Subgroups — C₂×C₅×#2522 #11957486

Order 1280.

2270 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2522	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₅×#399	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#398	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#398	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#396	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#389	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#388	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅×#388	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×#387	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×#387	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×#387	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅×#385	C₂	1	1	yes	maximal, characteristic
14	C₂×C₅×#380	C₂	1	1	yes	maximal, characteristic
15	C₂×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₄×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
17	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
18	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂²×C₅×C₄⋊C₄	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	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₂²×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₂²×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₂²×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
32	C₂²×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
33	C₂²×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
34	C₂²×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
35	C₂²×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
36	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
37	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
38	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
39	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
40	C₅×#399	C₂²	1	8	yes	—
41	C₅×#398	C₂²	1	8	yes	complemented
42	C₅×#398	C₂²	1	8	yes	complemented
43	C₅×#396	C₂²	1	8	yes	—
44	C₅×#389	C₂²	1	8	yes	complemented
45	C₅×#389	C₂²	1	8	yes	complemented
46	C₅×#388	C₂²	1	8	yes	—
47	C₅×#388	C₂²	1	8	yes	—
48	C₅×#387	C₂²	1	8	yes	complemented
49	C₅×#387	C₂²	1	8	yes	complemented
50	C₅×#387	C₂²	1	8	yes	complemented
51	C₅×#385	C₂²	1	8	yes	—
52	C₅×#380	C₂²	1	8	yes	—
53	C₄×C₅×C₄⋊C₄	C₂²	1	8	yes	complemented
54	C₄×C₅×C₂²⋊C₄	C₂²	1	8	yes	—
55	C₂×C₅×#95	C₂²	1	1	yes	characteristic
56	C₂×C₅×#95	C₂²	1	1	yes	characteristic
57	C₂×C₅×#95	C₂²	1	1	yes	characteristic
58	C₂×C₅×#95	C₂²	1	1	yes	characteristic
59	C₂×C₅×#95	C₂²	1	1	yes	characteristic
60	C₂×C₅×#95	C₂²	1	1	yes	characteristic
61	C₂×C₅×#95	C₂²	1	1	yes	characteristic
62	C₂×C₅×#95	C₂²	1	1	yes	characteristic
63	C₂×C₅×#95	C₂²	1	1	yes	characteristic
64	C₂×C₅×#95	C₂²	1	1	yes	characteristic
65	C₂×C₅×#95	C₂²	1	1	yes	characteristic
66	C₂×C₅×#95	C₂²	1	1	yes	characteristic
67	C₂×#2522	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂⁵×C₅	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₅	—	2	4	no	—
70	C₂³×C₄×C₅	—	2	4	no	—
71	C₂³×C₄×C₅	—	4	2	no	—
72	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
73	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
75	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
