Subgroups — C₅×#73771 #14327348

Order 1920.

1320 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#892	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×D₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic
4	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4987	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4987	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4940	C₂	1	2	yes	maximal, complemented
8	C₄×C₅×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal, complemented
9	C₅×#4936	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#4665	C₂	1	2	yes	maximal
11	C₅×#4653	C₂	1	2	yes	maximal, complemented
12	C₅×#2352	—	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	1	yes	characteristic
16	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
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×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
27	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
28	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
29	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	—
30	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
31	C₅×#892	—	2	4	no	—
32	C₅×D₃×C₄⋊C₄	—	2	4	no	—
33	C₄×C₅×D₁₂	C₄	1	8	yes	complemented
34	C₃×C₅×#95	C₂²	1	2	yes	—
35	C₅×C₃⋊#95	C₂²	1	2	yes	—
36	C₅×C₃⋊#95	C₂²	1	2	yes	—
37	C₅×C₃⋊#95	C₂²	1	2	yes	—
38	C₅×C₃⋊#95	C₂²	1	2	yes	—
39	#73771	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
40	C₂×C₄×C₅×D₄	—	3	1	no	—
41	C₂×C₅×#117	—	3	1	no	—
42	C₂²×C₅×C₄⋊C₄	—	3	1	no	—
43	C₅×#388	—	3	2	no	—
44	C₅×#387	—	3	2	no	—
45	C₅×#386	—	3	2	no	—
46	C₅×#382	—	3	1	no	—
47	C₅×#380	—	3	1	no	—
48	C₅×#380	—	3	1	no	—
49	C₄×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
50	C₄×C₅×C₂²⋊C₄	—	3	2	no	—
51	C₂³×C₅×D₃	C₂³	1	1	yes	characteristic
52	C₂³×C₅×D₃	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
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	2	yes	—
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	C₂×C₄	1	4	yes	—
65	C₂×C₄×C₅×D₃	—	2	2	no	—
66	C₂×C₄×C₅×D₃	—	2	2	no	—
67	C₂×C₄×C₅×D₃	—	2	2	no	—
68	C₂×C₄×C₅×D₃	—	2	2	no	—
69	C₂×C₄×C₅×D₃	—	2	4	no	—
70	C₂×C₄×C₅×D₃	—	2	4	no	—
71	C₂×C₄×C₅×D₃	—	4	2	no	—
72	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	—
73	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	complemented
74	C₃×C₅×C₄⋊C₄	—	2	2	no	—
75	C₃×C₅×C₄⋊C₄	D₄	1	4	yes	complemented
76	C₃×C₄²×C₅	—	4	2	no	—
77	C₃×C₄²×C₅	C₂×C₄	1	4	yes	complemented
78	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
79	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
80	C₅×(C₂×D₃)⋊C₄	—	4	2	no	—
81	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	—
82	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	complemented
83	C₅×C₄⋊Dic₃	—	2	2	no	—
84	C₅×C₄⋊Dic₃	—	2	2	no	—
