Subgroups — C₅×#34121 #11956841

Order 1280.

2554 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₅×#2357	C₂	1	2	yes	maximal, complemented
3	C₅×#2356	C₂	1	2	yes	maximal, complemented
4	C₅×#2355	C₂	1	2	yes	maximal, complemented
5	C₅×#2354	C₂	1	2	yes	maximal, complemented
6	C₅×#2353	C₂	1	4	yes	maximal, complemented
7	C₅×#2352	C₂	1	4	yes	maximal, complemented
8	C₅×#2298	C₂	1	4	yes	maximal, complemented
9	C₅×#2297	C₂	1	2	yes	maximal
10	C₅×#2285	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#2283	C₂	1	2	yes	maximal, complemented
12	C₄×C₅×#117	C₂	1	2	yes	maximal, complemented
13	C₄×C₅×#117	C₂	1	2	yes	maximal, complemented
14	C₂×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
16	C₄×C₅×#41	C₄	1	16	yes	complemented
17	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
20	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
21	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
23	C₂×C₅×#117	C₂²	1	2	yes	—
24	C₂×C₅×#117	C₂²	1	4	yes	—
25	C₂×C₅×#117	C₂²	1	1	yes	characteristic
26	C₂×C₅×#117	C₂²	1	2	yes	—
27	C₂²×C₅×C₄⋊C₄	C₂²	1	2	yes	—
28	C₂²×C₄²×C₅	C₂²	1	2	yes	—
29	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₅	C₂²	1	2	yes	—
31	C₅×#391	C₂²	1	4	yes	—
32	C₅×#390	C₂²	1	4	yes	—
33	C₅×#388	C₂²	1	8	yes	—
34	C₅×#388	C₂²	1	8	yes	—
35	C₅×#387	C₂²	1	8	yes	—
36	C₅×#387	C₂²	1	8	yes	—
37	C₅×#386	C₂²	1	8	yes	—
38	C₅×#384	C₂²	1	2	yes	—
39	C₅×#383	C₂²	1	4	yes	—
40	C₅×#382	C₂²	1	4	yes	—
41	C₅×#382	C₂²	1	2	yes	—
42	C₅×#381	C₂²	1	4	yes	—
43	C₅×#380	C₂²	1	4	yes	—
44	C₅×#380	C₂²	1	2	yes	—
45	C₅×#380	C₂²	1	2	yes	—
46	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
47	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
48	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
49	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
50	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
51	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
52	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
53	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
54	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
55	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
56	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
57	C₄×C₅×C₂²⋊C₄	C₂²	1	8	yes	—
58	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
59	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
60	C₅×#376	C₂²	1	4	yes	—
61	C₅×#376	C₂²	1	4	yes	—
62	C₂×C₅×#95	C₂²	1	2	yes	—
63	C₄³×C₅	C₂²	1	4	yes	—
64	#34121	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×C₅×#41	—	2	4	no	—
66	C₂×C₅×#41	—	2	4	no	—
67	C₂×C₅×#41	C₂×C₄	1	8	yes	—
68	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
69	C₂²×C₅×D₄	C₂³	1	2	yes	—
70	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₅	C₂³	1	4	yes	—
72	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₅	C₂³	1	2	yes	—
74	C₂³×C₄×C₅	C₂³	1	2	yes	—
75	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₅	C₂³	1	2	yes	—
77	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
78	C₄×C₅×Q₈	C₂×C₄	1	8	yes	complemented
