Subgroups — C₃×#10605003 #14282228

Order 1920.

532 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#41×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₃×#65758	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×#704	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×Q₈⋊#170	C₂	1	2	yes	maximal, complemented
5	C₃×D₄⋊#170	C₂	1	2	yes	maximal, complemented
6	#10605003	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	C₄	1	1	yes	characteristic
8	C₂×C₃×#704	—	2	1	no	—
9	C₂×C₃×#704	—	2	1	no	—
10	C₃×Q₈×Dic₅	C₄	1	1	yes	characteristic
11	C₃×Q₈×Dic₅	C₂²	1	1	yes	characteristic
12	C₃×D₄×Dic₅	—	2	1	no	—
13	C₃×D₄×Dic₅	—	2	1	no	—
14	C₃×D₄×Dic₅	C₄	1	1	yes	characteristic
15	C₃×D₄×Dic₅	C₂²	1	1	yes	characteristic
16	C₃×#4072	—	2	1	no	—
17	C₃×#4072	C₄	1	1	yes	characteristic
18	C₂×C₃×C₄×Dic₅	—	2	1	no	—
19	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
20	C₃×#4004	—	2	1	no	—
21	C₃×#4002	—	2	1	no	—
22	C₃×C₄⋊#170	C₂²	1	2	yes	—
23	C₃×C₄×#170	C₂²	1	2	yes	—
24	C₃×#1489	—	5	1	no	maximal, Hall
25	#41×Dic₅	C₂×C₃	1	1	yes	characteristic
26	#65758	C₂×C₃	1	1	yes	characteristic, complemented
27	C₄×#704	C₂×C₃	1	1	yes	characteristic, complemented
28	Q₈⋊#170	C₂×C₃	1	2	yes	complemented
29	D₄⋊#170	C₂×C₃	1	2	yes	complemented
30	C₃×C₅×#41	—	2	2	no	—
31	C₃×C₅×#41	—	2	2	no	—
32	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	—	2	1	no	—
36	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×Dic₅	—	2	1	no	—
38	C₂²×C₃×Dic₅	—	2	1	no	—
39	C₂²×C₃×Dic₅	D₄	1	1	yes	characteristic
40	C₂²×C₃×Dic₅	D₄	1	1	yes	characteristic
41	C₃×#704	—	2	4	no	—
42	C₃×#704	—	4	2	no	—
43	C₂×C₃×#170	—	2	2	no	—
44	C₂×C₃×#170	—	2	2	no	—
45	C₃×C₂²⋊Dic₅	—	2	1	no	—
46	C₃×C₂²⋊Dic₅	—	2	1	no	—
47	C₃×C₂²⋊Dic₅	—	2	1	no	—
48	C₃×C₄⋊Dic₅	—	2	1	no	—
49	C₃×C₄⋊Dic₅	—	2	1	no	—
50	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
51	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
52	C₃×C₄×Dic₅	—	2	1	no	—
53	C₃×C₄×Dic₅	—	2	1	no	—
54	C₃×C₄×Dic₅	—	2	1	no	—
55	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
56	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
57	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
58	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
59	C₃×C₄×#41	—	5	1	no	—
60	C₃×#423	—	5	1	no	—
61	C₃×C₄×M₁₆	—	5	1	no	—
62	C₃×Q₈⋊C₈	—	5	2	no	—
63	C₃×D₄⋊C₈	—	5	2	no	—
64	C₂×C₅×#41	C₃×C₄	1	1	yes	characteristic
65	C₂×#704	—	2	1	no	—
66	C₂×#704	—	2	1	no	—
67	Q₈×Dic₅	C₃×C₄	1	1	yes	characteristic
68	Q₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
69	D₄×Dic₅	—	2	1	no	—
70	D₄×Dic₅	—	2	1	no	—
71	D₄×Dic₅	C₃×C₄	1	1	yes	characteristic
72	D₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
73	#4072	—	2	1	no	—
74	#4072	C₃×C₄	1	1	yes	characteristic
75	C₂×C₄×Dic₅	—	2	1	no	—
