Subgroups — C₅×#62851 #11985571

Order 1280.

7066 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3524	C₂	1	8	yes	maximal, complemented
2	C₅×#3522	C₂	1	8	yes	maximal, complemented
3	C₅×#3517	C₂	1	8	yes	maximal, complemented
4	C₅×#3516	C₂	1	8	yes	maximal, complemented
5	C₅×#3506	C₂	1	6	yes	maximal, complemented
6	C₅×#3502	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#3497	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#3492	C₂	1	6	yes	maximal, complemented
9	C₅×#3487	C₂	1	2	yes	maximal, complemented
10	C₅×#3460	C₂	1	6	yes	maximal, complemented
11	C₅×#3455	C₂	1	6	yes	maximal, complemented
12	C₄×C₅×2₋¹⁺⁴	C₂	1	2	yes	maximal, complemented
13	C₂×C₅×#520	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₅×2₋¹⁺⁴	C₂²	1	2	yes	—
15	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
16	C₅×#559	C₂²	1	24	yes	complemented
17	C₅×Q₈²	C₂²	1	4	yes	complemented
18	C₅×#557	C₂²	1	8	yes	complemented
19	C₅×#557	C₂²	1	4	yes	complemented
20	C₅×#556	C₂²	1	12	yes	complemented
21	C₅×#555	C₂²	1	12	yes	complemented
22	C₅×#554	C₂²	1	24	yes	complemented
23	C₅×#553	C₂²	1	24	yes	complemented
24	C₅×#553	C₂²	1	24	yes	complemented
25	C₅×#552	C₂²	1	24	yes	complemented
26	C₅×#550	C₂²	1	8	yes	—
27	C₅×#550	C₂²	1	8	yes	—
28	C₅×#550	C₂²	1	4	yes	complemented
29	C₅×#550	C₂²	1	12	yes	complemented
30	C₅×D₄×Q₈	C₂²	1	4	yes	complemented
31	C₅×#548	C₂²	1	8	yes	—
32	C₅×#548	C₂²	1	8	yes	—
33	C₅×#548	C₂²	1	8	yes	complemented
34	C₅×#547	C₂²	1	24	yes	complemented
35	C₅×#547	C₂²	1	12	yes	complemented
36	C₅×#544	C₂²	1	6	yes	—
37	C₅×#542	C₂²	1	12	yes	complemented
38	C₅×#542	C₂²	1	12	yes	complemented
39	C₅×#542	C₂²	1	24	yes	complemented
40	C₅×#541	C₂²	1	12	yes	complemented
41	C₅×#540	C₂²	1	12	yes	complemented
42	C₅×#540	C₂²	1	24	yes	complemented
43	C₅×#539	C₂²	1	4	yes	—
44	C₅×#539	C₂²	1	12	yes	—
45	C₅×#538	C₂²	1	12	yes	—
46	C₅×#538	C₂²	1	12	yes	—
47	C₅×#537	C₂²	1	6	yes	—
48	C₅×#535	C₂²	1	12	yes	—
49	C₅×#533	C₂²	1	6	yes	—
50	C₅×#532	C₂²	1	6	yes	—
51	C₅×#532	C₂²	1	24	yes	complemented
52	C₅×#529	C₂²	1	24	yes	complemented
53	C₅×#529	C₂²	1	24	yes	complemented
54	C₅×#529	C₂²	1	24	yes	complemented
55	C₅×#525	C₂²	1	12	yes	—
56	C₅×#525	C₂²	1	24	yes	—
57	C₅×#520	C₂²	1	8	yes	—
58	C₅×#520	C₂²	1	8	yes	—
59	C₅×#520	C₂²	1	12	yes	—
60	C₅×#520	C₂²	1	8	yes	complemented
61	C₅×#519	C₂²	1	2	yes	—
62	C₅×#519	C₂²	1	8	yes	complemented
63	C₄×C₅×#41	C₂²	1	12	yes	—
64	C₄×C₅×#41	C₂²	1	8	yes	complemented
65	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
66	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
67	C₂×C₄×C₅×Q₈	C₂²	1	8	yes	complemented
68	C₂×C₄×C₅×D₄	C₂²	1	3	yes	—
69	C₂×C₄×C₅×D₄	C₂²	1	3	yes	—
70	C₂×C₅×#117	C₂²	1	3	yes	—
