Subgroups — C₄×#87874 #13105036

Order 1600.

714 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87874	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₅⋊D₅)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄²×C₅×D₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717022	C₂	1	2	yes	maximal, complemented
6	C₄×#6038	C₂	1	2	yes	maximal, complemented
7	#11717013	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#6036	C₂	1	2	yes	maximal, complemented
9	Dic₅×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716959	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716954	C₂	1	2	yes	maximal, complemented
12	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅⋊D₅	C₂²	1	2	yes	—
14	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
17	C₂×C₅×Dic₁₀	C₂²	1	1	yes	characteristic
18	C₂×#6038	C₂²	1	2	yes	—
19	C₂×#6036	C₂²	1	2	yes	—
20	C₂×#6036	C₂²	1	2	yes	—
21	#87874	C₄	1	8	yes	direct product factor
22	C₄²×C₅²	C₂²	1	1	yes	characteristic
23	#87808	C₂²	1	2	yes	—
24	C₄⋊#1039	C₂²	1	1	yes	characteristic
25	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic
26	C₅×#683	C₂²	1	2	yes	—
27	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
28	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
29	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
30	C₄×C₅×Dic₅	C₂²	1	2	yes	—
31	#87780	C₂²	1	2	yes	—
32	#87779	C₂²	1	2	yes	—
33	#87778	C₂²	1	2	yes	—
34	Dic₅²	C₂²	1	2	yes	—
35	Dic₅²	C₂²	1	2	yes	—
36	C₄×#713	—	5	1	no	maximal
37	C₄×#709	—	5	1	no	maximal
38	C₂²×C₅⋊D₅	C₂³	1	2	yes	—
39	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
43	C₂×#1039	C₂³	1	2	yes	—
44	C₄⋊(C₅⋊D₅)	C₂×C₄	1	4	yes	complemented
45	C₄×C₅⋊D₅	—	2	4	no	—
46	C₂×C₅×Dic₅	C₂³	1	2	yes	—
47	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
48	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
49	C₂×C₅×Dic₅	C₂³	1	2	yes	—
50	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
51	C₄×C₅×D₅	—	2	2	no	—
52	C₄×C₅×D₅	—	2	2	no	—
53	C₄×C₅×D₅	C₂×C₄	1	4	yes	complemented
54	C₅×Dic₁₀	C₂×C₄	1	4	yes	complemented
55	#6038	C₂×C₄	1	8	yes	complemented
56	#6036	—	2	4	no	—
57	#6036	C₂×C₄	1	8	yes	complemented
58	C₂×#713	—	5	1	no	—
59	C₂×#709	—	5	1	no	—
60	C₄×C₅×Q₈	—	5	1	no	—
61	C₂×C₄²×C₅	—	5	1	no	—
62	Q₈×Dic₅	—	5	1	no	—
63	C₄×C₂²⋊D₅	—	5	2	no	—
64	C₄×C₂²⋊D₅	—	5	2	no	—
65	#4072	—	5	1	no	—
66	#4039	—	5	2	no	—
67	#4039	—	5	1	no	—
68	#4038	—	5	1	no	—
69	#4038	—	5	2	no	—
70	C₄×D₂₀	—	5	2	no	—
71	C₄×D₂₀	—	5	1	no	—
72	C₄×D₂₀	—	5	1	no	—
73	C₄×D₂₀	—	5	1	no	—
74	C₄×D₂₀	—	10	2	no	—
75	#4018	—	5	2	no	—
76	C₄²×D₅	—	5	2	no	—
77	C₄²×D₅	—	5	2	no	—
78	C₄²×D₅	D₅	1	1	yes	characteristic, complemented
79	C₄×Dic₁₀	—	5	1	no	—
80	C₄×Dic₁₀	D₅	1	1	yes	characteristic, complemented
81	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
