Subgroups — C₂×C₅×#5681 #14484699

Order 1920.

2480 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#947	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×Q₈×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#940	C₂	1	1	yes	maximal, characteristic
5	C₂×C₅×D₄×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#5681	C₂	1	16	yes	maximal, direct product factor
7	C₂×C₅×#886	C₂	1	2	yes	maximal, complemented
8	C₂×C₅×#885	C₂	1	2	yes	maximal, complemented
9	C₂×C₅×#883	C₂	1	2	yes	maximal, complemented
10	C₂×C₅×#67⋊C₄	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₅×#559	—	3	1	no	maximal, Hall
14	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₅×D₄	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	2	yes	—
19	C₂²×C₅×D₁₂	C₂²	1	1	yes	characteristic
20	C₂²×C₄×C₅×D₃	C₂²	1	2	yes	—
21	C₂²×C₅×Dic₆	C₂²	1	1	yes	characteristic
22	C₃×C₅×#124	C₂²	1	8	yes	complemented
23	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
26	C₂×C₅×#217	C₂²	1	2	yes	—
27	C₅×#947	C₂²	1	8	yes	complemented
28	C₅×Q₈×Dic₃	C₂²	1	8	yes	complemented
29	C₅×#940	C₂²	1	8	yes	—
30	C₅×D₄×Dic₃	C₂²	1	8	yes	complemented
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	2	yes	—
34	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
35	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
36	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
37	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
38	C₂×C₅×Dic₃⋊C₄	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
40	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₅×Dic₃	C₂²	1	2	yes	—
42	C₅×#886	C₂²	1	16	yes	complemented
43	C₅×#885	C₂²	1	16	yes	complemented
44	C₅×#883	C₂²	1	16	yes	complemented
45	C₅×#67⋊C₄	C₂²	1	16	yes	complemented
46	C₄×C₅×D₁₂	C₂²	1	8	yes	complemented
47	C₄×C₅×Dic₆	C₂²	1	8	yes	complemented
48	C₂×#5681	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
49	C₅×#559	—	3	16	no	—
50	C₂×C₅×#127	—	3	1	no	—
51	C₂×C₅×#124	—	3	1	no	—
52	C₂×C₅×#124	—	3	2	no	—
53	C₂×C₅×#124	D₃	1	1	yes	characteristic, complemented
54	C₂×C₅×#123	—	3	2	no	—
55	C₂×C₅×#123	—	3	2	no	—
56	C₂×C₄×C₅×Q₈	—	3	1	no	—
57	C₂×C₄×C₅×Q₈	—	3	1	no	—
58	C₂×C₄×C₅×D₄	—	3	1	no	—
59	C₂×C₄×C₅×D₄	—	3	2	no	—
60	C₂×C₄×C₅×D₄	—	3	1	no	—
61	C₂⁴×C₃×C₅	C₂³	1	2	yes	—
62	C₂³×C₅×D₃	C₂³	1	2	yes	—
63	C₂×C₃×C₅×Q₈	—	2	4	no	—
64	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
65	C₂×C₃×C₅×D₄	—	2	4	no	—
66	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
67	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
68	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
69	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
70	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
71	C₂×C₅×#67	—	2	8	no	—
72	C₂×C₅×#67	—	4	4	no	—
73	C₂×C₅×#67	C₂³	1	8	yes	—
74	C₂×C₅×#67	C₂³	1	8	yes	—
