Subgroups — C₅×#35277 #11957997

Order 1280.

2370 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#542	C₂	1	2	yes	maximal, complemented
2	C₅×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2858	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2857	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2856	C₂	1	1	yes	maximal, characteristic
6	C₅×#2640	C₂	1	2	yes	maximal, complemented
7	C₅×#2620	C₂	1	2	yes	maximal, complemented
8	C₅×#2512	C₂	1	4	yes	maximal, complemented
9	C₅×#2491	C₂	1	4	yes	maximal, complemented
10	C₅×#2452	C₂	1	4	yes	maximal, complemented
11	C₅×#2422	C₂	1	4	yes	maximal, complemented
12	C₅×#2326	C₂	1	2	yes	maximal, complemented
13	C₄×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅×#542	—	2	16	no	—
17	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₅×C₄⋊Q₈	C₂²	1	4	yes	—
20	C₂×C₅×#127	C₂²	1	1	yes	characteristic
21	C₂×C₅×#126	C₂²	1	4	yes	—
22	C₂×C₅×#124	C₂²	1	4	yes	—
23	C₂×C₅×#124	C₂²	1	2	yes	—
24	C₂×C₅×#124	C₂²	1	2	yes	—
25	C₂×C₅×#123	C₂²	1	4	yes	—
26	C₂×C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
28	C₂×C₅×#121	C₂²	1	2	yes	—
29	C₂×C₅×#121	C₂²	1	4	yes	—
30	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
31	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
32	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
33	C₂×C₅×#117	C₂²	1	1	yes	characteristic
34	C₂×C₅×#117	C₂²	1	1	yes	characteristic
35	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
36	C₅×#397	C₂²	1	8	yes	—
37	C₅×#396	C₂²	1	8	yes	—
38	C₅×#394	C₂²	1	8	yes	—
39	C₅×#393	C₂²	1	8	yes	—
40	C₅×#391	C₂²	1	2	yes	—
41	C₅×#391	C₂²	1	2	yes	—
42	C₅×#390	C₂²	1	4	yes	—
43	C₅×#390	C₂²	1	4	yes	—
44	C₅×#390	C₂²	1	4	yes	—
45	C₅×#390	C₂²	1	4	yes	—
46	C₅×#390	C₂²	1	2	yes	—
47	C₅×#390	C₂²	1	2	yes	—
48	C₅×#389	C₂²	1	4	yes	—
49	C₅×#389	C₂²	1	4	yes	—
50	C₅×#388	C₂²	1	8	yes	—
51	C₅×#385	C₂²	1	8	yes	—
52	C₅×#384	C₂²	1	4	yes	—
53	C₅×#384	C₂²	1	1	yes	characteristic
54	C₅×#384	C₂²	1	1	yes	characteristic
55	C₅×#384	C₂²	1	2	yes	—
56	C₅×#380	C₂²	1	4	yes	—
57	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
58	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
59	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
60	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
61	C₄×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
62	C₅×#376	C₂²	1	1	yes	characteristic
63	C₅×#376	C₂²	1	1	yes	characteristic
64	C₄³×C₅	C₂²	1	1	yes	characteristic
65	C₄³×C₅	C₂²	1	1	yes	characteristic
66	#35277	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂²×C₅×Q₈	C₂³	1	2	yes	—
68	C₂²×C₅×Q₈	C₂³	1	4	yes	—
69	C₂²×C₅×D₄	C₂³	1	4	yes	—
70	C₂²×C₅×D₄	C₂³	1	2	yes	—
71	C₂²×C₅×D₄	C₂³	1	4	yes	—
72	C₂³×C₄×C₅	C₂³	1	2	yes	—
73	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₅	C₂³	1	4	yes	—
75	C₅×C₄⋊Q₈	—	2	8	no	—
76	C₅×C₄⋊Q₈	—	2	4	no	—
77	C₅×C₄⋊Q₈	—	4	8	no	—
