Subgroups — C₂×C₅×#5617 #14480764

Order 1920.

2672 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#939	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#937	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×D₄×Dic₃	C₂	1	2	yes	maximal, complemented
5	C₂²×C₅×(C₂×D₃)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#924	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#5617	C₂	1	16	yes	maximal, direct product factor
8	C₂×C₅×#898	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#886	C₂	1	2	yes	maximal, complemented
10	C₂×C₅×#885	C₂	1	2	yes	maximal, complemented
11	C₂×C₅×#538	—	3	1	no	maximal, Hall
12	C₂⁴×C₅×D₃	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
14	C₂²×C₃×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
15	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
16	C₂²×C₅×#67	C₂²	1	2	yes	—
17	C₂²×C₅×#67	C₂²	1	2	yes	—
18	C₂³×C₅×Dic₃	C₂²	1	1	yes	characteristic
19	C₂³×C₅×Dic₃	C₂²	1	2	yes	complemented
20	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic, complemented
21	C₃×C₅×#121	C₂²	1	8	yes	complemented
22	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	complemented
24	C₂×C₅×#217	C₂²	1	2	yes	—
25	C₂×C₅×#217	C₂²	1	2	yes	complemented
26	C₅×#939	C₂²	1	16	yes	complemented
27	C₅×#937	C₂²	1	16	yes	complemented
28	C₅×D₄×Dic₃	C₂²	1	16	yes	complemented
29	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
30	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
31	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
32	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
33	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	8	yes	complemented
34	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	complemented
36	C₅×#924	C₂²	1	8	yes	complemented
37	C₂×C₅×Dic₃⋊C₄	C₂²	1	2	yes	complemented
38	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	complemented
39	C₅×#898	C₂²	1	16	yes	complemented
40	C₅×#886	C₂²	1	16	yes	complemented
41	C₅×#885	C₂²	1	16	yes	complemented
42	C₂×#5617	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
43	C₅×#538	—	3	16	no	—
44	C₂×C₅×#126	—	3	2	no	—
45	C₂×C₅×#124	—	3	2	no	—
46	C₂×C₅×#123	—	3	2	no	—
47	C₂×C₅×C₂²⋊Q₈	—	3	1	no	—
48	C₂×C₅×#121	—	3	2	no	—
49	C₂×C₅×#121	D₃	1	1	yes	characteristic, complemented
50	C₂×C₅×#120	—	3	2	no	—
51	C₂×C₄×C₅×D₄	—	3	2	no	—
52	C₂²×C₅×C₂²⋊C₄	—	3	1	no	—
53	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
54	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
55	C₂³×C₅×D₃	—	2	4	no	—
56	C₂³×C₅×D₃	—	4	2	no	—
57	C₂³×C₅×D₃	—	4	2	no	—
58	C₂³×C₅×D₃	C₂³	1	2	yes	—
59	C₂³×C₅×D₃	C₂³	1	4	yes	—
60	C₂×C₃×C₅×D₄	—	2	8	no	—
61	C₂×C₃×C₅×D₄	—	4	2	no	—
62	C₂×C₃×C₅×D₄	C₂³	1	8	yes	—
63	C₂×C₃×C₅×D₄	C₂³	1	4	yes	complemented
64	C₂²×C₃×C₄×C₅	—	4	2	no	—
65	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
66	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
67	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
68	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
69	C₂×C₅×#67	—	4	4	no	—
70	C₂×C₅×#67	—	4	4	no	—
