Subgroups — C₂×C₅×#3510 #11985528

Order 1280.

7302 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	C₅×#3510	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₅×#559	C₂	1	4	yes	maximal, complemented
3	C₂×C₅×#556	C₂	1	4	yes	maximal, complemented
4	C₂×C₅×#551	C₂	1	8	yes	maximal, complemented
5	C₂×C₅×#542	C₂	1	4	yes	maximal, complemented
6	C₂×C₅×#536	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#535	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×#533	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#529	C₂	1	4	yes	maximal, complemented
10	C₂²×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×#519	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×#518	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
15	C₂³×C₅×Q₈	C₂²	1	1	yes	characteristic
16	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
17	C₅×#559	C₂²	1	64	yes	complemented
18	C₅×#556	C₂²	1	64	yes	complemented
19	C₅×#551	C₂²	1	128	yes	complemented
20	C₅×#542	C₂²	1	64	yes	complemented
21	C₅×#536	C₂²	1	16	yes	complemented
22	C₅×#535	C₂²	1	16	yes	—
23	C₅×#533	C₂²	1	16	yes	complemented
24	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
25	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
26	C₂×C₅×#127	C₂²	1	2	yes	—
27	C₅×#529	C₂²	1	64	yes	complemented
28	C₂×C₅×#126	C₂²	1	8	yes	complemented
29	C₂×C₅×#126	C₂²	1	8	yes	complemented
30	C₂×C₅×#125	C₂²	1	2	yes	complemented
31	C₂×C₅×#124	C₂²	1	2	yes	—
32	C₂×C₅×#124	C₂²	1	2	yes	—
33	C₂×C₅×#124	C₂²	1	4	yes	—
34	C₂×C₅×#124	C₂²	1	2	yes	—
35	C₂×C₅×#124	C₂²	1	16	yes	complemented
36	C₂×C₅×#124	C₂²	1	2	yes	complemented
37	C₂×C₅×#124	C₂²	1	8	yes	complemented
38	C₂×C₅×#123	C₂²	1	4	yes	—
39	C₂×C₅×#123	C₂²	1	8	yes	complemented
40	C₂×C₅×#123	C₂²	1	8	yes	complemented
41	C₂×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
42	C₂×C₅×C₂²⋊Q₈	C₂²	1	4	yes	complemented
43	C₂×C₅×C₂²⋊Q₈	C₂²	1	8	yes	complemented
44	C₂×C₅×#121	C₂²	1	4	yes	—
45	C₂×C₅×#120	C₂²	1	4	yes	—
46	C₅×#519	C₂²	1	16	yes	complemented
47	C₅×#518	C₂²	1	16	yes	complemented
48	C₄×C₅×#41	C₂²	1	16	yes	complemented
49	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	complemented
50	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	complemented
51	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	complemented
52	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	complemented
53	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
54	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
55	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
56	C₂×C₄×C₅×D₄	C₂²	1	2	yes	complemented
57	C₂×C₄×C₅×D₄	C₂²	1	8	yes	complemented
58	C₂×C₅×#117	C₂²	1	1	yes	characteristic
59	C₂×C₅×#117	C₂²	1	1	yes	characteristic
60	C₂×C₅×#117	C₂²	1	1	yes	characteristic
61	C₂×C₅×#117	C₂²	1	4	yes	—
62	C₂×C₅×#117	C₂²	1	2	yes	complemented
63	C₂×C₅×#117	C₂²	1	8	yes	complemented
64	C₂²×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
65	C₂²×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
66	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
67	C₂²×C₄²×C₅	C₂²	1	2	yes	complemented
68	C₂×#3510	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂⁵×C₅	C₂³	1	2	yes	—
70	C₂×C₅×#41	—	2	8	no	—
71	C₂×C₅×#41	C₂³	1	8	yes	—
