Subgroups — C₅×#35264 #11957984

Order 1280.

2654 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#544	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#535	C₂	1	1	yes	maximal, characteristic
3	C₅×#2861	C₂	1	2	yes	maximal, complemented
4	C₅×#2858	C₂	1	2	yes	maximal, complemented
5	C₅×#2641	C₂	1	2	yes	maximal, complemented
6	C₅×#2640	C₂	1	2	yes	maximal, complemented
7	C₅×#2491	C₂	1	4	yes	maximal, complemented
8	C₅×#2490	C₂	1	4	yes	maximal, complemented
9	C₅×#2472	C₂	1	8	yes	maximal, complemented
10	C₅×#2311	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#2309	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#2297	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅×#390	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
16	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
17	C₅×#544	—	2	8	no	—
18	C₅×#535	—	2	8	no	—
19	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
20	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
21	C₂×C₅×#127	C₂²	1	2	yes	—
22	C₂×C₅×#125	C₂²	1	2	yes	complemented
23	C₂×C₅×#125	C₂²	1	2	yes	complemented
24	C₂×C₅×#124	C₂²	1	2	yes	—
25	C₂×C₅×#124	C₂²	1	8	yes	complemented
26	C₂×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
27	C₂×C₅×#121	C₂²	1	2	yes	—
28	C₂×C₅×#121	C₂²	1	2	yes	—
29	C₂×C₅×#120	C₂²	1	4	yes	—
30	C₂×C₅×#117	C₂²	1	1	yes	characteristic
31	C₂×C₅×#117	C₂²	1	1	yes	characteristic
32	C₂×C₅×#117	C₂²	1	1	yes	characteristic
33	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂²×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
35	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
36	C₅×#397	C₂²	1	8	yes	complemented
37	C₅×#394	C₂²	1	8	yes	complemented
38	C₅×#394	C₂²	1	8	yes	complemented
39	C₅×#393	C₂²	1	8	yes	complemented
40	C₅×#391	C₂²	1	1	yes	characteristic, complemented
41	C₅×#391	C₂²	1	1	yes	characteristic, complemented
42	C₅×#391	C₂²	1	2	yes	complemented
43	C₅×#390	C₂²	1	4	yes	—
44	C₅×#390	C₂²	1	4	yes	—
45	C₅×#390	C₂²	1	2	yes	—
46	C₅×#390	C₂²	1	2	yes	—
47	C₅×#390	C₂²	1	2	yes	—
48	C₅×#390	C₂²	1	2	yes	—
49	C₅×#390	C₂²	1	8	yes	complemented
50	C₅×#390	C₂²	1	1	yes	characteristic, complemented
51	C₅×#390	C₂²	1	1	yes	characteristic, complemented
52	C₅×#390	C₂²	1	2	yes	complemented
53	C₅×#388	C₂²	1	8	yes	complemented
54	C₅×#388	C₂²	1	8	yes	complemented
55	C₅×#386	C₂²	1	4	yes	—
56	C₅×#386	C₂²	1	4	yes	—
57	C₅×#380	C₂²	1	4	yes	complemented
58	C₅×#379	C₂²	1	4	yes	—
59	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
60	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
61	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
62	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
63	C₅×#376	C₂²	1	2	yes	complemented
64	C₄³×C₅	C₂²	1	2	yes	complemented
65	#35264	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂⁵×C₅	C₂³	1	2	yes	—
