Subgroups — C₅×#68240 #14282405

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5578	C₂	1	1	yes	maximal, characteristic
2	C₅×#4837	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×D₃×C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4535	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4484	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4456	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1493	—	3	1	no	maximal, Hall
9	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₅×#946	C₄	1	1	yes	characteristic
11	C₅×#944	C₄	1	1	yes	characteristic
12	C₂×C₅×ΓL(1,𝔽₂₅)	—	2	1	no	—
13	C₂×C₅×D₃×C₈	—	2	1	no	—
14	C₅×#897	C₄	1	1	yes	characteristic
15	C₅×#896	—	2	1	no	—
16	C₅×#889	—	2	1	no	—
17	C₅×#888	C₄	1	1	yes	characteristic
18	C₅×#872	—	2	1	no	—
19	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
20	C₄×C₅×Dic₆	—	2	1	no	—
21	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
22	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×Dic₃⋊C₈	—	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	#68240	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
29	C₅×#533	—	3	1	no	—
30	C₅×#423	—	3	1	no	—
31	C₂×C₄×C₅×C₈	—	3	1	no	—
32	C₅×Q₈⋊C₈	—	3	1	no	—
33	C₅×Q₈⋊C₈	—	3	1	no	—
34	C₅×Q₈⋊C₈	—	3	1	no	—
35	C₅×Q₈⋊C₈	D₃	1	1	yes	characteristic, complemented
36	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
39	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
40	C₂×C₅×Dic₆	—	2	1	no	—
41	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₂×C₃×C₅×C₈	—	2	1	no	—
44	C₃×C₅×C₄⋊C₄	—	2	1	no	—
45	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
46	C₃×C₄²×C₅	—	2	1	no	—
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
49	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
50	C₅×C₄⋊Dic₃	—	2	1	no	—
51	C₅×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
52	C₅×Dic₃⋊C₄	—	2	1	no	—
53	C₅×Dic₃⋊C₄	—	2	1	no	—
54	C₅×Dic₃⋊C₄	—	2	1	no	—
55	C₅×Dic₃⋊C₄	—	2	1	no	—
56	C₅×Dic₃⋊C₄	—	2	1	no	—
57	C₄×C₅×Dic₃	—	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×ΓL(1,𝔽₂₅)	—	4	2	no	—
64	C₅×D₃×C₈	—	2	4	no	—
65	#5578	C₂×C₅	1	1	yes	characteristic
66	#4837	C₂×C₅	1	1	yes	characteristic
67	C₄×D₃×C₈	C₂×C₅	1	1	yes	characteristic
68	C₃×Q₈⋊C₈	C₂×C₅	1	1	yes	characteristic, complemented
69	#4535	C₂×C₅	1	1	yes	characteristic, complemented
70	#4484	C₂×C₅	1	1	yes	characteristic, complemented
71	#4456	C₂×C₅	1	1	yes	characteristic, complemented
72	C₂×C₅×M₁₆	—	6	1	no	—
73	C₂²×C₅×C₈	—	6	1	no	—
74	C₅×C₄⋊Q₈	—	3	1	no	—
75	C₅×C₄⋊Q₈	—	3	1	no	—
76	C₅×#125	—	6	1	no	—
77	C₅×C₂²⋊Q₈	—	3	1	no	—
78	C₅×C₂²⋊Q₈	—	3	1	no	—
79	C₅×C₂²⋊Q₈	—	6	1	no	—
80	C₄×C₅×Q₈	—	3	1	no	—
81	C₄×C₅×Q₈	—	6	1	no	—
82	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
83	C₅×#117	—	6	1	no	—
84	C₂×C₄²×C₅	—	3	1	no	—
85	C₅×C₄⋊C₈	—	3	1	no	—
86	C₅×C₄⋊C₈	—	6	1	no	—
87	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
88	C₄×C₅×C₈	—	3	1	no	—
89	C₄×C₅×C₈	—	6	1	no	—
90	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
91	#1493	—	3	1	no	Sylow
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	—	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	C₄×C₅×D₃	—	2	1	no	—
106	C₄×C₅×D₃	—	2	1	no	—
107	C₄×C₅×D₃	—	2	2	no	—
108	C₄×C₅×D₃	SD₁₆	1	2	yes	—
109	C₄×C₅×D₃	Q₁₆	1	2	yes	—