72	C₂²×C₅×Q₈	—	2	8	no	—
73	C₂²×C₅×Q₈	C₂³	1	2	yes	—
74	C₂²×C₅×Q₈	C₂³	1	2	yes	—
75	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
76	C₂²×C₅×Q₈	C₂³	1	8	yes	—
77	C₂²×C₅×Q₈	C₂³	1	4	yes	—
78	C₂²×C₅×D₄	—	2	8	no	—
79	C₂²×C₅×D₄	C₂³	1	2	yes	—
80	C₂²×C₅×D₄	C₂³	1	2	yes	—
81	C₂²×C₅×D₄	C₂³	1	2	yes	—
82	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
83	C₂²×C₅×D₄	C₂³	1	8	yes	—
84	C₂²×C₅×D₄	C₂³	1	8	yes	—
85	C₂³×C₄×C₅	C₂³	1	2	yes	—
86	C₂³×C₄×C₅	C₂³	1	2	yes	—
87	C₂³×C₄×C₅	C₂³	1	2	yes	—
88	C₂³×C₄×C₅	C₂³	1	4	yes	—
89	C₂³×C₄×C₅	C₂³	1	8	yes	—
90	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
91	C₅×C₄⋊Q₈	C₂³	1	16	yes	complemented
92	C₅×C₄⋊Q₈	C₂³	1	16	yes	complemented
93	C₅×#127	C₂³	1	16	yes	—
94	C₅×#126	C₂³	1	64	yes	complemented
95	C₅×#126	C₂³	1	64	yes	complemented
96	C₅×#125	C₂³	1	16	yes	complemented
97	C₅×#124	C₂³	1	16	yes	—
98	C₅×#124	C₂³	1	16	yes	—
99	C₅×#124	C₂³	1	32	yes	—
100	C₅×#124	C₂³	1	16	yes	—
101	C₅×#124	C₂³	1	16	yes	complemented
102	C₅×#124	C₂³	1	64	yes	complemented
103	C₅×#123	C₂³	1	32	yes	—
104	C₅×#123	C₂³	1	64	yes	complemented
105	C₅×#123	C₂³	1	64	yes	complemented
106	C₅×C₂²⋊Q₈	C₂³	1	32	yes	—
107	C₅×C₂²⋊Q₈	C₂³	1	32	yes	complemented
108	C₅×C₂²⋊Q₈	C₂³	1	64	yes	complemented
109	C₅×#121	C₂³	1	32	yes	—
110	C₅×#120	C₂³	1	32	yes	—
111	C₄×C₅×Q₈	C₂³	1	16	yes	complemented
112	C₄×C₅×Q₈	C₂³	1	32	yes	complemented
113	C₄×C₅×Q₈	C₂³	1	32	yes	complemented
114	C₄×C₅×Q₈	C₂³	1	32	yes	complemented
115	C₄×C₅×D₄	C₂³	1	32	yes	—
116	C₄×C₅×D₄	C₂³	1	32	yes	—
117	C₄×C₅×D₄	C₂³	1	32	yes	—
118	C₄×C₅×D₄	C₂³	1	16	yes	complemented
119	C₄×C₅×D₄	C₂³	1	64	yes	complemented
120	C₅×#117	C₂³	1	8	yes	—
121	C₅×#117	C₂³	1	8	yes	—
122	C₅×#117	C₂³	1	8	yes	—
123	C₅×#117	C₂³	1	32	yes	—
124	C₅×#117	C₂³	1	16	yes	complemented
125	C₅×#117	C₂³	1	64	yes	complemented
126	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
127	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
128	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
129	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
130	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
131	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
132	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
133	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
134	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
135	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
136	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
137	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
138	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
139	C₂×C₅×C₂²⋊C₄	C₂³	1	32	yes	—
140	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
141	C₂×C₅×C₂²⋊C₄	C₂³	1	32	yes	—
142	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
143	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
144	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
145	C₂×C₅×C₂²⋊C₄	C₂³	1	8	yes	—
146	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
147	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
148	C₂×C₄²×C₅	C₂³	1	2	yes	—
149	C₂×C₄²×C₅	C₂³	1	8	yes	—
150	C₂×C₄²×C₅	C₂³	1	2	yes	—
151	C₂×C₄²×C₅	C₂³	1	4	yes	—
152	C₂×C₄²×C₅	C₂³	1	8	yes	—
153	C₂×C₄²×C₅	C₂³	1	2	yes	—
154	C₂×C₄²×C₅	C₂³	1	2	yes	—