71	C₂×C₅×#67	C₂³	1	8	yes	—
72	C₂×C₅×#67	C₂³	1	8	yes	—
73	C₂²×C₅×Dic₃	—	4	2	no	—
74	C₂²×C₅×Dic₃	—	4	4	no	—
75	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
76	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
77	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
78	C₂²×C₅×Dic₃	C₂³	1	4	yes	—
79	C₂²×C₅×Dic₃	C₂³	1	8	yes	complemented
80	C₂×C₅×Dic₆	—	4	2	no	—
81	C₂×C₅×Dic₆	C₂³	1	4	yes	complemented
82	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	complemented
83	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	complemented
84	C₅×#217	C₂³	1	8	yes	—
85	C₅×#217	C₂³	1	8	yes	complemented
86	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
87	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
88	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
89	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
90	C₅×C₄⋊Dic₃	C₂³	1	4	yes	complemented
91	C₅×C₄⋊Dic₃	C₂³	1	8	yes	complemented
92	C₅×Dic₃⋊C₄	C₂³	1	8	yes	complemented
93	C₄×C₅×Dic₃	C₂³	1	8	yes	complemented
94	C₂×C₃×#121	C₂×C₅	1	1	yes	characteristic, complemented
95	C₂×#939	C₂×C₅	1	2	yes	complemented
96	C₂×#937	C₂×C₅	1	2	yes	complemented
97	C₂×D₄×Dic₃	C₂×C₅	1	2	yes	complemented
98	C₂²×(C₂×D₃)⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
99	C₂×#924	C₂×C₅	1	1	yes	characteristic, complemented
100	#5617	C₂×C₅	1	16	yes	direct product factor
101	C₂×#898	C₂×C₅	1	2	yes	complemented
102	C₂×#886	C₂×C₅	1	2	yes	complemented
103	C₂×#885	C₂×C₅	1	2	yes	complemented
104	C₂⁵×C₅	—	3	1	no	—
105	C₂²×C₅×Q₈	—	3	1	no	—
106	C₂²×C₅×D₄	—	3	2	no	—
107	C₂²×C₅×D₄	—	3	2	no	—
108	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
109	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic, complemented
110	C₂³×C₄×C₅	—	3	1	no	—
111	C₂³×C₄×C₅	—	3	2	no	—
112	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
113	C₅×#126	—	3	16	no	—
114	C₅×#124	—	3	16	no	—
115	C₅×#123	—	3	16	no	—
116	C₅×C₂²⋊Q₈	—	3	8	no	—
117	C₅×#121	—	3	16	no	—
118	C₅×#121	C₂×D₃	1	8	yes	complemented
119	C₅×#120	—	3	16	no	—
120	C₄×C₅×D₄	—	3	16	no	—
121	C₂×C₅×C₄⋊C₄	—	3	2	no	—
122	C₂×C₅×C₄⋊C₄	—	3	2	no	—
123	C₂×C₅×C₄⋊C₄	—	3	1	no	—
124	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic, complemented
125	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
126	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
127	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
128	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
129	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
130	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
131	C₂×C₅×C₂²⋊C₄	—	3	8	no	—
132	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	complemented
133	C₂×C₄²×C₅	—	3	2	no	—
134	C₂×#538	—	3	1	no	Sylow
135	C₂³×C₃×C₅	—	2	2	no	—
136	C₂³×C₃×C₅	—	2	2	no	—
137	C₂³×C₃×C₅	—	2	4	no	—
138	C₂³×C₃×C₅	—	2	4	no	—
139	C₂³×C₃×C₅	—	2	4	no	—
140	C₂³×C₃×C₅	#41	1	4	yes	—
141	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
142	C₂³×C₃×C₅	C₂⁴	1	2	yes	—
143	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
