Subgroups — C₅×#82536 #14412914

Order 1920.

1476 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5964	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5819	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#5005	C₂	1	6	yes	maximal, complemented
5	C₂×C₅×#805	C₂	1	6	yes	maximal, complemented
6	C₅×#2901	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₅×#1010	C₄	1	4	yes	complemented
9	C₅×#1009	C₄	1	4	yes	complemented
10	C₅×D₃×#41	—	2	4	no	—
11	C₅×#1007	—	2	1	no	—
12	C₅×#1004	—	2	3	no	—
13	C₂×C₅×#235	C₂²	1	6	yes	—
14	C₃×C₄×C₅×Q₈	—	2	1	no	—
15	C₃×C₄×C₅×D₄	—	2	3	no	—
16	C₃×C₅×#117	C₂²	1	3	yes	complemented
17	C₂×C₃×C₄²×C₅	C₂²	1	3	yes	complemented
18	C₅×#948	—	2	4	no	—
19	C₂×C₅×#208	C₂²	1	6	yes	complemented
20	C₅×#805	C₂²	1	4	yes	complemented
21	C₅×#805	C₂²	1	12	yes	complemented
22	#82536	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₅×#585	—	3	1	no	—
24	C₅×#568	—	3	1	no	—
25	C₄×C₅×#41	D₃	1	1	yes	characteristic, complemented
26	C₅×#421	—	3	6	no	—
27	C₂×C₅×#104	—	3	6	no	—
28	C₃×C₅×#41	C₂×C₄	1	4	yes	—
29	C₃×C₅×#41	D₄	1	4	yes	—
30	C₂×C₃×C₅×Q₈	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×D₄	D₄	1	3	yes	—
32	C₂²×C₃×C₄×C₅	—	2	3	no	—
33	C₂²×C₃×C₄×C₅	C₂³	1	3	yes	—
34	C₂×C₅×#67	—	2	6	no	—
35	C₅×#239	—	2	4	no	—
36	C₅×D₃×Q₈	—	2	4	no	—
37	C₅×#237	—	2	12	no	—
38	C₅×D₃×D₄	—	2	12	no	—
39	C₅×#235	—	2	6	no	—
40	C₅×#235	—	2	6	no	—
41	C₅×#235	C₂×C₄	1	12	yes	complemented
42	C₅×#235	C₂³	1	4	yes	—
43	C₂×C₅×D₁₂	C₂×C₄	1	6	yes	complemented
44	C₂×C₄×C₅×D₃	—	2	6	no	—
45	C₂×C₅×Dic₆	C₂×C₄	1	6	yes	complemented
46	C₃×C₅×C₄⋊C₄	—	2	3	no	—
47	C₃×C₅×C₂²⋊C₄	—	2	3	no	—
48	C₃×C₄²×C₅	—	2	3	no	—
49	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₃×C₄²×C₅	C₂³	1	3	yes	—
51	C₅×#208	—	2	6	no	—
52	C₅×#208	C₂³	1	4	yes	—
53	C₂×C₅×C₃⋊C₈	—	2	6	no	—
54	#5964	C₂×C₅	1	1	yes	characteristic
55	C₃×C₄×#41	C₂×C₅	1	1	yes	characteristic, complemented
56	#5819	C₂×C₅	1	1	yes	characteristic, complemented
57	#5005	C₂×C₅	1	6	yes	complemented
58	C₂×#805	C₂×C₅	1	6	yes	complemented
59	C₅×2₋¹⁺⁴	—	3	4	no	—
60	C₅×2₋¹⁺⁴	—	6	1	no	—
61	C₅×H(5,𝔽₂)	—	3	4	no	—
62	C₅×H(5,𝔽₂)	—	6	3	no	—
63	C₂×C₅×#41	—	3	6	no	—
64	C₂×C₅×#41	—	6	4	no	—
65	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
66	C₅×#131	—	6	4	no	—
67	C₂×C₅×M₁₆	—	3	6	no	—
68	C₄×C₅×Q₈	—	2	1	no	—
69	C₄×C₅×D₄	—	2	3	no	—
70	C₅×#117	C₂×D₃	1	3	yes	complemented
71	C₂×C₄²×C₅	C₂×D₃	1	3	yes	complemented