67	C₂×C₅×#41	—	2	4	no	—
68	C₂×C₅×#41	—	2	4	no	—
69	C₂×C₅×#41	—	2	4	no	—
70	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
71	C₂²×C₅×Q₈	C₂³	1	4	yes	—
72	C₂²×C₅×D₄	—	2	4	no	—
73	C₂²×C₅×D₄	—	2	4	no	—
74	C₂²×C₅×D₄	—	2	4	no	—
75	C₂²×C₅×D₄	C₂³	1	2	yes	—
76	C₂²×C₅×D₄	C₂³	1	2	yes	—
77	C₂²×C₅×D₄	C₂³	1	2	yes	—
78	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
79	C₂²×C₅×D₄	C₂³	1	4	yes	—
80	C₂²×C₅×D₄	C₂³	1	4	yes	—
81	C₂³×C₄×C₅	C₂³	1	2	yes	—
82	C₂³×C₄×C₅	C₂³	1	2	yes	—
83	C₂³×C₄×C₅	C₂³	1	2	yes	—
84	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
85	C₅×C₄⋊Q₈	—	2	8	no	—
86	C₅×C₄⋊Q₈	—	2	8	no	—
87	C₅×#127	—	2	8	no	—
88	C₅×#125	—	2	8	no	—
89	C₅×#125	—	2	8	no	—
90	C₅×#124	—	2	8	no	—
91	C₅×#124	—	4	16	no	—
92	C₅×C₂²⋊Q₈	—	2	16	no	—
93	C₅×#121	—	2	8	no	—
94	C₅×#121	—	2	8	no	—
95	C₅×#120	—	2	16	no	—
96	C₅×#117	—	2	4	no	—
97	C₅×#117	—	2	4	no	—
98	C₅×#117	D₄	1	8	yes	—
99	C₂×C₅×C₄⋊C₄	—	2	4	no	—
100	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
101	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
102	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
103	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
104	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
105	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
106	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
107	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
108	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
109	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
110	C₂×C₅×C₂²⋊C₄	—	4	8	no	—
111	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
112	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
113	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
114	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
115	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
116	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
117	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
118	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
119	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
120	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
121	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
122	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
123	C₂×C₄²×C₅	—	2	4	no	—
124	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
125	C₂×C₄²×C₅	C₂³	1	2	yes	—
126	C₂×C₄²×C₅	C₂³	1	4	yes	—
127	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₅	C₂³	1	2	yes	—
129	C₂×C₄²×C₅	C₂³	1	2	yes	—
130	C₂×C₄²×C₅	C₂³	1	4	yes	—
131	C₂×C₄²×C₅	C₂³	1	2	yes	—
132	C₅×#95	C₂³	1	4	yes	—
133	C₅×#95	C₂³	1	4	yes	—
134	C₅×#95	C₂³	1	4	yes	—
135	C₅×#95	C₂³	1	4	yes	—
136	C₅×#95	C₂³	1	8	yes	—
137	C₂×#544	C₂×C₅	1	1	yes	characteristic, complemented
138	C₂×#535	C₂×C₅	1	1	yes	characteristic
139	#2861	C₂×C₅	1	2	yes	complemented