144	C₂²×C₅×D₃	—	2	8	no	—
145	C₂²×C₅×D₃	—	2	4	no	—
146	C₂²×C₅×D₃	—	2	4	no	—
147	C₂²×C₅×D₃	—	2	4	no	—
148	C₂²×C₅×D₃	—	4	4	no	—
149	C₂²×C₅×D₃	—	4	4	no	—
150	C₂²×C₅×D₃	—	4	16	no	—
151	C₂²×C₅×D₃	C₂⁴	1	4	yes	—
152	C₃×C₅×D₄	—	2	16	no	—
153	C₃×C₅×D₄	—	4	4	no	—
154	C₂×C₃×C₄×C₅	—	2	4	no	—
155	C₂×C₃×C₄×C₅	—	2	2	no	—
156	C₂×C₃×C₄×C₅	—	2	2	no	—
157	C₂×C₃×C₄×C₅	—	4	4	no	—
158	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
159	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
160	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
161	C₅×#67	—	4	8	no	—
162	C₅×#67	—	4	8	no	—
163	C₂×C₅×Dic₃	—	2	4	no	—
164	C₂×C₅×Dic₃	—	2	4	no	—
165	C₂×C₅×Dic₃	—	2	4	no	—
166	C₂×C₅×Dic₃	—	4	4	no	—
167	C₂×C₅×Dic₃	—	4	8	no	—
168	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
169	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
170	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
171	C₅×Dic₆	—	4	4	no	—
172	C₂⁴×D₃	C₂²×C₅	1	1	yes	characteristic
173	C₂²×C₃×D₄	C₂²×C₅	1	2	yes	—
174	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
175	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
176	C₂²×#67	C₂²×C₅	1	2	yes	—
177	C₂²×#67	C₂²×C₅	1	2	yes	—
178	C₂³×Dic₃	C₂²×C₅	1	1	yes	characteristic
179	C₂³×Dic₃	C₂²×C₅	1	2	yes	complemented
180	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic, complemented
181	C₃×#121	C₂²×C₅	1	8	yes	complemented
182	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic, complemented
183	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	complemented
184	C₂×#217	C₂²×C₅	1	2	yes	—
185	C₂×#217	C₂²×C₅	1	2	yes	complemented
186	#939	C₂²×C₅	1	16	yes	complemented
187	#937	C₂²×C₅	1	16	yes	complemented
188	D₄×Dic₃	C₂²×C₅	1	16	yes	complemented
189	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
190	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
191	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
192	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
193	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	8	yes	complemented
194	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
195	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	complemented
196	#924	C₂²×C₅	1	8	yes	complemented
197	C₂×Dic₃⋊C₄	C₂²×C₅	1	2	yes	complemented
198	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	complemented
199	#898	C₂²×C₅	1	16	yes	complemented
200	#886	C₂²×C₅	1	16	yes	complemented
201	#885	C₂²×C₅	1	16	yes	complemented
202	C₂⁴×C₅	—	3	2	no	—
203	C₂⁴×C₅	—	3	4	no	—
204	C₂⁴×C₅	—	6	4	no	—
205	C₂⁴×C₅	—	12	2	no	—
206	C₂⁴×C₅	—	12	2	no	—
207	C₂⁴×C₅	C₂²×D₃	1	2	yes	—
208	C₂⁴×C₅	C₂²×D₃	1	1	yes	characteristic
209	C₂×C₅×Q₈	—	3	4	no	—
210	C₂×C₅×Q₈	—	12	2	no	—
211	C₂×C₅×D₄	—	2	8	no	—
212	C₂×C₅×D₄	—	3	8	no	—
213	C₂×C₅×D₄	—	3	8	no	—
214	C₂×C₅×D₄	—	4	2	no	—
215	C₂×C₅×D₄	—	12	4	no	—
216	C₂×C₅×D₄	—	12	4	no	—
217	C₂×C₅×D₄	C₂²×D₃	1	8	yes	—
218	C₂×C₅×D₄	C₂²×D₃	1	4	yes	complemented
219	C₂²×C₄×C₅	—	3	2	no	—
220	C₂²×C₄×C₅	—	3	2	no	—
221	C₂²×C₄×C₅	—	3	2	no	—
222	C₂²×C₄×C₅	—	3	4	no	—
223	C₂²×C₄×C₅	—	3	8	no	—
224	C₂²×C₄×C₅	—	4	2	no	—
