Subgroups — C₂×C₅×#2031 #11946191

Order 1280.

1234 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#421	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×#104	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#410	C₂	1	1	yes	maximal, characteristic
5	C₅×#2031	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×#344	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#339	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×#41	C₂²	1	1	yes	characteristic
10	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₅×#533	—	2	2	no	—
13	C₅×#533	—	2	2	no	—
14	C₅×#533	—	2	2	no	—
15	C₅×#533	C₂²	1	4	yes	complemented
16	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
17	C₂×C₅×#125	—	2	1	no	—
18	C₂×C₅×C₂²⋊Q₈	—	2	2	no	—
19	C₂×C₄×C₅×Q₈	—	2	2	no	—
20	C₂×C₅×#117	C₂²	1	1	yes	characteristic
21	C₂×C₅×#117	C₂²	1	1	yes	characteristic
22	C₂×C₅×#117	C₂²	1	1	yes	characteristic
23	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₅×#421	—	2	2	no	—
25	C₅×#421	C₂²	1	4	yes	—
26	C₂×C₅×#104	—	2	1	no	—
27	C₂×C₅×#104	—	2	1	no	—
28	C₂×C₅×#104	—	2	2	no	—
29	C₂×C₅×#104	—	2	2	no	—
30	C₂×C₅×#104	C₂²	1	4	yes	—
31	C₅×#410	—	2	2	no	—
32	C₅×#410	C₂²	1	4	yes	—
33	C₂×C₅×#99	—	2	2	no	—
34	C₅×#344	C₂²	1	4	yes	complemented
35	C₅×#339	C₂²	1	4	yes	complemented
36	C₅×H(3,ℤ/4)	C₂²	1	4	yes	complemented
37	C₂×#2031	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂×C₅×#41	—	2	1	no	—
39	C₂×C₅×#41	—	2	1	no	—
40	C₂×C₅×#41	—	2	1	no	—
41	C₂×C₅×#41	—	2	1	no	—
42	C₂×C₅×#41	—	2	1	no	—
43	C₂×C₅×#41	—	2	1	no	—
44	C₂×C₅×#41	—	2	1	no	—
45	C₂×C₅×#41	—	2	1	no	—
46	C₂×C₅×#41	—	2	1	no	—
47	C₂×C₅×#41	—	2	1	no	—
48	C₂×C₅×#41	Q₈	1	2	yes	complemented
49	C₂×C₅×#41	C₂³	1	2	yes	—
50	C₂²×C₅×Q₈	—	2	2	no	—
51	C₂²×C₅×Q₈	Q₈	1	1	yes	characteristic, complemented
52	C₂²×C₅×D₄	—	2	1	no	—
53	C₂²×C₅×D₄	Q₈	1	1	yes	characteristic, complemented
54	C₂³×C₄×C₅	—	2	1	no	—
55	C₂³×C₄×C₅	—	2	1	no	—
56	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₅×M₁₆	—	2	1	no	—
58	C₂×C₅×M₁₆	—	2	1	no	—
59	C₂×C₅×M₁₆	—	2	1	no	—
60	C₂×C₅×M₁₆	—	2	1	no	—
61	C₂×C₅×M₁₆	—	2	1	no	—
62	C₂×C₅×M₁₆	—	2	1	no	—
63	C₂×C₅×M₁₆	—	2	1	no	—
64	C₂×C₅×M₁₆	—	2	1	no	—
65	C₂×C₅×M₁₆	—	2	1	no	—
66	C₂×C₅×M₁₆	—	2	1	no	—
67	C₂×C₅×M₁₆	C₂³	1	2	yes	—
68	C₂×C₅×M₁₆	C₂³	1	2	yes	—
69	C₂²×C₅×C₈	—	2	1	no	—
70	C₂²×C₅×C₈	—	2	1	no	—
71	C₅×C₄⋊Q₈	—	2	4	no	—
72	C₅×C₄⋊Q₈	—	2	4	no	—
73	C₅×#125	—	2	4	no	—
74	C₅×#125	—	2	4	no	—
75	C₅×C₂²⋊Q₈	—	2	8	no	—
76	C₅×C₂²⋊Q₈	—	2	8	no	—
77	C₄×C₅×Q₈	—	2	8	no	—
78	C₄×C₅×Q₈	—	2	8	no	—
79	C₅×#117	—	2	1	no	—
80	C₅×#117	—	2	1	no	—
81	C₅×#117	—	2	1	no	—
82	C₅×#117	—	2	1	no	—
83	C₅×#117	—	2	1	no	—
84	C₅×#117	—	2	1	no	—
85	C₅×#117	—	2	1	no	—