75	C₂²×C₅×Dic₃	—	4	4	no	—
76	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
77	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
78	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
79	C₂²×C₅×Dic₃	C₂³	1	2	yes	—
80	C₂²×C₅×Dic₃	C₂³	1	8	yes	—
81	C₂²×C₅×Dic₃	C₂³	1	1	yes	characteristic
82	C₂×C₅×D₁₂	—	2	4	no	—
83	C₂×C₅×D₁₂	C₂³	1	4	yes	—
84	C₂×C₄×C₅×D₃	—	4	4	no	—
85	C₂×C₄×C₅×D₃	C₂³	1	8	yes	—
86	C₂×C₅×Dic₆	—	2	4	no	—
87	C₂×C₅×Dic₆	C₂³	1	4	yes	—
88	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
89	C₃×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
90	C₃×C₄²×C₅	C₂³	1	4	yes	—
91	C₅×#217	C₂³	1	8	yes	—
92	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
93	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
94	C₅×(C₂×D₃)⋊C₄	C₂³	1	8	yes	—
95	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
96	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
97	C₅×C₄⋊Dic₃	C₂³	1	8	yes	—
98	C₅×Dic₃⋊C₄	C₂³	1	4	yes	—
99	C₅×Dic₃⋊C₄	C₂³	1	4	yes	—
100	C₄×C₅×Dic₃	C₂³	1	4	yes	—
101	C₄×C₅×Dic₃	C₂³	1	4	yes	—
102	C₄×C₅×Dic₃	C₂³	1	8	yes	—
103	C₂×C₃×#124	C₂×C₅	1	1	yes	characteristic, complemented
104	C₂×#947	C₂×C₅	1	1	yes	characteristic, complemented
105	C₂×Q₈×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
106	C₂×#940	C₂×C₅	1	1	yes	characteristic
107	C₂×D₄×Dic₃	C₂×C₅	1	1	yes	characteristic, complemented
108	#5681	C₂×C₅	1	16	yes	direct product factor
109	C₂×#886	C₂×C₅	1	2	yes	complemented
110	C₂×#885	C₂×C₅	1	2	yes	complemented
111	C₂×#883	C₂×C₅	1	2	yes	complemented
112	C₂×#67⋊C₄	C₂×C₅	1	2	yes	complemented
113	C₂×C₄×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
114	C₂×C₄×Dic₆	C₂×C₅	1	1	yes	characteristic, complemented
115	C₂²×C₅×Q₈	—	3	1	no	—
116	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
117	C₂²×C₅×D₄	—	3	2	no	—
118	C₂²×C₅×D₄	—	3	2	no	—
119	C₂²×C₅×D₄	—	3	1	no	—
120	C₂²×C₅×D₄	C₂×D₃	1	1	yes	characteristic
121	C₂³×C₄×C₅	—	3	2	no	—
122	C₂³×C₄×C₅	—	3	2	no	—
123	C₅×#127	—	3	8	no	—
124	C₅×#124	—	3	8	no	—
125	C₅×#124	—	3	16	no	—
126	C₅×#124	C₂×D₃	1	8	yes	complemented
127	C₅×#123	—	3	16	no	—
128	C₅×#123	—	3	16	no	—
129	C₄×C₅×Q₈	—	3	8	no	—
130	C₄×C₅×Q₈	—	3	8	no	—
131	C₄×C₅×D₄	—	3	8	no	—
132	C₄×C₅×D₄	—	3	16	no	—
133	C₄×C₅×D₄	—	3	8	no	—
134	C₂×C₅×C₄⋊C₄	—	3	1	no	—
135	C₂×C₅×C₄⋊C₄	—	3	1	no	—
136	C₂×C₅×C₄⋊C₄	—	3	1	no	—
137	C₂×C₅×C₄⋊C₄	—	3	2	no	—
138	C₂×C₅×C₄⋊C₄	—	3	1	no	—
139	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
140	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
141	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
142	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
143	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
144	C₂×C₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
145	C₂×C₄²×C₅	—	3	2	no	—