79	C₄×C₅×Q₈	C₂×C₄	1	8	yes	complemented
80	C₄×C₅×D₄	C₂×C₄	1	16	yes	—
81	C₄×C₅×D₄	C₂×C₄	1	16	yes	—
82	C₄×C₅×D₄	C₂×C₄	1	8	yes	complemented
83	C₄×C₅×D₄	C₂×C₄	1	8	yes	complemented
84	C₅×#117	—	2	8	no	—
85	C₅×#117	—	2	16	no	—
86	C₅×#117	C₂×C₄	1	8	yes	—
87	C₅×#117	C₂×C₄	1	16	yes	complemented
88	C₂×C₅×C₄⋊C₄	—	2	8	no	—
89	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
90	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
91	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
92	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
93	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
94	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
95	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
96	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
97	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
98	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
99	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
100	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
101	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
102	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
103	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
104	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
105	C₂×C₄²×C₅	—	2	8	no	—
106	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
107	C₂×C₄²×C₅	C₂×C₄	1	16	yes	complemented
108	C₂×C₄²×C₅	C₂³	1	8	yes	—
109	C₂×C₄²×C₅	C₂³	1	4	yes	—
110	C₂×C₄²×C₅	C₂³	1	2	yes	—
111	C₂×C₄²×C₅	C₂³	1	4	yes	—
112	C₂×C₄²×C₅	C₂³	1	2	yes	—
113	C₂×C₄²×C₅	C₂³	1	8	yes	—
114	C₂×C₄²×C₅	C₂³	1	2	yes	—
115	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₅	C₂³	1	2	yes	—
117	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₅	C₂³	1	2	yes	—
119	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
121	C₅×#95	C₂³	1	2	yes	—
122	C₅×#95	C₂³	1	2	yes	—
123	C₅×#95	C₂³	1	4	yes	—
124	C₅×#95	C₂³	1	4	yes	—
125	C₅×#95	C₂³	1	8	yes	—
126	C₅×#95	C₂³	1	8	yes	—
127	C₅×#95	C₂³	1	8	yes	—
128	C₅×#95	C₂³	1	8	yes	—
129	C₅×#95	C₂³	1	4	yes	—
130	C₂×C₄×#41	C₂×C₅	1	1	yes	characteristic
131	#2357	C₂×C₅	1	2	yes	complemented
132	#2356	C₂×C₅	1	2	yes	complemented
133	#2355	C₂×C₅	1	2	yes	complemented
134	#2354	C₂×C₅	1	2	yes	complemented
135	#2353	C₂×C₅	1	4	yes	complemented
136	#2352	C₂×C₅	1	4	yes	complemented
137	#2298	C₂×C₅	1	4	yes	complemented
138	#2297	C₂×C₅	1	2	yes	—
139	#2285	C₂×C₅	1	1	yes	characteristic, complemented
140	#2283	C₂×C₅	1	2	yes	complemented
141	C₄×#117	C₂×C₅	1	2	yes	complemented
142	C₄×#117	C₂×C₅	1	2	yes	complemented
143	C₂×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
144	C₂⁴×C₅	C₂⁴	1	2	yes	—
145	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
146	C₅×#41	—	2	32	no	—
147	C₂×C₅×Q₈	—	2	2	no	—
148	C₂×C₅×Q₈	—	2	2	no	—
149	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
150	C₂×C₅×D₄	—	2	4	no	—
151	C₂×C₅×D₄	—	2	4	no	—
152	C₂×C₅×D₄	—	2	2	no	—
153	C₂×C₅×D₄	—	2	2	no	—
154	C₂×C₅×D₄	C₂²×C₄	1	8	yes	—
155	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
156	C₂²×C₄×C₅	—	2	4	no	—
157	C₂²×C₄×C₅	—	2	8	no	—
158	C₂²×C₄×C₅	—	2	4	no	—