140	#2858	C₂×C₅	1	2	yes	complemented
141	#2641	C₂×C₅	1	2	yes	complemented
142	#2640	C₂×C₅	1	2	yes	complemented
143	#2491	C₂×C₅	1	4	yes	complemented
144	#2490	C₂×C₅	1	4	yes	complemented
145	#2472	C₂×C₅	1	8	yes	complemented
146	#2311	C₂×C₅	1	1	yes	characteristic, complemented
147	#2309	C₂×C₅	1	1	yes	characteristic, complemented
148	#2297	C₂×C₅	1	1	yes	characteristic, complemented
149	C₂×#390	C₂×C₅	1	1	yes	characteristic, complemented
150	C₄×#117	C₂×C₅	1	1	yes	characteristic, complemented
151	C₂⁴×C₅	—	2	4	no	—
152	C₂⁴×C₅	—	2	4	no	—
153	C₂⁴×C₅	—	2	4	no	—
154	C₂⁴×C₅	—	4	4	no	—
155	C₂⁴×C₅	—	4	4	no	—
156	C₂⁴×C₅	C₂⁴	1	4	yes	—
157	C₂⁴×C₅	C₂⁴	1	2	yes	—
158	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
159	C₅×#41	—	4	16	no	—
160	C₂×C₅×Q₈	—	2	2	no	—
161	C₂×C₅×Q₈	—	2	8	no	—
162	C₂×C₅×Q₈	—	2	2	no	—
163	C₂×C₅×Q₈	—	2	2	no	—
164	C₂×C₅×Q₈	—	4	8	no	—
165	C₂×C₅×D₄	—	2	4	no	—
166	C₂×C₅×D₄	—	2	4	no	—
167	C₂×C₅×D₄	—	2	4	no	—
168	C₂×C₅×D₄	—	2	4	no	—
169	C₂×C₅×D₄	—	2	4	no	—
170	C₂×C₅×D₄	—	2	2	no	—
171	C₂×C₅×D₄	—	2	8	no	—
172	C₂×C₅×D₄	—	2	8	no	—
173	C₂×C₅×D₄	—	2	4	no	—
174	C₂×C₅×D₄	—	2	4	no	—
175	C₂×C₅×D₄	—	2	4	no	—
176	C₂×C₅×D₄	—	2	4	no	—
177	C₂×C₅×D₄	—	2	2	no	—
178	C₂×C₅×D₄	—	2	2	no	—
179	C₂×C₅×D₄	—	4	8	no	—
180	C₂×C₅×D₄	—	4	8	no	—
181	C₂×C₅×D₄	—	4	16	no	—
182	C₂²×C₄×C₅	—	2	4	no	—
183	C₂²×C₄×C₅	—	2	4	no	—
184	C₂²×C₄×C₅	—	2	4	no	—
185	C₂²×C₄×C₅	—	2	4	no	—
186	C₂²×C₄×C₅	—	2	4	no	—
187	C₂²×C₄×C₅	—	2	4	no	—
188	C₂²×C₄×C₅	—	4	2	no	—
189	C₂²×C₄×C₅	—	4	4	no	—
190	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
191	C₂²×C₄×C₅	#41	1	4	yes	—
192	C₂²×C₄×C₅	#41	1	4	yes	—
193	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
194	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
195	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
196	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
197	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
198	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
201	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
203	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
204	C₅×C₄⋊C₄	—	2	4	no	—
205	C₅×C₄⋊C₄	—	2	4	no	—
206	C₅×C₄⋊C₄	—	2	2	no	—
207	C₅×C₄⋊C₄	—	2	2	no	—
208	C₅×C₄⋊C₄	—	2	8	no	—
209	C₅×C₄⋊C₄	—	2	8	no	—
210	C₅×C₄⋊C₄	—	2	8	no	—
211	C₅×C₄⋊C₄	—	2	4	no	—
212	C₅×C₄⋊C₄	C₂×D₄	1	4	yes	—
213	C₅×C₄⋊C₄	C₂×D₄	1	4	yes	—
214	C₅×C₂²⋊C₄	—	2	4	no	—
215	C₅×C₂²⋊C₄	—	2	4	no	—
216	C₅×C₂²⋊C₄	—	2	8	no	—
217	C₅×C₂²⋊C₄	—	2	8	no	—
218	C₅×C₂²⋊C₄	—	2	16	no	—
219	C₅×C₂²⋊C₄	—	2	8	no	—
220	C₅×C₂²⋊C₄	—	4	8	no	—
221	C₅×C₂²⋊C₄	—	4	8	no	—
222	C₅×C₂²⋊C₄	—	4	4	no	—
223	C₅×C₂²⋊C₄	—	4	8	no	—
224	C₅×C₂²⋊C₄	—	4	8	no	—
225	C₅×C₂²⋊C₄	C₂×D₄	1	8	yes	—
226	C₄²×C₅	—	2	4	no	—
227	C₄²×C₅	—	2	2	no	—
228	C₄²×C₅	—	2	4	no	—
229	C₄²×C₅	—	2	2	no	—
230	C₄²×C₅	—	2	4	no	—
231	C₄²×C₅	—	2	8	no	—
232	C₄²×C₅	—	4	4	no	—