155	C₂×C₄²×C₅	C₂³	1	16	yes	complemented
156	#3510	C₂×C₅	1	32	yes	direct product factor
157	C₂×#559	C₂×C₅	1	4	yes	complemented
158	C₂×#556	C₂×C₅	1	4	yes	complemented
159	C₂×#551	C₂×C₅	1	8	yes	complemented
160	C₂×#542	C₂×C₅	1	4	yes	complemented
161	C₂×#536	C₂×C₅	1	1	yes	characteristic, complemented
162	C₂×#535	C₂×C₅	1	1	yes	characteristic
163	C₂×#533	C₂×C₅	1	1	yes	characteristic, complemented
164	C₂×#529	C₂×C₅	1	4	yes	complemented
165	C₂²×#124	C₂×C₅	1	1	yes	characteristic, complemented
166	C₂×#519	C₂×C₅	1	1	yes	characteristic, complemented
167	C₂×#518	C₂×C₅	1	1	yes	characteristic, complemented
168	C₂×C₄×#41	C₂×C₅	1	1	yes	characteristic, complemented
169	C₂⁴×C₅	—	2	8	no	—
170	C₂⁴×C₅	—	4	8	no	—
171	C₂⁴×C₅	C₂⁴	1	4	yes	—
172	C₂⁴×C₅	C₂⁴	1	2	yes	—
173	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
174	C₂⁴×C₅	C₂⁴	1	8	yes	—
175	C₅×#41	—	2	32	no	—
176	C₂×C₅×Q₈	—	2	8	no	—
177	C₂×C₅×Q₈	—	2	8	no	—
178	C₂×C₅×Q₈	—	2	32	no	—
179	C₂×C₅×Q₈	—	2	4	no	—
180	C₂×C₅×Q₈	—	4	8	no	—
181	C₂×C₅×Q₈	C₂⁴	1	8	yes	—
182	C₂×C₅×Q₈	C₂⁴	1	8	yes	—
183	C₂×C₅×Q₈	C₂⁴	1	4	yes	—
184	C₂×C₅×Q₈	C₂⁴	1	16	yes	—
185	C₂×C₅×D₄	—	2	8	no	—
186	C₂×C₅×D₄	—	2	8	no	—
187	C₂×C₅×D₄	—	2	8	no	—
188	C₂×C₅×D₄	—	2	32	no	—
189	C₂×C₅×D₄	—	2	4	no	—
190	C₂×C₅×D₄	—	4	16	no	—
191	C₂×C₅×D₄	C₂⁴	1	8	yes	—
192	C₂×C₅×D₄	C₂⁴	1	8	yes	—
193	C₂×C₅×D₄	C₂⁴	1	8	yes	—
194	C₂×C₅×D₄	C₂⁴	1	4	yes	—
195	C₂×C₅×D₄	C₂⁴	1	32	yes	—
196	C₂²×C₄×C₅	—	2	8	no	—
197	C₂²×C₄×C₅	—	2	8	no	—
198	C₂²×C₄×C₅	—	4	8	no	—
199	C₂²×C₄×C₅	—	4	16	no	—
200	C₂²×C₄×C₅	—	4	4	no	—
201	C₂²×C₄×C₅	#41	1	8	yes	—
202	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
203	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
204	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
205	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
206	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
207	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
208	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
209	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
210	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
211	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
212	C₂²×C₄×C₅	C₂⁴	1	16	yes	—
213	C₂²×C₄×C₅	C₂⁴	1	32	yes	—
214	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
215	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
216	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
217	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
218	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
219	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
220	C₅×C₄⋊C₄	C₂⁴	1	32	yes	—
221	C₅×C₄⋊C₄	C₂⁴	1	16	yes	—
222	C₅×C₄⋊C₄	C₂⁴	1	16	yes	—
223	C₅×C₄⋊C₄	C₂⁴	1	32	yes	—
224	C₅×C₄⋊C₄	C₂⁴	1	16	yes	—
225	C₅×C₄⋊C₄	C₂⁴	1	16	yes	—
226	C₅×C₂²⋊C₄	C₂⁴	1	8	yes	—
227	C₅×C₂²⋊C₄	C₂⁴	1	8	yes	—
228	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	—
229	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	—
230	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	—
231	C₅×C₂²⋊C₄	C₂⁴	1	8	yes	—
232	C₅×C₂²⋊C₄	C₂⁴	1	16	yes	—
233	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	—
234	C₅×C₂²⋊C₄	C₂⁴	1	32	yes	—
235	C₄²×C₅	C₂⁴	1	4	yes	—
236	C₄²×C₅	C₂⁴	1	4	yes	—
237	C₄²×C₅	C₂⁴	1	8	yes	—
238	C₄²×C₅	C₂⁴	1	8	yes	—