86	C₅×#117	D₄	1	2	yes	—
87	C₅×#117	D₄	1	2	yes	—
88	C₅×#117	C₂³	1	2	yes	—
89	C₅×#117	C₂³	1	2	yes	—
90	C₅×#117	C₂³	1	2	yes	—
91	C₂×C₅×C₄⋊C₄	—	2	4	no	—
92	C₂×C₅×C₄⋊C₄	—	2	2	no	—
93	C₂×C₅×C₄⋊C₄	—	2	1	no	—
94	C₂×C₅×C₄⋊C₄	—	2	1	no	—
95	C₂×C₅×C₄⋊C₄	—	2	1	no	—
96	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
97	C₂×C₅×C₂²⋊C₄	—	2	1	no	—
98	C₂×C₅×C₂²⋊C₄	D₄	1	2	yes	—
99	C₂×C₄²×C₅	—	2	1	no	—
100	C₂×C₄²×C₅	—	2	1	no	—
101	C₂×C₄²×C₅	—	2	1	no	—
102	C₂×C₄²×C₅	—	2	1	no	—
103	C₂×C₄²×C₅	—	2	1	no	—
104	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
105	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
106	C₂×C₄²×C₅	D₄	1	2	yes	—
107	C₂×C₄²×C₅	D₄	1	2	yes	—
108	C₂×C₄²×C₅	C₂³	1	2	yes	—
109	C₅×#104	—	2	4	no	—
110	C₅×#104	—	2	4	no	—
111	C₅×#104	—	2	8	no	—
112	C₅×#99	—	2	8	no	—
113	C₂×#533	C₂×C₅	1	1	yes	characteristic, complemented
114	C₂×#421	C₂×C₅	1	1	yes	characteristic
115	C₂²×#104	C₂×C₅	1	1	yes	characteristic
116	C₂×#410	C₂×C₅	1	1	yes	characteristic
117	#2031	C₂×C₅	1	8	yes	direct product factor
118	C₂×#344	C₂×C₅	1	1	yes	characteristic, complemented
119	C₂×#339	C₂×C₅	1	1	yes	characteristic, complemented
120	C₂×H(3,ℤ/4)	C₂×C₅	1	1	yes	characteristic, complemented
121	C₂⁴×C₅	—	2	1	no	—
122	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
123	C₅×#41	—	2	2	no	—
124	C₅×#41	—	2	2	no	—
125	C₅×#41	—	2	2	no	—
126	C₅×#41	—	2	2	no	—
127	C₅×#41	—	2	2	no	—
128	C₅×#41	—	2	2	no	—
129	C₅×#41	—	2	2	no	—
130	C₅×#41	—	2	2	no	—
131	C₅×#41	—	4	1	no	—
132	C₅×#41	—	4	1	no	—
133	C₅×#41	—	4	1	no	—
134	C₅×#41	—	4	1	no	—
135	C₅×#41	—	4	1	no	—
136	C₅×#41	—	4	1	no	—
137	C₅×#41	—	4	1	no	—
138	C₅×#41	—	4	1	no	—
139	C₂×C₅×Q₈	—	2	8	no	—
140	C₂×C₅×Q₈	—	2	1	no	—
141	C₂×C₅×Q₈	—	2	1	no	—
142	C₂×C₅×Q₈	—	2	1	no	—
143	C₂×C₅×Q₈	—	2	1	no	—
144	C₂×C₅×Q₈	—	2	1	no	—
145	C₂×C₅×Q₈	—	4	8	no	—
146	C₂×C₅×Q₈	C₂×Q₈	1	2	yes	complemented
147	C₂×C₅×D₄	—	2	2	no	—
148	C₂×C₅×D₄	—	2	2	no	—
149	C₂×C₅×D₄	—	2	1	no	—
150	C₂×C₅×D₄	—	2	1	no	—
151	C₂×C₅×D₄	—	2	1	no	—
152	C₂×C₅×D₄	—	2	1	no	—
153	C₂×C₅×D₄	—	2	1	no	—
154	C₂×C₅×D₄	—	4	1	no	—
155	C₂×C₅×D₄	—	4	1	no	—
156	C₂×C₅×D₄	—	4	1	no	—
157	C₂×C₅×D₄	—	4	1	no	—
158	C₂×C₅×D₄	C₂×Q₈	1	2	yes	complemented
159	C₂²×C₄×C₅	—	2	1	no	—
160	C₂²×C₄×C₅	—	2	1	no	—
161	C₂²×C₄×C₅	—	2	2	no	—
162	C₂²×C₄×C₅	—	2	2	no	—
163	C₂²×C₄×C₅	—	2	2	no	—
164	C₂²×C₄×C₅	—	2	1	no	—
165	C₂²×C₄×C₅	—	2	2	no	—
166	C₂²×C₄×C₅	—	2	2	no	—
167	C₂²×C₄×C₅	—	2	2	no	—
168	C₂²×C₄×C₅	—	4	4	no	—
169	C₂²×C₄×C₅	—	4	1	no	—
170	C₂²×C₄×C₅	—	4	1	no	—
171	C₂²×C₄×C₅	—	4	1	no	—
172	C₂²×C₄×C₅	—	4	1	no	—
173	C₂²×C₄×C₅	—	4	1	no	—
174	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