110	C₅×Dic₆	—	2	1	no	—
111	C₅×Dic₆	—	2	1	no	—
112	C₅×Dic₆	—	4	2	no	—
113	C₃×C₅×C₈	—	2	2	no	—
114	C₃×C₅×C₈	—	4	1	no	—
115	C₅×C₃⋊C₈	—	2	2	no	—
116	C₅×C₃⋊C₈	—	4	1	no	—
117	C₃×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
118	#946	C₄×C₅	1	1	yes	characteristic
119	#944	C₄×C₅	1	1	yes	characteristic
120	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
121	C₂×D₃×C₈	—	2	1	no	—
122	#897	C₄×C₅	1	1	yes	characteristic
123	#896	—	2	1	no	—
124	#889	—	2	1	no	—
125	#888	C₄×C₅	1	1	yes	characteristic
126	#872	—	2	1	no	—
127	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
128	C₄×Dic₆	—	2	1	no	—
129	C₄×Dic₆	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	Dic₃⋊C₈	—	2	1	no	—
133	C₈×Dic₃	—	2	1	no	—
134	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
135	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
136	C₂×C₅×Q₈	—	3	1	no	—
137	C₂×C₅×Q₈	—	6	1	no	—
138	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
139	C₂²×C₄×C₅	—	3	1	no	—
140	C₂²×C₄×C₅	—	3	1	no	—
141	C₂²×C₄×C₅	—	3	1	no	—
142	C₅×M₁₆	—	12	2	no	—
143	C₂×C₅×C₈	—	2	1	no	—
144	C₂×C₅×C₈	—	2	1	no	—
145	C₂×C₅×C₈	—	6	4	no	—
146	C₂×C₅×C₈	—	6	1	no	—
147	C₂×C₅×C₈	—	6	1	no	—
148	C₅×C₄⋊C₄	—	2	1	no	—
149	C₅×C₄⋊C₄	—	3	1	no	—
150	C₅×C₄⋊C₄	—	6	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	—	6	1	no	—
155	C₅×C₄⋊C₄	—	6	1	no	—
156	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
157	C₅×C₂²⋊C₄	—	6	1	no	—
158	C₅×C₂²⋊C₄	—	6	1	no	—
159	C₄²×C₅	—	2	1	no	—
160	C₄²×C₅	—	3	1	no	—
161	C₄²×C₅	—	3	1	no	—
162	C₄²×C₅	—	3	1	no	—
163	C₄²×C₅	—	6	1	no	—
164	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
165	#533	—	3	1	no	—
166	#423	—	3	1	no	—
167	C₂×C₄×C₈	—	3	1	no	—
168	Q₈⋊C₈	—	3	1	no	—
169	Q₈⋊C₈	—	3	1	no	—
170	Q₈⋊C₈	—	3	1	no	—
171	Q₈⋊C₈	C₅×D₃	1	1	yes	characteristic, complemented
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	#103	1	2	yes	—
176	C₃×C₄×C₅	—	2	1	no	—
177	C₃×C₄×C₅	—	4	1	no	—
178	C₃×C₄×C₅	—	4	1	no	—
179	C₃×C₄×C₅	#131	1	1	yes	characteristic
180	C₃×C₄×C₅	#131	1	1	yes	characteristic
181	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
182	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
183	C₅×Dic₃	—	2	1	no	—
184	C₅×Dic₃	—	4	1	no	—
185	C₅×Dic₃	—	4	1	no	—
186	C₅×Dic₃	#103	1	2	yes	—
187	C₅×Dic₃	#104	1	2	yes	—
188	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
189	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
190	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
191	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
192	C₂×Dic₆	—	2	1	no	—
193	C₂×Dic₆	C₂×C₄×C₅	1	1	yes	characteristic
194	C₂×C₃×C₈	—	2	1	no	—
195	C₂×C₃×C₈	—	2	1	no	—
196	C₃×C₄⋊C₄	—	2	1	no	—
197	C₃×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
198	C₃×C₄²	—	2	1	no	—
199	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
200	(C₂×D₃)⋊C₄	—	2	1	no	—
201	(C₂×D₃)⋊C₄	—	2	1	no	—
202	C₄⋊Dic₃	—	2	1	no	—
203	C₄⋊Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
204	Dic₃⋊C₄	—	2	1	no	—
205	Dic₃⋊C₄	—	2	1	no	—
206	Dic₃⋊C₄	—	2	1	no	—
207	Dic₃⋊C₄	—	2	1	no	—
208	Dic₃⋊C₄	—	2	1	no	—
209	C₄×Dic₃	—	2	1	no	—
210	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
211	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
