Subgroups — C₅×#23126 #11945846

Order 1280.

994 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#465	C₂	1	1	yes	maximal, characteristic
2	C₅×#2361	C₂	1	1	yes	maximal, characteristic
3	C₅×#2360	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1934	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1933	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1920	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1919	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1880	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1877	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1874	C₂	1	1	yes	maximal, characteristic
11	C₅×#1872	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×C₅×#103	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×C₅×#102	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×C₈×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₅×#340	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₅×SD₁₆	C₂²	1	1	yes	characteristic
17	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₅×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₅×#121	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₅×D₄	—	2	1	no	—
22	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
23	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₅×#465	C₄	1	8	yes	complemented
26	C₂×C₅×#106	C₂²	1	1	yes	characteristic
27	C₂×C₅×#103	C₂²	1	1	yes	characteristic
28	C₂×C₅×#103	C₂²	1	1	yes	characteristic
29	C₂×C₅×#103	C₂²	1	1	yes	characteristic
30	C₂×C₅×#102	C₂²	1	1	yes	characteristic
31	C₂×C₅×#102	C₂²	1	1	yes	characteristic
32	C₂×C₅×#102	C₂²	1	1	yes	characteristic
33	C₂×C₅×C₂²⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
35	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
36	C₅×#391	C₂²	1	1	yes	characteristic
37	C₅×#390	C₂²	1	1	yes	characteristic
38	C₅×#389	—	2	1	no	—
39	C₅×#389	C₂²	1	1	yes	characteristic
40	C₅×#389	C₂²	1	1	yes	characteristic
41	C₅×#388	—	2	1	no	—
42	C₅×#387	—	2	1	no	—
43	C₅×#386	—	2	1	no	—
44	C₅×#385	—	2	1	no	—
45	C₅×#385	—	2	1	no	—
46	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
47	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
48	C₄×C₅×C₂²⋊C₄	—	2	1	no	—
49	C₄×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
50	C₅×#340	C₂²	1	1	yes	characteristic
51	C₅×#340	C₂²	1	1	yes	characteristic
52	C₅×#340	C₂²	1	1	yes	characteristic
53	C₅×#340	C₂²	1	1	yes	characteristic
54	C₅×#340	C₂²	1	1	yes	characteristic
55	C₅×#340	C₂²	1	1	yes	characteristic
56	C₅×#340	C₂²	1	1	yes	characteristic
57	C₅×#340	C₂²	1	1	yes	characteristic
58	C₅×#336	C₂²	1	1	yes	characteristic
59	C₅×#336	C₂²	1	1	yes	characteristic
60	#23126	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
62	C₂²×C₅×D₄	—	2	1	no	—