159	C₂²×C₄×C₅	—	2	4	no	—
160	C₂²×C₄×C₅	—	2	8	no	—
161	C₂²×C₄×C₅	—	2	8	no	—
162	C₂²×C₄×C₅	—	4	4	no	—
163	C₂²×C₄×C₅	—	4	4	no	—
164	C₂²×C₄×C₅	—	4	2	no	—
165	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
166	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
167	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
168	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
169	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
170	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
171	C₂²×C₄×C₅	#41	1	4	yes	—
172	C₂²×C₄×C₅	#41	1	4	yes	—
173	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
174	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
177	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
182	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
184	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
185	C₅×C₄⋊C₄	—	2	8	no	—
186	C₅×C₄⋊C₄	—	2	4	no	—
187	C₅×C₄⋊C₄	—	2	8	no	—
188	C₅×C₄⋊C₄	—	2	8	no	—
189	C₅×C₄⋊C₄	—	4	4	no	—
190	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
191	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
192	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
193	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
194	C₅×C₄⋊C₄	#41	1	8	yes	complemented
195	C₅×C₂²⋊C₄	—	2	8	no	—
196	C₅×C₂²⋊C₄	—	2	16	no	—
197	C₅×C₂²⋊C₄	—	4	8	no	—
198	C₅×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
199	C₅×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
200	C₅×C₂²⋊C₄	C₂²×C₄	1	8	yes	—
201	C₄²×C₅	—	2	4	no	—
202	C₄²×C₅	—	2	8	no	—
203	C₄²×C₅	—	2	4	no	—
204	C₄²×C₅	—	2	16	no	—
205	C₄²×C₅	—	2	8	no	—
206	C₄²×C₅	—	4	8	no	—
207	C₄²×C₅	C₂²×C₄	1	4	yes	—
208	C₄²×C₅	C₂²×C₄	1	4	yes	—
209	C₄²×C₅	C₂²×C₄	1	4	yes	—
210	C₄²×C₅	C₂²×C₄	1	4	yes	—
211	C₄²×C₅	C₂²×C₄	1	8	yes	—
212	C₄²×C₅	C₂²×C₄	1	8	yes	—
213	C₄²×C₅	C₂²×C₄	1	8	yes	—
214	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
215	C₄×#41	C₄×C₅	1	16	yes	complemented
216	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
217	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
218	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
219	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
220	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
221	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
222	C₂×#117	C₂²×C₅	1	4	yes	—
223	C₂×#117	C₂²×C₅	1	2	yes	—
224	C₂×#117	C₂²×C₅	1	1	yes	characteristic
225	C₂×#117	C₂²×C₅	1	2	yes	—
226	C₂²×C₄⋊C₄	C₂²×C₅	1	2	yes	—
227	C₂²×C₄²	C₂²×C₅	1	2	yes	—
228	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
229	C₂²×C₄²	C₂²×C₅	1	2	yes	—
230	#391	C₂²×C₅	1	4	yes	—
231	#390	C₂²×C₅	1	4	yes	—
232	#388	C₂²×C₅	1	8	yes	—
233	#388	C₂²×C₅	1	8	yes	—
234	#387	C₂²×C₅	1	8	yes	—
235	#387	C₂²×C₅	1	8	yes	—
236	#386	C₂²×C₅	1	8	yes	—
237	#384	C₂²×C₅	1	2	yes	—
238	#383	C₂²×C₅	1	4	yes	—
239	#382	C₂²×C₅	1	4	yes	—
240	#382	C₂²×C₅	1	2	yes	—
241	#381	C₂²×C₅	1	4	yes	—
242	#380	C₂²×C₅	1	2	yes	—
243	#380	C₂²×C₅	1	4	yes	—
244	#380	C₂²×C₅	1	2	yes	—
245	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
