Subgroups — C₃×#10617441 #14417743

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66266	C₂	1	2	yes	maximal, complemented
2	C₃×D₅×#135	C₂	1	1	yes	maximal, characteristic
3	C₃×#65793	C₂	1	2	yes	maximal, complemented
4	C₃×#65739	C₂	1	2	yes	maximal, complemented
5	C₃×SD₁₆×AGL(1,𝔽₅)	C₂	1	4	yes	maximal, complemented
6	C₃×#65731	C₂	1	2	yes	maximal, complemented
7	C₂×C₃×#3993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#65718	C₂	1	1	yes	maximal, characteristic, complemented
9	#10617441	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₃×D₅×#41	C₂²	1	2	yes	—
11	C₃×Q₈×AGL(1,𝔽₅)	C₂²	1	4	yes	—
12	C₃×D₄×AGL(1,𝔽₅)	—	2	4	no	—
13	C₃×D₄×AGL(1,𝔽₅)	C₂²	1	4	yes	—
14	C₃×#4130	—	2	2	no	—
15	C₃×#4130	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
17	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
18	C₃×D₅⋊M₁₆	C₂²	1	1	yes	characteristic
19	C₃×C₅×#135	C₄	1	1	yes	characteristic, complemented
20	C₃×#4096	C₄	1	2	yes	complemented
21	C₃×#4065	C₄	1	1	yes	characteristic, complemented
22	C₃×D₅×Q₁₆	C₂²	1	1	yes	characteristic
23	C₃×#4062	C₄	1	2	yes	complemented
24	C₃×D₅×SD₁₆	C₂²	1	2	yes	—
25	C₃×#4058	C₄	1	1	yes	characteristic, complemented
26	C₃×D₅×D₈	C₂²	1	1	yes	characteristic
27	C₃×#4050	C₄	1	1	yes	characteristic, complemented
28	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
29	C₃×#4009	C₂²	1	4	yes	—
30	C₃×#4007	C₂²	1	4	yes	—
31	C₃×#3993	C₂²	1	2	yes	—
32	C₃×#3993	C₂²	1	2	yes	—
33	C₃×#3992	C₂²	1	2	yes	—
34	C₃×C₈×AGL(1,𝔽₅)	C₂²	1	2	yes	—
35	C₃×#2952	—	5	1	no	maximal, Hall
36	#66266	C₂×C₃	1	2	yes	complemented
37	D₅×#135	C₂×C₃	1	1	yes	characteristic
38	#65793	C₂×C₃	1	2	yes	complemented
39	#65739	C₂×C₃	1	2	yes	complemented
40	SD₁₆×AGL(1,𝔽₅)	C₂×C₃	1	4	yes	complemented
41	#65731	C₂×C₃	1	2	yes	complemented
42	C₂×#3993	C₂×C₃	1	1	yes	characteristic, complemented
43	#65718	C₂×C₃	1	1	yes	characteristic, complemented
44	C₂²×C₃×AGL(1,𝔽₅)	—	2	2	no	—
45	C₂²×C₃×AGL(1,𝔽₅)	—	2	4	no	—
46	C₃×C₅×#41	C₂×C₄	1	2	yes	complemented
47	C₃×#713	C₂×C₄	1	2	yes	complemented
48	C₃×Q₈×D₅	C₂³	1	2	yes	—
49	C₃×#711	—	2	2	no	—
50	C₃×#711	C₂×C₄	1	2	yes	complemented
51	C₃×D₄×D₅	—	2	2	no	—
52	C₃×D₄×D₅	C₂³	1	2	yes	—
53	C₃×#709	—	2	2	no	—
54	C₃×#709	C₂×C₄	1	2	yes	complemented
55	C₂×C₃×C₄×D₅	—	2	2	no	—
56	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
57	C₃×C₂²⋊AGL(1,𝔽₅)	—	2	4	no	—
58	C₃×C₂²⋊AGL(1,𝔽₅)	D₄	1	2	yes	—
59	C₃×#704	—	2	1	no	—
60	C₃×C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
61	C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
