Subgroups — C₅×#78688 #14411965

Order 1920.

1600 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#1004	C₂	1	1	yes	maximal, characteristic
2	C₅×#5486	C₂	1	2	yes	maximal, complemented
3	C₂×C₃×C₅×#99	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#834	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#806	C₂	1	2	yes	maximal, complemented
6	C₅×D₃×#99	C₂	1	4	yes	maximal, complemented
7	C₅×#4740	C₂	1	4	yes	maximal, complemented
8	C₅×#2883	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₂²×C₅×#67	C₄	1	2	yes	complemented
11	C₂²×C₅×#67	C₂²	1	1	yes	characteristic
12	C₂²×C₅×#67	C₂²	1	1	yes	characteristic
13	C₅×#1004	—	2	4	no	—
14	C₅×#1004	—	2	4	no	—
15	C₂×C₅×#237	C₄	1	2	yes	complemented
16	C₂×C₅×#237	C₂²	1	2	yes	—
17	C₂×C₅×D₃×D₄	C₄	1	2	yes	complemented
18	C₂×C₅×D₃×D₄	C₂²	1	2	yes	—
19	C₂×C₅×#235	C₄	1	2	yes	complemented
20	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
21	C₂×C₅×#217	C₂²	1	2	yes	complemented
22	C₄×C₅×#67	—	2	4	no	—
23	C₅×#67⋊C₄	—	2	4	no	—
24	C₅×D₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
25	C₅×#877	C₂²	1	4	yes	complemented
26	C₃×C₅×#99	C₂²	1	4	yes	complemented
27	C₅×#834	C₂²	1	4	yes	complemented
28	C₅×#806	C₂²	1	8	yes	complemented
29	#78688	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₂×C₅×H(5,𝔽₂)	—	3	1	no	—
31	C₅×#518	—	3	2	no	—
32	C₅×#410	—	3	4	no	—
33	C₂×C₅×#99	—	3	4	no	—
34	C₂×C₅×#99	—	3	1	no	—
35	C₂×C₅×#99	—	3	2	no	—
36	C₂×C₅×#99	D₃	1	1	yes	characteristic, complemented
37	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic, complemented
38	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
39	C₂³×C₅×D₃	C₂×C₄	1	1	yes	characteristic, complemented
40	C₂³×C₅×D₃	C₂×C₄	1	2	yes	complemented
41	C₂³×C₅×D₃	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₅×D₄	—	2	2	no	—
43	C₂×C₃×C₅×D₄	—	2	2	no	—
44	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	complemented
45	C₂×C₃×C₅×D₄	C₂³	1	2	yes	—
46	C₂²×C₃×C₄×C₅	—	2	2	no	—
47	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic, complemented
48	C₂×C₅×#67	—	2	2	no	—
49	C₂×C₅×#67	—	2	2	no	—
50	C₂×C₅×#67	—	2	4	no	—
51	C₂×C₅×#67	—	2	4	no	—
52	C₂×C₅×#67	C₂×C₄	1	2	yes	—
53	C₂×C₅×#67	C₂×C₄	1	2	yes	—
54	C₂×C₅×#67	C₂×C₄	1	2	yes	complemented
55	C₂×C₅×#67	C₂×C₄	1	2	yes	complemented
56	C₂×C₅×#67	C₂×C₄	1	2	yes	complemented
57	C₂×C₅×#67	D₄	1	4	yes	—
58	C₂×C₅×#67	D₄	1	4	yes	—
59	C₂×C₅×#67	C₂³	1	2	yes	—
60	C₂²×C₅×Dic₃	—	2	2	no	—
61	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic, complemented
62	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	complemented
63	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
64	C₅×#237	—	2	8	no	—
65	C₅×#237	—	2	8	no	—
66	C₅×D₃×D₄	—	2	8	no	—
67	C₅×D₃×D₄	—	2	8	no	—
68	C₅×#235	—	2	4	no	—
69	C₅×#235	—	2	4	no	—
