Subgroups — C₅×#68241 #14282454

Order 1920.

460 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5580	C₂	1	1	yes	maximal, characteristic
2	C₅×#4835	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₅×D₃×C₈	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×#1494	—	3	1	no	maximal, Hall
9	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₅×#947	C₄	1	1	yes	characteristic
11	C₅×#946	C₄	1	1	yes	characteristic
12	C₂×C₅×D₃×C₈	—	2	1	no	—
13	C₅×#898	—	2	1	no	—
14	C₅×#895	C₄	1	1	yes	characteristic
15	C₅×#894	—	2	1	no	—
16	C₅×#890	C₄	1	1	yes	characteristic
17	C₄×C₅×D₁₂	—	2	1	no	—
18	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
19	C₅×#872	—	2	1	no	—
20	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
21	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
23	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
24	C₅×#815	—	2	1	no	—
25	C₅×C₈×Dic₃	—	2	1	no	—
26	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
27	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
28	#68241	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₅×#529	—	3	1	no	—
30	C₅×#432	—	3	1	no	—
31	C₂×C₄×C₅×C₈	—	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	C₂×C₅×D₁₂	—	2	1	no	—
38	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₅×D₃	—	2	1	no	—
40	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
42	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
43	C₂×C₃×C₅×C₈	—	2	1	no	—
44	C₂×C₃×C₅×C₈	—	2	1	no	—
45	C₃×C₅×C₄⋊C₄	—	2	1	no	—
46	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄²×C₅	—	2	1	no	—
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
50	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
51	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
52	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
53	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
54	C₅×C₄⋊Dic₃	—	2	1	no	—
55	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
56	C₅×Dic₃⋊C₄	—	2	1	no	—
57	C₅×Dic₃⋊C₄	—	2	1	no	—
58	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
59	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
60	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
61	C₂×C₅×C₃⋊C₈	—	2	1	no	—
62	C₂×C₅×C₃⋊C₈	—	2	1	no	—
63	C₅×D₃×C₈	—	2	4	no	—
64	#5580	C₂×C₅	1	1	yes	characteristic
65	#4835	C₂×C₅	1	1	yes	characteristic, complemented
66	C₄×D₃×C₈	C₂×C₅	1	1	yes	characteristic, complemented
67	C₃×Q₈⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
68	#4535	C₂×C₅	1	1	yes	characteristic, complemented
69	#4483	C₂×C₅	1	1	yes	characteristic, complemented
70	#4459	C₂×C₅	1	1	yes	characteristic, complemented
71	C₂²×C₅×C₈	—	6	1	no	—
72	C₅×#126	—	6	1	no	—
73	C₅×#125	—	3	1	no	—
74	C₅×#124	—	3	1	no	—
75	C₅×#123	—	6	1	no	—
76	C₅×C₂²⋊Q₈	—	3	1	no	—