82	C₂×C₅⋊D₅	—	2	2	no	—
83	C₂×C₅⋊D₅	—	2	2	no	—
84	C₂×C₅⋊D₅	C₂²×C₄	1	4	yes	—
85	C₂×C₅×D₅	—	2	1	no	—
86	C₂×C₅×D₅	—	2	1	no	—
87	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
88	C₄×C₅²	—	2	1	no	—
89	C₄×C₅²	C₂²×C₄	1	2	yes	—
90	C₄×C₅²	#41	1	2	yes	complemented
91	#1039	—	2	2	no	—
92	C₅×Dic₅	—	2	2	no	—
93	C₅×Dic₅	—	2	1	no	—
94	C₅×Dic₅	C₂²×C₄	1	2	yes	—
95	C₅×Dic₅	C₂²×C₄	1	4	yes	—
96	C₅×Dic₅	#41	1	2	yes	complemented
97	C₂×C₅×Q₈	—	5	1	no	—
98	C₂²×C₄×C₅	—	5	1	no	—
99	C₂²×C₄×C₅	—	5	1	no	—
100	C₂²×C₄×C₅	—	5	1	no	—
101	C₂×C₂²⋊D₅	—	5	2	no	—
102	#713	—	5	8	no	—
103	#709	—	5	8	no	—
104	C₂×D₂₀	—	5	2	no	—
105	C₂×D₂₀	—	5	1	no	—
106	C₂×D₂₀	—	5	1	no	—
107	C₂×D₂₀	—	10	2	no	—
108	C₂×C₄×D₅	—	5	2	no	—
109	C₂×C₄×D₅	—	5	2	no	—
110	C₂×C₄×D₅	—	5	2	no	—
111	C₂×C₄×D₅	—	5	2	no	—
112	C₂×C₄×D₅	—	5	2	no	—
113	C₂×C₄×D₅	—	5	2	no	—
114	C₂×C₄×D₅	—	10	4	no	—
115	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
116	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
117	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
118	C₂×Dic₁₀	C₂×D₅	1	1	yes	characteristic
119	C₅×C₄⋊C₄	—	5	2	no	—
120	C₅×C₄⋊C₄	—	5	1	no	—
121	C₄²×C₅	—	2	2	no	—
122	C₄²×C₅	—	5	1	no	—
123	C₄²×C₅	—	5	1	no	—
124	C₄²×C₅	—	5	1	no	—
125	C₄²×C₅	—	5	2	no	—
126	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
127	C₄²×C₅	C₂×D₅	1	1	yes	characteristic
128	C₂²⋊Dic₅	—	5	2	no	—
129	#685	—	5	2	no	—
130	#685	—	5	2	no	—
131	#685	—	5	2	no	—
132	#685	—	5	2	no	—
133	#685	—	5	2	no	—
134	#685	—	10	4	no	—
135	C₄⋊Dic₅	—	5	1	no	—
136	C₄⋊Dic₅	—	5	1	no	—
137	C₄⋊Dic₅	—	5	2	no	—
138	C₄⋊Dic₅	—	10	2	no	—
139	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic
140	#683	—	5	2	no	—
141	#683	C₂×D₅	1	2	yes	—
142	C₄×Dic₅	—	5	2	no	—
143	C₄×Dic₅	—	5	2	no	—
144	C₄×Dic₅	—	5	2	no	—
145	C₄×Dic₅	—	5	2	no	—
146	C₄×Dic₅	C₂×D₅	1	2	yes	—
147	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
148	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
149	C₄×Dic₅	C₂×D₅	1	1	yes	characteristic
150	C₄×#41	—	25	1	no	Sylow
151	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₅²	C₂×#41	1	1	yes	characteristic
153	C₂×C₅²	C₂×#41	1	1	yes	characteristic
154	C₅⋊D₅	—	2	4	no	—
155	C₅×D₅	—	2	2	no	—
156	C₂³×C₅	—	5	1	no	—
157	C₂²×D₅	—	5	2	no	—
158	C₂²×D₅	—	5	2	no	—
159	C₂²×D₅	—	10	4	no	—
160	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
161	C₅×Q₈	—	5	4	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	5	1	no	—
166	C₂×C₄×C₅	—	5	1	no	—
167	C₂×C₄×C₅	—	5	2	no	—
168	C₂×C₄×C₅	—	5	4	no	—
169	C₂×C₄×C₅	—	5	2	no	—
170	C₂×C₄×C₅	—	5	1	no	—
171	C₂×C₄×C₅	—	10	2	no	—
172	C₂×C₄×C₅	—	10	2	no	—
173	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
179	C₂²⋊D₅	—	5	8	no	—
180	C₂×Dic₅	—	5	2	no	—
181	C₂×Dic₅	—	5	2	no	—
182	C₂×Dic₅	—	10	4	no	—
183	C₂×Dic₅	C₂²×D₅	1	2	yes	—
