no	—
209	C₂²×C₄×C₅	—	6	4	no	—
210	C₂²×C₄×C₅	—	6	4	no	—
211	C₂²×C₄×C₅	—	6	4	no	—
212	C₂²×C₄×C₅	—	6	4	no	—
213	C₂²×C₄×C₅	—	6	4	no	—
214	C₂²×C₄×C₅	—	12	2	no	—
215	C₂²×C₄×C₅	—	12	2	no	—
216	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
217	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
218	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
219	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
220	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
221	C₅×C₄⋊C₄	—	2	4	no	—
222	C₅×C₄⋊C₄	—	2	2	no	—
223	C₅×C₄⋊C₄	—	3	4	no	—
224	C₅×C₄⋊C₄	—	6	4	no	—
225	C₅×C₄⋊C₄	—	6	4	no	—
226	C₅×C₄⋊C₄	—	6	2	no	—
227	C₅×C₄⋊C₄	—	6	2	no	—
228	C₅×C₄⋊C₄	—	12	2	no	—
229	C₅×C₄⋊C₄	C₄×D₃	1	4	yes	complemented
230	C₅×C₂²⋊C₄	—	3	8	no	—
231	C₅×C₂²⋊C₄	—	6	4	no	—
232	C₅×C₂²⋊C₄	—	6	4	no	—
233	C₄²×C₅	—	2	2	no	—
234	C₄²×C₅	—	3	4	no	—
235	C₄²×C₅	—	6	2	no	—
236	C₂×C₄×D₄	—	3	1	no	—
237	C₂×C₄×D₄	—	3	1	no	—
238	C₂×C₄×D₄	—	3	1	no	—
239	C₂²×C₄⋊C₄	—	3	2	no	—
240	#389	—	3	1	no	—
241	#389	C₅×D₃	1	1	yes	characteristic, complemented
242	#385	—	3	2	no	—
243	#385	—	3	2	no	—
244	#384	—	3	1	no	—
245	#380	—	3	2	no	—
246	C₄×C₄⋊C₄	—	3	1	no	—
247	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
248	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
249	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
250	C₂²×C₃×C₅	C₂²×Q₈	1	1	yes	characteristic
251	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
252	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
253	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
254	C₂×C₅×D₃	—	2	4	no	—
255	C₂×C₅×D₃	—	2	4	no	—
256	C₂×C₅×D₃	—	2	4	no	—
257	C₂×C₅×D₃	—	2	4	no	—
258	C₂×C₅×D₃	—	2	4	no	—
259	C₂×C₅×D₃	—	2	4	no	—
260	C₂×C₅×D₃	C₂×C₄⋊C₄	1	8	yes	complemented
261	C₃×C₄×C₅	—	2	4	no	—
262	C₃×C₄×C₅	—	2	2	no	—
263	C₃×C₄×C₅	—	4	2	no	—
264	C₃×C₄×C₅	—	4	1	no	—
265	C₃×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	complemented
266	C₅×Dic₃	—	2	2	no	—
267	C₅×Dic₃	—	2	2	no	—
268	C₅×Dic₃	—	4	2	no	—
269	C₅×Dic₃	—	4	1	no	—
270	C₅×Dic₃	C₄×D₄	1	4	yes	complemented
271	C₂³×D₃	C₂³×C₅	1	2	yes	—
272	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
273	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
274	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
275	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
276	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
277	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
278	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
279	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
280	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
281	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
282	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	—
283	C₂×D₁₂	C₅×D₄	1	4	yes	—
284	C₂×D₁₂	C₅×Q₈	1	4	yes	—
285	C₂×C₄×D₃	—	2	4	no	—
286	C₂×C₄×D₃	—	2	4	no	—
287	C₂×C₄×D₃	—	2	4	no	—
288	C₂×C₄×D₃	—	2	4	no	—
