Subgroups — C₅×#33834 #11956554

Order 1280.

702 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#508	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3287	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3280	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2250	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2220	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2215	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2211	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2206	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#2189	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#2186	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#2186	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#2184	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#2175	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#2168	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₅×#135	C₂²	1	1	yes	characteristic
17	C₂×C₅×#135	C₂²	1	1	yes	characteristic
18	C₂×C₅×#135	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
20	C₅×#550	—	2	1	no	—
21	C₅×#546	—	2	1	no	—
22	C₅×#509	—	2	4	no	—
23	C₅×#508	—	2	4	no	—
24	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₅×SD₃₂	C₂²	1	1	yes	characteristic
26	C₂×C₅×SD₃₂	C₂²	1	1	yes	characteristic
27	C₂×C₅×SD₃₂	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₅×SD₃₂	C₂²	1	1	yes	characteristic, complemented
29	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
30	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
31	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
32	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
33	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
34	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
35	C₅×#495	C₂²	1	1	yes	characteristic
36	C₅×#493	C₂²	1	1	yes	characteristic, complemented
37	C₅×#467	C₂²	1	1	yes	characteristic
38	C₅×#466	C₂²	1	1	yes	characteristic
39	C₅×#466	C₂²	1	1	yes	characteristic
40	C₅×#466	C₂²	1	1	yes	characteristic, complemented
41	C₅×#463	—	2	1	no	—
42	C₅×#460	—	2	1	no	—
43	C₅×#450	—	2	1	no	—
44	C₅×#449	—	2	1	no	—
45	C₅×#449	—	2	1	no	—
46	C₅×#449	—	2	1	no	—
47	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic, complemented
48	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
49	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
50	C₅×#366	C₂²	1	1	yes	characteristic, complemented
51	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
52	C₅×#358	C₂²	1	1	yes	characteristic, complemented
53	C₅×#358	C₂²	1	1	yes	characteristic, complemented
54	C₅×#357	C₂²	1	1	yes	characteristic
55	C₅×#357	C₂²	1	1	yes	characteristic
56	C₅×#357	C₂²	1	1	yes	characteristic, complemented
57	C₅×#357	C₂²	1	1	yes	characteristic, complemented
58	C₅×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
59	C₅×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
60	C₅×#346	C₂²	1	1	yes	characteristic, complemented
61	#33834	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×C₅×#41	—	2	1	no	—
