Subgroups — C₂×C₅×#3508 #11985518

Order 1280.

8622 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3508	C₂	1	32	yes	maximal, direct product factor
2	C₂×C₅×#559	C₂	1	4	yes	maximal, complemented
3	C₂×C₅×#550	C₂	1	4	yes	maximal, complemented
4	C₂×C₅×#546	C₂	1	8	yes	maximal, complemented
5	C₂×C₅×#540	C₂	1	4	yes	maximal, complemented
6	C₂×C₅×#535	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#533	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#532	C₂	1	4	yes	maximal, complemented
9	C₂²×C₅×#127	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#518	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
13	C₂²×C₅×#41	C₂²	1	8	yes	complemented
14	C₂³×C₅×D₄	C₂²	1	2	yes	—
15	C₂³×C₅×D₄	C₂²	1	4	yes	—
16	C₅×#559	C₂²	1	64	yes	complemented
17	C₅×#550	C₂²	1	64	yes	complemented
18	C₅×#546	C₂²	1	128	yes	complemented
19	C₅×#540	C₂²	1	64	yes	complemented
20	C₅×#535	C₂²	1	32	yes	complemented
21	C₅×#533	C₂²	1	16	yes	complemented
22	C₅×#532	C₂²	1	64	yes	complemented
23	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	complemented
24	C₂×C₅×#127	C₂²	1	2	yes	—
25	C₂×C₅×#127	C₂²	1	2	yes	—
26	C₂×C₅×#127	C₂²	1	4	yes	—
27	C₂×C₅×#127	C₂²	1	16	yes	complemented
28	C₂×C₅×#127	C₂²	1	4	yes	complemented
29	C₂×C₅×#125	C₂²	1	2	yes	complemented
30	C₂×C₅×#124	C₂²	1	8	yes	complemented
31	C₂×C₅×#124	C₂²	1	4	yes	complemented
32	C₂×C₅×#123	C₂²	1	16	yes	complemented
33	C₂×C₅×C₂²⋊Q₈	C₂²	1	4	yes	—
34	C₂×C₅×#121	C₂²	1	8	yes	complemented
35	C₂×C₅×#121	C₂²	1	8	yes	complemented
36	C₂×C₅×#121	C₂²	1	16	yes	complemented
37	C₂×C₅×#120	C₂²	1	8	yes	—
38	C₅×#518	C₂²	1	32	yes	complemented
39	C₄×C₅×#41	C₂²	1	16	yes	complemented
40	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	complemented
41	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	complemented
42	C₂×C₄×C₅×D₄	C₂²	1	4	yes	—
43	C₂×C₄×C₅×D₄	C₂²	1	2	yes	complemented
44	C₂×C₄×C₅×D₄	C₂²	1	8	yes	complemented
45	C₂×C₄×C₅×D₄	C₂²	1	8	yes	complemented
46	C₂×C₄×C₅×D₄	C₂²	1	8	yes	complemented
47	C₂×C₅×#117	C₂²	1	1	yes	characteristic
48	C₂×C₅×#117	C₂²	1	2	yes	—
49	C₂×C₅×#117	C₂²	1	2	yes	complemented
50	C₂²×C₅×C₂²⋊C₄	C₂²	1	8	yes	—
51	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
52	C₂²×C₄²×C₅	C₂²	1	2	yes	complemented
53	C₂×#3508	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂⁵×C₅	C₂³	1	4	yes	—
55	C₂×C₅×#41	—	2	8	no	—
56	C₂×C₅×#41	—	4	32	no	—
57	C₂×C₅×#41	C₂³	1	8	yes	—
58	C₂×C₅×#41	C₂³	1	64	yes	complemented
59	C₂²×C₅×Q₈	C₂³	1	1	yes	characteristic
60	C₂²×C₅×Q₈	C₂³	1	4	yes	—
61	C₂²×C₅×D₄	—	2	16	no	—
62	C₂²×C₅×D₄	—	4	16	no	—
