Subgroups — C₅×#5591 #11755976

Order 960.

668 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#67	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₅×#120	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#217	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#939	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#939	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#938	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#937	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#886	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#885	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅×#883	C₂	1	1	yes	maximal, characteristic, complemented
13	C₅×#67⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×D₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#878	C₂	1	1	yes	maximal, characteristic, complemented
16	C₅×#546	—	3	1	no	maximal, Hall
17	C₂⁴×C₃×C₅	C₂²	1	1	yes	characteristic
18	C₂³×C₅×D₃	C₂²	1	1	yes	characteristic
19	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₅×#67	—	2	4	no	—
23	C₂×C₅×#67	C₂²	1	1	yes	characteristic
24	C₂×C₅×#67	C₂²	1	1	yes	characteristic
25	C₂×C₅×#67	C₂²	1	1	yes	characteristic
26	C₂×C₅×#67	C₂²	1	1	yes	characteristic
27	C₂×C₅×#67	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₅×#67	C₂²	1	1	yes	characteristic, complemented
29	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic
30	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
31	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
32	C₂²×C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
33	C₅×#237	—	2	4	no	—
34	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₄×C₅×D₃	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₅×Dic₆	C₂²	1	1	yes	characteristic, complemented
37	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
38	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
39	C₃×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
40	C₅×#217	C₂²	1	1	yes	characteristic
41	C₅×#217	C₂²	1	1	yes	characteristic
42	C₅×#217	C₂²	1	1	yes	characteristic, complemented
43	C₅×#217	C₂²	1	1	yes	characteristic, complemented
44	C₅×#217	C₂²	1	1	yes	characteristic, complemented
45	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic, complemented
46	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic, complemented
47	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic, complemented
48	C₅×(C₂×D₃)⋊C₄	C₂²	1	1	yes	characteristic, complemented
49	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
50	C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
51	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic, complemented
