Subgroups — C₅×#25197 #11947917

Order 1280.

968 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3068	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3068	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3020	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#454	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2889	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2070	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2050	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2048	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#2003	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#1967	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#1932	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#1915	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#1903	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#1861	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×2₋¹⁺⁴	C₂²	1	1	yes	characteristic
17	C₅×#575	C₂²	1	1	yes	characteristic
18	C₅×#575	C₂²	1	1	yes	characteristic
19	C₂×C₅×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
20	C₂×C₅×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
21	C₂×C₅×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
22	C₅×#568	C₂²	1	1	yes	characteristic
23	C₅×#568	C₂²	1	1	yes	characteristic
24	C₅×#568	C₂²	1	1	yes	characteristic
25	C₅×#536	C₂²	1	1	yes	characteristic
26	C₂×C₅×#124	C₂²	1	1	yes	characteristic
27	C₅×#472	—	2	2	no	—
28	C₅×#472	—	2	2	no	—
29	C₅×#471	—	2	2	no	—
30	C₅×#471	—	2	2	no	—
31	C₅×#456	—	2	2	no	—
32	C₅×#454	—	2	2	no	—
33	C₅×#454	—	2	2	no	—
34	C₅×#454	—	2	2	no	—
35	C₅×#444	—	2	2	no	—
36	C₅×#443	—	2	2	no	—
37	C₅×#430	C₂²	1	1	yes	characteristic
38	C₅×#430	C₂²	1	1	yes	characteristic
39	C₅×#430	C₂²	1	1	yes	characteristic
40	C₅×#423	C₂²	1	1	yes	characteristic
41	C₅×#421	C₂²	1	1	yes	characteristic
42	C₅×#421	C₂²	1	1	yes	characteristic
43	C₂×C₅×#104	C₂²	1	1	yes	characteristic
44	C₂×C₅×#104	C₂²	1	1	yes	characteristic
45	C₂×C₅×#104	C₂²	1	1	yes	characteristic
46	C₅×#418	C₂²	1	1	yes	characteristic
47	C₅×#413	—	2	2	no	—
48	C₅×#413	C₂²	1	1	yes	characteristic
49	C₅×#413	C₂²	1	1	yes	characteristic
50	C₅×#413	C₂²	1	1	yes	characteristic
51	C₂×C₅×#101	—	2	2	no	—
52	C₂×C₅×#101	C₂²	1	1	yes	characteristic
53	C₂×C₅×#101	C₂²	1	1	yes	characteristic
54	C₂×C₅×#101	C₂²	1	1	yes	characteristic
55	C₂×C₅×#101	C₂²	1	1	yes	characteristic
56	C₂×C₅×#100	C₂²	1	1	yes	characteristic
57	C₂×C₅×#100	C₂²	1	1	yes	characteristic
58	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
59	C₅×#344	C₂²	1	1	yes	characteristic
60	C₅×#344	C₂²	1	1	yes	characteristic
61	C₅×#339	C₂²	1	1	yes	characteristic
62	C₅×#338	C₂²	1	1	yes	characteristic
63	#25197	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₅×2₋¹⁺⁴	—	2	2	no	—