70	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	complemented
71	C₂×C₄×C₅×D₃	—	2	4	no	—
72	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	complemented
73	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	complemented
74	C₂×C₅×Dic₆	C₂×C₄	1	2	yes	complemented
75	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
76	C₃×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
77	C₅×#217	—	2	2	no	—
78	C₅×#217	C₂³	1	4	yes	—
79	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
80	C₅×Dic₃⋊C₄	—	2	4	no	—
81	C₄×C₅×Dic₃	—	2	4	no	—
82	C₂×#1004	C₂×C₅	1	1	yes	characteristic
83	#5486	C₂×C₅	1	2	yes	complemented
84	C₂×C₃×#99	C₂×C₅	1	1	yes	characteristic, complemented
85	C₂×#834	C₂×C₅	1	1	yes	characteristic, complemented
86	C₂×#806	C₂×C₅	1	2	yes	complemented
87	D₃×#99	C₂×C₅	1	4	yes	complemented
88	#4740	C₂×C₅	1	4	yes	complemented
89	C₅×H(5,𝔽₂)	—	6	4	no	—
90	C₅×H(5,𝔽₂)	—	6	4	no	—
91	C₂×C₅×#41	—	3	2	no	—
92	C₂×C₅×#41	—	3	2	no	—
93	C₂×C₅×#41	—	3	2	no	—
94	C₂²×C₅×D₄	—	3	1	no	—
95	C₂²×C₅×D₄	—	3	2	no	—
96	C₂²×C₅×D₄	—	3	1	no	—
97	C₂²×C₅×D₄	—	3	2	no	—
98	C₂²×C₅×D₄	—	3	2	no	—
99	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
100	C₄×C₅×D₄	—	6	4	no	—
101	C₄×C₅×D₄	—	6	4	no	—
102	C₅×#117	—	3	4	no	—
103	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
104	C₂×C₅×C₂²⋊C₄	—	3	4	no	—
105	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	complemented
106	C₅×#99	—	3	4	no	—
107	C₅×#99	—	3	8	no	—
108	C₅×#99	C₂×D₃	1	4	yes	complemented
109	#2883	—	3	1	no	Sylow
110	C₂³×C₃×C₅	—	2	1	no	—
111	C₂³×C₃×C₅	—	2	1	no	—
112	C₂³×C₃×C₅	—	8	1	no	—
113	C₂³×C₃×C₅	—	8	1	no	—
114	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
115	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
116	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
117	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
118	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
119	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
120	C₂²×C₅×D₃	—	2	2	no	—
121	C₂²×C₅×D₃	—	2	4	no	—
122	C₂²×C₅×D₃	—	8	1	no	—
123	C₂²×C₅×D₃	—	8	2	no	—
124	C₂²×C₅×D₃	—	8	1	no	—
125	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
126	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
127	C₂²×C₅×D₃	C₂×D₄	1	4	yes	—
128	C₃×C₅×D₄	—	2	4	no	—
129	C₃×C₅×D₄	—	2	4	no	—
130	C₂×C₃×C₄×C₅	—	2	1	no	—
131	C₂×C₃×C₄×C₅	—	2	1	no	—
132	C₂×C₃×C₄×C₅	—	2	4	no	—
133	C₂×C₃×C₄×C₅	—	8	2	no	—
134	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
135	C₅×#67	—	2	2	no	—
136	C₅×#67	—	2	2	no	—
137	C₅×#67	—	2	4	no	—
138	C₅×#67	—	2	4	no	—
139	C₅×#67	—	2	4	no	—
140	C₅×#67	C₂²⋊C₄	1	4	yes	—
141	C₅×#67	C₂²⋊C₄	1	4	yes	—
142	C₅×#67	C₂²⋊C₄	1	8	yes	—
143	C₂×C₅×Dic₃	—	2	2	no	—
144	C₂×C₅×Dic₃	—	2	4	no	—
145	C₂×C₅×Dic₃	—	2	4	no	—
146	C₂×C₅×Dic₃	—	8	2	no	—