146	C₂×C₄²×C₅	—	3	1	no	—
147	C₂×C₄²×C₅	—	3	1	no	—
148	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
149	C₂×#559	—	3	1	no	Sylow
150	C₂³×C₃×C₅	—	2	4	no	—
151	C₂³×C₃×C₅	—	2	4	no	—
152	C₂³×C₃×C₅	—	2	4	no	—
153	C₂³×C₃×C₅	C₂⁴	1	4	yes	—
154	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
155	C₂²×C₅×D₃	—	2	4	no	—
156	C₂²×C₅×D₃	—	2	4	no	—
157	C₂²×C₅×D₃	—	2	4	no	—
158	C₂²×C₅×D₃	C₂⁴	1	4	yes	—
159	C₃×C₅×Q₈	—	2	8	no	—
160	C₃×C₅×D₄	—	2	8	no	—
161	C₂×C₃×C₄×C₅	—	2	2	no	—
162	C₂×C₃×C₄×C₅	—	2	4	no	—
163	C₂×C₃×C₄×C₅	—	2	2	no	—
164	C₂×C₃×C₄×C₅	#41	1	4	yes	—
165	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
166	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
167	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
168	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
169	C₅×#67	—	2	16	no	—
170	C₅×#67	—	4	8	no	—
171	C₂×C₅×Dic₃	—	2	2	no	—
172	C₂×C₅×Dic₃	—	2	4	no	—
173	C₂×C₅×Dic₃	—	2	2	no	—
174	C₂×C₅×Dic₃	—	2	2	no	—
175	C₂×C₅×Dic₃	—	4	8	no	—
176	C₂×C₅×Dic₃	#41	1	4	yes	—
177	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
178	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
179	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
180	C₂×C₅×Dic₃	C₂⁴	1	4	yes	—
181	C₂×C₅×Dic₃	C₂⁴	1	2	yes	—
182	C₅×D₁₂	—	2	8	no	—
183	C₄×C₅×D₃	—	4	8	no	—
184	C₅×Dic₆	—	2	8	no	—
185	C₂²×C₃×Q₈	C₂²×C₅	1	1	yes	characteristic
186	C₂²×C₃×D₄	C₂²×C₅	1	1	yes	characteristic
187	C₂²×#67	C₂²×C₅	1	2	yes	—
188	C₂²×#67	C₂²×C₅	1	2	yes	—
189	C₂³×Dic₃	C₂²×C₅	1	2	yes	—
190	C₂²×D₁₂	C₂²×C₅	1	1	yes	characteristic
191	C₂²×C₄×D₃	C₂²×C₅	1	2	yes	—
192	C₂²×Dic₆	C₂²×C₅	1	1	yes	characteristic
193	C₃×#124	C₂²×C₅	1	8	yes	complemented
194	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
195	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
196	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
197	C₂×#217	C₂²×C₅	1	2	yes	—
198	#947	C₂²×C₅	1	8	yes	complemented
199	Q₈×Dic₃	C₂²×C₅	1	8	yes	complemented
200	#940	C₂²×C₅	1	8	yes	—
201	D₄×Dic₃	C₂²×C₅	1	8	yes	complemented
202	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
203	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
204	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	2	yes	—
205	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
206	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
207	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
208	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
209	C₂×Dic₃⋊C₄	C₂²×C₅	1	1	yes	characteristic
210	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
211	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
212	C₂×C₄×Dic₃	C₂²×C₅	1	2	yes	—
213	#886	C₂²×C₅	1	16	yes	complemented
214	#885	C₂²×C₅	1	16	yes	complemented
215	#883	C₂²×C₅	1	16	yes	complemented
216	#67⋊C₄	C₂²×C₅	1	16	yes	complemented