63	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₅	—	2	1	no	—
65	C₂³×C₄×C₅	—	2	1	no	—
66	C₂³×C₄×C₅	—	2	1	no	—
67	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
68	C₂×C₅×SD₁₆	—	4	1	no	—
69	C₂×C₅×SD₁₆	—	4	1	no	—
70	C₂×C₅×SD₁₆	C₂×C₄	1	4	yes	complemented
71	C₂²×C₅×C₈	—	4	1	no	—
72	C₂²×C₅×C₈	—	4	1	no	—
73	C₂²×C₅×C₈	C₂×C₄	1	4	yes	complemented
74	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
75	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
77	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
78	C₅×C₂²⋊Q₈	—	2	2	no	—
79	C₅×C₂²⋊Q₈	C₂×C₄	1	4	yes	complemented
80	C₅×#121	—	2	2	no	—
81	C₅×#121	C₂×C₄	1	4	yes	—
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	—	4	1	no	—
86	C₄×C₅×D₄	—	4	4	no	—
87	C₂×C₅×C₄⋊C₄	—	2	1	no	—
88	C₂×C₅×C₄⋊C₄	—	2	2	no	—
89	C₂×C₅×C₄⋊C₄	—	2	2	no	—
90	C₂×C₅×C₄⋊C₄	—	2	1	no	—
91	C₂×C₅×C₄⋊C₄	—	2	1	no	—
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	—	4	1	no	—
96	C₂×C₅×C₄⋊C₄	—	4	1	no	—
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×C₄⋊C₄	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
109	C₂×C₅×C₂²⋊C₄	D₄	1	1	yes	characteristic
110	C₂×C₄²×C₅	—	2	1	no	—
111	C₂×C₄²×C₅	—	2	1	no	—
112	C₂×C₄²×C₅	—	2	1	no	—
113	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
115	C₅×#106	C₂×C₄	1	4	yes	complemented
116	C₅×#103	—	2	2	no	—
117	C₅×#103	—	4	1	no	—
118	C₅×#103	C₂×C₄	1	4	yes	complemented
119	C₅×#102	—	2	2	no	—
120	C₅×#102	—	4	1	no	—
121	C₅×#102	C₂×C₄	1	4	yes	complemented
122	C₅×C₂²⋊C₈	—	2	2	no	—
123	C₄×C₅×C₈	—	4	1	no	—
124	C₄×C₅×C₈	—	4	1	no	—
125	C₅×#95	—	2	1	no	—
126	C₅×#95	—	2	1	no	—
127	C₅×#95	—	2	1	no	—
128	C₅×#95	—	2	1	no	—
129	C₅×#95	—	2	1	no	—
130	C₅×#95	—	2	1	no	—
131	C₅×#95	D₄	1	1	yes	characteristic
132	C₅×#95	D₄	1	1	yes	characteristic
133	C₂×#465	C₂×C₅	1	1	yes	characteristic
134	#2361	C₂×C₅	1	1	yes	characteristic
135	#2360	C₂×C₅	1	1	yes	characteristic, complemented
136	#1934	C₂×C₅	1	1	yes	characteristic, complemented
137	#1933	C₂×C₅	1	1	yes	characteristic, complemented
138	#1920	C₂×C₅	1	1	yes	characteristic, complemented
139	#1919	C₂×C₅	1	1	yes	characteristic, complemented
140	#1880	C₂×C₅	1	1	yes	characteristic, complemented
141	#1877	C₂×C₅	1	1	yes	characteristic, complemented
142	#1874	C₂×C₅	1	1	yes	characteristic
143	#1872	C₂×C₅	1	1	yes	characteristic, complemented
144	C₄×#103	C₂×C₅	1	1	yes	characteristic, complemented
145	C₄×#102	C₂×C₅	1	1	yes	characteristic, complemented
146	C₈×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
147	C₂×#340	C₂×C₅	1	1	yes	characteristic, complemented
148	C₂⁴×C₅	—	2	1	no	—
149	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
150	C₂×C₅×Q₈	—	2	1	no	—
151	C₂×C₅×Q₈	—	2	1	no	—
152	C₂×C₅×Q₈	—	2	1	no	—
153	C₂×C₅×Q₈	—	2	1	no	—
