Subgroups — C₅×#68237 #14282156

Order 1920.

492 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#239	C₂	1	1	yes	maximal, characteristic
2	C₅×#4835	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4685	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4535	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4483	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1488	—	3	1	no	maximal, Hall
9	C₂×C₅×#239	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₅×Q₈×Dic₃	C₄	1	1	yes	characteristic
12	C₅×#893	—	2	1	no	—
13	C₅×#893	C₄	1	1	yes	characteristic
14	C₅×#892	—	2	1	no	—
15	C₅×#892	C₄	1	1	yes	characteristic
16	C₄×C₅×D₁₂	—	2	1	no	—
17	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
18	C₄²×C₅×D₃	—	2	1	no	—
19	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
20	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
23	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
24	C₅×#815	—	2	1	no	—
25	C₅×Dic₃⋊C₈	—	2	1	no	—
26	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
28	#68237	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₄×C₅×#41	—	3	1	no	—
30	C₅×#432	—	3	1	no	—
31	C₅×#431	—	3	1	no	—
32	C₅×Q₈⋊C₈	—	3	1	no	—
33	C₅×Q₈⋊C₈	D₃	1	1	yes	characteristic, complemented
34	C₅×D₄⋊C₈	—	3	1	no	—
35	C₅×D₄⋊C₈	—	3	1	no	—
36	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
37	C₅×#239	—	2	2	no	—
38	C₅×#239	—	2	2	no	—
39	C₂×C₅×D₁₂	—	2	1	no	—
40	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅×D₃	—	2	1	no	—
42	C₂×C₄×C₅×D₃	—	2	1	no	—
43	C₂×C₄×C₅×D₃	—	2	1	no	—
44	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
46	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
47	C₂×C₃×C₅×C₈	—	2	1	no	—
48	C₂×C₃×C₅×C₈	—	2	1	no	—
49	C₃×C₅×C₄⋊C₄	—	2	1	no	—
50	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₃×C₄²×C₅	—	2	1	no	—
52	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
54	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
55	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
56	C₅×C₄⋊Dic₃	—	2	1	no	—
57	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
58	C₄×C₅×Dic₃	—	2	1	no	—
59	C₄×C₅×Dic₃	—	2	1	no	—
60	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
61	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
62	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
63	C₂×C₅×C₃⋊C₈	—	2	1	no	—
64	C₂×C₅×C₃⋊C₈	—	2	1	no	—
65	C₄×#239	C₂×C₅	1	1	yes	characteristic
66	#4835	C₂×C₅	1	1	yes	characteristic, complemented
67	#4685	C₂×C₅	1	1	yes	characteristic, complemented
68	C₃×Q₈⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
69	#4535	C₂×C₅	1	1	yes	characteristic, complemented
70	#4483	C₂×C₅	1	1	yes	characteristic, complemented
71	#4459	C₂×C₅	1	1	yes	characteristic, complemented
72	C₂×C₅×#41	—	3	1	no	—
73	C₄×C₅×Q₈	—	3	1	no	—
74	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
75	C₄×C₅×D₄	—	3	1	no	—
76	C₄×C₅×D₄	—	3	1	no	—
77	C₄×C₅×D₄	—	6	1	no	—