246	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
247	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
248	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
249	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
250	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
251	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
252	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
253	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
254	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
255	C₄×C₂²⋊C₄	C₂²×C₅	1	8	yes	—
256	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
257	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
258	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
259	#376	C₂²×C₅	1	4	yes	—
260	#376	C₂²×C₅	1	4	yes	—
261	C₂×#95	C₂²×C₅	1	2	yes	—
262	C₄³	C₂²×C₅	1	4	yes	—
263	C₂³×C₅	—	2	2	no	—
264	C₂³×C₅	—	2	2	no	—
265	C₂³×C₅	—	2	4	no	—
266	C₂³×C₅	—	2	2	no	—
267	C₂³×C₅	—	2	2	no	—
268	C₂³×C₅	—	2	4	no	—
269	C₂³×C₅	C₂³×C₄	1	4	yes	—
270	C₂³×C₅	C₂³×C₄	1	2	yes	—
271	C₂³×C₅	C₂×#41	1	2	yes	—
272	C₂³×C₅	C₂×#41	1	2	yes	—
273	C₂³×C₅	C₂⁵	1	1	yes	characteristic
274	C₅×Q₈	—	2	8	no	—
275	C₅×D₄	—	2	16	no	—
276	C₅×D₄	—	2	8	no	—
277	C₂×C₄×C₅	—	2	4	no	—
278	C₂×C₄×C₅	—	2	4	no	—
279	C₂×C₄×C₅	—	2	2	no	—
280	C₂×C₄×C₅	—	2	2	no	—
281	C₂×C₄×C₅	—	2	2	no	—
282	C₂×C₄×C₅	—	2	4	no	—
283	C₂×C₄×C₅	—	2	4	no	—
284	C₂×C₄×C₅	—	2	2	no	—
285	C₂×C₄×C₅	—	2	2	no	—
286	C₂×C₄×C₅	—	2	16	no	—
287	C₂×C₄×C₅	—	2	8	no	—
288	C₂×C₄×C₅	—	2	16	no	—
289	C₂×C₄×C₅	—	2	4	no	—
290	C₂×C₄×C₅	—	2	8	no	—
291	C₂×C₄×C₅	—	2	8	no	—
292	C₂×C₄×C₅	—	2	4	no	—
293	C₂×C₄×C₅	—	2	8	no	—
294	C₂×C₄×C₅	—	2	4	no	—
295	C₂×C₄×C₅	—	4	4	no	—
296	C₂×C₄×C₅	—	4	8	no	—
297	C₂×C₄×C₅	—	4	16	no	—
298	C₂×C₄×C₅	#117	1	16	yes	complemented
299	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
300	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
301	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
302	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
303	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
304	C₂×C₄×C₅	C₂³×C₄	1	4	yes	—
305	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
306	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
307	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
308	C₂×C₄×C₅	C₂×#41	1	8	yes	—
309	C₂×C₄×C₅	C₂×#41	1	2	yes	—
310	C₂×C₄×C₅	C₂×#41	1	2	yes	—
311	C₂×C₄×C₅	C₂×#41	1	2	yes	—
312	C₂×C₄×C₅	C₂×#41	1	2	yes	—
313	C₂×C₄×C₅	C₂×#41	1	4	yes	—
314	C₂×#41	—	2	4	no	—
315	C₂×#41	—	2	4	no	—
316	C₂×#41	C₂×C₄×C₅	1	8	yes	—
317	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
318	C₂²×D₄	C₂³×C₅	1	2	yes	—
319	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₅	1	4	yes	—
321	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
322	C₂³×C₄	C₂³×C₅	1	2	yes	—
323	C₂³×C₄	C₂³×C₅	1	2	yes	—
324	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
325	C₂³×C₄	C₂³×C₅	1	2	yes	—
326	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
327	C₄×Q₈	C₂×C₄×C₅	1	8	yes	complemented
328	C₄×Q₈	C₂×C₄×C₅	1	8	yes	complemented
329	C₄×D₄	C₂×C₄×C₅	1	16	yes	—