154	C₂×C₅×Q₈	—	2	1	no	—
155	C₂×C₅×Q₈	C₂²×C₄	1	2	yes	—
156	C₂×C₅×D₄	—	2	4	no	—
157	C₂×C₅×D₄	—	2	1	no	—
158	C₂×C₅×D₄	—	2	1	no	—
159	C₂×C₅×D₄	—	2	1	no	—
160	C₂×C₅×D₄	—	2	1	no	—
161	C₂×C₅×D₄	—	2	1	no	—
162	C₂×C₅×D₄	—	4	2	no	—
163	C₂×C₅×D₄	—	4	2	no	—
164	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
165	C₂²×C₄×C₅	—	2	1	no	—
166	C₂²×C₄×C₅	—	2	1	no	—
167	C₂²×C₄×C₅	—	2	1	no	—
168	C₂²×C₄×C₅	—	2	1	no	—
169	C₂²×C₄×C₅	—	2	1	no	—
170	C₂²×C₄×C₅	—	2	1	no	—
171	C₂²×C₄×C₅	—	2	1	no	—
172	C₂²×C₄×C₅	—	2	1	no	—
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	—	4	2	no	—
177	C₂²×C₄×C₅	—	4	2	no	—
178	C₂²×C₄×C₅	—	4	2	no	—
179	C₂²×C₄×C₅	—	4	2	no	—
180	C₂²×C₄×C₅	—	4	2	no	—
181	C₂²×C₄×C₅	—	4	2	no	—
182	C₂²×C₄×C₅	—	8	1	no	—
183	C₂²×C₄×C₅	—	8	1	no	—
184	C₂²×C₄×C₅	—	8	1	no	—
185	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
186	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
190	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₅	#41	1	2	yes	—
192	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
193	C₅×SD₁₆	—	4	4	no	—
194	C₂×C₅×C₈	—	2	1	no	—
195	C₂×C₅×C₈	—	2	1	no	—
196	C₂×C₅×C₈	—	2	1	no	—
197	C₂×C₅×C₈	—	2	1	no	—
198	C₂×C₅×C₈	—	2	1	no	—
199	C₂×C₅×C₈	—	2	1	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	—	4	4	no	—
205	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
206	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
207	C₅×C₄⋊C₄	—	2	4	no	—
208	C₅×C₄⋊C₄	—	2	1	no	—
209	C₅×C₄⋊C₄	—	2	1	no	—
210	C₅×C₄⋊C₄	—	2	1	no	—
211	C₅×C₄⋊C₄	—	2	1	no	—
212	C₅×C₄⋊C₄	—	2	1	no	—
213	C₅×C₄⋊C₄	—	2	1	no	—
214	C₅×C₄⋊C₄	—	2	1	no	—
215	C₅×C₄⋊C₄	—	2	1	no	—
216	C₅×C₄⋊C₄	—	4	2	no	—
217	C₅×C₄⋊C₄	—	4	2	no	—
218	C₅×C₄⋊C₄	—	4	2	no	—
219	C₅×C₄⋊C₄	—	4	2	no	—
220	C₅×C₄⋊C₄	—	8	1	no	—
221	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
222	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
223	C₅×C₄⋊C₄	#41	1	2	yes	—
224	C₅×C₄⋊C₄	#41	1	2	yes	—
225	C₅×C₂²⋊C₄	—	2	4	no	—
226	C₅×C₂²⋊C₄	—	2	2	no	—
227	C₅×C₂²⋊C₄	—	2	4	no	—
228	C₅×C₂²⋊C₄	—	4	2	no	—
229	C₅×C₂²⋊C₄	—	4	2	no	—
230	C₅×C₂²⋊C₄	—	8	1	no	—
231	C₅×C₂²⋊C₄	SD₁₆	1	4	yes	complemented
232	C₄²×C₅	—	2	1	no	—
233	C₄²×C₅	—	2	1	no	—
234	C₄²×C₅	—	2	1	no	—
235	C₄²×C₅	—	2	1	no	—
236	C₄²×C₅	—	4	2	no	—
237	C₄²×C₅	—	4	2	no	—
238	C₄²×C₅	—	8	1	no	—
239	C₂²×SD₁₆	C₂²×C₅	1	1	yes	characteristic
240	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
241	C₂×C₂²⋊Q₈	C₂²×C₅	1	1	yes	characteristic
242	C₂×#121	C₂²×C₅	1	1	yes	characteristic
243	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
244	C₂×C₄×D₄	—	2	1	no	—
245	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
246	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
247	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
248	#465	C₄×C₅	1	8	yes	complemented