63	C₂×C₅×#41	—	2	1	no	—
64	C₂×C₅×#41	—	2	1	no	—
65	C₂²×C₅×D₄	—	2	1	no	—
66	C₅×#135	—	2	2	no	—
67	C₅×#135	—	2	2	no	—
68	C₅×#135	—	2	2	no	—
69	C₅×#135	—	4	1	no	—
70	C₅×#135	—	4	1	no	—
71	C₅×#135	—	4	1	no	—
72	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
74	C₂×C₅×SD₁₆	—	2	1	no	—
75	C₂×C₅×SD₁₆	—	2	1	no	—
76	C₂×C₅×SD₁₆	—	2	1	no	—
77	C₂×C₅×SD₁₆	—	2	1	no	—
78	C₂×C₅×D₈	—	2	2	no	—
79	C₂×C₅×D₈	—	2	1	no	—
80	C₂×C₅×D₈	—	2	1	no	—
81	C₂×C₅×D₈	—	4	1	no	—
82	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
83	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
84	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
85	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
86	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
87	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
88	C₅×#127	—	2	1	no	—
89	C₅×#127	—	4	1	no	—
90	C₅×#124	—	2	1	no	—
91	C₅×#123	—	4	1	no	—
92	C₅×C₂²⋊Q₈	—	2	1	no	—
93	C₅×#121	—	2	1	no	—
94	C₅×#121	—	2	1	no	—
95	C₅×#121	—	2	1	no	—
96	C₅×#121	—	4	1	no	—
97	C₅×#121	—	4	1	no	—
98	C₅×#121	—	4	1	no	—
99	C₅×#120	—	4	1	no	—
100	C₄×C₅×Q₈	—	2	1	no	—
101	C₄×C₅×D₄	—	2	1	no	—
102	C₄×C₅×D₄	—	4	1	no	—
103	C₄×C₅×D₄	D₄	1	1	yes	characteristic
104	C₂×C₅×C₂²⋊C₄	—	4	1	no	—
105	C₅×Q₃₂	—	2	2	no	—
106	C₅×SD₃₂	—	2	2	no	—
107	C₅×SD₃₂	—	2	2	no	—
108	C₅×SD₃₂	—	4	1	no	—
109	C₅×SD₃₂	D₄	1	4	yes	complemented
110	C₅×D₁₆	—	2	2	no	—
111	C₅×D₁₆	—	2	2	no	—
112	C₅×D₁₆	—	2	2	no	—
113	C₅×D₁₆	—	4	1	no	—
114	C₅×M₃₂	—	2	2	no	—
115	C₂×C₅×C₁₆	—	2	2	no	—
116	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
117	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
118	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
119	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
120	C₅×#107	C₂³	1	1	yes	characteristic
121	C₅×#107	C₂³	1	1	yes	characteristic
122	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
123	C₅×#103	—	2	1	no	—
124	C₅×#103	—	2	1	no	—
125	C₅×#102	—	2	1	no	—
126	C₅×#102	—	2	1	no	—
127	C₅×#102	—	2	1	no	—
128	C₅×#102	—	2	1	no	—
129	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
130	C₅×C₂²⋊C₈	D₄	1	1	yes	characteristic
131	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
132	C₂×#509	C₂×C₅	1	1	yes	characteristic, complemented
133	C₂×#508	C₂×C₅	1	1	yes	characteristic, complemented
134	#3287	C₂×C₅	1	1	yes	characteristic, complemented
135	#3280	C₂×C₅	1	1	yes	characteristic, complemented
136	#2250	C₂×C₅	1	1	yes	characteristic, complemented
137	#2220	C₂×C₅	1	1	yes	characteristic, complemented
138	#2215	C₂×C₅	1	1	yes	characteristic, complemented
139	#2211	C₂×C₅	1	1	yes	characteristic, complemented
140	#2206	C₂×C₅	1	1	yes	characteristic, complemented
141	#2189	C₂×C₅	1	1	yes	characteristic, complemented
142	#2186	C₂×C₅	1	1	yes	characteristic, complemented
143	#2186	C₂×C₅	1	1	yes	characteristic, complemented
144	#2184	C₂×C₅	1	1	yes	characteristic, complemented
145	#2175	C₂×C₅	1	1	yes	characteristic, complemented
146	#2168	C₂×C₅	1	1	yes	characteristic, complemented
147	C₂⁴×C₅	—	4	1	no	—
148	C₅×#41	—	4	2	no	—