239	C₄²×C₅	C₂⁴	1	16	yes	—
240	C₄²×C₅	C₂⁴	1	32	yes	—
241	C₄²×C₅	C₂⁴	1	8	yes	—
242	C₄²×C₅	C₂⁴	1	8	yes	—
243	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
244	C₂³×Q₈	C₂²×C₅	1	1	yes	characteristic
245	C₂³×D₄	C₂²×C₅	1	1	yes	characteristic
246	#559	C₂²×C₅	1	64	yes	complemented
247	#556	C₂²×C₅	1	64	yes	complemented
248	#551	C₂²×C₅	1	128	yes	complemented
249	#542	C₂²×C₅	1	64	yes	complemented
250	#536	C₂²×C₅	1	16	yes	complemented
251	#535	C₂²×C₅	1	16	yes	—
252	#533	C₂²×C₅	1	16	yes	complemented
253	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
254	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
255	C₂×#127	C₂²×C₅	1	2	yes	—
256	#529	C₂²×C₅	1	64	yes	complemented
257	C₂×#126	C₂²×C₅	1	8	yes	complemented
258	C₂×#126	C₂²×C₅	1	8	yes	complemented
259	C₂×#125	C₂²×C₅	1	2	yes	complemented
260	C₂×#124	C₂²×C₅	1	2	yes	—
261	C₂×#124	C₂²×C₅	1	2	yes	—
262	C₂×#124	C₂²×C₅	1	4	yes	—
263	C₂×#124	C₂²×C₅	1	2	yes	—
264	C₂×#124	C₂²×C₅	1	16	yes	complemented
265	C₂×#124	C₂²×C₅	1	2	yes	complemented
266	C₂×#124	C₂²×C₅	1	8	yes	complemented
267	C₂×#123	C₂²×C₅	1	4	yes	—
268	C₂×#123	C₂²×C₅	1	8	yes	complemented
269	C₂×#123	C₂²×C₅	1	8	yes	complemented
270	C₂×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
271	C₂×C₂²⋊Q₈	C₂²×C₅	1	4	yes	complemented
272	C₂×C₂²⋊Q₈	C₂²×C₅	1	8	yes	complemented
273	C₂×#121	C₂²×C₅	1	4	yes	—
274	C₂×#120	C₂²×C₅	1	4	yes	—
275	#519	C₂²×C₅	1	16	yes	complemented
276	#518	C₂²×C₅	1	16	yes	complemented
277	C₄×#41	C₂²×C₅	1	16	yes	complemented
278	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	complemented
279	C₂×C₄×Q₈	C₂²×C₅	1	4	yes	complemented
280	C₂×C₄×Q₈	C₂²×C₅	1	4	yes	complemented
281	C₂×C₄×Q₈	C₂²×C₅	1	4	yes	complemented
282	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
283	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
284	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
285	C₂×C₄×D₄	C₂²×C₅	1	2	yes	complemented
286	C₂×C₄×D₄	C₂²×C₅	1	8	yes	complemented
287	C₂×#117	C₂²×C₅	1	1	yes	characteristic
288	C₂×#117	C₂²×C₅	1	1	yes	characteristic
289	C₂×#117	C₂²×C₅	1	1	yes	characteristic
290	C₂×#117	C₂²×C₅	1	4	yes	—
291	C₂×#117	C₂²×C₅	1	2	yes	complemented
292	C₂×#117	C₂²×C₅	1	8	yes	complemented
293	C₂²×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
294	C₂²×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
295	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
296	C₂²×C₄²	C₂²×C₅	1	2	yes	complemented
297	C₂³×C₅	—	2	4	no	—
298	C₂³×C₅	—	2	4	no	—
299	C₂³×C₅	—	2	8	no	—
300	C₂³×C₅	—	2	8	no	—
301	C₂³×C₅	—	2	16	no	—
302	C₂³×C₅	—	2	16	no	—
303	C₂³×C₅	—	4	16	no	—
304	C₂³×C₅	—	8	16	no	—
305	C₂³×C₅	C₂×#41	1	4	yes	—
306	C₂³×C₅	C₂⁵	1	8	yes	—
307	C₂³×C₅	C₂⁵	1	4	yes	—
308	C₂³×C₅	C₂⁵	1	2	yes	—
309	C₂³×C₅	C₂⁵	1	1	yes	characteristic
310	C₅×Q₈	—	2	16	no	—
311	C₅×Q₈	—	2	16	no	—
312	C₅×Q₈	—	2	8	no	—
313	C₅×Q₈	—	4	16	no	—
314	C₅×D₄	—	2	16	no	—
315	C₅×D₄	—	2	16	no	—
316	C₅×D₄	—	2	16	no	—
317	C₅×D₄	—	2	8	no	—
318	C₅×D₄	—	4	32	no	—
319	C₂×C₄×C₅	—	2	4	no	—
320	C₂×C₄×C₅	—	2	16	no	—
321	C₂×C₄×C₅	—	2	8	no	—
322	C₂×C₄×C₅	—	2	4	no	—
323	C₂×C₄×C₅	—	2	8	no	—
324	C₂×C₄×C₅	—	2	8	no	—
325	C₂×C₄×C₅	—	2	4	no	—
326	C₂×C₄×C₅	—	2	8	no	—
327	C₂×C₄×C₅	—	2	16	no	—
328	C₂×C₄×C₅	—	4	16	no	—
329	C₂×C₄×C₅	—	4	32	no	—
