Subgroups — C₅×#22999 #11945719

Order 1280.

1034 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#463	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
4	C₅×#2367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1958	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1949	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1934	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1933	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1879	C₂	1	1	yes	maximal, characteristic
11	C₅×#1877	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1864	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1863	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₅×#102	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₅×SD₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
18	C₂×C₅×#124	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
24	C₅×#463	C₄	1	8	yes	complemented
25	C₄×C₅×SD₁₆	—	2	4	no	—
26	C₄×C₅×D₈	—	2	4	no	—
27	C₂×C₅×#107	C₂²	1	1	yes	characteristic
28	C₂×C₅×#106	C₂²	1	1	yes	characteristic
29	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₂×C₅×#103	C₂²	1	1	yes	characteristic
31	C₂×C₅×#103	C₂²	1	1	yes	characteristic
32	C₂×C₅×#103	C₂²	1	1	yes	characteristic
33	C₂×C₅×#102	C₂²	1	1	yes	characteristic
34	C₂×C₅×#102	C₂²	1	1	yes	characteristic
35	C₂×C₅×#102	C₂²	1	1	yes	characteristic
36	C₂×C₅×#102	C₂²	1	1	yes	characteristic
37	C₂×C₅×#102	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
41	C₅×#391	C₂²	1	1	yes	characteristic
42	C₅×#390	C₂²	1	1	yes	characteristic
43	C₅×#389	C₂²	1	1	yes	characteristic
44	C₅×#389	C₂²	1	1	yes	characteristic
45	C₅×#388	—	2	1	no	—
46	C₅×#388	—	2	1	no	—
47	C₅×#385	—	2	1	no	—
48	C₅×#385	—	2	1	no	—
49	C₅×#385	—	2	1	no	—
50	C₅×#382	C₂²	1	1	yes	characteristic
51	C₅×#380	—	2	1	no	—
52	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
53	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
54	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
55	C₂×C₅×#95	—	2	1	no	—
56	C₅×#340	C₂²	1	1	yes	characteristic
57	C₅×#340	C₂²	1	1	yes	characteristic
58	C₅×#340	C₂²	1	1	yes	characteristic
59	C₅×#340	C₂²	1	1	yes	characteristic
60	C₅×#336	C₂²	1	1	yes	characteristic
61	C₅×#336	C₂²	1	1	yes	characteristic
62	#22999	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
64	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
65	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₅	—	2	1	no	—
67	C₂³×C₄×C₅	—	2	1	no	—
68	C₂³×C₄×C₅	—	2	1	no	—
69	C₂³×C₄×C₅	—	2	1	no	—
70	C₂×C₅×SD₁₆	—	2	2	no	—
71	C₂×C₅×SD₁₆	—	2	2	no	—
72	C₂×C₅×SD₁₆	C₂×C₄	1	4	yes	complemented
73	C₂×C₅×D₈	—	2	2	no	—
74	C₂×C₅×D₈	—	2	2	no	—
75	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
76	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
77	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
78	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