65	C₅×2₋¹⁺⁴	—	2	2	no	—
66	C₅×2₋¹⁺⁴	—	4	1	no	—
67	C₅×2₋¹⁺⁴	—	4	1	no	—
68	C₂×C₅×#41	—	2	1	no	—
69	C₂×C₅×#41	—	2	1	no	—
70	C₂×C₅×#41	—	2	2	no	—
71	C₂×C₅×#41	C₂³	1	1	yes	characteristic
72	C₂×C₅×#41	C₂³	1	1	yes	characteristic
73	C₂×C₅×#41	C₂³	1	1	yes	characteristic
74	C₂²×C₅×Q₈	—	2	1	no	—
75	C₂²×C₅×Q₈	—	2	1	no	—
76	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
77	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
78	C₅×#137	—	2	1	no	—
79	C₅×#137	—	2	1	no	—
80	C₅×#137	—	2	1	no	—
81	C₅×#137	D₄	1	2	yes	—
82	C₅×AGL(1,ℤ/8)	—	2	1	no	—
83	C₅×AGL(1,ℤ/8)	—	2	2	no	—
84	C₅×AGL(1,ℤ/8)	—	2	2	no	—
85	C₅×AGL(1,ℤ/8)	—	2	1	no	—
86	C₅×AGL(1,ℤ/8)	—	2	1	no	—
87	C₅×AGL(1,ℤ/8)	—	2	2	no	—
88	C₅×AGL(1,ℤ/8)	—	2	2	no	—
89	C₅×AGL(1,ℤ/8)	—	4	1	no	—
90	C₅×AGL(1,ℤ/8)	—	4	1	no	—
91	C₅×AGL(1,ℤ/8)	D₄	1	2	yes	—
92	C₅×#135	—	2	2	no	—
93	C₅×#135	—	4	1	no	—
94	C₂×C₅×SD₁₆	—	2	1	no	—
95	C₂×C₅×SD₁₆	—	2	1	no	—
96	C₂×C₅×SD₁₆	—	2	1	no	—
97	C₂×C₅×D₈	—	2	1	no	—
98	C₂×C₅×D₈	—	2	1	no	—
99	C₂×C₅×D₈	—	2	1	no	—
100	C₅×#131	—	2	2	no	—
101	C₅×#131	—	2	2	no	—
102	C₅×#131	—	2	2	no	—
103	C₅×#131	—	2	2	no	—
104	C₅×#131	—	4	1	no	—
105	C₅×#131	—	4	1	no	—
106	C₂×C₅×M₁₆	—	2	2	no	—
107	C₂×C₅×M₁₆	—	2	2	no	—
108	C₂×C₅×M₁₆	—	4	1	no	—
109	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
110	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
111	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
112	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
113	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
114	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
115	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
116	C₂×C₅×M₁₆	C₂³	1	1	yes	characteristic
117	C₅×C₄⋊Q₈	—	2	1	no	—
118	C₅×#124	—	2	1	no	—
119	C₅×#124	—	2	1	no	—
120	C₅×#124	—	2	1	no	—
121	C₅×#124	—	4	1	no	—
122	C₅×#123	—	2	2	no	—
123	C₅×C₂²⋊Q₈	—	4	1	no	—
124	C₅×#117	C₂³	1	1	yes	characteristic
125	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
126	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
127	C₅×#108	—	2	2	no	—
128	C₅×#108	—	2	2	no	—
129	C₅×#108	—	2	2	no	—
130	C₅×C₄⋊C₈	—	4	1	no	—
131	C₅×#104	—	2	2	no	—
132	C₅×#104	—	2	2	no	—
133	C₅×#104	—	2	1	no	—
134	C₅×#104	—	2	1	no	—
135	C₅×#104	—	2	1	no	—
136	C₅×#104	—	2	1	no	—
137	C₅×#104	D₄	1	2	yes	—
138	C₅×#104	D₄	1	2	yes	—
139	C₅×#102	—	4	1	no	—
140	C₅×#101	—	2	1	no	—
141	C₅×#101	—	2	1	no	—
142	C₅×#101	—	2	2	no	—
143	C₅×#101	—	2	2	no	—
144	C₅×#101	—	2	2	no	—
145	C₅×#101	—	2	2	no	—
146	C₅×#101	D₄	1	2	yes	—
147	C₅×#100	—	2	1	no	—
148	C₅×#100	—	2	1	no	—
149	C₅×#100	—	2	2	no	—
150	C₅×#100	—	2	2	no	—
151	C₅×#100	D₄	1	2	yes	—
152	C₅×#97	—	2	1	no	—
153	C₄×C₅×C₈	—	2	1	no	—
154	#3068	C₂×C₅	1	1	yes	characteristic, complemented
155	#3068	C₂×C₅	1	1	yes	characteristic, complemented