180	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
181	C₂²×C₄×C₅	#41	1	1	yes	characteristic
182	C₂²×C₄×C₅	#41	1	1	yes	characteristic
183	C₂²×C₄×C₅	#41	1	1	yes	characteristic
184	C₂²×C₄×C₅	#41	1	1	yes	characteristic
185	C₂²×C₄×C₅	#41	1	1	yes	characteristic
186	C₂²×C₄×C₅	#41	1	1	yes	characteristic
187	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
188	C₅×M₁₆	—	2	2	no	—
189	C₅×M₁₆	—	2	2	no	—
190	C₅×M₁₆	—	2	2	no	—
191	C₅×M₁₆	—	2	2	no	—
192	C₅×M₁₆	—	4	1	no	—
193	C₅×M₁₆	—	4	1	no	—
194	C₅×M₁₆	—	4	1	no	—
195	C₅×M₁₆	—	4	1	no	—
196	C₂×C₅×C₈	—	2	2	no	—
197	C₂×C₅×C₈	—	2	2	no	—
198	C₂×C₅×C₈	—	4	1	no	—
199	C₂×C₅×C₈	—	4	1	no	—
200	C₂×C₅×C₈	—	4	1	no	—
201	C₂×C₅×C₈	—	4	1	no	—
202	C₅×C₄⋊C₄	—	2	16	no	—
203	C₅×C₄⋊C₄	—	2	8	no	—
204	C₅×C₄⋊C₄	—	2	2	no	—
205	C₅×C₄⋊C₄	—	2	2	no	—
206	C₅×C₄⋊C₄	—	2	2	no	—
207	C₅×C₄⋊C₄	—	2	2	no	—
208	C₅×C₄⋊C₄	—	2	2	no	—
209	C₅×C₄⋊C₄	—	4	1	no	—
210	C₅×C₂²⋊C₄	—	2	2	no	—
211	C₅×C₂²⋊C₄	—	2	2	no	—
212	C₅×C₂²⋊C₄	—	2	2	no	—
213	C₅×C₂²⋊C₄	—	2	2	no	—
214	C₅×C₂²⋊C₄	—	4	1	no	—
215	C₅×C₂²⋊C₄	C₂×D₄	1	4	yes	—
216	C₄²×C₅	—	2	2	no	—
217	C₄²×C₅	—	2	1	no	—
218	C₄²×C₅	—	2	1	no	—
219	C₄²×C₅	—	2	2	no	—
220	C₄²×C₅	—	2	2	no	—
221	C₄²×C₅	—	2	2	no	—
222	C₄²×C₅	—	2	2	no	—
223	C₄²×C₅	—	2	2	no	—
224	C₄²×C₅	—	4	1	no	—
225	C₄²×C₅	C₂×D₄	1	2	yes	—
226	C₄²×C₅	C₂×D₄	1	2	yes	—
227	C₄²×C₅	C₂×D₄	1	4	yes	—
228	C₂²×#41	C₂²×C₅	1	1	yes	characteristic
229	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
230	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
231	#533	—	2	2	no	—
232	#533	—	2	2	no	—
233	#533	—	2	2	no	—
234	#533	C₂²×C₅	1	4	yes	complemented
235	C₂×C₄⋊Q₈	—	2	1	no	—
236	C₂×#125	—	2	1	no	—
237	C₂×C₂²⋊Q₈	—	2	2	no	—
238	C₂×C₄×Q₈	—	2	2	no	—
239	C₂×#117	C₂²×C₅	1	1	yes	characteristic
240	C₂×#117	C₂²×C₅	1	1	yes	characteristic
241	C₂×#117	C₂²×C₅	1	1	yes	characteristic
242	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
243	#421	—	2	2	no	—
244	#421	C₂²×C₅	1	4	yes	—
245	C₂×#104	—	2	1	no	—
246	C₂×#104	—	2	1	no	—
247	C₂×#104	—	2	2	no	—
248	C₂×#104	—	2	2	no	—
249	C₂×#104	C₂²×C₅	1	4	yes	—
250	#410	—	2	2	no	—
251	#410	C₂²×C₅	1	4	yes	—
252	C₂×#99	—	2	2	no	—
253	#344	C₂²×C₅	1	4	yes	complemented
254	#339	C₂²×C₅	1	4	yes	complemented
255	H(3,ℤ/4)	C₂²×C₅	1	4	yes	complemented
256	C₂³×C₅	—	2	2	no	—
257	C₂³×C₅	—	4	2	no	—
258	C₂³×C₅	—	4	1	no	—
259	C₂³×C₅	—	4	1	no	—
260	C₂³×C₅	—	8	2	no	—
261	C₂³×C₅	#120	1	1	yes	characteristic
262	C₂³×C₅	#120	1	1	yes	characteristic
263	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
264	C₂³×C₅	C₂²⋊Q₈	1	1	yes	characteristic
265	C₂³×C₅	#124	1	1	yes	characteristic
266	C₂³×C₅	#124	1	1	yes	characteristic
267	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
