Subgroups — C₅×#18556 #11941276

Order 1280.

954 conjugacy classes of subgroups.

The subgroup lattice has 744 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3182	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3115	C₂	1	1	yes	maximal, characteristic
4	C₅×#3037	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1649	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1648	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1645	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1644	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1627	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#1626	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1623	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1622	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1559	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1491	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1489	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×#137	—	2	1	no	—
17	C₅×Q₈²	C₂²	1	1	yes	characteristic
18	C₅×#557	—	2	1	no	—
19	C₅×#557	C₂²	1	1	yes	characteristic
20	C₅×#554	—	2	2	no	—
21	C₅×#554	—	2	1	no	—
22	C₅×#554	—	2	1	no	—
23	C₅×#554	C₂²	1	1	yes	characteristic
24	C₅×D₄×Q₈	—	2	1	no	—
25	C₅×D₄×Q₈	C₂²	1	1	yes	characteristic
26	C₅×#533	—	2	2	no	—
27	C₅×#533	—	2	1	no	—
28	C₅×#533	—	2	1	no	—
29	C₅×#533	C₂²	1	1	yes	characteristic
30	C₅×#532	C₂²	1	1	yes	characteristic
31	C₄×C₅×#41	—	2	1	no	—
32	C₄×C₅×#41	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₅×Q₈	—	2	1	no	—
34	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
35	C₅×#500	C₂²	1	1	yes	characteristic
36	C₅×#492	C₂²	1	1	yes	characteristic
37	C₅×#487	C₂²	1	1	yes	characteristic
38	C₅×#486	C₂²	1	1	yes	characteristic
39	C₅×#482	—	2	1	no	—
40	C₅×#475	C₂²	1	1	yes	characteristic
41	C₅×#474	C₂²	1	1	yes	characteristic
42	C₅×#468	—	2	2	no	—
43	C₅×#464	—	2	1	no	—
44	C₅×#464	—	2	1	no	—
45	C₅×#462	C₂²	1	1	yes	characteristic
46	C₅×#462	C₂²	1	1	yes	characteristic
47	C₅×#460	C₂²	1	1	yes	characteristic
48	C₅×#460	C₂²	1	1	yes	characteristic
49	C₅×#460	C₂²	1	1	yes	characteristic
50	C₅×#459	C₂²	1	1	yes	characteristic
51	C₅×#449	—	2	1	no	—
52	C₅×#442	—	2	1	no	—
53	C₅×#440	—	2	1	no	—
54	C₅×#440	—	2	2	no	—
55	C₅×#432	C₂²	1	1	yes	characteristic
56	C₅×#423	C₂²	1	1	yes	characteristic
57	C₅×#417	—	2	1	no	—
58	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
59	C₅×#333	C₂²	1	1	yes	characteristic
60	C₅×#333	C₂²	1	1	yes	characteristic
61	C₅×#332	C₂²	1	1	yes	characteristic
62	C₅×#332	C₂²	1	1	yes	characteristic
63	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
65	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
66	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
67	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
68	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
69	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
70	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
71	#18556	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂×C₅×#41	—	2	1	no	—