63	C₂²×C₅×D₄	C₂³	1	2	yes	—
64	C₂²×C₅×D₄	C₂³	1	4	yes	—
65	C₂²×C₅×D₄	C₂³	1	4	yes	—
66	C₂²×C₅×D₄	C₂³	1	1	yes	characteristic
67	C₂²×C₅×D₄	C₂³	1	8	yes	—
68	C₂²×C₅×D₄	C₂³	1	16	yes	—
69	C₂²×C₅×D₄	C₂³	1	16	yes	—
70	C₂²×C₅×D₄	C₂³	1	8	yes	—
71	C₂²×C₅×D₄	C₂³	1	32	yes	—
72	C₂³×C₄×C₅	C₂³	1	2	yes	—
73	C₂³×C₄×C₅	C₂³	1	2	yes	—
74	C₂³×C₄×C₅	C₂³	1	4	yes	—
75	C₂³×C₄×C₅	C₂³	1	16	yes	—
76	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
77	C₅×C₄⋊Q₈	C₂³	1	16	yes	complemented
78	C₅×#127	C₂³	1	16	yes	—
79	C₅×#127	C₂³	1	16	yes	—
80	C₅×#127	C₂³	1	32	yes	—
81	C₅×#127	C₂³	1	32	yes	complemented
82	C₅×#125	C₂³	1	16	yes	complemented
83	C₅×#124	C₂³	1	64	yes	complemented
84	C₅×#124	C₂³	1	32	yes	complemented
85	C₅×#123	C₂³	1	128	yes	complemented
86	C₅×C₂²⋊Q₈	C₂³	1	32	yes	—
87	C₅×#121	C₂³	1	64	yes	complemented
88	C₅×#121	C₂³	1	64	yes	complemented
89	C₅×#121	C₂³	1	128	yes	complemented
90	C₅×#120	C₂³	1	64	yes	—
91	C₄×C₅×Q₈	C₂³	1	16	yes	complemented
92	C₄×C₅×Q₈	C₂³	1	32	yes	complemented
93	C₄×C₅×D₄	C₂³	1	32	yes	—
94	C₄×C₅×D₄	C₂³	1	16	yes	complemented
95	C₄×C₅×D₄	C₂³	1	64	yes	complemented
96	C₄×C₅×D₄	C₂³	1	64	yes	complemented
97	C₄×C₅×D₄	C₂³	1	64	yes	complemented
98	C₅×#117	C₂³	1	8	yes	—
99	C₅×#117	C₂³	1	16	yes	—
100	C₅×#117	C₂³	1	16	yes	complemented
101	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
102	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
103	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
104	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
105	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
106	C₂×C₅×C₂²⋊C₄	C₂³	1	2	yes	—
107	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
108	C₂×C₅×C₂²⋊C₄	C₂³	1	16	yes	—
109	C₂×C₅×C₂²⋊C₄	C₂³	1	64	yes	—
110	C₂×C₅×C₂²⋊C₄	C₂³	1	4	yes	—
111	C₂×C₅×C₂²⋊C₄	C₂³	1	16	yes	—
112	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
113	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₅	C₂³	1	2	yes	—
115	C₂×C₄²×C₅	C₂³	1	8	yes	—
116	C₂×C₄²×C₅	C₂³	1	2	yes	—
117	C₂×C₄²×C₅	C₂³	1	4	yes	—
118	C₂×C₄²×C₅	C₂³	1	4	yes	—
119	C₂×C₄²×C₅	C₂³	1	16	yes	complemented
120	#3508	C₂×C₅	1	32	yes	direct product factor
121	C₂×#559	C₂×C₅	1	4	yes	complemented
122	C₂×#550	C₂×C₅	1	4	yes	complemented
123	C₂×#546	C₂×C₅	1	8	yes	complemented
124	C₂×#540	C₂×C₅	1	4	yes	complemented
125	C₂×#535	C₂×C₅	1	2	yes	complemented
126	C₂×#533	C₂×C₅	1	1	yes	characteristic, complemented
127	C₂×#532	C₂×C₅	1	4	yes	complemented
128	C₂²×#127	C₂×C₅	1	1	yes	characteristic, complemented
129	C₂×#518	C₂×C₅	1	2	yes	complemented
130	C₂×C₄×#41	C₂×C₅	1	1	yes	characteristic, complemented