249	C₂×#106	C₂²×C₅	1	1	yes	characteristic
250	C₂×#103	C₂²×C₅	1	1	yes	characteristic
251	C₂×#103	C₂²×C₅	1	1	yes	characteristic
252	C₂×#103	C₂²×C₅	1	1	yes	characteristic
253	C₂×#102	C₂²×C₅	1	1	yes	characteristic
254	C₂×#102	C₂²×C₅	1	1	yes	characteristic
255	C₂×#102	C₂²×C₅	1	1	yes	characteristic
256	C₂×C₂²⋊C₈	C₂²×C₅	1	1	yes	characteristic
257	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
258	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
259	#391	C₂²×C₅	1	1	yes	characteristic
260	#390	C₂²×C₅	1	1	yes	characteristic
261	#389	—	2	1	no	—
262	#389	C₂²×C₅	1	1	yes	characteristic
263	#389	C₂²×C₅	1	1	yes	characteristic
264	#388	—	2	1	no	—
265	#387	—	2	1	no	—
266	#386	—	2	1	no	—
267	#385	—	2	1	no	—
268	#385	—	2	1	no	—
269	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
270	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
271	C₄×C₂²⋊C₄	—	2	1	no	—
272	C₄×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
273	#340	C₂²×C₅	1	1	yes	characteristic
274	#340	C₂²×C₅	1	1	yes	characteristic
275	#340	C₂²×C₅	1	1	yes	characteristic
276	#340	C₂²×C₅	1	1	yes	characteristic
277	#340	C₂²×C₅	1	1	yes	characteristic
278	#340	C₂²×C₅	1	1	yes	characteristic
279	#340	C₂²×C₅	1	1	yes	characteristic
280	#340	C₂²×C₅	1	1	yes	characteristic
281	#336	C₂²×C₅	1	1	yes	characteristic
282	#336	C₂²×C₅	1	1	yes	characteristic
283	C₂³×C₅	—	2	1	no	—
284	C₂³×C₅	—	2	1	no	—
285	C₂³×C₅	—	2	1	no	—
286	C₂³×C₅	—	2	1	no	—
287	C₂³×C₅	—	2	2	no	—
288	C₂³×C₅	—	4	1	no	—
289	C₂³×C₅	—	4	1	no	—
290	C₂³×C₅	—	4	1	no	—
291	C₂³×C₅	—	4	1	no	—
292	C₂³×C₅	—	4	1	no	—
293	C₂³×C₅	—	4	1	no	—
294	C₂³×C₅	C₄×D₄	1	2	yes	—
295	C₂³×C₅	C₂×SD₁₆	1	2	yes	—
296	C₂³×C₅	#135	1	2	yes	—
297	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
298	C₅×Q₈	—	2	2	no	—
299	C₅×Q₈	—	2	2	no	—
300	C₅×Q₈	—	4	1	no	—
301	C₅×Q₈	—	4	1	no	—
302	C₅×D₄	—	2	2	no	—
303	C₅×D₄	—	2	2	no	—
304	C₅×D₄	—	4	8	no	—
305	C₅×D₄	—	4	1	no	—
306	C₅×D₄	—	4	1	no	—
307	C₂×C₄×C₅	—	2	1	no	—
308	C₂×C₄×C₅	—	2	1	no	—
309	C₂×C₄×C₅	—	2	1	no	—
310	C₂×C₄×C₅	—	2	1	no	—
311	C₂×C₄×C₅	—	2	1	no	—
312	C₂×C₄×C₅	—	2	1	no	—
313	C₂×C₄×C₅	—	2	2	no	—
314	C₂×C₄×C₅	—	2	2	no	—
315	C₂×C₄×C₅	—	2	2	no	—
316	C₂×C₄×C₅	—	2	2	no	—
317	C₂×C₄×C₅	—	2	2	no	—
318	C₂×C₄×C₅	—	2	2	no	—
319	C₂×C₄×C₅	—	2	2	no	—
320	C₂×C₄×C₅	—	2	1	no	—
321	C₂×C₄×C₅	—	2	1	no	—
322	C₂×C₄×C₅	—	2	2	no	—
323	C₂×C₄×C₅	—	2	2	no	—
324	C₂×C₄×C₅	—	2	1	no	—
325	C₂×C₄×C₅	—	2	1	no	—
326	C₂×C₄×C₅	—	2	1	no	—
327	C₂×C₄×C₅	—	2	1	no	—
328	C₂×C₄×C₅	—	4	1	no	—
329	C₂×C₄×C₅	—	4	1	no	—
330	C₂×C₄×C₅	—	4	1	no	—
331	C₂×C₄×C₅	—	4	1	no	—
332	C₂×C₄×C₅	—	4	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	—	8	4	no	—
345	C₂×C₄×C₅	—	8	4	no	—
346	C₂×C₄×C₅	—	8	4	no	—
347	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