79	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
80	C₅×#124	—	2	2	no	—
81	C₅×#124	C₂×C₄	1	4	yes	complemented
82	C₄×C₅×Q₈	—	2	2	no	—
83	C₄×C₅×Q₈	—	4	1	no	—
84	C₄×C₅×D₄	—	2	2	no	—
85	C₄×C₅×D₄	—	2	2	no	—
86	C₄×C₅×D₄	—	2	2	no	—
87	C₄×C₅×D₄	—	2	2	no	—
88	C₄×C₅×D₄	—	4	1	no	—
89	C₄×C₅×D₄	—	4	1	no	—
90	C₄×C₅×D₄	—	4	1	no	—
91	C₄×C₅×D₄	C₂×C₄	1	4	yes	complemented
92	C₂×C₅×C₄⋊C₄	—	2	1	no	—
93	C₂×C₅×C₄⋊C₄	—	2	1	no	—
94	C₂×C₅×C₄⋊C₄	—	2	1	no	—
95	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
96	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
97	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
98	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
99	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
100	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
101	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
102	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
103	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
104	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
105	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
106	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
107	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
108	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
109	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
110	C₂×C₄²×C₅	—	2	1	no	—
111	C₂×C₄²×C₅	—	2	1	no	—
112	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
115	C₅×#107	—	2	2	no	—
116	C₅×#106	—	2	2	no	—
117	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
118	C₅×#103	—	2	2	no	—
119	C₅×#103	—	4	1	no	—
120	C₅×#103	C₂×C₄	1	4	yes	complemented
121	C₅×#102	—	2	2	no	—
122	C₅×#102	—	2	2	no	—
123	C₅×#102	—	4	1	no	—
124	C₅×#102	C₂×C₄	1	4	yes	complemented
125	C₅×#102	D₄	1	4	yes	—
126	C₄×C₅×C₈	—	2	2	no	—
127	C₄×C₅×C₈	—	2	2	no	—
128	C₄×C₅×C₈	—	2	2	no	—
129	C₅×#95	—	2	1	no	—
130	C₅×#95	—	2	1	no	—
131	C₅×#95	—	2	1	no	—
132	C₅×#95	—	2	1	no	—
133	C₅×#95	—	2	1	no	—
134	C₅×#95	—	2	1	no	—
135	C₅×#95	—	2	1	no	—
136	C₅×#95	D₄	1	1	yes	characteristic
137	C₅×#95	D₄	1	1	yes	characteristic
138	C₂×#463	C₂×C₅	1	1	yes	characteristic
139	C₂×C₄×SD₁₆	C₂×C₅	1	1	yes	characteristic, complemented
140	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic
141	#2367	C₂×C₅	1	1	yes	characteristic, complemented
142	#2349	C₂×C₅	1	1	yes	characteristic, complemented
143	#1958	C₂×C₅	1	1	yes	characteristic, complemented
144	#1949	C₂×C₅	1	1	yes	characteristic, complemented
145	#1934	C₂×C₅	1	1	yes	characteristic, complemented
146	#1933	C₂×C₅	1	1	yes	characteristic, complemented
147	#1879	C₂×C₅	1	1	yes	characteristic
148	#1877	C₂×C₅	1	1	yes	characteristic, complemented
149	#1864	C₂×C₅	1	1	yes	characteristic, complemented
150	#1863	C₂×C₅	1	1	yes	characteristic, complemented
151	C₈×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
152	C₄×#102	C₂×C₅	1	1	yes	characteristic, complemented