330	C₂×C₄×C₅	—	4	8	no	—
331	C₂×C₄×C₅	C₂×#41	1	4	yes	—
332	C₂×C₄×C₅	C₂×#41	1	4	yes	—
333	C₂×C₄×C₅	C₂×#41	1	16	yes	—
334	C₂×C₄×C₅	C₂⁵	1	4	yes	—
335	C₂×C₄×C₅	C₂⁵	1	8	yes	—
336	C₂×C₄×C₅	C₂⁵	1	8	yes	—
337	C₂×C₄×C₅	C₂⁵	1	8	yes	—
338	C₂×C₄×C₅	C₂⁵	1	4	yes	—
339	C₂×C₄×C₅	C₂⁵	1	4	yes	—
340	C₂×C₄×C₅	C₂⁵	1	8	yes	—
341	C₂×C₄×C₅	C₂⁵	1	2	yes	—
342	C₂×C₄×C₅	C₂⁵	1	2	yes	—
343	C₂⁵	C₂³×C₅	1	2	yes	—
344	C₂×#41	—	2	8	no	—
345	C₂×#41	C₂³×C₅	1	8	yes	—
346	C₂²×Q₈	—	2	8	no	—
347	C₂²×Q₈	C₂³×C₅	1	2	yes	—
348	C₂²×Q₈	C₂³×C₅	1	2	yes	—
349	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
350	C₂²×Q₈	C₂³×C₅	1	8	yes	—
351	C₂²×Q₈	C₂³×C₅	1	4	yes	—
352	C₂²×D₄	—	2	8	no	—
353	C₂²×D₄	C₂³×C₅	1	2	yes	—
354	C₂²×D₄	C₂³×C₅	1	2	yes	—
355	C₂²×D₄	C₂³×C₅	1	2	yes	—
356	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
357	C₂²×D₄	C₂³×C₅	1	8	yes	—
358	C₂²×D₄	C₂³×C₅	1	8	yes	—
359	C₂³×C₄	C₂³×C₅	1	2	yes	—
360	C₂³×C₄	C₂³×C₅	1	2	yes	—
361	C₂³×C₄	C₂³×C₅	1	2	yes	—
362	C₂³×C₄	C₂³×C₅	1	8	yes	—
363	C₂³×C₄	C₂³×C₅	1	4	yes	—
364	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
365	C₄⋊Q₈	C₂³×C₅	1	16	yes	complemented
366	C₄⋊Q₈	C₂³×C₅	1	16	yes	complemented
367	#127	C₂³×C₅	1	16	yes	—
368	#126	C₂³×C₅	1	64	yes	complemented
369	#126	C₂³×C₅	1	64	yes	complemented
370	#125	C₂³×C₅	1	16	yes	complemented
371	#124	C₂³×C₅	1	16	yes	—
372	#124	C₂³×C₅	1	16	yes	—
373	#124	C₂³×C₅	1	32	yes	—
374	#124	C₂³×C₅	1	16	yes	—
375	#124	C₂³×C₅	1	16	yes	complemented
376	#124	C₂³×C₅	1	64	yes	complemented
377	#123	C₂³×C₅	1	32	yes	—
378	#123	C₂³×C₅	1	64	yes	complemented
379	#123	C₂³×C₅	1	64	yes	complemented
380	C₂²⋊Q₈	C₂³×C₅	1	32	yes	—
381	C₂²⋊Q₈	C₂³×C₅	1	32	yes	complemented
382	C₂²⋊Q₈	C₂³×C₅	1	64	yes	complemented
383	#121	C₂³×C₅	1	32	yes	—
384	#120	C₂³×C₅	1	32	yes	—
385	C₄×Q₈	C₂³×C₅	1	16	yes	complemented
386	C₄×Q₈	C₂³×C₅	1	32	yes	complemented
387	C₄×Q₈	C₂³×C₅	1	32	yes	complemented
388	C₄×Q₈	C₂³×C₅	1	32	yes	complemented
389	C₄×D₄	C₂³×C₅	1	32	yes	—
390	C₄×D₄	C₂³×C₅	1	32	yes	—
391	C₄×D₄	C₂³×C₅	1	32	yes	—
392	C₄×D₄	C₂³×C₅	1	16	yes	complemented
393	C₄×D₄	C₂³×C₅	1	64	yes	complemented
394	#117	C₂³×C₅	1	8	yes	—
395	#117	C₂³×C₅	1	8	yes	—
396	#117	C₂³×C₅	1	8	yes	—
397	#117	C₂³×C₅	1	32	yes	—
398	#117	C₂³×C₅	1	16	yes	complemented
399	#117	C₂³×C₅	1	64	yes	complemented
400	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
401	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
402	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
403	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
404	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
405	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
406	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
407	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
408	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
409	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
410	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
411	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₅	1	32	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₅	1	32	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₅	1	8	yes	—
420	C₂×C₄²	C₂³×C₅	1	2	yes	—
421	C₂×C₄²	C₂³×C₅	1	8	yes	—
422	C₂×C₄²	C₂³×C₅	1	2	yes	—
423	C₂×C₄²	C₂³×C₅	1	2	yes	—