52	C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic, complemented
53	C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
54	#5591	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
55	C₂×C₅×#41	—	3	1	no	—
56	C₂²×C₅×D₄	—	3	1	no	—
57	C₅×#124	—	3	1	no	—
58	C₅×#123	—	3	1	no	—
59	C₅×#123	—	3	1	no	—
60	C₅×C₂²⋊Q₈	—	3	1	no	—
61	C₅×#121	—	3	1	no	—
62	C₅×#121	—	3	1	no	—
63	C₅×#121	—	3	1	no	—
64	C₅×#120	—	3	1	no	—
65	C₅×#120	D₃	1	1	yes	characteristic, complemented
66	C₄×C₅×D₄	—	3	1	no	—
67	C₄×C₅×D₄	—	3	1	no	—
68	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
69	C₂×C₅×C₂²⋊C₄	—	3	1	no	—
70	C₂³×C₃×C₅	—	2	2	no	—
71	C₂³×C₃×C₅	—	2	2	no	—
72	C₂³×C₃×C₅	—	4	1	no	—
73	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
74	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
75	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
76	C₂³×C₃×C₅	C₂³	1	1	yes	characteristic
77	C₂²×C₅×D₃	—	2	2	no	—
78	C₂²×C₅×D₃	—	2	2	no	—
79	C₂²×C₅×D₃	—	4	1	no	—
80	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
81	C₂²×C₅×D₃	C₂³	1	1	yes	characteristic
82	C₃×C₅×D₄	—	2	2	no	—
83	C₃×C₅×D₄	—	2	2	no	—
84	C₃×C₅×D₄	—	4	1	no	—
85	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
88	C₅×#67	—	2	2	no	—
89	C₅×#67	—	2	2	no	—
90	C₅×#67	—	2	2	no	—
91	C₅×#67	—	2	2	no	—
92	C₅×#67	—	4	1	no	—
93	C₅×#67	D₄	1	4	yes	complemented
94	C₂×C₅×Dic₃	—	2	2	no	—
95	C₂×C₅×Dic₃	—	2	2	no	—
96	C₂×C₅×Dic₃	—	2	2	no	—
97	C₂×C₅×Dic₃	—	4	1	no	—
98	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
99	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
100	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
101	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
102	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
103	C₂×C₅×Dic₃	C₂³	1	1	yes	characteristic
104	C₄×C₅×D₃	—	2	2	no	—
105	C₄×C₅×D₃	—	4	1	no	—
106	C₅×Dic₆	—	2	2	no	—
107	C₂²×#67	C₂×C₅	1	1	yes	characteristic, complemented
108	C₂×#237	C₂×C₅	1	1	yes	characteristic, complemented
109	C₃×#120	C₂×C₅	1	1	yes	characteristic, complemented
110	C₂×#217	C₂×C₅	1	1	yes	characteristic, complemented
111	#939	C₂×C₅	1	1	yes	characteristic, complemented
112	#939	C₂×C₅	1	1	yes	characteristic, complemented
113	#938	C₂×C₅	1	1	yes	characteristic, complemented
114	#937	C₂×C₅	1	1	yes	characteristic, complemented
115	D₄×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
116	#886	C₂×C₅	1	1	yes	characteristic, complemented
117	#885	C₂×C₅	1	1	yes	characteristic, complemented
118	#883	C₂×C₅	1	1	yes	characteristic, complemented
119	#67⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
120	D₃×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