289	C₂×C₄×D₃	—	2	4	no	—
290	C₂×C₄×D₃	—	2	4	no	—
291	C₂×C₄×D₃	—	4	2	no	—
292	C₂×C₄×D₃	—	4	2	no	—
293	C₂×C₄×D₃	C₂×C₄×C₅	1	8	yes	complemented
294	C₃×C₄⋊C₄	—	2	4	no	—
295	C₃×C₄⋊C₄	—	2	2	no	—
296	C₃×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	complemented
297	C₃×C₄²	—	2	2	no	—
298	(C₂×D₃)⋊C₄	—	2	4	no	—
299	(C₂×D₃)⋊C₄	—	2	4	no	—
300	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	8	yes	complemented
301	C₄⋊Dic₃	—	2	2	no	—
302	C₄⋊Dic₃	—	2	4	no	—
303	Dic₃⋊C₄	—	2	4	no	—
304	Dic₃⋊C₄	—	2	2	no	—
305	Dic₃⋊C₄	—	4	2	no	—
306	Dic₃⋊C₄	C₅×D₄	1	4	yes	complemented
307	C₄×Dic₃	—	2	2	no	—
308	C₄×Dic₃	C₂×C₄×C₅	1	4	yes	complemented
309	C₂³×C₅	—	3	4	no	—
310	C₂³×C₅	—	3	4	no	—
311	C₂³×C₅	—	3	4	no	—
312	C₂³×C₅	—	6	2	no	—
313	C₂³×C₅	—	6	2	no	—
314	C₂³×C₅	—	6	2	no	—
315	C₂³×C₅	—	6	2	no	—
316	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
317	C₅×D₄	—	3	16	no	—
318	C₂×C₄×C₅	—	2	1	no	—
319	C₂×C₄×C₅	—	2	2	no	—
320	C₂×C₄×C₅	—	2	2	no	—
321	C₂×C₄×C₅	—	2	1	no	—
322	C₂×C₄×C₅	—	2	1	no	—
323	C₂×C₄×C₅	—	2	2	no	—
324	C₂×C₄×C₅	—	2	2	no	—
325	C₂×C₄×C₅	—	2	1	no	—
326	C₂×C₄×C₅	—	2	1	no	—
327	C₂×C₄×C₅	—	2	2	no	—
328	C₂×C₄×C₅	—	3	2	no	—
329	C₂×C₄×C₅	—	3	2	no	—
330	C₂×C₄×C₅	—	3	2	no	—
331	C₂×C₄×C₅	—	3	2	no	—
332	C₂×C₄×C₅	—	3	2	no	—
333	C₂×C₄×C₅	—	6	16	no	—
334	C₂×C₄×C₅	—	6	1	no	—
335	C₂×C₄×C₅	—	6	1	no	—
336	C₂×C₄×C₅	—	6	1	no	—
337	C₂×C₄×C₅	—	6	1	no	—
338	C₂×C₄×C₅	—	6	1	no	—
339	C₂×C₄×C₅	—	6	2	no	—
340	C₂×C₄×C₅	—	6	2	no	—
341	C₂×C₄×C₅	—	6	1	no	—
342	C₂×C₄×C₅	—	6	1	no	—
343	C₂×C₄×C₅	—	6	2	no	—
344	C₂×C₄×C₅	—	12	8	no	—
345	C₂×C₄×C₅	—	12	8	no	—
346	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
347	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
348	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
349	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
350	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
351	C₂²×D₄	—	3	1	no	—
352	C₂³×C₄	—	3	2	no	—
353	C₂³×C₄	—	3	2	no	—
354	C₂³×C₄	—	3	2	no	—
355	C₄×D₄	—	3	8	no	—
356	C₄×D₄	—	6	4	no	—
357	C₄×D₄	—	6	4	no	—
358	C₂×C₄⋊C₄	—	3	1	no	—
359	C₂×C₄⋊C₄	—	3	2	no	—
360	C₂×C₄⋊C₄	—	3	2	no	—
361	C₂×C₄⋊C₄	—	3	1	no	—
362	C₂×C₄⋊C₄	—	3	1	no	—
363	C₂×C₄⋊C₄	—	3	2	no	—
364	C₂×C₄⋊C₄	—	6	8	no	—
365	C₂×C₄⋊C₄	C₂×C₅×D₃	1	2	yes	—
366	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
367	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
368	C₂×C₂²⋊C₄	—	3	2	no	—
369	C₂×C₂²⋊C₄	—	3	2	no	—
370	C₂×C₂²⋊C₄	—	3	2	no	—
371	C₂×C₄²	—	3	1	no	—
372	C₂×C₄²	—	3	1	no	—
373	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
374	#95	—	3	1	no	—
375	#95	—	3	1	no	—
376	#95	—	3	2	no	—
377	#95	C₂×C₅×D₃	1	2	yes	—
378	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
379	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
380	C₂×C₃×C₅	#520	1	1	yes	characteristic
381	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
382	C₂×C₃×C₅	#547	1	1	yes	characteristic