153	C₂⁴×C₅	—	2	1	no	—
154	C₂⁴×C₅	—	2	1	no	—
155	C₂×C₅×Q₈	—	2	1	no	—
156	C₂×C₅×Q₈	—	2	1	no	—
157	C₂×C₅×Q₈	—	2	1	no	—
158	C₂×C₅×Q₈	—	2	1	no	—
159	C₂×C₅×Q₈	—	2	1	no	—
160	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
161	C₂×C₅×D₄	—	2	1	no	—
162	C₂×C₅×D₄	—	2	1	no	—
163	C₂×C₅×D₄	—	2	1	no	—
164	C₂×C₅×D₄	—	2	1	no	—
165	C₂×C₅×D₄	—	2	1	no	—
166	C₂×C₅×D₄	—	2	1	no	—
167	C₂×C₅×D₄	—	2	1	no	—
168	C₂×C₅×D₄	—	2	1	no	—
169	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
170	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
171	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
172	C₂×C₅×D₄	#41	1	2	yes	—
173	C₂²×C₄×C₅	—	2	1	no	—
174	C₂²×C₄×C₅	—	2	1	no	—
175	C₂²×C₄×C₅	—	2	1	no	—
176	C₂²×C₄×C₅	—	2	4	no	—
177	C₂²×C₄×C₅	—	2	1	no	—
178	C₂²×C₄×C₅	—	2	1	no	—
179	C₂²×C₄×C₅	—	2	1	no	—
180	C₂²×C₄×C₅	—	4	2	no	—
181	C₂²×C₄×C₅	—	4	2	no	—
182	C₂²×C₄×C₅	—	4	2	no	—
183	C₂²×C₄×C₅	—	4	2	no	—
184	C₂²×C₄×C₅	—	4	2	no	—
185	C₂²×C₄×C₅	—	4	2	no	—
186	C₂²×C₄×C₅	—	8	1	no	—
187	C₂²×C₄×C₅	—	8	1	no	—
188	C₂²×C₄×C₅	—	8	1	no	—
189	C₂²×C₄×C₅	—	8	1	no	—
190	C₂²×C₄×C₅	—	8	1	no	—
191	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
195	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
198	C₅×SD₁₆	—	2	8	no	—
199	C₅×D₈	—	2	8	no	—
200	C₂×C₅×C₈	—	2	1	no	—
201	C₂×C₅×C₈	—	2	1	no	—
202	C₂×C₅×C₈	—	2	1	no	—
203	C₂×C₅×C₈	—	2	1	no	—
204	C₂×C₅×C₈	—	2	1	no	—
205	C₂×C₅×C₈	—	2	1	no	—
206	C₂×C₅×C₈	—	2	1	no	—
207	C₂×C₅×C₈	—	2	1	no	—
208	C₂×C₅×C₈	—	2	1	no	—
209	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
210	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
211	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
212	C₅×C₄⋊C₄	—	2	2	no	—
213	C₅×C₄⋊C₄	—	2	2	no	—
214	C₅×C₄⋊C₄	—	2	1	no	—
215	C₅×C₄⋊C₄	—	2	1	no	—
216	C₅×C₄⋊C₄	—	2	1	no	—
217	C₅×C₄⋊C₄	—	2	1	no	—
218	C₅×C₄⋊C₄	—	2	1	no	—
219	C₅×C₄⋊C₄	—	2	1	no	—
220	C₅×C₄⋊C₄	—	2	1	no	—
221	C₅×C₄⋊C₄	—	2	1	no	—
222	C₅×C₄⋊C₄	—	4	2	no	—
223	C₅×C₄⋊C₄	—	8	1	no	—
224	C₅×C₄⋊C₄	—	8	1	no	—
225	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
226	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
227	C₅×C₂²⋊C₄	—	2	4	no	—
228	C₅×C₂²⋊C₄	—	2	4	no	—
229	C₅×C₂²⋊C₄	—	2	4	no	—
230	C₅×C₂²⋊C₄	—	4	2	no	—
231	C₅×C₂²⋊C₄	—	4	2	no	—
232	C₅×C₂²⋊C₄	—	4	2	no	—
233	C₅×C₂²⋊C₄	—	4	2	no	—
234	C₅×C₂²⋊C₄	—	8	1	no	—
235	C₄²×C₅	—	2	1	no	—
236	C₄²×C₅	—	2	1	no	—
237	C₄²×C₅	—	2	1	no	—
238	C₄²×C₅	—	2	1	no	—
239	C₄²×C₅	—	4	2	no	—
240	C₄²×C₅	—	8	1	no	—
241	C₄²×C₅	C₂²×C₄	1	2	yes	—
242	C₄²×C₅	#41	1	2	yes	—
243	C₂²×SD₁₆	C₂²×C₅	1	1	yes	characteristic
244	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
245	C₂×#124	C₂²×C₅	1	1	yes	characteristic
246	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
247	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
248	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
249	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