78	C₅×#127	—	2	4	no	—
79	C₅×#126	—	2	16	no	—
80	C₅×#124	—	2	16	no	—
81	C₅×#124	—	2	8	no	—
82	C₅×#124	—	2	8	no	—
83	C₅×#123	—	2	16	no	—
84	C₅×C₂²⋊Q₈	—	2	8	no	—
85	C₅×C₂²⋊Q₈	—	2	16	no	—
86	C₅×#121	—	2	8	no	—
87	C₅×#121	—	4	8	no	—
88	C₄×C₅×Q₈	—	2	8	no	—
89	C₄×C₅×D₄	—	2	8	no	—
90	C₄×C₅×D₄	—	4	8	no	—
91	C₅×#117	—	2	4	no	—
92	C₅×#117	D₄	1	8	yes	complemented
93	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
94	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
95	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
96	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
97	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
98	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
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	8	yes	—
103	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
104	C₂×C₅×C₄⋊C₄	C₂³	1	1	yes	characteristic
105	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
106	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
107	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
108	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
109	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
110	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
111	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
112	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
113	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
114	C₂×C₅×C₂²⋊C₄	C₂³	1	1	yes	characteristic
115	C₂×C₅×C₂²⋊C₄	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₅	—	2	4	no	—
117	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₅	C₂³	1	2	yes	—
119	C₂×C₄²×C₅	C₂³	1	2	yes	—
120	C₂×C₄²×C₅	C₂³	1	2	yes	—
121	C₂×C₄²×C₅	C₂³	1	2	yes	—
122	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
123	C₂×C₄²×C₅	C₂³	1	4	yes	—
124	C₂×C₄²×C₅	C₂³	1	2	yes	—
125	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
126	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
128	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
129	C₅×#95	C₂³	1	2	yes	—
130	C₅×#95	C₂³	1	2	yes	—
131	C₅×#95	C₂³	1	2	yes	—
132	C₅×#95	C₂³	1	2	yes	—
133	C₅×#95	C₂³	1	8	yes	—
134	C₅×#95	C₂³	1	8	yes	—
135	C₂×#542	C₂×C₅	1	2	yes	complemented
136	#2868	C₂×C₅	1	1	yes	characteristic, complemented
137	#2858	C₂×C₅	1	1	yes	characteristic, complemented
138	#2857	C₂×C₅	1	1	yes	characteristic, complemented
139	#2856	C₂×C₅	1	1	yes	characteristic
140	#2640	C₂×C₅	1	2	yes	complemented
141	#2620	C₂×C₅	1	2	yes	complemented
142	#2512	C₂×C₅	1	4	yes	complemented
143	#2491	C₂×C₅	1	4	yes	complemented
144	#2452	C₂×C₅	1	4	yes	complemented
145	#2422	C₂×C₅	1	4	yes	complemented
146	#2326	C₂×C₅	1	2	yes	complemented
147	C₄×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
148	C₄×#127	C₂×C₅	1	1	yes	characteristic, complemented
149	C₄×#117	C₂×C₅	1	1	yes	characteristic, complemented
150	C₂⁴×C₅	C₂⁴	1	4	yes	—