330	C₄×D₄	C₂×C₄×C₅	1	16	yes	—
331	C₄×D₄	C₂×C₄×C₅	1	8	yes	complemented
332	C₄×D₄	C₂×C₄×C₅	1	8	yes	complemented
333	#117	—	2	8	no	—
334	#117	—	2	16	no	—
335	#117	C₂×C₄×C₅	1	8	yes	—
336	#117	C₂×C₄×C₅	1	16	yes	complemented
337	C₂×C₄⋊C₄	—	2	8	no	—
338	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
339	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
340	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
341	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
342	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
343	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
344	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
345	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
346	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
347	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
349	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
354	C₂×C₄²	—	2	8	no	—
355	C₂×C₄²	C₂×C₄×C₅	1	8	yes	—
356	C₂×C₄²	C₂×C₄×C₅	1	16	yes	complemented
357	C₂×C₄²	C₂³×C₅	1	4	yes	—
358	C₂×C₄²	C₂³×C₅	1	8	yes	—
359	C₂×C₄²	C₂³×C₅	1	2	yes	—
360	C₂×C₄²	C₂³×C₅	1	2	yes	—
361	C₂×C₄²	C₂³×C₅	1	4	yes	—
362	C₂×C₄²	C₂³×C₅	1	8	yes	—
363	C₂×C₄²	C₂³×C₅	1	2	yes	—
364	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
365	C₂×C₄²	C₂³×C₅	1	2	yes	—
366	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
367	C₂×C₄²	C₂³×C₅	1	2	yes	—
368	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
369	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
370	#95	C₂³×C₅	1	2	yes	—
371	#95	C₂³×C₅	1	2	yes	—
372	#95	C₂³×C₅	1	4	yes	—
373	#95	C₂³×C₅	1	4	yes	—
374	#95	C₂³×C₅	1	8	yes	—
375	#95	C₂³×C₅	1	8	yes	—
376	#95	C₂³×C₅	1	8	yes	—
377	#95	C₂³×C₅	1	8	yes	—
378	#95	C₂³×C₅	1	4	yes	—
379	C₂²×C₅	—	2	2	no	—
380	C₂²×C₅	—	2	2	no	—
381	C₂²×C₅	—	2	4	no	—
382	C₂²×C₅	—	2	4	no	—
383	C₂²×C₅	—	2	4	no	—
384	C₂²×C₅	—	2	4	no	—
385	C₂²×C₅	—	2	4	no	—
386	C₂²×C₅	—	4	4	no	—
387	C₂²×C₅	—	4	4	no	—
388	C₂²×C₅	C₂×#117	1	4	yes	—
389	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
390	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
391	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
392	C₂²×C₅	C₂²×#41	1	2	yes	—
393	C₂²×C₅	#585	1	2	yes	—
394	C₄×C₅	—	2	4	no	—
395	C₄×C₅	—	2	4	no	—
396	C₄×C₅	—	2	2	no	—
397	C₄×C₅	—	2	2	no	—
398	C₄×C₅	—	2	4	no	—
399	C₄×C₅	—	2	2	no	—
400	C₄×C₅	—	2	8	no	—
401	C₄×C₅	—	4	4	no	—
402	C₄×C₅	—	4	8	no	—
403	C₄×C₅	C₂×#117	1	4	yes	—
404	C₄×C₅	C₂×#117	1	4	yes	—
405	C₄×C₅	C₄×#41	1	4	yes	—
406	C₂⁴	C₂⁴×C₅	1	2	yes	—
407	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
408	#41	—	2	32	no	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	2	no	—
411	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	4	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	C₂²×C₄×C₅	1	8	yes	—
417	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	8	no	—
420	C₂²×C₄	—	2	4	no	—
421	C₂²×C₄	—	2	4	no	—
422	C₂²×C₄	—	2	8	no	—
423	C₂²×C₄	—	2	8	no	—
424	C₂²×C₄	—	4	4	no	—
425	C₂²×C₄	—	4	4	no	—
426	C₂²×C₄	—	4	2	no	—
427	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