76	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
77	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
78	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₅	C₂³	1	4	yes	—
81	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
83	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
84	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
85	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
86	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
87	C₂×C₅×C₄⋊C₄	—	2	4	no	—
88	C₂×C₅×C₄⋊C₄	—	2	4	no	—
89	C₂×C₅×C₄⋊C₄	—	2	4	no	—
90	C₂×C₅×C₄⋊C₄	—	2	4	no	—
91	C₂×C₅×C₄⋊C₄	—	2	4	no	—
92	C₂×C₅×C₄⋊C₄	—	2	4	no	—
93	C₂×C₅×C₄⋊C₄	—	2	4	no	—
94	C₂×C₅×C₄⋊C₄	—	4	2	no	—
95	C₂×C₅×C₄⋊C₄	—	4	2	no	—
96	C₂×C₅×C₄⋊C₄	—	4	2	no	—
97	C₂×C₅×C₄⋊C₄	—	4	2	no	—
98	C₂×C₅×C₄⋊C₄	—	4	2	no	—
99	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
100	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
101	C₂×C₅×C₄⋊C₄	C₂³	1	4	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	4	yes	—
105	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
106	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
107	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
108	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
109	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
110	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
111	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
112	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
113	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
114	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
115	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
116	C₂×C₅×C₂²⋊C₄	Q₈	1	8	yes	—
117	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
118	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
119	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
120	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
121	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
122	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
123	C₂×C₄²×C₅	—	2	4	no	—
124	C₂×C₄²×C₅	—	4	2	no	—
125	C₂×C₄²×C₅	—	4	2	no	—
126	C₂×C₄²×C₅	—	4	2	no	—
127	C₂×C₄²×C₅	C₂³	1	4	yes	—
128	C₂×C₄²×C₅	C₂³	1	4	yes	—
129	C₂×C₄²×C₅	C₂³	1	4	yes	—
130	C₂×C₄²×C₅	C₂³	1	4	yes	—
131	C₅×#95	C₂³	1	4	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	4	yes	—
137	C₅×#95	C₂³	1	4	yes	—
138	C₅×#95	C₂³	1	4	yes	—
139	C₅×#95	C₂³	1	4	yes	—
140	C₅×#95	C₂³	1	4	yes	—
141	C₅×#95	C₂³	1	4	yes	—
142	C₅×#95	C₂³	1	4	yes	—
143	#2522	C₂×C₅	1	16	yes	direct product factor
144	C₂×#399	C₂×C₅	1	1	yes	characteristic
145	C₂×#398	C₂×C₅	1	1	yes	characteristic, complemented
146	C₂×#398	C₂×C₅	1	1	yes	characteristic, complemented
147	C₂×#396	C₂×C₅	1	1	yes	characteristic
148	C₂×#389	C₂×C₅	1	1	yes	characteristic, complemented
149	C₂×#389	C₂×C₅	1	1	yes	characteristic, complemented
150	C₂×#388	C₂×C₅	1	1	yes	characteristic
151	C₂×#388	C₂×C₅	1	1	yes	characteristic
152	C₂×#387	C₂×C₅	1	1	yes	characteristic, complemented
153	C₂×#387	C₂×C₅	1	1	yes	characteristic, complemented
154	C₂×#387	C₂×C₅	1	1	yes	characteristic, complemented
155	C₂×#385	C₂×C₅	1	1	yes	characteristic