424	C₂×C₄²	C₂³×C₅	1	4	yes	—
425	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
426	C₂×C₄²	C₂³×C₅	1	2	yes	—
427	C₂×C₄²	C₂³×C₅	1	8	yes	—
428	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
429	C₂×C₄²	C₂³×C₅	1	16	yes	complemented
430	C₂²×C₅	—	2	4	no	—
431	C₂²×C₅	—	2	8	no	—
432	C₂²×C₅	—	2	4	no	—
433	C₂²×C₅	—	2	8	no	—
434	C₂²×C₅	—	2	8	no	—
435	C₂²×C₅	—	2	16	no	—
436	C₂²×C₅	—	4	8	no	—
437	C₂²×C₅	—	4	16	no	—
438	C₂²×C₅	—	8	16	no	—
439	C₂²×C₅	C₂²×#41	1	4	yes	—
440	C₂²×C₅	C₂²×#41	1	2	yes	—
441	C₂²×C₅	#585	1	4	yes	—
442	C₂²×C₅	C₂⁶	1	1	yes	characteristic
443	C₄×C₅	—	2	4	no	—
444	C₄×C₅	—	2	8	no	—
445	C₄×C₅	—	2	4	no	—
446	C₄×C₅	—	2	8	no	—
447	C₄×C₅	—	2	8	no	—
448	C₄×C₅	—	2	4	no	—
449	C₄×C₅	—	2	8	no	—
450	C₄×C₅	C₂²×#41	1	4	yes	—
451	C₄×C₅	C₂²×#41	1	4	yes	—
452	C₂⁴	—	2	8	no	—
453	C₂⁴	—	4	8	no	—
454	C₂⁴	C₂⁴×C₅	1	4	yes	—
455	C₂⁴	C₂⁴×C₅	1	2	yes	—
456	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
457	C₂⁴	C₂⁴×C₅	1	8	yes	—
458	#41	—	2	32	no	—
459	C₂×Q₈	—	2	8	no	—
460	C₂×Q₈	—	2	8	no	—
461	C₂×Q₈	—	2	32	no	—
462	C₂×Q₈	—	2	4	no	—
463	C₂×Q₈	—	4	8	no	—
464	C₂×Q₈	C₂⁴×C₅	1	8	yes	—
465	C₂×Q₈	C₂⁴×C₅	1	8	yes	—
466	C₂×Q₈	C₂⁴×C₅	1	4	yes	—
467	C₂×Q₈	C₂⁴×C₅	1	16	yes	—
468	C₂×D₄	—	2	8	no	—
469	C₂×D₄	—	2	8	no	—
470	C₂×D₄	—	2	8	no	—
471	C₂×D₄	—	2	32	no	—
472	C₂×D₄	—	2	4	no	—
473	C₂×D₄	—	4	16	no	—
474	C₂×D₄	C₂⁴×C₅	1	8	yes	—
475	C₂×D₄	C₂⁴×C₅	1	8	yes	—
476	C₂×D₄	C₂⁴×C₅	1	8	yes	—
477	C₂×D₄	C₂⁴×C₅	1	4	yes	—
478	C₂×D₄	C₂⁴×C₅	1	32	yes	—
479	C₂²×C₄	—	2	8	no	—
480	C₂²×C₄	—	2	8	no	—
481	C₂²×C₄	—	4	8	no	—
482	C₂²×C₄	—	4	16	no	—
483	C₂²×C₄	—	4	4	no	—
484	C₂²×C₄	C₅×#41	1	8	yes	—
485	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
486	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
487	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
488	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
489	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
490	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
491	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
492	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
493	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
494	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
495	C₂²×C₄	C₂⁴×C₅	1	16	yes	—
496	C₂²×C₄	C₂⁴×C₅	1	32	yes	—
497	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
498	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
499	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
500	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
501	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
502	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
503	C₄⋊C₄	C₂⁴×C₅	1	32	yes	—
504	C₄⋊C₄	C₂⁴×C₅	1	16	yes	—
505	C₄⋊C₄	C₂⁴×C₅	1	16	yes	—
506	C₄⋊C₄	C₂⁴×C₅	1	32	yes	—
507	C₄⋊C₄	C₂⁴×C₅	1	16	yes	—
508	C₄⋊C₄	C₂⁴×C₅	1	16	yes	—
509	C₂²⋊C₄	C₂⁴×C₅	1	8	yes	—
510	C₂²⋊C₄	C₂⁴×C₅	1	8	yes	—
511	C₂²⋊C₄	C₂⁴×C₅	1	32	yes	—
512	C₂²⋊C₄	C₂⁴×C₅	1	32	yes	—
513	C₂²⋊C₄	C₂⁴×C₅	1	32	yes	—
514	C₂²⋊C₄	C₂⁴×C₅	1	8	yes	—
515	C₂²⋊C₄	C₂⁴×C₅	1	16	yes	—
516	C₂²⋊C₄	C₂⁴×C₅	1	32	yes	—
517	C₂²⋊C₄	C₂⁴×C₅	1	32	yes	—
518	C₄²	C₂⁴×C₅	1	4	yes	—
519	C₄²	C₂⁴×C₅	1	4	yes	—
520	C₄²	C₂⁴×C₅	1	8	yes	—
521	C₄²	C₂⁴×C₅	1	8	yes	—
522	C₄²	C₂⁴×C₅	1	16	yes	—
523	C₄²	C₂⁴×C₅	1	32	yes	—