151	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
152	C₂×C₅×Q₈	—	2	4	no	—
153	C₂×C₅×Q₈	—	2	8	no	—
154	C₂×C₅×Q₈	—	2	4	no	—
155	C₂×C₅×Q₈	—	2	4	no	—
156	C₂×C₅×Q₈	—	4	8	no	—
157	C₂×C₅×D₄	—	2	8	no	—
158	C₂×C₅×D₄	—	2	4	no	—
159	C₂×C₅×D₄	—	2	4	no	—
160	C₂×C₅×D₄	—	2	4	no	—
161	C₂×C₅×D₄	—	2	8	no	—
162	C₂×C₅×D₄	—	2	8	no	—
163	C₂×C₅×D₄	—	2	8	no	—
164	C₂×C₅×D₄	—	4	8	no	—
165	C₂²×C₄×C₅	—	2	2	no	—
166	C₂²×C₄×C₅	—	2	2	no	—
167	C₂²×C₄×C₅	—	2	4	no	—
168	C₂²×C₄×C₅	—	2	4	no	—
169	C₂²×C₄×C₅	—	2	4	no	—
170	C₂²×C₄×C₅	—	2	8	no	—
171	C₂²×C₄×C₅	—	4	8	no	—
172	C₂²×C₄×C₅	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
180	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
181	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
183	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
185	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
186	C₅×C₄⋊C₄	—	2	4	no	—
187	C₅×C₄⋊C₄	—	2	4	no	—
188	C₅×C₄⋊C₄	—	2	8	no	—
189	C₅×C₄⋊C₄	—	2	8	no	—
190	C₅×C₄⋊C₄	—	2	8	no	—
191	C₅×C₄⋊C₄	—	2	4	no	—
192	C₅×C₄⋊C₄	—	2	8	no	—
193	C₅×C₄⋊C₄	—	2	8	no	—
194	C₅×C₄⋊C₄	—	2	8	no	—
195	C₅×C₄⋊C₄	—	4	8	no	—
196	C₅×C₄⋊C₄	C₂×D₄	1	4	yes	—
197	C₅×C₄⋊C₄	C₂×D₄	1	4	yes	—
198	C₅×C₂²⋊C₄	—	2	8	no	—
199	C₅×C₂²⋊C₄	—	2	8	no	—
200	C₅×C₂²⋊C₄	—	2	8	no	—
201	C₅×C₂²⋊C₄	—	2	8	no	—
202	C₅×C₂²⋊C₄	—	2	8	no	—
203	C₅×C₂²⋊C₄	—	2	4	no	—
204	C₅×C₂²⋊C₄	—	2	16	no	—
205	C₅×C₂²⋊C₄	—	4	8	no	—
206	C₅×C₂²⋊C₄	—	4	8	no	—
207	C₅×C₂²⋊C₄	C₂×D₄	1	4	yes	—
208	C₅×C₂²⋊C₄	C₂×D₄	1	4	yes	—
209	C₄²×C₅	—	2	8	no	—
210	C₄²×C₅	—	2	4	no	—
211	C₄²×C₅	—	2	2	no	—
212	C₄²×C₅	—	2	2	no	—
213	C₄²×C₅	—	2	4	no	—
214	C₄²×C₅	—	2	4	no	—
215	C₄²×C₅	—	2	4	no	—
216	C₄²×C₅	—	2	4	no	—
217	C₄²×C₅	C₂×D₄	1	4	yes	—
218	C₄²×C₅	C₂×D₄	1	4	yes	—
219	C₄²×C₅	#41	1	4	yes	—
220	C₄²×C₅	#41	1	4	yes	—
221	#542	—	2	16	no	—
222	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
223	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
224	C₂×C₄⋊Q₈	C₂²×C₅	1	4	yes	—
225	C₂×#127	C₂²×C₅	1	1	yes	characteristic
226	C₂×#126	C₂²×C₅	1	4	yes	—
227	C₂×#124	C₂²×C₅	1	2	yes	—
228	C₂×#124	C₂²×C₅	1	2	yes	—
229	C₂×#124	C₂²×C₅	1	4	yes	—
230	C₂×#123	C₂²×C₅	1	4	yes	—
231	C₂×C₂²⋊Q₈	C₂²×C₅	1	2	yes	—
232	C₂×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
233	C₂×#121	C₂²×C₅	1	2	yes	—
234	C₂×#121	C₂²×C₅	1	4	yes	—
235	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
236	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
237	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
238	C₂×#117	C₂²×C₅	1	1	yes	characteristic
239	C₂×#117	C₂²×C₅	1	1	yes	characteristic
240	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
241	#397	C₂²×C₅	1	8	yes	—
242	#396	C₂²×C₅	1	8	yes	—
243	#394	C₂²×C₅	1	8	yes	—
244	#393	C₂²×C₅	1	8	yes	—
245	#391	C₂²×C₅	1	2	yes	—
246	#391	C₂²×C₅	1	2	yes	—
247	#390	C₂²×C₅	1	2	yes	—
248	#390	C₂²×C₅	1	4	yes	—
249	#390	C₂²×C₅	1	2	yes	—
250	#390	C₂²×C₅	1	4	yes	—