156	C₂×#380	C₂×C₅	1	1	yes	characteristic
157	C₂×C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
158	C₂×C₄×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
159	C₂⁴×C₅	—	2	2	no	—
160	C₂⁴×C₅	—	2	2	no	—
161	C₂⁴×C₅	—	2	2	no	—
162	C₂⁴×C₅	—	2	2	no	—
163	C₂⁴×C₅	—	2	2	no	—
164	C₂⁴×C₅	—	2	2	no	—
165	C₂⁴×C₅	C₂×Q₈	1	4	yes	—
166	C₂⁴×C₅	C₂⁴	1	2	yes	—
167	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
168	C₂²×C₄×C₅	—	2	2	no	—
169	C₂²×C₄×C₅	—	2	2	no	—
170	C₂²×C₄×C₅	—	2	2	no	—
171	C₂²×C₄×C₅	—	2	2	no	—
172	C₂²×C₄×C₅	—	2	2	no	—
173	C₂²×C₄×C₅	—	2	2	no	—
174	C₂²×C₄×C₅	—	2	2	no	—
175	C₂²×C₄×C₅	—	2	2	no	—
176	C₂²×C₄×C₅	—	2	2	no	—
177	C₂²×C₄×C₅	—	2	2	no	—
178	C₂²×C₄×C₅	—	2	2	no	—
179	C₂²×C₄×C₅	—	2	2	no	—
180	C₂²×C₄×C₅	—	2	2	no	—
181	C₂²×C₄×C₅	—	2	2	no	—
182	C₂²×C₄×C₅	—	2	2	no	—
183	C₂²×C₄×C₅	—	2	2	no	—
184	C₂²×C₄×C₅	—	2	2	no	—
185	C₂²×C₄×C₅	—	2	2	no	—
186	C₂²×C₄×C₅	—	2	2	no	—
187	C₂²×C₄×C₅	—	2	2	no	—
188	C₂²×C₄×C₅	—	2	2	no	—
189	C₂²×C₄×C₅	—	2	2	no	—
190	C₂²×C₄×C₅	—	2	2	no	—
191	C₂²×C₄×C₅	—	2	2	no	—
192	C₂²×C₄×C₅	—	2	2	no	—
193	C₂²×C₄×C₅	—	2	2	no	—
194	C₂²×C₄×C₅	—	2	2	no	—
195	C₂²×C₄×C₅	—	2	2	no	—
196	C₂²×C₄×C₅	—	2	2	no	—
197	C₂²×C₄×C₅	—	2	2	no	—
198	C₂²×C₄×C₅	—	2	2	no	—
199	C₂²×C₄×C₅	—	2	2	no	—
200	C₂²×C₄×C₅	—	2	2	no	—
201	C₂²×C₄×C₅	—	2	2	no	—
202	C₂²×C₄×C₅	—	2	2	no	—
203	C₂²×C₄×C₅	—	2	2	no	—
204	C₂²×C₄×C₅	—	2	8	no	—
205	C₂²×C₄×C₅	—	2	8	no	—
206	C₂²×C₄×C₅	—	4	4	no	—
207	C₂²×C₄×C₅	—	4	16	no	—
208	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
209	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
210	C₂²×C₄×C₅	#41	1	4	yes	—
211	C₂²×C₄×C₅	#41	1	4	yes	—
212	C₂²×C₄×C₅	#41	1	4	yes	—
213	C₂²×C₄×C₅	#41	1	4	yes	—
214	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
215	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
216	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
217	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
218	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
219	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
220	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
221	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
222	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
223	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
224	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
225	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
226	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
227	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
228	C₅×C₄⋊C₄	—	2	8	no	—
229	C₅×C₄⋊C₄	—	2	8	no	—
230	C₅×C₄⋊C₄	—	2	8	no	—
231	C₅×C₄⋊C₄	—	2	8	no	—
232	C₅×C₄⋊C₄	—	2	8	no	—
233	C₅×C₄⋊C₄	—	2	8	no	—
234	C₅×C₄⋊C₄	—	2	8	no	—
235	C₅×C₄⋊C₄	—	4	4	no	—
236	C₅×C₄⋊C₄	—	4	4	no	—
237	C₅×C₄⋊C₄	—	4	4	no	—
238	C₅×C₄⋊C₄	—	4	4	no	—
239	C₅×C₄⋊C₄	—	4	4	no	—
240	C₅×C₂²⋊C₄	—	2	8	no	—
241	C₅×C₂²⋊C₄	—	2	8	no	—
242	C₅×C₂²⋊C₄	—	4	4	no	—
243	C₅×C₂²⋊C₄	—	4	4	no	—
244	C₅×C₂²⋊C₄	—	4	4	no	—
245	C₅×C₂²⋊C₄	C₂×Q₈	1	16	yes	—
246	C₄²×C₅	—	2	8	no	—
247	C₄²×C₅	—	4	4	no	—
248	C₄²×C₅	—	4	4	no	—
249	C₄²×C₅	—	4	4	no	—
250	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
251	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