250	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
251	#463	C₄×C₅	1	8	yes	complemented
252	C₄×SD₁₆	—	2	4	no	—
253	C₄×D₈	—	2	4	no	—
254	C₂×#107	C₂²×C₅	1	1	yes	characteristic
255	C₂×#106	C₂²×C₅	1	1	yes	characteristic
256	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
257	C₂×#103	C₂²×C₅	1	1	yes	characteristic
258	C₂×#103	C₂²×C₅	1	1	yes	characteristic
259	C₂×#103	C₂²×C₅	1	1	yes	characteristic
260	C₂×#102	C₂²×C₅	1	1	yes	characteristic
261	C₂×#102	C₂²×C₅	1	1	yes	characteristic
262	C₂×#102	C₂²×C₅	1	1	yes	characteristic
263	C₂×#102	C₂²×C₅	1	1	yes	characteristic
264	C₂×#102	C₂²×C₅	1	1	yes	characteristic
265	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
266	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
267	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
268	#391	C₂²×C₅	1	1	yes	characteristic
269	#390	C₂²×C₅	1	1	yes	characteristic
270	#389	C₂²×C₅	1	1	yes	characteristic
271	#389	C₂²×C₅	1	1	yes	characteristic
272	#388	—	2	1	no	—
273	#388	—	2	1	no	—
274	#385	—	2	1	no	—
275	#385	—	2	1	no	—
276	#385	—	2	1	no	—
277	#382	C₂²×C₅	1	1	yes	characteristic
278	#380	—	2	1	no	—
279	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
280	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
281	C₄×C₂²⋊C₄	—	2	1	no	—
282	C₂×#95	—	2	1	no	—
283	#340	C₂²×C₅	1	1	yes	characteristic
284	#340	C₂²×C₅	1	1	yes	characteristic
285	#340	C₂²×C₅	1	1	yes	characteristic
286	#340	C₂²×C₅	1	1	yes	characteristic
287	#336	C₂²×C₅	1	1	yes	characteristic
288	#336	C₂²×C₅	1	1	yes	characteristic
289	C₂³×C₅	—	2	2	no	—
290	C₂³×C₅	—	2	2	no	—
291	C₂³×C₅	—	2	2	no	—
292	C₂³×C₅	—	2	2	no	—
293	C₂³×C₅	—	4	1	no	—
294	C₂³×C₅	—	4	1	no	—
295	C₂³×C₅	—	4	1	no	—
296	C₂³×C₅	—	4	1	no	—
297	C₂³×C₅	—	4	1	no	—
298	C₂³×C₅	—	4	1	no	—
299	C₂³×C₅	—	4	1	no	—
300	C₂³×C₅	—	4	1	no	—
301	C₂³×C₅	—	4	1	no	—
302	C₂³×C₅	—	4	1	no	—
303	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
304	C₅×Q₈	—	2	2	no	—
305	C₅×Q₈	—	2	2	no	—
306	C₅×Q₈	—	4	1	no	—
307	C₅×Q₈	—	4	1	no	—
308	C₅×D₄	—	2	2	no	—
309	C₅×D₄	—	2	2	no	—
310	C₅×D₄	—	2	2	no	—
311	C₅×D₄	—	2	2	no	—
312	C₅×D₄	—	2	2	no	—
313	C₅×D₄	—	4	1	no	—
314	C₅×D₄	—	4	1	no	—
315	C₅×D₄	C₄×D₄	1	4	yes	—
316	C₂×C₄×C₅	—	2	1	no	—
317	C₂×C₄×C₅	—	2	1	no	—
318	C₂×C₄×C₅	—	2	1	no	—
319	C₂×C₄×C₅	—	2	1	no	—
320	C₂×C₄×C₅	—	2	1	no	—
321	C₂×C₄×C₅	—	2	1	no	—
322	C₂×C₄×C₅	—	2	1	no	—
323	C₂×C₄×C₅	—	2	1	no	—
324	C₂×C₄×C₅	—	2	2	no	—
325	C₂×C₄×C₅	—	2	2	no	—
326	C₂×C₄×C₅	—	2	1	no	—
327	C₂×C₄×C₅	—	2	1	no	—
328	C₂×C₄×C₅	—	2	2	no	—
329	C₂×C₄×C₅	—	2	1	no	—
330	C₂×C₄×C₅	—	2	1	no	—
331	C₂×C₄×C₅	—	2	1	no	—
332	C₂×C₄×C₅	—	2	1	no	—
333	C₂×C₄×C₅	—	4	1	no	—
334	C₂×C₄×C₅	—	4	1	no	—
335	C₂×C₄×C₅	—	4	1	no	—
336	C₂×C₄×C₅	—	4	1	no	—
337	C₂×C₄×C₅	—	4	1	no	—
338	C₂×C₄×C₅	—	4	1	no	—
339	C₂×C₄×C₅	—	4	1	no	—
340	C₂×C₄×C₅	—	4	1	no	—
341	C₂×C₄×C₅	—	4	1	no	—
342	C₂×C₄×C₅	—	4	1	no	—
343	C₂×C₄×C₅	—	4	1	no	—
344	C₂×C₄×C₅	—	4	1	no	—