149	C₅×#41	—	4	2	no	—
150	C₅×#41	—	4	2	no	—
151	C₅×#41	—	8	1	no	—
152	C₅×#41	—	8	1	no	—
153	C₅×#41	—	8	1	no	—
154	C₂×C₅×Q₈	—	2	1	no	—
155	C₂×C₅×Q₈	—	2	1	no	—
156	C₂×C₅×D₄	—	2	1	no	—
157	C₂×C₅×D₄	—	2	1	no	—
158	C₂×C₅×D₄	—	2	1	no	—
159	C₂×C₅×D₄	—	2	1	no	—
160	C₂×C₅×D₄	—	4	2	no	—
161	C₂×C₅×D₄	—	4	1	no	—
162	C₂×C₅×D₄	—	4	1	no	—
163	C₂×C₅×D₄	—	4	1	no	—
164	C₂×C₅×D₄	—	4	1	no	—
165	C₂×C₅×D₄	—	4	1	no	—
166	C₂×C₅×D₄	—	4	1	no	—
167	C₂×C₅×D₄	—	4	1	no	—
168	C₂×C₅×D₄	—	4	1	no	—
169	C₂×C₅×D₄	—	4	1	no	—
170	C₂×C₅×D₄	—	8	1	no	—
171	C₂×C₅×D₄	D₈	1	1	yes	characteristic
172	C₂²×C₄×C₅	—	4	1	no	—
173	C₂²×C₄×C₅	—	4	1	no	—
174	C₂²×C₄×C₅	—	4	1	no	—
175	C₂²×C₄×C₅	—	4	1	no	—
176	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
178	C₅×Q₁₆	—	2	1	no	—
179	C₅×Q₁₆	—	2	1	no	—
180	C₅×Q₁₆	—	2	1	no	—
181	C₅×Q₁₆	C₂×D₄	1	2	yes	—
182	C₅×SD₁₆	—	4	2	no	—
183	C₅×SD₁₆	—	4	2	no	—
184	C₅×SD₁₆	—	4	2	no	—
185	C₅×SD₁₆	—	8	1	no	—
186	C₅×D₈	—	2	1	no	—
187	C₅×D₈	—	2	1	no	—
188	C₅×D₈	—	2	1	no	—
189	C₅×D₈	—	2	1	no	—
190	C₅×D₈	—	2	1	no	—
191	C₅×D₈	—	2	1	no	—
192	C₅×D₈	—	2	1	no	—
193	C₅×D₈	—	4	2	no	—
194	C₅×D₈	—	8	1	no	—
195	C₅×D₈	C₂×D₄	1	2	yes	—
196	C₂×C₅×C₈	—	2	1	no	—
197	C₂×C₅×C₈	—	2	1	no	—
198	C₂×C₅×C₈	—	2	1	no	—
199	C₂×C₅×C₈	—	2	1	no	—
200	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
201	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
202	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
203	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
204	C₅×C₄⋊C₄	—	2	1	no	—
205	C₅×C₄⋊C₄	—	2	1	no	—
206	C₅×C₄⋊C₄	—	4	1	no	—
207	C₅×C₄⋊C₄	—	4	1	no	—
208	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
209	C₅×C₂²⋊C₄	—	4	1	no	—
210	C₅×C₂²⋊C₄	—	4	1	no	—
211	C₅×C₂²⋊C₄	—	4	1	no	—
212	C₅×C₂²⋊C₄	—	4	1	no	—
213	C₅×C₂²⋊C₄	—	4	1	no	—
214	C₅×C₂²⋊C₄	—	4	1	no	—
215	C₅×C₂²⋊C₄	—	4	1	no	—
216	C₅×C₂²⋊C₄	D₈	1	1	yes	characteristic
217	C₅×C₂²⋊C₄	D₈	1	1	yes	characteristic
218	C₄²×C₅	—	4	1	no	—
219	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
220	C₅×C₁₆	—	2	1	no	—
221	C₅×C₁₆	—	2	1	no	—
222	C₅×C₁₆	—	2	1	no	—
223	C₅×C₁₆	C₂×D₄	1	2	yes	—
224	C₂×#135	C₂²×C₅	1	1	yes	characteristic
225	C₂×#135	C₂²×C₅	1	1	yes	characteristic
226	C₂×#135	C₂²×C₅	1	1	yes	characteristic, complemented
227	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
228	#550	—	2	1	no	—
229	#546	—	2	1	no	—
230	#509	—	2	4	no	—
231	#508	—	2	4	no	—
232	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic, complemented
233	C₂×SD₃₂	C₂²×C₅	1	1	yes	characteristic
234	C₂×SD₃₂	C₂²×C₅	1	1	yes	characteristic
235	C₂×SD₃₂	C₂²×C₅	1	1	yes	characteristic, complemented
236	C₂×SD₃₂	C₂²×C₅	1	1	yes	characteristic, complemented
237	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic
238	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic
239	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic
240	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic
241	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
242	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
243	#495	C₂²×C₅	1	1	yes	characteristic