78	C₄×C₅×D₄	—	6	1	no	—
79	C₅×#117	—	3	1	no	—
80	C₅×#117	—	6	1	no	—
81	C₂×C₄²×C₅	—	3	1	no	—
82	C₂×C₄²×C₅	—	6	1	no	—
83	C₅×C₄⋊C₈	—	3	1	no	—
84	C₅×C₄⋊C₈	—	6	1	no	—
85	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
86	C₅×C₂²⋊C₈	—	6	1	no	—
87	C₅×C₂²⋊C₈	—	6	1	no	—
88	C₅×#97	—	6	1	no	—
89	C₄×C₅×C₈	—	3	1	no	—
90	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
91	#1488	—	3	1	no	Sylow
92	C₂²×C₅×D₃	—	2	1	no	—
93	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₃×C₅×Q₈	—	2	1	no	—
95	C₃×C₅×Q₈	M₁₆	1	2	yes	—
96	C₂×C₃×C₄×C₅	—	2	1	no	—
97	C₂×C₃×C₄×C₅	—	2	1	no	—
98	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
101	C₂×C₅×Dic₃	—	2	1	no	—
102	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
105	C₅×D₁₂	—	2	4	no	—
106	C₅×D₁₂	—	2	1	no	—
107	C₅×D₁₂	M₁₆	1	2	yes	—
108	C₄×C₅×D₃	—	2	4	no	—
109	C₄×C₅×D₃	—	2	1	no	—
110	C₄×C₅×D₃	—	2	1	no	—
111	C₄×C₅×D₃	—	2	1	no	—
112	C₄×C₅×D₃	—	2	1	no	—
113	C₄×C₅×D₃	—	4	1	no	—
114	C₄×C₅×D₃	—	4	2	no	—
115	C₄×C₅×D₃	—	4	1	no	—
116	C₄×C₅×D₃	—	4	1	no	—
117	C₃×C₅×C₈	—	4	1	no	—
118	C₃×C₅×C₈	—	4	1	no	—
119	C₅×C₃⋊C₈	—	4	1	no	—
120	C₅×C₃⋊C₈	—	4	1	no	—
121	C₂×#239	C₄×C₅	1	1	yes	characteristic
122	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
123	Q₈×Dic₃	C₄×C₅	1	1	yes	characteristic
124	#893	—	2	1	no	—
125	#893	C₄×C₅	1	1	yes	characteristic
126	#892	—	2	1	no	—
127	#892	C₄×C₅	1	1	yes	characteristic
128	C₄×D₁₂	—	2	1	no	—
129	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
130	C₄²×D₃	—	2	1	no	—
131	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
132	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
133	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
134	(C₂×D₃)⋊C₈	—	2	1	no	—
135	(C₂×D₃)⋊C₈	—	2	1	no	—
136	#815	—	2	1	no	—
137	Dic₃⋊C₈	—	2	1	no	—
138	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
139	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
140	C₅×#41	—	6	2	no	—
141	C₅×#41	—	6	2	no	—
142	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
143	C₂×C₅×D₄	—	3	1	no	—
144	C₂×C₅×D₄	—	6	1	no	—
145	C₂²×C₄×C₅	—	3	1	no	—
146	C₂²×C₄×C₅	—	3	1	no	—
147	C₂²×C₄×C₅	—	3	1	no	—
148	C₂²×C₄×C₅	—	6	1	no	—
149	C₂²×C₄×C₅	—	6	1	no	—
150	C₂²×C₄×C₅	—	6	1	no	—
151	C₂×C₅×C₈	—	2	1	no	—
152	C₂×C₅×C₈	—	2	1	no	—
153	C₂×C₅×C₈	—	6	1	no	—
154	C₂×C₅×C₈	—	6	1	no	—
155	C₅×C₄⋊C₄	—	2	1	no	—
156	C₅×C₄⋊C₄	—	3	1	no	—
157	C₅×C₄⋊C₄	—	6	1	no	—
158	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
159	C₅×C₂²⋊C₄	—	6	1	no	—
160	C₅×C₂²⋊C₄	—	6	1	no	—
161	C₅×C₂²⋊C₄	—	6	1	no	—
162	C₄²×C₅	—	2	1	no	—
163	C₄²×C₅	—	3	1	no	—
164	C₄²×C₅	—	3	1	no	—
165	C₄²×C₅	—	3	1	no	—
166	C₄²×C₅	—	6	1	no	—
167	C₄²×C₅	—	6	1	no	—
168	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
169	C₄×#41	—	3	1	no	—
170	#432	—	3	1	no	—
171	#431	—	3	1	no	—
172	Q₈⋊C₈	—	3	1	no	—
173	Q₈⋊C₈	C₅×D₃	1	1	yes	characteristic, complemented
174	D₄⋊C₈	—	3	1	no	—
175	D₄⋊C₈	—	3	1	no	—