225	C₂²×C₄×C₅	—	12	2	no	—
226	C₂²×C₄×C₅	—	12	4	no	—
227	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
228	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
229	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
230	C₂²×C₄×C₅	C₂²×D₃	1	4	yes	—
231	C₅×C₄⋊C₄	—	3	8	no	—
232	C₅×C₄⋊C₄	—	3	8	no	—
233	C₅×C₄⋊C₄	—	3	4	no	—
234	C₅×C₄⋊C₄	C₂²×D₃	1	4	yes	complemented
235	C₅×C₂²⋊C₄	—	3	8	no	—
236	C₅×C₂²⋊C₄	—	3	8	no	—
237	C₅×C₂²⋊C₄	—	3	8	no	—
238	C₅×C₂²⋊C₄	—	3	8	no	—
239	C₅×C₂²⋊C₄	—	3	8	no	—
240	C₅×C₂²⋊C₄	—	3	8	no	—
241	C₅×C₂²⋊C₄	C₂²×D₃	1	8	yes	complemented
242	C₄²×C₅	—	3	8	no	—
243	#538	—	3	16	no	—
244	C₂×#126	—	3	2	no	—
245	C₂×#124	—	3	2	no	—
246	C₂×#123	—	3	2	no	—
247	C₂×C₂²⋊Q₈	—	3	1	no	—
248	C₂×#121	—	3	2	no	—
249	C₂×#121	C₅×D₃	1	1	yes	characteristic, complemented
250	C₂×#120	—	3	2	no	—
251	C₂×C₄×D₄	—	3	2	no	—
252	C₂²×C₂²⋊C₄	—	3	1	no	—
253	C₂²×C₃×C₅	—	2	8	no	—
254	C₂²×C₃×C₅	—	2	2	no	—
255	C₂²×C₃×C₅	—	2	2	no	—
256	C₂²×C₃×C₅	—	2	4	no	—
257	C₂²×C₃×C₅	—	2	4	no	—
258	C₂²×C₃×C₅	—	2	4	no	—
259	C₂²×C₃×C₅	—	4	8	no	—
260	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
261	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
262	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
263	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
264	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
265	C₂×C₅×D₃	—	2	4	no	—
266	C₂×C₅×D₃	—	2	4	no	—
267	C₂×C₅×D₃	—	2	4	no	—
268	C₂×C₅×D₃	—	4	16	no	—
269	C₂×C₅×D₃	—	4	8	no	—
270	C₃×C₄×C₅	—	2	4	no	—
271	C₃×C₄×C₅	—	2	2	no	—
272	C₃×C₄×C₅	—	2	2	no	—
273	C₅×Dic₃	—	2	4	no	—
274	C₅×Dic₃	—	2	4	no	—
275	C₅×Dic₃	—	2	4	no	—
276	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
277	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
278	C₂³×D₃	—	2	4	no	—
279	C₂³×D₃	—	4	2	no	—
280	C₂³×D₃	—	4	2	no	—
281	C₂³×D₃	C₂³×C₅	1	2	yes	—
282	C₂³×D₃	C₂³×C₅	1	4	yes	—
283	C₂×C₃×D₄	—	2	8	no	—
284	C₂×C₃×D₄	—	4	2	no	—
285	C₂×C₃×D₄	C₂³×C₅	1	8	yes	—
286	C₂×C₃×D₄	C₂³×C₅	1	4	yes	complemented
287	C₂²×C₃×C₄	—	4	2	no	—
288	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
289	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
290	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
291	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
292	C₂×#67	—	4	4	no	—
293	C₂×#67	—	4	4	no	—
294	C₂×#67	C₂³×C₅	1	8	yes	—
295	C₂×#67	C₂³×C₅	1	8	yes	—
296	C₂²×Dic₃	—	4	2	no	—
297	C₂²×Dic₃	—	4	4	no	—
298	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
299	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
300	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
301	C₂²×Dic₃	C₂³×C₅	1	4	yes	—
302	C₂²×Dic₃	C₂³×C₅	1	8	yes	complemented
303	C₂×Dic₆	—	4	2	no	—
304	C₂×Dic₆	C₂³×C₅	1	4	yes	complemented
305	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	complemented
306	C₃×C₂²⋊C₄	C₂³×C₅	1	8	yes	complemented
307	#217	C₂³×C₅	1	8	yes	—
308	#217	C₂³×C₅	1	8	yes	complemented
309	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
310	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