147	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
148	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
149	C₂×C₅×Dic₃	C₂×D₄	1	4	yes	—
150	C₅×D₁₂	—	2	4	no	—
151	C₄×C₅×D₃	—	2	4	no	—
152	C₄×C₅×D₃	—	2	4	no	—
153	C₄×C₅×D₃	—	8	4	no	—
154	C₅×Dic₆	—	2	4	no	—
155	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
156	C₂²×#67	C₄×C₅	1	2	yes	complemented
157	C₂²×#67	C₂²×C₅	1	1	yes	characteristic
158	C₂²×#67	C₂²×C₅	1	1	yes	characteristic
159	#1004	—	2	4	no	—
160	#1004	—	2	4	no	—
161	C₂×#237	C₄×C₅	1	2	yes	complemented
162	C₂×#237	C₂²×C₅	1	2	yes	—
163	C₂×D₃×D₄	C₄×C₅	1	2	yes	complemented
164	C₂×D₃×D₄	C₂²×C₅	1	2	yes	—
165	C₂×#235	C₄×C₅	1	2	yes	complemented
166	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	complemented
167	C₂×#217	C₂²×C₅	1	2	yes	complemented
168	C₄×#67	—	2	4	no	—
169	#67⋊C₄	—	2	4	no	—
170	D₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	complemented
171	#877	C₂²×C₅	1	4	yes	complemented
172	C₃×#99	C₂²×C₅	1	4	yes	complemented
173	#834	C₂²×C₅	1	4	yes	complemented
174	#806	C₂²×C₅	1	8	yes	complemented
175	C₂⁴×C₅	—	3	1	no	—
176	C₂⁴×C₅	—	3	2	no	—
177	C₂⁴×C₅	—	3	1	no	—
178	C₂⁴×C₅	C₄×D₃	1	1	yes	characteristic, complemented
179	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
180	C₅×#41	—	6	4	no	—
181	C₅×#41	—	6	4	no	—
182	C₅×#41	—	6	8	no	—
183	C₅×#41	—	6	8	no	—
184	C₂×C₅×Q₈	—	3	2	no	—
185	C₂×C₅×D₄	—	2	2	no	—
186	C₂×C₅×D₄	—	2	2	no	—
187	C₂×C₅×D₄	—	3	2	no	—
188	C₂×C₅×D₄	—	3	2	no	—
189	C₂×C₅×D₄	—	3	4	no	—
190	C₂×C₅×D₄	—	3	4	no	—
191	C₂×C₅×D₄	—	3	2	no	—
192	C₂×C₅×D₄	—	3	2	no	—
193	C₂×C₅×D₄	—	3	2	no	—
194	C₂×C₅×D₄	—	3	2	no	—
195	C₂×C₅×D₄	—	3	2	no	—
196	C₂×C₅×D₄	—	6	8	no	—
197	C₂×C₅×D₄	—	6	8	no	—
198	C₂×C₅×D₄	—	6	4	no	—
199	C₂×C₅×D₄	—	6	4	no	—
200	C₂×C₅×D₄	—	6	2	no	—
201	C₂×C₅×D₄	—	6	2	no	—
202	C₂×C₅×D₄	C₄×D₃	1	2	yes	complemented
203	C₂×C₅×D₄	C₂²×D₃	1	2	yes	—
204	C₂²×C₄×C₅	—	2	2	no	—
205	C₂²×C₄×C₅	—	3	1	no	—
206	C₂²×C₄×C₅	—	3	2	no	—
207	C₂²×C₄×C₅	—	3	1	no	—
208	C₂²×C₄×C₅	—	3	2	no	—
209	C₂²×C₄×C₅	—	3	2	no	—
210	C₂²×C₄×C₅	—	6	4	no	—
211	C₂²×C₄×C₅	—	6	2	no	—
212	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic, complemented
213	C₅×C₄⋊C₄	—	6	4	no	—
214	C₅×C₂²⋊C₄	—	2	2	no	—
215	C₅×C₂²⋊C₄	—	3	4	no	—
216	C₅×C₂²⋊C₄	—	6	4	no	—
217	C₅×C₂²⋊C₄	—	6	2	no	—
218	C₅×C₂²⋊C₄	C₂²×D₃	1	4	yes	—
219	C₄²×C₅	—	6	4	no	—
220	C₂×H(5,𝔽₂)	—	3	1	no	—
221	#518	—	3	2	no	—
222	#410	—	3	4	no	—
223	C₂×#99	—	3	1	no	—
224	C₂×#99	—	3	2	no	—
225	C₂×#99	—	3	4	no	—
226	C₂×#99	C₅×D₃	1	1	yes	characteristic, complemented
227	C₂²×C₃×C₅	—	2	2	no	—
228	C₂²×C₃×C₅	—	4	2	no	—
229	C₂²×C₃×C₅	—	4	2	no	—
230	C₂²×C₃×C₅	—	4	1	no	—
231	C₂²×C₃×C₅	—	4	1	no	—
232	C₂²×C₃×C₅	—	4	1	no	—
233	C₂²×C₃×C₅	—	4	2	no	—
234	C₂²×C₃×C₅	—	8	1	no	—
235	C₂²×C₃×C₅	—	8	1	no	—