71	C₂²×C₅×C₄⋊C₄	C₂²	1	3	yes	—
72	#62851	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₅×2₋¹⁺⁴	—	2	16	no	—
74	C₂×C₅×#41	—	2	8	no	—
75	C₂×C₅×#41	C₂³	1	12	yes	—
76	C₂×C₅×#41	C₂³	1	8	yes	—
77	C₂×C₅×#41	C₂³	1	8	yes	—
78	C₂×C₅×#41	C₂³	1	24	yes	—
79	C₂²×C₅×Q₈	C₂³	1	8	yes	—
80	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
81	C₂²×C₅×D₄	C₂³	1	3	yes	—
82	C₂³×C₄×C₅	C₂³	1	3	yes	—
83	C₂³×C₄×C₅	C₂³	1	6	yes	—
84	C₅×C₄⋊Q₈	C₂³	1	6	yes	—
85	C₅×C₄⋊Q₈	C₂³	1	24	yes	—
86	C₅×C₄⋊Q₈	C₂³	1	6	yes	—
87	C₅×C₄⋊Q₈	C₂³	1	6	yes	—
88	C₅×#127	C₂³	1	6	yes	—
89	C₅×#127	C₂³	1	24	yes	—
90	C₅×#126	C₂³	1	24	yes	—
91	C₅×#126	C₂³	1	8	yes	—
92	C₅×#126	C₂³	1	24	yes	—
93	C₅×#126	C₂³	1	24	yes	—
94	C₅×#125	C₂³	1	8	yes	—
95	C₅×#125	C₂³	1	24	yes	—
96	C₅×#125	C₂³	1	24	yes	—
97	C₅×#125	C₂³	1	12	yes	—
98	C₅×#124	C₂³	1	24	yes	—
99	C₅×#124	C₂³	1	24	yes	—
100	C₅×#124	C₂³	1	24	yes	—
101	C₅×#124	C₂³	1	12	yes	—
102	C₅×#123	C₂³	1	8	yes	—
103	C₅×#123	C₂³	1	48	yes	—
104	C₅×#123	C₂³	1	24	yes	—
105	C₅×#123	C₂³	1	8	yes	—
106	C₅×#123	C₂³	1	24	yes	—
107	C₅×#123	C₂³	1	24	yes	—
108	C₅×C₂²⋊Q₈	C₂³	1	12	yes	—
109	C₅×C₂²⋊Q₈	C₂³	1	12	yes	—
110	C₅×C₂²⋊Q₈	C₂³	1	12	yes	—
111	C₅×C₂²⋊Q₈	C₂³	1	24	yes	—
112	C₅×C₂²⋊Q₈	C₂³	1	8	yes	—
113	C₅×C₂²⋊Q₈	C₂³	1	24	yes	—
114	C₅×C₂²⋊Q₈	C₂³	1	24	yes	—
115	C₅×C₂²⋊Q₈	C₂³	1	4	yes	—
116	C₅×#121	C₂³	1	12	yes	—
117	C₅×#121	C₂³	1	24	yes	—
118	C₅×#121	C₂³	1	12	yes	—
119	C₅×#121	C₂³	1	12	yes	—
120	C₅×#121	C₂³	1	24	yes	—
121	C₅×#121	C₂³	1	24	yes	—
122	C₅×#121	C₂³	1	24	yes	—
123	C₅×#121	C₂³	1	4	yes	—
124	C₅×#121	C₂³	1	24	yes	—
125	C₅×#121	C₂³	1	24	yes	—
126	C₅×#120	C₂³	1	24	yes	—
127	C₄×C₅×Q₈	C₂³	1	8	yes	—
128	C₄×C₅×Q₈	C₂³	1	4	yes	—
129	C₄×C₅×Q₈	C₂³	1	4	yes	—
130	C₄×C₅×Q₈	C₂³	1	12	yes	—
131	C₄×C₅×Q₈	C₂³	1	8	yes	—
132	C₄×C₅×Q₈	C₂³	1	24	yes	—
133	C₄×C₅×Q₈	C₂³	1	12	yes	—
134	C₄×C₅×Q₈	C₂³	1	8	yes	—
135	C₄×C₅×Q₈	C₂³	1	12	yes	—
136	C₄×C₅×Q₈	C₂³	1	12	yes	—
137	C₄×C₅×D₄	C₂³	1	24	yes	—
138	C₄×C₅×D₄	C₂³	1	12	yes	—
139	C₄×C₅×D₄	C₂³	1	12	yes	—
140	C₄×C₅×D₄	C₂³	1	8	yes	—
141	C₄×C₅×D₄	C₂³	1	12	yes	—
142	C₄×C₅×D₄	C₂³	1	24	yes	—
143	C₄×C₅×D₄	C₂³	1	12	yes	—
144	C₄×C₅×D₄	C₂³	1	24	yes	—
145	C₄×C₅×D₄	C₂³	1	24	yes	—
146	C₄×C₅×D₄	C₂³	1	24	yes	—
147	C₄×C₅×D₄	C₂³	1	48	yes	—
148	C₄×C₅×D₄	C₂³	1	12	yes	—
149	C₄×C₅×D₄	C₂³	1	12	yes	—
150	C₅×#117	C₂³	1	12	yes	—
151	C₅×#117	C₂³	1	12	yes	—
152	C₅×#117	C₂³	1	12	yes	—
153	C₅×#117	C₂³	1	24	yes	—
154	C₅×#117	C₂³	1	24	yes	—
155	C₅×#117	C₂³	1	24	yes	—
156	C₂×C₅×C₄⋊C₄	C₂³	1	3	yes	—
157	C₂×C₅×C₄⋊C₄	C₂³	1	3	yes	—
158	C₂×C₅×C₄⋊C₄	C₂³	1	3	yes	—
159	C₂×C₅×C₄⋊C₄	C₂³	1	24	yes	—
160	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
161	C₂×C₅×C₄⋊C₄	C₂³	1	24	yes	—
162	C₂×C₅×C₂²⋊C₄	C₂³	1	3	yes	—