348	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
349	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
350	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
351	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
352	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
353	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
354	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
355	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
356	C₅×C₈	—	2	2	no	—
357	C₅×C₈	—	2	2	no	—
358	C₅×C₈	—	4	1	no	—
359	C₅×C₈	—	4	1	no	—
360	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
361	C₂²×D₄	—	2	1	no	—
362	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
363	C₂³×C₄	—	2	1	no	—
364	C₂³×C₄	—	2	1	no	—
365	C₂³×C₄	—	2	1	no	—
366	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
367	C₂×SD₁₆	—	4	1	no	—
368	C₂×SD₁₆	—	4	1	no	—
369	C₂×SD₁₆	C₂×C₄×C₅	1	4	yes	complemented
370	C₂²×C₈	—	4	1	no	—
371	C₂²×C₈	—	4	1	no	—
372	C₂²×C₈	C₂×C₄×C₅	1	4	yes	complemented
373	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
374	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
375	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
376	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
377	C₂²⋊Q₈	—	2	2	no	—
378	C₂²⋊Q₈	C₂×C₄×C₅	1	4	yes	complemented
379	#121	—	2	2	no	—
380	#121	C₂×C₄×C₅	1	4	yes	—
381	C₄×Q₈	—	2	2	no	—
382	C₄×Q₈	—	4	1	no	—
383	C₄×D₄	—	2	2	no	—
384	C₄×D₄	—	4	1	no	—
385	C₄×D₄	—	4	4	no	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	2	2	no	—
388	C₂×C₄⋊C₄	—	2	2	no	—
389	C₂×C₄⋊C₄	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	—	2	1	no	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	C₂×C₄⋊C₄	—	2	1	no	—
394	C₂×C₄⋊C₄	—	4	1	no	—
395	C₂×C₄⋊C₄	—	4	1	no	—
396	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
397	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
398	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂³×C₅	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×C₂²⋊C₄	—	2	1	no	—
403	C₂×C₂²⋊C₄	—	2	1	no	—
404	C₂×C₂²⋊C₄	—	2	1	no	—
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₂²⋊C₄	C₅×D₄	1	1	yes	characteristic
408	C₂×C₂²⋊C₄	C₅×D₄	1	1	yes	characteristic
409	C₂×C₄²	—	2	1	no	—
410	C₂×C₄²	—	2	1	no	—
411	C₂×C₄²	—	2	1	no	—
412	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
413	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
414	#106	C₂×C₄×C₅	1	4	yes	complemented
415	#103	—	2	2	no	—
416	#103	—	4	1	no	—
417	#103	C₂×C₄×C₅	1	4	yes	complemented
418	#102	—	2	2	no	—
419	#102	—	4	1	no	—
420	#102	C₂×C₄×C₅	1	4	yes	complemented
421	C₂²⋊C₈	—	2	2	no	—
422	C₄×C₈	—	4	1	no	—
423	C₄×C₈	—	4	1	no	—
424	#95	—	2	1	no	—
425	#95	—	2	1	no	—
426	#95	—	2	1	no	—
427	#95	—	2	1	no	—