76	C₂×C₄×Dic₅	C₂²×C₃	1	1	yes	characteristic
77	#4004	—	2	1	no	—
78	#4002	—	2	1	no	—
79	C₄⋊#170	C₂²×C₃	1	2	yes	—
80	C₄×#170	C₂²×C₃	1	2	yes	—
81	#1489	—	5	1	no	Sylow
82	C₂³×C₃×C₅	—	2	1	no	—
83	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
84	C₃×C₅×Q₈	—	2	1	no	—
85	C₃×C₅×Q₈	M₁₆	1	2	yes	complemented
86	C₃×C₅×D₄	—	2	4	no	—
87	C₃×C₅×D₄	—	2	1	no	—
88	C₃×C₅×D₄	M₁₆	1	2	yes	complemented
89	C₂×C₃×C₄×C₅	—	2	4	no	—
90	C₂×C₃×C₄×C₅	—	2	1	no	—
91	C₂×C₃×C₄×C₅	—	2	1	no	—
92	C₂×C₃×C₄×C₅	—	2	1	no	—
93	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₃×Dic₅	—	2	2	no	—
96	C₂×C₃×Dic₅	—	2	1	no	—
97	C₂×C₃×Dic₅	—	2	1	no	—
98	C₂×C₃×Dic₅	—	2	1	no	—
99	C₂×C₃×Dic₅	—	2	1	no	—
100	C₂×C₃×Dic₅	—	4	2	no	—
101	C₂×C₃×Dic₅	—	4	1	no	—
102	C₂×C₃×Dic₅	—	4	1	no	—
103	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
106	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
107	C₃×#170	—	2	4	no	—
108	C₃×#170	—	4	2	no	—
109	C₂×C₃×#41	AGL(1,𝔽₅)	1	1	yes	characteristic
110	C₂×C₃×M₁₆	—	10	1	no	—
111	C₂×C₃×M₁₆	—	10	1	no	—
112	C₃×C₄×Q₈	—	5	1	no	—
113	C₃×C₄×Q₈	—	5	1	no	—
114	C₃×C₄×D₄	—	5	1	no	—
115	C₃×C₄×D₄	—	5	1	no	—
116	C₃×C₄×D₄	—	10	1	no	—
117	C₃×C₄×D₄	—	10	1	no	—
118	C₃×#117	—	5	1	no	—
119	C₃×#117	—	10	1	no	—
120	C₂×C₃×C₄²	—	5	1	no	—
121	C₂×C₃×C₄²	—	10	1	no	—
122	C₃×C₄⋊C₈	—	5	2	no	—
123	C₃×C₄⋊C₈	—	10	1	no	—
124	C₃×#97	—	10	1	no	—
125	C₃×C₄×C₈	—	5	2	no	—
126	C₅×#41	—	2	2	no	—
127	C₅×#41	—	2	2	no	—
128	C₂×C₅×Q₈	C₂×C₃×C₄	1	1	yes	characteristic
129	C₂×C₅×D₄	—	2	1	no	—
130	C₂×C₅×D₄	C₂×C₃×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₅	—	2	1	no	—
132	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
133	C₂²×Dic₅	—	2	1	no	—
134	C₂²×Dic₅	—	2	1	no	—
135	C₂²×Dic₅	C₃×D₄	1	1	yes	characteristic
136	C₂²×Dic₅	C₃×D₄	1	1	yes	characteristic
137	#704	—	2	4	no	—
138	#704	—	4	2	no	—
139	C₂×#170	—	2	2	no	—
140	C₂×#170	—	2	2	no	—
141	C₂²⋊Dic₅	—	2	1	no	—
142	C₂²⋊Dic₅	—	2	1	no	—
143	C₂²⋊Dic₅	—	2	1	no	—
144	C₄⋊Dic₅	—	2	1	no	—
145	C₄⋊Dic₅	—	2	1	no	—
146	C₄⋊Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
147	C₄⋊Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
148	C₄×Dic₅	—	2	1	no	—
149	C₄×Dic₅	—	2	1	no	—
150	C₄×Dic₅	—	2	1	no	—
151	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
152	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
153	C₄×Dic₅	C₃×D₄	1	1	yes	characteristic
154	C₄×Dic₅	C₂³×C₃	1	1	yes	characteristic
155	C₄×#41	—	5	1	no	—
156	#423	—	5	1	no	—
157	C₄×M₁₆	—	5	1	no	—
158	Q₈⋊C₈	—	5	2	no	—
159	D₄⋊C₈	—	5	2	no	—
160	C₂²×C₃×C₅	—	2	1	no	—
161	C₂²×C₃×C₅	—	2	2	no	—
162	C₂²×C₃×C₅	—	2	1	no	—
163	C₂²×C₃×C₅	—	2	1	no	—
164	C₂²×C₃×C₅	—	4	1	no	—
165	C₂²×C₃×C₅	—	4	1	no	—
166	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₃×C₄×C₅	—	2	1	no	—
168	C₃×C₄×C₅	—	2	2	no	—