428	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
429	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
430	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
431	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
432	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
433	C₂²×C₄	C₅×#41	1	4	yes	—
434	C₂²×C₄	C₅×#41	1	4	yes	—
435	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
436	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
437	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
438	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
440	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
441	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
443	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
444	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
446	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
447	C₄⋊C₄	—	2	8	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	8	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	4	4	no	—
452	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
453	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
454	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
455	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
456	C₄⋊C₄	C₅×#41	1	8	yes	complemented
457	C₂²⋊C₄	—	2	8	no	—
458	C₂²⋊C₄	—	2	16	no	—
459	C₂²⋊C₄	—	4	8	no	—
460	C₂²⋊C₄	C₂²×C₄×C₅	1	8	yes	—
461	C₂²⋊C₄	C₂²×C₄×C₅	1	8	yes	—
462	C₂²⋊C₄	C₂²×C₄×C₅	1	8	yes	—
463	C₄²	—	2	4	no	—
464	C₄²	—	2	8	no	—
465	C₄²	—	2	4	no	—
466	C₄²	—	2	16	no	—
467	C₄²	—	2	8	no	—
468	C₄²	—	4	8	no	—
469	C₄²	C₂²×C₄×C₅	1	4	yes	—
470	C₄²	C₂²×C₄×C₅	1	4	yes	—
471	C₄²	C₂²×C₄×C₅	1	4	yes	—
472	C₄²	C₂²×C₄×C₅	1	4	yes	—
473	C₄²	C₂²×C₄×C₅	1	8	yes	—
474	C₄²	C₂²×C₄×C₅	1	8	yes	—
475	C₄²	C₂²×C₄×C₅	1	8	yes	—
476	C₂×C₅	—	2	4	no	—
477	C₂×C₅	—	4	4	no	—
478	C₂×C₅	C₂²×#117	1	1	yes	characteristic
479	C₂×C₅	C₂×C₄×#41	1	1	yes	characteristic
480	C₂×C₅	#3428	1	1	yes	characteristic
481	C₂×C₅	#3475	1	2	yes	—
482	C₂×C₅	#3480	1	2	yes	—
483	C₂³	—	2	2	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	4	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	4	no	—
489	C₂³	C₂³×C₄×C₅	1	4	yes	—
490	C₂³	C₂³×C₄×C₅	1	2	yes	—
491	C₂³	C₂×C₅×#41	1	2	yes	—
492	C₂³	C₂×C₅×#41	1	2	yes	—
493	C₂³	C₂⁵×C₅	1	1	yes	characteristic
494	Q₈	—	2	8	no	—
495	D₄	—	2	16	no	—
496	D₄	—	2	8	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	4	no	—
503	C₂×C₄	—	2	4	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	no	—
506	C₂×C₄	—	2	16	no	—
507	C₂×C₄	—	2	8	no	—
508	C₂×C₄	—	2	16	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	8	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	4	4	no	—
516	C₂×C₄	—	4	8	no	—
517	C₂×C₄	—	4	16	no	—
518	C₂×C₄	C₅×#117	1	16	yes	complemented
519	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
520	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
521	C₂×C₄	C₂³×C₄×C₅	1	4	yes	—
522	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
523	C₂×C₄	C₂³×C₄×C₅	1	4	yes	—
524	C₂×C₄	C₂³×C₄×C₅	1	4	yes	—