268	C₅×Q₈	—	2	2	no	—
269	C₅×Q₈	—	2	2	no	—
270	C₅×Q₈	—	4	16	no	—
271	C₅×Q₈	—	4	1	no	—
272	C₅×Q₈	—	4	1	no	—
273	C₅×D₄	—	2	2	no	—
274	C₅×D₄	—	2	2	no	—
275	C₅×D₄	—	4	2	no	—
276	C₅×D₄	—	4	2	no	—
277	C₅×D₄	—	4	2	no	—
278	C₅×D₄	—	4	2	no	—
279	C₅×D₄	—	4	1	no	—
280	C₅×D₄	—	4	1	no	—
281	C₂×C₄×C₅	—	2	4	no	—
282	C₂×C₄×C₅	—	2	2	no	—
283	C₂×C₄×C₅	—	2	2	no	—
284	C₂×C₄×C₅	—	2	4	no	—
285	C₂×C₄×C₅	—	2	4	no	—
286	C₂×C₄×C₅	—	2	2	no	—
287	C₂×C₄×C₅	—	4	4	no	—
288	C₂×C₄×C₅	—	4	4	no	—
289	C₂×C₄×C₅	—	4	2	no	—
290	C₂×C₄×C₅	—	4	4	no	—
291	C₂×C₄×C₅	—	4	4	no	—
292	C₂×C₄×C₅	—	4	1	no	—
293	C₂×C₄×C₅	—	4	1	no	—
294	C₂×C₄×C₅	—	4	2	no	—
295	C₂×C₄×C₅	—	4	2	no	—
296	C₂×C₄×C₅	—	4	2	no	—
297	C₂×C₄×C₅	—	4	2	no	—
298	C₂×C₄×C₅	—	4	2	no	—
299	C₂×C₄×C₅	—	8	2	no	—
300	C₂×C₄×C₅	#120	1	1	yes	characteristic
301	C₂×C₄×C₅	#120	1	1	yes	characteristic
302	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
303	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
304	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
305	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
306	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
307	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
308	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
309	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
310	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
311	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
312	C₂×C₄×C₅	#124	1	1	yes	characteristic
313	C₂×C₄×C₅	#124	1	1	yes	characteristic
314	C₂×C₄×C₅	#124	1	1	yes	characteristic
315	C₂×C₄×C₅	#124	1	1	yes	characteristic
316	C₂×C₄×C₅	#124	1	1	yes	characteristic
317	C₂×C₄×C₅	#124	1	1	yes	characteristic
318	C₂×C₄×C₅	#124	1	1	yes	characteristic
319	C₂×C₄×C₅	#124	1	1	yes	characteristic
320	C₂×C₄×C₅	#124	1	1	yes	characteristic
321	C₂×C₄×C₅	#124	1	1	yes	characteristic
322	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
323	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
324	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
325	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
326	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
327	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
328	C₅×C₈	—	4	2	no	—
329	C₅×C₈	—	4	2	no	—
330	C₂×#41	—	2	1	no	—
331	C₂×#41	—	2	1	no	—
332	C₂×#41	—	2	1	no	—
333	C₂×#41	—	2	1	no	—
334	C₂×#41	—	2	1	no	—
335	C₂×#41	—	2	1	no	—
336	C₂×#41	—	2	1	no	—
337	C₂×#41	—	2	1	no	—
338	C₂×#41	—	2	1	no	—
339	C₂×#41	—	2	1	no	—
340	C₂×#41	C₅×Q₈	1	2	yes	complemented
341	C₂×#41	C₂³×C₅	1	2	yes	—
342	C₂²×Q₈	—	2	2	no	—
343	C₂²×Q₈	C₅×Q₈	1	1	yes	characteristic, complemented
344	C₂²×D₄	—	2	1	no	—