236	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	2	yes	—
237	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
238	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
239	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
240	C₂²×C₃×C₅	C₂²×D₄	1	2	yes	—
241	C₂×C₅×D₃	—	2	2	no	—
242	C₂×C₅×D₃	—	4	4	no	—
243	C₂×C₅×D₃	—	4	2	no	—
244	C₂×C₅×D₃	—	4	2	no	—
245	C₂×C₅×D₃	—	8	4	no	—
246	C₂×C₅×D₃	—	8	2	no	—
247	C₂×C₅×D₃	C₂×C₂²⋊C₄	1	4	yes	—
248	C₃×C₄×C₅	—	2	2	no	—
249	C₃×C₄×C₅	—	4	4	no	—
250	C₅×Dic₃	—	2	2	no	—
251	C₅×Dic₃	—	4	4	no	—
252	C₅×Dic₃	C₂×C₂²⋊C₄	1	4	yes	—
253	C₂⁴×C₃	C₂×C₄×C₅	1	1	yes	characteristic, complemented
254	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
255	C₂³×D₃	C₂×C₄×C₅	1	1	yes	characteristic, complemented
256	C₂³×D₃	C₂×C₄×C₅	1	2	yes	complemented
257	C₂³×D₃	C₂³×C₅	1	1	yes	characteristic
258	C₂×C₃×D₄	—	2	2	no	—
259	C₂×C₃×D₄	—	2	2	no	—
260	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	complemented
261	C₂×C₃×D₄	C₂³×C₅	1	2	yes	—
262	C₂²×C₃×C₄	—	2	2	no	—
263	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic, complemented
264	C₂×#67	—	2	2	no	—
265	C₂×#67	—	2	2	no	—
266	C₂×#67	—	2	4	no	—
267	C₂×#67	—	2	4	no	—
268	C₂×#67	C₂×C₄×C₅	1	2	yes	—
269	C₂×#67	C₂×C₄×C₅	1	2	yes	—
270	C₂×#67	C₂×C₄×C₅	1	2	yes	complemented
271	C₂×#67	C₂×C₄×C₅	1	2	yes	complemented
272	C₂×#67	C₂×C₄×C₅	1	2	yes	complemented
273	C₂×#67	C₅×D₄	1	4	yes	—
274	C₂×#67	C₅×D₄	1	4	yes	—
275	C₂×#67	C₂³×C₅	1	2	yes	—
276	C₂²×Dic₃	—	2	2	no	—
277	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic, complemented
278	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	complemented
279	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
280	#237	—	2	8	no	—
281	#237	—	2	8	no	—
282	D₃×D₄	—	2	8	no	—
283	D₃×D₄	—	2	8	no	—
284	#235	—	2	4	no	—
285	#235	—	2	4	no	—
286	C₂×D₁₂	C₂×C₄×C₅	1	2	yes	complemented
287	C₂×C₄×D₃	—	2	4	no	—
288	C₂×C₄×D₃	C₂×C₄×C₅	1	2	yes	complemented
289	C₂×C₄×D₃	C₂×C₄×C₅	1	2	yes	complemented
290	C₂×Dic₆	C₂×C₄×C₅	1	2	yes	complemented
291	C₃×C₂²⋊C₄	—	2	2	no	—
292	C₃×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
293	#217	—	2	2	no	—
294	#217	C₂³×C₅	1	4	yes	—
295	(C₂×D₃)⋊C₄	—	2	4	no	—
296	Dic₃⋊C₄	—	2	4	no	—
297	C₄×Dic₃	—	2	4	no	—
298	C₂³×C₅	—	2	1	no	—
299	C₂³×C₅	—	2	1	no	—
300	C₂³×C₅	—	3	2	no	—
301	C₂³×C₅	—	3	4	no	—
302	C₂³×C₅	—	3	2	no	—
303	C₂³×C₅	—	6	4	no	—
304	C₂³×C₅	—	6	2	no	—
305	C₂³×C₅	—	8	1	no	—
306	C₂³×C₅	—	8	1	no	—
307	C₂³×C₅	—	24	1	no	—
308	C₂³×C₅	—	24	1	no	—
309	C₂³×C₅	—	24	2	no	—
310	C₂³×C₅	C₂×C₄×D₃	1	2	yes	—
311	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
312	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
313	C₂³×C₅	D₃×D₄	1	2	yes	—
314	C₂³×C₅	D₃×D₄	1	2	yes	—
315	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
316	C₅×Q₈	—	6	4	no	—