169	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
170	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
171	C₃×Dic₅	—	2	1	no	—
172	C₃×Dic₅	—	2	1	no	—
173	C₃×Dic₅	—	4	1	no	—
174	C₃×Dic₅	—	4	1	no	—
175	C₃×Dic₅	#104	1	2	yes	—
176	C₃×Dic₅	AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₃×Dic₅	#137	1	1	yes	characteristic
178	C₃×#41	—	2	2	no	—
179	C₃×#41	—	2	2	no	—
180	C₂×C₃×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
181	C₂×C₃×D₄	—	2	1	no	—
182	C₂×C₃×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
183	C₂²×C₃×C₄	—	2	1	no	—
184	C₂²×C₃×C₄	—	5	1	no	—
185	C₂²×C₃×C₄	—	5	1	no	—
186	C₂²×C₃×C₄	—	10	1	no	—
187	C₂²×C₃×C₄	—	10	1	no	—
188	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
189	C₃×M₁₆	—	10	4	no	—
190	C₃×M₁₆	—	20	2	no	—
191	C₂×C₃×C₈	—	10	2	no	—
192	C₂×C₃×C₈	—	10	2	no	—
193	C₃×C₄⋊C₄	—	5	1	no	—
194	C₃×C₄⋊C₄	—	5	1	no	—
195	C₃×C₄⋊C₄	—	10	1	no	—
196	C₃×C₄⋊C₄	—	10	1	no	—
197	C₃×C₂²⋊C₄	—	10	1	no	—
198	C₃×C₂²⋊C₄	—	10	1	no	—
199	C₃×C₂²⋊C₄	—	10	1	no	—
200	C₃×C₄²	—	5	1	no	—
201	C₃×C₄²	—	5	1	no	—
202	C₃×C₄²	—	5	1	no	—
203	C₃×C₄²	—	5	1	no	—
204	C₃×C₄²	—	10	1	no	—
205	C₃×C₄²	—	10	1	no	—
206	C₃×C₄²	—	10	1	no	—
207	C₂³×C₅	—	2	1	no	—
208	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
209	C₅×Q₈	—	2	1	no	—
210	C₅×Q₈	C₃×M₁₆	1	2	yes	complemented
211	C₅×D₄	—	2	4	no	—
212	C₅×D₄	—	2	1	no	—
213	C₅×D₄	C₃×M₁₆	1	2	yes	complemented
214	C₂×C₄×C₅	—	2	4	no	—
215	C₂×C₄×C₅	—	2	1	no	—
216	C₂×C₄×C₅	—	2	1	no	—
217	C₂×C₄×C₅	—	2	1	no	—
218	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
219	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
220	C₂×Dic₅	—	2	2	no	—
221	C₂×Dic₅	—	2	1	no	—
222	C₂×Dic₅	—	2	1	no	—
223	C₂×Dic₅	—	2	1	no	—
224	C₂×Dic₅	—	2	1	no	—
225	C₂×Dic₅	—	4	2	no	—
226	C₂×Dic₅	—	4	1	no	—
227	C₂×Dic₅	—	4	1	no	—
228	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
229	C₂×Dic₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
230	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
231	C₂×Dic₅	C₂×C₃×D₄	1	1	yes	characteristic
232	#170	—	2	4	no	—
233	#170	—	4	2	no	—
234	C₂×#41	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
235	C₂×M₁₆	—	10	1	no	—
236	C₂×M₁₆	—	10	1	no	—
237	C₄×Q₈	—	5	1	no	—
238	C₄×Q₈	—	5	1	no	—
239	C₄×D₄	—	5	1	no	—
240	C₄×D₄	—	5	1	no	—
241	C₄×D₄	—	10	1	no	—
242	C₄×D₄	—	10	1	no	—
243	#117	—	5	1	no	—
244	#117	—	10	1	no	—
245	C₂×C₄²	—	5	1	no	—
246	C₂×C₄²	—	10	1	no	—
247	C₄⋊C₈	—	5	2	no	—
248	C₄⋊C₈	—	10	1	no	—
249	#97	—	10	1	no	—
250	C₄×C₈	—	5	2	no	—
251	C₂×C₃×C₅	—	4	1	no	—
252	C₂×C₃×C₅	—	4	2	no	—
253	C₂×C₃×C₅	#407	1	1	yes	characteristic
254	C₂×C₃×C₅	#417	1	1	yes	characteristic
255	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
256	C₂³×C₃	—	2	1	no	—
257	C₂³×C₃	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