345	C₂×C₄×C₅	—	4	1	no	—
346	C₂×C₄×C₅	—	4	1	no	—
347	C₂×C₄×C₅	—	4	1	no	—
348	C₂×C₄×C₅	—	8	4	no	—
349	C₂×C₄×C₅	—	8	4	no	—
350	C₂×C₄×C₅	—	8	4	no	—
351	C₂×C₄×C₅	—	8	4	no	—
352	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
353	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
354	C₂×C₄×C₅	#135	1	2	yes	—
355	C₂×C₄×C₅	#135	1	2	yes	—
356	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
357	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
358	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
359	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
360	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
361	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
362	C₅×C₈	—	2	2	no	—
363	C₅×C₈	—	2	2	no	—
364	C₅×C₈	—	2	2	no	—
365	C₅×C₈	—	4	1	no	—
366	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
367	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
368	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
369	C₂³×C₄	—	2	1	no	—
370	C₂³×C₄	—	2	1	no	—
371	C₂³×C₄	—	2	1	no	—
372	C₂³×C₄	—	2	1	no	—
373	C₂×SD₁₆	—	2	2	no	—
374	C₂×SD₁₆	—	2	2	no	—
375	C₂×SD₁₆	C₂×C₄×C₅	1	4	yes	complemented
376	C₂×D₈	—	2	2	no	—
377	C₂×D₈	—	2	2	no	—
378	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
379	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
380	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
381	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
382	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
383	#124	—	2	2	no	—
384	#124	C₂×C₄×C₅	1	4	yes	complemented
385	C₄×Q₈	—	2	2	no	—
386	C₄×Q₈	—	4	1	no	—
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	—	2	2	no	—
389	C₄×D₄	—	2	2	no	—
390	C₄×D₄	—	2	2	no	—
391	C₄×D₄	—	4	1	no	—
392	C₄×D₄	—	4	1	no	—
393	C₄×D₄	—	4	1	no	—
394	C₄×D₄	C₂×C₄×C₅	1	4	yes	complemented
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	—	2	1	no	—
398	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
400	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
403	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
404	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₂²⋊C₄	—	2	1	no	—
408	C₂×C₂²⋊C₄	—	2	1	no	—
409	C₂×C₂²⋊C₄	—	2	1	no	—
410	C₂×C₂²⋊C₄	—	2	1	no	—
411	C₂×C₂²⋊C₄	—	2	1	no	—
412	C₂×C₂²⋊C₄	—	2	1	no	—
413	C₂×C₄²	—	2	1	no	—
414	C₂×C₄²	—	2	1	no	—
415	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
418	#107	—	2	2	no	—
419	#106	—	2	2	no	—
420	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	complemented
421	#103	—	2	2	no	—
422	#103	—	4	1	no	—
423	#103	C₂×C₄×C₅	1	4	yes	complemented
424	#102	—	2	2	no	—
425	#102	—	2	2	no	—
426	#102	—	4	1	no	—
427	#102	C₂×C₄×C₅	1	4	yes	complemented
428	#102	C₅×D₄	1	4	yes	—
429	C₄×C₈	—	2	2	no	—
430	C₄×C₈	—	2	2	no	—
431	C₄×C₈	—	2	2	no	—
432	#95	—	2	1	no	—
433	#95	—	2	1	no	—
434	#95	—	2	1	no	—
435	#95	—	2	1	no	—
436	#95	—	2	1	no	—
437	#95	—	2	1	no	—
438	#95	—	2	1	no	—