251	#390	C₂²×C₅	1	4	yes	—
252	#390	C₂²×C₅	1	4	yes	—
253	#389	C₂²×C₅	1	4	yes	—
254	#389	C₂²×C₅	1	4	yes	—
255	#388	C₂²×C₅	1	8	yes	—
256	#385	C₂²×C₅	1	8	yes	—
257	#384	C₂²×C₅	1	1	yes	characteristic
258	#384	C₂²×C₅	1	1	yes	characteristic
259	#384	C₂²×C₅	1	2	yes	—
260	#384	C₂²×C₅	1	4	yes	—
261	#380	C₂²×C₅	1	4	yes	—
262	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
263	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
264	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
265	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
266	C₄×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
267	#376	C₂²×C₅	1	1	yes	characteristic
268	#376	C₂²×C₅	1	1	yes	characteristic
269	C₄³	C₂²×C₅	1	1	yes	characteristic
270	C₄³	C₂²×C₅	1	1	yes	characteristic
271	C₂³×C₅	—	2	1	no	—
272	C₂³×C₅	—	2	1	no	—
273	C₂³×C₅	—	2	2	no	—
274	C₂³×C₅	—	2	2	no	—
275	C₂³×C₅	—	2	4	no	—
276	C₂³×C₅	—	2	4	no	—
277	C₂³×C₅	—	2	4	no	—
278	C₂³×C₅	—	2	8	no	—
279	C₂³×C₅	—	2	8	no	—
280	C₂³×C₅	C₂²×D₄	1	2	yes	—
281	C₂³×C₅	C₂⁵	1	1	yes	characteristic
282	C₅×Q₈	—	4	8	no	—
283	C₅×Q₈	—	8	8	no	—
284	C₅×D₄	—	4	8	no	—
285	C₅×D₄	—	4	16	no	—
286	C₅×D₄	—	8	8	no	—
287	C₂×C₄×C₅	—	2	1	no	—
288	C₂×C₄×C₅	—	2	1	no	—
289	C₂×C₄×C₅	—	2	2	no	—
290	C₂×C₄×C₅	—	2	2	no	—
291	C₂×C₄×C₅	—	2	2	no	—
292	C₂×C₄×C₅	—	2	2	no	—
293	C₂×C₄×C₅	—	2	2	no	—
294	C₂×C₄×C₅	—	2	2	no	—
295	C₂×C₄×C₅	—	2	8	no	—
296	C₂×C₄×C₅	—	2	4	no	—
297	C₂×C₄×C₅	—	2	4	no	—
298	C₂×C₄×C₅	—	2	4	no	—
299	C₂×C₄×C₅	—	2	2	no	—
300	C₂×C₄×C₅	—	2	2	no	—
301	C₂×C₄×C₅	—	2	4	no	—
302	C₂×C₄×C₅	—	2	4	no	—
303	C₂×C₄×C₅	—	2	2	no	—
304	C₂×C₄×C₅	—	2	2	no	—
305	C₂×C₄×C₅	—	2	8	no	—
306	C₂×C₄×C₅	—	2	8	no	—
307	C₂×C₄×C₅	—	2	2	no	—
308	C₂×C₄×C₅	—	2	2	no	—
309	C₂×C₄×C₅	—	4	8	no	—
310	C₂×C₄×C₅	—	8	8	no	—
311	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
312	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
313	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
314	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
315	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
316	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
317	C₂×C₄×C₅	C₂×#41	1	4	yes	—
318	C₂×C₄×C₅	C₂×#41	1	2	yes	—
319	C₂×C₄×C₅	C₂×#41	1	2	yes	—
320	C₂×C₄×C₅	C₂×#41	1	4	yes	—
321	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
322	C₂×C₄×C₅	2₋¹⁺⁴	1	4	yes	—
323	C₂²×Q₈	C₂³×C₅	1	2	yes	—
324	C₂²×Q₈	C₂³×C₅	1	4	yes	—
325	C₂²×D₄	C₂³×C₅	1	2	yes	—
326	C₂²×D₄	C₂³×C₅	1	4	yes	—
327	C₂²×D₄	C₂³×C₅	1	4	yes	—
328	C₂³×C₄	C₂³×C₅	1	2	yes	—
329	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
330	C₂³×C₄	C₂³×C₅	1	4	yes	—
331	C₄⋊Q₈	—	2	8	no	—
332	C₄⋊Q₈	—	2	4	no	—
333	C₄⋊Q₈	—	4	8	no	—
334	#127	—	2	4	no	—
335	#126	—	2	16	no	—
336	#124	—	2	16	no	—
337	#124	—	2	8	no	—
338	#124	—	2	8	no	—
339	#123	—	2	16	no	—
340	C₂²⋊Q₈	—	2	8	no	—
341	C₂²⋊Q₈	—	2	16	no	—
342	#121	—	2	8	no	—
343	#121	—	4	8	no	—
344	C₄×Q₈	—	2	8	no	—