73	C₂×C₅×#41	D₄	1	1	yes	characteristic
74	C₂²×C₅×Q₈	—	2	1	no	—
75	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
76	C₅×#137	—	4	4	no	—
77	C₂×C₅×Q₁₆	—	2	1	no	—
78	C₂×C₅×Q₁₆	—	2	1	no	—
79	C₂×C₅×SD₁₆	—	2	1	no	—
80	C₂×C₅×SD₁₆	—	2	2	no	—
81	C₂×C₅×SD₁₆	—	2	1	no	—
82	C₂×C₅×SD₁₆	—	2	1	no	—
83	C₂×C₅×D₈	—	2	1	no	—
84	C₂×C₅×M₁₆	—	2	1	no	—
85	C₂×C₅×M₁₆	—	2	1	no	—
86	C₅×C₄⋊Q₈	—	2	2	no	—
87	C₅×C₄⋊Q₈	—	2	1	no	—
88	C₅×C₄⋊Q₈	—	2	2	no	—
89	C₅×C₄⋊Q₈	—	2	1	no	—
90	C₅×C₄⋊Q₈	—	2	1	no	—
91	C₅×C₄⋊Q₈	—	2	1	no	—
92	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
93	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
94	C₅×C₄⋊Q₈	C₂³	1	1	yes	characteristic
95	C₅×#127	C₂³	1	1	yes	characteristic
96	C₅×#125	—	2	2	no	—
97	C₅×#125	—	2	2	no	—
98	C₅×#125	—	2	2	no	—
99	C₅×#125	—	2	1	no	—
100	C₅×#125	—	2	1	no	—
101	C₅×#125	—	2	1	no	—
102	C₅×#124	—	2	1	no	—
103	C₅×C₂²⋊Q₈	—	2	2	no	—
104	C₅×C₂²⋊Q₈	—	2	2	no	—
105	C₅×C₂²⋊Q₈	—	2	2	no	—
106	C₅×C₂²⋊Q₈	—	2	2	no	—
107	C₅×C₂²⋊Q₈	—	2	2	no	—
108	C₅×C₂²⋊Q₈	—	2	1	no	—
109	C₅×C₂²⋊Q₈	—	2	2	no	—
110	C₅×C₂²⋊Q₈	—	2	1	no	—
111	C₅×C₂²⋊Q₈	—	2	1	no	—
112	C₅×C₂²⋊Q₈	—	2	2	no	—
113	C₅×C₂²⋊Q₈	D₄	1	2	yes	—
114	C₅×#121	—	2	1	no	—
115	C₅×#121	—	2	1	no	—
116	C₄×C₅×Q₈	—	2	2	no	—
117	C₄×C₅×Q₈	—	2	2	no	—
118	C₄×C₅×Q₈	—	2	2	no	—
119	C₄×C₅×Q₈	—	2	1	no	—
120	C₄×C₅×Q₈	—	2	1	no	—
121	C₄×C₅×Q₈	—	2	1	no	—
122	C₄×C₅×Q₈	—	2	1	no	—
123	C₄×C₅×Q₈	—	2	1	no	—
124	C₄×C₅×Q₈	—	2	1	no	—
125	C₄×C₅×Q₈	—	2	1	no	—
126	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
127	C₄×C₅×Q₈	D₄	1	2	yes	—
128	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
129	C₄×C₅×Q₈	D₄	1	1	yes	characteristic
130	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
131	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
132	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
133	C₄×C₅×D₄	—	2	2	no	—
134	C₄×C₅×D₄	—	2	2	no	—
135	C₄×C₅×D₄	—	2	1	no	—
136	C₄×C₅×D₄	—	2	1	no	—
137	C₄×C₅×D₄	—	2	1	no	—
138	C₄×C₅×D₄	—	2	1	no	—
139	C₄×C₅×D₄	—	2	1	no	—
140	C₄×C₅×D₄	—	2	1	no	—
141	C₄×C₅×D₄	D₄	1	1	yes	characteristic
142	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
143	C₅×#117	—	2	2	no	—
144	C₅×#117	—	2	1	no	—
145	C₅×#117	—	2	1	no	—
146	C₅×#117	D₄	1	1	yes	characteristic
147	C₂×C₅×C₄⋊C₄	—	2	2	no	—
148	C₂×C₅×C₄⋊C₄	—	2	1	no	—
149	C₂×C₅×C₄⋊C₄	—	2	1	no	—
150	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
151	C₂×C₄²×C₅	—	2	2	no	—
152	C₂×C₄²×C₅	—	2	1	no	—
153	C₂×C₄²×C₅	—	2	1	no	—
154	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
155	C₅×#107	—	2	2	no	—
156	C₅×#107	—	2	1	no	—
157	C₅×#106	—	2	1	no	—
158	C₅×C₄⋊C₈	—	2	1	no	—
159	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
160	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
161	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
162	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
163	C₅×#103	—	2	1	no	—
164	C₅×#103	—	2	2	no	—
165	C₅×#103	—	2	1	no	—
166	C₅×#103	—	2	1	no	—
167	C₅×#102	—	2	1	no	—
168	C₅×#102	—	2	1	no	—
169	C₅×#102	—	2	1	no	—