439	#95	C₅×D₄	1	1	yes	characteristic
440	#95	C₅×D₄	1	1	yes	characteristic
441	C₂²×C₅	—	2	4	no	—
442	C₂²×C₅	—	4	2	no	—
443	C₂²×C₅	—	4	2	no	—
444	C₂²×C₅	—	4	2	no	—
445	C₂²×C₅	—	4	2	no	—
446	C₂²×C₅	—	4	2	no	—
447	C₂²×C₅	—	4	2	no	—
448	C₂²×C₅	—	4	2	no	—
449	C₂²×C₅	—	4	2	no	—
450	C₂²×C₅	—	4	2	no	—
451	C₂²×C₅	—	8	1	no	—
452	C₂²×C₅	—	8	1	no	—
453	C₂²×C₅	—	8	1	no	—
454	C₂²×C₅	—	8	1	no	—
455	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
456	C₂²×C₅	D₄²	1	1	yes	characteristic
457	C₂²×C₅	#548	1	1	yes	characteristic
458	C₂²×C₅	C₂×#135	1	1	yes	characteristic
459	C₂²×C₅	C₂×#135	1	1	yes	characteristic
460	C₂²×C₅	#576	1	1	yes	characteristic
461	C₂²×C₅	#577	1	1	yes	characteristic
462	C₄×C₅	—	2	2	no	—
463	C₄×C₅	—	2	2	no	—
464	C₄×C₅	—	2	2	no	—
465	C₄×C₅	—	2	2	no	—
466	C₄×C₅	—	4	2	no	—
467	C₄×C₅	—	4	2	no	—
468	C₄×C₅	—	4	1	no	—
469	C₄×C₅	—	4	2	no	—
470	C₄×C₅	—	4	1	no	—
471	C₄×C₅	—	8	1	no	—
472	C₄×C₅	—	8	1	no	—
473	C₄×C₅	—	8	1	no	—
474	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
475	C₄×C₅	#520	1	1	yes	characteristic
476	C₄×C₅	#547	1	1	yes	characteristic
477	C₄×C₅	#553	1	1	yes	characteristic
478	C₂⁴	—	2	1	no	—
479	C₂⁴	—	2	1	no	—
480	C₂×Q₈	—	2	1	no	—
481	C₂×Q₈	—	2	1	no	—
482	C₂×Q₈	—	2	1	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
486	C₂×D₄	—	2	1	no	—
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	—	2	1	no	—
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	2	1	no	—
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
495	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
496	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
497	C₂×D₄	C₅×#41	1	2	yes	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	2	1	no	—
500	C₂²×C₄	—	2	1	no	—
501	C₂²×C₄	—	2	4	no	—
502	C₂²×C₄	—	2	1	no	—
503	C₂²×C₄	—	2	1	no	—
504	C₂²×C₄	—	2	1	no	—
505	C₂²×C₄	—	4	2	no	—
506	C₂²×C₄	—	4	2	no	—
507	C₂²×C₄	—	4	2	no	—
508	C₂²×C₄	—	4	2	no	—
509	C₂²×C₄	—	4	2	no	—
510	C₂²×C₄	—	4	2	no	—
511	C₂²×C₄	—	8	1	no	—
512	C₂²×C₄	—	8	1	no	—
513	C₂²×C₄	—	8	1	no	—
514	C₂²×C₄	—	8	1	no	—
515	C₂²×C₄	—	8	1	no	—
516	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
517	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
518	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
519	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
520	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
521	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
522	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
523	SD₁₆	—	2	8	no	—
524	D₈	—	2	8	no	—
525	C₂×C₈	—	2	1	no	—
526	C₂×C₈	—	2	1	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	2	1	no	—
529	C₂×C₈	—	2	1	no	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	1	no	—
532	C₂×C₈	—	2	1	no	—
533	C₂×C₈	—	2	1	no	—
534	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
535	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
536	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