345	C₄×D₄	—	2	8	no	—
346	C₄×D₄	—	4	8	no	—
347	#117	—	2	4	no	—
348	#117	C₅×D₄	1	8	yes	complemented
349	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
350	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
351	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
352	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
353	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
354	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
355	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
356	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
357	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
358	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
359	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
360	C₂×C₄⋊C₄	C₂³×C₅	1	1	yes	characteristic
361	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
362	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
363	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
364	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
365	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
366	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
367	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
368	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
369	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
370	C₂×C₂²⋊C₄	C₂³×C₅	1	1	yes	characteristic
371	C₂×C₂²⋊C₄	C₂³×C₅	1	1	yes	characteristic
372	C₂×C₄²	—	2	4	no	—
373	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
374	C₂×C₄²	C₂³×C₅	1	2	yes	—
375	C₂×C₄²	C₂³×C₅	1	4	yes	—
376	C₂×C₄²	C₂³×C₅	1	2	yes	—
377	C₂×C₄²	C₂³×C₅	1	2	yes	—
378	C₂×C₄²	C₂³×C₅	1	2	yes	—
379	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
380	C₂×C₄²	C₂³×C₅	1	2	yes	—
381	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
382	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
383	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
384	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
385	#95	C₂³×C₅	1	2	yes	—
386	#95	C₂³×C₅	1	2	yes	—
387	#95	C₂³×C₅	1	2	yes	—
388	#95	C₂³×C₅	1	2	yes	—
389	#95	C₂³×C₅	1	8	yes	—
390	#95	C₂³×C₅	1	8	yes	—
391	C₂²×C₅	—	2	2	no	—
392	C₂²×C₅	—	2	2	no	—
393	C₂²×C₅	—	2	2	no	—
394	C₂²×C₅	—	4	2	no	—
395	C₂²×C₅	—	4	2	no	—
396	C₂²×C₅	—	4	4	no	—
397	C₂²×C₅	—	4	8	no	—
398	C₂²×C₅	—	4	4	no	—
399	C₂²×C₅	—	4	4	no	—
400	C₂²×C₅	—	4	8	no	—
401	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
402	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
403	C₂²×C₅	C₂²×#41	1	1	yes	characteristic
404	C₂²×C₅	C₂×H(5,𝔽₂)	1	2	yes	—
405	C₂²×C₅	C₂×2₋¹⁺⁴	1	2	yes	—
406	C₄×C₅	—	2	2	no	—
407	C₄×C₅	—	2	2	no	—
408	C₄×C₅	—	2	2	no	—
409	C₄×C₅	—	2	2	no	—
410	C₄×C₅	—	2	2	no	—
411	C₄×C₅	—	2	4	no	—
412	C₄×C₅	—	4	4	no	—
413	C₄×C₅	—	4	2	no	—
414	C₄×C₅	—	4	4	no	—
415	C₄×C₅	—	4	4	no	—
416	C₄×C₅	—	4	2	no	—
417	C₄×C₅	#532	1	4	yes	—
418	C₄×C₅	#542	1	8	yes	complemented