176	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₅×D₃	—	2	1	no	—
178	C₂×C₅×D₃	—	2	2	no	—
179	C₂×C₅×D₃	—	2	1	no	—
180	C₂×C₅×D₃	—	2	1	no	—
181	C₂×C₅×D₃	—	4	1	no	—
182	C₂×C₅×D₃	—	4	1	no	—
183	C₃×C₄×C₅	—	2	2	no	—
184	C₃×C₄×C₅	—	2	1	no	—
185	C₃×C₄×C₅	—	4	1	no	—
186	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
187	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
188	C₃×C₄×C₅	#135	1	1	yes	characteristic
189	C₃×C₄×C₅	#135	1	1	yes	characteristic
190	C₅×Dic₃	—	2	1	no	—
191	C₅×Dic₃	—	2	1	no	—
192	C₅×Dic₃	—	2	1	no	—
193	C₅×Dic₃	—	4	1	no	—
194	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
195	#239	—	2	2	no	—
196	#239	—	2	2	no	—
197	C₂×D₁₂	—	2	1	no	—
198	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
199	C₂×C₄×D₃	—	2	1	no	—
200	C₂×C₄×D₃	—	2	1	no	—
201	C₂×C₄×D₃	—	2	1	no	—
202	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
203	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
204	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
205	C₂×C₃×C₈	—	2	1	no	—
206	C₂×C₃×C₈	—	2	1	no	—
207	C₃×C₄⋊C₄	—	2	1	no	—
208	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
209	C₃×C₄²	—	2	1	no	—
210	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
211	(C₂×D₃)⋊C₄	—	2	1	no	—
212	(C₂×D₃)⋊C₄	—	2	1	no	—
213	(C₂×D₃)⋊C₄	—	2	1	no	—
214	C₄⋊Dic₃	—	2	1	no	—
215	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
216	C₄×Dic₃	—	2	1	no	—
217	C₄×Dic₃	—	2	1	no	—
218	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
219	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
220	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
221	C₂×C₃⋊C₈	—	2	1	no	—
222	C₂×C₃⋊C₈	—	2	1	no	—
223	C₂³×C₅	—	3	1	no	—
224	C₂³×C₅	—	6	1	no	—
225	C₅×Q₈	—	2	1	no	—
226	C₅×Q₈	ΓL(1,𝔽₂₅)	1	2	yes	—
227	C₅×D₄	—	3	2	no	—
228	C₅×D₄	—	6	4	no	—
229	C₅×D₄	—	6	1	no	—
230	C₂×C₄×C₅	—	2	1	no	—
231	C₂×C₄×C₅	—	2	1	no	—
232	C₂×C₄×C₅	—	3	1	no	—
233	C₂×C₄×C₅	—	3	1	no	—
234	C₂×C₄×C₅	—	3	1	no	—
235	C₂×C₄×C₅	—	6	4	no	—
236	C₂×C₄×C₅	—	6	1	no	—
237	C₂×C₄×C₅	—	6	1	no	—
238	C₂×C₄×C₅	—	6	1	no	—
239	C₂×C₄×C₅	—	6	1	no	—
240	C₂×C₄×C₅	—	6	1	no	—
241	C₂×C₄×C₅	—	12	1	no	—
242	C₂×C₄×C₅	—	12	2	no	—
243	C₂×C₄×C₅	—	12	1	no	—
244	C₂×C₄×C₅	—	12	1	no	—
245	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
246	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
247	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
248	C₅×C₈	—	4	1	no	—
249	C₅×C₈	—	4	1	no	—
250	C₅×C₈	—	12	1	no	—
251	C₅×C₈	—	12	1	no	—
252	C₂×#41	—	3	1	no	—
253	C₄×Q₈	—	3	1	no	—
254	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
255	C₄×D₄	—	3	1	no	—
256	C₄×D₄	—	3	1	no	—
257	C₄×D₄	—	6	1	no	—
258	C₄×D₄	—	6	1	no	—
259	#117	—	3	1	no	—
260	#117	—	6	1	no	—
261	C₂×C₄²	—	3	1	no	—
262	C₂×C₄²	—	6	1	no	—
263	C₄⋊C₈	—	3	1	no	—
264	C₄⋊C₈	—	6	1	no	—
265	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
266	C₂²⋊C₈	—	6	1	no	—
267	C₂²⋊C₈	—	6	1	no	—
268	#97	—	6	1	no	—
269	C₄×C₈	—	3	1	no	—