170	C₅×#102	—	2	1	no	—
171	C₅×#102	—	2	2	no	—
172	C₅×#102	—	2	1	no	—
173	C₅×C₂²⋊C₈	—	2	1	no	—
174	C₅×#97	—	2	1	no	—
175	C₅×#97	—	2	1	no	—
176	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
177	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
178	Q₈×#41	C₂×C₅	1	1	yes	characteristic, complemented
179	#3182	C₂×C₅	1	1	yes	characteristic, complemented
180	#3115	C₂×C₅	1	1	yes	characteristic
181	#3037	C₂×C₅	1	1	yes	characteristic, complemented
182	#1649	C₂×C₅	1	1	yes	characteristic, complemented
183	#1648	C₂×C₅	1	1	yes	characteristic, complemented
184	#1645	C₂×C₅	1	1	yes	characteristic, complemented
185	#1644	C₂×C₅	1	1	yes	characteristic, complemented
186	#1627	C₂×C₅	1	1	yes	characteristic, complemented
187	#1626	C₂×C₅	1	1	yes	characteristic, complemented
188	#1623	C₂×C₅	1	1	yes	characteristic, complemented
189	#1622	C₂×C₅	1	1	yes	characteristic, complemented
190	#1559	C₂×C₅	1	1	yes	characteristic, complemented
191	#1491	C₂×C₅	1	1	yes	characteristic, complemented
192	#1489	C₂×C₅	1	1	yes	characteristic, complemented
193	C₅×#41	—	2	2	no	—
194	C₅×#41	—	2	2	no	—
195	C₅×#41	—	4	2	no	—
196	C₅×#41	—	4	2	no	—
197	C₂×C₅×Q₈	—	2	2	no	—
198	C₂×C₅×Q₈	—	2	2	no	—
199	C₂×C₅×Q₈	—	2	1	no	—
200	C₂×C₅×Q₈	—	2	1	no	—
201	C₂×C₅×Q₈	—	2	2	no	—
202	C₂×C₅×Q₈	—	2	1	no	—
203	C₂×C₅×Q₈	—	4	1	no	—
204	C₂×C₅×Q₈	—	4	2	no	—
205	C₂×C₅×Q₈	—	4	2	no	—
206	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
207	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
208	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
209	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
210	C₂×C₅×D₄	—	2	1	no	—
211	C₂×C₅×D₄	—	2	1	no	—
212	C₂×C₅×D₄	—	2	1	no	—
213	C₂×C₅×D₄	—	2	1	no	—
214	C₂×C₅×D₄	—	4	1	no	—
215	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
216	C₂²×C₄×C₅	—	2	1	no	—
217	C₂²×C₄×C₅	—	2	2	no	—
218	C₂²×C₄×C₅	—	2	1	no	—
219	C₂²×C₄×C₅	—	2	2	no	—
220	C₂²×C₄×C₅	—	2	1	no	—
221	C₂²×C₄×C₅	—	2	1	no	—
222	C₂²×C₄×C₅	—	2	1	no	—
223	C₂²×C₄×C₅	—	4	1	no	—
224	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
225	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
226	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
227	C₅×Q₁₆	—	4	2	no	—
228	C₅×Q₁₆	—	4	2	no	—
229	C₅×SD₁₆	—	4	4	no	—
230	C₅×SD₁₆	—	4	2	no	—
231	C₅×SD₁₆	—	4	2	no	—
232	C₅×SD₁₆	—	4	2	no	—
233	C₅×D₈	—	4	2	no	—
234	C₅×M₁₆	—	4	2	no	—
235	C₅×M₁₆	—	4	2	no	—
236	C₂×C₅×C₈	—	2	1	no	—
237	C₂×C₅×C₈	—	2	1	no	—
238	C₂×C₅×C₈	—	2	1	no	—
239	C₂×C₅×C₈	—	2	1	no	—
240	C₂×C₅×C₈	—	2	1	no	—
241	C₂×C₅×C₈	—	2	1	no	—
242	C₅×C₄⋊C₄	—	2	2	no	—
243	C₅×C₄⋊C₄	—	2	2	no	—
244	C₅×C₄⋊C₄	—	2	2	no	—
245	C₅×C₄⋊C₄	—	2	2	no	—
246	C₅×C₄⋊C₄	—	2	2	no	—
247	C₅×C₄⋊C₄	—	2	2	no	—
248	C₅×C₄⋊C₄	—	2	2	no	—
249	C₅×C₄⋊C₄	—	2	2	no	—
250	C₅×C₄⋊C₄	—	2	1	no	—
251	C₅×C₄⋊C₄	—	2	2	no	—
252	C₅×C₄⋊C₄	—	2	1	no	—
253	C₅×C₄⋊C₄	—	2	1	no	—
254	C₅×C₄⋊C₄	—	2	1	no	—
255	C₅×C₄⋊C₄	—	2	2	no	—
256	C₅×C₄⋊C₄	—	2	1	no	—
257	C₅×C₄⋊C₄	—	2	1	no	—
258	C₅×C₄⋊C₄	—	2	1	no	—
259	C₅×C₄⋊C₄	—	2	1	no	—
260	C₅×C₄⋊C₄	—	2	1	no	—
261	C₅×C₄⋊C₄	—	2	1	no	—