62	C₃×C₄⋊AGL(1,𝔽₅)	D₄	1	2	yes	—
63	C₃×C₄⋊AGL(1,𝔽₅)	C₂³	1	4	yes	—
64	C₃×C₄×AGL(1,𝔽₅)	—	2	4	no	—
65	C₃×C₄×AGL(1,𝔽₅)	C₂³	1	2	yes	—
66	C₃×#700	—	2	1	no	—
67	C₃×D₅⋊C₈	C₂³	1	2	yes	—
68	C₃×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
69	C₃×C₅×SD₁₆	C₂×C₄	1	2	yes	complemented
70	C₃×C₅×D₈	C₂×C₄	1	1	yes	characteristic, complemented
71	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic, complemented
72	C₃×#689	C₂×C₄	1	2	yes	complemented
73	C₃×#688	C₂×C₄	1	2	yes	complemented
74	C₃×#687	C₂×C₄	1	2	yes	complemented
75	C₃×#686	C₂×C₄	1	2	yes	complemented
76	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic, complemented
77	C₃×Dic₂₀	C₂×C₄	1	1	yes	characteristic, complemented
78	C₃×D₄₀	C₂×C₄	1	1	yes	characteristic, complemented
79	C₃×#677	C₂×C₄	1	2	yes	complemented
80	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic, complemented
81	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic, complemented
82	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
83	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
84	C₂×C₃×#135	—	5	1	no	—
85	C₃×#520	—	5	2	no	—
86	C₃×#442	—	5	2	no	—
87	C₃×#441	—	5	2	no	—
88	C₃×C₄×SD₁₆	—	5	4	no	—
89	C₂×C₃×#106	—	5	1	no	—
90	C₃×#417	—	5	2	no	—
91	C₃×#405	—	5	1	no	—
92	D₅×#41	C₂²×C₃	1	2	yes	—
93	Q₈×AGL(1,𝔽₅)	C₂²×C₃	1	4	yes	—
94	D₄×AGL(1,𝔽₅)	—	2	4	no	—
95	D₄×AGL(1,𝔽₅)	C₂²×C₃	1	4	yes	—
96	#4130	—	2	2	no	—
97	#4130	C₂²×C₃	1	1	yes	characteristic
98	C₂×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
99	C₂×C₄⋊AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
100	D₅⋊M₁₆	C₂²×C₃	1	1	yes	characteristic
101	C₅×#135	C₃×C₄	1	1	yes	characteristic, complemented
102	#4096	C₃×C₄	1	2	yes	complemented
103	#4065	C₃×C₄	1	1	yes	characteristic, complemented
104	D₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
105	#4062	C₃×C₄	1	2	yes	complemented
106	D₅×SD₁₆	C₂²×C₃	1	2	yes	—
107	#4058	C₃×C₄	1	1	yes	characteristic, complemented
108	D₅×D₈	C₂²×C₃	1	1	yes	characteristic
109	#4050	C₃×C₄	1	1	yes	characteristic, complemented
110	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
111	#4009	C₂²×C₃	1	4	yes	—
112	#4007	C₂²×C₃	1	4	yes	—
113	#3993	C₂²×C₃	1	2	yes	—
114	#3993	C₂²×C₃	1	2	yes	—
115	#3992	C₂²×C₃	1	2	yes	—
116	C₈×AGL(1,𝔽₅)	C₂²×C₃	1	2	yes	—
117	#2952	—	5	1	no	Sylow
118	C₂²×C₃×D₅	—	2	2	no	—
119	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
120	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
121	C₂×C₃×AGL(1,𝔽₅)	—	8	4	no	—
122	C₂×C₃×AGL(1,𝔽₅)	—	8	2	no	—