77	C₅×#121	—	3	1	no	—
78	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
79	C₄×C₅×D₄	—	3	1	no	—
80	C₄×C₅×D₄	—	6	1	no	—
81	C₅×#117	—	6	1	no	—
82	C₂×C₄²×C₅	—	3	1	no	—
83	C₅×C₄⋊C₈	—	3	1	no	—
84	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
85	C₅×C₂²⋊C₈	—	6	1	no	—
86	C₅×#97	—	6	1	no	—
87	C₄×C₅×C₈	—	3	1	no	—
88	C₄×C₅×C₈	—	6	1	no	—
89	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
90	#1494	—	3	1	no	Sylow
91	C₂²×C₅×D₃	—	2	1	no	—
92	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₃×C₅×Q₈	—	2	1	no	—
94	C₃×C₅×Q₈	—	2	1	no	—
95	C₂×C₃×C₄×C₅	—	2	1	no	—
96	C₂×C₃×C₄×C₅	—	2	1	no	—
97	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
100	C₂×C₅×Dic₃	—	2	1	no	—
101	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
104	C₅×D₁₂	—	2	1	no	—
105	C₅×D₁₂	—	2	1	no	—
106	C₅×D₁₂	—	4	2	no	—
107	C₄×C₅×D₃	—	2	2	no	—
108	C₄×C₅×D₃	—	2	1	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	—	4	1	no	—
113	C₄×C₅×D₃	—	4	1	no	—
114	C₃×C₅×C₈	—	2	2	no	—
115	C₃×C₅×C₈	—	4	1	no	—
116	C₅×C₃⋊C₈	—	2	2	no	—
117	C₅×C₃⋊C₈	—	4	1	no	—
118	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
119	#947	C₄×C₅	1	1	yes	characteristic
120	#946	C₄×C₅	1	1	yes	characteristic
121	C₂×D₃×C₈	—	2	1	no	—
122	#898	—	2	1	no	—
123	#895	C₄×C₅	1	1	yes	characteristic
124	#894	—	2	1	no	—
125	#890	C₄×C₅	1	1	yes	characteristic
126	C₄×D₁₂	—	2	1	no	—
127	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
128	#872	—	2	1	no	—
129	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
130	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
131	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
132	(C₂×D₃)⋊C₈	—	2	1	no	—
133	#815	—	2	1	no	—
134	C₈×Dic₃	—	2	1	no	—
135	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
136	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
137	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
138	C₂×C₅×D₄	—	3	1	no	—
139	C₂×C₅×D₄	—	6	1	no	—
140	C₂²×C₄×C₅	—	3	1	no	—
141	C₂²×C₄×C₅	—	3	1	no	—
142	C₂²×C₄×C₅	—	3	1	no	—
143	C₂²×C₄×C₅	—	6	1	no	—
144	C₂×C₅×C₈	—	2	1	no	—
145	C₂×C₅×C₈	—	2	1	no	—
146	C₂×C₅×C₈	—	6	4	no	—
147	C₂×C₅×C₈	—	6	1	no	—
148	C₂×C₅×C₈	—	6	1	no	—
149	C₅×C₄⋊C₄	—	2	1	no	—
150	C₅×C₄⋊C₄	—	3	1	no	—
151	C₅×C₄⋊C₄	—	6	1	no	—
152	C₅×C₄⋊C₄	—	6	1	no	—
153	C₅×C₄⋊C₄	—	6	1	no	—
154	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
155	C₅×C₂²⋊C₄	—	6	1	no	—
156	C₅×C₂²⋊C₄	—	6	1	no	—
157	C₅×C₂²⋊C₄	—	6	1	no	—
158	C₅×C₂²⋊C₄	—	6	1	no	—
159	C₅×C₂²⋊C₄	—	6	1	no	—
160	C₄²×C₅	—	2	1	no	—
161	C₄²×C₅	—	3	1	no	—
162	C₄²×C₅	—	3	1	no	—
163	C₄²×C₅	—	3	1	no	—
164	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
165	#529	—	3	1	no	—
166	#432	—	3	1	no	—
167	C₂×C₄×C₈	—	3	1	no	—
168	Q₈⋊C₈	—	3	1	no	—
169	Q₈⋊C₈	C₅×D₃	1	1	yes	characteristic, complemented
170	D₄⋊C₈	—	3	1	no	—