217	C₄×D₁₂	C₂²×C₅	1	8	yes	complemented
218	C₄×Dic₆	C₂²×C₅	1	8	yes	complemented
219	C₂⁴×C₅	—	3	2	no	—
220	C₂⁴×C₅	C₂²×D₃	1	2	yes	—
221	C₂×C₅×Q₈	—	2	4	no	—
222	C₂×C₅×Q₈	—	3	4	no	—
223	C₂×C₅×Q₈	—	6	4	no	—
224	C₂×C₅×Q₈	C₂²×D₃	1	4	yes	—
225	C₂×C₅×D₄	—	2	4	no	—
226	C₂×C₅×D₄	—	3	8	no	—
227	C₂×C₅×D₄	—	3	8	no	—
228	C₂×C₅×D₄	—	3	4	no	—
229	C₂×C₅×D₄	—	6	8	no	—
230	C₂×C₅×D₄	—	6	4	no	—
231	C₂×C₅×D₄	—	12	4	no	—
232	C₂×C₅×D₄	C₂²×D₃	1	4	yes	—
233	C₂²×C₄×C₅	—	3	1	no	—
234	C₂²×C₄×C₅	—	3	1	no	—
235	C₂²×C₄×C₅	—	3	2	no	—
236	C₂²×C₄×C₅	—	3	1	no	—
237	C₂²×C₄×C₅	—	3	1	no	—
238	C₂²×C₄×C₅	—	3	8	no	—
239	C₂²×C₄×C₅	—	3	8	no	—
240	C₂²×C₄×C₅	—	12	4	no	—
241	C₂²×C₄×C₅	—	12	4	no	—
242	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
243	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
244	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
245	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
246	C₅×C₄⋊C₄	—	3	4	no	—
247	C₅×C₄⋊C₄	—	3	4	no	—
248	C₅×C₄⋊C₄	—	3	4	no	—
249	C₅×C₄⋊C₄	—	3	8	no	—
250	C₅×C₄⋊C₄	—	3	4	no	—
251	C₅×C₂²⋊C₄	—	3	8	no	—
252	C₅×C₂²⋊C₄	—	3	8	no	—
253	C₅×C₂²⋊C₄	—	3	8	no	—
254	C₅×C₂²⋊C₄	—	3	8	no	—
255	C₅×C₂²⋊C₄	C₂²×D₃	1	8	yes	—
256	C₅×C₂²⋊C₄	C₂²×D₃	1	8	yes	—
257	C₄²×C₅	—	3	8	no	—
258	C₄²×C₅	—	3	4	no	—
259	C₄²×C₅	—	3	4	no	—
260	C₄²×C₅	C₂²×D₃	1	4	yes	—
261	#559	—	3	16	no	—
262	C₂×#127	—	3	1	no	—
263	C₂×#124	—	3	1	no	—
264	C₂×#124	—	3	2	no	—
265	C₂×#124	C₅×D₃	1	1	yes	characteristic, complemented
266	C₂×#123	—	3	2	no	—
267	C₂×#123	—	3	2	no	—
268	C₂×C₄×Q₈	—	3	1	no	—
269	C₂×C₄×Q₈	—	3	1	no	—
270	C₂×C₄×D₄	—	3	1	no	—
271	C₂×C₄×D₄	—	3	2	no	—
272	C₂×C₄×D₄	—	3	1	no	—
273	C₂²×C₃×C₅	—	2	4	no	—
274	C₂²×C₃×C₅	—	2	4	no	—
275	C₂²×C₃×C₅	—	2	4	no	—
276	C₂²×C₃×C₅	—	4	8	no	—
277	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
278	C₂²×C₃×C₅	C₂×#41	1	2	yes	—
279	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
280	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
281	C₂×C₅×D₃	—	2	4	no	—
282	C₂×C₅×D₃	—	2	4	no	—
283	C₂×C₅×D₃	—	2	4	no	—
284	C₂×C₅×D₃	—	4	8	no	—
285	C₃×C₄×C₅	—	2	2	no	—
286	C₃×C₄×C₅	—	2	4	no	—
287	C₃×C₄×C₅	—	2	2	no	—
288	C₃×C₄×C₅	C₂×#41	1	4	yes	—
289	C₅×Dic₃	—	2	2	no	—
290	C₅×Dic₃	—	2	4	no	—
291	C₅×Dic₃	—	2	2	no	—
292	C₅×Dic₃	—	2	2	no	—
293	C₅×Dic₃	C₂×#41	1	4	yes	—
294	C₂⁴×C₃	C₂³×C₅	1	2	yes	—
295	C₂³×D₃	C₂³×C₅	1	2	yes	—
296	C₂×C₃×Q₈	—	2	4	no	—
297	C₂×C₃×Q₈	C₂³×C₅	1	4	yes	—
298	C₂×C₃×D₄	—	2	4	no	—
299	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
300	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
301	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
302	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
303	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
304	C₂×#67	—	2	8	no	—
305	C₂×#67	—	4	4	no	—
306	C₂×#67	C₂³×C₅	1	8	yes	—