121	#878	C₂×C₅	1	1	yes	characteristic, complemented
122	C₂⁴×C₅	—	3	1	no	—
123	C₂⁴×C₅	C₂×D₃	1	1	yes	characteristic
124	C₅×#41	—	6	4	no	—
125	C₂×C₅×Q₈	—	3	1	no	—
126	C₂×C₅×D₄	—	3	1	no	—
127	C₂×C₅×D₄	—	3	1	no	—
128	C₂×C₅×D₄	—	3	1	no	—
129	C₂×C₅×D₄	—	3	1	no	—
130	C₂×C₅×D₄	—	3	1	no	—
131	C₂×C₅×D₄	—	3	1	no	—
132	C₂×C₅×D₄	—	6	4	no	—
133	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic
134	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic, complemented
135	C₂×C₅×D₄	C₂×D₃	1	1	yes	characteristic, complemented
136	C₂²×C₄×C₅	—	3	1	no	—
137	C₂²×C₄×C₅	—	3	1	no	—
138	C₂²×C₄×C₅	—	3	1	no	—
139	C₂²×C₄×C₅	—	3	1	no	—
140	C₂²×C₄×C₅	—	3	1	no	—
141	C₂²×C₄×C₅	—	3	1	no	—
142	C₅×C₄⋊C₄	—	3	1	no	—
143	C₅×C₄⋊C₄	—	3	1	no	—
144	C₅×C₄⋊C₄	—	3	1	no	—
145	C₅×C₄⋊C₄	—	3	1	no	—
146	C₅×C₂²⋊C₄	—	3	1	no	—
147	C₅×C₂²⋊C₄	—	3	1	no	—
148	C₅×C₂²⋊C₄	—	3	1	no	—
149	C₅×C₂²⋊C₄	—	3	1	no	—
150	C₅×C₂²⋊C₄	—	3	1	no	—
151	C₅×C₂²⋊C₄	—	3	1	no	—
152	C₅×C₂²⋊C₄	—	3	1	no	—
153	C₅×C₂²⋊C₄	—	3	1	no	—
154	C₅×C₂²⋊C₄	—	3	1	no	—
155	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
156	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
157	C₅×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
158	C₄²×C₅	—	3	1	no	—
159	#546	—	3	1	no	Sylow
160	C₂²×C₃×C₅	—	2	1	no	—
161	C₂²×C₃×C₅	—	2	1	no	—
162	C₂²×C₃×C₅	—	2	1	no	—
163	C₂²×C₃×C₅	—	2	1	no	—
164	C₂²×C₃×C₅	—	2	1	no	—
165	C₂²×C₃×C₅	—	2	1	no	—
166	C₂²×C₃×C₅	—	2	1	no	—
167	C₂²×C₃×C₅	—	2	1	no	—
168	C₂²×C₃×C₅	—	2	1	no	—
169	C₂²×C₃×C₅	—	2	1	no	—
170	C₂²×C₃×C₅	—	4	4	no	—
171	C₂²×C₃×C₅	C₂×D₄	1	2	yes	—
172	C₂²×C₃×C₅	#41	1	2	yes	complemented
173	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
174	C₂×C₅×D₃	—	2	1	no	—
175	C₂×C₅×D₃	—	2	1	no	—
176	C₂×C₅×D₃	—	2	1	no	—
177	C₂×C₅×D₃	—	2	1	no	—
178	C₂×C₅×D₃	—	2	1	no	—
179	C₂×C₅×D₃	—	4	4	no	—
180	C₂×C₅×D₃	C₂×D₄	1	2	yes	—
181	C₃×C₄×C₅	—	2	1	no	—
182	C₃×C₄×C₅	—	2	1	no	—
183	C₃×C₄×C₅	—	2	1	no	—
184	C₅×Dic₃	—	2	1	no	—
185	C₅×Dic₃	—	2	1	no	—
186	C₅×Dic₃	—	2	1	no	—
187	C₅×Dic₃	—	2	1	no	—
188	C₅×Dic₃	—	2	1	no	—
189	C₅×Dic₃	C₂×D₄	1	2	yes	—
190	C₂⁴×C₃	C₂²×C₅	1	1	yes	characteristic
191	C₂³×D₃	C₂²×C₅	1	1	yes	characteristic
192	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
193	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
194	C₂×C₃×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
195	C₂×#67	—	2	4	no	—
196	C₂×#67	C₂²×C₅	1	1	yes	characteristic
197	C₂×#67	C₂²×C₅	1	1	yes	characteristic
198	C₂×#67	C₂²×C₅	1	1	yes	characteristic
199	C₂×#67	C₂²×C₅	1	1	yes	characteristic
200	C₂×#67	C₂²×C₅	1	1	yes	characteristic, complemented
201	C₂×#67	C₂²×C₅	1	1	yes	characteristic, complemented
202	C₂²×Dic₃	C₂²×C₅	1	1	yes	characteristic