212	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
213	C₂×C₃⋊C₈	—	2	1	no	—
214	C₂×C₃⋊C₈	—	2	1	no	—
215	ΓL(1,𝔽₂₅)	—	4	2	no	—
216	D₃×C₈	—	2	4	no	—
217	C₂³×C₅	—	3	1	no	—
218	C₅×Q₈	—	2	1	no	—
219	C₅×Q₈	—	2	1	no	—
220	C₅×Q₈	—	6	1	no	—
221	C₅×Q₈	—	6	1	no	—
222	C₅×Q₈	—	12	2	no	—
223	C₂×C₄×C₅	—	2	1	no	—
224	C₂×C₄×C₅	—	2	1	no	—
225	C₂×C₄×C₅	—	3	1	no	—
226	C₂×C₄×C₅	—	3	1	no	—
227	C₂×C₄×C₅	—	3	2	no	—
228	C₂×C₄×C₅	—	3	2	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	C₂×C₄×D₃	1	1	yes	characteristic
236	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
237	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
238	C₅×C₈	—	2	2	no	—
239	C₅×C₈	—	4	1	no	—
240	C₅×C₈	—	6	2	no	—
241	C₅×C₈	—	12	1	no	—
242	C₂×M₁₆	—	6	1	no	—
243	C₂²×C₈	—	6	1	no	—
244	C₄⋊Q₈	—	3	1	no	—
245	C₄⋊Q₈	—	3	1	no	—
246	#125	—	6	1	no	—
247	C₂²⋊Q₈	—	3	1	no	—
248	C₂²⋊Q₈	—	3	1	no	—
249	C₂²⋊Q₈	—	6	1	no	—
250	C₄×Q₈	—	3	1	no	—
251	C₄×Q₈	—	6	1	no	—
252	C₄×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
253	#117	—	6	1	no	—
254	C₂×C₄²	—	3	1	no	—
255	C₄⋊C₈	—	3	1	no	—
256	C₄⋊C₈	—	6	1	no	—
257	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
258	C₄×C₈	—	3	1	no	—
259	C₄×C₈	—	6	1	no	—
260	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
261	C₂×C₃×C₅	#408	1	1	yes	characteristic
262	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
263	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
264	C₅×D₃	—	2	2	no	—
265	C₂²×D₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
266	C₃×Q₈	—	2	1	no	—
267	C₃×Q₈	—	2	1	no	—
268	C₂×C₃×C₄	—	2	1	no	—
269	C₂×C₃×C₄	—	2	1	no	—
270	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
271	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
272	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
273	C₂×Dic₃	—	2	1	no	—
274	C₂×Dic₃	—	2	1	no	—
275	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
276	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
277	C₂×Dic₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
278	C₄×D₃	—	2	1	no	—
279	C₄×D₃	—	2	1	no	—
280	C₄×D₃	—	2	2	no	—
281	C₄×D₃	C₅×SD₁₆	1	2	yes	—
282	C₄×D₃	C₅×Q₁₆	1	2	yes	—
283	Dic₆	—	2	1	no	—
284	Dic₆	—	2	1	no	—
285	Dic₆	—	4	2	no	—
286	C₃×C₈	—	2	2	no	—
287	C₃×C₈	—	4	1	no	—
288	C₃⋊C₈	—	2	2	no	—
289	C₃⋊C₈	—	4	1	no	—
290	C₂²×C₅	—	3	2	no	—
291	C₂²×C₅	—	6	1	no	—
292	C₂²×C₅	—	6	1	no	—
293	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
294	C₄×C₅	—	2	1	no	—
295	C₄×C₅	—	3	2	no	—
296	C₄×C₅	—	3	2	no	—
297	C₄×C₅	—	4	1	no	—
298	C₄×C₅	—	4	1	no	—
299	C₄×C₅	—	6	1	no	—
300	C₄×C₅	—	12	1	no	—
301	C₄×C₅	—	12	1	no	—
302	C₄×C₅	#901	1	1	yes	characteristic
303	C₄×C₅	#907	1	1	yes	characteristic
304	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
305	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
306	C₂×Q₈	—	3	1	no	—
307	C₂×Q₈	—	6	1	no	—
308	C₂×Q₈	C₄×C₅×D₃	1	1	yes	characteristic
309	C₂²×C₄	—	3	1	no	—
310	C₂²×C₄	—	3	1	no	—
311	C₂²×C₄	—	3	1	no	—
312	M₁₆	—	12	2	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	6	4	no	—
316	C₂×C₈	—	6	1	no	—
317	C₂×C₈	—	6	1	no	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	3	1	no	—
320	C₄⋊C₄	—	6	1	no	—
321	C₄⋊C₄	—	6	1	no	—
322	C₄⋊C₄	—	6	1	no	—
323	C₄⋊C₄	—	6	1	no	—
324	C₄⋊C₄	—	6	1	no	—
325	C₄⋊C₄	—	6	1	no	—