317	C₅×D₄	—	2	4	no	—
318	C₅×D₄	—	2	4	no	—
319	C₅×D₄	—	3	4	no	—
320	C₅×D₄	—	3	4	no	—
321	C₅×D₄	—	3	8	no	—
322	C₅×D₄	—	6	4	no	—
323	C₅×D₄	—	6	4	no	—
324	C₅×D₄	—	6	4	no	—
325	C₅×D₄	—	6	2	no	—
326	C₅×D₄	—	6	2	no	—
327	C₅×D₄	—	6	4	no	—
328	C₂×C₄×C₅	—	2	1	no	—
329	C₂×C₄×C₅	—	2	1	no	—
330	C₂×C₄×C₅	—	2	4	no	—
331	C₂×C₄×C₅	—	3	2	no	—
332	C₂×C₄×C₅	—	3	4	no	—
333	C₂×C₄×C₅	—	3	2	no	—
334	C₂×C₄×C₅	—	6	4	no	—
335	C₂×C₄×C₅	—	6	4	no	—
336	C₂×C₄×C₅	—	6	4	no	—
337	C₂×C₄×C₅	—	6	2	no	—
338	C₂×C₄×C₅	—	6	4	no	—
339	C₂×C₄×C₅	—	8	2	no	—
340	C₂×C₄×C₅	—	24	4	no	—
341	C₂×C₄×C₅	—	24	2	no	—
342	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
343	H(5,𝔽₂)	—	6	4	no	—
344	H(5,𝔽₂)	—	6	4	no	—
345	C₂×#41	—	3	2	no	—
346	C₂×#41	—	3	2	no	—
347	C₂×#41	—	3	2	no	—
348	C₂²×D₄	—	3	1	no	—
349	C₂²×D₄	—	3	2	no	—
350	C₂²×D₄	—	3	1	no	—
351	C₂²×D₄	—	3	2	no	—
352	C₂²×D₄	—	3	2	no	—
353	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
354	C₄×D₄	—	6	4	no	—
355	C₄×D₄	—	6	4	no	—
356	#117	—	3	4	no	—
357	C₂×C₂²⋊C₄	—	3	2	no	—
358	C₂×C₂²⋊C₄	—	3	4	no	—
359	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	complemented
360	#99	—	3	4	no	—
361	#99	—	3	8	no	—
362	#99	C₂×C₅×D₃	1	4	yes	complemented
363	C₂×C₃×C₅	—	2	2	no	—
364	C₂×C₃×C₅	—	2	1	no	—
365	C₂×C₃×C₅	—	2	2	no	—
366	C₂×C₃×C₅	—	2	1	no	—
367	C₂×C₃×C₅	—	2	1	no	—
368	C₂×C₃×C₅	—	4	2	no	—
369	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
370	C₅×D₃	—	2	4	no	—
371	C₅×D₃	—	4	2	no	—
372	C₂³×C₃	—	2	1	no	—
373	C₂³×C₃	—	2	1	no	—
374	C₂³×C₃	—	8	1	no	—
375	C₂³×C₃	—	8	1	no	—
376	C₂³×C₃	C₂²×C₄×C₅	1	2	yes	—
377	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
378	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
379	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
380	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
381	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
382	C₂²×D₃	—	2	2	no	—
383	C₂²×D₃	—	2	4	no	—
384	C₂²×D₃	—	8	1	no	—
385	C₂²×D₃	—	8	2	no	—
386	C₂²×D₃	—	8	1	no	—
387	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
388	C₂²×D₃	C₂²×C₄×C₅	1	2	yes	—
389	C₂²×D₃	C₂×C₅×D₄	1	4	yes	—
390	C₃×D₄	—	2	4	no	—
391	C₃×D₄	—	2	4	no	—
392	C₂×C₃×C₄	—	2	1	no	—
393	C₂×C₃×C₄	—	2	1	no	—
394	C₂×C₃×C₄	—	2	4	no	—
395	C₂×C₃×C₄	—	8	2	no	—
396	C₂×C₃×C₄	C₂²×C₄×C₅	1	2	yes	—
397	#67	—	2	2	no	—
398	#67	—	2	2	no	—
399	#67	—	2	4	no	—
400	#67	—	2	4	no	—
401	#67	—	2	4	no	—
402	#67	C₅×C₂²⋊C₄	1	4	yes	—
403	#67	C₅×C₂²⋊C₄	1	4	yes	—
404	#67	C₅×C₂²⋊C₄	1	8	yes	—
405	C₂×Dic₃	—	2	2	no	—
406	C₂×Dic₃	—	2	4	no	—
407	C₂×Dic₃	—	2	4	no	—
408	C₂×Dic₃	—	8	2	no	—
409	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
410	C₂×Dic₃	C₂²×C₄×C₅	1	2	yes	—
411	C₂×Dic₃	C₂×C₅×D₄	1	4	yes	—
412	D₁₂	—	2	4	no	—
413	C₄×D₃	—	2	4	no	—
414	C₄×D₃	—	2	4	no	—
415	C₄×D₃	—	8	4	no	—