123	C₂×C₃×AGL(1,𝔽₅)	C₂×D₄	1	4	yes	—
124	C₃×C₅×Q₈	C₂²×C₄	1	2	yes	—
125	C₃×C₅×D₄	—	2	2	no	—
126	C₃×C₅×D₄	C₂²×C₄	1	2	yes	—
127	C₂×C₃×C₄×C₅	—	2	2	no	—
128	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
129	C₃×C₂²⋊D₅	—	2	2	no	—
130	C₃×C₂²⋊D₅	—	2	2	no	—
131	C₃×C₂²⋊D₅	—	2	2	no	—
132	C₂×C₃×Dic₅	—	2	2	no	—
133	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
134	C₃×D₂₀	—	2	2	no	—
135	C₃×D₂₀	C₂²×C₄	1	2	yes	—
136	C₃×C₄×D₅	—	2	2	no	—
137	C₃×C₄×D₅	—	2	2	no	—
138	C₃×C₄×D₅	—	2	2	no	—
139	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
140	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
141	C₃×C₄×D₅	#41	1	1	yes	characteristic
142	C₃×C₄×D₅	C₂⁴	1	1	yes	characteristic
143	C₃×Dic₁₀	—	2	2	no	—
144	C₃×Dic₁₀	C₂²×C₄	1	2	yes	—
145	C₃×#170	—	2	2	no	—
146	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
147	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
148	C₃×#168	C₂²×C₄	1	1	yes	characteristic
149	C₃×#168	C₂²×C₄	1	1	yes	characteristic
150	C₂×C₃×#41	—	5	2	no	—
151	C₃×#135	—	5	2	no	—
152	C₃×#135	—	5	1	no	—
153	C₃×#135	—	5	2	no	—
154	C₃×#135	—	5	1	no	—
155	C₃×#135	—	5	1	no	—
156	C₃×#135	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
157	C₂×C₃×Q₁₆	—	5	1	no	—
158	C₂×C₃×SD₁₆	—	5	2	no	—
159	C₂×C₃×D₈	—	5	1	no	—
160	C₂×C₃×M₁₆	—	5	1	no	—
161	C₂²×C₃×C₈	—	5	1	no	—
162	C₃×C₄×Q₈	—	5	4	no	—
163	C₃×C₄×D₄	—	5	4	no	—
164	C₃×C₄×D₄	—	10	4	no	—
165	C₃×#117	—	5	1	no	—
166	C₃×#117	—	10	2	no	—
167	C₂×C₃×C₄⋊C₄	—	5	2	no	—
168	C₂×C₃×C₄⋊C₄	—	10	2	no	—
169	C₃×#106	—	5	2	no	—
170	C₃×#106	—	5	2	no	—
171	C₃×#103	—	5	4	no	—
172	C₃×#102	—	5	4	no	—
173	C₃×#97	—	5	2	no	—
174	C₃×C₄×C₈	—	5	2	no	—
175	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
176	C₂²×AGL(1,𝔽₅)	—	2	4	no	—
177	C₅×#41	C₂×C₃×C₄	1	2	yes	complemented
178	#713	C₂×C₃×C₄	1	2	yes	complemented
179	Q₈×D₅	C₂³×C₃	1	2	yes	—
180	#711	—	2	2	no	—
181	#711	C₂×C₃×C₄	1	2	yes	complemented
182	D₄×D₅	—	2	2	no	—
183	D₄×D₅	C₂³×C₃	1	2	yes	—
184	#709	—	2	2	no	—
185	#709	C₂×C₃×C₄	1	2	yes	complemented
186	C₂×C₄×D₅	—	2	2	no	—
187	C₂×C₄×D₅	C₂³×C₃	1	1	yes	characteristic
188	C₂²⋊AGL(1,𝔽₅)	—	2	4	no	—
189	C₂²⋊AGL(1,𝔽₅)	C₃×D₄	1	2	yes	—
190	#704	—	2	1	no	—
191	C₄⋊AGL(1,𝔽₅)	—	2	4	no	—
192	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
193	C₄⋊AGL(1,𝔽₅)	C₃×D₄	1	2	yes	—
194	C₄⋊AGL(1,𝔽₅)	C₂³×C₃	1	4	yes	—
195	C₄×AGL(1,𝔽₅)	—	2	4	no	—
196	C₄×AGL(1,𝔽₅)	C₂³×C₃	1	2	yes	—
197	#700	—	2	1	no	—
198	D₅⋊C₈	C₂³×C₃	1	2	yes	—
199	C₅×Q₁₆	C₂×C₃×C₄	1	1	yes	characteristic, complemented
200	C₅×SD₁₆	C₂×C₃×C₄	1	2	yes	complemented