163	C₂×C₅×C₂²⋊C₄	C₂³	1	3	yes	—
164	C₂×C₄²×C₅	C₂³	1	3	yes	—
165	C₂×C₄²×C₅	C₂³	1	3	yes	—
166	C₂×C₄²×C₅	C₂³	1	24	yes	—
167	#3524	C₂×C₅	1	8	yes	complemented
168	#3522	C₂×C₅	1	8	yes	complemented
169	#3517	C₂×C₅	1	8	yes	complemented
170	#3516	C₂×C₅	1	8	yes	complemented
171	#3506	C₂×C₅	1	6	yes	complemented
172	#3502	C₂×C₅	1	1	yes	characteristic, complemented
173	#3497	C₂×C₅	1	1	yes	characteristic, complemented
174	#3492	C₂×C₅	1	6	yes	complemented
175	#3487	C₂×C₅	1	2	yes	complemented
176	#3460	C₂×C₅	1	6	yes	complemented
177	#3455	C₂×C₅	1	6	yes	complemented
178	C₄×2₋¹⁺⁴	C₂×C₅	1	2	yes	complemented
179	C₂×#520	C₂×C₅	1	1	yes	characteristic, complemented
180	C₂⁴×C₅	C₂⁴	1	3	yes	—
181	C₅×#41	—	2	32	no	—
182	C₅×#41	—	2	32	no	—
183	C₅×#41	—	2	48	no	—
184	C₅×#41	—	8	24	no	—
185	C₂×C₅×Q₈	—	2	32	no	—
186	C₂×C₅×Q₈	#41	1	8	yes	complemented
187	C₂×C₅×Q₈	C₂⁴	1	12	yes	—
188	C₂×C₅×Q₈	C₂⁴	1	4	yes	—
189	C₂×C₅×Q₈	C₂⁴	1	12	yes	—
190	C₂×C₅×Q₈	C₂⁴	1	4	yes	—
191	C₂×C₅×D₄	—	2	12	no	—
192	C₂×C₅×D₄	C₂⁴	1	8	yes	—
193	C₂×C₅×D₄	C₂⁴	1	12	yes	—
194	C₂×C₅×D₄	C₂⁴	1	12	yes	—
195	C₂×C₅×D₄	C₂⁴	1	24	yes	—
196	C₂×C₅×D₄	C₂⁴	1	4	yes	—
197	C₂×C₅×D₄	C₂⁴	1	24	yes	—
198	C₂×C₅×D₄	C₂⁴	1	12	yes	—
199	C₂²×C₄×C₅	—	2	12	no	—
200	C₂²×C₄×C₅	—	8	6	no	—
201	C₂²×C₄×C₅	C₂⁴	1	24	yes	—
202	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
203	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
204	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
205	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
206	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
207	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
208	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
209	C₂²×C₄×C₅	C₂⁴	1	24	yes	—
210	C₂²×C₄×C₅	C₂⁴	1	24	yes	—
211	C₂²×C₄×C₅	C₂⁴	1	3	yes	—
212	C₅×C₄⋊C₄	C₂⁴	1	12	yes	—
213	C₅×C₄⋊C₄	C₂⁴	1	12	yes	—
214	C₅×C₄⋊C₄	C₂⁴	1	12	yes	—
215	C₅×C₄⋊C₄	C₂⁴	1	12	yes	—
216	C₅×C₄⋊C₄	C₂⁴	1	24	yes	—
217	C₅×C₄⋊C₄	C₂⁴	1	2	yes	—
218	C₅×C₄⋊C₄	C₂⁴	1	2	yes	—
219	C₅×C₄⋊C₄	C₂⁴	1	12	yes	—
220	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
221	C₅×C₄⋊C₄	C₂⁴	1	12	yes	—
222	C₅×C₄⋊C₄	C₂⁴	1	12	yes	—
223	C₅×C₄⋊C₄	C₂⁴	1	4	yes	—
224	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
225	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
226	C₅×C₄⋊C₄	C₂⁴	1	24	yes	—
227	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
228	C₅×C₄⋊C₄	C₂⁴	1	4	yes	—
229	C₅×C₂²⋊C₄	C₂⁴	1	6	yes	—
230	C₅×C₂²⋊C₄	C₂⁴	1	12	yes	—
231	C₅×C₂²⋊C₄	C₂⁴	1	6	yes	—
232	C₅×C₂²⋊C₄	C₂⁴	1	8	yes	—
233	C₅×C₂²⋊C₄	C₂⁴	1	24	yes	—
234	C₅×C₂²⋊C₄	C₂⁴	1	8	yes	—
235	C₅×C₂²⋊C₄	C₂⁴	1	24	yes	—
236	C₅×C₂²⋊C₄	C₂⁴	1	24	yes	—
237	C₅×C₂²⋊C₄	C₂⁴	1	8	yes	—
238	C₅×C₂²⋊C₄	C₂⁴	1	24	yes	—
239	C₅×C₂²⋊C₄	C₂⁴	1	8	yes	—