383	C₂×C₃×C₅	D₄×Q₈	1	1	yes	characteristic
384	C₂×C₃×C₅	#554	1	1	yes	characteristic
385	C₅×D₃	—	2	8	no	—
386	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
387	C₂²×D₃	—	2	2	no	—
388	C₂²×D₃	—	2	2	no	—
389	C₂²×D₃	—	2	2	no	—
390	C₂²×D₃	—	2	2	no	—
391	C₂²×D₃	C₂²×C₄×C₅	1	4	yes	—
392	C₂²×D₃	C₂×C₅×D₄	1	4	yes	—
393	C₂²×D₃	C₂×C₅×Q₈	1	4	yes	—
394	C₂×C₃×C₄	—	2	2	no	—
395	C₂×C₃×C₄	—	2	1	no	—
396	C₂×C₃×C₄	—	2	2	no	—
397	C₂×C₃×C₄	—	2	1	no	—
398	C₂×C₃×C₄	—	2	2	no	—
399	C₂×C₃×C₄	—	2	1	no	—
400	C₂×C₃×C₄	—	2	2	no	—
401	C₂×C₃×C₄	—	2	1	no	—
402	C₂×C₃×C₄	—	2	2	no	—
403	C₂×C₃×C₄	—	2	1	no	—
404	C₂×C₃×C₄	C₂²×C₄×C₅	1	4	yes	—
405	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
406	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
407	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
408	C₂×C₃×C₄	C₂×C₅×Q₈	1	2	yes	—
409	C₂×Dic₃	—	2	1	no	—
410	C₂×Dic₃	—	2	1	no	—
411	C₂×Dic₃	—	2	1	no	—
412	C₂×Dic₃	—	2	1	no	—
413	C₂×Dic₃	—	2	1	no	—
414	C₂×Dic₃	—	2	2	no	—
415	C₂×Dic₃	—	2	2	no	—
416	C₂×Dic₃	—	2	1	no	—
417	C₂×Dic₃	—	2	2	no	—
418	C₂×Dic₃	—	2	1	no	—
419	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
420	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
421	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
422	C₂×Dic₃	C₂×C₅×D₄	1	2	yes	—
423	C₂×Dic₃	C₅×#41	1	2	yes	—
424	D₁₂	C₅×C₄⋊C₄	1	16	yes	complemented
425	C₄×D₃	—	2	16	no	—
426	C₄×D₃	—	4	8	no	—
427	C₄×D₃	—	4	8	no	—
428	C₂²×C₅	—	3	8	no	—
429	C₂²×C₅	—	6	4	no	—
430	C₂²×C₅	—	6	4	no	—
431	C₂²×C₅	—	6	4	no	—
432	C₂²×C₅	—	6	4	no	—
433	C₂²×C₅	—	6	4	no	—
434	C₂²×C₅	—	6	4	no	—
435	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
436	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
437	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
438	C₂²×C₅	C₂×D₃×Q₈	1	1	yes	characteristic
439	C₂²×C₅	#1007	1	1	yes	characteristic
440	C₂²×C₅	D₃×#41	1	1	yes	characteristic
441	C₂²×C₅	#1009	1	1	yes	characteristic
442	C₄×C₅	—	2	4	no	—
443	C₄×C₅	—	2	2	no	—
444	C₄×C₅	—	3	4	no	—
445	C₄×C₅	—	4	2	no	—
446	C₄×C₅	—	4	1	no	—
447	C₄×C₅	—	6	2	no	—
448	C₄×C₅	—	6	2	no	—
449	C₄×C₅	—	12	2	no	—
450	C₄×C₅	—	12	1	no	—
451	C₄×C₅	D₃×C₄⋊C₄	1	4	yes	complemented
452	C₂⁴	—	3	2	no	—
453	C₂×D₄	—	3	4	no	—
454	C₂×D₄	—	3	4	no	—
455	C₂×D₄	—	3	4	no	—
456	C₂²×C₄	—	3	1	no	—
457	C₂²×C₄	—	3	1	no	—
458	C₂²×C₄	—	3	1	no	—
459	C₂²×C₄	—	3	8	no	—
460	C₂²×C₄	—	3	1	no	—
461	C₂²×C₄	—	3	2	no	—
462	C₂²×C₄	—	6	4	no	—
463	C₂²×C₄	—	6	4	no	—
464	C₂²×C₄	—	6	4	no	—
465	C₂²×C₄	—	6	4	no	—
466	C₂²×C₄	—	6	4	no	—
467	C₂²×C₄	—	6	4	no	—
468	C₂²×C₄	—	12	2	no	—
469	C₂²×C₄	—	12	2	no	—
470	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
471	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
472	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
473	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
474	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	—	3	4	no	—
478	C₄⋊C₄	—	6	4	no	—