428	#95	—	2	1	no	—
429	#95	—	2	1	no	—
430	#95	C₅×D₄	1	1	yes	characteristic
431	#95	C₅×D₄	1	1	yes	characteristic
432	C₂²×C₅	—	2	2	no	—
433	C₂²×C₅	—	2	2	no	—
434	C₂²×C₅	—	2	2	no	—
435	C₂²×C₅	—	2	2	no	—
436	C₂²×C₅	—	2	2	no	—
437	C₂²×C₅	—	2	2	no	—
438	C₂²×C₅	—	4	2	no	—
439	C₂²×C₅	—	4	2	no	—
440	C₂²×C₅	—	4	2	no	—
441	C₂²×C₅	—	8	1	no	—
442	C₂²×C₅	—	8	1	no	—
443	C₂²×C₅	—	8	1	no	—
444	C₂²×C₅	—	8	1	no	—
445	C₂²×C₅	C₄×SD₁₆	1	4	yes	—
446	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
447	C₂²×C₅	#546	1	1	yes	characteristic
448	C₂²×C₅	#546	1	1	yes	characteristic
449	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
450	C₂²×C₅	C₂×#135	1	1	yes	characteristic
451	C₂²×C₅	#576	1	1	yes	characteristic
452	C₂²×C₅	#578	1	1	yes	characteristic
453	C₄×C₅	—	2	2	no	—
454	C₄×C₅	—	2	2	no	—
455	C₄×C₅	—	2	2	no	—
456	C₄×C₅	—	4	2	no	—
457	C₄×C₅	—	4	2	no	—
458	C₄×C₅	—	4	2	no	—
459	C₄×C₅	—	4	2	no	—
460	C₄×C₅	—	4	1	no	—
461	C₄×C₅	—	4	2	no	—
462	C₄×C₅	—	4	1	no	—
463	C₄×C₅	—	8	1	no	—
464	C₄×C₅	—	8	1	no	—
465	C₄×C₅	C₄×#41	1	1	yes	characteristic
466	C₄×C₅	#518	1	1	yes	characteristic
467	C₄×C₅	#553	1	1	yes	characteristic
468	C₄×C₅	#553	1	1	yes	characteristic
469	C₂⁴	—	2	1	no	—
470	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	1	no	—
475	C₂×Q₈	—	2	1	no	—
476	C₂×Q₈	C₂²×C₄×C₅	1	2	yes	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	—	2	1	no	—
481	C₂×D₄	—	2	1	no	—
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	—	4	2	no	—
484	C₂×D₄	—	4	2	no	—
485	C₂×D₄	C₂²×C₄×C₅	1	2	yes	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	4	2	no	—
498	C₂²×C₄	—	4	2	no	—
499	C₂²×C₄	—	4	2	no	—
500	C₂²×C₄	—	4	2	no	—
501	C₂²×C₄	—	4	2	no	—
502	C₂²×C₄	—	4	2	no	—
503	C₂²×C₄	—	8	1	no	—
504	C₂²×C₄	—	8	1	no	—
505	C₂²×C₄	—	8	1	no	—
506	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
507	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
508	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
509	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
510	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
511	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
512	C₂²×C₄	C₅×#41	1	2	yes	—
513	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
514	SD₁₆	—	4	4	no	—
515	C₂×C₈	—	2	1	no	—
516	C₂×C₈	—	2	1	no	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	—	2	1	no	—
520	C₂×C₈	—	2	1	no	—
521	C₂×C₈	—	2	1	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	1	no	—
525	C₂×C₈	—	4	4	no	—
526	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
527	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
528	C₄⋊C₄	—	2	4	no	—
529	C₄⋊C₄	—	2	1	no	—
530	C₄⋊C₄	—	2	1	no	—
531	C₄⋊C₄	—	2	1	no	—
532	C₄⋊C₄	—	2	1	no	—
533	C₄⋊C₄	—	2	1	no	—
534	C₄⋊C₄	—	2	1	no	—
535	C₄⋊C₄	—	2	1	no	—