240	C₅×C₂²⋊C₄	C₂⁴	1	8	yes	—
241	C₄²×C₅	C₂⁴	1	12	yes	—
242	C₄²×C₅	C₂⁴	1	8	yes	—
243	C₄²×C₅	C₂⁴	1	24	yes	—
244	C₄²×C₅	C₂⁴	1	6	yes	—
245	C₄²×C₅	C₂⁴	1	6	yes	—
246	C₄²×C₅	C₂⁴	1	24	yes	—
247	C₂×2₋¹⁺⁴	C₂²×C₅	1	2	yes	—
248	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
249	#559	C₂²×C₅	1	24	yes	complemented
250	Q₈²	C₂²×C₅	1	4	yes	complemented
251	#557	C₂²×C₅	1	8	yes	complemented
252	#557	C₂²×C₅	1	4	yes	complemented
253	#556	C₂²×C₅	1	12	yes	complemented
254	#555	C₂²×C₅	1	12	yes	complemented
255	#554	C₂²×C₅	1	24	yes	complemented
256	#553	C₂²×C₅	1	24	yes	complemented
257	#553	C₂²×C₅	1	24	yes	complemented
258	#552	C₂²×C₅	1	24	yes	complemented
259	#550	C₂²×C₅	1	8	yes	—
260	#550	C₂²×C₅	1	8	yes	—
261	#550	C₂²×C₅	1	4	yes	complemented
262	#550	C₂²×C₅	1	12	yes	complemented
263	D₄×Q₈	C₂²×C₅	1	4	yes	complemented
264	#548	C₂²×C₅	1	8	yes	—
265	#548	C₂²×C₅	1	8	yes	—
266	#548	C₂²×C₅	1	8	yes	complemented
267	#547	C₂²×C₅	1	24	yes	complemented
268	#547	C₂²×C₅	1	12	yes	complemented
269	#544	C₂²×C₅	1	6	yes	—
270	#542	C₂²×C₅	1	12	yes	complemented
271	#542	C₂²×C₅	1	12	yes	complemented
272	#542	C₂²×C₅	1	24	yes	complemented
273	#541	C₂²×C₅	1	12	yes	complemented
274	#540	C₂²×C₅	1	12	yes	complemented
275	#540	C₂²×C₅	1	24	yes	complemented
276	#539	C₂²×C₅	1	4	yes	—
277	#539	C₂²×C₅	1	12	yes	—
278	#538	C₂²×C₅	1	12	yes	—
279	#538	C₂²×C₅	1	12	yes	—
280	#537	C₂²×C₅	1	6	yes	—
281	#535	C₂²×C₅	1	12	yes	—
282	#533	C₂²×C₅	1	6	yes	—
283	#532	C₂²×C₅	1	6	yes	—
284	#532	C₂²×C₅	1	24	yes	complemented
285	#529	C₂²×C₅	1	24	yes	complemented
286	#529	C₂²×C₅	1	24	yes	complemented
287	#529	C₂²×C₅	1	24	yes	complemented
288	#525	C₂²×C₅	1	12	yes	—
289	#525	C₂²×C₅	1	24	yes	—
290	#520	C₂²×C₅	1	8	yes	—
291	#520	C₂²×C₅	1	8	yes	—
292	#520	C₂²×C₅	1	12	yes	—
293	#520	C₂²×C₅	1	8	yes	complemented
294	#519	C₂²×C₅	1	2	yes	—
295	#519	C₂²×C₅	1	8	yes	complemented
296	C₄×#41	C₂²×C₅	1	12	yes	—
297	C₄×#41	C₂²×C₅	1	8	yes	complemented
298	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
299	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
300	C₂×C₄×Q₈	C₂²×C₅	1	8	yes	complemented
301	C₂×C₄×D₄	C₂²×C₅	1	3	yes	—
302	C₂×C₄×D₄	C₂²×C₅	1	3	yes	—
303	C₂×#117	C₂²×C₅	1	3	yes	—
304	C₂²×C₄⋊C₄	C₂²×C₅	1	3	yes	—
305	C₂³×C₅	—	2	6	no	—
306	C₂³×C₅	—	4	3	no	—
307	C₂³×C₅	—	4	3	no	—
308	C₂³×C₅	C₂⁵	1	6	yes	—
309	C₂³×C₅	C₂⁵	1	8	yes	—
310	C₂³×C₅	C₂⁵	1	1	yes	characteristic
311	C₅×Q₈	—	2	24	no	—
312	C₅×Q₈	—	4	4	no	—
313	C₅×Q₈	—	4	12	no	—
314	C₅×Q₈	C₂×#41	1	16	yes	—
315	C₅×D₄	—	2	16	no	—
316	C₅×D₄	—	2	24	no	—
317	C₅×D₄	—	2	24	no	—
318	C₅×D₄	—	4	4	no	—
319	C₅×D₄	—	4	24	no	—
320	C₅×D₄	—	4	12	no	—
321	C₅×D₄	—	4	24	no	—
322	C₂×C₄×C₅	—	2	2	no	—
323	C₂×C₄×C₅	—	2	24	no	—