203	C₂²×Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
204	C₂²×Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
205	C₂²×Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
206	#237	—	2	4	no	—
207	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic, complemented
208	C₂×C₄×D₃	C₂²×C₅	1	1	yes	characteristic, complemented
209	C₂×Dic₆	C₂²×C₅	1	1	yes	characteristic, complemented
210	C₃×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
211	C₃×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
212	C₃×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
213	#217	C₂²×C₅	1	1	yes	characteristic
214	#217	C₂²×C₅	1	1	yes	characteristic
215	#217	C₂²×C₅	1	1	yes	characteristic, complemented
216	#217	C₂²×C₅	1	1	yes	characteristic, complemented
217	#217	C₂²×C₅	1	1	yes	characteristic, complemented
218	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
219	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
220	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
221	(C₂×D₃)⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
222	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
223	C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
224	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
225	Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
226	C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
227	C₂³×C₅	—	2	2	no	—
228	C₂³×C₅	—	2	2	no	—
229	C₂³×C₅	—	3	1	no	—
230	C₂³×C₅	—	3	1	no	—
231	C₂³×C₅	—	4	1	no	—
232	C₂³×C₅	—	6	2	no	—
233	C₂³×C₅	—	6	2	no	—
234	C₂³×C₅	—	12	1	no	—
235	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
236	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
237	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
238	C₂³×C₅	C₂²×D₃	1	1	yes	characteristic
239	C₅×Q₈	—	6	2	no	—
240	C₅×D₄	—	2	2	no	—
241	C₅×D₄	—	2	2	no	—
242	C₅×D₄	—	3	4	no	—
243	C₅×D₄	—	4	1	no	—
244	C₅×D₄	—	6	2	no	—
245	C₅×D₄	—	6	2	no	—
246	C₅×D₄	—	6	2	no	—
247	C₅×D₄	—	6	2	no	—
248	C₅×D₄	—	12	1	no	—
249	C₂×C₄×C₅	—	3	1	no	—
250	C₂×C₄×C₅	—	3	1	no	—
251	C₂×C₄×C₅	—	3	1	no	—
252	C₂×C₄×C₅	—	3	1	no	—
253	C₂×C₄×C₅	—	3	1	no	—
254	C₂×C₄×C₅	—	3	1	no	—
255	C₂×C₄×C₅	—	6	2	no	—
256	C₂×C₄×C₅	—	6	2	no	—
257	C₂×C₄×C₅	—	6	2	no	—
258	C₂×C₄×C₅	—	6	2	no	—
259	C₂×C₄×C₅	—	12	1	no	—
260	C₂×C₄×C₅	—	12	1	no	—
261	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
262	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
263	C₂×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
264	C₂×#41	—	3	1	no	—
265	C₂²×D₄	—	3	1	no	—
266	#124	—	3	1	no	—
267	#123	—	3	1	no	—
268	#123	—	3	1	no	—
269	C₂²⋊Q₈	—	3	1	no	—
270	#121	—	3	1	no	—
271	#121	—	3	1	no	—