244	#493	C₂²×C₅	1	1	yes	characteristic, complemented
245	#467	C₂²×C₅	1	1	yes	characteristic
246	#466	C₂²×C₅	1	1	yes	characteristic
247	#466	C₂²×C₅	1	1	yes	characteristic
248	#466	C₂²×C₅	1	1	yes	characteristic, complemented
249	#463	—	2	1	no	—
250	#460	—	2	1	no	—
251	#450	—	2	1	no	—
252	#449	—	2	1	no	—
253	#449	—	2	1	no	—
254	#449	—	2	1	no	—
255	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
256	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
257	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
258	#366	C₂²×C₅	1	1	yes	characteristic, complemented
259	C₄⋊C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
260	#358	C₂²×C₅	1	1	yes	characteristic, complemented
261	#358	C₂²×C₅	1	1	yes	characteristic, complemented
262	#357	C₂²×C₅	1	1	yes	characteristic
263	#357	C₂²×C₅	1	1	yes	characteristic
264	#357	C₂²×C₅	1	1	yes	characteristic, complemented
265	#357	C₂²×C₅	1	1	yes	characteristic, complemented
266	C₂²⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
267	C₂²⋊C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
268	#346	C₂²×C₅	1	1	yes	characteristic, complemented
269	C₂³×C₅	—	4	1	no	—
270	C₂³×C₅	—	4	1	no	—
271	C₂³×C₅	—	4	1	no	—
272	C₂³×C₅	—	4	1	no	—
273	C₂³×C₅	—	8	2	no	—
274	C₂³×C₅	—	8	2	no	—
275	C₂³×C₅	—	16	1	no	—
276	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
277	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
278	C₅×Q₈	—	2	2	no	—
279	C₅×Q₈	—	4	1	no	—
280	C₅×Q₈	—	4	1	no	—
281	C₅×Q₈	—	4	1	no	—
282	C₅×D₄	—	2	2	no	—
283	C₅×D₄	—	4	1	no	—
284	C₅×D₄	—	4	1	no	—
285	C₅×D₄	—	4	1	no	—
286	C₅×D₄	—	4	1	no	—
287	C₅×D₄	—	4	1	no	—
288	C₅×D₄	—	4	1	no	—
289	C₅×D₄	—	4	1	no	—
290	C₅×D₄	—	4	1	no	—
291	C₅×D₄	—	8	2	no	—
292	C₅×D₄	—	8	2	no	—
293	C₅×D₄	—	8	2	no	—
294	C₅×D₄	—	8	2	no	—
295	C₅×D₄	—	8	2	no	—
296	C₅×D₄	—	16	1	no	—
297	C₅×D₄	—	16	1	no	—
298	C₅×D₄	—	16	1	no	—
299	C₅×D₄	—	16	1	no	—
300	C₂×C₄×C₅	—	2	1	no	—
301	C₂×C₄×C₅	—	2	1	no	—
302	C₂×C₄×C₅	—	2	1	no	—
303	C₂×C₄×C₅	—	2	1	no	—
304	C₂×C₄×C₅	—	4	1	no	—
305	C₂×C₄×C₅	—	4	1	no	—
306	C₂×C₄×C₅	—	4	1	no	—
307	C₂×C₄×C₅	—	4	1	no	—
308	C₂×C₄×C₅	—	8	2	no	—
309	C₂×C₄×C₅	—	8	2	no	—
310	C₂×C₄×C₅	—	8	2	no	—
311	C₂×C₄×C₅	—	16	1	no	—
312	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
313	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
314	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
315	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
316	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
317	C₅×C₈	—	2	1	no	—
318	C₅×C₈	—	2	1	no	—
319	C₅×C₈	—	2	1	no	—
320	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
321	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
322	C₂×#41	—	2	1	no	—
323	C₂×#41	—	2	1	no	—
324	C₂×#41	—	2	1	no	—
325	C₂²×D₄	—	2	1	no	—
326	#135	—	2	2	no	—
327	#135	—	2	2	no	—
328	#135	—	2	2	no	—
329	#135	—	4	1	no	—
330	#135	—	4	1	no	—
331	#135	—	4	1	no	—
332	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
333	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
334	C₂×SD₁₆	—	2	1	no	—
335	C₂×SD₁₆	—	2	1	no	—
336	C₂×SD₁₆	—	2	1	no	—