262	C₅×C₄⋊C₄	—	2	1	no	—
263	C₅×C₄⋊C₄	—	2	1	no	—
264	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
265	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
266	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
267	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
268	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
269	C₅×C₂²⋊C₄	—	2	2	no	—
270	C₅×C₂²⋊C₄	—	2	2	no	—
271	C₅×C₂²⋊C₄	—	2	2	no	—
272	C₅×C₂²⋊C₄	—	2	1	no	—
273	C₅×C₂²⋊C₄	—	2	1	no	—
274	C₅×C₂²⋊C₄	—	2	1	no	—
275	C₅×C₂²⋊C₄	—	4	1	no	—
276	C₅×C₂²⋊C₄	—	4	1	no	—
277	C₄²×C₅	—	2	2	no	—
278	C₄²×C₅	—	2	2	no	—
279	C₄²×C₅	—	2	1	no	—
280	C₄²×C₅	—	2	2	no	—
281	C₄²×C₅	—	2	1	no	—
282	C₄²×C₅	—	2	1	no	—
283	C₄²×C₅	—	2	1	no	—
284	C₄²×C₅	—	2	1	no	—
285	C₄²×C₅	—	2	1	no	—
286	C₄²×C₅	—	2	1	no	—
287	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
288	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
289	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
290	C₄²×C₅	C₂⁴	1	1	yes	characteristic
291	C₂×#137	—	2	1	no	—
292	Q₈²	C₂²×C₅	1	1	yes	characteristic
293	#557	—	2	1	no	—
294	#557	C₂²×C₅	1	1	yes	characteristic
295	#554	—	2	2	no	—
296	#554	—	2	1	no	—
297	#554	—	2	1	no	—
298	#554	C₂²×C₅	1	1	yes	characteristic
299	D₄×Q₈	—	2	1	no	—
300	D₄×Q₈	C₂²×C₅	1	1	yes	characteristic
301	#533	—	2	2	no	—
302	#533	—	2	1	no	—
303	#533	—	2	1	no	—
304	#533	C₂²×C₅	1	1	yes	characteristic
305	#532	C₂²×C₅	1	1	yes	characteristic
306	C₄×#41	—	2	1	no	—
307	C₄×#41	C₂²×C₅	1	1	yes	characteristic
308	C₂×C₄×Q₈	—	2	1	no	—
309	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
310	#500	C₂²×C₅	1	1	yes	characteristic
311	#492	C₂²×C₅	1	1	yes	characteristic
312	#487	C₂²×C₅	1	1	yes	characteristic
313	#486	C₂²×C₅	1	1	yes	characteristic
314	#482	—	2	1	no	—
315	#475	C₂²×C₅	1	1	yes	characteristic
316	#474	C₂²×C₅	1	1	yes	characteristic
317	#468	—	2	2	no	—
318	#464	—	2	1	no	—
319	#464	—	2	1	no	—
320	#462	C₂²×C₅	1	1	yes	characteristic
321	#462	C₂²×C₅	1	1	yes	characteristic
322	#460	C₂²×C₅	1	1	yes	characteristic
323	#460	C₂²×C₅	1	1	yes	characteristic
324	#460	C₂²×C₅	1	1	yes	characteristic
325	#459	C₂²×C₅	1	1	yes	characteristic
326	#449	—	2	1	no	—
327	#442	—	2	1	no	—
328	#440	—	2	1	no	—
329	#440	—	2	2	no	—
330	#432	C₂²×C₅	1	1	yes	characteristic
331	#423	C₂²×C₅	1	1	yes	characteristic
332	#417	—	2	1	no	—
333	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
334	#333	C₂²×C₅	1	1	yes	characteristic
335	#333	C₂²×C₅	1	1	yes	characteristic
336	#332	C₂²×C₅	1	1	yes	characteristic
337	#332	C₂²×C₅	1	1	yes	characteristic
338	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
339	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
340	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
341	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
342	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
343	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
344	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
345	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
346	C₂³×C₅	—	2	1	no	—