524	C₄²	C₂⁴×C₅	1	8	yes	—
525	C₄²	C₂⁴×C₅	1	8	yes	—
526	C₂×C₅	—	2	4	no	—
527	C₂×C₅	—	4	4	no	—
528	C₂×C₅	—	4	8	no	—
529	C₂×C₅	#3510	1	4	yes	direct product factor
530	C₂×C₅	C₂³×#41	1	1	yes	characteristic
531	C₂×C₅	C₂×#585	1	2	yes	—
532	C₂³	—	2	4	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	2	8	no	—
535	C₂³	—	2	8	no	—
536	C₂³	—	2	16	no	—
537	C₂³	—	2	16	no	—
538	C₂³	—	4	16	no	—
539	C₂³	—	8	16	no	—
540	C₂³	C₂×C₅×#41	1	4	yes	—
541	C₂³	C₂⁵×C₅	1	8	yes	—
542	C₂³	C₂⁵×C₅	1	4	yes	—
543	C₂³	C₂⁵×C₅	1	2	yes	—
544	C₂³	C₂⁵×C₅	1	1	yes	characteristic
545	Q₈	—	2	16	no	—
546	Q₈	—	2	16	no	—
547	Q₈	—	2	8	no	—
548	Q₈	—	4	16	no	—
549	D₄	—	2	16	no	—
550	D₄	—	2	16	no	—
551	D₄	—	2	16	no	—
552	D₄	—	2	8	no	—
553	D₄	—	4	32	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	16	no	—
556	C₂×C₄	—	2	8	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	8	no	—
559	C₂×C₄	—	2	8	no	—
560	C₂×C₄	—	2	4	no	—
561	C₂×C₄	—	2	8	no	—
562	C₂×C₄	—	2	16	no	—
563	C₂×C₄	—	4	16	no	—
564	C₂×C₄	—	4	32	no	—
565	C₂×C₄	—	4	8	no	—
566	C₂×C₄	C₂×C₅×#41	1	4	yes	—
567	C₂×C₄	C₂×C₅×#41	1	4	yes	—
568	C₂×C₄	C₂×C₅×#41	1	16	yes	—
569	C₂×C₄	C₂⁵×C₅	1	4	yes	—
570	C₂×C₄	C₂⁵×C₅	1	8	yes	—
571	C₂×C₄	C₂⁵×C₅	1	8	yes	—
572	C₂×C₄	C₂⁵×C₅	1	8	yes	—
573	C₂×C₄	C₂⁵×C₅	1	4	yes	—
574	C₂×C₄	C₂⁵×C₅	1	4	yes	—
575	C₂×C₄	C₂⁵×C₅	1	8	yes	—
576	C₂×C₄	C₂⁵×C₅	1	2	yes	—
577	C₂×C₄	C₂⁵×C₅	1	2	yes	—
578	C₅	C₂×#3510	1	1	yes	characteristic, direct product factor, Sylow
579	C₂²	—	2	4	no	—
580	C₂²	—	2	8	no	—
581	C₂²	—	2	4	no	—
582	C₂²	—	2	8	no	—
583	C₂²	—	2	8	no	—
584	C₂²	—	2	16	no	—
585	C₂²	—	4	16	no	—
586	C₂²	—	4	8	no	—
587	C₂²	—	8	16	no	—
588	C₂²	C₂²×C₅×#41	1	4	yes	—
589	C₂²	C₂²×C₅×#41	1	2	yes	—
590	C₂²	C₅×#585	1	4	yes	—
591	C₂²	C₂⁶×C₅	1	1	yes	characteristic
592	C₄	—	2	4	no	—
593	C₄	—	2	8	no	—
594	C₄	—	2	4	no	—
595	C₄	—	2	8	no	—
596	C₄	—	2	8	no	—
597	C₄	—	2	4	no	—
598	C₄	—	2	8	no	—
599	C₄	C₂²×C₅×#41	1	4	yes	—
600	C₄	C₂²×C₅×#41	1	4	yes	—
601	C₂	—	2	4	no	—
602	C₂	—	4	8	no	—
603	C₂	—	4	4	no	—
604	C₂	C₅×#3510	1	4	yes	direct product factor
605	C₂	C₂³×C₅×#41	1	1	yes	characteristic
606	C₂	C₂×C₅×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅
prime radical	C₂×C₅×#535	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3510	C₂	direct (N × H)
2. C₂×C₅×#559	C₂	semidirect
3. C₂×C₅×#556	C₂	semidirect
4. C₂×C₅×#551	C₂	semidirect
5. C₂×C₅×#542	C₂	semidirect
6. C₂×C₅×#536	C₂	semidirect
8. C₂×C₅×#533	C₂	semidirect
9. C₂×C₅×#529	C₂	semidirect
10. C₂²×C₅×#124	C₂	semidirect
11. C₂×C₅×#519	C₂	semidirect
12. C₂×C₅×#518	C₂	semidirect
13. C₂×C₄×C₅×#41	C₂	semidirect
17. C₅×#559	C₂²	semidirect
18. C₅×#556	C₂²	semidirect
19. C₅×#551	C₂²	semidirect
20. C₅×#542	C₂²	semidirect
21. C₅×#536	C₂²	semidirect
23. C₅×#533	C₂²	semidirect
24. C₂×C₅×C₄⋊Q₈	C₂²	semidirect
25. C₂×C₅×C₄⋊Q₈	C₂²	semidirect
27. C₅×#529	C₂²	semidirect
28. C₂×C₅×#126	C₂²	semidirect
29. C₂×C₅×#126	C₂²	semidirect
30. C₂×C₅×#125	C₂²	semidirect
35. C₂×C₅×#124	C₂²	semidirect
36. C₂×C₅×#124	C₂²	semidirect
37. C₂×C₅×#124	C₂²	semidirect
39. C₂×C₅×#123	C₂²	semidirect
40. C₂×C₅×#123	C₂²	semidirect
42. C₂×C₅×C₂²⋊Q₈	C₂²	semidirect
43. C₂×C₅×C₂²⋊Q₈	C₂²	semidirect
46. C₅×#519	C₂²	semidirect
47. C₅×#518	C₂²	semidirect
48. C₄×C₅×#41	C₂²	semidirect
49. C₂×C₄×C₅×Q₈	C₂²	semidirect