337	C₂×SD₁₆	—	2	1	no	—
338	C₂×D₈	—	2	2	no	—
339	C₂×D₈	—	2	1	no	—
340	C₂×D₈	—	2	1	no	—
341	C₂×D₈	—	4	1	no	—
342	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
343	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
344	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
345	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
346	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
347	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
348	#127	—	2	1	no	—
349	#127	—	4	1	no	—
350	#124	—	2	1	no	—
351	#123	—	4	1	no	—
352	C₂²⋊Q₈	—	2	1	no	—
353	#121	—	2	1	no	—
354	#121	—	2	1	no	—
355	#121	—	2	1	no	—
356	#121	—	4	1	no	—
357	#121	—	4	1	no	—
358	#121	—	4	1	no	—
359	#120	—	4	1	no	—
360	C₄×Q₈	—	2	1	no	—
361	C₄×D₄	—	2	1	no	—
362	C₄×D₄	—	4	1	no	—
363	C₄×D₄	C₅×D₄	1	1	yes	characteristic
364	C₂×C₂²⋊C₄	—	4	1	no	—
365	Q₃₂	—	2	2	no	—
366	SD₃₂	—	2	2	no	—
367	SD₃₂	—	2	2	no	—
368	SD₃₂	—	4	1	no	—
369	SD₃₂	C₅×D₄	1	4	yes	complemented
370	D₁₆	—	2	2	no	—
371	D₁₆	—	2	2	no	—
372	D₁₆	—	2	2	no	—
373	D₁₆	—	4	1	no	—
374	M₃₂	—	2	2	no	—
375	C₂×C₁₆	—	2	2	no	—
376	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
377	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
378	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
379	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
380	#107	C₂³×C₅	1	1	yes	characteristic
381	#107	C₂³×C₅	1	1	yes	characteristic
382	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
383	#103	—	2	1	no	—
384	#103	—	2	1	no	—
385	#102	—	2	1	no	—
386	#102	—	2	1	no	—
387	#102	—	2	1	no	—
388	#102	—	2	1	no	—
389	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
390	C₂²⋊C₈	C₅×D₄	1	1	yes	characteristic
391	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
392	C₂²×C₅	—	2	1	no	—
393	C₂²×C₅	—	2	1	no	—
394	C₂²×C₅	—	2	1	no	—
395	C₂²×C₅	—	2	1	no	—
396	C₂²×C₅	—	2	1	no	—
397	C₂²×C₅	—	2	1	no	—
398	C₂²×C₅	—	4	2	no	—
399	C₂²×C₅	—	8	1	no	—
400	C₂²×C₅	—	8	1	no	—
401	C₂²×C₅	—	8	1	no	—
402	C₂²×C₅	—	8	1	no	—
403	C₂²×C₅	—	8	1	no	—
404	C₂²×C₅	—	8	1	no	—
405	C₂²×C₅	—	8	1	no	—
406	C₂²×C₅	—	8	1	no	—
407	C₂²×C₅	—	8	1	no	—
408	C₂²×C₅	—	8	1	no	—
409	C₂²×C₅	—	8	1	no	—
410	C₂²×C₅	—	16	4	no	—
411	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
412	C₄×C₅	—	2	1	no	—
413	C₄×C₅	—	2	1	no	—
414	C₄×C₅	—	2	1	no	—
415	C₄×C₅	—	2	1	no	—
416	C₄×C₅	—	4	2	no	—
417	C₄×C₅	—	8	1	no	—
418	C₄×C₅	—	8	1	no	—
419	C₄×C₅	—	8	1	no	—
420	C₄×C₅	D₄²	1	1	yes	characteristic
421	C₄×C₅	#576	1	1	yes	characteristic
422	C₂⁴	—	4	1	no	—
423	#41	—	4	2	no	—
424	#41	—	4	2	no	—
425	#41	—	4	2	no	—
426	#41	—	8	1	no	—
427	#41	—	8	1	no	—
428	#41	—	8	1	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	4	2	no	—
436	C₂×D₄	—	4	1	no	—
437	C₂×D₄	—	4	1	no	—
438	C₂×D₄	—	4	1	no	—
439	C₂×D₄	—	4	1	no	—
440	C₂×D₄	—	4	1	no	—
441	C₂×D₄	—	4	1	no	—
442	C₂×D₄	—	4	1	no	—
443	C₂×D₄	—	4	1	no	—
444	C₂×D₄	—	4	1	no	—
445	C₂×D₄	—	8	1	no	—
446	C₂×D₄	C₅×D₈	1	1	yes	characteristic
447	C₂²×C₄	—	4	1	no	—
448	C₂²×C₄	—	4	1	no	—
449	C₂²×C₄	—	4	1	no	—
450	C₂²×C₄	—	4	1	no	—