233	C₄²×C₅	C₂×D₄	1	8	yes	—
234	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
235	C₂³×D₄	C₂²×C₅	1	1	yes	characteristic
236	#544	—	2	8	no	—
237	#535	—	2	8	no	—
238	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
239	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
240	C₂×#127	C₂²×C₅	1	2	yes	—
241	C₂×#125	C₂²×C₅	1	2	yes	complemented
242	C₂×#125	C₂²×C₅	1	2	yes	complemented
243	C₂×#124	C₂²×C₅	1	2	yes	—
244	C₂×#124	C₂²×C₅	1	8	yes	complemented
245	C₂×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
246	C₂×#121	C₂²×C₅	1	2	yes	—
247	C₂×#121	C₂²×C₅	1	2	yes	—
248	C₂×#120	C₂²×C₅	1	4	yes	—
249	C₂×#117	C₂²×C₅	1	1	yes	characteristic
250	C₂×#117	C₂²×C₅	1	1	yes	characteristic
251	C₂×#117	C₂²×C₅	1	1	yes	characteristic
252	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
253	C₂²×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
254	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
255	#397	C₂²×C₅	1	8	yes	complemented
256	#394	C₂²×C₅	1	8	yes	complemented
257	#394	C₂²×C₅	1	8	yes	complemented
258	#393	C₂²×C₅	1	8	yes	complemented
259	#391	C₂²×C₅	1	2	yes	complemented
260	#391	C₂²×C₅	1	1	yes	characteristic, complemented
261	#391	C₂²×C₅	1	1	yes	characteristic, complemented
262	#390	C₂²×C₅	1	2	yes	—
263	#390	C₂²×C₅	1	2	yes	—
264	#390	C₂²×C₅	1	4	yes	—
265	#390	C₂²×C₅	1	2	yes	—
266	#390	C₂²×C₅	1	2	yes	—
267	#390	C₂²×C₅	1	4	yes	—
268	#390	C₂²×C₅	1	2	yes	complemented
269	#390	C₂²×C₅	1	1	yes	characteristic, complemented
270	#390	C₂²×C₅	1	8	yes	complemented
271	#390	C₂²×C₅	1	1	yes	characteristic, complemented
272	#388	C₂²×C₅	1	8	yes	complemented
273	#388	C₂²×C₅	1	8	yes	complemented
274	#386	C₂²×C₅	1	4	yes	—
275	#386	C₂²×C₅	1	4	yes	—
276	#380	C₂²×C₅	1	4	yes	complemented
277	#379	C₂²×C₅	1	4	yes	—
278	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
279	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	complemented
280	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
281	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
282	#376	C₂²×C₅	1	2	yes	complemented
283	C₄³	C₂²×C₅	1	2	yes	complemented
284	C₂³×C₅	—	2	2	no	—
285	C₂³×C₅	—	2	2	no	—
286	C₂³×C₅	—	2	4	no	—
287	C₂³×C₅	—	2	2	no	—
288	C₂³×C₅	—	2	2	no	—
289	C₂³×C₅	—	2	2	no	—
290	C₂³×C₅	—	2	4	no	—
291	C₂³×C₅	—	2	4	no	—
292	C₂³×C₅	—	2	4	no	—
293	C₂³×C₅	—	2	8	no	—
294	C₂³×C₅	—	2	4	no	—
295	C₂³×C₅	—	2	8	no	—
296	C₂³×C₅	—	4	16	no	—
297	C₂³×C₅	—	4	8	no	—
298	C₂³×C₅	—	8	8	no	—
299	C₂³×C₅	—	8	4	no	—
300	C₂³×C₅	—	8	4	no	—
301	C₂³×C₅	C₂²×D₄	1	2	yes	—
302	C₂³×C₅	C₂×#41	1	2	yes	—
303	C₂³×C₅	C₂×#41	1	2	yes	—
304	C₂³×C₅	C₂⁵	1	1	yes	characteristic
305	C₅×Q₈	—	4	4	no	—
306	C₅×Q₈	—	8	8	no	—
307	C₅×D₄	—	4	8	no	—
308	C₅×D₄	—	4	8	no	—
309	C₅×D₄	—	4	8	no	—
310	C₅×D₄	—	4	4	no	—
311	C₅×D₄	—	8	8	no	—
312	C₅×D₄	—	8	8	no	—
313	C₂×C₄×C₅	—	2	2	no	—
314	C₂×C₄×C₅	—	2	4	no	—