258	C₃×Q₈	—	2	1	no	—
259	C₃×Q₈	#704	1	2	yes	complemented
260	C₃×D₄	—	2	4	no	—
261	C₃×D₄	—	2	1	no	—
262	C₃×D₄	#704	1	2	yes	complemented
263	C₂×C₃×C₄	—	2	4	no	—
264	C₂×C₃×C₄	—	2	1	no	—
265	C₂×C₃×C₄	—	2	1	no	—
266	C₂×C₃×C₄	—	2	1	no	—
267	C₂×C₃×C₄	—	5	1	no	—
268	C₂×C₃×C₄	—	5	1	no	—
269	C₂×C₃×C₄	—	5	1	no	—
270	C₂×C₃×C₄	—	5	1	no	—
271	C₂×C₃×C₄	—	10	2	no	—
272	C₂×C₃×C₄	—	10	1	no	—
273	C₂×C₃×C₄	—	10	1	no	—
274	C₂×C₃×C₄	—	10	1	no	—
275	C₂×C₃×C₄	—	10	1	no	—
276	C₂×C₃×C₄	—	20	2	no	—
277	C₂×C₃×C₄	—	20	1	no	—
278	C₂×C₃×C₄	—	20	1	no	—
279	C₂×C₃×C₄	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
280	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
281	C₃×C₈	—	10	4	no	—
282	C₃×C₈	—	20	2	no	—
283	C₂²×C₅	—	2	1	no	—
284	C₂²×C₅	—	2	2	no	—
285	C₂²×C₅	—	2	1	no	—
286	C₂²×C₅	—	2	1	no	—
287	C₂²×C₅	—	4	1	no	—
288	C₂²×C₅	—	4	1	no	—
289	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
290	C₄×C₅	—	2	1	no	—
291	C₄×C₅	—	2	2	no	—
292	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
293	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
294	Dic₅	—	2	1	no	—
295	Dic₅	—	2	1	no	—
296	Dic₅	—	4	1	no	—
297	Dic₅	—	4	1	no	—
298	Dic₅	C₃×#104	1	2	yes	—
299	Dic₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
300	Dic₅	C₃×#137	1	1	yes	characteristic
301	#41	—	2	2	no	—
302	#41	—	2	2	no	—
303	C₂×Q₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	5	1	no	—
308	C₂²×C₄	—	5	1	no	—
309	C₂²×C₄	—	10	1	no	—
310	C₂²×C₄	—	10	1	no	—
311	C₂²×C₄	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
312	M₁₆	—	10	4	no	—
313	M₁₆	—	20	2	no	—
314	C₂×C₈	—	10	2	no	—
315	C₂×C₈	—	10	2	no	—
316	C₄⋊C₄	—	5	1	no	—
317	C₄⋊C₄	—	5	1	no	—
318	C₄⋊C₄	—	10	1	no	—
319	C₄⋊C₄	—	10	1	no	—
320	C₂²⋊C₄	—	10	1	no	—
321	C₂²⋊C₄	—	10	1	no	—
322	C₂²⋊C₄	—	10	1	no	—
323	C₄²	—	5	1	no	—
324	C₄²	—	5	1	no	—
325	C₄²	—	5	1	no	—
326	C₄²	—	5	1	no	—
327	C₄²	—	10	1	no	—
328	C₄²	—	10	1	no	—
329	C₄²	—	10	1	no	—
330	C₃×C₅	#1489	1	1	yes	characteristic, complemented, Hall
331	C₂²×C₃	—	2	1	no	—
332	C₂²×C₃	—	2	1	no	—
333	C₂²×C₃	—	2	1	no	—
334	C₂²×C₃	—	2	2	no	—
335	C₂²×C₃	—	4	1	no	—
336	C₂²×C₃	—	4	1	no	—
337	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
338	C₃×C₄	—	2	1	no	—
339	C₃×C₄	—	2	2	no	—
340	C₃×C₄	—	5	2	no	—
341	C₃×C₄	—	5	1	no	—
342	C₃×C₄	—	5	1	no	—
343	C₃×C₄	—	10	1	no	—
344	C₃×C₄	—	10	1	no	—
345	C₃×C₄	—	20	1	no	—
346	C₃×C₄	—	20	1	no	—
347	C₃×C₄	C₂×#704	1	1	yes	characteristic
348	C₃×C₄	C₂×#704	1	1	yes	characteristic
349	C₂×C₅	—	4	1	no	—
350	C₂×C₅	—	4	2	no	—
351	C₂×C₅	C₃×#407	1	1	yes	characteristic
352	C₂×C₅	C₃×#417	1	1	yes	characteristic
353	C₂×C₅	C₂×C₃×#104	1	1	yes	characteristic
354	C₂³	—	2	1	no	—
355	C₂³	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
356	Q₈	—	2	1	no	—