252	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
253	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
254	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
255	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
256	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
257	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
258	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
259	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
260	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
261	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
262	C₂²×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
263	C₂²×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
264	C₂²×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
265	C₂²×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
266	C₂²×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
267	C₂²×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
268	C₂²×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
269	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
270	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
271	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
272	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
273	#399	C₂²×C₅	1	8	yes	—
274	#398	C₂²×C₅	1	8	yes	complemented
275	#398	C₂²×C₅	1	8	yes	complemented
276	#396	C₂²×C₅	1	8	yes	—
277	#389	C₂²×C₅	1	8	yes	complemented
278	#389	C₂²×C₅	1	8	yes	complemented
279	#388	C₂²×C₅	1	8	yes	—
280	#388	C₂²×C₅	1	8	yes	—
281	#387	C₂²×C₅	1	8	yes	complemented
282	#387	C₂²×C₅	1	8	yes	complemented
283	#387	C₂²×C₅	1	8	yes	complemented
284	#385	C₂²×C₅	1	8	yes	—
285	#380	C₂²×C₅	1	8	yes	—
286	C₄×C₄⋊C₄	C₂²×C₅	1	8	yes	complemented
287	C₄×C₂²⋊C₄	C₂²×C₅	1	8	yes	—
288	C₂×#95	C₂²×C₅	1	1	yes	characteristic
289	C₂×#95	C₂²×C₅	1	1	yes	characteristic
290	C₂×#95	C₂²×C₅	1	1	yes	characteristic
291	C₂×#95	C₂²×C₅	1	1	yes	characteristic
292	C₂×#95	C₂²×C₅	1	1	yes	characteristic
293	C₂×#95	C₂²×C₅	1	1	yes	characteristic
294	C₂×#95	C₂²×C₅	1	1	yes	characteristic
295	C₂×#95	C₂²×C₅	1	1	yes	characteristic
296	C₂×#95	C₂²×C₅	1	1	yes	characteristic
297	C₂×#95	C₂²×C₅	1	1	yes	characteristic
298	C₂×#95	C₂²×C₅	1	1	yes	characteristic
299	C₂×#95	C₂²×C₅	1	1	yes	characteristic
300	C₂³×C₅	—	2	8	no	—
301	C₂³×C₅	—	2	8	no	—
302	C₂³×C₅	—	2	8	no	—
303	C₂³×C₅	—	2	2	no	—
304	C₂³×C₅	—	2	2	no	—
305	C₂³×C₅	—	2	2	no	—
306	C₂³×C₅	—	2	2	no	—
307	C₂³×C₅	—	2	2	no	—
308	C₂³×C₅	—	2	2	no	—
309	C₂³×C₅	—	4	4	no	—
310	C₂³×C₅	—	4	4	no	—
311	C₂³×C₅	—	4	4	no	—
312	C₂³×C₅	—	4	4	no	—
313	C₂³×C₅	C₂²×Q₈	1	4	yes	—
314	C₂³×C₅	C₂²×Q₈	1	2	yes	—
315	C₂³×C₅	C₂×#41	1	2	yes	—
316	C₂³×C₅	C₂×#41	1	2	yes	—
317	C₂³×C₅	C₂×#41	1	2	yes	—
318	C₂³×C₅	C₂×#41	1	2	yes	—
319	C₂³×C₅	H(5,𝔽₂)	1	2	yes	—
320	C₂³×C₅	H(5,𝔽₂)	1	2	yes	—
321	C₂³×C₅	C₂⁵	1	1	yes	characteristic
322	C₂×C₄×C₅	—	2	8	no	—
323	C₂×C₄×C₅	—	2	8	no	—
324	C₂×C₄×C₅	—	2	8	no	—
325	C₂×C₄×C₅	—	2	8	no	—
326	C₂×C₄×C₅	—	2	8	no	—
327	C₂×C₄×C₅	—	2	8	no	—
328	C₂×C₄×C₅	—	2	2	no	—
329	C₂×C₄×C₅	—	2	2	no	—
330	C₂×C₄×C₅	—	2	2	no	—
331	C₂×C₄×C₅	—	2	2	no	—
332	C₂×C₄×C₅	—	2	2	no	—
333	C₂×C₄×C₅	—	2	2	no	—
334	C₂×C₄×C₅	—	2	2	no	—
335	C₂×C₄×C₅	—	2	2	no	—
336	C₂×C₄×C₅	—	2	2	no	—