156	#3020	C₂×C₅	1	1	yes	characteristic, complemented
157	C₂×#454	C₂×C₅	1	1	yes	characteristic, complemented
158	#2956	C₂×C₅	1	1	yes	characteristic, complemented
159	#2889	C₂×C₅	1	1	yes	characteristic, complemented
160	#2070	C₂×C₅	1	1	yes	characteristic, complemented
161	#2050	C₂×C₅	1	1	yes	characteristic, complemented
162	#2048	C₂×C₅	1	1	yes	characteristic, complemented
163	#2003	C₂×C₅	1	1	yes	characteristic, complemented
164	#1967	C₂×C₅	1	1	yes	characteristic, complemented
165	#1932	C₂×C₅	1	1	yes	characteristic, complemented
166	#1915	C₂×C₅	1	1	yes	characteristic, complemented
167	#1903	C₂×C₅	1	1	yes	characteristic, complemented
168	#1861	C₂×C₅	1	1	yes	characteristic, complemented
169	C₂⁴×C₅	—	2	1	no	—
170	C₅×#41	—	2	1	no	—
171	C₅×#41	—	2	1	no	—
172	C₅×#41	—	2	1	no	—
173	C₅×#41	—	2	1	no	—
174	C₅×#41	—	2	1	no	—
175	C₅×#41	—	2	1	no	—
176	C₅×#41	—	2	1	no	—
177	C₅×#41	—	2	1	no	—
178	C₅×#41	—	2	1	no	—
179	C₅×#41	—	2	1	no	—
180	C₅×#41	—	4	2	no	—
181	C₅×#41	—	4	2	no	—
182	C₅×#41	—	4	4	no	—
183	C₅×#41	—	4	4	no	—
184	C₅×#41	—	4	2	no	—
185	C₅×#41	—	4	2	no	—
186	C₅×#41	C₂×D₄	1	2	yes	—
187	C₅×#41	C₂×D₄	1	2	yes	—
188	C₂×C₅×Q₈	—	2	2	no	—
189	C₂×C₅×Q₈	—	2	1	no	—
190	C₂×C₅×Q₈	—	2	1	no	—
191	C₂×C₅×Q₈	—	2	1	no	—
192	C₂×C₅×Q₈	—	2	1	no	—
193	C₂×C₅×Q₈	—	2	1	no	—
194	C₂×C₅×Q₈	—	2	1	no	—
195	C₂×C₅×Q₈	—	4	2	no	—
196	C₂×C₅×Q₈	—	4	2	no	—
197	C₂×C₅×Q₈	—	4	2	no	—
198	C₂×C₅×Q₈	—	4	2	no	—
199	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
200	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
201	C₂×C₅×Q₈	#41	1	1	yes	characteristic
202	C₂×C₅×D₄	—	2	1	no	—
203	C₂×C₅×D₄	—	2	2	no	—
204	C₂×C₅×D₄	—	2	1	no	—
205	C₂×C₅×D₄	—	2	1	no	—
206	C₂×C₅×D₄	—	2	1	no	—
207	C₂×C₅×D₄	—	2	1	no	—
208	C₂×C₅×D₄	—	2	1	no	—
209	C₂×C₅×D₄	—	2	1	no	—
210	C₂×C₅×D₄	—	2	1	no	—
211	C₂×C₅×D₄	—	2	1	no	—
212	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
213	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
214	C₂×C₅×D₄	#41	1	1	yes	characteristic
215	C₂²×C₄×C₅	—	2	1	no	—
216	C₂²×C₄×C₅	—	2	2	no	—
217	C₂²×C₄×C₅	—	2	2	no	—
218	C₂²×C₄×C₅	—	2	1	no	—
219	C₂²×C₄×C₅	—	2	1	no	—
220	C₂²×C₄×C₅	—	2	1	no	—
221	C₂²×C₄×C₅	#41	1	2	yes	—
222	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
223	C₅×Q₁₆	—	2	2	no	—
224	C₅×Q₁₆	—	4	1	no	—
225	C₅×SD₁₆	—	2	2	no	—
226	C₅×SD₁₆	—	2	2	no	—
227	C₅×SD₁₆	—	4	2	no	—
228	C₅×SD₁₆	—	4	2	no	—
229	C₅×SD₁₆	—	4	1	no	—
230	C₅×SD₁₆	—	4	1	no	—
231	C₅×SD₁₆	—	4	2	no	—
232	C₅×D₈	—	2	2	no	—
233	C₅×D₈	—	4	2	no	—
234	C₅×D₈	—	4	2	no	—
235	C₅×D₈	—	4	1	no	—
236	C₅×D₈	—	4	2	no	—
237	C₅×M₁₆	—	2	1	no	—
238	C₅×M₁₆	—	2	1	no	—
239	C₅×M₁₆	—	2	1	no	—
240	C₅×M₁₆	—	2	1	no	—
241	C₅×M₁₆	—	2	1	no	—
242	C₅×M₁₆	—	2	1	no	—
243	C₅×M₁₆	—	2	1	no	—
244	C₅×M₁₆	—	2	1	no	—
245	C₅×M₁₆	—	2	1	no	—
246	C₅×M₁₆	—	2	1	no	—
247	C₅×M₁₆	—	2	1	no	—
248	C₅×M₁₆	—	2	1	no	—