131	C₂⁴×C₅	—	2	16	no	—
132	C₂⁴×C₅	—	4	16	no	—
133	C₂⁴×C₅	C₂⁴	1	8	yes	—
134	C₂⁴×C₅	C₂⁴	1	2	yes	—
135	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
136	C₂⁴×C₅	C₂⁴	1	16	yes	—
137	C₅×#41	—	2	32	no	—
138	C₅×#41	—	4	128	no	—
139	C₂×C₅×Q₈	—	2	4	no	—
140	C₂×C₅×Q₈	—	2	16	no	—
141	C₂×C₅×Q₈	C₂⁴	1	4	yes	—
142	C₂×C₅×Q₈	C₂⁴	1	16	yes	—
143	C₂×C₅×D₄	—	2	8	no	—
144	C₂×C₅×D₄	—	2	16	no	—
145	C₂×C₅×D₄	—	2	16	no	—
146	C₂×C₅×D₄	—	2	4	no	—
147	C₂×C₅×D₄	—	2	64	no	—
148	C₂×C₅×D₄	—	2	32	no	—
149	C₂×C₅×D₄	—	4	64	no	—
150	C₂×C₅×D₄	—	4	16	no	—
151	C₂×C₅×D₄	—	4	32	no	—
152	C₂×C₅×D₄	C₂⁴	1	8	yes	—
153	C₂×C₅×D₄	C₂⁴	1	16	yes	—
154	C₂×C₅×D₄	C₂⁴	1	16	yes	—
155	C₂×C₅×D₄	C₂⁴	1	4	yes	—
156	C₂×C₅×D₄	C₂⁴	1	32	yes	—
157	C₂×C₅×D₄	C₂⁴	1	64	yes	—
158	C₂×C₅×D₄	C₂⁴	1	32	yes	—
159	C₂²×C₄×C₅	—	2	8	no	—
160	C₂²×C₄×C₅	—	2	8	no	—
161	C₂²×C₄×C₅	—	2	16	no	—
162	C₂²×C₄×C₅	—	4	32	no	—
163	C₂²×C₄×C₅	#41	1	8	yes	complemented
164	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
165	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
166	C₂²×C₄×C₅	C₂⁴	1	16	yes	—
167	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
169	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
171	C₂²×C₄×C₅	C₂⁴	1	64	yes	—
172	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
174	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
175	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
176	C₅×C₄⋊C₄	C₂⁴	1	8	yes	—
177	C₅×C₄⋊C₄	C₂⁴	1	16	yes	—
178	C₅×C₄⋊C₄	C₂⁴	1	16	yes	—
179	C₅×C₄⋊C₄	C₂⁴	1	16	yes	—
180	C₅×C₄⋊C₄	C₂⁴	1	32	yes	—
181	C₅×C₂²⋊C₄	C₂⁴	1	8	yes	—
182	C₅×C₂²⋊C₄	C₂⁴	1	16	yes	—
183	C₅×C₂²⋊C₄	C₂⁴	1	64	yes	—
184	C₅×C₂²⋊C₄	C₂⁴	1	16	yes	—
185	C₅×C₂²⋊C₄	C₂⁴	1	64	yes	—
186	C₄²×C₅	C₂⁴	1	4	yes	—
187	C₄²×C₅	C₂⁴	1	4	yes	—
188	C₄²×C₅	C₂⁴	1	8	yes	—
189	C₄²×C₅	C₂⁴	1	8	yes	—
190	C₄²×C₅	C₂⁴	1	16	yes	—
191	C₄²×C₅	C₂⁴	1	16	yes	—
192	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
193	C₂²×#41	C₂²×C₅	1	8	yes	complemented
194	C₂³×D₄	C₂²×C₅	1	2	yes	—
195	C₂³×D₄	C₂²×C₅	1	4	yes	—
196	#559	C₂²×C₅	1	64	yes	complemented
197	#550	C₂²×C₅	1	64	yes	complemented
198	#546	C₂²×C₅	1	128	yes	complemented
199	#540	C₂²×C₅	1	64	yes	complemented
200	#535	C₂²×C₅	1	32	yes	complemented
201	#533	C₂²×C₅	1	16	yes	complemented
202	#532	C₂²×C₅	1	64	yes	complemented
203	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	complemented
204	C₂×#127	C₂²×C₅	1	2	yes	—
205	C₂×#127	C₂²×C₅	1	2	yes	—
206	C₂×#127	C₂²×C₅	1	4	yes	—
207	C₂×#127	C₂²×C₅	1	16	yes	complemented
208	C₂×#127	C₂²×C₅	1	4	yes	complemented
209	C₂×#125	C₂²×C₅	1	2	yes	complemented