419	C₂⁴	C₂⁴×C₅	1	4	yes	—
420	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
421	C₂×Q₈	—	2	4	no	—
422	C₂×Q₈	—	2	8	no	—
423	C₂×Q₈	—	2	4	no	—
424	C₂×Q₈	—	2	4	no	—
425	C₂×Q₈	—	4	8	no	—
426	C₂×D₄	—	2	8	no	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	2	4	no	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	8	no	—
431	C₂×D₄	—	2	8	no	—
432	C₂×D₄	—	2	8	no	—
433	C₂×D₄	—	4	8	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	4	no	—
439	C₂²×C₄	—	2	8	no	—
440	C₂²×C₄	—	4	8	no	—
441	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
442	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
445	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
446	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
447	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
448	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
449	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
450	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
452	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
453	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
454	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	4	no	—
457	C₄⋊C₄	—	2	8	no	—
458	C₄⋊C₄	—	2	8	no	—
459	C₄⋊C₄	—	2	8	no	—
460	C₄⋊C₄	—	2	4	no	—
461	C₄⋊C₄	—	2	8	no	—
462	C₄⋊C₄	—	2	8	no	—
463	C₄⋊C₄	—	2	8	no	—
464	C₄⋊C₄	—	4	8	no	—
465	C₄⋊C₄	C₂×C₅×D₄	1	4	yes	—
466	C₄⋊C₄	C₂×C₅×D₄	1	4	yes	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	8	no	—
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	—	2	8	no	—
471	C₂²⋊C₄	—	2	8	no	—
472	C₂²⋊C₄	—	2	4	no	—
473	C₂²⋊C₄	—	2	16	no	—
474	C₂²⋊C₄	—	4	8	no	—
475	C₂²⋊C₄	—	4	8	no	—
476	C₂²⋊C₄	C₂×C₅×D₄	1	4	yes	—
477	C₂²⋊C₄	C₂×C₅×D₄	1	4	yes	—
478	C₄²	—	2	8	no	—
479	C₄²	—	2	4	no	—
480	C₄²	—	2	2	no	—
481	C₄²	—	2	2	no	—
482	C₄²	—	2	4	no	—
483	C₄²	—	2	4	no	—
484	C₄²	—	2	4	no	—
485	C₄²	—	2	4	no	—
486	C₄²	C₂×C₅×D₄	1	4	yes	—
487	C₄²	C₂×C₅×D₄	1	4	yes	—
488	C₄²	C₅×#41	1	4	yes	—
489	C₄²	C₅×#41	1	4	yes	—
490	C₂×C₅	—	4	2	no	—
491	C₂×C₅	—	8	4	no	—
492	C₂×C₅	C₂×#532	1	1	yes	characteristic
493	C₂×C₅	C₂×#542	1	2	yes	—
494	C₂×C₅	#3498	1	1	yes	characteristic
495	C₂×C₅	#3499	1	1	yes	characteristic
496	C₂×C₅	#3511	1	2	yes	—
497	C₂³	—	2	1	no	—
498	C₂³	—	2	1	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	2	4	no	—
502	C₂³	—	2	4	no	—
503	C₂³	—	2	4	no	—
504	C₂³	—	2	8	no	—
505	C₂³	—	2	8	no	—
506	C₂³	C₂²×C₅×D₄	1	2	yes	—
507	C₂³	C₂⁵×C₅	1	1	yes	characteristic
508	Q₈	—	4	8	no	—
509	Q₈	—	8	8	no	—
510	D₄	—	4	8	no	—
511	D₄	—	4	16	no	—
512	D₄	—	8	8	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	8	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	8	no	—
532	C₂×C₄	—	2	8	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	4	8	no	—
536	C₂×C₄	—	8	8	no	—
537	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
538	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