357	Q₈	C₃×#704	1	2	yes	complemented
358	D₄	—	2	4	no	—
359	D₄	—	2	1	no	—
360	D₄	C₃×#704	1	2	yes	complemented
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	5	1	no	—
366	C₂×C₄	—	5	1	no	—
367	C₂×C₄	—	5	1	no	—
368	C₂×C₄	—	5	1	no	—
369	C₂×C₄	—	10	2	no	—
370	C₂×C₄	—	10	1	no	—
371	C₂×C₄	—	10	1	no	—
372	C₂×C₄	—	10	1	no	—
373	C₂×C₄	—	10	1	no	—
374	C₂×C₄	—	20	2	no	—
375	C₂×C₄	—	20	1	no	—
376	C₂×C₄	—	20	1	no	—
377	C₂×C₄	C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
378	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
379	C₈	—	10	4	no	—
380	C₈	—	20	2	no	—
381	C₂×C₃	—	4	1	no	—
382	C₂×C₃	—	4	2	no	—
383	C₂×C₃	D₅⋊#104	1	1	yes	characteristic
384	C₂×C₃	#65793	1	1	yes	characteristic
385	C₂×C₃	#65798	1	1	yes	characteristic
386	C₅	C₃×#1489	1	1	yes	characteristic, complemented, Sylow
387	C₂²	—	2	1	no	—
388	C₂²	—	2	1	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	4	1	no	—
392	C₂²	—	4	1	no	—
393	C₂²	C₂×C₃×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
394	C₄	—	2	1	no	—
395	C₄	—	2	2	no	—
396	C₄	—	5	2	no	—
397	C₄	—	5	1	no	—
398	C₄	—	5	1	no	—
399	C₄	—	10	1	no	—
400	C₄	—	10	1	no	—
401	C₄	—	20	1	no	—
402	C₄	—	20	1	no	—
403	C₄	C₂×C₃×#704	1	1	yes	characteristic
404	C₄	C₂×C₃×#704	1	1	yes	characteristic
405	C₃	#10605003	1	1	yes	characteristic, direct product factor, Sylow
406	C₂	—	4	1	no	—
407	C₂	—	4	2	no	—
408	C₂	C₃×D₅⋊#104	1	1	yes	characteristic
409	C₂	C₃×#65793	1	1	yes	characteristic
410	C₂	C₃×#65798	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×C₂²⋊AGL(1,𝔽₅)
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)
Fitting, prime radical	C₂×C₃×C₅×#41	C₄
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
cosocle	C₄×Dic₅	C₂³×C₃
hypercenter	C₂×C₃×#41	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#1489
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#41

Complements

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

N	H ≅ G/N	type
2. C₃×#65758	C₂	semidirect
3. C₃×C₄×#704	C₂	semidirect
4. C₃×Q₈⋊#170	C₂	semidirect
5. C₃×D₄⋊#170	C₂	semidirect
6. #10605003	C₃	direct (N × H)
26. #65758	C₂×C₃	semidirect
27. C₄×#704	C₂×C₃	semidirect
28. Q₈⋊#170	C₂×C₃	semidirect
29. D₄⋊#170	C₂×C₃	semidirect
85. C₃×C₅×Q₈	M₁₆	semidirect
88. C₃×C₅×D₄	M₁₆	semidirect
210. C₅×Q₈	C₃×M₁₆	semidirect
213. C₅×D₄	C₃×M₁₆	semidirect
259. C₃×Q₈	#704	semidirect
262. C₃×D₄	#704	semidirect
330. C₃×C₅	#1489	semidirect
357. Q₈	C₃×#704	semidirect
360. D₄	C₃×#704	semidirect
386. C₅	C₃×#1489	semidirect
405. C₃	#10605003	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#41×Dic₅	2
C₃×#65758	2
C₃×C₄×#704	2
C₃×Q₈⋊#170	2
C₃×D₄⋊#170	2
#10605003	3
C₃×#1489	5

Subgroups — C3×#10605003 #14282228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10605003 #14282228