201	C₅×D₈	C₂×C₃×C₄	1	1	yes	characteristic, complemented
202	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic, complemented
203	#689	C₂×C₃×C₄	1	2	yes	complemented
204	#688	C₂×C₃×C₄	1	2	yes	complemented
205	#687	C₂×C₃×C₄	1	2	yes	complemented
206	#686	C₂×C₃×C₄	1	2	yes	complemented
207	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic, complemented
208	Dic₂₀	C₂×C₃×C₄	1	1	yes	characteristic, complemented
209	D₄₀	C₂×C₃×C₄	1	1	yes	characteristic, complemented
210	#677	C₂×C₃×C₄	1	2	yes	complemented
211	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
212	C₈×D₅	C₂×C₃×C₄	1	1	yes	characteristic, complemented
213	C₈×D₅	C₂³×C₃	1	1	yes	characteristic
214	C₈×D₅	C₂³×C₃	1	1	yes	characteristic
215	C₂×#135	—	5	1	no	—
216	#520	—	5	2	no	—
217	#442	—	5	2	no	—
218	#441	—	5	2	no	—
219	C₄×SD₁₆	—	5	4	no	—
220	C₂×#106	—	5	1	no	—
221	#417	—	5	2	no	—
222	#405	—	5	1	no	—
223	C₂²×C₃×C₅	—	2	2	no	—
224	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
225	C₂×C₃×D₅	—	2	2	no	—
226	C₂×C₃×D₅	—	2	2	no	—
227	C₂×C₃×D₅	—	4	2	no	—
228	C₂×C₃×D₅	—	4	2	no	—
229	C₂×C₃×D₅	C₄×D₄	1	1	yes	characteristic
230	C₂×C₃×D₅	C₂²×D₄	1	1	yes	characteristic
231	C₃×AGL(1,𝔽₅)	—	2	4	no	—
232	C₃×AGL(1,𝔽₅)	—	4	4	no	—
233	C₃×C₄×C₅	—	2	2	no	—
234	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
235	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
236	C₃×Dic₅	—	2	2	no	—
237	C₃×Dic₅	C₄×D₄	1	1	yes	characteristic
238	C₃×Dic₅	C₂×#41	1	1	yes	characteristic
239	C₃×#41	—	5	2	no	—
240	C₃×#41	—	5	2	no	—
241	C₃×#41	—	5	2	no	—
242	C₃×#41	—	10	2	no	—
243	C₃×#41	—	10	2	no	—
244	C₃×#41	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
245	C₂×C₃×Q₈	—	5	2	no	—
246	C₂×C₃×D₄	—	5	2	no	—
247	C₂×C₃×D₄	—	10	2	no	—
248	C₂²×C₃×C₄	—	5	1	no	—
249	C₂²×C₃×C₄	—	10	2	no	—
250	C₂²×C₃×C₄	—	10	4	no	—
251	C₂²×C₃×C₄	—	10	2	no	—
252	C₃×Q₁₆	—	5	2	no	—
253	C₃×Q₁₆	—	5	1	no	—
254	C₃×Q₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
255	C₃×SD₁₆	—	5	2	no	—
256	C₃×SD₁₆	—	5	2	no	—
257	C₃×SD₁₆	—	5	2	no	—
258	C₃×SD₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
259	C₃×D₈	—	5	2	no	—
260	C₃×D₈	—	5	1	no	—
261	C₃×D₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
262	C₃×M₁₆	—	10	1	no	—
263	C₃×M₁₆	—	10	1	no	—
264	C₂×C₃×C₈	—	5	1	no	—
265	C₂×C₃×C₈	—	5	1	no	—
266	C₂×C₃×C₈	—	5	1	no	—
267	C₂×C₃×C₈	—	5	1	no	—
268	C₂×C₃×C₈	—	5	1	no	—
269	C₂×C₃×C₈	—	5	2	no	—
270	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
271	C₃×C₄⋊C₄	—	5	2	no	—
272	C₃×C₄⋊C₄	—	5	4	no	—