72	C₅×#104	—	3	4	no	—
73	C₅×#104	—	3	12	no	—
74	#2901	—	3	1	no	Sylow
75	C₂³×C₃×C₅	C₂×D₄	1	3	yes	—
76	C₂²×C₅×D₃	—	2	6	no	—
77	C₃×C₅×Q₈	C₂²⋊C₄	1	4	yes	—
78	C₃×C₅×D₄	C₂²⋊C₄	1	12	yes	—
79	C₂×C₃×C₄×C₅	—	2	1	no	—
80	C₂×C₃×C₄×C₅	—	2	3	no	—
81	C₂×C₃×C₄×C₅	—	4	6	no	—
82	C₂×C₃×C₄×C₅	C₂²×C₄	1	6	yes	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	3	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	6	yes	—
85	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₅×#67	—	2	12	no	—
87	C₅×#67	—	2	4	no	—
88	C₂×C₅×Dic₃	—	2	6	no	—
89	C₅×D₁₂	—	2	6	no	—
90	C₅×D₁₂	C₂²×C₄	1	4	yes	—
91	C₄×C₅×D₃	—	2	12	no	—
92	C₄×C₅×D₃	—	2	4	no	—
93	C₅×Dic₆	—	2	6	no	—
94	C₅×Dic₆	C₂²×C₄	1	4	yes	—
95	C₅×C₃⋊C₈	—	2	4	no	—
96	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
97	#1010	C₄×C₅	1	4	yes	complemented
98	#1009	C₄×C₅	1	4	yes	complemented
99	D₃×#41	—	2	4	no	—
100	#1007	—	2	1	no	—
101	#1004	—	2	3	no	—
102	C₂×#235	C₂²×C₅	1	6	yes	—
103	C₃×C₄×Q₈	—	2	1	no	—
104	C₃×C₄×D₄	—	2	3	no	—
105	C₃×#117	C₂²×C₅	1	3	yes	complemented
106	C₂×C₃×C₄²	C₂²×C₅	1	3	yes	complemented
107	#948	—	2	4	no	—
108	C₂×#208	C₂²×C₅	1	6	yes	complemented
109	#805	C₂²×C₅	1	4	yes	complemented
110	#805	C₂²×C₅	1	12	yes	complemented
111	C₅×#41	—	3	12	no	—
112	C₅×#41	—	3	4	no	—
113	C₅×#41	—	6	4	no	—
114	C₅×#41	—	6	12	no	—
115	C₅×#41	—	6	6	no	—
116	C₅×#41	—	6	6	no	—
117	C₅×#41	C₄×D₃	1	4	yes	—
118	C₅×#41	D₁₂	1	4	yes	—
119	C₂×C₅×Q₈	—	3	6	no	—
120	C₂×C₅×Q₈	—	6	4	no	—
121	C₂×C₅×Q₈	#67	1	1	yes	characteristic
122	C₂×C₅×D₄	—	3	6	no	—
123	C₂×C₅×D₄	—	6	6	no	—
124	C₂×C₅×D₄	—	6	12	no	—
125	C₂×C₅×D₄	#67	1	3	yes	—
126	C₂²×C₄×C₅	—	2	3	no	—
127	C₂²×C₄×C₅	—	6	6	no	—
128	C₂²×C₄×C₅	C₂²×D₃	1	3	yes	—
129	C₅×M₁₆	—	3	4	no	—
130	C₅×M₁₆	—	6	6	no	—
131	C₂×C₅×C₈	—	6	6	no	—
132	C₅×C₄⋊C₄	—	2	3	no	—
133	C₅×C₂²⋊C₄	—	2	3	no	—
134	C₄²×C₅	—	2	3	no	—
135	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
136	C₄²×C₅	C₂²×D₃	1	3	yes	—
137	#585	—	3	1	no	—
138	#568	—	3	1	no	—
139	C₄×#41	C₅×D₃	1	1	yes	characteristic, complemented
140	#421	—	3	6	no	—
141	C₂×#104	—	3	6	no	—
142	C₂²×C₃×C₅	—	4	3	no	—
143	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	6	yes	—
144	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₅×D₃	—	2	4	no	—
146	C₂×C₅×D₃	—	8	6	no	—
147	C₃×C₄×C₅	—	2	2	no	—
148	C₃×C₄×C₅	—	4	3	no	—
149	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