536	C₄⋊C₄	—	2	1	no	—
537	C₄⋊C₄	—	4	2	no	—
538	C₄⋊C₄	—	4	2	no	—
539	C₄⋊C₄	—	4	2	no	—
540	C₄⋊C₄	—	4	2	no	—
541	C₄⋊C₄	—	8	1	no	—
542	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
543	C₄⋊C₄	C₂²×C₄×C₅	1	2	yes	—
544	C₄⋊C₄	C₅×#41	1	2	yes	—
545	C₄⋊C₄	C₅×#41	1	2	yes	—
546	C₂²⋊C₄	—	2	4	no	—
547	C₂²⋊C₄	—	2	2	no	—
548	C₂²⋊C₄	—	2	4	no	—
549	C₂²⋊C₄	—	4	2	no	—
550	C₂²⋊C₄	—	4	2	no	—
551	C₂²⋊C₄	—	8	1	no	—
552	C₂²⋊C₄	C₅×SD₁₆	1	4	yes	complemented
553	C₄²	—	2	1	no	—
554	C₄²	—	2	1	no	—
555	C₄²	—	2	1	no	—
556	C₄²	—	2	1	no	—
557	C₄²	—	4	2	no	—
558	C₄²	—	4	2	no	—
559	C₄²	—	8	1	no	—
560	C₂×C₅	—	2	4	no	—
561	C₂×C₅	—	8	2	no	—
562	C₂×C₅	#2364	1	1	yes	characteristic
563	C₂×C₅	C₂×C₄×SD₁₆	1	1	yes	characteristic
564	C₂×C₅	#2951	1	1	yes	characteristic
565	C₂×C₅	#3295	1	1	yes	characteristic
566	C₂×C₅	#3297	1	1	yes	characteristic
567	C₂×C₅	#3298	1	1	yes	characteristic
568	C₂×C₅	#3300	1	1	yes	characteristic
569	C₂³	—	2	1	no	—
570	C₂³	—	2	1	no	—
571	C₂³	—	2	1	no	—
572	C₂³	—	2	1	no	—
573	C₂³	—	2	2	no	—
574	C₂³	—	4	1	no	—
575	C₂³	—	4	1	no	—
576	C₂³	—	4	1	no	—
577	C₂³	—	4	1	no	—
578	C₂³	—	4	1	no	—
579	C₂³	—	4	1	no	—
580	C₂³	C₄×C₅×D₄	1	2	yes	—
581	C₂³	C₂×C₅×SD₁₆	1	2	yes	—
582	C₂³	C₅×#135	1	2	yes	—
583	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
584	Q₈	—	2	2	no	—
585	Q₈	—	2	2	no	—
586	Q₈	—	4	1	no	—
587	Q₈	—	4	1	no	—
588	D₄	—	2	2	no	—
589	D₄	—	2	2	no	—
590	D₄	—	4	8	no	—
591	D₄	—	4	1	no	—
592	D₄	—	4	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	1	no	—
596	C₂×C₄	—	2	1	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	1	no	—
612	C₂×C₄	—	2	1	no	—
613	C₂×C₄	—	2	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	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	—	8	4	no	—
631	C₂×C₄	—	8	4	no	—
632	C₂×C₄	—	8	4	no	—
633	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
634	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
635	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
636	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
637	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
638	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
639	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
640	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
641	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
642	C₈	—	2	2	no	—
643	C₈	—	2	2	no	—
644	C₈	—	4	1	no	—
645	C₈	—	4	1	no	—
646	C₅	#23126	1	1	yes	characteristic, direct product factor, Sylow
647	C₂²	—	2	2	no	—
648	C₂²	—	2	2	no	—
649	C₂²	—	2	2	no	—
650	C₂²	—	2	2	no	—
651	C₂²	—	2	2	no	—
652	C₂²	—	2	2	no	—
653	C₂²	—	4	2	no	—
654	C₂²	—	4	2	no	—
655	C₂²	—	4	2	no	—
656	C₂²	—	8	1	no	—
657	C₂²	—	8	1	no	—
658	C₂²	—	8	1	no	—
659	C₂²	—	8	1	no	—
660	C₂²	C₄×C₅×SD₁₆	1	4	yes	—