85	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	complemented
86	C₅×Dic₃⋊C₄	—	2	4	no	—
87	C₄×C₅×Dic₃	—	2	4	no	—
88	C₂×#892	C₂×C₅	1	1	yes	characteristic, complemented
89	C₂×D₃×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
90	C₂×C₄×D₁₂	C₂×C₅	1	1	yes	characteristic
91	C₃×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
92	#4987	C₂×C₅	1	1	yes	characteristic, complemented
93	#4987	C₂×C₅	1	1	yes	characteristic, complemented
94	#4940	C₂×C₅	1	2	yes	complemented
95	C₄×(C₂×D₃)⋊C₄	C₂×C₅	1	2	yes	complemented
96	#4936	C₂×C₅	1	1	yes	characteristic, complemented
97	#4665	C₂×C₅	1	2	yes	—
98	#4653	C₂×C₅	1	2	yes	complemented
99	C₂²×C₅×D₄	—	3	1	no	—
100	C₂³×C₄×C₅	—	3	1	no	—
101	C₂³×C₄×C₅	—	3	1	no	—
102	C₂³×C₄×C₅	—	3	2	no	—
103	C₄×C₅×D₄	—	3	8	no	—
104	C₅×#117	—	6	4	no	—
105	C₂×C₅×C₄⋊C₄	—	3	1	no	—
106	C₂×C₅×C₄⋊C₄	—	3	1	no	—
107	C₂×C₅×C₄⋊C₄	—	3	1	no	—
108	C₂×C₅×C₄⋊C₄	—	3	2	no	—
109	C₂×C₅×C₄⋊C₄	—	6	4	no	—
110	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
111	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
112	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
113	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
114	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
115	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
116	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
117	C₂×C₄²×C₅	—	3	2	no	—
118	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
119	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
120	C₅×#95	—	3	2	no	—
121	C₅×#95	—	3	2	no	—
122	C₅×#95	—	3	2	no	—
123	C₅×#95	—	3	2	no	—
124	C₅×#95	C₂×D₃	1	2	yes	—
125	#2352	—	3	1	no	Sylow
126	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
127	C₂²×C₅×D₃	—	2	1	no	—
128	C₂²×C₅×D₃	—	2	1	no	—
129	C₂²×C₅×D₃	—	2	1	no	—
130	C₂²×C₅×D₃	—	2	1	no	—
131	C₂²×C₅×D₃	—	2	1	no	—
132	C₂²×C₅×D₃	—	2	1	no	—
133	C₂²×C₅×D₃	—	2	1	no	—
134	C₂²×C₅×D₃	—	2	1	no	—
135	C₂²×C₅×D₃	—	2	1	no	—
136	C₂²×C₅×D₃	—	2	1	no	—
137	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
138	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
139	C₂²×C₅×D₃	#41	1	2	yes	—
140	C₂²×C₅×D₃	#41	1	2	yes	—
141	C₂×C₃×C₄×C₅	—	2	2	no	—
142	C₂×C₃×C₄×C₅	—	2	1	no	—
143	C₂×C₃×C₄×C₅	—	2	1	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	—	2	2	no	—
148	C₂×C₃×C₄×C₅	—	2	2	no	—
149	C₂×C₃×C₄×C₅	—	2	2	no	—
150	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
151	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
152	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
153	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
154	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
155	C₂×C₃×C₄×C₅	#41	1	2	yes	—
156	C₂×C₅×Dic₃	—	2	1	no	—
157	C₂×C₅×Dic₃	—	2	1	no	—
158	C₂×C₅×Dic₃	—	2	1	no	—
159	C₂×C₅×Dic₃	—	2	1	no	—
160	C₂×C₅×Dic₃	—	2	2	no	—
161	C₂×C₅×Dic₃	—	2	2	no	—