272	#121	—	3	1	no	—
273	#120	—	3	1	no	—
274	#120	C₅×D₃	1	1	yes	characteristic, complemented
275	C₄×D₄	—	3	1	no	—
276	C₄×D₄	—	3	1	no	—
277	C₂×C₂²⋊C₄	—	3	1	no	—
278	C₂×C₂²⋊C₄	—	3	1	no	—
279	C₂×C₃×C₅	—	2	2	no	—
280	C₂×C₃×C₅	—	2	2	no	—
281	C₂×C₃×C₅	—	4	1	no	—
282	C₂×C₃×C₅	—	4	1	no	—
283	C₂×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
284	C₂×C₃×C₅	C₂×#41	1	1	yes	characteristic
285	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
286	C₅×D₃	—	2	2	no	—
287	C₅×D₃	—	4	1	no	—
288	C₂³×C₃	—	2	2	no	—
289	C₂³×C₃	—	2	2	no	—
290	C₂³×C₃	—	4	1	no	—
291	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
292	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
293	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
294	C₂³×C₃	C₂³×C₅	1	1	yes	characteristic
295	C₂²×D₃	—	2	2	no	—
296	C₂²×D₃	—	2	2	no	—
297	C₂²×D₃	—	4	1	no	—
298	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
299	C₂²×D₃	C₂³×C₅	1	1	yes	characteristic
300	C₃×D₄	—	2	2	no	—
301	C₃×D₄	—	2	2	no	—
302	C₃×D₄	—	4	1	no	—
303	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
304	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
305	C₂×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
306	#67	—	2	2	no	—
307	#67	—	2	2	no	—
308	#67	—	2	2	no	—
309	#67	—	2	2	no	—
310	#67	—	4	1	no	—
311	#67	C₅×D₄	1	4	yes	complemented
312	C₂×Dic₃	—	2	2	no	—
313	C₂×Dic₃	—	2	2	no	—
314	C₂×Dic₃	—	2	2	no	—
315	C₂×Dic₃	—	4	1	no	—
316	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
317	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
318	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
319	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
320	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
321	C₂×Dic₃	C₂³×C₅	1	1	yes	characteristic
322	C₄×D₃	—	2	2	no	—
323	C₄×D₃	—	4	1	no	—
324	Dic₆	—	2	2	no	—
325	C₂²×C₅	—	2	1	no	—
326	C₂²×C₅	—	2	1	no	—
327	C₂²×C₅	—	2	1	no	—
328	C₂²×C₅	—	2	1	no	—
329	C₂²×C₅	—	2	1	no	—
330	C₂²×C₅	—	2	1	no	—
331	C₂²×C₅	—	2	1	no	—
332	C₂²×C₅	—	2	1	no	—
333	C₂²×C₅	—	2	1	no	—
334	C₂²×C₅	—	2	1	no	—
335	C₂²×C₅	—	3	2	no	—
336	C₂²×C₅	—	4	4	no	—
337	C₂²×C₅	—	6	1	no	—
338	C₂²×C₅	—	6	1	no	—
339	C₂²×C₅	—	6	1	no	—
340	C₂²×C₅	—	6	1	no	—
341	C₂²×C₅	—	6	1	no	—
342	C₂²×C₅	—	12	4	no	—
343	C₂²×C₅	D₃×D₄	1	2	yes	—
344	C₂²×C₅	#237	1	2	yes	complemented
345	C₂²×C₅	C₂³×D₃	1	1	yes	characteristic
346	C₄×C₅	—	2	1	no	—
347	C₄×C₅	—	2	1	no	—
348	C₄×C₅	—	2	1	no	—
349	C₄×C₅	—	3	2	no	—
350	C₄×C₅	—	6	1	no	—
351	C₄×C₅	—	6	1	no	—
352	C₄×C₅	—	6	1	no	—
353	C₄×C₅	—	6	1	no	—
354	C₄×C₅	—	6	1	no	—
355	C₂⁴	—	3	1	no	—
356	C₂⁴	C₂×C₅×D₃	1	1	yes	characteristic
357	#41	—	6	4	no	—
358	C₂×Q₈	—	3	1	no	—
359	C₂×D₄	—	3	1	no	—