50. C₂×C₄×C₅×Q₈	C₂²	semidirect
51. C₂×C₄×C₅×Q₈	C₂²	semidirect
52. C₂×C₄×C₅×Q₈	C₂²	semidirect
56. C₂×C₄×C₅×D₄	C₂²	semidirect
57. C₂×C₄×C₅×D₄	C₂²	semidirect
62. C₂×C₅×#117	C₂²	semidirect
63. C₂×C₅×#117	C₂²	semidirect
67. C₂²×C₄²×C₅	C₂²	semidirect
68. C₂×#3510	C₅	direct (N × H)
91. C₅×C₄⋊Q₈	C₂³	semidirect
92. C₅×C₄⋊Q₈	C₂³	semidirect
94. C₅×#126	C₂³	semidirect
95. C₅×#126	C₂³	semidirect
96. C₅×#125	C₂³	semidirect
101. C₅×#124	C₂³	semidirect
102. C₅×#124	C₂³	semidirect
104. C₅×#123	C₂³	semidirect
105. C₅×#123	C₂³	semidirect
107. C₅×C₂²⋊Q₈	C₂³	semidirect
108. C₅×C₂²⋊Q₈	C₂³	semidirect
111. C₄×C₅×Q₈	C₂³	semidirect
112. C₄×C₅×Q₈	C₂³	semidirect
113. C₄×C₅×Q₈	C₂³	semidirect
114. C₄×C₅×Q₈	C₂³	semidirect
118. C₄×C₅×D₄	C₂³	semidirect
119. C₄×C₅×D₄	C₂³	semidirect
124. C₅×#117	C₂³	semidirect
125. C₅×#117	C₂³	semidirect
155. C₂×C₄²×C₅	C₂³	semidirect
156. #3510	C₂×C₅	direct (N × H)
157. C₂×#559	C₂×C₅	semidirect
158. C₂×#556	C₂×C₅	semidirect
159. C₂×#551	C₂×C₅	semidirect
160. C₂×#542	C₂×C₅	semidirect
161. C₂×#536	C₂×C₅	semidirect
163. C₂×#533	C₂×C₅	semidirect
164. C₂×#529	C₂×C₅	semidirect
165. C₂²×#124	C₂×C₅	semidirect
166. C₂×#519	C₂×C₅	semidirect
167. C₂×#518	C₂×C₅	semidirect
168. C₂×C₄×#41	C₂×C₅	semidirect
246. #559	C₂²×C₅	semidirect
247. #556	C₂²×C₅	semidirect
248. #551	C₂²×C₅	semidirect
249. #542	C₂²×C₅	semidirect
250. #536	C₂²×C₅	semidirect
252. #533	C₂²×C₅	semidirect
253. C₂×C₄⋊Q₈	C₂²×C₅	semidirect
254. C₂×C₄⋊Q₈	C₂²×C₅	semidirect
256. #529	C₂²×C₅	semidirect
257. C₂×#126	C₂²×C₅	semidirect
258. C₂×#126	C₂²×C₅	semidirect
259. C₂×#125	C₂²×C₅	semidirect
264. C₂×#124	C₂²×C₅	semidirect
265. C₂×#124	C₂²×C₅	semidirect
266. C₂×#124	C₂²×C₅	semidirect
268. C₂×#123	C₂²×C₅	semidirect
269. C₂×#123	C₂²×C₅	semidirect
271. C₂×C₂²⋊Q₈	C₂²×C₅	semidirect
272. C₂×C₂²⋊Q₈	C₂²×C₅	semidirect
275. #519	C₂²×C₅	semidirect
276. #518	C₂²×C₅	semidirect
277. C₄×#41	C₂²×C₅	semidirect
278. C₂×C₄×Q₈	C₂²×C₅	semidirect
279. C₂×C₄×Q₈	C₂²×C₅	semidirect
280. C₂×C₄×Q₈	C₂²×C₅	semidirect
281. C₂×C₄×Q₈	C₂²×C₅	semidirect
285. C₂×C₄×D₄	C₂²×C₅	semidirect
286. C₂×C₄×D₄	C₂²×C₅	semidirect
291. C₂×#117	C₂²×C₅	semidirect
292. C₂×#117	C₂²×C₅	semidirect
296. C₂²×C₄²	C₂²×C₅	semidirect
365. C₄⋊Q₈	C₂³×C₅	semidirect
366. C₄⋊Q₈	C₂³×C₅	semidirect
368. #126	C₂³×C₅	semidirect
369. #126	C₂³×C₅	semidirect
370. #125	C₂³×C₅	semidirect
375. #124	C₂³×C₅	semidirect
376. #124	C₂³×C₅	semidirect
378. #123	C₂³×C₅	semidirect
379. #123	C₂³×C₅	semidirect
381. C₂²⋊Q₈	C₂³×C₅	semidirect
382. C₂²⋊Q₈	C₂³×C₅	semidirect
385. C₄×Q₈	C₂³×C₅	semidirect
386. C₄×Q₈	C₂³×C₅	semidirect
387. C₄×Q₈	C₂³×C₅	semidirect
388. C₄×Q₈	C₂³×C₅	semidirect
392. C₄×D₄	C₂³×C₅	semidirect
393. C₄×D₄	C₂³×C₅	semidirect
398. #117	C₂³×C₅	semidirect
399. #117	C₂³×C₅	semidirect
429. C₂×C₄²	C₂³×C₅	semidirect
529. C₂×C₅	#3510	direct (N × H)
578. C₅	C₂×#3510	direct (N × H)
604. C₂	C₅×#3510	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#3510	2
C₂×C₅×#559	2
C₂×C₅×#556	2
C₂×C₅×#551	2
C₂×C₅×#542	2
C₂×C₅×#536	2
C₂×C₅×#535	2
C₂×C₅×#533	2
C₂×C₅×#529	2
C₂²×C₅×#124	2
C₂×C₅×#519	2
C₂×C₅×#518	2
C₂×C₄×C₅×#41	2
C₂×#3510	5

Subgroups — C2×C5×#3510 #11985528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3510 #11985528