162	C₂×C₅×Dic₃	—	2	2	no	—
163	C₂×C₅×Dic₃	—	2	2	no	—
164	C₂×C₅×Dic₃	—	2	2	no	—
165	C₂×C₅×Dic₃	—	2	2	no	—
166	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
167	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
168	C₅×D₁₂	—	2	8	no	—
169	C₄×C₅×D₃	—	2	8	no	—
170	C₄×C₅×D₃	—	2	8	no	—
171	C₄×C₅×D₃	—	4	8	no	—
172	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
173	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
174	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
175	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
176	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
177	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
178	C₂×C₃×C₄²	C₂²×C₅	1	2	yes	—
179	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
180	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
181	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
182	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
183	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
184	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic
185	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
186	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
187	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
188	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	—
189	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
190	#892	—	2	4	no	—
191	D₃×C₄⋊C₄	—	2	4	no	—
192	C₄×D₁₂	C₄×C₅	1	8	yes	complemented
193	C₃×#95	C₂²×C₅	1	2	yes	—
194	C₃⋊#95	C₂²×C₅	1	2	yes	—
195	C₃⋊#95	C₂²×C₅	1	2	yes	—
196	C₃⋊#95	C₂²×C₅	1	2	yes	—
197	C₃⋊#95	C₂²×C₅	1	2	yes	—
198	C₂⁴×C₅	—	3	1	no	—
199	C₂⁴×C₅	—	3	1	no	—
200	C₂×C₅×D₄	—	3	4	no	—
201	C₂×C₅×D₄	—	6	2	no	—
202	C₂×C₅×D₄	—	6	2	no	—
203	C₂²×C₄×C₅	—	3	1	no	—
204	C₂²×C₄×C₅	—	3	1	no	—
205	C₂²×C₄×C₅	—	3	4	no	—
206	C₂²×C₄×C₅	—	3	4	no	—
207	C₂²×C₄×C₅	—	3	2	no	—
208	C₂²×C₄×C₅	—	3	2	no	—
209	C₂²×C₄×C₅	—	6	4	no	—
210	C₂²×C₄×C₅	—	6	2	no	—
211	C₂²×C₄×C₅	—	6	2	no	—
212	C₂²×C₄×C₅	—	6	2	no	—
213	C₂²×C₄×C₅	—	6	2	no	—
214	C₂²×C₄×C₅	—	6	4	no	—
215	C₂²×C₄×C₅	—	12	2	no	—
216	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
217	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
218	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
219	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
220	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
221	C₅×C₄⋊C₄	—	2	2	no	—
222	C₅×C₄⋊C₄	—	3	4	no	—
223	C₅×C₄⋊C₄	—	6	4	no	—
224	C₅×C₄⋊C₄	—	6	2	no	—
225	C₅×C₄⋊C₄	—	6	2	no	—
226	C₅×C₄⋊C₄	D₁₂	1	4	yes	complemented
227	C₅×C₂²⋊C₄	—	3	4	no	—
228	C₅×C₂²⋊C₄	—	3	4	no	—
229	C₅×C₂²⋊C₄	—	6	4	no	—
230	C₅×C₂²⋊C₄	—	6	4	no	—
231	C₅×C₂²⋊C₄	—	12	2	no	—
232	C₄²×C₅	—	4	2	no	—
233	C₄²×C₅	—	6	4	no	—
234	C₄²×C₅	C₄×D₃	1	4	yes	complemented
235	C₂×C₄×D₄	—	3	1	no	—
236	C₂×#117	—	3	1	no	—
237	C₂²×C₄⋊C₄	—	3	1	no	—