311	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
312	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
313	C₄⋊Dic₃	C₂³×C₅	1	4	yes	complemented
314	C₄⋊Dic₃	C₂³×C₅	1	8	yes	complemented
315	Dic₃⋊C₄	C₂³×C₅	1	8	yes	complemented
316	C₄×Dic₃	C₂³×C₅	1	8	yes	complemented
317	C₂³×C₅	—	2	2	no	—
318	C₂³×C₅	—	2	2	no	—
319	C₂³×C₅	—	2	4	no	—
320	C₂³×C₅	—	2	4	no	—
321	C₂³×C₅	—	2	4	no	—
322	C₂³×C₅	—	3	4	no	—
323	C₂³×C₅	—	6	8	no	—
324	C₂³×C₅	—	6	4	no	—
325	C₂³×C₅	—	6	4	no	—
326	C₂³×C₅	—	6	4	no	—
327	C₂³×C₅	—	12	4	no	—
328	C₂³×C₅	—	12	4	no	—
329	C₂³×C₅	—	12	16	no	—
330	C₂³×C₅	#237	1	4	yes	—
331	C₂³×C₅	C₂³×D₃	1	4	yes	—
332	C₂³×C₅	C₂³×D₃	1	2	yes	—
333	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
334	C₅×Q₈	—	12	4	no	—
335	C₅×D₄	—	2	16	no	—
336	C₅×D₄	—	4	4	no	—
337	C₅×D₄	—	12	8	no	—
338	C₅×D₄	—	12	8	no	—
339	C₂×C₄×C₅	—	2	4	no	—
340	C₂×C₄×C₅	—	2	2	no	—
341	C₂×C₄×C₅	—	2	2	no	—
342	C₂×C₄×C₅	—	3	4	no	—
343	C₂×C₄×C₅	—	3	4	no	—
344	C₂×C₄×C₅	—	3	4	no	—
345	C₂×C₄×C₅	—	4	4	no	—
346	C₂×C₄×C₅	—	6	4	no	—
347	C₂×C₄×C₅	—	6	4	no	—
348	C₂×C₄×C₅	—	6	4	no	—
349	C₂×C₄×C₅	—	12	4	no	—
350	C₂×C₄×C₅	—	12	8	no	—
351	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
352	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
353	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
354	C₂⁵	—	3	1	no	—
355	C₂²×Q₈	—	3	1	no	—
356	C₂²×D₄	—	3	2	no	—
357	C₂²×D₄	—	3	2	no	—
358	C₂²×D₄	C₂×C₅×D₃	1	2	yes	—
359	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic, complemented
360	C₂³×C₄	—	3	1	no	—
361	C₂³×C₄	—	3	2	no	—
362	C₂³×C₄	C₂×C₅×D₃	1	1	yes	characteristic
363	#126	—	3	16	no	—
364	#124	—	3	16	no	—
365	#123	—	3	16	no	—
366	C₂²⋊Q₈	—	3	8	no	—
367	#121	—	3	16	no	—
368	#121	C₂×C₅×D₃	1	8	yes	complemented
369	#120	—	3	16	no	—
370	C₄×D₄	—	3	16	no	—
371	C₂×C₄⋊C₄	—	3	2	no	—
372	C₂×C₄⋊C₄	—	3	2	no	—
373	C₂×C₄⋊C₄	—	3	1	no	—
374	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic, complemented
375	C₂×C₂²⋊C₄	—	3	2	no	—
376	C₂×C₂²⋊C₄	—	3	2	no	—
377	C₂×C₂²⋊C₄	—	3	2	no	—
378	C₂×C₂²⋊C₄	—	3	2	no	—
379	C₂×C₂²⋊C₄	—	3	8	no	—
380	C₂×C₂²⋊C₄	—	3	2	no	—
381	C₂×C₂²⋊C₄	—	3	2	no	—
382	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	complemented
383	C₂×C₄²	—	3	2	no	—
384	C₂×C₃×C₅	—	2	4	no	—
385	C₂×C₃×C₅	—	4	4	no	—
386	C₂×C₃×C₅	#538	1	4	yes	complemented
387	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
388	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
389	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
390	C₅×D₃	—	4	4	no	—
391	C₂³×C₃	—	2	2	no	—
392	C₂³×C₃	—	2	2	no	—
393	C₂³×C₃	—	2	4	no	—
394	C₂³×C₃	—	2	4	no	—
395	C₂³×C₃	—	2	4	no	—
396	C₂³×C₃	C₅×#41	1	4	yes	—
397	C₂³×C₃	C₂⁴×C₅	1	4	yes	—
398	C₂³×C₃	C₂⁴×C₅	1	2	yes	—
399	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
400	C₂²×D₃	—	2	8	no	—
401	C₂²×D₃	—	2	4	no	—
402	C₂²×D₃	—	2	4	no	—
403	C₂²×D₃	—	2	4	no	—
404	C₂²×D₃	—	4	4	no	—
405	C₂²×D₃	—	4	4	no	—
406	C₂²×D₃	—	4	16	no	—
407	C₂²×D₃	C₂⁴×C₅	1	4	yes	—
408	C₃×D₄	—	2	16	no	—
409	C₃×D₄	—	4	4	no	—
410	C₂×C₃×C₄	—	2	4	no	—
411	C₂×C₃×C₄	—	2	2	no	—