539	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
540	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
541	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
542	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
543	C₂×C₄	C₂×C₅×#41	1	4	yes	—
544	C₂×C₄	C₂×C₅×#41	1	2	yes	—
545	C₂×C₄	C₂×C₅×#41	1	2	yes	—
546	C₂×C₄	C₂×C₅×#41	1	4	yes	—
547	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
548	C₂×C₄	C₅×2₋¹⁺⁴	1	4	yes	—
549	C₅	#35277	1	1	yes	characteristic, direct product factor, Sylow
550	C₂²	—	2	2	no	—
551	C₂²	—	2	2	no	—
552	C₂²	—	2	2	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	4	2	no	—
555	C₂²	—	4	4	no	—
556	C₂²	—	4	8	no	—
557	C₂²	—	4	4	no	—
558	C₂²	—	4	4	no	—
559	C₂²	—	4	8	no	—
560	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
561	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
562	C₂²	C₂²×C₅×#41	1	1	yes	characteristic
563	C₂²	C₂×C₅×H(5,𝔽₂)	1	2	yes	—
564	C₂²	C₂×C₅×2₋¹⁺⁴	1	2	yes	—
565	C₄	—	2	2	no	—
566	C₄	—	2	2	no	—
567	C₄	—	2	2	no	—
568	C₄	—	2	2	no	—
569	C₄	—	2	2	no	—
570	C₄	—	2	4	no	—
571	C₄	—	4	4	no	—
572	C₄	—	4	2	no	—
573	C₄	—	4	4	no	—
574	C₄	—	4	4	no	—
575	C₄	—	4	2	no	—
576	C₄	C₅×#532	1	4	yes	—
577	C₄	C₅×#542	1	8	yes	complemented
578	C₂	—	4	2	no	—
579	C₂	—	8	4	no	—
580	C₂	C₂×C₅×#532	1	1	yes	characteristic
581	C₂	C₂×C₅×#542	1	2	yes	—
582	C₂	C₅×#3498	1	1	yes	characteristic
583	C₂	C₅×#3499	1	1	yes	characteristic
584	C₂	C₅×#3511	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₅
prime radical	C₅×#2856	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×#542	C₂	semidirect
2. C₅×#2868	C₂	semidirect
3. C₅×#2858	C₂	semidirect
4. C₅×#2857	C₂	semidirect
6. C₅×#2640	C₂	semidirect
7. C₅×#2620	C₂	semidirect
8. C₅×#2512	C₂	semidirect
9. C₅×#2491	C₂	semidirect
10. C₅×#2452	C₂	semidirect
11. C₅×#2422	C₂	semidirect
12. C₅×#2326	C₂	semidirect
13. C₄×C₅×C₄⋊Q₈	C₂	semidirect
14. C₄×C₅×#127	C₂	semidirect
15. C₄×C₅×#117	C₂	semidirect
66. #35277	C₅	direct (N × H)
92. C₅×#117	D₄	semidirect
135. C₂×#542	C₂×C₅	semidirect
136. #2868	C₂×C₅	semidirect
137. #2858	C₂×C₅	semidirect
138. #2857	C₂×C₅	semidirect
140. #2640	C₂×C₅	semidirect
141. #2620	C₂×C₅	semidirect
142. #2512	C₂×C₅	semidirect
143. #2491	C₂×C₅	semidirect
144. #2452	C₂×C₅	semidirect
145. #2422	C₂×C₅	semidirect
146. #2326	C₂×C₅	semidirect
147. C₄×C₄⋊Q₈	C₂×C₅	semidirect
148. C₄×#127	C₂×C₅	semidirect
149. C₄×#117	C₂×C₅	semidirect
348. #117	C₅×D₄	semidirect
418. C₄×C₅	#542	semidirect
549. C₅	#35277	direct (N × H)
577. C₄	C₅×#542	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#542	2
C₅×#2868	2
C₅×#2858	2
C₅×#2857	2
C₅×#2856	2
C₅×#2640	2
C₅×#2620	2
C₅×#2512	2
C₅×#2491	2
C₅×#2452	2
C₅×#2422	2
C₅×#2326	2
C₄×C₅×C₄⋊Q₈	2
C₄×C₅×#127	2
C₄×C₅×#117	2
#35277	5

Subgroups — C5×#35277 #11957997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#35277 #11957997