171	D₄⋊C₈	—	3	1	no	—
172	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₅×D₃	—	2	1	no	—
174	C₂×C₅×D₃	—	2	1	no	—
175	C₂×C₅×D₃	—	4	1	no	—
176	C₂×C₅×D₃	—	4	1	no	—
177	C₂×C₅×D₃	—	4	1	no	—
178	C₂×C₅×D₃	#104	1	2	yes	—
179	C₃×C₄×C₅	—	2	1	no	—
180	C₃×C₄×C₅	—	4	1	no	—
181	C₃×C₄×C₅	—	4	1	no	—
182	C₃×C₄×C₅	#131	1	1	yes	characteristic
183	C₃×C₄×C₅	#131	1	1	yes	characteristic
184	C₃×C₄×C₅	#135	1	1	yes	characteristic
185	C₃×C₄×C₅	#135	1	1	yes	characteristic
186	C₅×Dic₃	—	2	1	no	—
187	C₅×Dic₃	—	2	1	no	—
188	C₅×Dic₃	—	2	1	no	—
189	C₅×Dic₃	—	4	1	no	—
190	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
191	C₂×D₁₂	—	2	1	no	—
192	C₂×D₁₂	C₂×C₄×C₅	1	1	yes	characteristic
193	C₂×C₄×D₃	—	2	1	no	—
194	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
195	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
196	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
197	C₂×C₃×C₈	—	2	1	no	—
198	C₂×C₃×C₈	—	2	1	no	—
199	C₃×C₄⋊C₄	—	2	1	no	—
200	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
201	C₃×C₄²	—	2	1	no	—
202	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
203	(C₂×D₃)⋊C₄	—	2	1	no	—
204	(C₂×D₃)⋊C₄	—	2	1	no	—
205	(C₂×D₃)⋊C₄	—	2	1	no	—
206	(C₂×D₃)⋊C₄	—	2	1	no	—
207	(C₂×D₃)⋊C₄	—	2	1	no	—
208	C₄⋊Dic₃	—	2	1	no	—
209	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
210	Dic₃⋊C₄	—	2	1	no	—
211	Dic₃⋊C₄	—	2	1	no	—
212	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
213	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
214	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
215	C₂×C₃⋊C₈	—	2	1	no	—
216	C₂×C₃⋊C₈	—	2	1	no	—
217	D₃×C₈	—	2	4	no	—
218	C₂³×C₅	—	3	1	no	—
219	C₂³×C₅	—	6	1	no	—
220	C₅×Q₈	—	2	1	no	—
221	C₅×Q₈	—	2	1	no	—
222	C₅×D₄	—	6	1	no	—
223	C₅×D₄	—	6	1	no	—
224	C₅×D₄	—	12	2	no	—
225	C₂×C₄×C₅	—	2	1	no	—
226	C₂×C₄×C₅	—	2	1	no	—
227	C₂×C₄×C₅	—	3	1	no	—
228	C₂×C₄×C₅	—	3	1	no	—
229	C₂×C₄×C₅	—	3	1	no	—
230	C₂×C₄×C₅	—	6	2	no	—
231	C₂×C₄×C₅	—	6	1	no	—
232	C₂×C₄×C₅	—	6	1	no	—
233	C₂×C₄×C₅	—	6	1	no	—
234	C₂×C₄×C₅	—	6	1	no	—
235	C₂×C₄×C₅	—	6	1	no	—
236	C₂×C₄×C₅	—	12	1	no	—
237	C₂×C₄×C₅	—	12	1	no	—
238	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
239	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
240	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
241	C₅×C₈	—	2	2	no	—
242	C₅×C₈	—	4	1	no	—
243	C₅×C₈	—	6	2	no	—
244	C₅×C₈	—	12	1	no	—
245	C₂²×C₈	—	6	1	no	—
246	#126	—	6	1	no	—
247	#125	—	3	1	no	—
248	#124	—	3	1	no	—
249	#123	—	6	1	no	—
250	C₂²⋊Q₈	—	3	1	no	—
251	#121	—	3	1	no	—
252	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
253	C₄×D₄	—	3	1	no	—
254	C₄×D₄	—	6	1	no	—
255	#117	—	6	1	no	—
256	C₂×C₄²	—	3	1	no	—
257	C₄⋊C₈	—	3	1	no	—
258	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
259	C₂²⋊C₈	—	6	1	no	—
260	#97	—	6	1	no	—