238	#388	—	3	2	no	—
239	#387	—	3	2	no	—
240	#386	—	3	2	no	—
241	#382	—	3	1	no	—
242	#380	—	3	1	no	—
243	#380	—	3	1	no	—
244	C₄×C₄⋊C₄	C₅×D₃	1	1	yes	characteristic, complemented
245	C₄×C₂²⋊C₄	—	3	2	no	—
246	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
247	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
248	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
249	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
250	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
251	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
252	C₂²×C₃×C₅	2₋¹⁺⁴	1	1	yes	characteristic
253	C₂×C₅×D₃	—	2	2	no	—
254	C₂×C₅×D₃	—	2	2	no	—
255	C₂×C₅×D₃	—	2	2	no	—
256	C₂×C₅×D₃	—	2	2	no	—
257	C₂×C₅×D₃	—	2	2	no	—
258	C₂×C₅×D₃	—	2	2	no	—
259	C₂×C₅×D₃	—	2	2	no	—
260	C₂×C₅×D₃	—	2	2	no	—
261	C₂×C₅×D₃	—	2	2	no	—
262	C₂×C₅×D₃	—	4	1	no	—
263	C₂×C₅×D₃	—	4	1	no	—
264	C₂×C₅×D₃	—	4	1	no	—
265	C₂×C₅×D₃	—	4	1	no	—
266	C₂×C₅×D₃	#117	1	4	yes	complemented
267	C₃×C₄×C₅	—	2	2	no	—
268	C₃×C₄×C₅	—	2	2	no	—
269	C₃×C₄×C₅	—	2	4	no	—
270	C₃×C₄×C₅	—	4	2	no	—
271	C₃×C₄×C₅	C₄×D₄	1	4	yes	—
272	C₅×Dic₃	—	2	2	no	—
273	C₅×Dic₃	—	2	2	no	—
274	C₅×Dic₃	—	4	2	no	—
275	C₅×Dic₃	—	4	2	no	—
276	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
277	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
278	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
279	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
280	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
281	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
282	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
283	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
284	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
285	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
286	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
287	C₂×D₁₂	—	2	2	no	—
288	C₂×D₁₂	—	2	2	no	—
289	C₂×D₁₂	C₂×C₄×C₅	1	4	yes	—
290	C₂×C₄×D₃	—	2	2	no	—
291	C₂×C₄×D₃	—	2	2	no	—
292	C₂×C₄×D₃	—	2	2	no	—
293	C₂×C₄×D₃	—	2	2	no	—
294	C₂×C₄×D₃	—	2	4	no	—
295	C₂×C₄×D₃	—	2	4	no	—
296	C₂×C₄×D₃	—	4	2	no	—
297	C₂×C₄×D₃	C₂×C₄×C₅	1	4	yes	—
298	C₂×C₄×D₃	C₂×C₄×C₅	1	4	yes	complemented
299	C₃×C₄⋊C₄	—	2	2	no	—
300	C₃×C₄⋊C₄	C₅×D₄	1	4	yes	complemented
301	C₃×C₄²	—	4	2	no	—
302	C₃×C₄²	C₂×C₄×C₅	1	4	yes	complemented
303	(C₂×D₃)⋊C₄	—	2	4	no	—
304	(C₂×D₃)⋊C₄	—	2	4	no	—
305	(C₂×D₃)⋊C₄	—	4	2	no	—
306	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	4	yes	—
307	(C₂×D₃)⋊C₄	C₂×C₄×C₅	1	4	yes	complemented
308	C₄⋊Dic₃	—	2	2	no	—
309	C₄⋊Dic₃	—	2	2	no	—
310	C₄⋊Dic₃	C₂×C₄×C₅	1	4	yes	complemented
311	Dic₃⋊C₄	—	2	4	no	—
312	C₄×Dic₃	—	2	4	no	—
313	C₂³×C₅	—	3	2	no	—
314	C₂³×C₅	—	3	2	no	—
315	C₂³×C₅	—	3	2	no	—
316	C₂³×C₅	—	3	2	no	—
317	C₂³×C₅	—	6	1	no	—
318	C₂³×C₅	—	6	1	no	—
319	C₂³×C₅	—	6	1	no	—
320	C₂³×C₅	—	6	1	no	—