210	C₂×#124	C₂²×C₅	1	8	yes	complemented
211	C₂×#124	C₂²×C₅	1	4	yes	complemented
212	C₂×#123	C₂²×C₅	1	16	yes	complemented
213	C₂×C₂²⋊Q₈	C₂²×C₅	1	4	yes	—
214	C₂×#121	C₂²×C₅	1	8	yes	complemented
215	C₂×#121	C₂²×C₅	1	8	yes	complemented
216	C₂×#121	C₂²×C₅	1	16	yes	complemented
217	C₂×#120	C₂²×C₅	1	8	yes	—
218	#518	C₂²×C₅	1	32	yes	complemented
219	C₄×#41	C₂²×C₅	1	16	yes	complemented
220	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	complemented
221	C₂×C₄×Q₈	C₂²×C₅	1	4	yes	complemented
222	C₂×C₄×D₄	C₂²×C₅	1	4	yes	—
223	C₂×C₄×D₄	C₂²×C₅	1	2	yes	complemented
224	C₂×C₄×D₄	C₂²×C₅	1	8	yes	complemented
225	C₂×C₄×D₄	C₂²×C₅	1	8	yes	complemented
226	C₂×C₄×D₄	C₂²×C₅	1	8	yes	complemented
227	C₂×#117	C₂²×C₅	1	1	yes	characteristic
228	C₂×#117	C₂²×C₅	1	2	yes	—
229	C₂×#117	C₂²×C₅	1	2	yes	complemented
230	C₂²×C₂²⋊C₄	C₂²×C₅	1	8	yes	—
231	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
232	C₂²×C₄²	C₂²×C₅	1	2	yes	complemented
233	C₂³×C₅	—	2	4	no	—
234	C₂³×C₅	—	2	4	no	—
235	C₂³×C₅	—	2	16	no	—
236	C₂³×C₅	—	2	8	no	—
237	C₂³×C₅	—	2	32	no	—
238	C₂³×C₅	—	2	32	no	—
239	C₂³×C₅	—	4	32	no	—
240	C₂³×C₅	—	8	32	no	—
241	C₂³×C₅	C₂×#41	1	4	yes	—
242	C₂³×C₅	C₂⁵	1	16	yes	—
243	C₂³×C₅	C₂⁵	1	4	yes	—
244	C₂³×C₅	C₂⁵	1	2	yes	—
245	C₂³×C₅	C₂⁵	1	1	yes	characteristic
246	C₅×Q₈	—	2	8	no	—
247	C₅×Q₈	—	2	32	no	—
248	C₅×D₄	—	2	16	no	—
249	C₅×D₄	—	2	32	no	—
250	C₅×D₄	—	2	32	no	—
251	C₅×D₄	—	2	8	no	—
252	C₅×D₄	—	2	64	no	—
253	C₅×D₄	—	4	32	no	—
254	C₅×D₄	—	4	64	no	—
255	C₂×C₄×C₅	—	2	4	no	—
256	C₂×C₄×C₅	—	2	8	no	—
257	C₂×C₄×C₅	—	2	4	no	—
258	C₂×C₄×C₅	—	2	16	no	—
259	C₂×C₄×C₅	—	2	16	no	—
260	C₂×C₄×C₅	—	2	16	no	—
261	C₂×C₄×C₅	—	2	32	no	—
262	C₂×C₄×C₅	—	4	64	no	—
263	C₂×C₄×C₅	C₂×#41	1	4	yes	—
264	C₂×C₄×C₅	C₂×#41	1	4	yes	—
265	C₂×C₄×C₅	C₂×#41	1	16	yes	complemented
266	C₂×C₄×C₅	H(5,𝔽₂)	1	8	yes	—
267	C₂×C₄×C₅	C₂⁵	1	4	yes	—
268	C₂×C₄×C₅	C₂⁵	1	8	yes	—
269	C₂×C₄×C₅	C₂⁵	1	4	yes	—
270	C₂×C₄×C₅	C₂⁵	1	16	yes	—
271	C₂×C₄×C₅	C₂⁵	1	2	yes	—
272	C₂×C₄×C₅	C₂⁵	1	2	yes	—
273	C₂×C₄×C₅	C₂⁵	1	4	yes	—
274	C₂⁵	C₂³×C₅	1	4	yes	—
275	C₂×#41	—	2	8	no	—
276	C₂×#41	—	4	32	no	—
277	C₂×#41	C₂³×C₅	1	8	yes	—
278	C₂×#41	C₂³×C₅	1	64	yes	complemented
279	C₂²×Q₈	C₂³×C₅	1	1	yes	characteristic
280	C₂²×Q₈	C₂³×C₅	1	4	yes	—
281	C₂²×D₄	—	2	16	no	—
282	C₂²×D₄	—	4	16	no	—
283	C₂²×D₄	C₂³×C₅	1	2	yes	—
284	C₂²×D₄	C₂³×C₅	1	4	yes	—
285	C₂²×D₄	C₂³×C₅	1	4	yes	—
286	C₂²×D₄	C₂³×C₅	1	1	yes	characteristic
287	C₂²×D₄	C₂³×C₅	1	8	yes	—
288	C₂²×D₄	C₂³×C₅	1	16	yes	—
289	C₂²×D₄	C₂³×C₅	1	16	yes	—
290	C₂²×D₄	C₂³×C₅	1	8	yes	—
291	C₂²×D₄	C₂³×C₅	1	32	yes	—