451	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
452	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
453	Q₁₆	—	2	1	no	—
454	Q₁₆	—	2	1	no	—
455	Q₁₆	—	2	1	no	—
456	Q₁₆	C₂×C₅×D₄	1	2	yes	—
457	SD₁₆	—	4	2	no	—
458	SD₁₆	—	4	2	no	—
459	SD₁₆	—	4	2	no	—
460	SD₁₆	—	8	1	no	—
461	D₈	—	2	1	no	—
462	D₈	—	2	1	no	—
463	D₈	—	2	1	no	—
464	D₈	—	2	1	no	—
465	D₈	—	2	1	no	—
466	D₈	—	2	1	no	—
467	D₈	—	2	1	no	—
468	D₈	—	4	2	no	—
469	D₈	—	8	1	no	—
470	D₈	C₂×C₅×D₄	1	2	yes	—
471	C₂×C₈	—	2	1	no	—
472	C₂×C₈	—	2	1	no	—
473	C₂×C₈	—	2	1	no	—
474	C₂×C₈	—	2	1	no	—
475	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
476	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
477	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
478	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	4	1	no	—
482	C₄⋊C₄	—	4	1	no	—
483	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
484	C₂²⋊C₄	—	4	1	no	—
485	C₂²⋊C₄	—	4	1	no	—
486	C₂²⋊C₄	—	4	1	no	—
487	C₂²⋊C₄	—	4	1	no	—
488	C₂²⋊C₄	—	4	1	no	—
489	C₂²⋊C₄	—	4	1	no	—
490	C₂²⋊C₄	—	4	1	no	—
491	C₂²⋊C₄	C₅×D₈	1	1	yes	characteristic
492	C₂²⋊C₄	C₅×D₈	1	1	yes	characteristic
493	C₄²	—	4	1	no	—
494	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
495	C₁₆	—	2	1	no	—
496	C₁₆	—	2	1	no	—
497	C₁₆	—	2	1	no	—
498	C₁₆	C₂×C₅×D₄	1	2	yes	—
499	C₂×C₅	—	4	1	no	—
500	C₂×C₅	—	4	1	no	—
501	C₂×C₅	—	8	2	no	—
502	C₂×C₅	—	16	1	no	—
503	C₂×C₅	—	16	1	no	—
504	C₂×C₅	—	16	1	no	—
505	C₂×C₅	D₄×D₈	1	1	yes	characteristic
506	C₂×C₅	#3414	1	1	yes	characteristic
507	C₂×C₅	#3415	1	1	yes	characteristic
508	C₂³	—	4	1	no	—
509	C₂³	—	4	1	no	—
510	C₂³	—	4	1	no	—
511	C₂³	—	4	1	no	—
512	C₂³	—	8	2	no	—
513	C₂³	—	8	2	no	—
514	C₂³	—	16	1	no	—
515	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
516	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
517	Q₈	—	2	2	no	—
518	Q₈	—	4	1	no	—
519	Q₈	—	4	1	no	—
520	Q₈	—	4	1	no	—
521	D₄	—	2	2	no	—
522	D₄	—	4	1	no	—
523	D₄	—	4	1	no	—
524	D₄	—	4	1	no	—
525	D₄	—	4	1	no	—
526	D₄	—	4	1	no	—
527	D₄	—	4	1	no	—
528	D₄	—	4	1	no	—
529	D₄	—	4	1	no	—
530	D₄	—	8	2	no	—
531	D₄	—	8	2	no	—
532	D₄	—	8	2	no	—
533	D₄	—	8	2	no	—
534	D₄	—	8	2	no	—
535	D₄	—	16	1	no	—
536	D₄	—	16	1	no	—
537	D₄	—	16	1	no	—
538	D₄	—	16	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	8	2	no	—
548	C₂×C₄	—	8	2	no	—
549	C₂×C₄	—	8	2	no	—
550	C₂×C₄	—	16	1	no	—
551	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
552	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
553	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
554	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
555	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
556	C₈	—	2	1	no	—
557	C₈	—	2	1	no	—
558	C₈	—	2	1	no	—
559	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
560	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
561	C₅	#33834	1	1	yes	characteristic, direct product factor, Sylow
562	C₂²	—	2	1	no	—
563	C₂²	—	2	1	no	—
564	C₂²	—	2	1	no	—
565	C₂²	—	2	1	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	1	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	8	1	no	—