315	C₂×C₄×C₅	—	2	2	no	—
316	C₂×C₄×C₅	—	2	4	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	4	no	—
321	C₂×C₄×C₅	—	2	2	no	—
322	C₂×C₄×C₅	—	2	4	no	—
323	C₂×C₄×C₅	—	2	2	no	—
324	C₂×C₄×C₅	—	2	2	no	—
325	C₂×C₄×C₅	—	2	4	no	—
326	C₂×C₄×C₅	—	2	8	no	—
327	C₂×C₄×C₅	—	2	4	no	—
328	C₂×C₄×C₅	—	2	2	no	—
329	C₂×C₄×C₅	—	2	4	no	—
330	C₂×C₄×C₅	—	2	4	no	—
331	C₂×C₄×C₅	—	2	2	no	—
332	C₂×C₄×C₅	—	4	8	no	—
333	C₂×C₄×C₅	—	4	8	no	—
334	C₂×C₄×C₅	—	8	4	no	—
335	C₂×C₄×C₅	#124	1	16	yes	complemented
336	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
337	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
338	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
339	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
340	C₂×C₄×C₅	C₂×#41	1	2	yes	—
341	C₂×C₄×C₅	C₂×#41	1	2	yes	—
342	C₂×C₄×C₅	C₂×#41	1	2	yes	—
343	C₂×C₄×C₅	C₂×#41	1	2	yes	—
344	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
345	C₂⁵	C₂³×C₅	1	2	yes	—
346	C₂×#41	—	2	4	no	—
347	C₂×#41	—	2	4	no	—
348	C₂×#41	—	2	4	no	—
349	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
350	C₂²×Q₈	C₂³×C₅	1	4	yes	—
351	C₂²×D₄	—	2	4	no	—
352	C₂²×D₄	—	2	4	no	—
353	C₂²×D₄	—	2	4	no	—
354	C₂²×D₄	C₂³×C₅	1	2	yes	—
355	C₂²×D₄	C₂³×C₅	1	2	yes	—
356	C₂²×D₄	C₂³×C₅	1	2	yes	—
357	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
358	C₂²×D₄	C₂³×C₅	1	4	yes	—
359	C₂²×D₄	C₂³×C₅	1	4	yes	—
360	C₂³×C₄	C₂³×C₅	1	2	yes	—
361	C₂³×C₄	C₂³×C₅	1	2	yes	—
362	C₂³×C₄	C₂³×C₅	1	2	yes	—
363	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
364	C₄⋊Q₈	—	2	8	no	—
365	C₄⋊Q₈	—	2	8	no	—
366	#127	—	2	8	no	—
367	#125	—	2	8	no	—
368	#125	—	2	8	no	—
369	#124	—	2	8	no	—
370	#124	—	4	16	no	—
371	C₂²⋊Q₈	—	2	16	no	—
372	#121	—	2	8	no	—
373	#121	—	2	8	no	—
374	#120	—	2	16	no	—
375	#117	—	2	4	no	—
376	#117	—	2	4	no	—
377	#117	C₅×D₄	1	8	yes	—
378	C₂×C₄⋊C₄	—	2	4	no	—
379	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
380	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
381	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
383	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
384	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
385	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
386	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
387	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
389	C₂×C₂²⋊C₄	—	4	8	no	—
390	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
391	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
402	C₂×C₄²	—	2	4	no	—
403	C₂×C₄²	C₂³×C₅	1	2	yes	—
404	C₂×C₄²	C₂³×C₅	1	2	yes	—
405	C₂×C₄²	C₂³×C₅	1	4	yes	—
406	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
408	C₂×C₄²	C₂³×C₅	1	2	yes	—
409	C₂×C₄²	C₂³×C₅	1	4	yes	—
410	C₂×C₄²	C₂³×C₅	1	2	yes	—
411	#95	C₂³×C₅	1	4	yes	—
412	#95	C₂³×C₅	1	4	yes	—
413	#95	C₂³×C₅	1	4	yes	—
414	#95	C₂³×C₅	1	4	yes	—