273	C₃×C₄⋊C₄	—	10	2	no	—
274	C₃×C₄⋊C₄	—	10	4	no	—
275	C₃×C₂²⋊C₄	—	5	2	no	—
276	C₃×C₂²⋊C₄	—	10	4	no	—
277	C₃×C₄²	—	5	2	no	—
278	C₃×C₄²	—	10	4	no	—
279	C₂²×D₅	—	2	2	no	—
280	C₂²×D₅	C₂×C₃×D₄	1	1	yes	characteristic
281	C₂×AGL(1,𝔽₅)	—	2	4	no	—
282	C₂×AGL(1,𝔽₅)	—	8	4	no	—
283	C₂×AGL(1,𝔽₅)	—	8	2	no	—
284	C₂×AGL(1,𝔽₅)	C₂×C₃×D₄	1	4	yes	—
285	C₅×Q₈	C₂²×C₃×C₄	1	2	yes	—
286	C₅×D₄	—	2	2	no	—
287	C₅×D₄	C₂²×C₃×C₄	1	2	yes	—
288	C₂×C₄×C₅	—	2	2	no	—
289	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
290	C₂²⋊D₅	—	2	2	no	—
291	C₂²⋊D₅	—	2	2	no	—
292	C₂²⋊D₅	—	2	2	no	—
293	C₂×Dic₅	—	2	2	no	—
294	C₂×Dic₅	C₃×#41	1	1	yes	characteristic
295	D₂₀	—	2	2	no	—
296	D₂₀	C₂²×C₃×C₄	1	2	yes	—
297	C₄×D₅	—	2	2	no	—
298	C₄×D₅	—	2	2	no	—
299	C₄×D₅	—	2	2	no	—
300	C₄×D₅	C₂²×C₃×C₄	1	1	yes	characteristic
301	C₄×D₅	C₂×C₃×D₄	1	1	yes	characteristic
302	C₄×D₅	C₃×#41	1	1	yes	characteristic
303	C₄×D₅	C₂⁴×C₃	1	1	yes	characteristic
304	Dic₁₀	—	2	2	no	—
305	Dic₁₀	C₂²×C₃×C₄	1	2	yes	—
306	#170	—	2	2	no	—
307	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
308	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
309	#168	C₂²×C₃×C₄	1	1	yes	characteristic
310	#168	C₂²×C₃×C₄	1	1	yes	characteristic
311	C₂×#41	—	5	2	no	—
312	#135	—	5	2	no	—
313	#135	—	5	1	no	—
314	#135	—	5	2	no	—
315	#135	—	5	1	no	—
316	#135	—	5	1	no	—
317	#135	C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
318	C₂×Q₁₆	—	5	1	no	—
319	C₂×SD₁₆	—	5	2	no	—
320	C₂×D₈	—	5	1	no	—
321	C₂×M₁₆	—	5	1	no	—
322	C₂²×C₈	—	5	1	no	—
323	C₄×Q₈	—	5	4	no	—
324	C₄×D₄	—	5	4	no	—
325	C₄×D₄	—	10	4	no	—
326	#117	—	5	1	no	—
327	#117	—	10	2	no	—
328	C₂×C₄⋊C₄	—	5	2	no	—
329	C₂×C₄⋊C₄	—	10	2	no	—
330	#106	—	5	2	no	—
331	#106	—	5	2	no	—
332	#103	—	5	4	no	—
333	#102	—	5	4	no	—
334	#97	—	5	2	no	—
335	C₄×C₈	—	5	2	no	—
336	C₂×C₃×C₅	—	2	1	no	—
337	C₂×C₃×C₅	—	4	2	no	—
338	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
339	C₃×D₅	—	2	1	no	—
340	C₃×D₅	—	4	2	no	—
341	C₃×D₅	#577	1	2	yes	—
342	C₂³×C₃	—	5	1	no	—
343	C₂³×C₃	—	10	2	no	—
344	C₃×Q₈	—	5	2	no	—
345	C₃×Q₈	—	10	2	no	—
346	C₃×Q₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
347	C₃×D₄	—	2	2	no	—
348	C₃×D₄	—	5	2	no	—
349	C₃×D₄	—	10	2	no	—
350	C₃×D₄	—	10	2	no	—
351	C₃×D₄	—	10	2	no	—
352	C₃×D₄	—	10	2	no	—
353	C₃×D₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
354	C₂×C₃×C₄	—	2	2	no	—
355	C₂×C₃×C₄	—	5	1	no	—
356	C₂×C₃×C₄	—	5	1	no	—
357	C₂×C₃×C₄	—	5	1	no	—
358	C₂×C₃×C₄	—	5	1	no	—
359	C₂×C₃×C₄	—	5	1	no	—
360	C₂×C₃×C₄	—	5	4	no	—
361	C₂×C₃×C₄	—	10	2	no	—