292	C₂³×C₄	C₂³×C₅	1	2	yes	—
293	C₂³×C₄	C₂³×C₅	1	2	yes	—
294	C₂³×C₄	C₂³×C₅	1	4	yes	—
295	C₂³×C₄	C₂³×C₅	1	16	yes	—
296	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
297	C₄⋊Q₈	C₂³×C₅	1	16	yes	complemented
298	#127	C₂³×C₅	1	16	yes	—
299	#127	C₂³×C₅	1	16	yes	—
300	#127	C₂³×C₅	1	32	yes	—
301	#127	C₂³×C₅	1	32	yes	complemented
302	#125	C₂³×C₅	1	16	yes	complemented
303	#124	C₂³×C₅	1	64	yes	complemented
304	#124	C₂³×C₅	1	32	yes	complemented
305	#123	C₂³×C₅	1	128	yes	complemented
306	C₂²⋊Q₈	C₂³×C₅	1	32	yes	—
307	#121	C₂³×C₅	1	64	yes	complemented
308	#121	C₂³×C₅	1	64	yes	complemented
309	#121	C₂³×C₅	1	128	yes	complemented
310	#120	C₂³×C₅	1	64	yes	—
311	C₄×Q₈	C₂³×C₅	1	16	yes	complemented
312	C₄×Q₈	C₂³×C₅	1	32	yes	complemented
313	C₄×D₄	C₂³×C₅	1	32	yes	—
314	C₄×D₄	C₂³×C₅	1	16	yes	complemented
315	C₄×D₄	C₂³×C₅	1	64	yes	complemented
316	C₄×D₄	C₂³×C₅	1	64	yes	complemented
317	C₄×D₄	C₂³×C₅	1	64	yes	complemented
318	#117	C₂³×C₅	1	8	yes	—
319	#117	C₂³×C₅	1	16	yes	—
320	#117	C₂³×C₅	1	16	yes	complemented
321	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
322	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
323	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
324	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
325	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
326	C₂×C₂²⋊C₄	C₂³×C₅	1	2	yes	—
327	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
328	C₂×C₂²⋊C₄	C₂³×C₅	1	16	yes	—
329	C₂×C₂²⋊C₄	C₂³×C₅	1	64	yes	—
330	C₂×C₂²⋊C₄	C₂³×C₅	1	4	yes	—
331	C₂×C₂²⋊C₄	C₂³×C₅	1	16	yes	—
332	C₂×C₄²	C₂³×C₅	1	2	yes	—
333	C₂×C₄²	C₂³×C₅	1	4	yes	—
334	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
335	C₂×C₄²	C₂³×C₅	1	4	yes	—
336	C₂×C₄²	C₂³×C₅	1	2	yes	—
337	C₂×C₄²	C₂³×C₅	1	8	yes	—
338	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
339	C₂×C₄²	C₂³×C₅	1	16	yes	complemented
340	C₂²×C₅	—	2	4	no	—
341	C₂²×C₅	—	2	8	no	—
342	C₂²×C₅	—	2	4	no	—
343	C₂²×C₅	—	2	16	no	—
344	C₂²×C₅	—	2	8	no	—
345	C₂²×C₅	—	2	32	no	—
346	C₂²×C₅	—	4	8	no	—
347	C₂²×C₅	—	4	32	no	—
348	C₂²×C₅	—	8	32	no	—
349	C₂²×C₅	C₂²×#41	1	4	yes	—
350	C₂²×C₅	C₂²×#41	1	2	yes	—
351	C₂²×C₅	C₂×H(5,𝔽₂)	1	4	yes	—
352	C₂²×C₅	C₂⁶	1	1	yes	characteristic
353	C₄×C₅	—	2	4	no	—
354	C₄×C₅	—	2	8	no	—
355	C₄×C₅	—	2	4	no	—
356	C₄×C₅	—	2	16	no	—
357	C₄×C₅	C₂²×#41	1	4	yes	—
358	C₄×C₅	C₂²×#41	1	4	yes	—
359	C₄×C₅	C₂×H(5,𝔽₂)	1	8	yes	—
360	C₂⁴	—	2	16	no	—
361	C₂⁴	—	4	16	no	—
362	C₂⁴	C₂⁴×C₅	1	8	yes	—
363	C₂⁴	C₂⁴×C₅	1	2	yes	—
364	C₂⁴	C₂⁴×C₅	1	1	yes	characteristic
365	C₂⁴	C₂⁴×C₅	1	16	yes	—
366	#41	—	2	32	no	—