321	C₂³×C₅	—	6	1	no	—
322	C₂³×C₅	—	6	1	no	—
323	C₂³×C₅	—	6	1	no	—
324	C₂³×C₅	—	6	1	no	—
325	C₂³×C₅	—	6	1	no	—
326	C₂³×C₅	—	6	1	no	—
327	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
328	C₅×D₄	—	6	8	no	—
329	C₂×C₄×C₅	—	2	1	no	—
330	C₂×C₄×C₅	—	2	2	no	—
331	C₂×C₄×C₅	—	2	1	no	—
332	C₂×C₄×C₅	—	2	2	no	—
333	C₂×C₄×C₅	—	2	1	no	—
334	C₂×C₄×C₅	—	2	1	no	—
335	C₂×C₄×C₅	—	2	2	no	—
336	C₂×C₄×C₅	—	2	2	no	—
337	C₂×C₄×C₅	—	2	2	no	—
338	C₂×C₄×C₅	—	3	2	no	—
339	C₂×C₄×C₅	—	3	2	no	—
340	C₂×C₄×C₅	—	6	1	no	—
341	C₂×C₄×C₅	—	6	1	no	—
342	C₂×C₄×C₅	—	6	1	no	—
343	C₂×C₄×C₅	—	6	1	no	—
344	C₂×C₄×C₅	—	6	8	no	—
345	C₂×C₄×C₅	—	6	8	no	—
346	C₂×C₄×C₅	—	6	2	no	—
347	C₂×C₄×C₅	—	6	2	no	—
348	C₂×C₄×C₅	—	6	2	no	—
349	C₂×C₄×C₅	—	6	2	no	—
350	C₂×C₄×C₅	—	6	2	no	—
351	C₂×C₄×C₅	—	6	2	no	—
352	C₂×C₄×C₅	—	12	8	no	—
353	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
354	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
355	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
356	C₂×C₄×C₅	C₂×D₁₂	1	4	yes	—
357	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
358	C₂×C₄×C₅	#235	1	2	yes	—
359	C₂²×D₄	—	3	1	no	—
360	C₂³×C₄	—	3	1	no	—
361	C₂³×C₄	—	3	1	no	—
362	C₂³×C₄	—	3	2	no	—
363	C₄×D₄	—	3	8	no	—
364	#117	—	6	4	no	—
365	C₂×C₄⋊C₄	—	3	1	no	—
366	C₂×C₄⋊C₄	—	3	1	no	—
367	C₂×C₄⋊C₄	—	3	2	no	—
368	C₂×C₄⋊C₄	—	3	1	no	—
369	C₂×C₄⋊C₄	—	6	4	no	—
370	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
372	C₂×C₂²⋊C₄	—	3	1	no	—
373	C₂×C₂²⋊C₄	—	3	1	no	—
374	C₂×C₂²⋊C₄	—	3	2	no	—
375	C₂×C₂²⋊C₄	—	3	2	no	—
376	C₂×C₂²⋊C₄	—	3	2	no	—
377	C₂×C₄²	—	3	2	no	—
378	C₂×C₄²	C₂×C₅×D₃	1	2	yes	—
379	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
380	#95	—	3	2	no	—
381	#95	—	3	2	no	—
382	#95	—	3	2	no	—
383	#95	—	3	2	no	—
384	#95	C₂×C₅×D₃	1	2	yes	—
385	C₂×C₃×C₅	C₂×#117	1	1	yes	characteristic
386	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
387	C₂×C₃×C₅	#520	1	1	yes	characteristic
388	C₂×C₃×C₅	#546	1	1	yes	characteristic
389	C₂×C₃×C₅	#548	1	1	yes	characteristic
390	C₂×C₃×C₅	#552	1	1	yes	characteristic
391	C₂×C₃×C₅	#553	1	1	yes	characteristic
392	C₅×D₃	—	2	4	no	—
393	C₅×D₃	—	4	2	no	—
394	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
395	C₂²×D₃	—	2	1	no	—
396	C₂²×D₃	—	2	1	no	—
397	C₂²×D₃	—	2	1	no	—
398	C₂²×D₃	—	2	1	no	—
399	C₂²×D₃	—	2	1	no	—
400	C₂²×D₃	—	2	1	no	—
401	C₂²×D₃	—	2	1	no	—
402	C₂²×D₃	—	2	1	no	—
403	C₂²×D₃	—	2	1	no	—
404	C₂²×D₃	—	2	1	no	—
405	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
406	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
407	C₂²×D₃	C₅×#41	1	2	yes	—
408	C₂²×D₃	C₅×#41	1	2	yes	—
409	C₂×C₃×C₄	—	2	2	no	—
410	C₂×C₃×C₄	—	2	1	no	—
411	C₂×C₃×C₄	—	2	1	no	—
412	C₂×C₃×C₄	—	2	1	no	—
413	C₂×C₃×C₄	—	2	2	no	—
414	C₂×C₃×C₄	—	2	1	no	—
415	C₂×C₃×C₄	—	2	2	no	—
416	C₂×C₃×C₄	—	2	2	no	—