362	C₂×C₃×C₄	—	10	2	no	—
363	C₂×C₃×C₄	—	10	2	no	—
364	C₂×C₃×C₄	—	10	2	no	—
365	C₂×C₃×C₄	—	10	4	no	—
366	C₂×C₃×C₄	—	40	4	no	—
367	C₂×C₃×C₄	—	40	2	no	—
368	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
369	C₃×C₈	—	5	1	no	—
370	C₃×C₈	—	5	1	no	—
371	C₃×C₈	—	10	2	no	—
372	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
373	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
374	C₂²×C₅	—	2	2	no	—
375	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
376	C₂×D₅	—	2	2	no	—
377	C₂×D₅	—	2	2	no	—
378	C₂×D₅	—	4	2	no	—
379	C₂×D₅	—	4	2	no	—
380	C₂×D₅	C₃×C₄×D₄	1	1	yes	characteristic
381	C₂×D₅	C₂²×C₃×D₄	1	1	yes	characteristic
382	AGL(1,𝔽₅)	—	2	4	no	—
383	AGL(1,𝔽₅)	—	4	4	no	—
384	C₄×C₅	—	2	2	no	—
385	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
386	C₄×C₅	C₂³×C₃×C₄	1	1	yes	characteristic
387	Dic₅	—	2	2	no	—
388	Dic₅	C₃×C₄×D₄	1	1	yes	characteristic
389	Dic₅	C₂×C₃×#41	1	1	yes	characteristic
390	#41	—	5	2	no	—
391	#41	—	5	2	no	—
392	#41	—	5	2	no	—
393	#41	—	10	2	no	—
394	#41	—	10	2	no	—
395	#41	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
396	C₂×Q₈	—	5	2	no	—
397	C₂×D₄	—	5	2	no	—
398	C₂×D₄	—	10	2	no	—
399	C₂²×C₄	—	5	1	no	—
400	C₂²×C₄	—	10	2	no	—
401	C₂²×C₄	—	10	4	no	—
402	C₂²×C₄	—	10	2	no	—
403	Q₁₆	—	5	2	no	—
404	Q₁₆	—	5	1	no	—
405	Q₁₆	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
406	SD₁₆	—	5	2	no	—
407	SD₁₆	—	5	2	no	—
408	SD₁₆	—	5	2	no	—
409	SD₁₆	C₂×C₃×AGL(1,𝔽₅)	1	2	yes	complemented
410	D₈	—	5	2	no	—
411	D₈	—	5	1	no	—
412	D₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
413	M₁₆	—	10	1	no	—
414	M₁₆	—	10	1	no	—
415	C₂×C₈	—	5	1	no	—
416	C₂×C₈	—	5	1	no	—
417	C₂×C₈	—	5	1	no	—
418	C₂×C₈	—	5	1	no	—
419	C₂×C₈	—	5	1	no	—
420	C₂×C₈	—	5	2	no	—
421	C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
422	C₄⋊C₄	—	5	2	no	—
423	C₄⋊C₄	—	5	4	no	—
424	C₄⋊C₄	—	10	2	no	—
425	C₄⋊C₄	—	10	4	no	—
426	C₂²⋊C₄	—	5	2	no	—
427	C₂²⋊C₄	—	10	4	no	—
428	C₄²	—	5	2	no	—
429	C₄²	—	10	4	no	—
430	C₃×C₅	#2952	1	1	yes	characteristic, complemented, Hall
431	C₂²×C₃	—	2	2	no	—
432	C₂²×C₃	—	5	1	no	—
433	C₂²×C₃	—	5	1	no	—
434	C₂²×C₃	—	10	2	no	—
435	C₂²×C₃	—	10	2	no	—
436	C₂²×C₃	—	20	2	no	—
437	C₂²×C₃	—	20	2	no	—
438	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
439	C₃×C₄	—	2	2	no	—
440	C₃×C₄	—	5	1	no	—
441	C₃×C₄	—	5	1	no	—
442	C₃×C₄	—	10	4	no	—
443	C₃×C₄	—	10	2	no	—
444	C₃×C₄	—	20	4	no	—
445	C₃×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
446	C₃×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
447	C₂×C₅	—	2	1	no	—