249	C₅×M₁₆	—	4	2	no	—
250	C₅×M₁₆	—	4	2	no	—
251	C₅×M₁₆	—	4	2	no	—
252	C₅×M₁₆	C₂×D₄	1	2	yes	—
253	C₅×M₁₆	C₂×D₄	1	2	yes	—
254	C₅×M₁₆	C₂×D₄	1	2	yes	—
255	C₅×M₁₆	C₂×D₄	1	2	yes	—
256	C₂×C₅×C₈	—	2	1	no	—
257	C₂×C₅×C₈	—	2	1	no	—
258	C₂×C₅×C₈	—	2	1	no	—
259	C₂×C₅×C₈	—	2	1	no	—
260	C₂×C₅×C₈	—	2	1	no	—
261	C₂×C₅×C₈	—	2	1	no	—
262	C₂×C₅×C₈	—	2	1	no	—
263	C₂×C₅×C₈	—	2	1	no	—
264	C₂×C₅×C₈	—	4	2	no	—
265	C₂×C₅×C₈	—	4	2	no	—
266	C₂×C₅×C₈	—	4	2	no	—
267	C₅×C₄⋊C₄	—	2	1	no	—
268	C₅×C₄⋊C₄	—	4	2	no	—
269	C₅×C₂²⋊C₄	—	2	1	no	—
270	C₅×C₂²⋊C₄	—	4	2	no	—
271	C₅×C₂²⋊C₄	—	4	2	no	—
272	C₅×C₂²⋊C₄	—	4	2	no	—
273	C₄²×C₅	—	2	1	no	—
274	C₄²×C₅	—	2	1	no	—
275	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
276	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
277	C₂×2₋¹⁺⁴	C₂²×C₅	1	1	yes	characteristic
278	#575	C₂²×C₅	1	1	yes	characteristic
279	#575	C₂²×C₅	1	1	yes	characteristic
280	C₂×AGL(1,ℤ/8)	C₂²×C₅	1	1	yes	characteristic
281	C₂×AGL(1,ℤ/8)	C₂²×C₅	1	1	yes	characteristic
282	C₂×AGL(1,ℤ/8)	C₂²×C₅	1	1	yes	characteristic
283	#568	C₂²×C₅	1	1	yes	characteristic
284	#568	C₂²×C₅	1	1	yes	characteristic
285	#568	C₂²×C₅	1	1	yes	characteristic
286	#536	C₂²×C₅	1	1	yes	characteristic
287	C₂×#124	C₂²×C₅	1	1	yes	characteristic
288	#472	—	2	2	no	—
289	#472	—	2	2	no	—
290	#471	—	2	2	no	—
291	#471	—	2	2	no	—
292	#456	—	2	2	no	—
293	#454	—	2	2	no	—
294	#454	—	2	2	no	—
295	#454	—	2	2	no	—
296	#444	—	2	2	no	—
297	#443	—	2	2	no	—
298	#430	C₂²×C₅	1	1	yes	characteristic
299	#430	C₂²×C₅	1	1	yes	characteristic
300	#430	C₂²×C₅	1	1	yes	characteristic
301	#423	C₂²×C₅	1	1	yes	characteristic
302	#421	C₂²×C₅	1	1	yes	characteristic
303	#421	C₂²×C₅	1	1	yes	characteristic
304	C₂×#104	C₂²×C₅	1	1	yes	characteristic
305	C₂×#104	C₂²×C₅	1	1	yes	characteristic
306	C₂×#104	C₂²×C₅	1	1	yes	characteristic
307	#418	C₂²×C₅	1	1	yes	characteristic
308	#413	—	2	2	no	—
309	#413	C₂²×C₅	1	1	yes	characteristic
310	#413	C₂²×C₅	1	1	yes	characteristic
311	#413	C₂²×C₅	1	1	yes	characteristic
312	C₂×#101	—	2	2	no	—
313	C₂×#101	C₂²×C₅	1	1	yes	characteristic
314	C₂×#101	C₂²×C₅	1	1	yes	characteristic
315	C₂×#101	C₂²×C₅	1	1	yes	characteristic
316	C₂×#101	C₂²×C₅	1	1	yes	characteristic
317	C₂×#100	C₂²×C₅	1	1	yes	characteristic
318	C₂×#100	C₂²×C₅	1	1	yes	characteristic
319	C₄×M₁₆	C₂²×C₅	1	1	yes	characteristic
320	#344	C₂²×C₅	1	1	yes	characteristic
321	#344	C₂²×C₅	1	1	yes	characteristic
322	#339	C₂²×C₅	1	1	yes	characteristic
323	#338	C₂²×C₅	1	1	yes	characteristic
324	C₂³×C₅	—	2	1	no	—
325	C₂³×C₅	—	2	1	no	—
326	C₂³×C₅	—	2	1	no	—
327	C₂³×C₅	—	4	1	no	—
328	C₂³×C₅	—	4	1	no	—
329	C₂³×C₅	—	4	1	no	—
330	C₂³×C₅	—	8	2	no	—
331	C₂³×C₅	C₂×#41	1	1	yes	characteristic
332	C₅×Q₈	—	2	1	no	—
333	C₅×Q₈	—	2	1	no	—
334	C₅×Q₈	—	2	1	no	—
335	C₅×Q₈	—	2	1	no	—
336	C₅×Q₈	—	2	1	no	—