345	C₂²×D₄	C₅×Q₈	1	1	yes	characteristic, complemented
346	C₂³×C₄	—	2	1	no	—
347	C₂³×C₄	—	2	1	no	—
348	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
349	C₂×M₁₆	—	2	1	no	—
350	C₂×M₁₆	—	2	1	no	—
351	C₂×M₁₆	—	2	1	no	—
352	C₂×M₁₆	—	2	1	no	—
353	C₂×M₁₆	—	2	1	no	—
354	C₂×M₁₆	—	2	1	no	—
355	C₂×M₁₆	—	2	1	no	—
356	C₂×M₁₆	—	2	1	no	—
357	C₂×M₁₆	—	2	1	no	—
358	C₂×M₁₆	—	2	1	no	—
359	C₂×M₁₆	C₂³×C₅	1	2	yes	—
360	C₂×M₁₆	C₂³×C₅	1	2	yes	—
361	C₂²×C₈	—	2	1	no	—
362	C₂²×C₈	—	2	1	no	—
363	C₄⋊Q₈	—	2	4	no	—
364	C₄⋊Q₈	—	2	4	no	—
365	#125	—	2	4	no	—
366	#125	—	2	4	no	—
367	C₂²⋊Q₈	—	2	8	no	—
368	C₂²⋊Q₈	—	2	8	no	—
369	C₄×Q₈	—	2	8	no	—
370	C₄×Q₈	—	2	8	no	—
371	#117	—	2	1	no	—
372	#117	—	2	1	no	—
373	#117	—	2	1	no	—
374	#117	—	2	1	no	—
375	#117	—	2	1	no	—
376	#117	—	2	1	no	—
377	#117	—	2	1	no	—
378	#117	C₅×D₄	1	2	yes	—
379	#117	C₅×D₄	1	2	yes	—
380	#117	C₂³×C₅	1	2	yes	—
381	#117	C₂³×C₅	1	2	yes	—
382	#117	C₂³×C₅	1	2	yes	—
383	C₂×C₄⋊C₄	—	2	4	no	—
384	C₂×C₄⋊C₄	—	2	2	no	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₂²⋊C₄	—	2	1	no	—
389	C₂×C₂²⋊C₄	—	2	1	no	—
390	C₂×C₂²⋊C₄	C₅×D₄	1	2	yes	—
391	C₂×C₄²	—	2	1	no	—
392	C₂×C₄²	—	2	1	no	—
393	C₂×C₄²	—	2	1	no	—
394	C₂×C₄²	—	2	1	no	—
395	C₂×C₄²	—	2	1	no	—
396	C₂×C₄²	C₅×D₄	1	2	yes	—
397	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
398	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
399	C₂×C₄²	C₅×D₄	1	2	yes	—
400	C₂×C₄²	C₂³×C₅	1	2	yes	—
401	#104	—	2	4	no	—
402	#104	—	2	4	no	—
403	#104	—	2	8	no	—
404	#99	—	2	8	no	—
405	C₂²×C₅	—	2	2	no	—
406	C₂²×C₅	—	2	2	no	—
407	C₂²×C₅	—	2	2	no	—
408	C₂²×C₅	—	4	1	no	—
409	C₂²×C₅	—	4	1	no	—
410	C₂²×C₅	—	4	1	no	—
411	C₂²×C₅	—	4	1	no	—
412	C₂²×C₅	—	4	2	no	—
413	C₂²×C₅	—	8	2	no	—
414	C₂²×C₅	—	8	2	no	—
415	C₂²×C₅	—	8	2	no	—
416	C₂²×C₅	#393	1	1	yes	characteristic
417	C₂²×C₅	#393	1	1	yes	characteristic
418	C₂²×C₅	#393	1	1	yes	characteristic
419	C₂²×C₅	#393	1	1	yes	characteristic
420	C₂²×C₅	C₂×#120	1	1	yes	characteristic
421	C₂²×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
422	C₂²×C₅	C₂×#124	1	1	yes	characteristic
423	C₄×C₅	—	2	4	no	—
424	C₄×C₅	—	4	2	no	—
425	C₄×C₅	—	4	4	no	—
426	C₄×C₅	—	4	4	no	—
427	C₄×C₅	—	4	2	no	—
428	C₄×C₅	—	8	2	no	—
429	C₄×C₅	#393	1	1	yes	characteristic
430	C₄×C₅	#393	1	1	yes	characteristic
431	C₄×C₅	#393	1	1	yes	characteristic
432	C₄×C₅	#393	1	1	yes	characteristic
433	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
434	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
435	C₄×C₅	C₂×#124	1	1	yes	characteristic
436	C₄×C₅	C₂×#124	1	1	yes	characteristic
437	C₂⁴	—	2	1	no	—
438	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
439	#41	—	2	2	no	—