537	C₄⋊C₄	—	2	2	no	—
538	C₄⋊C₄	—	2	2	no	—
539	C₄⋊C₄	—	2	1	no	—
540	C₄⋊C₄	—	2	1	no	—
541	C₄⋊C₄	—	2	1	no	—
542	C₄⋊C₄	—	2	1	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	4	2	no	—
548	C₄⋊C₄	—	8	1	no	—
549	C₄⋊C₄	—	8	1	no	—
550	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
551	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
552	C₂²⋊C₄	—	2	4	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	4	2	no	—
556	C₂²⋊C₄	—	4	2	no	—
557	C₂²⋊C₄	—	4	2	no	—
558	C₂²⋊C₄	—	4	2	no	—
559	C₂²⋊C₄	—	8	1	no	—
560	C₄²	—	2	1	no	—
561	C₄²	—	2	1	no	—
562	C₄²	—	2	1	no	—
563	C₄²	—	2	1	no	—
564	C₄²	—	4	2	no	—
565	C₄²	—	8	1	no	—
566	C₄²	C₂²×C₄×C₅	1	2	yes	—
567	C₄²	C₅×#41	1	2	yes	—
568	C₂×C₅	—	4	4	no	—
569	C₂×C₅	—	8	2	no	—
570	C₂×C₅	#2361	1	1	yes	characteristic
571	C₂×C₅	C₄×#135	1	1	yes	characteristic
572	C₂×C₅	#2946	1	1	yes	characteristic
573	C₂×C₅	#3280	1	1	yes	characteristic
574	C₂×C₅	#3282	1	1	yes	characteristic
575	C₂×C₅	#3360	1	1	yes	characteristic
576	C₂×C₅	#3361	1	1	yes	characteristic
577	C₂³	—	2	2	no	—
578	C₂³	—	2	2	no	—
579	C₂³	—	2	2	no	—
580	C₂³	—	2	2	no	—
581	C₂³	—	4	1	no	—
582	C₂³	—	4	1	no	—
583	C₂³	—	4	1	no	—
584	C₂³	—	4	1	no	—
585	C₂³	—	4	1	no	—
586	C₂³	—	4	1	no	—
587	C₂³	—	4	1	no	—
588	C₂³	—	4	1	no	—
589	C₂³	—	4	1	no	—
590	C₂³	—	4	1	no	—
591	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
592	Q₈	—	2	2	no	—
593	Q₈	—	2	2	no	—
594	Q₈	—	4	1	no	—
595	Q₈	—	4	1	no	—
596	D₄	—	2	2	no	—
597	D₄	—	2	2	no	—
598	D₄	—	2	2	no	—
599	D₄	—	2	2	no	—
600	D₄	—	2	2	no	—
601	D₄	—	4	1	no	—
602	D₄	—	4	1	no	—
603	D₄	C₄×C₅×D₄	1	4	yes	—
604	C₂×C₄	—	2	1	no	—
605	C₂×C₄	—	2	1	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	1	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	1	no	—
620	C₂×C₄	—	2	1	no	—
621	C₂×C₄	—	4	1	no	—
622	C₂×C₄	—	4	1	no	—
623	C₂×C₄	—	4	1	no	—
624	C₂×C₄	—	4	1	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	1	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	—	4	1	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	—	4	1	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	4	1	no	—
635	C₂×C₄	—	4	1	no	—
636	C₂×C₄	—	8	4	no	—
637	C₂×C₄	—	8	4	no	—
638	C₂×C₄	—	8	4	no	—
639	C₂×C₄	—	8	4	no	—
640	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
641	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
642	C₂×C₄	C₅×#135	1	2	yes	—
643	C₂×C₄	C₅×#135	1	2	yes	—
644	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
645	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
646	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
647	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
648	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
649	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
650	C₈	—	2	2	no	—
651	C₈	—	2	2	no	—
652	C₈	—	2	2	no	—
653	C₈	—	4	1	no	—
654	C₅	#22999	1	1	yes	characteristic, direct product factor, Sylow