448	C₂×C₅	—	4	2	no	—
449	C₂×C₅	C₂×C₃×C₄×D₄	1	1	yes	characteristic
450	D₅	—	2	1	no	—
451	D₅	—	4	2	no	—
452	D₅	C₃×#577	1	2	yes	—
453	C₂³	—	5	1	no	—
454	C₂³	—	10	2	no	—
455	Q₈	—	5	2	no	—
456	Q₈	—	10	2	no	—
457	Q₈	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
458	D₄	—	2	2	no	—
459	D₄	—	5	2	no	—
460	D₄	—	10	2	no	—
461	D₄	—	10	2	no	—
462	D₄	—	10	2	no	—
463	D₄	—	10	2	no	—
464	D₄	C₂²×C₃×AGL(1,𝔽₅)	1	2	yes	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	5	1	no	—
467	C₂×C₄	—	5	1	no	—
468	C₂×C₄	—	5	1	no	—
469	C₂×C₄	—	5	1	no	—
470	C₂×C₄	—	5	1	no	—
471	C₂×C₄	—	5	4	no	—
472	C₂×C₄	—	10	2	no	—
473	C₂×C₄	—	10	2	no	—
474	C₂×C₄	—	10	2	no	—
475	C₂×C₄	—	10	2	no	—
476	C₂×C₄	—	10	4	no	—
477	C₂×C₄	—	40	4	no	—
478	C₂×C₄	—	40	2	no	—
479	C₂×C₄	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
480	C₈	—	5	1	no	—
481	C₈	—	5	1	no	—
482	C₈	—	10	2	no	—
483	C₈	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
484	C₈	C₂²×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
485	C₂×C₃	—	2	1	no	—
486	C₂×C₃	—	4	2	no	—
487	C₂×C₃	—	5	2	no	—
488	C₂×C₃	—	10	1	no	—
489	C₂×C₃	—	20	2	no	—
490	C₂×C₃	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
491	C₅	C₃×#2952	1	1	yes	characteristic, complemented, Sylow
492	C₂²	—	2	2	no	—
493	C₂²	—	5	1	no	—
494	C₂²	—	5	1	no	—
495	C₂²	—	10	2	no	—
496	C₂²	—	10	2	no	—
497	C₂²	—	20	2	no	—
498	C₂²	—	20	2	no	—
499	C₂²	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
500	C₄	—	2	2	no	—
501	C₄	—	5	1	no	—
502	C₄	—	5	1	no	—
503	C₄	—	10	4	no	—
504	C₄	—	10	2	no	—
505	C₄	—	20	4	no	—
506	C₄	C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
507	C₄	C₂³×C₃×AGL(1,𝔽₅)	1	1	yes	characteristic
508	C₃	#10617441	1	1	yes	characteristic, direct product factor, Sylow
509	C₂	—	2	1	no	—
510	C₂	—	4	2	no	—
511	C₂	—	5	2	no	—
512	C₂	—	10	1	no	—
513	C₂	—	20	2	no	—
514	C₂	C₂×C₃×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₂³×C₃×C₄
Frattini	C₄	C₂³×C₃×AGL(1,𝔽₅)
Fitting	C₃×C₅×#135	C₄
socle	C₂×C₃×C₅	C₂×C₄×D₄
prime radical	C₃×D₅×#135	C₂
cosocle	C₄×D₅	C₂⁴×C₃
hypercenter	C₃×#135	AGL(1,𝔽₅)
hypocenter	C₅	C₃×#2952
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#135

Complements

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

N	H ≅ G/N	type
1. C₃×#66266	C₂	semidirect
3. C₃×#65793	C₂	semidirect
4. C₃×#65739	C₂	semidirect
5. C₃×SD₁₆×AGL(1,𝔽₅)	C₂	semidirect
6. C₃×#65731	C₂	semidirect
7. C₂×C₃×#3993	C₂	semidirect
8. C₃×#65718	C₂	semidirect
9. #10617441	C₃	direct (N × H)
19. C₃×C₅×#135	C₄	semidirect