150	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
151	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
152	C₅×Dic₃	—	2	4	no	—
153	C₃×#41	C₂×C₄×C₅	1	4	yes	—
154	C₃×#41	C₅×D₄	1	4	yes	—
155	C₂×C₃×Q₈	C₅×D₄	1	1	yes	characteristic
156	C₂×C₃×D₄	C₅×D₄	1	3	yes	—
157	C₂²×C₃×C₄	—	2	3	no	—
158	C₂²×C₃×C₄	C₂³×C₅	1	3	yes	—
159	C₂×#67	—	2	6	no	—
160	#239	—	2	4	no	—
161	D₃×Q₈	—	2	4	no	—
162	#237	—	2	12	no	—
163	D₃×D₄	—	2	12	no	—
164	#235	—	2	6	no	—
165	#235	—	2	6	no	—
166	#235	C₂×C₄×C₅	1	12	yes	complemented
167	#235	C₂³×C₅	1	4	yes	—
168	C₂×D₁₂	C₂×C₄×C₅	1	6	yes	complemented
169	C₂×C₄×D₃	—	2	6	no	—
170	C₂×Dic₆	C₂×C₄×C₅	1	6	yes	complemented
171	C₃×C₄⋊C₄	—	2	3	no	—
172	C₃×C₂²⋊C₄	—	2	3	no	—
173	C₃×C₄²	—	2	3	no	—
174	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
175	C₃×C₄²	C₂³×C₅	1	3	yes	—
176	#208	—	2	6	no	—
177	#208	C₂³×C₅	1	4	yes	—
178	C₂×C₃⋊C₈	—	2	6	no	—
179	C₂³×C₅	—	6	6	no	—
180	C₂³×C₅	C₂×#67	1	3	yes	—
181	C₅×Q₈	—	3	4	no	—
182	C₅×Q₈	—	6	6	no	—
183	C₅×Q₈	(C₂×D₃)⋊C₄	1	4	yes	—
184	C₅×D₄	—	3	4	no	—
185	C₅×D₄	—	6	12	no	—
186	C₅×D₄	—	6	4	no	—
187	C₅×D₄	—	6	6	no	—
188	C₅×D₄	(C₂×D₃)⋊C₄	1	12	yes	—
189	C₂×C₄×C₅	—	2	1	no	—
190	C₂×C₄×C₅	—	2	3	no	—
191	C₂×C₄×C₅	—	4	6	no	—
192	C₂×C₄×C₅	—	6	6	no	—
193	C₂×C₄×C₅	—	6	12	no	—
194	C₂×C₄×C₅	—	6	4	no	—
195	C₂×C₄×C₅	C₂×C₄×D₃	1	6	yes	—
196	C₂×C₄×C₅	C₂×D₁₂	1	6	yes	—
197	C₂×C₄×C₅	C₂×#67	1	3	yes	—
198	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₅×C₈	—	6	4	no	—
200	2₋¹⁺⁴	—	3	4	no	—
201	2₋¹⁺⁴	—	6	1	no	—
202	H(5,𝔽₂)	—	3	4	no	—
203	H(5,𝔽₂)	—	6	3	no	—
204	C₂×#41	—	3	6	no	—
205	C₂×#41	—	6	4	no	—
206	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
207	#131	—	6	4	no	—
208	C₂×M₁₆	—	3	6	no	—
209	C₄×Q₈	—	2	1	no	—
210	C₄×D₄	—	2	3	no	—
211	#117	C₂×C₅×D₃	1	3	yes	complemented
212	C₂×C₄²	C₂×C₅×D₃	1	3	yes	complemented
213	#104	—	3	4	no	—
214	#104	—	3	12	no	—
215	C₂×C₃×C₅	—	2	1	no	—
216	C₂×C₃×C₅	—	2	6	no	—
217	C₂×C₃×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
218	C₅×D₃	—	4	4	no	—
219	C₂³×C₃	C₂×C₅×D₄	1	3	yes	—
220	C₂²×D₃	—	2	6	no	—
221	C₃×Q₈	C₅×C₂²⋊C₄	1	4	yes	—
222	C₃×D₄	C₅×C₂²⋊C₄	1	12	yes	—
223	C₂×C₃×C₄	—	2	1	no	—
224	C₂×C₃×C₄	—	2	3	no	—
225	C₂×C₃×C₄	—	4	6	no	—
226	C₂×C₃×C₄	C₂²×C₄×C₅	1	6	yes	—
227	C₂×C₃×C₄	C₂×C₅×D₄	1	6	yes	—
228	C₂×C₃×C₄	C₂×C₅×D₄	1	3	yes	—
229	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
230	#67	—	2	12	no	—
231	#67	—	2	4	no	—
232	C₂×Dic₃	—	2	6	no	—