415	#95	C₂³×C₅	1	8	yes	—
416	C₂²×C₅	—	2	4	no	—
417	C₂²×C₅	—	2	4	no	—
418	C₂²×C₅	—	2	2	no	—
419	C₂²×C₅	—	2	2	no	—
420	C₂²×C₅	—	4	2	no	—
421	C₂²×C₅	—	4	4	no	—
422	C₂²×C₅	—	4	4	no	—
423	C₂²×C₅	—	4	8	no	—
424	C₂²×C₅	—	4	4	no	—
425	C₂²×C₅	—	4	8	no	—
426	C₂²×C₅	—	4	2	no	—
427	C₂²×C₅	—	4	4	no	—
428	C₂²×C₅	—	4	2	no	—
429	C₂²×C₅	—	4	4	no	—
430	C₂²×C₅	—	8	16	no	—
431	C₂²×C₅	C₂×#124	1	4	yes	—
432	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
433	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
434	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
435	C₂²×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
436	C₂²×C₅	#585	1	2	yes	—
437	C₄×C₅	—	2	4	no	—
438	C₄×C₅	—	2	4	no	—
439	C₄×C₅	—	2	4	no	—
440	C₄×C₅	—	4	2	no	—
441	C₄×C₅	—	4	4	no	—
442	C₄×C₅	—	4	2	no	—
443	C₄×C₅	—	4	4	no	—
444	C₄×C₅	—	4	2	no	—
445	C₄×C₅	—	4	2	no	—
446	C₄×C₅	C₂×#124	1	4	yes	—
447	C₄×C₅	C₂×#124	1	4	yes	—
448	C₂⁴	—	2	4	no	—
449	C₂⁴	—	2	4	no	—
450	C₂⁴	—	2	4	no	—
451	C₂⁴	—	4	4	no	—
452	C₂⁴	—	4	4	no	—
453	C₂⁴	C₂⁴×C₅	1	4	yes	—
454	C₂⁴	C₂⁴×C₅	1	2	yes	—
455	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
456	#41	—	4	16	no	—
457	C₂×Q₈	—	2	2	no	—
458	C₂×Q₈	—	2	8	no	—
459	C₂×Q₈	—	2	2	no	—
460	C₂×Q₈	—	2	2	no	—
461	C₂×Q₈	—	4	8	no	—
462	C₂×D₄	—	2	4	no	—
463	C₂×D₄	—	2	4	no	—
464	C₂×D₄	—	2	4	no	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	8	no	—
469	C₂×D₄	—	2	8	no	—
470	C₂×D₄	—	2	4	no	—
471	C₂×D₄	—	2	4	no	—
472	C₂×D₄	—	2	4	no	—
473	C₂×D₄	—	2	4	no	—
474	C₂×D₄	—	2	2	no	—
475	C₂×D₄	—	2	2	no	—
476	C₂×D₄	—	4	8	no	—
477	C₂×D₄	—	4	8	no	—
478	C₂×D₄	—	4	16	no	—
479	C₂²×C₄	—	2	4	no	—
480	C₂²×C₄	—	2	4	no	—
481	C₂²×C₄	—	2	4	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	4	2	no	—
486	C₂²×C₄	—	4	4	no	—
487	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
488	C₂²×C₄	C₅×#41	1	4	yes	—
489	C₂²×C₄	C₅×#41	1	4	yes	—
490	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
491	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
492	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
493	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
494	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
495	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
496	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
498	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
499	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
500	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
501	C₄⋊C₄	—	2	4	no	—
502	C₄⋊C₄	—	2	4	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	8	no	—
506	C₄⋊C₄	—	2	8	no	—
507	C₄⋊C₄	—	2	8	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	C₂×C₅×D₄	1	4	yes	—
510	C₄⋊C₄	C₂×C₅×D₄	1	4	yes	—
511	C₂²⋊C₄	—	2	4	no	—
512	C₂²⋊C₄	—	2	4	no	—
513	C₂²⋊C₄	—	2	8	no	—
514	C₂²⋊C₄	—	2	8	no	—
515	C₂²⋊C₄	—	2	16	no	—
516	C₂²⋊C₄	—	2	8	no	—