655	C₂²	—	2	4	no	—
656	C₂²	—	4	2	no	—
657	C₂²	—	4	2	no	—
658	C₂²	—	4	2	no	—
659	C₂²	—	4	2	no	—
660	C₂²	—	4	2	no	—
661	C₂²	—	4	2	no	—
662	C₂²	—	4	2	no	—
663	C₂²	—	4	2	no	—
664	C₂²	—	4	2	no	—
665	C₂²	—	8	1	no	—
666	C₂²	—	8	1	no	—
667	C₂²	—	8	1	no	—
668	C₂²	—	8	1	no	—
669	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
670	C₂²	C₅×D₄²	1	1	yes	characteristic
671	C₂²	C₅×#548	1	1	yes	characteristic
672	C₂²	C₂×C₅×#135	1	1	yes	characteristic
673	C₂²	C₂×C₅×#135	1	1	yes	characteristic
674	C₂²	C₅×#576	1	1	yes	characteristic
675	C₂²	C₅×#577	1	1	yes	characteristic
676	C₄	—	2	2	no	—
677	C₄	—	2	2	no	—
678	C₄	—	2	2	no	—
679	C₄	—	2	2	no	—
680	C₄	—	4	1	no	—
681	C₄	—	4	2	no	—
682	C₄	—	4	1	no	—
683	C₄	—	4	2	no	—
684	C₄	—	4	2	no	—
685	C₄	—	8	1	no	—
686	C₄	—	8	1	no	—
687	C₄	—	8	1	no	—
688	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
689	C₄	C₅×#520	1	1	yes	characteristic
690	C₄	C₅×#547	1	1	yes	characteristic
691	C₄	C₅×#553	1	1	yes	characteristic
692	C₂	—	4	4	no	—
693	C₂	—	8	2	no	—
694	C₂	C₅×#2361	1	1	yes	characteristic
695	C₂	C₄×C₅×#135	1	1	yes	characteristic
696	C₂	C₅×#2946	1	1	yes	characteristic
697	C₂	C₅×#3280	1	1	yes	characteristic
698	C₂	C₅×#3282	1	1	yes	characteristic
699	C₂	C₅×#3360	1	1	yes	characteristic
700	C₂	C₅×#3361	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄×C₅×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₅×SD₁₆	C₂	semidirect
4. C₅×#2367	C₂	semidirect
5. C₅×#2349	C₂	semidirect
6. C₅×#1958	C₂	semidirect
7. C₅×#1949	C₂	semidirect
8. C₅×#1934	C₂	semidirect
9. C₅×#1933	C₂	semidirect
11. C₅×#1877	C₂	semidirect
12. C₅×#1864	C₂	semidirect
13. C₅×#1863	C₂	semidirect
14. C₅×C₈×C₄⋊C₄	C₂	semidirect
15. C₄×C₅×#102	C₂	semidirect
24. C₅×#463	C₄	semidirect
62. #22999	C₅	direct (N × H)
72. C₂×C₅×SD₁₆	C₂×C₄	semidirect
81. C₅×#124	C₂×C₄	semidirect
91. C₄×C₅×D₄	C₂×C₄	semidirect
117. C₅×C₄⋊C₈	C₂×C₄	semidirect
120. C₅×#103	C₂×C₄	semidirect
124. C₅×#102	C₂×C₄	semidirect
139. C₂×C₄×SD₁₆	C₂×C₅	semidirect
141. #2367	C₂×C₅	semidirect
142. #2349	C₂×C₅	semidirect
143. #1958	C₂×C₅	semidirect
144. #1949	C₂×C₅	semidirect
145. #1934	C₂×C₅	semidirect
146. #1933	C₂×C₅	semidirect
148. #1877	C₂×C₅	semidirect
149. #1864	C₂×C₅	semidirect
150. #1863	C₂×C₅	semidirect
151. C₈×C₄⋊C₄	C₂×C₅	semidirect
152. C₄×#102	C₂×C₅	semidirect
251. #463	C₄×C₅	semidirect
375. C₂×SD₁₆	C₂×C₄×C₅	semidirect
384. #124	C₂×C₄×C₅	semidirect
394. C₄×D₄	C₂×C₄×C₅	semidirect
420. C₄⋊C₈	C₂×C₄×C₅	semidirect
423. #103	C₂×C₄×C₅	semidirect
427. #102	C₂×C₄×C₅	semidirect
654. C₅	#22999	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×C₅×#463	2
C₂×C₄×C₅×SD₁₆	2
C₂×C₄×C₅×D₈	2
C₅×#2367	2
C₅×#2349	2
C₅×#1958	2
C₅×#1949	2
C₅×#1934	2
C₅×#1933	2
C₅×#1879	2
C₅×#1877	2
C₅×#1864	2
C₅×#1863	2
C₅×C₈×C₄⋊C₄	2
C₄×C₅×#102	2
#22999	5

Subgroups — C5×#22999 #11945719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22999 #11945719