347	C₂³×C₅	—	4	1	no	—
348	C₂³×C₅	#120	1	1	yes	characteristic
349	C₅×Q₈	—	2	1	no	—
350	C₅×Q₈	—	2	1	no	—
351	C₅×Q₈	—	2	1	no	—
352	C₅×Q₈	—	2	1	no	—
353	C₅×Q₈	—	4	2	no	—
354	C₅×Q₈	—	4	2	no	—
355	C₅×Q₈	—	4	4	no	—
356	C₅×Q₈	—	4	1	no	—
357	C₅×Q₈	—	4	1	no	—
358	C₅×Q₈	—	4	1	no	—
359	C₅×Q₈	—	4	1	no	—
360	C₅×Q₈	—	8	2	no	—
361	C₅×Q₈	AGL(1,ℤ/8)	1	2	yes	—
362	C₅×Q₈	#137	1	2	yes	complemented
363	C₅×D₄	—	2	4	no	—
364	C₅×D₄	—	2	1	no	—
365	C₅×D₄	—	4	2	no	—
366	C₅×D₄	—	4	1	no	—
367	C₅×D₄	—	4	1	no	—
368	C₅×D₄	—	4	1	no	—
369	C₅×D₄	—	4	1	no	—
370	C₅×D₄	—	8	2	no	—
371	C₅×D₄	#137	1	2	yes	complemented
372	C₂×C₄×C₅	—	2	4	no	—
373	C₂×C₄×C₅	—	2	1	no	—
374	C₂×C₄×C₅	—	2	1	no	—
375	C₂×C₄×C₅	—	2	1	no	—
376	C₂×C₄×C₅	—	2	1	no	—
377	C₂×C₄×C₅	—	2	1	no	—
378	C₂×C₄×C₅	—	2	1	no	—
379	C₂×C₄×C₅	—	2	1	no	—
380	C₂×C₄×C₅	—	2	1	no	—
381	C₂×C₄×C₅	—	2	2	no	—
382	C₂×C₄×C₅	—	2	2	no	—
383	C₂×C₄×C₅	—	2	1	no	—
384	C₂×C₄×C₅	—	2	1	no	—
385	C₂×C₄×C₅	—	2	1	no	—
386	C₂×C₄×C₅	—	4	1	no	—
387	C₂×C₄×C₅	—	4	2	no	—
388	C₂×C₄×C₅	—	4	4	no	—
389	C₂×C₄×C₅	—	4	1	no	—
390	C₂×C₄×C₅	—	4	1	no	—
391	C₂×C₄×C₅	—	4	1	no	—
392	C₂×C₄×C₅	—	8	1	no	—
393	C₂×C₄×C₅	—	8	2	no	—
394	C₂×C₄×C₅	—	8	1	no	—
395	C₂×C₄×C₅	—	8	2	no	—
396	C₂×C₄×C₅	#120	1	1	yes	characteristic
397	C₂×C₄×C₅	#120	1	1	yes	characteristic
398	C₂×C₄×C₅	#120	1	1	yes	characteristic
399	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
400	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
401	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
402	C₅×C₈	—	2	2	no	—
403	C₅×C₈	—	4	1	no	—
404	C₅×C₈	—	4	1	no	—
405	C₅×C₈	—	4	1	no	—
406	C₅×C₈	—	4	1	no	—
407	C₅×C₈	—	4	1	no	—
408	C₂×#41	—	2	1	no	—
409	C₂×#41	C₅×D₄	1	1	yes	characteristic
410	C₂²×Q₈	—	2	1	no	—
411	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
412	#137	—	4	4	no	—
413	C₂×Q₁₆	—	2	1	no	—
414	C₂×Q₁₆	—	2	1	no	—
415	C₂×SD₁₆	—	2	1	no	—
416	C₂×SD₁₆	—	2	1	no	—
417	C₂×SD₁₆	—	2	1	no	—
418	C₂×SD₁₆	—	2	2	no	—
419	C₂×D₈	—	2	1	no	—
420	C₂×M₁₆	—	2	1	no	—
421	C₂×M₁₆	—	2	1	no	—
422	C₄⋊Q₈	—	2	2	no	—
423	C₄⋊Q₈	—	2	1	no	—
424	C₄⋊Q₈	—	2	2	no	—
425	C₄⋊Q₈	—	2	1	no	—
426	C₄⋊Q₈	—	2	1	no	—
427	C₄⋊Q₈	—	2	1	no	—
428	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
429	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
430	C₄⋊Q₈	C₂³×C₅	1	1	yes	characteristic
431	#127	C₂³×C₅	1	1	yes	characteristic
432	#125	—	2	2	no	—
433	#125	—	2	2	no	—
434	#125	—	2	2	no	—
435	#125	—	2	1	no	—
436	#125	—	2	1	no	—
437	#125	—	2	1	no	—
438	#124	—	2	1	no	—
439	C₂²⋊Q₈	—	2	2	no	—
440	C₂²⋊Q₈	—	2	2	no	—
441	C₂²⋊Q₈	—	2	2	no	—
442	C₂²⋊Q₈	—	2	2	no	—
443	C₂²⋊Q₈	—	2	2	no	—
444	C₂²⋊Q₈	—	2	1	no	—
445	C₂²⋊Q₈	—	2	2	no	—
446	C₂²⋊Q₈	—	2	1	no	—
447	C₂²⋊Q₈	—	2	1	no	—
448	C₂²⋊Q₈	—	2	2	no	—
449	C₂²⋊Q₈	C₅×D₄	1	2	yes	—
450	#121	—	2	1	no	—
451	#121	—	2	1	no	—
452	C₄×Q₈	—	2	2	no	—
453	C₄×Q₈	—	2	2	no	—
454	C₄×Q₈	—	2	2	no	—
455	C₄×Q₈	—	2	1	no	—
456	C₄×Q₈	—	2	1	no	—
457	C₄×Q₈	—	2	1	no	—
458	C₄×Q₈	—	2	1	no	—
459	C₄×Q₈	—	2	1	no	—
460	C₄×Q₈	—	2	1	no	—
461	C₄×Q₈	—	2	1	no	—
462	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