417	C₂×C₃×C₄	—	2	2	no	—
418	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
419	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
420	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
421	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
422	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
423	C₂×C₃×C₄	C₅×#41	1	2	yes	—
424	C₂×Dic₃	—	2	1	no	—
425	C₂×Dic₃	—	2	1	no	—
426	C₂×Dic₃	—	2	1	no	—
427	C₂×Dic₃	—	2	1	no	—
428	C₂×Dic₃	—	2	2	no	—
429	C₂×Dic₃	—	2	2	no	—
430	C₂×Dic₃	—	2	2	no	—
431	C₂×Dic₃	—	2	2	no	—
432	C₂×Dic₃	—	2	2	no	—
433	C₂×Dic₃	—	2	2	no	—
434	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
435	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
436	D₁₂	—	2	8	no	—
437	C₄×D₃	—	2	8	no	—
438	C₄×D₃	—	2	8	no	—
439	C₄×D₃	—	4	8	no	—
440	C₂²×C₅	—	3	4	no	—
441	C₂²×C₅	—	6	2	no	—
442	C₂²×C₅	—	6	2	no	—
443	C₂²×C₅	—	6	2	no	—
444	C₂²×C₅	—	6	2	no	—
445	C₂²×C₅	—	6	2	no	—
446	C₂²×C₅	—	6	2	no	—
447	C₂²×C₅	—	6	2	no	—
448	C₂²×C₅	—	6	2	no	—
449	C₂²×C₅	—	6	2	no	—
450	C₂²×C₅	—	12	1	no	—
451	C₂²×C₅	—	12	1	no	—
452	C₂²×C₅	—	12	1	no	—
453	C₂²×C₅	—	12	1	no	—
454	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
455	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
456	C₂²×C₅	C₂×#235	1	1	yes	characteristic
457	C₂²×C₅	#1004	1	1	yes	characteristic
458	C₂²×C₅	#1007	1	1	yes	characteristic
459	C₂²×C₅	D₃×#41	1	1	yes	characteristic
460	C₂²×C₅	D₃×#41	1	1	yes	characteristic
461	C₄×C₅	—	2	2	no	—
462	C₄×C₅	—	2	2	no	—
463	C₄×C₅	—	2	4	no	—
464	C₄×C₅	—	4	2	no	—
465	C₄×C₅	—	6	2	no	—
466	C₄×C₅	—	6	2	no	—
467	C₄×C₅	—	12	2	no	—
468	C₄×C₅	—	12	2	no	—
469	C₄×C₅	C₄×D₁₂	1	4	yes	—
470	C₂⁴	—	3	1	no	—
471	C₂⁴	—	3	1	no	—
472	C₂×D₄	—	3	4	no	—
473	C₂×D₄	—	6	2	no	—
474	C₂×D₄	—	6	2	no	—
475	C₂²×C₄	—	3	1	no	—
476	C₂²×C₄	—	3	1	no	—
477	C₂²×C₄	—	3	4	no	—
478	C₂²×C₄	—	3	4	no	—
479	C₂²×C₄	—	3	2	no	—
480	C₂²×C₄	—	3	2	no	—
481	C₂²×C₄	—	6	4	no	—
482	C₂²×C₄	—	6	2	no	—
483	C₂²×C₄	—	6	2	no	—
484	C₂²×C₄	—	6	2	no	—
485	C₂²×C₄	—	6	2	no	—
486	C₂²×C₄	—	6	4	no	—
487	C₂²×C₄	—	12	2	no	—
488	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
489	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
490	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
491	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
492	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	3	4	no	—
495	C₄⋊C₄	—	6	4	no	—
496	C₄⋊C₄	—	6	2	no	—
497	C₄⋊C₄	—	6	2	no	—
498	C₄⋊C₄	C₅×D₁₂	1	4	yes	complemented
499	C₂²⋊C₄	—	3	4	no	—
500	C₂²⋊C₄	—	3	4	no	—
501	C₂²⋊C₄	—	6	4	no	—
502	C₂²⋊C₄	—	6	4	no	—
503	C₂²⋊C₄	—	12	2	no	—
504	C₄²	—	4	2	no	—
505	C₄²	—	6	4	no	—
506	C₄²	C₄×C₅×D₃	1	4	yes	complemented
507	C₃×C₅	#2352	1	1	yes	characteristic, complemented, Hall
508	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
509	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