307	C₂×#67	C₂³×C₅	1	8	yes	—
308	C₂²×Dic₃	—	4	4	no	—
309	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
310	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
311	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
312	C₂²×Dic₃	C₂³×C₅	1	2	yes	—
313	C₂²×Dic₃	C₂³×C₅	1	8	yes	—
314	C₂²×Dic₃	C₂³×C₅	1	1	yes	characteristic
315	C₂×D₁₂	—	2	4	no	—
316	C₂×D₁₂	C₂³×C₅	1	4	yes	—
317	C₂×C₄×D₃	—	4	4	no	—
318	C₂×C₄×D₃	C₂³×C₅	1	8	yes	—
319	C₂×Dic₆	—	2	4	no	—
320	C₂×Dic₆	C₂³×C₅	1	4	yes	—
321	C₃×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
322	C₃×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
323	C₃×C₄²	C₂³×C₅	1	4	yes	—
324	#217	C₂³×C₅	1	8	yes	—
325	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
326	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
327	(C₂×D₃)⋊C₄	C₂³×C₅	1	8	yes	—
328	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
329	C₄⋊Dic₃	C₂³×C₅	1	4	yes	—
330	C₄⋊Dic₃	C₂³×C₅	1	8	yes	—
331	Dic₃⋊C₄	C₂³×C₅	1	4	yes	—
332	Dic₃⋊C₄	C₂³×C₅	1	4	yes	—
333	C₄×Dic₃	C₂³×C₅	1	4	yes	—
334	C₄×Dic₃	C₂³×C₅	1	4	yes	—
335	C₄×Dic₃	C₂³×C₅	1	8	yes	—
336	C₂³×C₅	—	2	4	no	—
337	C₂³×C₅	—	2	4	no	—
338	C₂³×C₅	—	2	4	no	—
339	C₂³×C₅	—	3	4	no	—
340	C₂³×C₅	—	6	4	no	—
341	C₂³×C₅	—	6	4	no	—
342	C₂³×C₅	—	6	4	no	—
343	C₂³×C₅	C₂³×D₃	1	4	yes	—
344	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
345	C₅×Q₈	—	2	8	no	—
346	C₅×Q₈	—	6	8	no	—
347	C₅×D₄	—	2	8	no	—
348	C₅×D₄	—	6	16	no	—
349	C₅×D₄	—	6	8	no	—
350	C₅×D₄	—	12	8	no	—
351	C₂×C₄×C₅	—	2	4	no	—
352	C₂×C₄×C₅	—	2	2	no	—
353	C₂×C₄×C₅	—	2	2	no	—
354	C₂×C₄×C₅	—	3	4	no	—
355	C₂×C₄×C₅	—	3	2	no	—
356	C₂×C₄×C₅	—	3	2	no	—
357	C₂×C₄×C₅	—	3	4	no	—
358	C₂×C₄×C₅	—	3	2	no	—
359	C₂×C₄×C₅	—	3	2	no	—
360	C₂×C₄×C₅	—	6	2	no	—
361	C₂×C₄×C₅	—	6	4	no	—
362	C₂×C₄×C₅	—	6	2	no	—
363	C₂×C₄×C₅	—	6	2	no	—
364	C₂×C₄×C₅	—	12	8	no	—
365	C₂×C₄×C₅	—	12	8	no	—
366	C₂×C₄×C₅	#237	1	4	yes	—
367	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
368	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
369	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
370	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
371	C₂²×Q₈	—	3	1	no	—
372	C₂²×Q₈	C₂×C₅×D₃	1	1	yes	characteristic
373	C₂²×D₄	—	3	2	no	—
374	C₂²×D₄	—	3	2	no	—
375	C₂²×D₄	—	3	1	no	—
376	C₂²×D₄	C₂×C₅×D₃	1	1	yes	characteristic
377	C₂³×C₄	—	3	2	no	—
378	C₂³×C₄	—	3	2	no	—
379	#127	—	3	8	no	—
380	#124	—	3	8	no	—
381	#124	—	3	16	no	—
382	#124	C₂×C₅×D₃	1	8	yes	complemented
383	#123	—	3	16	no	—
384	#123	—	3	16	no	—
385	C₄×Q₈	—	3	8	no	—
386	C₄×Q₈	—	3	8	no	—
387	C₄×D₄	—	3	8	no	—
388	C₄×D₄	—	3	16	no	—
389	C₄×D₄	—	3	8	no	—
390	C₂×C₄⋊C₄	—	3	1	no	—
391	C₂×C₄⋊C₄	—	3	1	no	—
392	C₂×C₄⋊C₄	—	3	1	no	—
393	C₂×C₄⋊C₄	—	3	2	no	—
394	C₂×C₄⋊C₄	—	3	1	no	—
395	C₂×C₂²⋊C₄	—	3	2	no	—
396	C₂×C₂²⋊C₄	—	3	2	no	—
397	C₂×C₂²⋊C₄	—	3	2	no	—