463	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
464	C₄×Q₈	C₅×D₄	1	1	yes	characteristic
465	C₄×Q₈	C₅×D₄	1	2	yes	—
466	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
467	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
468	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
469	C₄×D₄	—	2	2	no	—
470	C₄×D₄	—	2	2	no	—
471	C₄×D₄	—	2	1	no	—
472	C₄×D₄	—	2	1	no	—
473	C₄×D₄	—	2	1	no	—
474	C₄×D₄	—	2	1	no	—
475	C₄×D₄	—	2	1	no	—
476	C₄×D₄	—	2	1	no	—
477	C₄×D₄	C₅×D₄	1	1	yes	characteristic
478	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
479	#117	—	2	2	no	—
480	#117	—	2	1	no	—
481	#117	—	2	1	no	—
482	#117	C₅×D₄	1	1	yes	characteristic
483	C₂×C₄⋊C₄	—	2	2	no	—
484	C₂×C₄⋊C₄	—	2	1	no	—
485	C₂×C₄⋊C₄	—	2	1	no	—
486	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
487	C₂×C₄²	—	2	2	no	—
488	C₂×C₄²	—	2	1	no	—
489	C₂×C₄²	—	2	1	no	—
490	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
491	#107	—	2	2	no	—
492	#107	—	2	1	no	—
493	#106	—	2	1	no	—
494	C₄⋊C₈	—	2	1	no	—
495	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
496	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
497	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
498	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
499	#103	—	2	1	no	—
500	#103	—	2	2	no	—
501	#103	—	2	1	no	—
502	#103	—	2	1	no	—
503	#102	—	2	1	no	—
504	#102	—	2	1	no	—
505	#102	—	2	1	no	—
506	#102	—	2	1	no	—
507	#102	—	2	2	no	—
508	#102	—	2	1	no	—
509	C₂²⋊C₈	—	2	1	no	—
510	#97	—	2	1	no	—
511	#97	—	2	1	no	—
512	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
513	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
514	C₂²×C₅	—	2	1	no	—
515	C₂²×C₅	—	2	2	no	—
516	C₂²×C₅	—	2	1	no	—
517	C₂²×C₅	—	2	1	no	—
518	C₂²×C₅	—	4	1	no	—
519	C₂²×C₅	—	4	1	no	—
520	C₂²×C₅	—	8	1	no	—
521	C₂²×C₅	—	8	1	no	—
522	C₂²×C₅	—	8	1	no	—
523	C₂²×C₅	C₂×#120	1	1	yes	characteristic
524	C₄×C₅	—	2	1	no	—
525	C₄×C₅	—	2	2	no	—
526	C₄×C₅	—	2	1	no	—
527	C₄×C₅	—	2	2	no	—
528	C₄×C₅	—	2	1	no	—
529	C₄×C₅	—	2	1	no	—
530	C₄×C₅	—	4	1	no	—
531	C₄×C₅	—	4	2	no	—
532	C₄×C₅	—	4	1	no	—
533	C₄×C₅	—	4	1	no	—
534	C₄×C₅	—	4	2	no	—
535	C₄×C₅	—	4	1	no	—
536	C₄×C₅	—	4	1	no	—
537	C₄×C₅	—	8	1	no	—
538	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
539	C₄×C₅	C₂×#137	1	1	yes	characteristic
540	C₄×C₅	C₂×#137	1	1	yes	characteristic
541	C₄×C₅	#575	1	1	yes	characteristic
542	#41	—	2	2	no	—
543	#41	—	2	2	no	—
544	#41	—	4	2	no	—
545	#41	—	4	2	no	—
546	C₂×Q₈	—	2	2	no	—
547	C₂×Q₈	—	2	2	no	—
548	C₂×Q₈	—	2	1	no	—
549	C₂×Q₈	—	2	1	no	—
550	C₂×Q₈	—	2	2	no	—
551	C₂×Q₈	—	2	1	no	—
552	C₂×Q₈	—	4	1	no	—
553	C₂×Q₈	—	4	2	no	—
554	C₂×Q₈	—	4	2	no	—
555	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
556	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
557	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
558	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
559	C₂×D₄	—	2	1	no	—
560	C₂×D₄	—	2	1	no	—
561	C₂×D₄	—	2	1	no	—
562	C₂×D₄	—	2	1	no	—
563	C₂×D₄	—	4	1	no	—