184	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
185	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
186	C₂×Dic₅	C₂²×D₅	1	2	yes	—
187	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
188	D₂₀	—	5	8	no	—
189	D₂₀	—	5	4	no	—
190	D₂₀	—	5	4	no	—
191	D₂₀	—	10	8	no	—
192	C₄×D₅	—	2	2	no	—
193	C₄×D₅	—	2	2	no	—
194	C₄×D₅	—	5	8	no	—
195	C₄×D₅	—	5	8	no	—
196	C₄×D₅	—	10	4	no	—
197	C₄×D₅	—	10	4	no	—
198	C₄×D₅	—	10	4	no	—
199	C₄×D₅	—	10	4	no	—
200	C₄×D₅	—	20	8	no	—
201	C₄×D₅	C₄×D₅	1	4	yes	complemented
202	Dic₁₀	C₄×D₅	1	4	yes	complemented
203	C₂×#41	—	25	1	no	—
204	C₄×Q₈	—	5	1	no	—
205	C₄×Q₈	—	25	1	no	—
206	C₄×D₄	—	25	1	no	—
207	C₄×D₄	—	25	2	no	—
208	C₄×D₄	—	25	2	no	—
209	C₄×D₄	—	25	1	no	—
210	#117	—	25	2	no	—
211	#117	—	25	1	no	—
212	C₂×C₄²	—	5	1	no	—
213	C₂×C₄²	—	25	2	no	—
214	C₅²	C₄×#41	1	1	yes	characteristic, complemented, Sylow
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	—	5	2	no	—
217	C₂²×C₅	—	10	1	no	—
218	C₂²×C₅	—	10	1	no	—
219	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
220	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
221	C₂×D₅	—	2	1	no	—
222	C₂×D₅	—	2	1	no	—
223	C₂×D₅	—	5	4	no	—
224	C₂×D₅	—	5	4	no	—
225	C₂×D₅	—	10	8	no	—
226	C₂×D₅	—	10	2	no	—
227	C₂×D₅	—	10	2	no	—
228	C₂×D₅	—	10	2	no	—
229	C₂×D₅	—	10	2	no	—
230	C₂×D₅	—	20	4	no	—
231	C₂×D₅	—	20	4	no	—
232	C₂×D₅	C₂×C₄×D₅	1	2	yes	—
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	4	no	—
236	C₄×C₅	—	2	4	no	—
237	C₄×C₅	—	4	2	no	—
238	C₄×C₅	—	5	2	no	—
239	C₄×C₅	—	5	4	no	—
240	C₄×C₅	—	5	2	no	—
241	C₄×C₅	—	10	2	no	—
242	C₄×C₅	—	10	1	no	—
243	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
244	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
245	C₄×C₅	#709	1	2	yes	complemented
246	C₄×C₅	#713	1	2	yes	complemented
247	Dic₅	—	2	1	no	—
248	Dic₅	—	2	2	no	—
249	Dic₅	—	10	2	no	—
250	Dic₅	—	10	2	no	—
251	Dic₅	—	20	4	no	—
252	Dic₅	C₂×C₄×D₅	1	4	yes	—
253	Dic₅	C₂×C₄×D₅	1	2	yes	—
254	Dic₅	#709	1	2	yes	complemented
255	#41	—	25	8	no	—
256	C₂×Q₈	—	5	1	no	—
257	C₂×D₄	—	25	2	no	—
258	C₂×D₄	—	25	1	no	—
259	C₂²×C₄	—	5	1	no	—
260	C₂²×C₄	—	5	1	no	—
261	C₂²×C₄	—	5	1	no	—
262	C₂²×C₄	—	25	2	no	—
263	C₂²×C₄	—	25	2	no	—
264	C₂²×C₄	—	25	2	no	—
265	C₄⋊C₄	—	5	2	no	—
266	C₄⋊C₄	—	5	1	no	—
267	C₄⋊C₄	—	25	2	no	—
268	C₄⋊C₄	—	25	1	no	—
269	C₂²⋊C₄	—	25	2	no	—
270	C₂²⋊C₄	—	25	2	no	—
271	C₂²⋊C₄	—	25	2	no	—
272	C₄²	—	5	1	no	—
273	C₄²	—	5	1	no	—
274	C₄²	—	5	1	no	—
275	C₄²	—	5	2	no	—
276	C₄²	—	25	2	no	—
277	C₄²	—	25	2	no	—
278	C₄²	D₅²	1	1	yes	characteristic
279	C₂×C₅	—	2	2	no	—
280	C₂×C₅	—	2	2	no	—
281	C₂×C₅	—	2	2	no	—
282	C₂×C₅	—	10	2	no	—
283	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
284	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
285	C₂×C₅	C₂×#709	1	1	yes	characteristic