367	#41	—	4	128	no	—
368	C₂×Q₈	—	2	4	no	—
369	C₂×Q₈	—	2	16	no	—
370	C₂×Q₈	C₂⁴×C₅	1	4	yes	—
371	C₂×Q₈	C₂⁴×C₅	1	16	yes	—
372	C₂×D₄	—	2	8	no	—
373	C₂×D₄	—	2	16	no	—
374	C₂×D₄	—	2	16	no	—
375	C₂×D₄	—	2	4	no	—
376	C₂×D₄	—	2	64	no	—
377	C₂×D₄	—	2	32	no	—
378	C₂×D₄	—	4	64	no	—
379	C₂×D₄	—	4	16	no	—
380	C₂×D₄	—	4	32	no	—
381	C₂×D₄	C₂⁴×C₅	1	8	yes	—
382	C₂×D₄	C₂⁴×C₅	1	16	yes	—
383	C₂×D₄	C₂⁴×C₅	1	16	yes	—
384	C₂×D₄	C₂⁴×C₅	1	4	yes	—
385	C₂×D₄	C₂⁴×C₅	1	32	yes	—
386	C₂×D₄	C₂⁴×C₅	1	64	yes	—
387	C₂×D₄	C₂⁴×C₅	1	32	yes	—
388	C₂²×C₄	—	2	8	no	—
389	C₂²×C₄	—	2	8	no	—
390	C₂²×C₄	—	2	16	no	—
391	C₂²×C₄	—	4	32	no	—
392	C₂²×C₄	C₅×#41	1	8	yes	complemented
393	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
394	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
395	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
396	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
397	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
398	C₂²×C₄	C₂⁴×C₅	1	8	yes	—
399	C₂²×C₄	C₂⁴×C₅	1	16	yes	—
400	C₂²×C₄	C₂⁴×C₅	1	64	yes	—
401	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
402	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
403	C₂²×C₄	C₂⁴×C₅	1	4	yes	—
404	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
405	C₄⋊C₄	C₂⁴×C₅	1	8	yes	—
406	C₄⋊C₄	C₂⁴×C₅	1	16	yes	—
407	C₄⋊C₄	C₂⁴×C₅	1	16	yes	—
408	C₄⋊C₄	C₂⁴×C₅	1	16	yes	—
409	C₄⋊C₄	C₂⁴×C₅	1	32	yes	—
410	C₂²⋊C₄	C₂⁴×C₅	1	8	yes	—
411	C₂²⋊C₄	C₂⁴×C₅	1	16	yes	—
412	C₂²⋊C₄	C₂⁴×C₅	1	64	yes	—
413	C₂²⋊C₄	C₂⁴×C₅	1	16	yes	—
414	C₂²⋊C₄	C₂⁴×C₅	1	64	yes	—
415	C₄²	C₂⁴×C₅	1	4	yes	—
416	C₄²	C₂⁴×C₅	1	4	yes	—
417	C₄²	C₂⁴×C₅	1	8	yes	—
418	C₄²	C₂⁴×C₅	1	8	yes	—
419	C₄²	C₂⁴×C₅	1	16	yes	—
420	C₄²	C₂⁴×C₅	1	16	yes	—
421	C₂×C₅	—	2	4	no	—
422	C₂×C₅	—	4	4	no	—
423	C₂×C₅	—	4	16	no	—
424	C₂×C₅	#3508	1	4	yes	direct product factor
425	C₂×C₅	C₂³×#41	1	1	yes	characteristic
426	C₂×C₅	C₂²×H(5,𝔽₂)	1	2	yes	—
427	C₂³	—	2	4	no	—
428	C₂³	—	2	4	no	—
429	C₂³	—	2	16	no	—
430	C₂³	—	2	8	no	—
431	C₂³	—	2	32	no	—
432	C₂³	—	2	32	no	—
433	C₂³	—	4	32	no	—
434	C₂³	—	8	32	no	—
435	C₂³	C₂×C₅×#41	1	4	yes	—
436	C₂³	C₂⁵×C₅	1	16	yes	—
437	C₂³	C₂⁵×C₅	1	4	yes	—
438	C₂³	C₂⁵×C₅	1	2	yes	—
439	C₂³	C₂⁵×C₅	1	1	yes	characteristic
440	Q₈	—	2	8	no	—
441	Q₈	—	2	32	no	—
442	D₄	—	2	16	no	—
443	D₄	—	2	32	no	—
444	D₄	—	2	32	no	—
445	D₄	—	2	8	no	—
446	D₄	—	2	64	no	—
447	D₄	—	4	32	no	—
448	D₄	—	4	64	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	8	no	—
451	C₂×C₄	—	2	4	no	—
452	C₂×C₄	—	2	16	no	—
453	C₂×C₄	—	2	16	no	—
454	C₂×C₄	—	2	16	no	—
455	C₂×C₄	—	2	32	no	—
456	C₂×C₄	—	4	64	no	—