564	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
565	C₂²×C₄	—	2	1	no	—
566	C₂²×C₄	—	2	2	no	—
567	C₂²×C₄	—	2	1	no	—
568	C₂²×C₄	—	2	2	no	—
569	C₂²×C₄	—	2	1	no	—
570	C₂²×C₄	—	2	1	no	—
571	C₂²×C₄	—	2	1	no	—
572	C₂²×C₄	—	4	1	no	—
573	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
574	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
575	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
576	Q₁₆	—	4	2	no	—
577	Q₁₆	—	4	2	no	—
578	SD₁₆	—	4	4	no	—
579	SD₁₆	—	4	2	no	—
580	SD₁₆	—	4	2	no	—
581	SD₁₆	—	4	2	no	—
582	D₈	—	4	2	no	—
583	M₁₆	—	4	2	no	—
584	M₁₆	—	4	2	no	—
585	C₂×C₈	—	2	1	no	—
586	C₂×C₈	—	2	1	no	—
587	C₂×C₈	—	2	1	no	—
588	C₂×C₈	—	2	1	no	—
589	C₂×C₈	—	2	1	no	—
590	C₂×C₈	—	2	1	no	—
591	C₄⋊C₄	—	2	2	no	—
592	C₄⋊C₄	—	2	2	no	—
593	C₄⋊C₄	—	2	2	no	—
594	C₄⋊C₄	—	2	2	no	—
595	C₄⋊C₄	—	2	2	no	—
596	C₄⋊C₄	—	2	2	no	—
597	C₄⋊C₄	—	2	2	no	—
598	C₄⋊C₄	—	2	2	no	—
599	C₄⋊C₄	—	2	1	no	—
600	C₄⋊C₄	—	2	2	no	—
601	C₄⋊C₄	—	2	1	no	—
602	C₄⋊C₄	—	2	1	no	—
603	C₄⋊C₄	—	2	1	no	—
604	C₄⋊C₄	—	2	2	no	—
605	C₄⋊C₄	—	2	1	no	—
606	C₄⋊C₄	—	2	1	no	—
607	C₄⋊C₄	—	2	1	no	—
608	C₄⋊C₄	—	2	1	no	—
609	C₄⋊C₄	—	2	1	no	—
610	C₄⋊C₄	—	2	1	no	—
611	C₄⋊C₄	—	2	1	no	—
612	C₄⋊C₄	—	2	1	no	—
613	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
614	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
615	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
616	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
617	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
618	C₂²⋊C₄	—	2	2	no	—
619	C₂²⋊C₄	—	2	2	no	—
620	C₂²⋊C₄	—	2	2	no	—
621	C₂²⋊C₄	—	2	1	no	—
622	C₂²⋊C₄	—	2	1	no	—
623	C₂²⋊C₄	—	2	1	no	—
624	C₂²⋊C₄	—	4	1	no	—
625	C₂²⋊C₄	—	4	1	no	—
626	C₄²	—	2	2	no	—
627	C₄²	—	2	2	no	—
628	C₄²	—	2	1	no	—
629	C₄²	—	2	2	no	—
630	C₄²	—	2	1	no	—
631	C₄²	—	2	1	no	—
632	C₄²	—	2	1	no	—
633	C₄²	—	2	1	no	—
634	C₄²	—	2	1	no	—
635	C₄²	—	2	1	no	—
636	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
637	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
638	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
639	C₄²	C₂⁴×C₅	1	1	yes	characteristic
640	C₂×C₅	—	4	1	no	—
641	C₂×C₅	—	4	2	no	—
642	C₂×C₅	—	16	1	no	—
643	C₂×C₅	#3004	1	1	yes	characteristic
644	C₂×C₅	#3006	1	1	yes	characteristic
645	C₂×C₅	#3018	1	1	yes	characteristic
646	C₂³	—	2	1	no	—
647	C₂³	—	4	1	no	—
648	C₂³	C₅×#120	1	1	yes	characteristic
649	Q₈	—	2	1	no	—
650	Q₈	—	2	1	no	—
651	Q₈	—	2	1	no	—
652	Q₈	—	2	1	no	—
653	Q₈	—	4	2	no	—
654	Q₈	—	4	2	no	—
655	Q₈	—	4	4	no	—
656	Q₈	—	4	1	no	—
657	Q₈	—	4	1	no	—
658	Q₈	—	4	1	no	—
659	Q₈	—	4	1	no	—
660	Q₈	—	8	2	no	—
661	Q₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
662	Q₈	C₅×#137	1	2	yes	complemented
663	D₄	—	2	4	no	—
664	D₄	—	2	1	no	—
665	D₄	—	4	2	no	—
666	D₄	—	4	1	no	—
667	D₄	—	4	1	no	—
668	D₄	—	4	1	no	—
669	D₄	—	4	1	no	—
670	D₄	—	8	2	no	—
671	D₄	C₅×#137	1	2	yes	complemented
672	C₂×C₄	—	2	4	no	—
673	C₂×C₄	—	2	1	no	—
674	C₂×C₄	—	2	1	no	—
675	C₂×C₄	—	2	1	no	—
676	C₂×C₄	—	2	1	no	—
677	C₂×C₄	—	2	1	no	—
678	C₂×C₄	—	2	1	no	—
679	C₂×C₄	—	2	1	no	—