360	C₂×D₄	—	3	1	no	—
361	C₂×D₄	—	3	1	no	—
362	C₂×D₄	—	3	1	no	—
363	C₂×D₄	—	3	1	no	—
364	C₂×D₄	—	3	1	no	—
365	C₂×D₄	—	6	4	no	—
366	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic
367	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic, complemented
368	C₂×D₄	C₂×C₅×D₃	1	1	yes	characteristic, complemented
369	C₂²×C₄	—	3	1	no	—
370	C₂²×C₄	—	3	1	no	—
371	C₂²×C₄	—	3	1	no	—
372	C₂²×C₄	—	3	1	no	—
373	C₂²×C₄	—	3	1	no	—
374	C₂²×C₄	—	3	1	no	—
375	C₄⋊C₄	—	3	1	no	—
376	C₄⋊C₄	—	3	1	no	—
377	C₄⋊C₄	—	3	1	no	—
378	C₄⋊C₄	—	3	1	no	—
379	C₂²⋊C₄	—	3	1	no	—
380	C₂²⋊C₄	—	3	1	no	—
381	C₂²⋊C₄	—	3	1	no	—
382	C₂²⋊C₄	—	3	1	no	—
383	C₂²⋊C₄	—	3	1	no	—
384	C₂²⋊C₄	—	3	1	no	—
385	C₂²⋊C₄	—	3	1	no	—
386	C₂²⋊C₄	—	3	1	no	—
387	C₂²⋊C₄	—	3	1	no	—
388	C₂²⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
389	C₂²⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic, complemented
390	C₂²⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic, complemented
391	C₄²	—	3	1	no	—
392	C₃×C₅	#546	1	1	yes	characteristic, complemented, Hall
393	C₂²×C₃	—	2	1	no	—
394	C₂²×C₃	—	2	1	no	—
395	C₂²×C₃	—	2	1	no	—
396	C₂²×C₃	—	2	1	no	—
397	C₂²×C₃	—	2	1	no	—
398	C₂²×C₃	—	2	1	no	—
399	C₂²×C₃	—	2	1	no	—
400	C₂²×C₃	—	2	1	no	—
401	C₂²×C₃	—	2	1	no	—
402	C₂²×C₃	—	2	1	no	—
403	C₂²×C₃	—	4	4	no	—
404	C₂²×C₃	C₂×C₅×D₄	1	2	yes	—
405	C₂²×C₃	C₅×#41	1	2	yes	complemented
406	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
407	C₂×D₃	—	2	1	no	—
408	C₂×D₃	—	2	1	no	—
409	C₂×D₃	—	2	1	no	—
410	C₂×D₃	—	2	1	no	—
411	C₂×D₃	—	2	1	no	—
412	C₂×D₃	—	4	4	no	—
413	C₂×D₃	C₂×C₅×D₄	1	2	yes	—
414	C₃×C₄	—	2	1	no	—
415	C₃×C₄	—	2	1	no	—
416	C₃×C₄	—	2	1	no	—
417	Dic₃	—	2	1	no	—
418	Dic₃	—	2	1	no	—
419	Dic₃	—	2	1	no	—
420	Dic₃	—	2	1	no	—
421	Dic₃	—	2	1	no	—
422	Dic₃	C₂×C₅×D₄	1	2	yes	—
423	C₂×C₅	—	2	2	no	—
424	C₂×C₅	—	2	2	no	—
425	C₂×C₅	—	4	1	no	—
426	C₂×C₅	—	4	1	no	—
427	C₂×C₅	—	6	2	no	—
428	C₂×C₅	—	12	1	no	—
429	C₂×C₅	C₂×D₃×D₄	1	1	yes	characteristic
430	C₂×C₅	C₂×#237	1	1	yes	characteristic
431	C₂×C₅	#1004	1	1	yes	characteristic
432	C₂³	—	2	2	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	3	1	no	—
435	C₂³	—	3	1	no	—
436	C₂³	—	4	1	no	—
437	C₂³	—	6	2	no	—
438	C₂³	—	6	2	no	—
439	C₂³	—	12	1	no	—
440	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
441	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
442	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
443	C₂³	C₂²×C₅×D₃	1	1	yes	characteristic
444	Q₈	—	6	2	no	—
445	D₄	—	2	2	no	—
446	D₄	—	2	2	no	—
447	D₄	—	3	4	no	—
448	D₄	—	4	1	no	—
449	D₄	—	6	2	no	—
450	D₄	—	6	2	no	—
451	D₄	—	6	2	no	—
452	D₄	—	6	2	no	—
453	D₄	—	12	1	no	—
454	C₂×C₄	—	3	1	no	—
455	C₂×C₄	—	3	1	no	—