510	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
511	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
512	C₂²×C₃	C₂×C₅×#41	1	1	yes	characteristic
513	C₂²×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
514	C₂²×C₃	C₅×2₋¹⁺⁴	1	1	yes	characteristic
515	C₂×D₃	—	2	2	no	—
516	C₂×D₃	—	2	2	no	—
517	C₂×D₃	—	2	2	no	—
518	C₂×D₃	—	2	2	no	—
519	C₂×D₃	—	2	2	no	—
520	C₂×D₃	—	2	2	no	—
521	C₂×D₃	—	2	2	no	—
522	C₂×D₃	—	2	2	no	—
523	C₂×D₃	—	2	2	no	—
524	C₂×D₃	—	4	1	no	—
525	C₂×D₃	—	4	1	no	—
526	C₂×D₃	—	4	1	no	—
527	C₂×D₃	—	4	1	no	—
528	C₂×D₃	C₅×#117	1	4	yes	complemented
529	C₃×C₄	—	2	2	no	—
530	C₃×C₄	—	2	2	no	—
531	C₃×C₄	—	2	4	no	—
532	C₃×C₄	—	4	2	no	—
533	C₃×C₄	C₄×C₅×D₄	1	4	yes	—
534	Dic₃	—	2	2	no	—
535	Dic₃	—	2	2	no	—
536	Dic₃	—	4	2	no	—
537	Dic₃	—	4	2	no	—
538	C₂×C₅	—	6	4	no	—
539	C₂×C₅	—	12	2	no	—
540	C₂×C₅	C₂×C₄×D₁₂	1	1	yes	characteristic
541	C₂×C₅	#5504	1	1	yes	characteristic
542	C₂×C₅	D₃×#117	1	1	yes	characteristic
543	C₂×C₅	#5554	1	1	yes	characteristic
544	C₂×C₅	#5558	1	1	yes	characteristic
545	C₂×C₅	#5576	1	1	yes	characteristic
546	C₂×C₅	#5581	1	1	yes	characteristic
547	C₂³	—	3	2	no	—
548	C₂³	—	3	2	no	—
549	C₂³	—	3	2	no	—
550	C₂³	—	3	2	no	—
551	C₂³	—	6	1	no	—
552	C₂³	—	6	1	no	—
553	C₂³	—	6	1	no	—
554	C₂³	—	6	1	no	—
555	C₂³	—	6	1	no	—
556	C₂³	—	6	1	no	—
557	C₂³	—	6	1	no	—
558	C₂³	—	6	1	no	—
559	C₂³	—	6	1	no	—
560	C₂³	—	6	1	no	—
561	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
562	D₄	—	6	8	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	3	2	no	—
573	C₂×C₄	—	3	2	no	—
574	C₂×C₄	—	6	1	no	—
575	C₂×C₄	—	6	1	no	—
576	C₂×C₄	—	6	1	no	—
577	C₂×C₄	—	6	1	no	—
578	C₂×C₄	—	6	8	no	—
579	C₂×C₄	—	6	8	no	—
580	C₂×C₄	—	6	2	no	—
581	C₂×C₄	—	6	2	no	—
582	C₂×C₄	—	6	2	no	—
583	C₂×C₄	—	6	2	no	—
584	C₂×C₄	—	6	2	no	—
585	C₂×C₄	—	6	2	no	—
586	C₂×C₄	—	12	8	no	—
587	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
588	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
589	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
590	C₂×C₄	C₂×C₅×D₁₂	1	4	yes	—
591	C₂×C₄	C₂×C₅×D₁₂	1	2	yes	—
592	C₂×C₄	C₅×#235	1	2	yes	—
593	C₂×C₃	C₂×C₅×#117	1	1	yes	characteristic
594	C₂×C₃	C₂×C₄×C₅×D₄	1	1	yes	characteristic
595	C₂×C₃	C₅×#520	1	1	yes	characteristic
596	C₂×C₃	C₅×#546	1	1	yes	characteristic
597	C₂×C₃	C₅×#548	1	1	yes	characteristic
598	C₂×C₃	C₅×#552	1	1	yes	characteristic
599	C₂×C₃	C₅×#553	1	1	yes	characteristic
600	D₃	—	2	4	no	—
601	D₃	—	4	2	no	—
602	C₅	#73771	1	1	yes	characteristic, direct product factor, Sylow
603	C₂²	—	3	4	no	—
604	C₂²	—	6	2	no	—
605	C₂²	—	6	2	no	—
606	C₂²	—	6	2	no	—
607	C₂²	—	6	2	no	—
608	C₂²	—	6	2	no	—
609	C₂²	—	6	2	no	—
610	C₂²	—	6	2	no	—
611	C₂²	—	6	2	no	—
612	C₂²	—	6	2	no	—
613	C₂²	—	12	1	no	—
614	C₂²	—	12	1	no	—
615	C₂²	—	12	1	no	—
616	C₂²	—	12	1	no	—