412	C₂×C₃×C₄	—	2	2	no	—
413	C₂×C₃×C₄	—	4	4	no	—
414	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
415	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
416	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
417	#67	—	4	8	no	—
418	#67	—	4	8	no	—
419	C₂×Dic₃	—	2	4	no	—
420	C₂×Dic₃	—	2	4	no	—
421	C₂×Dic₃	—	2	4	no	—
422	C₂×Dic₃	—	4	4	no	—
423	C₂×Dic₃	—	4	8	no	—
424	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
425	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
426	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
427	Dic₆	—	4	4	no	—
428	C₂²×C₅	—	2	2	no	—
429	C₂²×C₅	—	2	2	no	—
430	C₂²×C₅	—	2	8	no	—
431	C₂²×C₅	—	2	4	no	—
432	C₂²×C₅	—	2	4	no	—
433	C₂²×C₅	—	2	4	no	—
434	C₂²×C₅	—	4	8	no	—
435	C₂²×C₅	—	6	4	no	—
436	C₂²×C₅	—	6	4	no	—
437	C₂²×C₅	—	6	4	no	—
438	C₂²×C₅	—	12	16	no	—
439	C₂²×C₅	—	12	8	no	—
440	C₂²×C₅	C₂×#237	1	4	yes	—
441	C₂²×C₅	C₂×#237	1	2	yes	—
442	C₂²×C₅	#1004	1	2	yes	—
443	C₂²×C₅	#1009	1	2	yes	—
444	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
445	C₄×C₅	—	2	4	no	—
446	C₄×C₅	—	2	2	no	—
447	C₄×C₅	—	2	2	no	—
448	C₄×C₅	—	6	4	no	—
449	C₄×C₅	—	6	4	no	—
450	C₄×C₅	—	6	4	no	—
451	C₂⁴	—	3	2	no	—
452	C₂⁴	—	3	4	no	—
453	C₂⁴	—	6	4	no	—
454	C₂⁴	—	12	2	no	—
455	C₂⁴	—	12	2	no	—
456	C₂⁴	C₂²×C₅×D₃	1	2	yes	—
457	C₂⁴	C₂²×C₅×D₃	1	1	yes	characteristic
458	C₂×Q₈	—	3	4	no	—
459	C₂×Q₈	—	12	2	no	—
460	C₂×D₄	—	2	8	no	—
461	C₂×D₄	—	3	8	no	—
462	C₂×D₄	—	3	8	no	—
463	C₂×D₄	—	4	2	no	—
464	C₂×D₄	—	12	4	no	—
465	C₂×D₄	—	12	4	no	—
466	C₂×D₄	C₂²×C₅×D₃	1	8	yes	—
467	C₂×D₄	C₂²×C₅×D₃	1	4	yes	complemented
468	C₂²×C₄	—	3	2	no	—
469	C₂²×C₄	—	3	2	no	—
470	C₂²×C₄	—	3	2	no	—
471	C₂²×C₄	—	3	4	no	—
472	C₂²×C₄	—	3	8	no	—
473	C₂²×C₄	—	4	2	no	—
474	C₂²×C₄	—	12	2	no	—
475	C₂²×C₄	—	12	4	no	—
476	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
477	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
478	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
479	C₂²×C₄	C₂²×C₅×D₃	1	4	yes	—
480	C₄⋊C₄	—	3	8	no	—
481	C₄⋊C₄	—	3	8	no	—
482	C₄⋊C₄	—	3	4	no	—
483	C₄⋊C₄	C₂²×C₅×D₃	1	4	yes	complemented
484	C₂²⋊C₄	—	3	8	no	—
485	C₂²⋊C₄	—	3	8	no	—
486	C₂²⋊C₄	—	3	8	no	—
487	C₂²⋊C₄	—	3	8	no	—
488	C₂²⋊C₄	—	3	8	no	—
489	C₂²⋊C₄	—	3	8	no	—
490	C₂²⋊C₄	C₂²×C₅×D₃	1	8	yes	complemented
491	C₄²	—	3	8	no	—
492	C₃×C₅	C₂×#538	1	1	yes	characteristic, complemented, Hall
493	C₂²×C₃	—	2	8	no	—
494	C₂²×C₃	—	2	2	no	—
495	C₂²×C₃	—	2	2	no	—
496	C₂²×C₃	—	2	4	no	—
497	C₂²×C₃	—	2	4	no	—
498	C₂²×C₃	—	2	4	no	—
499	C₂²×C₃	—	4	8	no	—
500	C₂²×C₃	C₂×C₅×#41	1	4	yes	—
501	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
502	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
503	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
504	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
505	C₂×D₃	—	2	4	no	—
506	C₂×D₃	—	2	4	no	—
507	C₂×D₃	—	2	4	no	—
508	C₂×D₃	—	4	16	no	—
509	C₂×D₃	—	4	8	no	—
510	C₃×C₄	—	2	4	no	—
511	C₃×C₄	—	2	2	no	—
512	C₃×C₄	—	2	2	no	—
513	Dic₃	—	2	4	no	—
514	Dic₃	—	2	4	no	—
515	Dic₃	—	2	4	no	—
516	C₂×C₅	—	2	4	no	—
517	C₂×C₅	—	4	4	no	—