570	C₂²	—	8	1	no	—
571	C₂²	—	8	1	no	—
572	C₂²	—	8	1	no	—
573	C₂²	—	8	1	no	—
574	C₂²	—	8	1	no	—
575	C₂²	—	8	1	no	—
576	C₂²	—	8	1	no	—
577	C₂²	—	8	1	no	—
578	C₂²	—	8	1	no	—
579	C₂²	—	8	1	no	—
580	C₂²	—	16	4	no	—
581	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
582	C₄	—	2	1	no	—
583	C₄	—	2	1	no	—
584	C₄	—	2	1	no	—
585	C₄	—	2	1	no	—
586	C₄	—	4	2	no	—
587	C₄	—	8	1	no	—
588	C₄	—	8	1	no	—
589	C₄	—	8	1	no	—
590	C₄	C₅×D₄²	1	1	yes	characteristic
591	C₄	C₅×#576	1	1	yes	characteristic
592	C₂	—	4	1	no	—
593	C₂	—	4	1	no	—
594	C₂	—	8	2	no	—
595	C₂	—	16	1	no	—
596	C₂	—	16	1	no	—
597	C₂	—	16	1	no	—
598	C₂	C₅×D₄×D₈	1	1	yes	characteristic
599	C₂	C₅×#3414	1	1	yes	characteristic
600	C₂	C₅×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄ ◁ C₅×C₈×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#509	C₂	semidirect
2. C₂×C₅×#508	C₂	semidirect
3. C₅×#3287	C₂	semidirect
4. C₅×#3280	C₂	semidirect
5. C₅×#2250	C₂	semidirect
6. C₅×#2220	C₂	semidirect
7. C₅×#2215	C₂	semidirect
8. C₅×#2211	C₂	semidirect
9. C₅×#2206	C₂	semidirect
10. C₅×#2189	C₂	semidirect
11. C₅×#2186	C₂	semidirect
12. C₅×#2186	C₂	semidirect
13. C₅×#2184	C₂	semidirect
14. C₅×#2175	C₂	semidirect
15. C₅×#2168	C₂	semidirect
18. C₂×C₅×#135	C₂²	semidirect
24. C₂×C₅×Q₃₂	C₂²	semidirect
27. C₂×C₅×SD₃₂	C₂²	semidirect
28. C₂×C₅×SD₃₂	C₂²	semidirect
34. C₂²×C₅×C₁₆	C₂²	semidirect
36. C₅×#493	C₂²	semidirect
40. C₅×#466	C₂²	semidirect
47. C₄×C₅×Q₁₆	C₂²	semidirect
50. C₅×#366	C₂²	semidirect
51. C₅×C₄⋊C₁₆	C₂²	semidirect
52. C₅×#358	C₂²	semidirect
53. C₅×#358	C₂²	semidirect
56. C₅×#357	C₂²	semidirect
57. C₅×#357	C₂²	semidirect
59. C₅×C₂²⋊C₁₆	C₂²	semidirect
60. C₅×#346	C₂²	semidirect
61. #33834	C₅	direct (N × H)
109. C₅×SD₃₂	D₄	semidirect
132. C₂×#509	C₂×C₅	semidirect
133. C₂×#508	C₂×C₅	semidirect
134. #3287	C₂×C₅	semidirect
135. #3280	C₂×C₅	semidirect
136. #2250	C₂×C₅	semidirect
137. #2220	C₂×C₅	semidirect
138. #2215	C₂×C₅	semidirect
139. #2211	C₂×C₅	semidirect
140. #2206	C₂×C₅	semidirect
141. #2189	C₂×C₅	semidirect
142. #2186	C₂×C₅	semidirect
143. #2186	C₂×C₅	semidirect
144. #2184	C₂×C₅	semidirect
145. #2175	C₂×C₅	semidirect
146. #2168	C₂×C₅	semidirect
226. C₂×#135	C₂²×C₅	semidirect
232. C₂×Q₃₂	C₂²×C₅	semidirect
235. C₂×SD₃₂	C₂²×C₅	semidirect
236. C₂×SD₃₂	C₂²×C₅	semidirect
242. C₂²×C₁₆	C₂²×C₅	semidirect
244. #493	C₂²×C₅	semidirect
248. #466	C₂²×C₅	semidirect
255. C₄×Q₁₆	C₂²×C₅	semidirect
258. #366	C₂²×C₅	semidirect
259. C₄⋊C₁₆	C₂²×C₅	semidirect
260. #358	C₂²×C₅	semidirect
261. #358	C₂²×C₅	semidirect
264. #357	C₂²×C₅	semidirect
265. #357	C₂²×C₅	semidirect
267. C₂²⋊C₁₆	C₂²×C₅	semidirect
268. #346	C₂²×C₅	semidirect
369. SD₃₂	C₅×D₄	semidirect
561. C₅	#33834	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#509	2
C₂×C₅×#508	2
C₅×#3287	2
C₅×#3280	2
C₅×#2250	2
C₅×#2220	2
C₅×#2215	2
C₅×#2211	2
C₅×#2206	2
C₅×#2189	2
C₅×#2186	2
C₅×#2184	2
C₅×#2175	2
C₅×#2168	2
#33834	5

Subgroups — C5×#33834 #11956554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33834 #11956554