398	C₂×C₂²⋊C₄	—	3	2	no	—
399	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
400	C₂×C₂²⋊C₄	C₂×C₅×D₃	1	2	yes	—
401	C₂×C₄²	—	3	2	no	—
402	C₂×C₄²	—	3	1	no	—
403	C₂×C₄²	—	3	1	no	—
404	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
405	C₂×C₃×C₅	—	4	4	no	—
406	C₂×C₃×C₅	#559	1	4	yes	complemented
407	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
408	C₂×C₃×C₅	C₂²×#41	1	1	yes	characteristic
409	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
410	C₅×D₃	—	4	4	no	—
411	C₂³×C₃	—	2	4	no	—
412	C₂³×C₃	—	2	4	no	—
413	C₂³×C₃	—	2	4	no	—
414	C₂³×C₃	C₂⁴×C₅	1	4	yes	—
415	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
416	C₂²×D₃	—	2	4	no	—
417	C₂²×D₃	—	2	4	no	—
418	C₂²×D₃	—	2	4	no	—
419	C₂²×D₃	C₂⁴×C₅	1	4	yes	—
420	C₃×Q₈	—	2	8	no	—
421	C₃×D₄	—	2	8	no	—
422	C₂×C₃×C₄	—	2	2	no	—
423	C₂×C₃×C₄	—	2	4	no	—
424	C₂×C₃×C₄	—	2	2	no	—
425	C₂×C₃×C₄	C₅×#41	1	4	yes	—
426	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
427	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
428	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
429	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
430	#67	—	2	16	no	—
431	#67	—	4	8	no	—
432	C₂×Dic₃	—	2	2	no	—
433	C₂×Dic₃	—	2	4	no	—
434	C₂×Dic₃	—	2	2	no	—
435	C₂×Dic₃	—	2	2	no	—
436	C₂×Dic₃	—	4	8	no	—
437	C₂×Dic₃	C₅×#41	1	4	yes	—
438	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
439	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
440	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
441	C₂×Dic₃	C₂⁴×C₅	1	4	yes	—
442	C₂×Dic₃	C₂⁴×C₅	1	2	yes	—
443	D₁₂	—	2	8	no	—
444	C₄×D₃	—	4	8	no	—
445	Dic₆	—	2	8	no	—
446	C₂²×C₅	—	2	4	no	—
447	C₂²×C₅	—	2	4	no	—
448	C₂²×C₅	—	2	4	no	—
449	C₂²×C₅	—	4	8	no	—
450	C₂²×C₅	—	6	4	no	—
451	C₂²×C₅	—	6	4	no	—
452	C₂²×C₅	—	6	4	no	—
453	C₂²×C₅	—	12	8	no	—
454	C₂²×C₅	C₂×#237	1	2	yes	—
455	C₂²×C₅	D₃×#41	1	2	yes	—
456	C₂²×C₅	#1009	1	2	yes	—
457	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
458	C₄×C₅	—	2	4	no	—
459	C₄×C₅	—	2	2	no	—
460	C₄×C₅	—	2	2	no	—
461	C₄×C₅	—	3	4	no	—
462	C₄×C₅	—	6	2	no	—
463	C₄×C₅	—	6	4	no	—
464	C₄×C₅	—	6	2	no	—
465	C₄×C₅	—	6	2	no	—
466	C₄×C₅	C₂×#237	1	4	yes	—
467	C₂⁴	—	3	2	no	—
468	C₂⁴	C₂²×C₅×D₃	1	2	yes	—
469	C₂×Q₈	—	2	4	no	—
470	C₂×Q₈	—	3	4	no	—
471	C₂×Q₈	—	6	4	no	—
472	C₂×Q₈	C₂²×C₅×D₃	1	4	yes	—
473	C₂×D₄	—	2	4	no	—
474	C₂×D₄	—	3	8	no	—
475	C₂×D₄	—	3	8	no	—
476	C₂×D₄	—	3	4	no	—
477	C₂×D₄	—	6	8	no	—
478	C₂×D₄	—	6	4	no	—
479	C₂×D₄	—	12	4	no	—
480	C₂×D₄	C₂²×C₅×D₃	1	4	yes	—
481	C₂²×C₄	—	3	1	no	—
482	C₂²×C₄	—	3	1	no	—
483	C₂²×C₄	—	3	2	no	—
484	C₂²×C₄	—	3	1	no	—
485	C₂²×C₄	—	3	1	no	—
486	C₂²×C₄	—	3	8	no	—
487	C₂²×C₄	—	3	8	no	—
488	C₂²×C₄	—	12	4	no	—
489	C₂²×C₄	—	12	4	no	—
490	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