440	#41	—	2	2	no	—
441	#41	—	2	2	no	—
442	#41	—	2	2	no	—
443	#41	—	2	2	no	—
444	#41	—	2	2	no	—
445	#41	—	2	2	no	—
446	#41	—	2	2	no	—
447	#41	—	4	1	no	—
448	#41	—	4	1	no	—
449	#41	—	4	1	no	—
450	#41	—	4	1	no	—
451	#41	—	4	1	no	—
452	#41	—	4	1	no	—
453	#41	—	4	1	no	—
454	#41	—	4	1	no	—
455	C₂×Q₈	—	2	8	no	—
456	C₂×Q₈	—	2	1	no	—
457	C₂×Q₈	—	2	1	no	—
458	C₂×Q₈	—	2	1	no	—
459	C₂×Q₈	—	2	1	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	—	4	8	no	—
462	C₂×Q₈	C₂×C₅×Q₈	1	2	yes	complemented
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	2	1	no	—
470	C₂×D₄	—	4	1	no	—
471	C₂×D₄	—	4	1	no	—
472	C₂×D₄	—	4	1	no	—
473	C₂×D₄	—	4	1	no	—
474	C₂×D₄	C₂×C₅×Q₈	1	2	yes	complemented
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	2	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	4	4	no	—
485	C₂²×C₄	—	4	1	no	—
486	C₂²×C₄	—	4	1	no	—
487	C₂²×C₄	—	4	1	no	—
488	C₂²×C₄	—	4	1	no	—
489	C₂²×C₄	—	4	1	no	—
490	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
491	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
492	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
493	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
494	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
495	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
496	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
497	C₂²×C₄	C₅×#41	1	1	yes	characteristic
498	C₂²×C₄	C₅×#41	1	1	yes	characteristic
499	C₂²×C₄	C₅×#41	1	1	yes	characteristic
500	C₂²×C₄	C₅×#41	1	1	yes	characteristic
501	C₂²×C₄	C₅×#41	1	1	yes	characteristic
502	C₂²×C₄	C₅×#41	1	1	yes	characteristic
503	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
504	M₁₆	—	2	2	no	—
505	M₁₆	—	2	2	no	—
506	M₁₆	—	2	2	no	—
507	M₁₆	—	2	2	no	—
508	M₁₆	—	4	1	no	—
509	M₁₆	—	4	1	no	—
510	M₁₆	—	4	1	no	—
511	M₁₆	—	4	1	no	—
512	C₂×C₈	—	2	2	no	—
513	C₂×C₈	—	2	2	no	—
514	C₂×C₈	—	4	1	no	—
515	C₂×C₈	—	4	1	no	—
516	C₂×C₈	—	4	1	no	—
517	C₂×C₈	—	4	1	no	—
518	C₄⋊C₄	—	2	16	no	—
519	C₄⋊C₄	—	2	8	no	—
520	C₄⋊C₄	—	2	2	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	2	2	no	—
523	C₄⋊C₄	—	2	2	no	—
524	C₄⋊C₄	—	2	2	no	—
525	C₄⋊C₄	—	4	1	no	—
526	C₂²⋊C₄	—	2	2	no	—
527	C₂²⋊C₄	—	2	2	no	—
528	C₂²⋊C₄	—	2	2	no	—
529	C₂²⋊C₄	—	2	2	no	—
530	C₂²⋊C₄	—	4	1	no	—
531	C₂²⋊C₄	C₂×C₅×D₄	1	4	yes	—
532	C₄²	—	2	2	no	—
533	C₄²	—	2	1	no	—
534	C₄²	—	2	1	no	—
535	C₄²	—	2	2	no	—
536	C₄²	—	2	2	no	—
537	C₄²	—	2	2	no	—
538	C₄²	—	2	2	no	—
539	C₄²	—	2	2	no	—
540	C₄²	—	4	1	no	—
541	C₄²	C₂×C₅×D₄	1	2	yes	—
542	C₄²	C₂×C₅×D₄	1	2	yes	—
543	C₄²	C₂×C₅×D₄	1	4	yes	—
544	C₂×C₅	—	2	1	no	—
545	C₂×C₅	—	2	1	no	—
546	C₂×C₅	—	2	1	no	—
547	C₂×C₅	—	2	1	no	—
548	C₂×C₅	—	2	1	no	—
549	C₂×C₅	—	2	1	no	—