416	Dic₆	—	2	4	no	—
417	C₂²×C₅	—	2	2	no	—
418	C₂²×C₅	—	3	4	no	—
419	C₂²×C₅	—	4	2	no	—
420	C₂²×C₅	—	4	2	no	—
421	C₂²×C₅	—	4	1	no	—
422	C₂²×C₅	—	4	1	no	—
423	C₂²×C₅	—	4	1	no	—
424	C₂²×C₅	—	4	2	no	—
425	C₂²×C₅	—	6	2	no	—
426	C₂²×C₅	—	8	1	no	—
427	C₂²×C₅	—	8	1	no	—
428	C₂²×C₅	—	12	4	no	—
429	C₂²×C₅	—	12	2	no	—
430	C₂²×C₅	—	12	2	no	—
431	C₂²×C₅	—	24	2	no	—
432	C₂²×C₅	—	24	4	no	—
433	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
434	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
435	C₂²×C₅	D₃×C₂²⋊C₄	1	2	yes	—
436	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
437	C₂²×C₅	C₂×D₃×D₄	1	2	yes	—
438	C₄×C₅	—	2	2	no	—
439	C₄×C₅	—	3	4	no	—
440	C₄×C₅	—	4	4	no	—
441	C₄×C₅	—	6	2	no	—
442	C₄×C₅	—	12	4	no	—
443	C₂⁴	—	3	1	no	—
444	C₂⁴	—	3	2	no	—
445	C₂⁴	—	3	1	no	—
446	C₂⁴	C₄×C₅×D₃	1	1	yes	characteristic, complemented
447	C₂⁴	C₂²×C₅×D₃	1	1	yes	characteristic
448	#41	—	6	4	no	—
449	#41	—	6	4	no	—
450	#41	—	6	8	no	—
451	#41	—	6	8	no	—
452	C₂×Q₈	—	3	2	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	2	no	—
455	C₂×D₄	—	3	2	no	—
456	C₂×D₄	—	3	2	no	—
457	C₂×D₄	—	3	4	no	—
458	C₂×D₄	—	3	4	no	—
459	C₂×D₄	—	3	2	no	—
460	C₂×D₄	—	3	2	no	—
461	C₂×D₄	—	3	2	no	—
462	C₂×D₄	—	3	2	no	—
463	C₂×D₄	—	3	2	no	—
464	C₂×D₄	—	6	8	no	—
465	C₂×D₄	—	6	8	no	—
466	C₂×D₄	—	6	4	no	—
467	C₂×D₄	—	6	4	no	—
468	C₂×D₄	—	6	2	no	—
469	C₂×D₄	—	6	2	no	—
470	C₂×D₄	C₄×C₅×D₃	1	2	yes	complemented
471	C₂×D₄	C₂²×C₅×D₃	1	2	yes	—
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	3	1	no	—
474	C₂²×C₄	—	3	2	no	—
475	C₂²×C₄	—	3	1	no	—
476	C₂²×C₄	—	3	2	no	—
477	C₂²×C₄	—	3	2	no	—
478	C₂²×C₄	—	6	4	no	—
479	C₂²×C₄	—	6	2	no	—
480	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic, complemented
481	C₄⋊C₄	—	6	4	no	—
482	C₂²⋊C₄	—	2	2	no	—
483	C₂²⋊C₄	—	3	4	no	—
484	C₂²⋊C₄	—	6	4	no	—
485	C₂²⋊C₄	—	6	2	no	—
486	C₂²⋊C₄	C₂²×C₅×D₃	1	4	yes	—
487	C₄²	—	6	4	no	—
488	C₃×C₅	#2883	1	1	yes	characteristic, complemented, Hall
489	C₂²×C₃	—	2	2	no	—
490	C₂²×C₃	—	4	2	no	—
491	C₂²×C₃	—	4	2	no	—
492	C₂²×C₃	—	4	1	no	—
493	C₂²×C₃	—	4	1	no	—
494	C₂²×C₃	—	4	1	no	—
495	C₂²×C₃	—	4	2	no	—
496	C₂²×C₃	—	8	1	no	—
497	C₂²×C₃	—	8	1	no	—
498	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	2	yes	—
499	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
500	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
501	C₂²×C₃	C₂³×C₄×C₅	1	1	yes	characteristic
502	C₂²×C₃	C₂²×C₅×D₄	1	2	yes	—
503	C₂×D₃	—	2	2	no	—
504	C₂×D₃	—	4	4	no	—
505	C₂×D₃	—	4	2	no	—
506	C₂×D₃	—	4	2	no	—
507	C₂×D₃	—	8	4	no	—
508	C₂×D₃	—	8	2	no	—
509	C₂×D₃	C₂×C₅×C₂²⋊C₄	1	4	yes	—
510	C₃×C₄	—	2	2	no	—
511	C₃×C₄	—	4	4	no	—
512	Dic₃	—	2	2	no	—
513	Dic₃	—	4	4	no	—
514	Dic₃	C₂×C₅×C₂²⋊C₄	1	4	yes	—
515	C₂×C₅	—	2	2	no	—
516	C₂×C₅	—	2	1	no	—
517	C₂×C₅	—	2	1	no	—