270	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
271	C₂×C₃×C₅	#407	1	1	yes	characteristic
272	C₂×C₃×C₅	#416	1	1	yes	characteristic
273	C₂×C₃×C₅	#421	1	1	yes	characteristic
274	C₅×D₃	—	4	1	no	—
275	C₅×D₃	—	4	2	no	—
276	C₂²×D₃	—	2	1	no	—
277	C₂²×D₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
278	C₃×Q₈	—	2	1	no	—
279	C₃×Q₈	C₅×M₁₆	1	2	yes	—
280	C₂×C₃×C₄	—	2	1	no	—
281	C₂×C₃×C₄	—	2	1	no	—
282	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
283	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
284	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
285	C₂×Dic₃	—	2	1	no	—
286	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
287	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
288	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
289	D₁₂	—	2	4	no	—
290	D₁₂	—	2	1	no	—
291	D₁₂	C₅×M₁₆	1	2	yes	—
292	C₄×D₃	—	2	4	no	—
293	C₄×D₃	—	2	1	no	—
294	C₄×D₃	—	2	1	no	—
295	C₄×D₃	—	2	1	no	—
296	C₄×D₃	—	2	1	no	—
297	C₄×D₃	—	4	1	no	—
298	C₄×D₃	—	4	2	no	—
299	C₄×D₃	—	4	1	no	—
300	C₄×D₃	—	4	1	no	—
301	C₃×C₈	—	4	1	no	—
302	C₃×C₈	—	4	1	no	—
303	C₃⋊C₈	—	4	1	no	—
304	C₃⋊C₈	—	4	1	no	—
305	C₂²×C₅	—	6	2	no	—
306	C₂²×C₅	—	6	1	no	—
307	C₂²×C₅	—	6	1	no	—
308	C₂²×C₅	—	6	1	no	—
309	C₂²×C₅	—	12	1	no	—
310	C₂²×C₅	—	12	1	no	—
311	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
312	C₄×C₅	—	2	2	no	—
313	C₄×C₅	—	2	1	no	—
314	C₄×C₅	—	4	1	no	—
315	C₄×C₅	—	6	1	no	—
316	C₄×C₅	—	6	1	no	—
317	C₄×C₅	—	6	1	no	—
318	C₄×C₅	—	12	1	no	—
319	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
320	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
321	C₄×C₅	#916	1	1	yes	characteristic
322	C₄×C₅	#919	1	1	yes	characteristic
323	#41	—	6	2	no	—
324	#41	—	6	2	no	—
325	C₂×Q₈	C₄×C₅×D₃	1	1	yes	characteristic
326	C₂×D₄	—	3	1	no	—
327	C₂×D₄	—	6	1	no	—
328	C₂²×C₄	—	3	1	no	—
329	C₂²×C₄	—	3	1	no	—
330	C₂²×C₄	—	3	1	no	—
331	C₂²×C₄	—	6	1	no	—
332	C₂²×C₄	—	6	1	no	—
333	C₂²×C₄	—	6	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	6	1	no	—
337	C₂×C₈	—	6	1	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	3	1	no	—
340	C₄⋊C₄	—	6	1	no	—
341	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
342	C₂²⋊C₄	—	6	1	no	—
343	C₂²⋊C₄	—	6	1	no	—
344	C₂²⋊C₄	—	6	1	no	—
345	C₄²	—	2	1	no	—
346	C₄²	—	3	1	no	—
347	C₄²	—	3	1	no	—
348	C₄²	—	3	1	no	—
349	C₄²	—	6	1	no	—
350	C₄²	—	6	1	no	—
351	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
352	C₃×C₅	#1488	1	1	yes	characteristic, complemented, Hall
353	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
354	C₂×D₃	—	2	1	no	—
355	C₂×D₃	—	2	2	no	—
356	C₂×D₃	—	2	1	no	—
357	C₂×D₃	—	2	1	no	—
358	C₂×D₃	—	4	1	no	—
359	C₂×D₃	—	4	1	no	—
360	C₃×C₄	—	2	2	no	—
361	C₃×C₄	—	2	1	no	—
362	C₃×C₄	—	4	1	no	—
363	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
364	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
365	C₃×C₄	C₅×#135	1	1	yes	characteristic
366	C₃×C₄	C₅×#135	1	1	yes	characteristic