550	C₂×C₅	—	8	2	no	—
551	C₂×C₅	#2031	1	2	yes	direct product factor
552	C₂×C₅	C₂×#393	1	1	yes	characteristic
553	C₂³	—	2	2	no	—
554	C₂³	—	4	2	no	—
555	C₂³	—	4	1	no	—
556	C₂³	—	4	1	no	—
557	C₂³	—	8	2	no	—
558	C₂³	C₅×#120	1	1	yes	characteristic
559	C₂³	C₅×#120	1	1	yes	characteristic
560	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
561	C₂³	C₅×C₂²⋊Q₈	1	1	yes	characteristic
562	C₂³	C₅×#124	1	1	yes	characteristic
563	C₂³	C₅×#124	1	1	yes	characteristic
564	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
565	Q₈	—	2	2	no	—
566	Q₈	—	2	2	no	—
567	Q₈	—	4	16	no	—
568	Q₈	—	4	1	no	—
569	Q₈	—	4	1	no	—
570	D₄	—	2	2	no	—
571	D₄	—	2	2	no	—
572	D₄	—	4	2	no	—
573	D₄	—	4	2	no	—
574	D₄	—	4	2	no	—
575	D₄	—	4	2	no	—
576	D₄	—	4	1	no	—
577	D₄	—	4	1	no	—
578	C₂×C₄	—	2	4	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	4	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	4	4	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	4	no	—
588	C₂×C₄	—	4	4	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	—	4	2	no	—
593	C₂×C₄	—	4	2	no	—
594	C₂×C₄	—	4	2	no	—
595	C₂×C₄	—	4	2	no	—
596	C₂×C₄	—	8	2	no	—
597	C₂×C₄	C₅×#120	1	1	yes	characteristic
598	C₂×C₄	C₅×#120	1	1	yes	characteristic
599	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
600	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
601	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
602	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
603	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
604	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
605	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
606	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
607	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
608	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
609	C₂×C₄	C₅×#124	1	1	yes	characteristic
610	C₂×C₄	C₅×#124	1	1	yes	characteristic
611	C₂×C₄	C₅×#124	1	1	yes	characteristic
612	C₂×C₄	C₅×#124	1	1	yes	characteristic
613	C₂×C₄	C₅×#124	1	1	yes	characteristic
614	C₂×C₄	C₅×#124	1	1	yes	characteristic
615	C₂×C₄	C₅×#124	1	1	yes	characteristic
616	C₂×C₄	C₅×#124	1	1	yes	characteristic
617	C₂×C₄	C₅×#124	1	1	yes	characteristic
618	C₂×C₄	C₅×#124	1	1	yes	characteristic
619	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
620	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
621	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
622	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
623	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
624	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
625	C₈	—	4	2	no	—
626	C₈	—	4	2	no	—