286	C₂×C₅	C₂×#709	1	1	yes	characteristic
287	C₂×C₅	C₂×#713	1	1	yes	characteristic
288	C₂×C₅	D₅×#41	1	1	yes	characteristic
289	D₅	—	2	2	no	—
290	D₅	—	10	4	no	—
291	D₅	—	10	4	no	—
292	D₅	—	20	8	no	—
293	C₂³	—	5	1	no	—
294	C₂³	—	25	2	no	—
295	Q₈	—	5	4	no	—
296	D₄	—	25	8	no	—
297	D₄	—	25	4	no	—
298	C₂×C₄	—	5	1	no	—
299	C₂×C₄	—	5	1	no	—
300	C₂×C₄	—	5	2	no	—
301	C₂×C₄	—	5	4	no	—
302	C₂×C₄	—	5	1	no	—
303	C₂×C₄	—	5	2	no	—
304	C₂×C₄	—	10	2	no	—
305	C₂×C₄	—	10	2	no	—
306	C₂×C₄	—	25	8	no	—
307	C₂×C₄	—	25	2	no	—
308	C₂×C₄	—	50	4	no	—
309	C₂×C₄	—	50	4	no	—
310	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
311	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
312	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
313	C₅	—	2	2	no	—
314	C₅	C₄×#709	1	1	yes	characteristic, complemented
315	C₅	C₄×#713	1	1	yes	characteristic, complemented
316	C₂²	—	5	2	no	—
317	C₂²	—	10	1	no	—
318	C₂²	—	10	1	no	—
319	C₂²	—	25	4	no	—
320	C₂²	—	50	2	no	—
321	C₂²	—	50	2	no	—
322	C₂²	C₂²×D₅²	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	5	2	no	—
325	C₄	—	5	4	no	—
326	C₄	—	5	2	no	—
327	C₄	—	10	1	no	—
328	C₄	—	10	2	no	—
329	C₄	—	50	2	no	—
330	C₄	#87874	1	2	yes	direct product factor
331	C₄	C₄×D₅²	1	2	yes	—
332	C₂	—	10	2	no	—
333	C₂	—	50	4	no	—
334	C₂	C₂×#87874	1	1	yes	characteristic
335	C₂	C₂×C₄×D₅²	1	1	yes	characteristic
336	C₂	D₅×#709	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅²
commutator	C₂×C₅²	C₂³×C₄
Frattini	C₂²	C₂²×D₅²
Fitting	C₄²×C₅²	C₂²
socle, cosocle	C₂²×C₅²	C₂⁴
prime radical	C₂×#87874	C₂
hypercenter	C₄²	D₅²
hypocenter	C₅²	C₄×#41
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. C₄×C₄⋊(C₅⋊D₅)	C₂	semidirect
3. C₄²×C₅×D₅	C₂	semidirect
4. C₄×C₅×Dic₁₀	C₂	semidirect
5. #11717022	C₂	semidirect
6. C₄×#6038	C₂	semidirect
7. #11717013	C₂	semidirect
8. C₄×#6036	C₂	semidirect
9. Dic₅×Dic₁₀	C₂	semidirect
10. #11716959	C₂	semidirect
11. #11716954	C₂	semidirect
21. #87874	C₄	direct (N × H)
44. C₄⋊(C₅⋊D₅)	C₂×C₄	semidirect
53. C₄×C₅×D₅	C₂×C₄	semidirect
54. C₅×Dic₁₀	C₂×C₄	semidirect
55. #6038	C₂×C₄	semidirect
57. #6036	C₂×C₄	semidirect
78. C₄²×D₅	D₅	semidirect
80. C₄×Dic₁₀	D₅	semidirect
90. C₄×C₅²	#41	semidirect
96. C₅×Dic₅	#41	semidirect
201. C₄×D₅	C₄×D₅	semidirect
202. Dic₁₀	C₄×D₅	semidirect
214. C₅²	C₄×#41	semidirect
245. C₄×C₅	#709	semidirect
246. C₄×C₅	#713	semidirect
254. Dic₅	#709	semidirect
314. C₅	C₄×#709	semidirect
315. C₅	C₄×#713	semidirect
330. C₄	#87874	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×#87874	2
C₄×C₄⋊(C₅⋊D₅)	2
C₄²×C₅×D₅	2
C₄×C₅×Dic₁₀	2
#11717022	2
C₄×#6038	2
#11717013	2
C₄×#6036	2
Dic₅×Dic₁₀	2
#11716959	2
#11716954	2
C₄×#713	5
C₄×#709	5

Subgroups — C4×#87874 #13105036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#87874 #13105036