326	C₄⋊C₄	C₄×C₅×D₃	1	1	yes	characteristic
327	C₂²⋊C₄	—	6	1	no	—
328	C₂²⋊C₄	—	6	1	no	—
329	C₄²	—	2	1	no	—
330	C₄²	—	3	1	no	—
331	C₄²	—	3	1	no	—
332	C₄²	—	3	1	no	—
333	C₄²	—	6	1	no	—
334	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
335	C₃×C₅	#1493	1	1	yes	characteristic, complemented, Hall
336	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
337	C₂×D₃	—	2	1	no	—
338	C₂×D₃	—	2	1	no	—
339	C₂×D₃	C₅×#103	1	2	yes	—
340	C₃×C₄	—	2	1	no	—
341	C₃×C₄	—	4	1	no	—
342	C₃×C₄	—	4	1	no	—
343	C₃×C₄	C₅×#131	1	1	yes	characteristic
344	C₃×C₄	C₅×#131	1	1	yes	characteristic
345	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
346	C₃×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
347	Dic₃	—	2	1	no	—
348	Dic₃	—	4	1	no	—
349	Dic₃	—	4	1	no	—
350	Dic₃	C₅×#103	1	2	yes	—
351	Dic₃	C₅×#104	1	2	yes	—
352	C₂×C₅	—	6	2	no	—
353	C₂×C₅	#4727	1	1	yes	characteristic
354	C₂×C₅	D₃×#103	1	1	yes	characteristic
355	C₂×C₅	D₃×#104	1	1	yes	characteristic
356	C₂³	—	3	1	no	—
357	Q₈	—	2	1	no	—
358	Q₈	—	2	1	no	—
359	Q₈	—	6	1	no	—
360	Q₈	—	6	1	no	—
361	Q₈	—	12	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	3	1	no	—
365	C₂×C₄	—	3	1	no	—
366	C₂×C₄	—	3	2	no	—
367	C₂×C₄	—	3	2	no	—
368	C₂×C₄	—	3	1	no	—
369	C₂×C₄	—	6	2	no	—
370	C₂×C₄	—	6	1	no	—
371	C₂×C₄	—	6	1	no	—
372	C₂×C₄	—	6	1	no	—
373	C₂×C₄	—	6	1	no	—
374	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
375	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
376	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
377	C₈	—	2	2	no	—
378	C₈	—	4	1	no	—
379	C₈	—	6	2	no	—
380	C₈	—	12	1	no	—
381	C₂×C₃	C₅×#408	1	1	yes	characteristic
382	C₂×C₃	C₂×C₅×#103	1	1	yes	characteristic
383	C₂×C₃	C₂×C₅×#104	1	1	yes	characteristic
384	D₃	—	2	2	no	—
385	C₅	#68240	1	1	yes	characteristic, direct product factor, Sylow
386	C₂²	—	3	2	no	—
387	C₂²	—	6	1	no	—
388	C₂²	—	6	1	no	—
389	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
390	C₄	—	2	1	no	—
391	C₄	—	3	2	no	—
392	C₄	—	3	2	no	—
393	C₄	—	4	1	no	—
394	C₄	—	4	1	no	—
395	C₄	—	6	1	no	—
396	C₄	—	12	1	no	—
397	C₄	—	12	1	no	—
398	C₄	C₅×#901	1	1	yes	characteristic
399	C₄	C₅×#907	1	1	yes	characteristic
400	C₄	C₅×D₃×SD₁₆	1	1	yes	characteristic
401	C₄	C₅×D₃×Q₁₆	1	1	yes	characteristic
402	C₃	C₅×#1493	1	1	yes	characteristic, complemented, Sylow
403	C₂	—	6	2	no	—
404	C₂	C₅×#4727	1	1	yes	characteristic
405	C₂	C₅×D₃×#103	1	1	yes	characteristic
406	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	C₂²×C₅×D₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×Q₈⋊C₈	D₃
hypocenter	C₃	C₅×#1493
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
4. C₃×C₅×Q₈⋊C₈	C₂	semidirect
5. C₅×#4535	C₂	semidirect
6. C₅×#4484	C₂	semidirect
7. C₅×#4456	C₂	semidirect
28. #68240	C₅	direct (N × H)
35. C₅×Q₈⋊C₈	D₃	semidirect
68. C₃×Q₈⋊C₈	C₂×C₅	semidirect
69. #4535	C₂×C₅	semidirect
70. #4484	C₂×C₅	semidirect
71. #4456	C₂×C₅	semidirect
171. Q₈⋊C₈	C₅×D₃	semidirect
335. C₃×C₅	#1493	semidirect
385. C₅	#68240	direct (N × H)
402. C₃	C₅×#1493	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5578	2
C₅×#4837	2
C₄×C₅×D₃×C₈	2
C₃×C₅×Q₈⋊C₈	2
C₅×#4535	2
C₅×#4484	2
C₅×#4456	2
C₅×#1493	3
#68240	5

Subgroups — C5×#68240 #14282405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68240 #14282405