518	C₂×C₅	—	12	4	no	—
519	C₂×C₅	#5617	1	4	yes	direct product factor
520	C₂×C₅	C₂²×#237	1	1	yes	characteristic
521	C₂×C₅	C₂×#1004	1	1	yes	characteristic
522	C₂×C₅	C₂×#1009	1	1	yes	characteristic
523	C₂³	—	2	2	no	—
524	C₂³	—	2	2	no	—
525	C₂³	—	2	4	no	—
526	C₂³	—	2	4	no	—
527	C₂³	—	2	4	no	—
528	C₂³	—	3	4	no	—
529	C₂³	—	6	8	no	—
530	C₂³	—	6	4	no	—
531	C₂³	—	6	4	no	—
532	C₂³	—	6	4	no	—
533	C₂³	—	12	4	no	—
534	C₂³	—	12	4	no	—
535	C₂³	—	12	16	no	—
536	C₂³	C₅×#237	1	4	yes	—
537	C₂³	C₂³×C₅×D₃	1	4	yes	—
538	C₂³	C₂³×C₅×D₃	1	2	yes	—
539	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
540	Q₈	—	12	4	no	—
541	D₄	—	2	16	no	—
542	D₄	—	4	4	no	—
543	D₄	—	12	8	no	—
544	D₄	—	12	8	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	3	4	no	—
549	C₂×C₄	—	3	4	no	—
550	C₂×C₄	—	3	4	no	—
551	C₂×C₄	—	4	4	no	—
552	C₂×C₄	—	6	4	no	—
553	C₂×C₄	—	6	4	no	—
554	C₂×C₄	—	6	4	no	—
555	C₂×C₄	—	12	4	no	—
556	C₂×C₄	—	12	8	no	—
557	C₂×C₄	C₂³×C₅×D₃	1	4	yes	—
558	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
559	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
560	C₂×C₃	—	2	4	no	—
561	C₂×C₃	—	4	4	no	—
562	C₂×C₃	C₅×#538	1	4	yes	complemented
563	C₂×C₃	C₂²×C₅×#41	1	1	yes	characteristic
564	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
565	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
566	D₃	—	4	4	no	—
567	C₅	C₂×#5617	1	1	yes	characteristic, direct product factor, Sylow
568	C₂²	—	2	2	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	8	no	—
571	C₂²	—	2	4	no	—
572	C₂²	—	2	4	no	—
573	C₂²	—	2	4	no	—
574	C₂²	—	4	8	no	—
575	C₂²	—	6	4	no	—
576	C₂²	—	6	4	no	—
577	C₂²	—	6	4	no	—
578	C₂²	—	12	16	no	—
579	C₂²	—	12	8	no	—
580	C₂²	C₂×C₅×#237	1	4	yes	—
581	C₂²	C₂×C₅×#237	1	2	yes	—
582	C₂²	C₅×#1004	1	2	yes	—
583	C₂²	C₅×#1009	1	2	yes	—
584	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
585	C₄	—	2	4	no	—
586	C₄	—	2	2	no	—
587	C₄	—	2	2	no	—
588	C₄	—	6	4	no	—
589	C₄	—	6	4	no	—
590	C₄	—	6	4	no	—
591	C₃	C₂×C₅×#538	1	1	yes	characteristic, complemented, Sylow
592	C₂	—	2	4	no	—
593	C₂	—	4	4	no	—
594	C₂	—	12	4	no	—
595	C₂	C₅×#5617	1	4	yes	direct product factor
596	C₂	C₂²×C₅×#237	1	1	yes	characteristic
597	C₂	C₂×C₅×#1004	1	1	yes	characteristic
598	C₂	C₂×C₅×#1009	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₅
Frattini	C₂²	C₂⁴×C₅×D₃
Fitting	C₂×C₃×C₅×#121	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₅×#121	D₃
hypocenter	C₃	C₂×C₅×#538
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#121

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#121	C₂	semidirect
2. C₂×C₅×#939	C₂	semidirect
3. C₂×C₅×#937	C₂	semidirect
4. C₂×C₅×D₄×Dic₃	C₂	semidirect
5. C₂²×C₅×(C₂×D₃)⋊C₄	C₂	semidirect
6. C₂×C₅×#924	C₂	semidirect
7. C₅×#5617	C₂	direct (N × H)
8. C₂×C₅×#898	C₂	semidirect
9. C₂×C₅×#886	C₂	semidirect
10. C₂×C₅×#885	C₂	semidirect
14. C₂²×C₃×C₅×D₄	C₂²	semidirect
19. C₂³×C₅×Dic₃	C₂²	semidirect
20. C₂²×C₅×Dic₆	C₂²	semidirect
21. C₃×C₅×#121	C₂²	semidirect
22. C₂×C₃×C₅×C₄⋊C₄	C₂²	semidirect
23. C₂×C₃×C₅×C₂²⋊C₄	C₂²	semidirect