324	C₂×C₄×C₅	—	2	48	no	—
325	C₂×C₄×C₅	—	4	3	no	—
326	C₂×C₄×C₅	—	4	3	no	—
327	C₂×C₄×C₅	—	4	24	no	—
328	C₂×C₄×C₅	—	4	1	no	—
329	C₂×C₄×C₅	—	4	8	no	—
330	C₂×C₄×C₅	—	4	1	no	—
331	C₂×C₄×C₅	—	4	24	no	—
332	C₂×C₄×C₅	C₂×#41	1	12	yes	—
333	C₂×C₄×C₅	C₂⁵	1	2	yes	—
334	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
335	C₂×C₄×C₅	C₂⁵	1	8	yes	—
336	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
337	C₂×C₄×C₅	C₂⁵	1	6	yes	—
338	C₂×C₄×C₅	C₂⁵	1	8	yes	—
339	C₂×C₄×C₅	C₂⁵	1	8	yes	—
340	C₂×C₄×C₅	C₂⁵	1	6	yes	—
341	C₂×C₄×C₅	C₂⁵	1	2	yes	—
342	C₂×C₄×C₅	C₂⁵	1	6	yes	—
343	2₋¹⁺⁴	—	2	16	no	—
344	C₂×#41	—	2	8	no	—
345	C₂×#41	C₂³×C₅	1	12	yes	—
346	C₂×#41	C₂³×C₅	1	8	yes	—
347	C₂×#41	C₂³×C₅	1	8	yes	—
348	C₂×#41	C₂³×C₅	1	24	yes	—
349	C₂²×Q₈	C₂³×C₅	1	8	yes	—
350	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
351	C₂²×D₄	C₂³×C₅	1	3	yes	—
352	C₂³×C₄	C₂³×C₅	1	3	yes	—
353	C₂³×C₄	C₂³×C₅	1	6	yes	—
354	C₄⋊Q₈	C₂³×C₅	1	6	yes	—
355	C₄⋊Q₈	C₂³×C₅	1	24	yes	—
356	C₄⋊Q₈	C₂³×C₅	1	6	yes	—
357	C₄⋊Q₈	C₂³×C₅	1	6	yes	—
358	#127	C₂³×C₅	1	6	yes	—
359	#127	C₂³×C₅	1	24	yes	—
360	#126	C₂³×C₅	1	24	yes	—
361	#126	C₂³×C₅	1	8	yes	—
362	#126	C₂³×C₅	1	24	yes	—
363	#126	C₂³×C₅	1	24	yes	—
364	#125	C₂³×C₅	1	8	yes	—
365	#125	C₂³×C₅	1	24	yes	—
366	#125	C₂³×C₅	1	24	yes	—
367	#125	C₂³×C₅	1	12	yes	—
368	#124	C₂³×C₅	1	24	yes	—
369	#124	C₂³×C₅	1	24	yes	—
370	#124	C₂³×C₅	1	24	yes	—
371	#124	C₂³×C₅	1	12	yes	—
372	#123	C₂³×C₅	1	8	yes	—
373	#123	C₂³×C₅	1	48	yes	—
374	#123	C₂³×C₅	1	24	yes	—
375	#123	C₂³×C₅	1	8	yes	—
376	#123	C₂³×C₅	1	24	yes	—
377	#123	C₂³×C₅	1	24	yes	—
378	C₂²⋊Q₈	C₂³×C₅	1	12	yes	—
379	C₂²⋊Q₈	C₂³×C₅	1	12	yes	—
380	C₂²⋊Q₈	C₂³×C₅	1	12	yes	—
381	C₂²⋊Q₈	C₂³×C₅	1	24	yes	—
382	C₂²⋊Q₈	C₂³×C₅	1	8	yes	—
383	C₂²⋊Q₈	C₂³×C₅	1	24	yes	—
384	C₂²⋊Q₈	C₂³×C₅	1	24	yes	—
385	C₂²⋊Q₈	C₂³×C₅	1	4	yes	—
386	#121	C₂³×C₅	1	12	yes	—
387	#121	C₂³×C₅	1	24	yes	—
388	#121	C₂³×C₅	1	12	yes	—
389	#121	C₂³×C₅	1	12	yes	—
390	#121	C₂³×C₅	1	24	yes	—
391	#121	C₂³×C₅	1	24	yes	—
392	#121	C₂³×C₅	1	24	yes	—
393	#121	C₂³×C₅	1	4	yes	—
394	#121	C₂³×C₅	1	24	yes	—
395	#121	C₂³×C₅	1	24	yes	—
396	#120	C₂³×C₅	1	24	yes	—
397	C₄×Q₈	C₂³×C₅	1	8	yes	—
398	C₄×Q₈	C₂³×C₅	1	4	yes	—
399	C₄×Q₈	C₂³×C₅	1	4	yes	—
400	C₄×Q₈	C₂³×C₅	1	12	yes	—
401	C₄×Q₈	C₂³×C₅	1	8	yes	—
402	C₄×Q₈	C₂³×C₅	1	24	yes	—
403	C₄×Q₈	C₂³×C₅	1	12	yes	—
404	C₄×Q₈	C₂³×C₅	1	8	yes	—
405	C₄×Q₈	C₂³×C₅	1	12	yes	—
406	C₄×Q₈	C₂³×C₅	1	12	yes	—
407	C₄×D₄	C₂³×C₅	1	24	yes	—
408	C₄×D₄	C₂³×C₅	1	12	yes	—
409	C₄×D₄	C₂³×C₅	1	12	yes	—
410	C₄×D₄	C₂³×C₅	1	8	yes	—
411	C₄×D₄	C₂³×C₅	1	12	yes	—
412	C₄×D₄	C₂³×C₅	1	24	yes	—
413	C₄×D₄	C₂³×C₅	1	12	yes	—
414	C₄×D₄	C₂³×C₅	1	24	yes	—
415	C₄×D₄	C₂³×C₅	1	24	yes	—
416	C₄×D₄	C₂³×C₅	1	24	yes	—