337	C₅×Q₈	—	4	4	no	—
338	C₅×Q₈	—	4	1	no	—
339	C₅×Q₈	—	4	1	no	—
340	C₅×Q₈	—	4	1	no	—
341	C₅×Q₈	—	4	1	no	—
342	C₅×Q₈	#121	1	2	yes	—
343	C₅×D₄	—	2	2	no	—
344	C₅×D₄	—	2	2	no	—
345	C₅×D₄	—	2	1	no	—
346	C₅×D₄	—	2	1	no	—
347	C₅×D₄	—	2	1	no	—
348	C₅×D₄	—	2	1	no	—
349	C₅×D₄	—	2	1	no	—
350	C₅×D₄	—	4	4	no	—
351	C₅×D₄	—	4	1	no	—
352	C₅×D₄	—	4	1	no	—
353	C₅×D₄	—	4	1	no	—
354	C₅×D₄	—	4	1	no	—
355	C₅×D₄	—	4	1	no	—
356	C₅×D₄	—	4	1	no	—
357	C₅×D₄	—	4	1	no	—
358	C₅×D₄	—	4	1	no	—
359	C₅×D₄	#121	1	2	yes	—
360	C₂×C₄×C₅	—	2	1	no	—
361	C₂×C₄×C₅	—	2	2	no	—
362	C₂×C₄×C₅	—	2	2	no	—
363	C₂×C₄×C₅	—	2	1	no	—
364	C₂×C₄×C₅	—	2	2	no	—
365	C₂×C₄×C₅	—	2	1	no	—
366	C₂×C₄×C₅	—	2	1	no	—
367	C₂×C₄×C₅	—	2	2	no	—
368	C₂×C₄×C₅	—	2	2	no	—
369	C₂×C₄×C₅	—	2	1	no	—
370	C₂×C₄×C₅	—	4	4	no	—
371	C₂×C₄×C₅	—	4	4	no	—
372	C₂×C₄×C₅	—	4	1	no	—
373	C₂×C₄×C₅	—	4	1	no	—
374	C₂×C₄×C₅	—	4	1	no	—
375	C₂×C₄×C₅	—	4	1	no	—
376	C₂×C₄×C₅	—	4	1	no	—
377	C₂×C₄×C₅	—	8	1	no	—
378	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
379	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
380	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
381	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
382	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
383	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
384	C₅×C₈	—	2	2	no	—
385	C₅×C₈	—	2	2	no	—
386	C₅×C₈	—	2	2	no	—
387	C₅×C₈	—	2	2	no	—
388	C₅×C₈	—	4	1	no	—
389	C₅×C₈	—	4	1	no	—
390	C₅×C₈	—	4	1	no	—
391	C₅×C₈	—	4	1	no	—
392	2₋¹⁺⁴	—	2	2	no	—
393	2₋¹⁺⁴	—	2	2	no	—
394	2₋¹⁺⁴	—	4	1	no	—
395	2₋¹⁺⁴	—	4	1	no	—
396	C₂×#41	—	2	1	no	—
397	C₂×#41	—	2	1	no	—
398	C₂×#41	—	2	2	no	—
399	C₂×#41	C₂³×C₅	1	1	yes	characteristic
400	C₂×#41	C₂³×C₅	1	1	yes	characteristic
401	C₂×#41	C₂³×C₅	1	1	yes	characteristic
402	C₂²×Q₈	—	2	1	no	—
403	C₂²×Q₈	—	2	1	no	—
404	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
405	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
406	#137	—	2	1	no	—
407	#137	—	2	1	no	—
408	#137	—	2	1	no	—
409	#137	C₅×D₄	1	2	yes	—
410	AGL(1,ℤ/8)	—	2	1	no	—
411	AGL(1,ℤ/8)	—	2	2	no	—
412	AGL(1,ℤ/8)	—	2	2	no	—
413	AGL(1,ℤ/8)	—	2	1	no	—
414	AGL(1,ℤ/8)	—	2	1	no	—
415	AGL(1,ℤ/8)	—	2	2	no	—
416	AGL(1,ℤ/8)	—	2	2	no	—
417	AGL(1,ℤ/8)	—	4	1	no	—
418	AGL(1,ℤ/8)	—	4	1	no	—
419	AGL(1,ℤ/8)	C₅×D₄	1	2	yes	—
420	#135	—	2	2	no	—
421	#135	—	4	1	no	—
422	C₂×SD₁₆	—	2	1	no	—
423	C₂×SD₁₆	—	2	1	no	—
424	C₂×SD₁₆	—	2	1	no	—
425	C₂×D₈	—	2	1	no	—
426	C₂×D₈	—	2	1	no	—
427	C₂×D₈	—	2	1	no	—
428	#131	—	2	2	no	—
429	#131	—	2	2	no	—
430	#131	—	2	2	no	—
431	#131	—	2	2	no	—
432	#131	—	4	1	no	—
433	#131	—	4	1	no	—
434	C₂×M₁₆	—	2	2	no	—
435	C₂×M₁₆	—	2	2	no	—
436	C₂×M₁₆	—	4	1	no	—
437	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
438	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
439	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