518	C₂×C₅	—	2	1	no	—
519	C₂×C₅	—	2	2	no	—
520	C₂×C₅	—	4	2	no	—
521	C₂×C₅	—	6	4	no	—
522	C₂×C₅	—	12	2	no	—
523	C₂×C₅	C₂×D₃×C₂²⋊C₄	1	1	yes	characteristic
524	C₂³	—	2	1	no	—
525	C₂³	—	2	1	no	—
526	C₂³	—	3	2	no	—
527	C₂³	—	3	4	no	—
528	C₂³	—	3	2	no	—
529	C₂³	—	6	4	no	—
530	C₂³	—	6	2	no	—
531	C₂³	—	8	1	no	—
532	C₂³	—	8	1	no	—
533	C₂³	—	24	1	no	—
534	C₂³	—	24	1	no	—
535	C₂³	—	24	2	no	—
536	C₂³	C₂×C₄×C₅×D₃	1	2	yes	—
537	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
538	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
539	C₂³	C₅×D₃×D₄	1	2	yes	—
540	C₂³	C₅×D₃×D₄	1	2	yes	—
541	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
542	Q₈	—	6	4	no	—
543	D₄	—	2	4	no	—
544	D₄	—	2	4	no	—
545	D₄	—	3	4	no	—
546	D₄	—	3	4	no	—
547	D₄	—	3	8	no	—
548	D₄	—	6	4	no	—
549	D₄	—	6	4	no	—
550	D₄	—	6	4	no	—
551	D₄	—	6	2	no	—
552	D₄	—	6	2	no	—
553	D₄	—	6	4	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	1	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	3	2	no	—
558	C₂×C₄	—	3	4	no	—
559	C₂×C₄	—	3	2	no	—
560	C₂×C₄	—	6	4	no	—
561	C₂×C₄	—	6	4	no	—
562	C₂×C₄	—	6	4	no	—
563	C₂×C₄	—	6	2	no	—
564	C₂×C₄	—	6	4	no	—
565	C₂×C₄	—	8	2	no	—
566	C₂×C₄	—	24	4	no	—
567	C₂×C₄	—	24	2	no	—
568	C₂×C₄	C₂×C₄×C₅×D₃	1	2	yes	—
569	C₂×C₃	—	2	2	no	—
570	C₂×C₃	—	2	1	no	—
571	C₂×C₃	—	2	2	no	—
572	C₂×C₃	—	2	1	no	—
573	C₂×C₃	—	2	1	no	—
574	C₂×C₃	—	4	2	no	—
575	C₂×C₃	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
576	D₃	—	2	4	no	—
577	D₃	—	4	2	no	—
578	C₅	#78688	1	1	yes	characteristic, direct product factor, Sylow
579	C₂²	—	2	2	no	—
580	C₂²	—	3	4	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	—	4	2	no	—
587	C₂²	—	6	2	no	—
588	C₂²	—	8	1	no	—
589	C₂²	—	8	1	no	—
590	C₂²	—	12	4	no	—
591	C₂²	—	12	2	no	—
592	C₂²	—	12	2	no	—
593	C₂²	—	24	2	no	—
594	C₂²	—	24	4	no	—
595	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
596	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
597	C₂²	C₅×D₃×C₂²⋊C₄	1	2	yes	—
598	C₂²	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
599	C₂²	C₂×C₅×D₃×D₄	1	2	yes	—
600	C₄	—	2	2	no	—
601	C₄	—	3	4	no	—
602	C₄	—	4	4	no	—
603	C₄	—	6	2	no	—
604	C₄	—	12	4	no	—
605	C₃	C₅×#2883	1	1	yes	characteristic, complemented, Sylow
606	C₂	—	2	2	no	—
607	C₂	—	2	1	no	—
608	C₂	—	2	1	no	—
609	C₂	—	2	1	no	—
610	C₂	—	2	2	no	—
611	C₂	—	4	2	no	—
612	C₂	—	6	4	no	—
613	C₂	—	12	2	no	—
614	C₂	C₂×C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂×D₃×C₂²⋊C₄
commutator	C₂²×C₃	C₂³×C₄×C₅
Frattini	C₂³	C₂³×C₅×D₃
Fitting	C₂×C₃×C₅×#99	C₂
socle	C₂×C₃×C₅	C₂²×C₂²⋊C₄
prime radical	C₂×C₅×#1004	C₂
cosocle	C₂³×C₃	C₂⁴×C₅
hypercenter	C₂×C₅×#99	D₃
hypocenter	C₃	C₅×#2883
LCS	G ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅ ◁ C₂⁴×C₅ ◁ C₂×C₅×#99

Complements

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