457	C₂×C₄	C₂×C₅×#41	1	4	yes	—
458	C₂×C₄	C₂×C₅×#41	1	4	yes	—
459	C₂×C₄	C₂×C₅×#41	1	16	yes	complemented
460	C₂×C₄	C₅×H(5,𝔽₂)	1	8	yes	—
461	C₂×C₄	C₂⁵×C₅	1	4	yes	—
462	C₂×C₄	C₂⁵×C₅	1	8	yes	—
463	C₂×C₄	C₂⁵×C₅	1	4	yes	—
464	C₂×C₄	C₂⁵×C₅	1	16	yes	—
465	C₂×C₄	C₂⁵×C₅	1	2	yes	—
466	C₂×C₄	C₂⁵×C₅	1	2	yes	—
467	C₂×C₄	C₂⁵×C₅	1	4	yes	—
468	C₅	C₂×#3508	1	1	yes	characteristic, direct product factor, Sylow
469	C₂²	—	2	4	no	—
470	C₂²	—	2	8	no	—
471	C₂²	—	2	4	no	—
472	C₂²	—	2	16	no	—
473	C₂²	—	2	8	no	—
474	C₂²	—	2	32	no	—
475	C₂²	—	4	32	no	—
476	C₂²	—	4	8	no	—
477	C₂²	—	8	32	no	—
478	C₂²	C₂²×C₅×#41	1	4	yes	—
479	C₂²	C₂²×C₅×#41	1	2	yes	—
480	C₂²	C₂×C₅×H(5,𝔽₂)	1	4	yes	—
481	C₂²	C₂⁶×C₅	1	1	yes	characteristic
482	C₄	—	2	4	no	—
483	C₄	—	2	8	no	—
484	C₄	—	2	4	no	—
485	C₄	—	2	16	no	—
486	C₄	C₂²×C₅×#41	1	4	yes	—
487	C₄	C₂²×C₅×#41	1	4	yes	—
488	C₄	C₂×C₅×H(5,𝔽₂)	1	8	yes	—
489	C₂	—	2	4	no	—
490	C₂	—	4	16	no	—
491	C₂	—	4	4	no	—
492	C₂	C₅×#3508	1	4	yes	direct product factor
493	C₂	C₂³×C₅×#41	1	1	yes	characteristic
494	C₂	C₂²×C₅×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁶×C₅
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×#3508	C₂	direct (N × H)
2. C₂×C₅×#559	C₂	semidirect
3. C₂×C₅×#550	C₂	semidirect
4. C₂×C₅×#546	C₂	semidirect
5. C₂×C₅×#540	C₂	semidirect
6. C₂×C₅×#535	C₂	semidirect
7. C₂×C₅×#533	C₂	semidirect
8. C₂×C₅×#532	C₂	semidirect
9. C₂²×C₅×#127	C₂	semidirect
10. C₂×C₅×#518	C₂	semidirect
11. C₂×C₄×C₅×#41	C₂	semidirect
13. C₂²×C₅×#41	C₂²	semidirect
16. C₅×#559	C₂²	semidirect
17. C₅×#550	C₂²	semidirect
18. C₅×#546	C₂²	semidirect
19. C₅×#540	C₂²	semidirect
20. C₅×#535	C₂²	semidirect
21. C₅×#533	C₂²	semidirect
22. C₅×#532	C₂²	semidirect
23. C₂×C₅×C₄⋊Q₈	C₂²	semidirect
27. C₂×C₅×#127	C₂²	semidirect
28. C₂×C₅×#127	C₂²	semidirect
29. C₂×C₅×#125	C₂²	semidirect
30. C₂×C₅×#124	C₂²	semidirect
31. C₂×C₅×#124	C₂²	semidirect
32. C₂×C₅×#123	C₂²	semidirect
34. C₂×C₅×#121	C₂²	semidirect
35. C₂×C₅×#121	C₂²	semidirect
36. C₂×C₅×#121	C₂²	semidirect
38. C₅×#518	C₂²	semidirect
39. C₄×C₅×#41	C₂²	semidirect
40. C₂×C₄×C₅×Q₈	C₂²	semidirect
41. C₂×C₄×C₅×Q₈	C₂²	semidirect
43. C₂×C₄×C₅×D₄	C₂²	semidirect
44. C₂×C₄×C₅×D₄	C₂²	semidirect
45. C₂×C₄×C₅×D₄	C₂²	semidirect
46. C₂×C₄×C₅×D₄	C₂²	semidirect
49. C₂×C₅×#117	C₂²	semidirect
52. C₂²×C₄²×C₅	C₂²	semidirect
53. C₂×#3508	C₅	direct (N × H)
58. C₂×C₅×#41	C₂³	semidirect
77. C₅×C₄⋊Q₈	C₂³	semidirect
81. C₅×#127	C₂³	semidirect
82. C₅×#125	C₂³	semidirect
83. C₅×#124	C₂³	semidirect
84. C₅×#124	C₂³	semidirect
85. C₅×#123	C₂³	semidirect
87. C₅×#121	C₂³	semidirect