337	C₂×C₄×C₅	—	2	2	no	—
338	C₂×C₄×C₅	—	2	2	no	—
339	C₂×C₄×C₅	—	2	2	no	—
340	C₂×C₄×C₅	—	2	2	no	—
341	C₂×C₄×C₅	—	2	2	no	—
342	C₂×C₄×C₅	—	2	2	no	—
343	C₂×C₄×C₅	—	2	2	no	—
344	C₂×C₄×C₅	—	2	2	no	—
345	C₂×C₄×C₅	—	2	2	no	—
346	C₂×C₄×C₅	—	2	2	no	—
347	C₂×C₄×C₅	—	2	2	no	—
348	C₂×C₄×C₅	—	2	2	no	—
349	C₂×C₄×C₅	—	2	2	no	—
350	C₂×C₄×C₅	—	2	2	no	—
351	C₂×C₄×C₅	—	2	2	no	—
352	C₂×C₄×C₅	—	2	2	no	—
353	C₂×C₄×C₅	—	2	2	no	—
354	C₂×C₄×C₅	—	2	2	no	—
355	C₂×C₄×C₅	—	2	2	no	—
356	C₂×C₄×C₅	—	2	2	no	—
357	C₂×C₄×C₅	—	2	2	no	—
358	C₂×C₄×C₅	—	2	2	no	—
359	C₂×C₄×C₅	—	2	2	no	—
360	C₂×C₄×C₅	—	2	2	no	—
361	C₂×C₄×C₅	—	2	2	no	—
362	C₂×C₄×C₅	—	2	2	no	—
363	C₂×C₄×C₅	—	2	2	no	—
364	C₂×C₄×C₅	—	4	16	no	—
365	C₂×C₄×C₅	—	4	4	no	—
366	C₂×C₄×C₅	—	4	4	no	—
367	C₂×C₄×C₅	—	4	4	no	—
368	C₂×C₄×C₅	—	4	4	no	—
369	C₂×C₄×C₅	—	4	4	no	—
370	C₂×C₄×C₅	—	4	4	no	—
371	C₂×C₄×C₅	—	4	4	no	—
372	C₂×C₄×C₅	—	4	4	no	—
373	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
374	C₂×C₄×C₅	C₂²×Q₈	1	4	yes	—
375	C₂×C₄×C₅	C₂×#41	1	4	yes	—
376	C₂×C₄×C₅	C₂×#41	1	4	yes	—
377	C₂×C₄×C₅	C₂×#41	1	4	yes	—
378	C₂×C₄×C₅	C₂×#41	1	4	yes	—
379	C₂⁵	C₂³×C₅	1	1	yes	characteristic
380	C₂³×C₄	—	2	4	no	—
381	C₂³×C₄	—	2	4	no	—
382	C₂³×C₄	—	4	2	no	—
383	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
384	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
385	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
386	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
387	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
388	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
389	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
390	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
391	C₂³×C₄	C₂³×C₅	1	4	yes	—
392	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
393	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
394	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
395	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
396	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
397	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
398	C₂×C₄⋊C₄	—	2	4	no	—
399	C₂×C₄⋊C₄	—	2	4	no	—
400	C₂×C₄⋊C₄	—	2	4	no	—
401	C₂×C₄⋊C₄	—	2	4	no	—
402	C₂×C₄⋊C₄	—	2	4	no	—
403	C₂×C₄⋊C₄	—	2	4	no	—
404	C₂×C₄⋊C₄	—	2	4	no	—
405	C₂×C₄⋊C₄	—	4	2	no	—
406	C₂×C₄⋊C₄	—	4	2	no	—
407	C₂×C₄⋊C₄	—	4	2	no	—
408	C₂×C₄⋊C₄	—	4	2	no	—
409	C₂×C₄⋊C₄	—	4	2	no	—
410	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
411	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
412	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
413	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
414	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
415	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
416	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
417	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
418	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
419	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
420	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
421	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
422	C₂×C₂²⋊C₄	—	2	4	no	—
423	C₂×C₂²⋊C₄	—	2	4	no	—
424	C₂×C₂²⋊C₄	—	4	2	no	—
425	C₂×C₂²⋊C₄	—	4	2	no	—