N	H ≅ G/N	type
2. C₅×#5486	C₂	semidirect
3. C₂×C₃×C₅×#99	C₂	semidirect
4. C₂×C₅×#834	C₂	semidirect
5. C₂×C₅×#806	C₂	semidirect
6. C₅×D₃×#99	C₂	semidirect
7. C₅×#4740	C₂	semidirect
10. C₂²×C₅×#67	C₄	semidirect
15. C₂×C₅×#237	C₄	semidirect
17. C₂×C₅×D₃×D₄	C₄	semidirect
19. C₂×C₅×#235	C₄	semidirect
20. C₂×C₃×C₅×C₂²⋊C₄	C₂²	semidirect
21. C₂×C₅×#217	C₂²	semidirect
24. C₅×D₃×C₂²⋊C₄	C₂²	semidirect
25. C₅×#877	C₂²	semidirect
26. C₃×C₅×#99	C₂²	semidirect
27. C₅×#834	C₂²	semidirect
28. C₅×#806	C₂²	semidirect
29. #78688	C₅	direct (N × H)
36. C₂×C₅×#99	D₃	semidirect
37. C₂⁴×C₃×C₅	C₂×C₄	semidirect
39. C₂³×C₅×D₃	C₂×C₄	semidirect
40. C₂³×C₅×D₃	C₂×C₄	semidirect
44. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
47. C₂²×C₃×C₄×C₅	C₂×C₄	semidirect
54. C₂×C₅×#67	C₂×C₄	semidirect
55. C₂×C₅×#67	C₂×C₄	semidirect
56. C₂×C₅×#67	C₂×C₄	semidirect
61. C₂²×C₅×Dic₃	C₂×C₄	semidirect
62. C₂²×C₅×Dic₃	C₂×C₄	semidirect
70. C₂×C₅×D₁₂	C₂×C₄	semidirect
72. C₂×C₄×C₅×D₃	C₂×C₄	semidirect
73. C₂×C₄×C₅×D₃	C₂×C₄	semidirect
74. C₂×C₅×Dic₆	C₂×C₄	semidirect
83. #5486	C₂×C₅	semidirect
84. C₂×C₃×#99	C₂×C₅	semidirect
85. C₂×#834	C₂×C₅	semidirect
86. C₂×#806	C₂×C₅	semidirect
87. D₃×#99	C₂×C₅	semidirect
88. #4740	C₂×C₅	semidirect
105. C₂×C₅×C₂²⋊C₄	C₂×D₃	semidirect
108. C₅×#99	C₂×D₃	semidirect
156. C₂²×#67	C₄×C₅	semidirect
161. C₂×#237	C₄×C₅	semidirect
163. C₂×D₃×D₄	C₄×C₅	semidirect
165. C₂×#235	C₄×C₅	semidirect
166. C₂×C₃×C₂²⋊C₄	C₂²×C₅	semidirect
167. C₂×#217	C₂²×C₅	semidirect
170. D₃×C₂²⋊C₄	C₂²×C₅	semidirect
171. #877	C₂²×C₅	semidirect
172. C₃×#99	C₂²×C₅	semidirect
173. #834	C₂²×C₅	semidirect
174. #806	C₂²×C₅	semidirect
178. C₂⁴×C₅	C₄×D₃	semidirect
202. C₂×C₅×D₄	C₄×D₃	semidirect
212. C₂²×C₄×C₅	C₄×D₃	semidirect
226. C₂×#99	C₅×D₃	semidirect
253. C₂⁴×C₃	C₂×C₄×C₅	semidirect
255. C₂³×D₃	C₂×C₄×C₅	semidirect
256. C₂³×D₃	C₂×C₄×C₅	semidirect
260. C₂×C₃×D₄	C₂×C₄×C₅	semidirect
263. C₂²×C₃×C₄	C₂×C₄×C₅	semidirect
270. C₂×#67	C₂×C₄×C₅	semidirect
271. C₂×#67	C₂×C₄×C₅	semidirect
272. C₂×#67	C₂×C₄×C₅	semidirect
277. C₂²×Dic₃	C₂×C₄×C₅	semidirect
278. C₂²×Dic₃	C₂×C₄×C₅	semidirect
286. C₂×D₁₂	C₂×C₄×C₅	semidirect
288. C₂×C₄×D₃	C₂×C₄×C₅	semidirect
289. C₂×C₄×D₃	C₂×C₄×C₅	semidirect
290. C₂×Dic₆	C₂×C₄×C₅	semidirect
359. C₂×C₂²⋊C₄	C₂×C₅×D₃	semidirect
362. #99	C₂×C₅×D₃	semidirect
446. C₂⁴	C₄×C₅×D₃	semidirect
470. C₂×D₄	C₄×C₅×D₃	semidirect
480. C₂²×C₄	C₄×C₅×D₃	semidirect
488. C₃×C₅	#2883	semidirect
578. C₅	#78688	direct (N × H)
605. C₃	C₅×#2883	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#1004	2
C₅×#5486	2
C₂×C₃×C₅×#99	2
C₂×C₅×#834	2
C₂×C₅×#806	2
C₅×D₃×#99	2
C₅×#4740	2
C₅×#2883	3
#78688	5

Subgroups — C5×#78688 #14411965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#78688 #14411965