261	C₄×C₈	—	3	1	no	—
262	C₄×C₈	—	6	1	no	—
263	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
264	C₂×C₃×C₅	#408	1	1	yes	characteristic
265	C₂×C₃×C₅	#416	1	1	yes	characteristic
266	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
267	C₅×D₃	—	2	2	no	—
268	C₅×D₃	—	8	1	no	—
269	C₂²×D₃	—	2	1	no	—
270	C₂²×D₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
271	C₃×Q₈	—	2	1	no	—
272	C₃×Q₈	—	2	1	no	—
273	C₂×C₃×C₄	—	2	1	no	—
274	C₂×C₃×C₄	—	2	1	no	—
275	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
276	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
277	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
278	C₂×Dic₃	—	2	1	no	—
279	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
280	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
281	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
282	D₁₂	—	2	1	no	—
283	D₁₂	—	2	1	no	—
284	D₁₂	—	4	2	no	—
285	C₄×D₃	—	2	2	no	—
286	C₄×D₃	—	2	1	no	—
287	C₄×D₃	—	2	1	no	—
288	C₄×D₃	—	2	1	no	—
289	C₄×D₃	—	2	1	no	—
290	C₄×D₃	—	4	1	no	—
291	C₄×D₃	—	4	1	no	—
292	C₃×C₈	—	2	2	no	—
293	C₃×C₈	—	4	1	no	—
294	C₃⋊C₈	—	2	2	no	—
295	C₃⋊C₈	—	4	1	no	—
296	C₂²×C₅	—	3	2	no	—
297	C₂²×C₅	—	6	1	no	—
298	C₂²×C₅	—	6	1	no	—
299	C₂²×C₅	—	12	1	no	—
300	C₂²×C₅	—	12	1	no	—
301	C₂²×C₅	—	12	1	no	—
302	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
303	C₄×C₅	—	2	1	no	—
304	C₄×C₅	—	4	1	no	—
305	C₄×C₅	—	4	1	no	—
306	C₄×C₅	—	6	1	no	—
307	C₄×C₅	—	6	1	no	—
308	C₄×C₅	—	6	1	no	—
309	C₄×C₅	—	12	1	no	—
310	C₄×C₅	#901	1	1	yes	characteristic
311	C₄×C₅	#907	1	1	yes	characteristic
312	C₄×C₅	#916	1	1	yes	characteristic
313	C₄×C₅	#919	1	1	yes	characteristic
314	C₂×Q₈	C₄×C₅×D₃	1	1	yes	characteristic
315	C₂×D₄	—	3	1	no	—
316	C₂×D₄	—	6	1	no	—
317	C₂²×C₄	—	3	1	no	—
318	C₂²×C₄	—	3	1	no	—
319	C₂²×C₄	—	3	1	no	—
320	C₂²×C₄	—	6	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	6	4	no	—
324	C₂×C₈	—	6	1	no	—
325	C₂×C₈	—	6	1	no	—
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	3	1	no	—
328	C₄⋊C₄	—	6	1	no	—
329	C₄⋊C₄	—	6	1	no	—
330	C₄⋊C₄	—	6	1	no	—
331	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
332	C₂²⋊C₄	—	6	1	no	—
333	C₂²⋊C₄	—	6	1	no	—
334	C₂²⋊C₄	—	6	1	no	—
335	C₂²⋊C₄	—	6	1	no	—
336	C₂²⋊C₄	—	6	1	no	—
337	C₄²	—	2	1	no	—
338	C₄²	—	3	1	no	—
339	C₄²	—	3	1	no	—
340	C₄²	—	3	1	no	—
341	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
342	C₃×C₅	#1494	1	1	yes	characteristic, complemented, Hall
343	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
344	C₂×D₃	—	2	1	no	—
345	C₂×D₃	—	2	1	no	—
346	C₂×D₃	—	4	1	no	—
347	C₂×D₃	—	4	1	no	—
348	C₂×D₃	—	4	1	no	—
349	C₂×D₃	C₅×#104	1	2	yes	—
350	C₃×C₄	—	2	1	no	—
351	C₃×C₄	—	4	1	no	—
352	C₃×C₄	—	4	1	no	—
353	C₃×C₄	C₅×#131	1	1	yes	characteristic
354	C₃×C₄	C₅×#131	1	1	yes	characteristic