426	C₂×C₂²⋊C₄	—	4	2	no	—
427	C₂×C₂²⋊C₄	C₅×Q₈	1	8	yes	—
428	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
429	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
430	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
431	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
432	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
433	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
434	C₂×C₄²	—	2	4	no	—
435	C₂×C₄²	—	4	2	no	—
436	C₂×C₄²	—	4	2	no	—
437	C₂×C₄²	—	4	2	no	—
438	C₂×C₄²	C₂³×C₅	1	4	yes	—
439	C₂×C₄²	C₂³×C₅	1	4	yes	—
440	C₂×C₄²	C₂³×C₅	1	4	yes	—
441	C₂×C₄²	C₂³×C₅	1	4	yes	—
442	#95	C₂³×C₅	1	4	yes	—
443	#95	C₂³×C₅	1	4	yes	—
444	#95	C₂³×C₅	1	4	yes	—
445	#95	C₂³×C₅	1	4	yes	—
446	#95	C₂³×C₅	1	4	yes	—
447	#95	C₂³×C₅	1	4	yes	—
448	#95	C₂³×C₅	1	4	yes	—
449	#95	C₂³×C₅	1	4	yes	—
450	#95	C₂³×C₅	1	4	yes	—
451	#95	C₂³×C₅	1	4	yes	—
452	#95	C₂³×C₅	1	4	yes	—
453	#95	C₂³×C₅	1	4	yes	—
454	C₂²×C₅	—	2	4	no	—
455	C₂²×C₅	—	2	4	no	—
456	C₂²×C₅	—	2	4	no	—
457	C₂²×C₅	—	4	2	no	—
458	C₂²×C₅	—	4	2	no	—
459	C₂²×C₅	—	4	2	no	—
460	C₂²×C₅	—	4	2	no	—
461	C₂²×C₅	—	4	16	no	—
462	C₂²×C₅	#529	1	4	yes	—
463	C₂²×C₅	#533	1	4	yes	—
464	C₂²×C₅	#540	1	4	yes	—
465	C₂²×C₅	#551	1	4	yes	—
466	C₂²×C₅	#553	1	4	yes	—
467	C₂²×C₅	#554	1	4	yes	—
468	C₂²×C₅	#554	1	4	yes	—
469	C₂²×C₅	C₂³×Q₈	1	1	yes	characteristic
470	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
471	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
472	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
473	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
474	C₂²×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
475	C₂²×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
476	C₄×C₅	—	2	4	no	—
477	C₄×C₅	—	2	4	no	—
478	C₄×C₅	—	2	4	no	—
479	C₄×C₅	—	2	4	no	—
480	C₄×C₅	—	2	4	no	—
481	C₄×C₅	—	2	4	no	—
482	C₄×C₅	—	4	2	no	—
483	C₄×C₅	—	4	2	no	—
484	C₄×C₅	—	4	2	no	—
485	C₄×C₅	—	4	2	no	—
486	C₄×C₅	—	4	2	no	—
487	C₄×C₅	—	4	2	no	—
488	C₄×C₅	—	4	2	no	—
489	C₄×C₅	—	4	2	no	—
490	C₂⁴	—	2	2	no	—
491	C₂⁴	—	2	2	no	—
492	C₂⁴	—	2	2	no	—
493	C₂⁴	—	2	2	no	—
494	C₂⁴	—	2	2	no	—
495	C₂⁴	—	2	2	no	—
496	C₂⁴	C₂×C₅×Q₈	1	4	yes	—
497	C₂⁴	C₂⁴×C₅	1	2	yes	—
498	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	2	no	—
502	C₂²×C₄	—	2	2	no	—
503	C₂²×C₄	—	2	2	no	—
504	C₂²×C₄	—	2	2	no	—
505	C₂²×C₄	—	2	2	no	—
506	C₂²×C₄	—	2	2	no	—
507	C₂²×C₄	—	2	2	no	—
508	C₂²×C₄	—	2	2	no	—
509	C₂²×C₄	—	2	2	no	—
510	C₂²×C₄	—	2	2	no	—
511	C₂²×C₄	—	2	2	no	—
512	C₂²×C₄	—	2	2	no	—
513	C₂²×C₄	—	2	2	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	2	2	no	—
516	C₂²×C₄	—	2	2	no	—
517	C₂²×C₄	—	2	2	no	—
518	C₂²×C₄	—	2	2	no	—
519	C₂²×C₄	—	2	2	no	—
520	C₂²×C₄	—	2	2	no	—
521	C₂²×C₄	—	2	2	no	—
522	C₂²×C₄	—	2	2	no	—
523	C₂²×C₄	—	2	2	no	—
524	C₂²×C₄	—	2	2	no	—
525	C₂²×C₄	—	2	2	no	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	2	2	no	—
529	C₂²×C₄	—	2	2	no	—
530	C₂²×C₄	—	2	2	no	—
531	C₂²×C₄	—	2	2	no	—
532	C₂²×C₄	—	2	2	no	—
533	C₂²×C₄	—	2	2	no	—
534	C₂²×C₄	—	2	2	no	—
535	C₂²×C₄	—	2	8	no	—
536	C₂²×C₄	—	2	8	no	—
537	C₂²×C₄	—	4	4	no	—
538	C₂²×C₄	—	4	16	no	—
539	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
540	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