20. C₃×#4096	C₄	semidirect
21. C₃×#4065	C₄	semidirect
23. C₃×#4062	C₄	semidirect
25. C₃×#4058	C₄	semidirect
27. C₃×#4050	C₄	semidirect
36. #66266	C₂×C₃	semidirect
38. #65793	C₂×C₃	semidirect
39. #65739	C₂×C₃	semidirect
40. SD₁₆×AGL(1,𝔽₅)	C₂×C₃	semidirect
41. #65731	C₂×C₃	semidirect
42. C₂×#3993	C₂×C₃	semidirect
43. #65718	C₂×C₃	semidirect
46. C₃×C₅×#41	C₂×C₄	semidirect
47. C₃×#713	C₂×C₄	semidirect
50. C₃×#711	C₂×C₄	semidirect
54. C₃×#709	C₂×C₄	semidirect
68. C₃×C₅×Q₁₆	C₂×C₄	semidirect
69. C₃×C₅×SD₁₆	C₂×C₄	semidirect
70. C₃×C₅×D₈	C₂×C₄	semidirect
71. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
72. C₃×#689	C₂×C₄	semidirect
73. C₃×#688	C₂×C₄	semidirect
74. C₃×#687	C₂×C₄	semidirect
75. C₃×#686	C₂×C₄	semidirect
76. C₂×C₃×#168	C₂×C₄	semidirect
77. C₃×Dic₂₀	C₂×C₄	semidirect
78. C₃×D₄₀	C₂×C₄	semidirect
79. C₃×#677	C₂×C₄	semidirect
80. C₃×C₈×D₅	C₂×C₄	semidirect
81. C₃×C₈×D₅	C₂×C₄	semidirect
101. C₅×#135	C₃×C₄	semidirect
102. #4096	C₃×C₄	semidirect
103. #4065	C₃×C₄	semidirect
105. #4062	C₃×C₄	semidirect
107. #4058	C₃×C₄	semidirect
109. #4050	C₃×C₄	semidirect
156. C₃×#135	AGL(1,𝔽₅)	semidirect
177. C₅×#41	C₂×C₃×C₄	semidirect
178. #713	C₂×C₃×C₄	semidirect
181. #711	C₂×C₃×C₄	semidirect
185. #709	C₂×C₃×C₄	semidirect
199. C₅×Q₁₆	C₂×C₃×C₄	semidirect
200. C₅×SD₁₆	C₂×C₃×C₄	semidirect
201. C₅×D₈	C₂×C₃×C₄	semidirect
202. C₂×C₅×C₈	C₂×C₃×C₄	semidirect
203. #689	C₂×C₃×C₄	semidirect
204. #688	C₂×C₃×C₄	semidirect
205. #687	C₂×C₃×C₄	semidirect
206. #686	C₂×C₃×C₄	semidirect
207. C₂×#168	C₂×C₃×C₄	semidirect
208. Dic₂₀	C₂×C₃×C₄	semidirect
209. D₄₀	C₂×C₃×C₄	semidirect
210. #677	C₂×C₃×C₄	semidirect
211. C₈×D₅	C₂×C₃×C₄	semidirect
212. C₈×D₅	C₂×C₃×C₄	semidirect
244. C₃×#41	C₂×AGL(1,𝔽₅)	semidirect
254. C₃×Q₁₆	C₂×AGL(1,𝔽₅)	semidirect
258. C₃×SD₁₆	C₂×AGL(1,𝔽₅)	semidirect
261. C₃×D₈	C₂×AGL(1,𝔽₅)	semidirect
270. C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	semidirect
317. #135	C₃×AGL(1,𝔽₅)	semidirect
395. #41	C₂×C₃×AGL(1,𝔽₅)	semidirect
405. Q₁₆	C₂×C₃×AGL(1,𝔽₅)	semidirect
409. SD₁₆	C₂×C₃×AGL(1,𝔽₅)	semidirect
412. D₈	C₂×C₃×AGL(1,𝔽₅)	semidirect
421. C₂×C₈	C₂×C₃×AGL(1,𝔽₅)	semidirect
430. C₃×C₅	#2952	semidirect
491. C₅	C₃×#2952	semidirect
508. C₃	#10617441	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×#66266	2
C₃×D₅×#135	2
C₃×#65793	2
C₃×#65739	2
C₃×SD₁₆×AGL(1,𝔽₅)	2
C₃×#65731	2
C₂×C₃×#3993	2
C₃×#65718	2
#10617441	3
C₃×#2952	5

Subgroups — C3×#10617441 #14417743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10617441 #14417743