680	C₂×C₄	—	2	1	no	—
681	C₂×C₄	—	2	2	no	—
682	C₂×C₄	—	2	2	no	—
683	C₂×C₄	—	2	1	no	—
684	C₂×C₄	—	2	1	no	—
685	C₂×C₄	—	2	1	no	—
686	C₂×C₄	—	4	1	no	—
687	C₂×C₄	—	4	2	no	—
688	C₂×C₄	—	4	4	no	—
689	C₂×C₄	—	4	1	no	—
690	C₂×C₄	—	4	1	no	—
691	C₂×C₄	—	4	1	no	—
692	C₂×C₄	—	8	1	no	—
693	C₂×C₄	—	8	2	no	—
694	C₂×C₄	—	8	1	no	—
695	C₂×C₄	—	8	2	no	—
696	C₂×C₄	C₅×#120	1	1	yes	characteristic
697	C₂×C₄	C₅×#120	1	1	yes	characteristic
698	C₂×C₄	C₅×#120	1	1	yes	characteristic
699	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
700	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
701	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
702	C₈	—	2	2	no	—
703	C₈	—	4	1	no	—
704	C₈	—	4	1	no	—
705	C₈	—	4	1	no	—
706	C₈	—	4	1	no	—
707	C₈	—	4	1	no	—
708	C₅	#18556	1	1	yes	characteristic, direct product factor, Sylow
709	C₂²	—	2	1	no	—
710	C₂²	—	2	2	no	—
711	C₂²	—	2	1	no	—
712	C₂²	—	2	1	no	—
713	C₂²	—	4	1	no	—
714	C₂²	—	4	1	no	—
715	C₂²	—	8	1	no	—
716	C₂²	—	8	1	no	—
717	C₂²	—	8	1	no	—
718	C₂²	C₂×C₅×#120	1	1	yes	characteristic
719	C₄	—	2	1	no	—
720	C₄	—	2	2	no	—
721	C₄	—	2	1	no	—
722	C₄	—	2	2	no	—
723	C₄	—	2	1	no	—
724	C₄	—	2	1	no	—
725	C₄	—	4	1	no	—
726	C₄	—	4	2	no	—
727	C₄	—	4	1	no	—
728	C₄	—	4	1	no	—
729	C₄	—	4	2	no	—
730	C₄	—	4	1	no	—
731	C₄	—	4	1	no	—
732	C₄	—	8	1	no	—
733	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
734	C₄	C₂×C₅×#137	1	1	yes	characteristic
735	C₄	C₂×C₅×#137	1	1	yes	characteristic
736	C₄	C₅×#575	1	1	yes	characteristic
737	C₂	—	4	1	no	—
738	C₂	—	4	2	no	—
739	C₂	—	16	1	no	—
740	C₂	C₅×#3004	1	1	yes	characteristic
741	C₂	C₅×#3006	1	1	yes	characteristic
742	C₂	C₅×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₅×#3115	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×Q₈×#41	C₂	semidirect
2. C₅×#3182	C₂	semidirect
4. C₅×#3037	C₂	semidirect
5. C₅×#1649	C₂	semidirect
6. C₅×#1648	C₂	semidirect
7. C₅×#1645	C₂	semidirect
8. C₅×#1644	C₂	semidirect
9. C₅×#1627	C₂	semidirect
10. C₅×#1626	C₂	semidirect
11. C₅×#1623	C₂	semidirect
12. C₅×#1622	C₂	semidirect
13. C₅×#1559	C₂	semidirect
14. C₅×#1491	C₂	semidirect
15. C₅×#1489	C₂	semidirect
71. #18556	C₅	direct (N × H)
178. Q₈×#41	C₂×C₅	semidirect
179. #3182	C₂×C₅	semidirect
181. #3037	C₂×C₅	semidirect
182. #1649	C₂×C₅	semidirect
183. #1648	C₂×C₅	semidirect
184. #1645	C₂×C₅	semidirect
185. #1644	C₂×C₅	semidirect
186. #1627	C₂×C₅	semidirect
187. #1626	C₂×C₅	semidirect
188. #1623	C₂×C₅	semidirect
189. #1622	C₂×C₅	semidirect
190. #1559	C₂×C₅	semidirect
191. #1491	C₂×C₅	semidirect
192. #1489	C₂×C₅	semidirect
362. C₅×Q₈	#137	semidirect
371. C₅×D₄	#137	semidirect
662. Q₈	C₅×#137	semidirect
671. D₄	C₅×#137	semidirect
708. C₅	#18556	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×Q₈×#41	2
C₅×#3182	2
C₅×#3115	2
C₅×#3037	2
C₅×#1649	2
C₅×#1648	2
C₅×#1645	2
C₅×#1644	2
C₅×#1627	2
C₅×#1626	2
C₅×#1623	2
C₅×#1622	2
C₅×#1559	2
C₅×#1491	2
C₅×#1489	2
#18556	5

Subgroups — C5×#18556 #11941276

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#18556 #11941276