88. C₅×#121	C₂³	semidirect
89. C₅×#121	C₂³	semidirect
91. C₄×C₅×Q₈	C₂³	semidirect
92. C₄×C₅×Q₈	C₂³	semidirect
94. C₄×C₅×D₄	C₂³	semidirect
95. C₄×C₅×D₄	C₂³	semidirect
96. C₄×C₅×D₄	C₂³	semidirect
97. C₄×C₅×D₄	C₂³	semidirect
100. C₅×#117	C₂³	semidirect
119. C₂×C₄²×C₅	C₂³	semidirect
120. #3508	C₂×C₅	direct (N × H)
121. C₂×#559	C₂×C₅	semidirect
122. C₂×#550	C₂×C₅	semidirect
123. C₂×#546	C₂×C₅	semidirect
124. C₂×#540	C₂×C₅	semidirect
125. C₂×#535	C₂×C₅	semidirect
126. C₂×#533	C₂×C₅	semidirect
127. C₂×#532	C₂×C₅	semidirect
128. C₂²×#127	C₂×C₅	semidirect
129. C₂×#518	C₂×C₅	semidirect
130. C₂×C₄×#41	C₂×C₅	semidirect
163. C₂²×C₄×C₅	#41	semidirect
193. C₂²×#41	C₂²×C₅	semidirect
196. #559	C₂²×C₅	semidirect
197. #550	C₂²×C₅	semidirect
198. #546	C₂²×C₅	semidirect
199. #540	C₂²×C₅	semidirect
200. #535	C₂²×C₅	semidirect
201. #533	C₂²×C₅	semidirect
202. #532	C₂²×C₅	semidirect
203. C₂×C₄⋊Q₈	C₂²×C₅	semidirect
207. C₂×#127	C₂²×C₅	semidirect
208. C₂×#127	C₂²×C₅	semidirect
209. C₂×#125	C₂²×C₅	semidirect
210. C₂×#124	C₂²×C₅	semidirect
211. C₂×#124	C₂²×C₅	semidirect
212. C₂×#123	C₂²×C₅	semidirect
214. C₂×#121	C₂²×C₅	semidirect
215. C₂×#121	C₂²×C₅	semidirect
216. C₂×#121	C₂²×C₅	semidirect
218. #518	C₂²×C₅	semidirect
219. C₄×#41	C₂²×C₅	semidirect
220. C₂×C₄×Q₈	C₂²×C₅	semidirect
221. C₂×C₄×Q₈	C₂²×C₅	semidirect
223. C₂×C₄×D₄	C₂²×C₅	semidirect
224. C₂×C₄×D₄	C₂²×C₅	semidirect
225. C₂×C₄×D₄	C₂²×C₅	semidirect
226. C₂×C₄×D₄	C₂²×C₅	semidirect
229. C₂×#117	C₂²×C₅	semidirect
232. C₂²×C₄²	C₂²×C₅	semidirect
265. C₂×C₄×C₅	C₂×#41	semidirect
278. C₂×#41	C₂³×C₅	semidirect
297. C₄⋊Q₈	C₂³×C₅	semidirect
301. #127	C₂³×C₅	semidirect
302. #125	C₂³×C₅	semidirect
303. #124	C₂³×C₅	semidirect
304. #124	C₂³×C₅	semidirect
305. #123	C₂³×C₅	semidirect
307. #121	C₂³×C₅	semidirect
308. #121	C₂³×C₅	semidirect
309. #121	C₂³×C₅	semidirect
311. C₄×Q₈	C₂³×C₅	semidirect
312. C₄×Q₈	C₂³×C₅	semidirect
314. C₄×D₄	C₂³×C₅	semidirect
315. C₄×D₄	C₂³×C₅	semidirect
316. C₄×D₄	C₂³×C₅	semidirect
317. C₄×D₄	C₂³×C₅	semidirect
320. #117	C₂³×C₅	semidirect
339. C₂×C₄²	C₂³×C₅	semidirect
392. C₂²×C₄	C₅×#41	semidirect
424. C₂×C₅	#3508	direct (N × H)
459. C₂×C₄	C₂×C₅×#41	semidirect
468. C₅	C₂×#3508	direct (N × H)
492. C₂	C₅×#3508	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×#3508	2
C₂×C₅×#559	2
C₂×C₅×#550	2
C₂×C₅×#546	2
C₂×C₅×#540	2
C₂×C₅×#535	2
C₂×C₅×#533	2
C₂×C₅×#532	2
C₂²×C₅×#127	2
C₂×C₅×#518	2
C₂×C₄×C₅×#41	2
C₂×#3508	5

Subgroups — C2×C5×#3508 #11985518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3508 #11985518