479	C₄⋊C₄	—	6	4	no	—
480	C₄⋊C₄	—	6	2	no	—
481	C₄⋊C₄	—	6	2	no	—
482	C₄⋊C₄	—	12	2	no	—
483	C₄⋊C₄	C₄×C₅×D₃	1	4	yes	complemented
484	C₂²⋊C₄	—	3	8	no	—
485	C₂²⋊C₄	—	6	4	no	—
486	C₂²⋊C₄	—	6	4	no	—
487	C₄²	—	2	2	no	—
488	C₄²	—	3	4	no	—
489	C₄²	—	6	2	no	—
490	C₃×C₅	D₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
491	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
492	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
493	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
494	C₂²×C₃	C₂²×C₅×Q₈	1	1	yes	characteristic
495	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
496	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
497	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
498	C₂×D₃	—	2	4	no	—
499	C₂×D₃	—	2	4	no	—
500	C₂×D₃	—	2	4	no	—
501	C₂×D₃	—	2	4	no	—
502	C₂×D₃	—	2	4	no	—
503	C₂×D₃	—	2	4	no	—
504	C₂×D₃	C₂×C₅×C₄⋊C₄	1	8	yes	complemented
505	C₃×C₄	—	2	4	no	—
506	C₃×C₄	—	2	2	no	—
507	C₃×C₄	—	4	2	no	—
508	C₃×C₄	—	4	1	no	—
509	C₃×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	complemented
510	Dic₃	—	2	2	no	—
511	Dic₃	—	2	2	no	—
512	Dic₃	—	4	2	no	—
513	Dic₃	—	4	1	no	—
514	Dic₃	C₄×C₅×D₄	1	4	yes	complemented
515	C₂×C₅	—	6	8	no	—
516	C₂×C₅	C₂×D₃×C₄⋊C₄	1	1	yes	characteristic
517	C₂×C₅	C₄×D₃×D₄	1	1	yes	characteristic
518	C₂×C₅	#5574	1	1	yes	characteristic
519	C₂×C₅	#5609	1	1	yes	characteristic
520	C₂×C₅	#5631	1	1	yes	characteristic
521	C₂×C₅	#5690	1	1	yes	characteristic
522	C₂×C₅	#5727	1	1	yes	characteristic
523	C₂³	—	3	4	no	—
524	C₂³	—	3	4	no	—
525	C₂³	—	3	4	no	—
526	C₂³	—	6	2	no	—
527	C₂³	—	6	2	no	—
528	C₂³	—	6	2	no	—
529	C₂³	—	6	2	no	—
530	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
531	D₄	—	3	16	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	3	2	no	—
543	C₂×C₄	—	3	2	no	—
544	C₂×C₄	—	3	2	no	—
545	C₂×C₄	—	3	2	no	—
546	C₂×C₄	—	3	2	no	—
547	C₂×C₄	—	6	16	no	—
548	C₂×C₄	—	6	1	no	—
549	C₂×C₄	—	6	1	no	—
550	C₂×C₄	—	6	1	no	—
551	C₂×C₄	—	6	1	no	—
552	C₂×C₄	—	6	1	no	—
553	C₂×C₄	—	6	2	no	—
554	C₂×C₄	—	6	2	no	—
555	C₂×C₄	—	6	1	no	—
556	C₂×C₄	—	6	1	no	—
557	C₂×C₄	—	6	2	no	—
558	C₂×C₄	—	12	8	no	—
559	C₂×C₄	—	12	8	no	—
560	C₂×C₄	C₂×C₄×C₅×D₃	1	4	yes	—
561	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
562	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
563	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
564	C₂×C₄	C₅×D₃×Q₈	1	2	yes	—
565	C₂×C₃	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
566	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
567	C₂×C₃	C₅×#520	1	1	yes	characteristic
568	C₂×C₃	C₅×D₄²	1	1	yes	characteristic
569	C₂×C₃	C₅×#547	1	1	yes	characteristic
570	C₂×C₃	C₅×D₄×Q₈	1	1	yes	characteristic
571	C₂×C₃	C₅×#554	1	1	yes	characteristic
572	D₃	—	2	8	no	—
573	C₅	#76202	1	1	yes	characteristic, direct product factor, Sylow
574	C₂²	—	3	8	no	—
575	C₂²	—	6	4	no	—
576	C₂²	—	6	4	no	—
577	C₂²	—	6	4	no	—
578	C₂²	—	6	4	no	—