491	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
492	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
493	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
494	C₄⋊C₄	—	3	4	no	—
495	C₄⋊C₄	—	3	4	no	—
496	C₄⋊C₄	—	3	4	no	—
497	C₄⋊C₄	—	3	8	no	—
498	C₄⋊C₄	—	3	4	no	—
499	C₂²⋊C₄	—	3	8	no	—
500	C₂²⋊C₄	—	3	8	no	—
501	C₂²⋊C₄	—	3	8	no	—
502	C₂²⋊C₄	—	3	8	no	—
503	C₂²⋊C₄	C₂²×C₅×D₃	1	8	yes	—
504	C₂²⋊C₄	C₂²×C₅×D₃	1	8	yes	—
505	C₄²	—	3	8	no	—
506	C₄²	—	3	4	no	—
507	C₄²	—	3	4	no	—
508	C₄²	C₂²×C₅×D₃	1	4	yes	—
509	C₃×C₅	C₂×#559	1	1	yes	characteristic, complemented, Hall
510	C₂²×C₃	—	2	4	no	—
511	C₂²×C₃	—	2	4	no	—
512	C₂²×C₃	—	2	4	no	—
513	C₂²×C₃	—	4	8	no	—
514	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
515	C₂²×C₃	C₂×C₅×#41	1	2	yes	—
516	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
517	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
518	C₂×D₃	—	2	4	no	—
519	C₂×D₃	—	2	4	no	—
520	C₂×D₃	—	2	4	no	—
521	C₂×D₃	—	4	8	no	—
522	C₃×C₄	—	2	2	no	—
523	C₃×C₄	—	2	4	no	—
524	C₃×C₄	—	2	2	no	—
525	C₃×C₄	C₂×C₅×#41	1	4	yes	—
526	Dic₃	—	2	2	no	—
527	Dic₃	—	2	4	no	—
528	Dic₃	—	2	2	no	—
529	Dic₃	—	2	2	no	—
530	Dic₃	C₂×C₅×#41	1	4	yes	—
531	C₂×C₅	—	4	4	no	—
532	C₂×C₅	—	12	4	no	—
533	C₂×C₅	#5681	1	4	yes	direct product factor
534	C₂×C₅	C₂²×#237	1	1	yes	characteristic
535	C₂×C₅	C₂×D₃×#41	1	1	yes	characteristic
536	C₂×C₅	C₂×#1009	1	1	yes	characteristic
537	C₂³	—	2	4	no	—
538	C₂³	—	2	4	no	—
539	C₂³	—	2	4	no	—
540	C₂³	—	3	4	no	—
541	C₂³	—	6	4	no	—
542	C₂³	—	6	4	no	—
543	C₂³	—	6	4	no	—
544	C₂³	C₂³×C₅×D₃	1	4	yes	—
545	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
546	Q₈	—	2	8	no	—
547	Q₈	—	6	8	no	—
548	D₄	—	2	8	no	—
549	D₄	—	6	16	no	—
550	D₄	—	6	8	no	—
551	D₄	—	12	8	no	—
552	C₂×C₄	—	2	4	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	3	4	no	—
556	C₂×C₄	—	3	2	no	—
557	C₂×C₄	—	3	2	no	—
558	C₂×C₄	—	3	4	no	—
559	C₂×C₄	—	3	2	no	—
560	C₂×C₄	—	3	2	no	—
561	C₂×C₄	—	6	2	no	—
562	C₂×C₄	—	6	4	no	—
563	C₂×C₄	—	6	2	no	—
564	C₂×C₄	—	6	2	no	—
565	C₂×C₄	—	12	8	no	—
566	C₂×C₄	—	12	8	no	—
567	C₂×C₄	C₅×#237	1	4	yes	—
568	C₂×C₄	C₂³×C₅×D₃	1	4	yes	—
569	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
570	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
571	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
572	C₂×C₃	—	4	4	no	—
573	C₂×C₃	C₅×#559	1	4	yes	complemented
574	C₂×C₃	C₂²×C₅×#41	1	1	yes	characteristic
575	C₂×C₃	C₂²×C₅×#41	1	1	yes	characteristic
576	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
577	D₃	—	4	4	no	—
578	C₅	C₂×#5681	1	1	yes	characteristic, direct product factor, Sylow
579	C₂²	—	2	4	no	—
580	C₂²	—	2	4	no	—
581	C₂²	—	2	4	no	—
582	C₂²	—	4	8	no	—
583	C₂²	—	6	4	no	—
584	C₂²	—	6	4	no	—
585	C₂²	—	6	4	no	—
586	C₂²	—	12	8	no	—
587	C₂²	C₂×C₅×#237	1	2	yes	—
588	C₂²	C₅×D₃×#41	1	2	yes	—