541	C₂²×C₄	C₅×#41	1	4	yes	—
542	C₂²×C₄	C₅×#41	1	4	yes	—
543	C₂²×C₄	C₅×#41	1	4	yes	—
544	C₂²×C₄	C₅×#41	1	4	yes	—
545	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
546	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
547	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
548	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
549	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
550	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
551	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
552	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
553	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
554	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
555	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
556	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
557	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
558	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
559	C₄⋊C₄	—	2	8	no	—
560	C₄⋊C₄	—	2	8	no	—
561	C₄⋊C₄	—	2	8	no	—
562	C₄⋊C₄	—	2	8	no	—
563	C₄⋊C₄	—	2	8	no	—
564	C₄⋊C₄	—	2	8	no	—
565	C₄⋊C₄	—	2	8	no	—
566	C₄⋊C₄	—	4	4	no	—
567	C₄⋊C₄	—	4	4	no	—
568	C₄⋊C₄	—	4	4	no	—
569	C₄⋊C₄	—	4	4	no	—
570	C₄⋊C₄	—	4	4	no	—
571	C₂²⋊C₄	—	2	8	no	—
572	C₂²⋊C₄	—	2	8	no	—
573	C₂²⋊C₄	—	4	4	no	—
574	C₂²⋊C₄	—	4	4	no	—
575	C₂²⋊C₄	—	4	4	no	—
576	C₂²⋊C₄	C₂×C₅×Q₈	1	16	yes	—
577	C₄²	—	2	8	no	—
578	C₄²	—	4	4	no	—
579	C₄²	—	4	4	no	—
580	C₄²	—	4	4	no	—
581	C₂×C₅	—	4	4	no	—
582	C₂×C₅	#2522	1	8	yes	direct product factor
583	C₂×C₅	C₂×#529	1	1	yes	characteristic
584	C₂×C₅	C₂×#533	1	1	yes	characteristic
585	C₂×C₅	C₂×#540	1	1	yes	characteristic
586	C₂×C₅	C₂×#551	1	1	yes	characteristic
587	C₂×C₅	C₂×#553	1	1	yes	characteristic
588	C₂×C₅	C₂×#554	1	1	yes	characteristic
589	C₂×C₅	C₂×#554	1	1	yes	characteristic
590	C₂³	—	2	8	no	—
591	C₂³	—	2	8	no	—
592	C₂³	—	2	8	no	—
593	C₂³	—	2	2	no	—
594	C₂³	—	2	2	no	—
595	C₂³	—	2	2	no	—
596	C₂³	—	2	2	no	—
597	C₂³	—	2	2	no	—
598	C₂³	—	2	2	no	—
599	C₂³	—	4	4	no	—
600	C₂³	—	4	4	no	—
601	C₂³	—	4	4	no	—
602	C₂³	—	4	4	no	—
603	C₂³	C₂²×C₅×Q₈	1	4	yes	—
604	C₂³	C₂²×C₅×Q₈	1	2	yes	—
605	C₂³	C₂×C₅×#41	1	2	yes	—
606	C₂³	C₂×C₅×#41	1	2	yes	—
607	C₂³	C₂×C₅×#41	1	2	yes	—
608	C₂³	C₂×C₅×#41	1	2	yes	—
609	C₂³	C₅×H(5,𝔽₂)	1	2	yes	—
610	C₂³	C₅×H(5,𝔽₂)	1	2	yes	—
611	C₂³	C₂⁵×C₅	1	1	yes	characteristic
612	C₂×C₄	—	2	8	no	—
613	C₂×C₄	—	2	8	no	—
614	C₂×C₄	—	2	8	no	—
615	C₂×C₄	—	2	8	no	—
616	C₂×C₄	—	2	8	no	—
617	C₂×C₄	—	2	8	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	2	no	—
624	C₂×C₄	—	2	2	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	2	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	2	2	no	—
636	C₂×C₄	—	2	2	no	—
637	C₂×C₄	—	2	2	no	—
638	C₂×C₄	—	2	2	no	—
639	C₂×C₄	—	2	2	no	—
640	C₂×C₄	—	2	2	no	—
641	C₂×C₄	—	2	2	no	—
642	C₂×C₄	—	2	2	no	—
643	C₂×C₄	—	2	2	no	—
644	C₂×C₄	—	2	2	no	—
645	C₂×C₄	—	2	2	no	—
646	C₂×C₄	—	2	2	no	—
647	C₂×C₄	—	2	2	no	—
648	C₂×C₄	—	2	2	no	—
649	C₂×C₄	—	2	2	no	—
650	C₂×C₄	—	2	2	no	—
651	C₂×C₄	—	2	2	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	4	16	no	—
655	C₂×C₄	—	4	4	no	—
656	C₂×C₄	—	4	4	no	—
657	C₂×C₄	—	4	4	no	—
658	C₂×C₄	—	4	4	no	—
659	C₂×C₄	—	4	4	no	—
660	C₂×C₄	—	4	4	no	—