456	C₂×C₄	—	3	1	no	—
457	C₂×C₄	—	3	1	no	—
458	C₂×C₄	—	3	1	no	—
459	C₂×C₄	—	3	1	no	—
460	C₂×C₄	—	6	2	no	—
461	C₂×C₄	—	6	2	no	—
462	C₂×C₄	—	6	2	no	—
463	C₂×C₄	—	6	2	no	—
464	C₂×C₄	—	12	1	no	—
465	C₂×C₄	—	12	1	no	—
466	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
467	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
468	C₂×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
469	C₂×C₃	—	2	2	no	—
470	C₂×C₃	—	2	2	no	—
471	C₂×C₃	—	4	1	no	—
472	C₂×C₃	—	4	1	no	—
473	C₂×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
474	C₂×C₃	C₂×C₅×#41	1	1	yes	characteristic
475	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
476	D₃	—	2	2	no	—
477	D₃	—	4	1	no	—
478	C₅	#5591	1	1	yes	characteristic, direct product factor, Sylow
479	C₂²	—	2	1	no	—
480	C₂²	—	2	1	no	—
481	C₂²	—	2	1	no	—
482	C₂²	—	2	1	no	—
483	C₂²	—	2	1	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	2	1	no	—
487	C₂²	—	2	1	no	—
488	C₂²	—	2	1	no	—
489	C₂²	—	3	2	no	—
490	C₂²	—	4	4	no	—
491	C₂²	—	6	1	no	—
492	C₂²	—	6	1	no	—
493	C₂²	—	6	1	no	—
494	C₂²	—	6	1	no	—
495	C₂²	—	6	1	no	—
496	C₂²	—	12	4	no	—
497	C₂²	C₅×D₃×D₄	1	2	yes	—
498	C₂²	C₅×#237	1	2	yes	complemented
499	C₂²	C₂³×C₅×D₃	1	1	yes	characteristic
500	C₄	—	2	1	no	—
501	C₄	—	2	1	no	—
502	C₄	—	2	1	no	—
503	C₄	—	3	2	no	—
504	C₄	—	6	1	no	—
505	C₄	—	6	1	no	—
506	C₄	—	6	1	no	—
507	C₄	—	6	1	no	—
508	C₄	—	6	1	no	—
509	C₃	C₅×#546	1	1	yes	characteristic, complemented, Sylow
510	C₂	—	2	2	no	—
511	C₂	—	2	2	no	—
512	C₂	—	4	1	no	—
513	C₂	—	4	1	no	—
514	C₂	—	6	2	no	—
515	C₂	—	12	1	no	—
516	C₂	C₂×C₅×D₃×D₄	1	1	yes	characteristic
517	C₂	C₂×C₅×#237	1	1	yes	characteristic
518	C₂	C₅×#1004	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₃
Fitting	C₃×C₅×#120	C₂
socle	C₂²×C₃×C₅	C₂⁴
hypercenter	C₅×#120	D₃
hypocenter	C₃	C₅×#546
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#120

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#67	C₂	semidirect
2. C₂×C₅×#237	C₂	semidirect
3. C₃×C₅×#120	C₂	semidirect
4. C₂×C₅×#217	C₂	semidirect
5. C₅×#939	C₂	semidirect
6. C₅×#939	C₂	semidirect
7. C₅×#938	C₂	semidirect
8. C₅×#937	C₂	semidirect
9. C₅×D₄×Dic₃	C₂	semidirect
10. C₅×#886	C₂	semidirect
11. C₅×#885	C₂	semidirect
12. C₅×#883	C₂	semidirect
13. C₅×#67⋊C₄	C₂	semidirect
14. C₅×D₃×C₂²⋊C₄	C₂	semidirect
15. C₅×#878	C₂	semidirect
20. C₂×C₃×C₅×D₄	C₂²	semidirect
21. C₂×C₃×C₅×D₄	C₂²	semidirect
27. C₂×C₅×#67	C₂²	semidirect
28. C₂×C₅×#67	C₂²	semidirect
30. C₂²×C₅×Dic₃	C₂²	semidirect
31. C₂²×C₅×Dic₃	C₂²	semidirect
32. C₂²×C₅×Dic₃	C₂²	semidirect
34. C₂×C₄×C₅×D₃	C₂²	semidirect
35. C₂×C₄×C₅×D₃	C₂²	semidirect
36. C₂×C₅×Dic₆	C₂²	semidirect
38. C₃×C₅×C₂²⋊C₄	C₂²	semidirect