579	C₂²	—	6	4	no	—
580	C₂²	—	6	4	no	—
581	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
582	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
583	C₂²	C₂×C₅×D₃×D₄	1	1	yes	characteristic
584	C₂²	C₂×C₅×D₃×Q₈	1	1	yes	characteristic
585	C₂²	C₅×#1007	1	1	yes	characteristic
586	C₂²	C₅×D₃×#41	1	1	yes	characteristic
587	C₂²	C₅×#1009	1	1	yes	characteristic
588	C₄	—	2	4	no	—
589	C₄	—	2	2	no	—
590	C₄	—	3	4	no	—
591	C₄	—	4	2	no	—
592	C₄	—	4	1	no	—
593	C₄	—	6	2	no	—
594	C₄	—	6	2	no	—
595	C₄	—	12	2	no	—
596	C₄	—	12	1	no	—
597	C₄	C₅×D₃×C₄⋊C₄	1	4	yes	complemented
598	C₃	C₅×D₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
599	C₂	—	6	8	no	—
600	C₂	C₂×C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
601	C₂	C₄×C₅×D₃×D₄	1	1	yes	characteristic
602	C₂	C₅×#5574	1	1	yes	characteristic
603	C₂	C₅×#5609	1	1	yes	characteristic
604	C₂	C₅×#5631	1	1	yes	characteristic
605	C₂	C₅×#5690	1	1	yes	characteristic
606	C₂	C₅×#5727	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	C₂²×C₅×D₁₂	C₂²
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₅×#389	D₃
hypocenter	C₃	C₅×D₄×C₄⋊C₄
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
3. C₂×C₅×D₃×C₄⋊C₄	C₂	semidirect
5. C₃×C₅×#389	C₂	semidirect
6. C₅×#4989	C₂	semidirect
7. C₅×#4984	C₂	semidirect
8. C₅×#4975	C₂	semidirect
9. C₄×C₅×Dic₃⋊C₄	C₂	semidirect
10. C₅×#4668	C₂	semidirect
11. C₅×#4667	C₂	semidirect
33. C₅×#893	C₄	semidirect
40. #76202	C₅	direct (N × H)
46. C₅×#389	D₃	semidirect
74. C₂×C₄×C₅×D₃	C₂×C₄	semidirect
77. C₃×C₅×C₄⋊C₄	C₂×C₄	semidirect
81. C₅×(C₂×D₃)⋊C₄	C₂×C₄	semidirect
87. C₅×Dic₃⋊C₄	D₄	semidirect
89. C₄×C₅×Dic₃	C₂×C₄	semidirect
92. C₂×D₃×C₄⋊C₄	C₂×C₅	semidirect
94. C₃×#389	C₂×C₅	semidirect
95. #4989	C₂×C₅	semidirect
96. #4984	C₂×C₅	semidirect
97. #4975	C₂×C₅	semidirect
98. C₄×Dic₃⋊C₄	C₂×C₅	semidirect
99. #4668	C₂×C₅	semidirect
100. #4667	C₂×C₅	semidirect
167. C₅×D₁₂	C₄⋊C₄	semidirect
191. #893	C₄×C₅	semidirect
229. C₅×C₄⋊C₄	C₄×D₃	semidirect
241. #389	C₅×D₃	semidirect
260. C₂×C₅×D₃	C₂×C₄⋊C₄	semidirect
265. C₃×C₄×C₅	C₂×C₄⋊C₄	semidirect
270. C₅×Dic₃	C₄×D₄	semidirect
293. C₂×C₄×D₃	C₂×C₄×C₅	semidirect
296. C₃×C₄⋊C₄	C₂×C₄×C₅	semidirect
300. (C₂×D₃)⋊C₄	C₂×C₄×C₅	semidirect
306. Dic₃⋊C₄	C₅×D₄	semidirect
308. C₄×Dic₃	C₂×C₄×C₅	semidirect
424. D₁₂	C₅×C₄⋊C₄	semidirect
451. C₄×C₅	D₃×C₄⋊C₄	semidirect
483. C₄⋊C₄	C₄×C₅×D₃	semidirect
490. C₃×C₅	D₄×C₄⋊C₄	semidirect
504. C₂×D₃	C₂×C₅×C₄⋊C₄	semidirect
509. C₃×C₄	C₂×C₅×C₄⋊C₄	semidirect
514. Dic₃	C₄×C₅×D₄	semidirect
573. C₅	#76202	direct (N × H)
597. C₄	C₅×D₃×C₄⋊C₄	semidirect
598. C₃	C₅×D₄×C₄⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#893	2
C₂×C₅×D₃×C₄⋊C₄	2
C₂×C₄×C₅×D₁₂	2
C₃×C₅×#389	2
C₅×#4989	2
C₅×#4984	2
C₅×#4975	2
C₄×C₅×Dic₃⋊C₄	2
C₅×#4668	2
C₅×#4667	2
C₅×D₄×C₄⋊C₄	3
#76202	5

Subgroups — C5×#76202 #14327007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#76202 #14327007