367	Dic₃	—	2	1	no	—
368	Dic₃	—	2	1	no	—
369	Dic₃	—	2	1	no	—
370	Dic₃	—	4	1	no	—
371	C₂×C₅	—	12	1	no	—
372	C₂×C₅	—	12	2	no	—
373	C₂×C₅	#4726	1	1	yes	characteristic
374	C₂×C₅	#4804	1	1	yes	characteristic
375	C₂×C₅	#4821	1	1	yes	characteristic
376	C₂³	—	3	1	no	—
377	C₂³	—	6	1	no	—
378	Q₈	—	2	1	no	—
379	Q₈	C₅×ΓL(1,𝔽₂₅)	1	2	yes	—
380	D₄	—	3	2	no	—
381	D₄	—	6	4	no	—
382	D₄	—	6	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	3	1	no	—
386	C₂×C₄	—	3	1	no	—
387	C₂×C₄	—	3	1	no	—
388	C₂×C₄	—	6	4	no	—
389	C₂×C₄	—	6	1	no	—
390	C₂×C₄	—	6	1	no	—
391	C₂×C₄	—	6	1	no	—
392	C₂×C₄	—	6	1	no	—
393	C₂×C₄	—	6	1	no	—
394	C₂×C₄	—	12	1	no	—
395	C₂×C₄	—	12	2	no	—
396	C₂×C₄	—	12	1	no	—
397	C₂×C₄	—	12	1	no	—
398	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
399	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
400	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
401	C₈	—	4	1	no	—
402	C₈	—	4	1	no	—
403	C₈	—	12	1	no	—
404	C₈	—	12	1	no	—
405	C₂×C₃	C₅×#407	1	1	yes	characteristic
406	C₂×C₃	C₅×#416	1	1	yes	characteristic
407	C₂×C₃	C₅×#421	1	1	yes	characteristic
408	D₃	—	4	1	no	—
409	D₃	—	4	2	no	—
410	C₅	#68237	1	1	yes	characteristic, direct product factor, Sylow
411	C₂²	—	6	2	no	—
412	C₂²	—	6	1	no	—
413	C₂²	—	6	1	no	—
414	C₂²	—	6	1	no	—
415	C₂²	—	12	1	no	—
416	C₂²	—	12	1	no	—
417	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
418	C₄	—	2	2	no	—
419	C₄	—	2	1	no	—
420	C₄	—	4	1	no	—
421	C₄	—	6	1	no	—
422	C₄	—	6	1	no	—
423	C₄	—	6	1	no	—
424	C₄	—	12	1	no	—
425	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
426	C₄	C₅×D₃×M₁₆	1	1	yes	characteristic
427	C₄	C₅×#916	1	1	yes	characteristic
428	C₄	C₅×#919	1	1	yes	characteristic
429	C₃	C₅×#1488	1	1	yes	characteristic, complemented, Sylow
430	C₂	—	12	1	no	—
431	C₂	—	12	2	no	—
432	C₂	C₅×#4726	1	1	yes	characteristic
433	C₂	C₅×#4804	1	1	yes	characteristic
434	C₂	C₅×#4821	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×Q₈⋊C₈	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₅×#239	C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×Q₈⋊C₈	D₃
hypocenter	C₃	C₅×#1488
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₅×#4835	C₂	semidirect
3. C₅×#4685	C₂	semidirect
4. C₃×C₅×Q₈⋊C₈	C₂	semidirect
5. C₅×#4535	C₂	semidirect
6. C₅×#4483	C₂	semidirect
7. C₅×#4459	C₂	semidirect
28. #68237	C₅	direct (N × H)
33. C₅×Q₈⋊C₈	D₃	semidirect
66. #4835	C₂×C₅	semidirect
67. #4685	C₂×C₅	semidirect
68. C₃×Q₈⋊C₈	C₂×C₅	semidirect
69. #4535	C₂×C₅	semidirect
70. #4483	C₂×C₅	semidirect
71. #4459	C₂×C₅	semidirect
173. Q₈⋊C₈	C₅×D₃	semidirect
352. C₃×C₅	#1488	semidirect
410. C₅	#68237	direct (N × H)
429. C₃	C₅×#1488	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#239	2
C₅×#4835	2
C₅×#4685	2
C₃×C₅×Q₈⋊C₈	2
C₅×#4535	2
C₅×#4483	2
C₅×#4459	2
C₅×#1488	3
#68237	5

Subgroups — C5×#68237 #14282156

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68237 #14282156