417	C₄×D₄	C₂³×C₅	1	48	yes	—
418	C₄×D₄	C₂³×C₅	1	12	yes	—
419	C₄×D₄	C₂³×C₅	1	12	yes	—
420	#117	C₂³×C₅	1	12	yes	—
421	#117	C₂³×C₅	1	12	yes	—
422	#117	C₂³×C₅	1	12	yes	—
423	#117	C₂³×C₅	1	24	yes	—
424	#117	C₂³×C₅	1	24	yes	—
425	#117	C₂³×C₅	1	24	yes	—
426	C₂×C₄⋊C₄	C₂³×C₅	1	3	yes	—
427	C₂×C₄⋊C₄	C₂³×C₅	1	3	yes	—
428	C₂×C₄⋊C₄	C₂³×C₅	1	3	yes	—
429	C₂×C₄⋊C₄	C₂³×C₅	1	24	yes	—
430	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₅	1	24	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₅	1	3	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₅	1	3	yes	—
434	C₂×C₄²	C₂³×C₅	1	3	yes	—
435	C₂×C₄²	C₂³×C₅	1	3	yes	—
436	C₂×C₄²	C₂³×C₅	1	24	yes	—
437	C₂²×C₅	—	2	1	no	—
438	C₂²×C₅	—	2	1	no	—
439	C₂²×C₅	—	2	8	no	—
440	C₂²×C₅	—	2	6	no	—
441	C₂²×C₅	—	2	6	no	—
442	C₂²×C₅	—	2	8	no	—
443	C₂²×C₅	—	2	8	no	—
444	C₂²×C₅	—	2	6	no	—
445	C₂²×C₅	—	4	12	no	—
446	C₂²×C₅	C₂²×#41	1	2	yes	—
447	C₂²×C₅	C₂⁶	1	1	yes	characteristic
448	C₄×C₅	—	2	2	no	—
449	C₄×C₅	—	2	8	no	—
450	C₄×C₅	—	2	6	no	—
451	C₄×C₅	—	2	6	no	—
452	C₄×C₅	—	2	8	no	—
453	C₄×C₅	—	2	2	no	—
454	C₄×C₅	—	2	1	no	—
455	C₄×C₅	—	2	8	no	—
456	C₄×C₅	—	2	1	no	—
457	C₄×C₅	C₂²×#41	1	12	yes	—
458	C₂⁴	C₂⁴×C₅	1	3	yes	—
459	#41	—	2	32	no	—
460	#41	—	2	32	no	—
461	#41	—	2	48	no	—
462	#41	—	8	24	no	—
463	C₂×Q₈	—	2	32	no	—
464	C₂×Q₈	C₅×#41	1	8	yes	complemented
465	C₂×Q₈	C₂⁴×C₅	1	12	yes	—
466	C₂×Q₈	C₂⁴×C₅	1	4	yes	—
467	C₂×Q₈	C₂⁴×C₅	1	12	yes	—
468	C₂×Q₈	C₂⁴×C₅	1	4	yes	—
469	C₂×D₄	—	2	12	no	—
470	C₂×D₄	C₂⁴×C₅	1	8	yes	—
471	C₂×D₄	C₂⁴×C₅	1	12	yes	—
472	C₂×D₄	C₂⁴×C₅	1	12	yes	—
473	C₂×D₄	C₂⁴×C₅	1	24	yes	—
474	C₂×D₄	C₂⁴×C₅	1	4	yes	—
475	C₂×D₄	C₂⁴×C₅	1	24	yes	—
476	C₂×D₄	C₂⁴×C₅	1	12	yes	—
477	C₂²×C₄	—	2	12	no	—
478	C₂²×C₄	—	8	6	no	—
479	C₂²×C₄	C₂⁴×C₅	1	24	yes	—
480	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₅	1	12	yes	—
482	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
483	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
484	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
485	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
486	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
487	C₂²×C₄	C₂⁴×C₅	1	24	yes	—
488	C₂²×C₄	C₂⁴×C₅	1	24	yes	—
489	C₂²×C₄	C₂⁴×C₅	1	3	yes	—
490	C₄⋊C₄	C₂⁴×C₅	1	12	yes	—
491	C₄⋊C₄	C₂⁴×C₅	1	12	yes	—
492	C₄⋊C₄	C₂⁴×C₅	1	12	yes	—
493	C₄⋊C₄	C₂⁴×C₅	1	12	yes	—
494	C₄⋊C₄	C₂⁴×C₅	1	24	yes	—
495	C₄⋊C₄	C₂⁴×C₅	1	2	yes	—
496	C₄⋊C₄	C₂⁴×C₅	1	2	yes	—
497	C₄⋊C₄	C₂⁴×C₅	1	12	yes	—
498	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
499	C₄⋊C₄	C₂⁴×C₅	1	12	yes	—
500	C₄⋊C₄	C₂⁴×C₅	1	12	yes	—
501	C₄⋊C₄	C₂⁴×C₅	1	4	yes	—
502	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
503	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
504	C₄⋊C₄	C₂⁴×C₅	1	24	yes	—