233	D₁₂	—	2	6	no	—
234	D₁₂	C₂²×C₄×C₅	1	4	yes	—
235	C₄×D₃	—	2	12	no	—
236	C₄×D₃	—	2	4	no	—
237	Dic₆	—	2	6	no	—
238	Dic₆	C₂²×C₄×C₅	1	4	yes	—
239	C₃⋊C₈	—	2	4	no	—
240	C₂²×C₅	—	4	3	no	—
241	C₂²×C₅	—	6	4	no	—
242	C₂²×C₅	—	24	6	no	—
243	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	6	yes	—
244	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
245	C₄×C₅	—	2	2	no	—
246	C₄×C₅	—	4	3	no	—
247	C₄×C₅	—	6	4	no	—
248	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	6	yes	—
249	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
250	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
251	#41	—	3	12	no	—
252	#41	—	3	4	no	—
253	#41	—	6	4	no	—
254	#41	—	6	12	no	—
255	#41	—	6	6	no	—
256	#41	—	6	6	no	—
257	#41	C₄×C₅×D₃	1	4	yes	—
258	#41	C₅×D₁₂	1	4	yes	—
259	C₂×Q₈	—	3	6	no	—
260	C₂×Q₈	—	6	4	no	—
261	C₂×Q₈	C₅×#67	1	1	yes	characteristic
262	C₂×D₄	—	3	6	no	—
263	C₂×D₄	—	6	6	no	—
264	C₂×D₄	—	6	12	no	—
265	C₂×D₄	C₅×#67	1	3	yes	—
266	C₂²×C₄	—	2	3	no	—
267	C₂²×C₄	—	6	6	no	—
268	C₂²×C₄	C₂²×C₅×D₃	1	3	yes	—
269	M₁₆	—	3	4	no	—
270	M₁₆	—	6	6	no	—
271	C₂×C₈	—	6	6	no	—
272	C₄⋊C₄	—	2	3	no	—
273	C₂²⋊C₄	—	2	3	no	—
274	C₄²	—	2	3	no	—
275	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
276	C₄²	C₂²×C₅×D₃	1	3	yes	—
277	C₃×C₅	#2901	1	1	yes	characteristic, complemented, Hall
278	C₂²×C₃	—	4	3	no	—
279	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	6	yes	—
280	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
281	C₂×D₃	—	2	4	no	—
282	C₂×D₃	—	8	6	no	—
283	C₃×C₄	—	2	2	no	—
284	C₃×C₄	—	4	3	no	—
285	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
286	C₃×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
287	C₃×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
288	Dic₃	—	2	4	no	—
289	C₂×C₅	—	2	1	no	—
290	C₂×C₅	—	2	6	no	—
291	C₂×C₅	—	12	4	no	—
292	C₂×C₅	C₂²×(C₂×D₃)⋊C₄	1	1	yes	characteristic
293	C₂³	—	6	6	no	—
294	C₂³	C₂×C₅×#67	1	3	yes	—
295	Q₈	—	3	4	no	—
296	Q₈	—	6	6	no	—
297	Q₈	C₅×(C₂×D₃)⋊C₄	1	4	yes	—
298	D₄	—	3	4	no	—
299	D₄	—	6	12	no	—
300	D₄	—	6	4	no	—
301	D₄	—	6	6	no	—
302	D₄	C₅×(C₂×D₃)⋊C₄	1	12	yes	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	3	no	—
305	C₂×C₄	—	4	6	no	—
306	C₂×C₄	—	6	6	no	—
307	C₂×C₄	—	6	12	no	—
308	C₂×C₄	—	6	4	no	—
309	C₂×C₄	C₂×C₄×C₅×D₃	1	6	yes	—
310	C₂×C₄	C₂×C₅×D₁₂	1	6	yes	—
311	C₂×C₄	C₂×C₅×#67	1	3	yes	—
312	C₂×C₄	C₂³×C₅×D₃	1	1	yes	characteristic
313	C₈	—	6	4	no	—