355	C₃×C₄	C₅×#135	1	1	yes	characteristic
356	C₃×C₄	C₅×#135	1	1	yes	characteristic
357	Dic₃	—	2	1	no	—
358	Dic₃	—	2	1	no	—
359	Dic₃	—	2	1	no	—
360	Dic₃	—	4	1	no	—
361	C₂×C₅	—	6	2	no	—
362	C₂×C₅	—	24	1	no	—
363	C₂×C₅	#4727	1	1	yes	characteristic
364	C₂×C₅	#4804	1	1	yes	characteristic
365	C₂×C₅	D₃×#104	1	1	yes	characteristic
366	C₂³	—	3	1	no	—
367	C₂³	—	6	1	no	—
368	Q₈	—	2	1	no	—
369	Q₈	—	2	1	no	—
370	D₄	—	6	1	no	—
371	D₄	—	6	1	no	—
372	D₄	—	12	2	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	3	1	no	—
376	C₂×C₄	—	3	1	no	—
377	C₂×C₄	—	3	1	no	—
378	C₂×C₄	—	6	2	no	—
379	C₂×C₄	—	6	1	no	—
380	C₂×C₄	—	6	1	no	—
381	C₂×C₄	—	6	1	no	—
382	C₂×C₄	—	6	1	no	—
383	C₂×C₄	—	6	1	no	—
384	C₂×C₄	—	12	1	no	—
385	C₂×C₄	—	12	1	no	—
386	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
387	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
388	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
389	C₈	—	2	2	no	—
390	C₈	—	4	1	no	—
391	C₈	—	6	2	no	—
392	C₈	—	12	1	no	—
393	C₂×C₃	C₅×#408	1	1	yes	characteristic
394	C₂×C₃	C₅×#416	1	1	yes	characteristic
395	C₂×C₃	C₂×C₅×#104	1	1	yes	characteristic
396	D₃	—	2	2	no	—
397	D₃	—	8	1	no	—
398	C₅	#68241	1	1	yes	characteristic, direct product factor, Sylow
399	C₂²	—	3	2	no	—
400	C₂²	—	6	1	no	—
401	C₂²	—	6	1	no	—
402	C₂²	—	12	1	no	—
403	C₂²	—	12	1	no	—
404	C₂²	—	12	1	no	—
405	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
406	C₄	—	2	1	no	—
407	C₄	—	4	1	no	—
408	C₄	—	4	1	no	—
409	C₄	—	6	1	no	—
410	C₄	—	6	1	no	—
411	C₄	—	6	1	no	—
412	C₄	—	12	1	no	—
413	C₄	C₅×#901	1	1	yes	characteristic
414	C₄	C₅×#907	1	1	yes	characteristic
415	C₄	C₅×#916	1	1	yes	characteristic
416	C₄	C₅×#919	1	1	yes	characteristic
417	C₃	C₅×#1494	1	1	yes	characteristic, complemented, Sylow
418	C₂	—	6	2	no	—
419	C₂	—	24	1	no	—
420	C₂	C₅×#4727	1	1	yes	characteristic
421	C₂	C₅×#4804	1	1	yes	characteristic
422	C₂	C₅×D₃×#104	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	C₅×#895	C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×Q₈⋊C₈	D₃
hypocenter	C₃	C₅×#1494
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. C₄×C₅×D₃×C₈	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. #68241	C₅	direct (N × H)
33. C₅×Q₈⋊C₈	D₃	semidirect
65. #4835	C₂×C₅	semidirect
66. C₄×D₃×C₈	C₂×C₅	semidirect
67. C₃×Q₈⋊C₈	C₂×C₅	semidirect
68. #4535	C₂×C₅	semidirect
69. #4483	C₂×C₅	semidirect
70. #4459	C₂×C₅	semidirect
169. Q₈⋊C₈	C₅×D₃	semidirect
342. C₃×C₅	#1494	semidirect
398. C₅	#68241	direct (N × H)
417. C₃	C₅×#1494	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5580	2
C₅×#4835	2
C₄×C₅×D₃×C₈	2
C₃×C₅×Q₈⋊C₈	2
C₅×#4535	2
C₅×#4483	2
C₅×#4459	2
C₅×#1494	3
#68241	5

Subgroups — C5×#68241 #14282454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68241 #14282454