440	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
441	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
442	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
443	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
444	C₂×M₁₆	C₂³×C₅	1	1	yes	characteristic
445	C₄⋊Q₈	—	2	1	no	—
446	#124	—	2	1	no	—
447	#124	—	2	1	no	—
448	#124	—	2	1	no	—
449	#124	—	4	1	no	—
450	#123	—	2	2	no	—
451	C₂²⋊Q₈	—	4	1	no	—
452	#117	C₂³×C₅	1	1	yes	characteristic
453	C₂×C₂²⋊C₄	—	2	2	no	—
454	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
455	#108	—	2	2	no	—
456	#108	—	2	2	no	—
457	#108	—	2	2	no	—
458	C₄⋊C₈	—	4	1	no	—
459	#104	—	2	2	no	—
460	#104	—	2	2	no	—
461	#104	—	2	1	no	—
462	#104	—	2	1	no	—
463	#104	—	2	1	no	—
464	#104	—	2	1	no	—
465	#104	C₅×D₄	1	2	yes	—
466	#104	C₅×D₄	1	2	yes	—
467	#102	—	4	1	no	—
468	#101	—	2	1	no	—
469	#101	—	2	1	no	—
470	#101	—	2	2	no	—
471	#101	—	2	2	no	—
472	#101	—	2	2	no	—
473	#101	—	2	2	no	—
474	#101	C₅×D₄	1	2	yes	—
475	#100	—	2	1	no	—
476	#100	—	2	1	no	—
477	#100	—	2	2	no	—
478	#100	—	2	2	no	—
479	#100	C₅×D₄	1	2	yes	—
480	#97	—	2	1	no	—
481	C₄×C₈	—	2	1	no	—
482	C₂²×C₅	—	2	2	no	—
483	C₂²×C₅	—	4	1	no	—
484	C₂²×C₅	—	4	1	no	—
485	C₂²×C₅	—	4	1	no	—
486	C₂²×C₅	—	8	1	no	—
487	C₂²×C₅	—	8	1	no	—
488	C₂²×C₅	—	8	1	no	—
489	C₂²×C₅	—	8	2	no	—
490	C₂²×C₅	—	8	1	no	—
491	C₂²×C₅	—	16	1	no	—
492	C₂²×C₅	—	16	1	no	—
493	C₂²×C₅	#525	1	1	yes	characteristic
494	C₂²×C₅	#546	1	1	yes	characteristic
495	C₂²×C₅	#546	1	1	yes	characteristic
496	C₄×C₅	—	2	2	no	—
497	C₄×C₅	—	4	1	no	—
498	C₄×C₅	—	4	1	no	—
499	C₄×C₅	—	4	2	no	—
500	C₄×C₅	—	4	1	no	—
501	C₄×C₅	—	4	1	no	—
502	C₄×C₅	—	8	1	no	—
503	C₄×C₅	C₂×#121	1	1	yes	characteristic
504	C₄×C₅	#538	1	1	yes	characteristic
505	C₄×C₅	D₄²	1	1	yes	characteristic
506	C₄×C₅	D₄²	1	1	yes	characteristic
507	C₂⁴	—	2	1	no	—
508	#41	—	2	1	no	—
509	#41	—	2	1	no	—
510	#41	—	2	1	no	—
511	#41	—	2	1	no	—
512	#41	—	2	1	no	—
513	#41	—	2	1	no	—
514	#41	—	2	1	no	—
515	#41	—	2	1	no	—
516	#41	—	2	1	no	—
517	#41	—	2	1	no	—
518	#41	—	4	2	no	—
519	#41	—	4	2	no	—
520	#41	—	4	4	no	—
521	#41	—	4	4	no	—
522	#41	—	4	2	no	—
523	#41	—	4	2	no	—
524	#41	C₂×C₅×D₄	1	2	yes	—
525	#41	C₂×C₅×D₄	1	2	yes	—
526	C₂×Q₈	—	2	2	no	—
527	C₂×Q₈	—	2	1	no	—
528	C₂×Q₈	—	2	1	no	—
529	C₂×Q₈	—	2	1	no	—
530	C₂×Q₈	—	2	1	no	—
531	C₂×Q₈	—	2	1	no	—
532	C₂×Q₈	—	2	1	no	—
533	C₂×Q₈	—	4	2	no	—
534	C₂×Q₈	—	4	2	no	—
535	C₂×Q₈	—	4	2	no	—
536	C₂×Q₈	—	4	2	no	—
537	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
538	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
539	C₂×Q₈	C₅×#41	1	1	yes	characteristic
540	C₂×D₄	—	2	1	no	—
541	C₂×D₄	—	2	2	no	—
542	C₂×D₄	—	2	1	no	—
543	C₂×D₄	—	2	1	no	—
544	C₂×D₄	—	2	1	no	—
545	C₂×D₄	—	2	1	no	—
546	C₂×D₄	—	2	1	no	—
547	C₂×D₄	—	2	1	no	—