627	C₅	C₂×#2031	1	1	yes	characteristic, direct product factor, Sylow
628	C₂²	—	2	2	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	2	no	—
631	C₂²	—	4	1	no	—
632	C₂²	—	4	1	no	—
633	C₂²	—	4	1	no	—
634	C₂²	—	4	1	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	8	2	no	—
637	C₂²	—	8	2	no	—
638	C₂²	—	8	2	no	—
639	C₂²	C₅×#393	1	1	yes	characteristic
640	C₂²	C₅×#393	1	1	yes	characteristic
641	C₂²	C₅×#393	1	1	yes	characteristic
642	C₂²	C₅×#393	1	1	yes	characteristic
643	C₂²	C₂×C₅×#120	1	1	yes	characteristic
644	C₂²	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
645	C₂²	C₂×C₅×#124	1	1	yes	characteristic
646	C₄	—	2	4	no	—
647	C₄	—	4	2	no	—
648	C₄	—	4	4	no	—
649	C₄	—	4	4	no	—
650	C₄	—	4	2	no	—
651	C₄	—	8	2	no	—
652	C₄	C₅×#393	1	1	yes	characteristic
653	C₄	C₅×#393	1	1	yes	characteristic
654	C₄	C₅×#393	1	1	yes	characteristic
655	C₄	C₅×#393	1	1	yes	characteristic
656	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
657	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
658	C₄	C₂×C₅×#124	1	1	yes	characteristic
659	C₄	C₂×C₅×#124	1	1	yes	characteristic
660	C₂	—	2	1	no	—
661	C₂	—	2	1	no	—
662	C₂	—	2	1	no	—
663	C₂	—	2	1	no	—
664	C₂	—	2	1	no	—
665	C₂	—	2	1	no	—
666	C₂	—	8	2	no	—
667	C₂	C₅×#2031	1	2	yes	direct product factor
668	C₂	C₂×C₅×#393	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	#393
prime radical	C₂²×C₅×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#533	C₂	semidirect
5. C₅×#2031	C₂	direct (N × H)
6. C₂×C₅×#344	C₂	semidirect
7. C₂×C₅×#339	C₂	semidirect
8. C₂×C₅×H(3,ℤ/4)	C₂	semidirect
15. C₅×#533	C₂²	semidirect
34. C₅×#344	C₂²	semidirect
35. C₅×#339	C₂²	semidirect
36. C₅×H(3,ℤ/4)	C₂²	semidirect
37. C₂×#2031	C₅	direct (N × H)
48. C₂×C₅×#41	Q₈	semidirect
51. C₂²×C₅×Q₈	Q₈	semidirect
53. C₂²×C₅×D₄	Q₈	semidirect
113. C₂×#533	C₂×C₅	semidirect
117. #2031	C₂×C₅	direct (N × H)
118. C₂×#344	C₂×C₅	semidirect
119. C₂×#339	C₂×C₅	semidirect
120. C₂×H(3,ℤ/4)	C₂×C₅	semidirect
146. C₂×C₅×Q₈	C₂×Q₈	semidirect
158. C₂×C₅×D₄	C₂×Q₈	semidirect
234. #533	C₂²×C₅	semidirect
253. #344	C₂²×C₅	semidirect
254. #339	C₂²×C₅	semidirect
255. H(3,ℤ/4)	C₂²×C₅	semidirect
340. C₂×#41	C₅×Q₈	semidirect
343. C₂²×Q₈	C₅×Q₈	semidirect
345. C₂²×D₄	C₅×Q₈	semidirect
462. C₂×Q₈	C₂×C₅×Q₈	semidirect
474. C₂×D₄	C₂×C₅×Q₈	semidirect
551. C₂×C₅	#2031	direct (N × H)
627. C₅	C₂×#2031	direct (N × H)
667. C₂	C₅×#2031	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×C₅×#533	2
C₂×C₅×#421	2
C₂²×C₅×#104	2
C₂×C₅×#410	2
C₅×#2031	2
C₂×C₅×#344	2
C₂×C₅×#339	2
C₂×C₅×H(3,ℤ/4)	2
C₂×#2031	5

Subgroups — C2×C5×#2031 #11946191

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2031 #11946191