617	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
618	C₂²	C₂²×C₅×D₁₂	1	1	yes	characteristic
619	C₂²	C₂×C₅×#235	1	1	yes	characteristic
620	C₂²	C₅×#1004	1	1	yes	characteristic
621	C₂²	C₅×#1007	1	1	yes	characteristic
622	C₂²	C₅×D₃×#41	1	1	yes	characteristic
623	C₂²	C₅×D₃×#41	1	1	yes	characteristic
624	C₄	—	2	2	no	—
625	C₄	—	2	2	no	—
626	C₄	—	2	4	no	—
627	C₄	—	4	2	no	—
628	C₄	—	6	2	no	—
629	C₄	—	6	2	no	—
630	C₄	—	12	2	no	—
631	C₄	—	12	2	no	—
632	C₄	C₄×C₅×D₁₂	1	4	yes	—
633	C₃	C₅×#2352	1	1	yes	characteristic, complemented, Sylow
634	C₂	—	6	4	no	—
635	C₂	—	12	2	no	—
636	C₂	C₂×C₄×C₅×D₁₂	1	1	yes	characteristic
637	C₂	C₅×#5504	1	1	yes	characteristic
638	C₂	C₅×D₃×#117	1	1	yes	characteristic
639	C₂	C₅×#5554	1	1	yes	characteristic
640	C₂	C₅×#5558	1	1	yes	characteristic
641	C₂	C₅×#5576	1	1	yes	characteristic
642	C₂	C₅×#5581	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×C₄×C₅×C₄⋊C₄	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂²×C₅×D₁₂	C₂²
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₄×C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₅×#2352
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₄×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#892	C₂	semidirect
2. C₂×C₅×D₃×C₄⋊C₄	C₂	semidirect
4. C₃×C₄×C₅×C₄⋊C₄	C₂	semidirect
5. C₅×#4987	C₂	semidirect
6. C₅×#4987	C₂	semidirect
7. C₅×#4940	C₂	semidirect
8. C₄×C₅×(C₂×D₃)⋊C₄	C₂	semidirect
9. C₅×#4936	C₂	semidirect
11. C₅×#4653	C₂	semidirect
33. C₄×C₅×D₁₂	C₄	semidirect
39. #73771	C₅	direct (N × H)
49. C₄×C₅×C₄⋊C₄	D₃	semidirect
73. C₂×C₄×C₅×D₃	C₂×C₄	semidirect
75. C₃×C₅×C₄⋊C₄	D₄	semidirect
77. C₃×C₄²×C₅	C₂×C₄	semidirect
82. C₅×(C₂×D₃)⋊C₄	C₂×C₄	semidirect
85. C₅×C₄⋊Dic₃	C₂×C₄	semidirect
88. C₂×#892	C₂×C₅	semidirect
89. C₂×D₃×C₄⋊C₄	C₂×C₅	semidirect
91. C₃×C₄×C₄⋊C₄	C₂×C₅	semidirect
92. #4987	C₂×C₅	semidirect
93. #4987	C₂×C₅	semidirect
94. #4940	C₂×C₅	semidirect
95. C₄×(C₂×D₃)⋊C₄	C₂×C₅	semidirect
96. #4936	C₂×C₅	semidirect
98. #4653	C₂×C₅	semidirect
192. C₄×D₁₂	C₄×C₅	semidirect
226. C₅×C₄⋊C₄	D₁₂	semidirect
234. C₄²×C₅	C₄×D₃	semidirect
244. C₄×C₄⋊C₄	C₅×D₃	semidirect
266. C₂×C₅×D₃	#117	semidirect
298. C₂×C₄×D₃	C₂×C₄×C₅	semidirect
300. C₃×C₄⋊C₄	C₅×D₄	semidirect
302. C₃×C₄²	C₂×C₄×C₅	semidirect
307. (C₂×D₃)⋊C₄	C₂×C₄×C₅	semidirect
310. C₄⋊Dic₃	C₂×C₄×C₅	semidirect
498. C₄⋊C₄	C₅×D₁₂	semidirect
506. C₄²	C₄×C₅×D₃	semidirect
507. C₃×C₅	#2352	semidirect
528. C₂×D₃	C₅×#117	semidirect
602. C₅	#73771	direct (N × H)
633. C₃	C₅×#2352	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#892	2
C₂×C₅×D₃×C₄⋊C₄	2
C₂×C₄×C₅×D₁₂	2
C₃×C₄×C₅×C₄⋊C₄	2
C₅×#4987	2
C₅×#4940	2
C₄×C₅×(C₂×D₃)⋊C₄	2
C₅×#4936	2
C₅×#4665	2
C₅×#4653	2
C₅×#2352	3
#73771	5

Subgroups — C5×#73771 #14327348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#73771 #14327348