517	C₂²⋊C₄	—	4	8	no	—
518	C₂²⋊C₄	—	4	8	no	—
519	C₂²⋊C₄	—	4	8	no	—
520	C₂²⋊C₄	—	4	8	no	—
521	C₂²⋊C₄	—	4	4	no	—
522	C₂²⋊C₄	C₂×C₅×D₄	1	8	yes	—
523	C₄²	—	2	4	no	—
524	C₄²	—	2	2	no	—
525	C₄²	—	2	4	no	—
526	C₄²	—	2	2	no	—
527	C₄²	—	2	4	no	—
528	C₄²	—	2	8	no	—
529	C₄²	—	4	4	no	—
530	C₄²	C₂×C₅×D₄	1	8	yes	—
531	C₂×C₅	—	2	4	no	—
532	C₂×C₅	—	8	2	no	—
533	C₂×C₅	—	8	4	no	—
534	C₂×C₅	C₂²×#124	1	1	yes	characteristic
535	C₂×C₅	#3460	1	2	yes	—
536	C₂×C₅	#3494	1	2	yes	—
537	C₂×C₅	#3498	1	1	yes	characteristic
538	C₂×C₅	#3500	1	1	yes	characteristic
539	C₂³	—	2	2	no	—
540	C₂³	—	2	2	no	—
541	C₂³	—	2	4	no	—
542	C₂³	—	2	2	no	—
543	C₂³	—	2	2	no	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	4	no	—
546	C₂³	—	2	4	no	—
547	C₂³	—	2	4	no	—
548	C₂³	—	2	8	no	—
549	C₂³	—	2	4	no	—
550	C₂³	—	2	8	no	—
551	C₂³	—	4	16	no	—
552	C₂³	—	4	8	no	—
553	C₂³	—	8	8	no	—
554	C₂³	—	8	4	no	—
555	C₂³	—	8	4	no	—
556	C₂³	C₂²×C₅×D₄	1	2	yes	—
557	C₂³	C₂×C₅×#41	1	2	yes	—
558	C₂³	C₂×C₅×#41	1	2	yes	—
559	C₂³	C₂⁵×C₅	1	1	yes	characteristic
560	Q₈	—	4	4	no	—
561	Q₈	—	8	8	no	—
562	D₄	—	4	8	no	—
563	D₄	—	4	8	no	—
564	D₄	—	4	8	no	—
565	D₄	—	4	4	no	—
566	D₄	—	8	8	no	—
567	D₄	—	8	8	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	8	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	2	4	no	—
585	C₂×C₄	—	2	4	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	4	8	no	—
588	C₂×C₄	—	4	8	no	—
589	C₂×C₄	—	8	4	no	—
590	C₂×C₄	C₅×#124	1	16	yes	complemented
591	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
592	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
593	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
594	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
595	C₂×C₄	C₂×C₅×#41	1	2	yes	—
596	C₂×C₄	C₂×C₅×#41	1	2	yes	—
597	C₂×C₄	C₂×C₅×#41	1	2	yes	—
598	C₂×C₄	C₂×C₅×#41	1	2	yes	—
599	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
600	C₅	#35264	1	1	yes	characteristic, direct product factor, Sylow
601	C₂²	—	2	4	no	—
602	C₂²	—	2	4	no	—
603	C₂²	—	2	2	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	4	no	—
607	C₂²	—	4	8	no	—
608	C₂²	—	4	4	no	—
609	C₂²	—	4	8	no	—
610	C₂²	—	4	4	no	—
611	C₂²	—	4	2	no	—
612	C₂²	—	4	4	no	—
613	C₂²	—	4	2	no	—
614	C₂²	—	4	4	no	—
615	C₂²	—	8	16	no	—
616	C₂²	C₂×C₅×#124	1	4	yes	—
617	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
618	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
619	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
620	C₂²	C₂×C₅×H(5,𝔽₂)	1	2	yes	—
621	C₂²	C₅×#585	1	2	yes	—
622	C₄	—	2	4	no	—
623	C₄	—	2	4	no	—
624	C₄	—	2	4	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	4	no	—
627	C₄	—	4	2	no	—
628	C₄	—	4	4	no	—
629	C₄	—	4	2	no	—
630	C₄	—	4	2	no	—
631	C₄	C₂×C₅×#124	1	4	yes	—
632	C₄	C₂×C₅×#124	1	4	yes	—
633	C₂	—	2	4	no	—
634	C₂	—	8	4	no	—
635	C₂	—	8	2	no	—
636	C₂	C₂²×C₅×#124	1	1	yes	characteristic