661	C₂×C₄	—	4	4	no	—
662	C₂×C₄	—	4	4	no	—
663	C₂×C₄	C₂²×C₅×Q₈	1	4	yes	—
664	C₂×C₄	C₂²×C₅×Q₈	1	4	yes	—
665	C₂×C₄	C₂×C₅×#41	1	4	yes	—
666	C₂×C₄	C₂×C₅×#41	1	4	yes	—
667	C₂×C₄	C₂×C₅×#41	1	4	yes	—
668	C₂×C₄	C₂×C₅×#41	1	4	yes	—
669	C₅	C₂×#2522	1	1	yes	characteristic, direct product factor, Sylow
670	C₂²	—	2	4	no	—
671	C₂²	—	2	4	no	—
672	C₂²	—	2	4	no	—
673	C₂²	—	4	16	no	—
674	C₂²	—	4	2	no	—
675	C₂²	—	4	2	no	—
676	C₂²	—	4	2	no	—
677	C₂²	—	4	2	no	—
678	C₂²	C₅×#529	1	4	yes	—
679	C₂²	C₅×#533	1	4	yes	—
680	C₂²	C₅×#540	1	4	yes	—
681	C₂²	C₅×#551	1	4	yes	—
682	C₂²	C₅×#553	1	4	yes	—
683	C₂²	C₅×#554	1	4	yes	—
684	C₂²	C₅×#554	1	4	yes	—
685	C₂²	C₂³×C₅×Q₈	1	1	yes	characteristic
686	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
687	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
688	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
689	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
690	C₂²	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
691	C₂²	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
692	C₄	—	2	4	no	—
693	C₄	—	2	4	no	—
694	C₄	—	2	4	no	—
695	C₄	—	2	4	no	—
696	C₄	—	2	4	no	—
697	C₄	—	2	4	no	—
698	C₄	—	4	2	no	—
699	C₄	—	4	2	no	—
700	C₄	—	4	2	no	—
701	C₄	—	4	2	no	—
702	C₄	—	4	2	no	—
703	C₄	—	4	2	no	—
704	C₄	—	4	2	no	—
705	C₄	—	4	2	no	—
706	C₂	—	4	4	no	—
707	C₂	C₅×#2522	1	8	yes	direct product factor
708	C₂	C₂×C₅×#529	1	1	yes	characteristic
709	C₂	C₂×C₅×#533	1	1	yes	characteristic
710	C₂	C₂×C₅×#540	1	1	yes	characteristic
711	C₂	C₂×C₅×#551	1	1	yes	characteristic
712	C₂	C₂×C₅×#553	1	1	yes	characteristic
713	C₂	C₂×C₅×#554	1	1	yes	characteristic
714	C₂	C₂×C₅×#554	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₂⁵×C₅	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#2522	C₂	direct (N × H)
3. C₂×C₅×#398	C₂	semidirect
4. C₂×C₅×#398	C₂	semidirect
6. C₂×C₅×#389	C₂	semidirect
7. C₂×C₅×#389	C₂	semidirect
10. C₂×C₅×#387	C₂	semidirect
11. C₂×C₅×#387	C₂	semidirect
12. C₂×C₅×#387	C₂	semidirect
15. C₂×C₄×C₅×C₄⋊C₄	C₂	semidirect
41. C₅×#398	C₂²	semidirect
42. C₅×#398	C₂²	semidirect
44. C₅×#389	C₂²	semidirect
45. C₅×#389	C₂²	semidirect
48. C₅×#387	C₂²	semidirect
49. C₅×#387	C₂²	semidirect
50. C₅×#387	C₂²	semidirect
53. C₄×C₅×C₄⋊C₄	C₂²	semidirect
67. C₂×#2522	C₅	direct (N × H)
143. #2522	C₂×C₅	direct (N × H)
145. C₂×#398	C₂×C₅	semidirect
146. C₂×#398	C₂×C₅	semidirect
148. C₂×#389	C₂×C₅	semidirect
149. C₂×#389	C₂×C₅	semidirect
152. C₂×#387	C₂×C₅	semidirect
153. C₂×#387	C₂×C₅	semidirect
154. C₂×#387	C₂×C₅	semidirect
157. C₂×C₄×C₄⋊C₄	C₂×C₅	semidirect
274. #398	C₂²×C₅	semidirect
275. #398	C₂²×C₅	semidirect
277. #389	C₂²×C₅	semidirect
278. #389	C₂²×C₅	semidirect
281. #387	C₂²×C₅	semidirect
282. #387	C₂²×C₅	semidirect
283. #387	C₂²×C₅	semidirect
286. C₄×C₄⋊C₄	C₂²×C₅	semidirect
582. C₂×C₅	#2522	direct (N × H)
669. C₅	C₂×#2522	direct (N × H)
707. C₂	C₅×#2522	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×#2522	2
C₂×C₅×#399	2
C₂×C₅×#398	2
C₂×C₅×#396	2
C₂×C₅×#389	2
C₂×C₅×#388	2
C₂×C₅×#387	2
C₂×C₅×#385	2
C₂×C₅×#380	2
C₂×C₄×C₅×C₄⋊C₄	2
C₂×C₄×C₅×C₂²⋊C₄	2
C₂×#2522	5

Subgroups — C2×C5×#2522 #11957486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2522 #11957486