548	C₂×D₄	—	2	1	no	—
549	C₂×D₄	—	2	1	no	—
550	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
551	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
552	C₂×D₄	C₅×#41	1	1	yes	characteristic
553	C₂²×C₄	—	2	1	no	—
554	C₂²×C₄	—	2	2	no	—
555	C₂²×C₄	—	2	2	no	—
556	C₂²×C₄	—	2	1	no	—
557	C₂²×C₄	—	2	1	no	—
558	C₂²×C₄	—	2	1	no	—
559	C₂²×C₄	C₅×#41	1	2	yes	—
560	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
561	Q₁₆	—	2	2	no	—
562	Q₁₆	—	4	1	no	—
563	SD₁₆	—	2	2	no	—
564	SD₁₆	—	2	2	no	—
565	SD₁₆	—	4	2	no	—
566	SD₁₆	—	4	2	no	—
567	SD₁₆	—	4	1	no	—
568	SD₁₆	—	4	1	no	—
569	SD₁₆	—	4	2	no	—
570	D₈	—	2	2	no	—
571	D₈	—	4	2	no	—
572	D₈	—	4	2	no	—
573	D₈	—	4	1	no	—
574	D₈	—	4	2	no	—
575	M₁₆	—	2	1	no	—
576	M₁₆	—	2	1	no	—
577	M₁₆	—	2	1	no	—
578	M₁₆	—	2	1	no	—
579	M₁₆	—	2	1	no	—
580	M₁₆	—	2	1	no	—
581	M₁₆	—	2	1	no	—
582	M₁₆	—	2	1	no	—
583	M₁₆	—	2	1	no	—
584	M₁₆	—	2	1	no	—
585	M₁₆	—	2	1	no	—
586	M₁₆	—	2	1	no	—
587	M₁₆	—	4	2	no	—
588	M₁₆	—	4	2	no	—
589	M₁₆	—	4	2	no	—
590	M₁₆	C₂×C₅×D₄	1	2	yes	—
591	M₁₆	C₂×C₅×D₄	1	2	yes	—
592	M₁₆	C₂×C₅×D₄	1	2	yes	—
593	M₁₆	C₂×C₅×D₄	1	2	yes	—
594	C₂×C₈	—	2	1	no	—
595	C₂×C₈	—	2	1	no	—
596	C₂×C₈	—	2	1	no	—
597	C₂×C₈	—	2	1	no	—
598	C₂×C₈	—	2	1	no	—
599	C₂×C₈	—	2	1	no	—
600	C₂×C₈	—	2	1	no	—
601	C₂×C₈	—	2	1	no	—
602	C₂×C₈	—	4	2	no	—
603	C₂×C₈	—	4	2	no	—
604	C₂×C₈	—	4	2	no	—
605	C₄⋊C₄	—	2	1	no	—
606	C₄⋊C₄	—	4	2	no	—
607	C₂²⋊C₄	—	2	1	no	—
608	C₂²⋊C₄	—	4	2	no	—
609	C₂²⋊C₄	—	4	2	no	—
610	C₂²⋊C₄	—	4	2	no	—
611	C₄²	—	2	1	no	—
612	C₄²	—	2	1	no	—
613	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
614	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
615	C₂×C₅	—	2	1	no	—
616	C₂×C₅	—	2	1	no	—
617	C₂×C₅	—	2	1	no	—
618	C₂×C₅	—	4	2	no	—
619	C₂×C₅	—	8	1	no	—
620	C₂×C₅	—	8	1	no	—
621	C₂×C₅	—	16	1	no	—
622	C₂×C₅	#2424	1	1	yes	characteristic
623	C₂³	—	2	1	no	—
624	C₂³	—	2	1	no	—
625	C₂³	—	2	1	no	—
626	C₂³	—	4	1	no	—
627	C₂³	—	4	1	no	—
628	C₂³	—	4	1	no	—
629	C₂³	—	8	2	no	—
630	C₂³	C₂×C₅×#41	1	1	yes	characteristic
631	Q₈	—	2	1	no	—
632	Q₈	—	2	1	no	—
633	Q₈	—	2	1	no	—
634	Q₈	—	2	1	no	—
635	Q₈	—	2	1	no	—
636	Q₈	—	4	4	no	—
637	Q₈	—	4	1	no	—
638	Q₈	—	4	1	no	—
639	Q₈	—	4	1	no	—
640	Q₈	—	4	1	no	—
641	Q₈	C₅×#121	1	2	yes	—
642	D₄	—	2	2	no	—
643	D₄	—	2	2	no	—
644	D₄	—	2	1	no	—
645	D₄	—	2	1	no	—
646	D₄	—	2	1	no	—
647	D₄	—	2	1	no	—
648	D₄	—	2	1	no	—
649	D₄	—	4	4	no	—
650	D₄	—	4	1	no	—
651	D₄	—	4	1	no	—
652	D₄	—	4	1	no	—
653	D₄	—	4	1	no	—
654	D₄	—	4	1	no	—
655	D₄	—	4	1	no	—
656	D₄	—	4	1	no	—
657	D₄	—	4	1	no	—
658	D₄	C₅×#121	1	2	yes	—
659	C₂×C₄	—	2	1	no	—
660	C₂×C₄	—	2	2	no	—
661	C₂×C₄	—	2	2	no	—
662	C₂×C₄	—	2	1	no	—
663	C₂×C₄	—	2	2	no	—
664	C₂×C₄	—	2	1	no	—
665	C₂×C₄	—	2	1	no	—
666	C₂×C₄	—	2	2	no	—
667	C₂×C₄	—	2	2	no	—
668	C₂×C₄	—	2	1	no	—
669	C₂×C₄	—	4	4	no	—
670	C₂×C₄	—	4	4	no	—