589	C₂²	C₅×#1009	1	2	yes	—
590	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
591	C₄	—	2	4	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	3	4	no	—
595	C₄	—	6	2	no	—
596	C₄	—	6	4	no	—
597	C₄	—	6	2	no	—
598	C₄	—	6	2	no	—
599	C₄	C₂×C₅×#237	1	4	yes	—
600	C₃	C₂×C₅×#559	1	1	yes	characteristic, complemented, Sylow
601	C₂	—	4	4	no	—
602	C₂	—	12	4	no	—
603	C₂	C₅×#5681	1	4	yes	direct product factor
604	C₂	C₂²×C₅×#237	1	1	yes	characteristic
605	C₂	C₂×C₅×D₃×#41	1	1	yes	characteristic
606	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×#124	C₂
socle	C₂³×C₃×C₅	C₂⁴
prime radical	C₂×C₅×#940	C₂
hypercenter	C₂×C₅×#124	D₃
hypocenter	C₃	C₂×C₅×#559
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×#124

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×#124	C₂	semidirect
2. C₂×C₅×#947	C₂	semidirect
3. C₂×C₅×Q₈×Dic₃	C₂	semidirect
5. C₂×C₅×D₄×Dic₃	C₂	semidirect
6. C₅×#5681	C₂	direct (N × H)
7. C₂×C₅×#886	C₂	semidirect
8. C₂×C₅×#885	C₂	semidirect
9. C₂×C₅×#883	C₂	semidirect
10. C₂×C₅×#67⋊C₄	C₂	semidirect
11. C₂×C₄×C₅×D₁₂	C₂	semidirect
12. C₂×C₄×C₅×Dic₆	C₂	semidirect
22. C₃×C₅×#124	C₂²	semidirect
27. C₅×#947	C₂²	semidirect
28. C₅×Q₈×Dic₃	C₂²	semidirect
30. C₅×D₄×Dic₃	C₂²	semidirect
42. C₅×#886	C₂²	semidirect
43. C₅×#885	C₂²	semidirect
44. C₅×#883	C₂²	semidirect
45. C₅×#67⋊C₄	C₂²	semidirect
46. C₄×C₅×D₁₂	C₂²	semidirect
47. C₄×C₅×Dic₆	C₂²	semidirect
48. C₂×#5681	C₅	direct (N × H)
53. C₂×C₅×#124	D₃	semidirect
103. C₂×C₃×#124	C₂×C₅	semidirect
104. C₂×#947	C₂×C₅	semidirect
105. C₂×Q₈×Dic₃	C₂×C₅	semidirect
107. C₂×D₄×Dic₃	C₂×C₅	semidirect
108. #5681	C₂×C₅	direct (N × H)
109. C₂×#886	C₂×C₅	semidirect
110. C₂×#885	C₂×C₅	semidirect
111. C₂×#883	C₂×C₅	semidirect
112. C₂×#67⋊C₄	C₂×C₅	semidirect
113. C₂×C₄×D₁₂	C₂×C₅	semidirect
114. C₂×C₄×Dic₆	C₂×C₅	semidirect
126. C₅×#124	C₂×D₃	semidirect
193. C₃×#124	C₂²×C₅	semidirect
198. #947	C₂²×C₅	semidirect
199. Q₈×Dic₃	C₂²×C₅	semidirect
201. D₄×Dic₃	C₂²×C₅	semidirect
213. #886	C₂²×C₅	semidirect
214. #885	C₂²×C₅	semidirect
215. #883	C₂²×C₅	semidirect
216. #67⋊C₄	C₂²×C₅	semidirect
217. C₄×D₁₂	C₂²×C₅	semidirect
218. C₄×Dic₆	C₂²×C₅	semidirect
265. C₂×#124	C₅×D₃	semidirect
382. #124	C₂×C₅×D₃	semidirect
406. C₂×C₃×C₅	#559	semidirect
509. C₃×C₅	C₂×#559	semidirect
533. C₂×C₅	#5681	direct (N × H)
573. C₂×C₃	C₅×#559	semidirect
578. C₅	C₂×#5681	direct (N × H)
600. C₃	C₂×C₅×#559	semidirect
603. C₂	C₅×#5681	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×#124	2
C₂×C₅×#947	2
C₂×C₅×Q₈×Dic₃	2
C₂×C₅×#940	2
C₂×C₅×D₄×Dic₃	2
C₅×#5681	2
C₂×C₅×#886	2
C₂×C₅×#885	2
C₂×C₅×#883	2
C₂×C₅×#67⋊C₄	2
C₂×C₄×C₅×D₁₂	2
C₂×C₄×C₅×Dic₆	2
C₂×C₅×#559	3
C₂×#5681	5

Subgroups — C2×C5×#5681 #14484699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5681 #14484699