314	C₂×C₃	—	2	1	no	—
315	C₂×C₃	—	2	6	no	—
316	C₂×C₃	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
317	D₃	—	4	4	no	—
318	C₅	#82536	1	1	yes	characteristic, direct product factor, Sylow
319	C₂²	—	4	3	no	—
320	C₂²	—	6	4	no	—
321	C₂²	—	24	6	no	—
322	C₂²	C₂×C₅×(C₂×D₃)⋊C₄	1	6	yes	—
323	C₂²	C₂²×C₅×#67	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	4	3	no	—
326	C₄	—	6	4	no	—
327	C₄	C₂×C₅×(C₂×D₃)⋊C₄	1	6	yes	—
328	C₄	C₂²×C₄×C₅×D₃	1	1	yes	characteristic
329	C₄	C₂²×C₅×D₁₂	1	1	yes	characteristic
330	C₃	C₅×#2901	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	1	no	—
332	C₂	—	2	6	no	—
333	C₂	—	12	4	no	—
334	C₂	C₂²×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂²×D₁₂
commutator	C₃×C₄	C₂³×C₄×C₅
Frattini	C₂×C₄	C₂³×C₅×D₃
Fitting	C₃×C₄×C₅×#41	C₂
socle	C₂×C₃×C₅	C₂²×C₂²⋊C₄
prime radical	C₅×#5964	C₂
cosocle	C₂×C₃×C₄	C₂⁴×C₅
hypercenter	C₄×C₅×#41	D₃
hypocenter	C₃	C₅×#2901
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×#41 ◁ C₄×C₅×#41

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×C₅×#41	C₂	semidirect
3. C₅×#5819	C₂	semidirect
4. C₅×#5005	C₂	semidirect
5. C₂×C₅×#805	C₂	semidirect
8. C₅×#1010	C₄	semidirect
9. C₅×#1009	C₄	semidirect
16. C₃×C₅×#117	C₂²	semidirect
17. C₂×C₃×C₄²×C₅	C₂²	semidirect
19. C₂×C₅×#208	C₂²	semidirect
20. C₅×#805	C₂²	semidirect
21. C₅×#805	C₂²	semidirect
22. #82536	C₅	direct (N × H)
25. C₄×C₅×#41	D₃	semidirect
41. C₅×#235	C₂×C₄	semidirect
43. C₂×C₅×D₁₂	C₂×C₄	semidirect
45. C₂×C₅×Dic₆	C₂×C₄	semidirect
55. C₃×C₄×#41	C₂×C₅	semidirect
56. #5819	C₂×C₅	semidirect
57. #5005	C₂×C₅	semidirect
58. C₂×#805	C₂×C₅	semidirect
70. C₅×#117	C₂×D₃	semidirect
71. C₂×C₄²×C₅	C₂×D₃	semidirect
97. #1010	C₄×C₅	semidirect
98. #1009	C₄×C₅	semidirect
105. C₃×#117	C₂²×C₅	semidirect
106. C₂×C₃×C₄²	C₂²×C₅	semidirect
108. C₂×#208	C₂²×C₅	semidirect
109. #805	C₂²×C₅	semidirect
110. #805	C₂²×C₅	semidirect
139. C₄×#41	C₅×D₃	semidirect
166. #235	C₂×C₄×C₅	semidirect
168. C₂×D₁₂	C₂×C₄×C₅	semidirect
170. C₂×Dic₆	C₂×C₄×C₅	semidirect
211. #117	C₂×C₅×D₃	semidirect
212. C₂×C₄²	C₂×C₅×D₃	semidirect
277. C₃×C₅	#2901	semidirect
318. C₅	#82536	direct (N × H)
330. C₃	C₅×#2901	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5964	2
C₃×C₄×C₅×#41	2
C₅×#5819	2
C₅×#5005	2
C₂×C₅×#805	2
C₅×#2901	3
#82536	5

Subgroups — C5×#82536 #14412914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#82536 #14412914