661	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
662	C₂²	C₅×#546	1	1	yes	characteristic
663	C₂²	C₅×#546	1	1	yes	characteristic
664	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
665	C₂²	C₂×C₅×#135	1	1	yes	characteristic
666	C₂²	C₅×#576	1	1	yes	characteristic
667	C₂²	C₅×#578	1	1	yes	characteristic
668	C₄	—	2	2	no	—
669	C₄	—	2	2	no	—
670	C₄	—	2	2	no	—
671	C₄	—	4	2	no	—
672	C₄	—	4	1	no	—
673	C₄	—	4	2	no	—
674	C₄	—	4	1	no	—
675	C₄	—	4	2	no	—
676	C₄	—	4	2	no	—
677	C₄	—	4	2	no	—
678	C₄	—	8	1	no	—
679	C₄	—	8	1	no	—
680	C₄	C₄×C₅×#41	1	1	yes	characteristic
681	C₄	C₅×#518	1	1	yes	characteristic
682	C₄	C₅×#553	1	1	yes	characteristic
683	C₄	C₅×#553	1	1	yes	characteristic
684	C₂	—	2	4	no	—
685	C₂	—	8	2	no	—
686	C₂	C₅×#2364	1	1	yes	characteristic
687	C₂	C₂×C₄×C₅×SD₁₆	1	1	yes	characteristic
688	C₂	C₅×#2951	1	1	yes	characteristic
689	C₂	C₅×#3295	1	1	yes	characteristic
690	C₂	C₅×#3297	1	1	yes	characteristic
691	C₂	C₅×#3298	1	1	yes	characteristic
692	C₂	C₅×#3300	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	C₂×C₅×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄×C₅×C₂²⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2360	C₂	semidirect
4. C₅×#1934	C₂	semidirect
5. C₅×#1933	C₂	semidirect
6. C₅×#1920	C₂	semidirect
7. C₅×#1919	C₂	semidirect
8. C₅×#1880	C₂	semidirect
9. C₅×#1877	C₂	semidirect
11. C₅×#1872	C₂	semidirect
12. C₄×C₅×#103	C₂	semidirect
13. C₄×C₅×#102	C₂	semidirect
14. C₅×C₈×C₂²⋊C₄	C₂	semidirect
15. C₂×C₅×#340	C₂	semidirect
25. C₅×#465	C₄	semidirect
60. #23126	C₅	direct (N × H)
70. C₂×C₅×SD₁₆	C₂×C₄	semidirect
73. C₂²×C₅×C₈	C₂×C₄	semidirect
79. C₅×C₂²⋊Q₈	C₂×C₄	semidirect
115. C₅×#106	C₂×C₄	semidirect
118. C₅×#103	C₂×C₄	semidirect
121. C₅×#102	C₂×C₄	semidirect
135. #2360	C₂×C₅	semidirect
136. #1934	C₂×C₅	semidirect
137. #1933	C₂×C₅	semidirect
138. #1920	C₂×C₅	semidirect
139. #1919	C₂×C₅	semidirect
140. #1880	C₂×C₅	semidirect
141. #1877	C₂×C₅	semidirect
143. #1872	C₂×C₅	semidirect
144. C₄×#103	C₂×C₅	semidirect
145. C₄×#102	C₂×C₅	semidirect
146. C₈×C₂²⋊C₄	C₂×C₅	semidirect
147. C₂×#340	C₂×C₅	semidirect
231. C₅×C₂²⋊C₄	SD₁₆	semidirect
248. #465	C₄×C₅	semidirect
369. C₂×SD₁₆	C₂×C₄×C₅	semidirect
372. C₂²×C₈	C₂×C₄×C₅	semidirect
378. C₂²⋊Q₈	C₂×C₄×C₅	semidirect
414. #106	C₂×C₄×C₅	semidirect
417. #103	C₂×C₄×C₅	semidirect
420. #102	C₂×C₄×C₅	semidirect
552. C₂²⋊C₄	C₅×SD₁₆	semidirect
646. C₅	#23126	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×#465	2
C₅×#2361	2
C₅×#2360	2
C₅×#1934	2
C₅×#1933	2
C₅×#1920	2
C₅×#1919	2
C₅×#1880	2
C₅×#1877	2
C₅×#1874	2
C₅×#1872	2
C₄×C₅×#103	2
C₄×C₅×#102	2
C₅×C₈×C₂²⋊C₄	2
C₂×C₅×#340	2
#23126	5

Subgroups — C5×#23126 #11945846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#23126 #11945846