637	C₂	C₅×#3460	1	2	yes	—
638	C₂	C₅×#3494	1	2	yes	—
639	C₂	C₅×#3498	1	1	yes	characteristic
640	C₂	C₅×#3500	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂×C₅×#535	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#544	C₂	semidirect
3. C₅×#2861	C₂	semidirect
4. C₅×#2858	C₂	semidirect
5. C₅×#2641	C₂	semidirect
6. C₅×#2640	C₂	semidirect
7. C₅×#2491	C₂	semidirect
8. C₅×#2490	C₂	semidirect
9. C₅×#2472	C₂	semidirect
10. C₅×#2311	C₂	semidirect
11. C₅×#2309	C₂	semidirect
12. C₅×#2297	C₂	semidirect
13. C₂×C₅×#390	C₂	semidirect
14. C₄×C₅×#117	C₂	semidirect
19. C₂×C₅×C₄⋊Q₈	C₂²	semidirect
20. C₂×C₅×C₄⋊Q₈	C₂²	semidirect
22. C₂×C₅×#125	C₂²	semidirect
23. C₂×C₅×#125	C₂²	semidirect
25. C₂×C₅×#124	C₂²	semidirect
36. C₅×#397	C₂²	semidirect
37. C₅×#394	C₂²	semidirect
38. C₅×#394	C₂²	semidirect
39. C₅×#393	C₂²	semidirect
40. C₅×#391	C₂²	semidirect
41. C₅×#391	C₂²	semidirect
42. C₅×#391	C₂²	semidirect
49. C₅×#390	C₂²	semidirect
50. C₅×#390	C₂²	semidirect
51. C₅×#390	C₂²	semidirect
52. C₅×#390	C₂²	semidirect
53. C₅×#388	C₂²	semidirect
54. C₅×#388	C₂²	semidirect
57. C₅×#380	C₂²	semidirect
59. C₄×C₅×C₄⋊C₄	C₂²	semidirect
60. C₄×C₅×C₄⋊C₄	C₂²	semidirect
62. C₄×C₅×C₂²⋊C₄	C₂²	semidirect
63. C₅×#376	C₂²	semidirect
64. C₄³×C₅	C₂²	semidirect
65. #35264	C₅	direct (N × H)
137. C₂×#544	C₂×C₅	semidirect
139. #2861	C₂×C₅	semidirect
140. #2858	C₂×C₅	semidirect
141. #2641	C₂×C₅	semidirect
142. #2640	C₂×C₅	semidirect
143. #2491	C₂×C₅	semidirect
144. #2490	C₂×C₅	semidirect
145. #2472	C₂×C₅	semidirect
146. #2311	C₂×C₅	semidirect
147. #2309	C₂×C₅	semidirect
148. #2297	C₂×C₅	semidirect
149. C₂×#390	C₂×C₅	semidirect
150. C₄×#117	C₂×C₅	semidirect
238. C₂×C₄⋊Q₈	C₂²×C₅	semidirect
239. C₂×C₄⋊Q₈	C₂²×C₅	semidirect
241. C₂×#125	C₂²×C₅	semidirect
242. C₂×#125	C₂²×C₅	semidirect
244. C₂×#124	C₂²×C₅	semidirect
255. #397	C₂²×C₅	semidirect
256. #394	C₂²×C₅	semidirect
257. #394	C₂²×C₅	semidirect
258. #393	C₂²×C₅	semidirect
259. #391	C₂²×C₅	semidirect
260. #391	C₂²×C₅	semidirect
261. #391	C₂²×C₅	semidirect
268. #390	C₂²×C₅	semidirect
269. #390	C₂²×C₅	semidirect
270. #390	C₂²×C₅	semidirect
271. #390	C₂²×C₅	semidirect
272. #388	C₂²×C₅	semidirect
273. #388	C₂²×C₅	semidirect
276. #380	C₂²×C₅	semidirect
278. C₄×C₄⋊C₄	C₂²×C₅	semidirect
279. C₄×C₄⋊C₄	C₂²×C₅	semidirect
281. C₄×C₂²⋊C₄	C₂²×C₅	semidirect
282. #376	C₂²×C₅	semidirect
283. C₄³	C₂²×C₅	semidirect
335. C₂×C₄×C₅	#124	semidirect
590. C₂×C₄	C₅×#124	semidirect
600. C₅	#35264	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×#544	2
C₂×C₅×#535	2
C₅×#2861	2
C₅×#2858	2
C₅×#2641	2
C₅×#2640	2
C₅×#2491	2
C₅×#2490	2
C₅×#2472	2
C₅×#2311	2
C₅×#2309	2
C₅×#2297	2
C₂×C₅×#390	2
C₄×C₅×#117	2
#35264	5

Subgroups — C5×#35264 #11957984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#35264 #11957984