525	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
526	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
527	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
528	C₂×C₄	C₂×C₅×#41	1	8	yes	—
529	C₂×C₄	C₂×C₅×#41	1	2	yes	—
530	C₂×C₄	C₂×C₅×#41	1	2	yes	—
531	C₂×C₄	C₂×C₅×#41	1	2	yes	—
532	C₂×C₄	C₂×C₅×#41	1	2	yes	—
533	C₂×C₄	C₂×C₅×#41	1	4	yes	—
534	C₅	#34121	1	1	yes	characteristic, direct product factor, Sylow
535	C₂²	—	2	2	no	—
536	C₂²	—	2	2	no	—
537	C₂²	—	2	4	no	—
538	C₂²	—	2	4	no	—
539	C₂²	—	2	4	no	—
540	C₂²	—	2	4	no	—
541	C₂²	—	2	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	4	no	—
544	C₂²	C₂×C₅×#117	1	4	yes	—
545	C₂²	C₂⁴×C₄×C₅	1	1	yes	characteristic
546	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
547	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
548	C₂²	C₂²×C₅×#41	1	2	yes	—
549	C₂²	C₅×#585	1	2	yes	—
550	C₄	—	2	4	no	—
551	C₄	—	2	4	no	—
552	C₄	—	2	2	no	—
553	C₄	—	2	2	no	—
554	C₄	—	2	4	no	—
555	C₄	—	2	2	no	—
556	C₄	—	2	8	no	—
557	C₄	—	4	4	no	—
558	C₄	—	4	8	no	—
559	C₄	C₂×C₅×#117	1	4	yes	—
560	C₄	C₂×C₅×#117	1	4	yes	—
561	C₄	C₄×C₅×#41	1	4	yes	—
562	C₂	—	2	4	no	—
563	C₂	—	4	4	no	—
564	C₂	C₂²×C₅×#117	1	1	yes	characteristic
565	C₂	C₂×C₄×C₅×#41	1	1	yes	characteristic
566	C₂	C₅×#3428	1	1	yes	characteristic
567	C₂	C₅×#3475	1	2	yes	—
568	C₂	C₅×#3480	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₅
Frattini, cosocle	C₂³	C₂⁵×C₅
socle	C₂³×C₅	C₂⁵
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2357	C₂	semidirect
3. C₅×#2356	C₂	semidirect
4. C₅×#2355	C₂	semidirect
5. C₅×#2354	C₂	semidirect
6. C₅×#2353	C₂	semidirect
7. C₅×#2352	C₂	semidirect
8. C₅×#2298	C₂	semidirect
10. C₅×#2285	C₂	semidirect
11. C₅×#2283	C₂	semidirect
12. C₄×C₅×#117	C₂	semidirect
13. C₄×C₅×#117	C₂	semidirect
14. C₂×C₄×C₅×C₄⋊C₄	C₂	semidirect
16. C₄×C₅×#41	C₄	semidirect
64. #34121	C₅	direct (N × H)
78. C₄×C₅×Q₈	C₂×C₄	semidirect
79. C₄×C₅×Q₈	C₂×C₄	semidirect
82. C₄×C₅×D₄	C₂×C₄	semidirect
83. C₄×C₅×D₄	C₂×C₄	semidirect
87. C₅×#117	C₂×C₄	semidirect
107. C₂×C₄²×C₅	C₂×C₄	semidirect
131. #2357	C₂×C₅	semidirect
132. #2356	C₂×C₅	semidirect
133. #2355	C₂×C₅	semidirect
134. #2354	C₂×C₅	semidirect
135. #2353	C₂×C₅	semidirect
136. #2352	C₂×C₅	semidirect
137. #2298	C₂×C₅	semidirect
139. #2285	C₂×C₅	semidirect
140. #2283	C₂×C₅	semidirect
141. C₄×#117	C₂×C₅	semidirect
142. C₄×#117	C₂×C₅	semidirect
143. C₂×C₄×C₄⋊C₄	C₂×C₅	semidirect
194. C₅×C₄⋊C₄	#41	semidirect
215. C₄×#41	C₄×C₅	semidirect
298. C₂×C₄×C₅	#117	semidirect
327. C₄×Q₈	C₂×C₄×C₅	semidirect
328. C₄×Q₈	C₂×C₄×C₅	semidirect
331. C₄×D₄	C₂×C₄×C₅	semidirect
332. C₄×D₄	C₂×C₄×C₅	semidirect
336. #117	C₂×C₄×C₅	semidirect
356. C₂×C₄²	C₂×C₄×C₅	semidirect
456. C₄⋊C₄	C₅×#41	semidirect
518. C₂×C₄	C₅×#117	semidirect
534. C₅	#34121	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×#41	2
C₅×#2357	2
C₅×#2356	2
C₅×#2355	2
C₅×#2354	2
C₅×#2353	2
C₅×#2352	2
C₅×#2298	2
C₅×#2297	2
C₅×#2285	2
C₅×#2283	2
C₄×C₅×#117	2
C₂×C₄×C₅×C₄⋊C₄	2
#34121	5

Subgroups — C5×#34121 #11956841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#34121 #11956841