671	C₂×C₄	—	4	1	no	—
672	C₂×C₄	—	4	1	no	—
673	C₂×C₄	—	4	1	no	—
674	C₂×C₄	—	4	1	no	—
675	C₂×C₄	—	4	1	no	—
676	C₂×C₄	—	8	1	no	—
677	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
678	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
679	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
680	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
681	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
682	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
683	C₈	—	2	2	no	—
684	C₈	—	2	2	no	—
685	C₈	—	2	2	no	—
686	C₈	—	2	2	no	—
687	C₈	—	4	1	no	—
688	C₈	—	4	1	no	—
689	C₈	—	4	1	no	—
690	C₈	—	4	1	no	—
691	C₅	#25197	1	1	yes	characteristic, direct product factor, Sylow
692	C₂²	—	2	2	no	—
693	C₂²	—	4	1	no	—
694	C₂²	—	4	1	no	—
695	C₂²	—	4	1	no	—
696	C₂²	—	8	1	no	—
697	C₂²	—	8	1	no	—
698	C₂²	—	8	1	no	—
699	C₂²	—	8	2	no	—
700	C₂²	—	8	1	no	—
701	C₂²	—	16	1	no	—
702	C₂²	—	16	1	no	—
703	C₂²	C₅×#525	1	1	yes	characteristic
704	C₂²	C₅×#546	1	1	yes	characteristic
705	C₂²	C₅×#546	1	1	yes	characteristic
706	C₄	—	2	2	no	—
707	C₄	—	4	1	no	—
708	C₄	—	4	1	no	—
709	C₄	—	4	2	no	—
710	C₄	—	4	1	no	—
711	C₄	—	4	1	no	—
712	C₄	—	8	1	no	—
713	C₄	C₂×C₅×#121	1	1	yes	characteristic
714	C₄	C₅×#538	1	1	yes	characteristic
715	C₄	C₅×D₄²	1	1	yes	characteristic
716	C₄	C₅×D₄²	1	1	yes	characteristic
717	C₂	—	2	1	no	—
718	C₂	—	2	1	no	—
719	C₂	—	2	1	no	—
720	C₂	—	4	2	no	—
721	C₂	—	8	1	no	—
722	C₂	—	8	1	no	—
723	C₂	—	16	1	no	—
724	C₂	C₅×#2424	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	#2424
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3068	C₂	semidirect
2. C₅×#3068	C₂	semidirect
3. C₅×#3020	C₂	semidirect
4. C₂×C₅×#454	C₂	semidirect
5. C₅×#2956	C₂	semidirect
6. C₅×#2889	C₂	semidirect
7. C₅×#2070	C₂	semidirect
8. C₅×#2050	C₂	semidirect
9. C₅×#2048	C₂	semidirect
10. C₅×#2003	C₂	semidirect
11. C₅×#1967	C₂	semidirect
12. C₅×#1932	C₂	semidirect
13. C₅×#1915	C₂	semidirect
14. C₅×#1903	C₂	semidirect
15. C₅×#1861	C₂	semidirect
63. #25197	C₅	direct (N × H)
154. #3068	C₂×C₅	semidirect
155. #3068	C₂×C₅	semidirect
156. #3020	C₂×C₅	semidirect
157. C₂×#454	C₂×C₅	semidirect
158. #2956	C₂×C₅	semidirect
159. #2889	C₂×C₅	semidirect
160. #2070	C₂×C₅	semidirect
161. #2050	C₂×C₅	semidirect
162. #2048	C₂×C₅	semidirect
163. #2003	C₂×C₅	semidirect
164. #1967	C₂×C₅	semidirect
165. #1932	C₂×C₅	semidirect
166. #1915	C₂×C₅	semidirect
167. #1903	C₂×C₅	semidirect
168. #1861	C₂×C₅	semidirect
691. C₅	#25197	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×#3068	2
C₅×#3020	2
C₂×C₅×#454	2
C₅×#2956	2
C₅×#2889	2
C₅×#2070	2
C₅×#2050	2
C₅×#2048	2
C₅×#2003	2
C₅×#1967	2
C₅×#1932	2
C₅×#1915	2
C₅×#1903	2
C₅×#1861	2
#25197	5

Subgroups — C5×#25197 #11947917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#25197 #11947917