505	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
506	C₄⋊C₄	C₂⁴×C₅	1	4	yes	—
507	C₂²⋊C₄	C₂⁴×C₅	1	6	yes	—
508	C₂²⋊C₄	C₂⁴×C₅	1	12	yes	—
509	C₂²⋊C₄	C₂⁴×C₅	1	6	yes	—
510	C₂²⋊C₄	C₂⁴×C₅	1	8	yes	—
511	C₂²⋊C₄	C₂⁴×C₅	1	24	yes	—
512	C₂²⋊C₄	C₂⁴×C₅	1	8	yes	—
513	C₂²⋊C₄	C₂⁴×C₅	1	24	yes	—
514	C₂²⋊C₄	C₂⁴×C₅	1	24	yes	—
515	C₂²⋊C₄	C₂⁴×C₅	1	8	yes	—
516	C₂²⋊C₄	C₂⁴×C₅	1	24	yes	—
517	C₂²⋊C₄	C₂⁴×C₅	1	8	yes	—
518	C₂²⋊C₄	C₂⁴×C₅	1	8	yes	—
519	C₄²	C₂⁴×C₅	1	12	yes	—
520	C₄²	C₂⁴×C₅	1	8	yes	—
521	C₄²	C₂⁴×C₅	1	24	yes	—
522	C₄²	C₂⁴×C₅	1	6	yes	—
523	C₄²	C₂⁴×C₅	1	6	yes	—
524	C₄²	C₂⁴×C₅	1	24	yes	—
525	C₂×C₅	—	2	2	no	—
526	C₂×C₅	—	4	8	no	—
527	C₂×C₅	—	4	6	no	—
528	C₂×C₅	C₂³×#41	1	1	yes	characteristic
529	C₂×C₅	H(7,𝔽₂)	1	1	yes	characteristic
530	C₂×C₅	2₋¹⁺⁶	1	1	yes	characteristic
531	C₂³	—	2	6	no	—
532	C₂³	—	4	3	no	—
533	C₂³	—	4	3	no	—
534	C₂³	C₂⁵×C₅	1	6	yes	—
535	C₂³	C₂⁵×C₅	1	8	yes	—
536	C₂³	C₂⁵×C₅	1	1	yes	characteristic
537	Q₈	—	2	24	no	—
538	Q₈	—	4	12	no	—
539	Q₈	—	4	4	no	—
540	Q₈	C₂×C₅×#41	1	16	yes	—
541	D₄	—	2	16	no	—
542	D₄	—	2	24	no	—
543	D₄	—	2	24	no	—
544	D₄	—	4	12	no	—
545	D₄	—	4	24	no	—
546	D₄	—	4	4	no	—
547	D₄	—	4	24	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	24	no	—
550	C₂×C₄	—	2	48	no	—
551	C₂×C₄	—	4	3	no	—
552	C₂×C₄	—	4	3	no	—
553	C₂×C₄	—	4	24	no	—
554	C₂×C₄	—	4	24	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	8	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	C₂×C₅×#41	1	12	yes	—
559	C₂×C₄	C₂⁵×C₅	1	2	yes	—
560	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
561	C₂×C₄	C₂⁵×C₅	1	8	yes	—
562	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
563	C₂×C₄	C₂⁵×C₅	1	6	yes	—
564	C₂×C₄	C₂⁵×C₅	1	8	yes	—
565	C₂×C₄	C₂⁵×C₅	1	8	yes	—
566	C₂×C₄	C₂⁵×C₅	1	6	yes	—
567	C₂×C₄	C₂⁵×C₅	1	2	yes	—
568	C₂×C₄	C₂⁵×C₅	1	6	yes	—
569	C₅	#62851	1	1	yes	characteristic, direct product factor, Sylow
570	C₂²	—	2	1	no	—
571	C₂²	—	2	1	no	—
572	C₂²	—	2	8	no	—
573	C₂²	—	2	6	no	—
574	C₂²	—	2	6	no	—
575	C₂²	—	2	8	no	—
576	C₂²	—	2	8	no	—
577	C₂²	—	2	6	no	—
578	C₂²	—	4	12	no	—
579	C₂²	C₂²×C₅×#41	1	2	yes	—
580	C₂²	C₂⁶×C₅	1	1	yes	characteristic
581	C₄	—	2	2	no	—
582	C₄	—	2	8	no	—
583	C₄	—	2	6	no	—
584	C₄	—	2	6	no	—
585	C₄	—	2	8	no	—
586	C₄	—	2	2	no	—
587	C₄	—	2	1	no	—
588	C₄	—	2	8	no	—
589	C₄	—	2	1	no	—
590	C₄	C₂²×C₅×#41	1	12	yes	—
591	C₂	—	2	2	no	—
592	C₂	—	4	6	no	—
593	C₂	—	4	8	no	—
594	C₂	C₂³×C₅×#41	1	1	yes	characteristic
595	C₂	C₅×H(7,𝔽₂)	1	1	yes	characteristic
596	C₂	C₅×2₋¹⁺⁶	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3524	C₂	semidirect
2. C₅×#3522	C₂	semidirect
3. C₅×#3517	C₂	semidirect
4. C₅×#3516	C₂	semidirect
5. C₅×#3506	C₂	semidirect
6. C₅×#3502	C₂	semidirect