25. C₂×C₅×#217	C₂²	semidirect
26. C₅×#939	C₂²	semidirect
27. C₅×#937	C₂²	semidirect
28. C₅×D₄×Dic₃	C₂²	semidirect
33. C₂×C₅×(C₂×D₃)⋊C₄	C₂²	semidirect
34. C₂×C₅×C₄⋊Dic₃	C₂²	semidirect
35. C₂×C₅×C₄⋊Dic₃	C₂²	semidirect
36. C₅×#924	C₂²	semidirect
37. C₂×C₅×Dic₃⋊C₄	C₂²	semidirect
38. C₂×C₄×C₅×Dic₃	C₂²	semidirect
39. C₅×#898	C₂²	semidirect
40. C₅×#886	C₂²	semidirect
41. C₅×#885	C₂²	semidirect
42. C₂×#5617	C₅	direct (N × H)
49. C₂×C₅×#121	D₃	semidirect
63. C₂×C₃×C₅×D₄	C₂³	semidirect
79. C₂²×C₅×Dic₃	C₂³	semidirect
81. C₂×C₅×Dic₆	C₂³	semidirect
82. C₃×C₅×C₄⋊C₄	C₂³	semidirect
83. C₃×C₅×C₂²⋊C₄	C₂³	semidirect
85. C₅×#217	C₂³	semidirect
90. C₅×C₄⋊Dic₃	C₂³	semidirect
91. C₅×C₄⋊Dic₃	C₂³	semidirect
92. C₅×Dic₃⋊C₄	C₂³	semidirect
93. C₄×C₅×Dic₃	C₂³	semidirect
94. C₂×C₃×#121	C₂×C₅	semidirect
95. C₂×#939	C₂×C₅	semidirect
96. C₂×#937	C₂×C₅	semidirect
97. C₂×D₄×Dic₃	C₂×C₅	semidirect
98. C₂²×(C₂×D₃)⋊C₄	C₂×C₅	semidirect
99. C₂×#924	C₂×C₅	semidirect
100. #5617	C₂×C₅	direct (N × H)
101. C₂×#898	C₂×C₅	semidirect
102. C₂×#886	C₂×C₅	semidirect
103. C₂×#885	C₂×C₅	semidirect
109. C₂²×C₅×D₄	C₂×D₃	semidirect
118. C₅×#121	C₂×D₃	semidirect
124. C₂×C₅×C₄⋊C₄	C₂×D₃	semidirect
132. C₂×C₅×C₂²⋊C₄	C₂×D₃	semidirect
174. C₂²×C₃×D₄	C₂²×C₅	semidirect
179. C₂³×Dic₃	C₂²×C₅	semidirect
180. C₂²×Dic₆	C₂²×C₅	semidirect
181. C₃×#121	C₂²×C₅	semidirect
182. C₂×C₃×C₄⋊C₄	C₂²×C₅	semidirect
183. C₂×C₃×C₂²⋊C₄	C₂²×C₅	semidirect
185. C₂×#217	C₂²×C₅	semidirect
186. #939	C₂²×C₅	semidirect
187. #937	C₂²×C₅	semidirect
188. D₄×Dic₃	C₂²×C₅	semidirect
193. C₂×(C₂×D₃)⋊C₄	C₂²×C₅	semidirect
194. C₂×C₄⋊Dic₃	C₂²×C₅	semidirect
195. C₂×C₄⋊Dic₃	C₂²×C₅	semidirect
196. #924	C₂²×C₅	semidirect
197. C₂×Dic₃⋊C₄	C₂²×C₅	semidirect
198. C₂×C₄×Dic₃	C₂²×C₅	semidirect
199. #898	C₂²×C₅	semidirect
200. #886	C₂²×C₅	semidirect
201. #885	C₂²×C₅	semidirect
218. C₂×C₅×D₄	C₂²×D₃	semidirect
234. C₅×C₄⋊C₄	C₂²×D₃	semidirect
241. C₅×C₂²⋊C₄	C₂²×D₃	semidirect
249. C₂×#121	C₅×D₃	semidirect
286. C₂×C₃×D₄	C₂³×C₅	semidirect
302. C₂²×Dic₃	C₂³×C₅	semidirect
304. C₂×Dic₆	C₂³×C₅	semidirect
305. C₃×C₄⋊C₄	C₂³×C₅	semidirect
306. C₃×C₂²⋊C₄	C₂³×C₅	semidirect
308. #217	C₂³×C₅	semidirect
313. C₄⋊Dic₃	C₂³×C₅	semidirect
314. C₄⋊Dic₃	C₂³×C₅	semidirect
315. Dic₃⋊C₄	C₂³×C₅	semidirect
316. C₄×Dic₃	C₂³×C₅	semidirect
359. C₂²×D₄	C₂×C₅×D₃	semidirect
368. #121	C₂×C₅×D₃	semidirect
374. C₂×C₄⋊C₄	C₂×C₅×D₃	semidirect
382. C₂×C₂²⋊C₄	C₂×C₅×D₃	semidirect
386. C₂×C₃×C₅	#538	semidirect
467. C₂×D₄	C₂²×C₅×D₃	semidirect
483. C₄⋊C₄	C₂²×C₅×D₃	semidirect
490. C₂²⋊C₄	C₂²×C₅×D₃	semidirect
492. C₃×C₅	C₂×#538	semidirect
519. C₂×C₅	#5617	direct (N × H)
562. C₂×C₃	C₅×#538	semidirect
567. C₅	C₂×#5617	direct (N × H)
591. C₃	C₂×C₅×#538	semidirect
595. C₂	C₅×#5617	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#121	2
C₂×C₅×#939	2
C₂×C₅×#937	2
C₂×C₅×D₄×Dic₃	2
C₂²×C₅×(C₂×D₃)⋊C₄	2
C₂×C₅×#924	2
C₅×#5617	2
C₂×C₅×#898	2
C₂×C₅×#886	2
C₂×C₅×#885	2
C₂×C₅×#538	3
C₂×#5617	5

Subgroups — C2×C5×#5617 #14480764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5617 #14480764