39. C₃×C₅×C₂²⋊C₄	C₂²	semidirect
42. C₅×#217	C₂²	semidirect
43. C₅×#217	C₂²	semidirect
44. C₅×#217	C₂²	semidirect
45. C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
46. C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
47. C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
48. C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
49. C₅×C₄⋊Dic₃	C₂²	semidirect
50. C₅×C₄⋊Dic₃	C₂²	semidirect
51. C₅×Dic₃⋊C₄	C₂²	semidirect
52. C₅×Dic₃⋊C₄	C₂²	semidirect
53. C₄×C₅×Dic₃	C₂²	semidirect
54. #5591	C₅	direct (N × H)
65. C₅×#120	D₃	semidirect
93. C₅×#67	D₄	semidirect
107. C₂²×#67	C₂×C₅	semidirect
108. C₂×#237	C₂×C₅	semidirect
109. C₃×#120	C₂×C₅	semidirect
110. C₂×#217	C₂×C₅	semidirect
111. #939	C₂×C₅	semidirect
112. #939	C₂×C₅	semidirect
113. #938	C₂×C₅	semidirect
114. #937	C₂×C₅	semidirect
115. D₄×Dic₃	C₂×C₅	semidirect
116. #886	C₂×C₅	semidirect
117. #885	C₂×C₅	semidirect
118. #883	C₂×C₅	semidirect
119. #67⋊C₄	C₂×C₅	semidirect
120. D₃×C₂²⋊C₄	C₂×C₅	semidirect
121. #878	C₂×C₅	semidirect
134. C₂×C₅×D₄	C₂×D₃	semidirect
135. C₂×C₅×D₄	C₂×D₃	semidirect
156. C₅×C₂²⋊C₄	C₂×D₃	semidirect
157. C₅×C₂²⋊C₄	C₂×D₃	semidirect
172. C₂²×C₃×C₅	#41	semidirect
193. C₂×C₃×D₄	C₂²×C₅	semidirect
194. C₂×C₃×D₄	C₂²×C₅	semidirect
200. C₂×#67	C₂²×C₅	semidirect
201. C₂×#67	C₂²×C₅	semidirect
203. C₂²×Dic₃	C₂²×C₅	semidirect
204. C₂²×Dic₃	C₂²×C₅	semidirect
205. C₂²×Dic₃	C₂²×C₅	semidirect
207. C₂×C₄×D₃	C₂²×C₅	semidirect
208. C₂×C₄×D₃	C₂²×C₅	semidirect
209. C₂×Dic₆	C₂²×C₅	semidirect
211. C₃×C₂²⋊C₄	C₂²×C₅	semidirect
212. C₃×C₂²⋊C₄	C₂²×C₅	semidirect
215. #217	C₂²×C₅	semidirect
216. #217	C₂²×C₅	semidirect
217. #217	C₂²×C₅	semidirect
218. (C₂×D₃)⋊C₄	C₂²×C₅	semidirect
219. (C₂×D₃)⋊C₄	C₂²×C₅	semidirect
220. (C₂×D₃)⋊C₄	C₂²×C₅	semidirect
221. (C₂×D₃)⋊C₄	C₂²×C₅	semidirect
222. C₄⋊Dic₃	C₂²×C₅	semidirect
223. C₄⋊Dic₃	C₂²×C₅	semidirect
224. Dic₃⋊C₄	C₂²×C₅	semidirect
225. Dic₃⋊C₄	C₂²×C₅	semidirect
226. C₄×Dic₃	C₂²×C₅	semidirect
274. #120	C₅×D₃	semidirect
311. #67	C₅×D₄	semidirect
344. C₂²×C₅	#237	semidirect
367. C₂×D₄	C₂×C₅×D₃	semidirect
368. C₂×D₄	C₂×C₅×D₃	semidirect
389. C₂²⋊C₄	C₂×C₅×D₃	semidirect
390. C₂²⋊C₄	C₂×C₅×D₃	semidirect
392. C₃×C₅	#546	semidirect
405. C₂²×C₃	C₅×#41	semidirect
478. C₅	#5591	direct (N × H)
498. C₂²	C₅×#237	semidirect
509. C₃	C₅×#546	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×#67	2
C₂×C₅×#237	2
C₃×C₅×#120	2
C₂×C₅×#217	2
C₅×#939	2
C₅×#938	2
C₅×#937	2
C₅×D₄×Dic₃	2
C₅×#886	2
C₅×#885	2
C₅×#883	2
C₅×#67⋊C₄	2
C₅×D₃×C₂²⋊C₄	2
C₅×#878	2
C₅×#546	3
#5591	5

Subgroups — C5×#5591 #11755976

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#5591 #11755976