7. C₅×#3497	C₂	semidirect
8. C₅×#3492	C₂	semidirect
9. C₅×#3487	C₂	semidirect
10. C₅×#3460	C₂	semidirect
11. C₅×#3455	C₂	semidirect
12. C₄×C₅×2₋¹⁺⁴	C₂	semidirect
13. C₂×C₅×#520	C₂	semidirect
16. C₅×#559	C₂²	semidirect
17. C₅×Q₈²	C₂²	semidirect
18. C₅×#557	C₂²	semidirect
19. C₅×#557	C₂²	semidirect
20. C₅×#556	C₂²	semidirect
21. C₅×#555	C₂²	semidirect
22. C₅×#554	C₂²	semidirect
23. C₅×#553	C₂²	semidirect
24. C₅×#553	C₂²	semidirect
25. C₅×#552	C₂²	semidirect
28. C₅×#550	C₂²	semidirect
29. C₅×#550	C₂²	semidirect
30. C₅×D₄×Q₈	C₂²	semidirect
33. C₅×#548	C₂²	semidirect
34. C₅×#547	C₂²	semidirect
35. C₅×#547	C₂²	semidirect
37. C₅×#542	C₂²	semidirect
38. C₅×#542	C₂²	semidirect
39. C₅×#542	C₂²	semidirect
40. C₅×#541	C₂²	semidirect
41. C₅×#540	C₂²	semidirect
42. C₅×#540	C₂²	semidirect
51. C₅×#532	C₂²	semidirect
52. C₅×#529	C₂²	semidirect
53. C₅×#529	C₂²	semidirect
54. C₅×#529	C₂²	semidirect
60. C₅×#520	C₂²	semidirect
62. C₅×#519	C₂²	semidirect
64. C₄×C₅×#41	C₂²	semidirect
67. C₂×C₄×C₅×Q₈	C₂²	semidirect
72. #62851	C₅	direct (N × H)
167. #3524	C₂×C₅	semidirect
168. #3522	C₂×C₅	semidirect
169. #3517	C₂×C₅	semidirect
170. #3516	C₂×C₅	semidirect
171. #3506	C₂×C₅	semidirect
172. #3502	C₂×C₅	semidirect
173. #3497	C₂×C₅	semidirect
174. #3492	C₂×C₅	semidirect
175. #3487	C₂×C₅	semidirect
176. #3460	C₂×C₅	semidirect
177. #3455	C₂×C₅	semidirect
178. C₄×2₋¹⁺⁴	C₂×C₅	semidirect
179. C₂×#520	C₂×C₅	semidirect
186. C₂×C₅×Q₈	#41	semidirect
249. #559	C₂²×C₅	semidirect
250. Q₈²	C₂²×C₅	semidirect
251. #557	C₂²×C₅	semidirect
252. #557	C₂²×C₅	semidirect
253. #556	C₂²×C₅	semidirect
254. #555	C₂²×C₅	semidirect
255. #554	C₂²×C₅	semidirect
256. #553	C₂²×C₅	semidirect
257. #553	C₂²×C₅	semidirect
258. #552	C₂²×C₅	semidirect
261. #550	C₂²×C₅	semidirect
262. #550	C₂²×C₅	semidirect
263. D₄×Q₈	C₂²×C₅	semidirect
266. #548	C₂²×C₅	semidirect
267. #547	C₂²×C₅	semidirect
268. #547	C₂²×C₅	semidirect
270. #542	C₂²×C₅	semidirect
271. #542	C₂²×C₅	semidirect
272. #542	C₂²×C₅	semidirect
273. #541	C₂²×C₅	semidirect
274. #540	C₂²×C₅	semidirect
275. #540	C₂²×C₅	semidirect
284. #532	C₂²×C₅	semidirect
285. #529	C₂²×C₅	semidirect
286. #529	C₂²×C₅	semidirect
287. #529	C₂²×C₅	semidirect
293. #520	C₂²×C₅	semidirect
295. #519	C₂²×C₅	semidirect
297. C₄×#41	C₂²×C₅	semidirect
300. C₂×C₄×Q₈	C₂²×C₅	semidirect
464. C₂×Q₈	C₅×#41	semidirect
569. C₅	#62851	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3524	2
C₅×#3522	2
C₅×#3517	2
C₅×#3516	2
C₅×#3506	2
C₅×#3502	2
C₅×#3497	2
C₅×#3492	2
C₅×#3487	2
C₅×#3460	2
C₅×#3455	2
C₄×C₅×2₋¹⁺⁴	2
C₂×C₅×#520	2
#62851	5

Subgroups — C5×#62851 #11985571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#62851 #11985571