Subgroups — C₅×#71437 #14313002

Order 1920.

440 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#901	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4508	C₂	1	4	yes	maximal, complemented
5	C₅×#2117	—	3	1	no	maximal, Hall
6	C₂×C₅×#235	C₄	1	1	yes	characteristic
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#208	C₄	1	1	yes	characteristic
9	C₅×#901	—	2	1	no	—
10	C₅×#901	—	2	1	no	—
11	C₅×#901	C₄	1	2	yes	—
12	C₅×#901	C₂²	1	2	yes	—
13	C₂×C₅×ΓL(1,𝔽₂₅)	—	2	1	no	—
14	C₂×C₅×D₃×C₈	—	2	1	no	—
15	C₃×C₅×M₃₂	—	2	1	no	—
16	C₃×C₅×M₃₂	—	2	1	no	—
17	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	complemented
18	C₅×#812	—	2	1	no	—
19	C₅×#812	C₂²	1	2	yes	complemented
20	C₂×C₅×C₃⋊C₁₆	—	2	1	no	—
21	#71437	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂×C₅×#131	—	3	1	no	—
23	C₂×C₅×M₃₂	—	3	1	no	—
24	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic, complemented
25	C₅×#350	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₅×#67	—	2	1	no	—
28	C₅×#235	—	2	1	no	—
29	C₅×#235	—	2	1	no	—
30	C₅×#235	C₈	1	2	yes	—
31	C₅×#235	C₂×C₄	1	2	yes	—
32	C₂×C₅×D₁₂	C₈	1	1	yes	characteristic
33	C₂×C₄×C₅×D₃	—	2	1	no	—
34	C₂×C₅×Dic₆	C₈	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
40	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₅×#208	—	2	1	no	—
42	C₅×#208	C₂×C₄	1	2	yes	—
43	C₂×C₅×C₃⋊C₈	—	2	1	no	—
44	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
45	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
46	C₅×D₃×C₈	—	2	2	no	—
47	C₅×D₃×C₈	—	2	2	no	—
48	C₃×C₅×C₁₆	—	2	2	no	—
49	C₅×C₃⋊C₁₆	—	2	2	no	—
50	C₂×#901	C₂×C₅	1	1	yes	characteristic
51	C₂×C₃×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
52	C₂×#812	C₂×C₅	1	1	yes	characteristic, complemented
53	#4508	C₂×C₅	1	4	yes	complemented
54	C₂×C₅×#41	—	3	1	no	—
55	C₅×#131	—	3	2	no	—
56	C₅×#131	—	3	2	no	—
57	C₅×#131	—	6	1	no	—
58	C₅×#131	—	6	1	no	—
59	C₂×C₅×M₁₆	—	3	1	no	—
60	C₂×C₅×M₁₆	—	6	1	no	—
61	C₂²×C₅×C₈	—	6	1	no	—
62	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
63	C₅×M₃₂	—	2	1	no	—
64	C₅×M₃₂	—	2	1	no	—
65	C₅×M₃₂	—	3	2	no	—
66	C₅×M₃₂	—	6	1	no	—
67	C₂×C₅×C₁₆	—	6	1	no	—
68	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	complemented
69	#2117	—	3	1	no	Sylow
70	C₂³×C₃×C₅	M₁₆	1	1	yes	characteristic
71	C₂²×C₅×D₃	—	2	1	no	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	M₁₆	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₅×#67	—	2	2	no	—
79	C₅×#67	—	2	2	no	—
80	C₂×C₅×Dic₃	—	2	1	no	—
81	C₅×D₁₂	—	2	1	no	—
82	C₅×D₁₂	C₂×C₈	1	2	yes	—
83	C₄×C₅×D₃	—	2	2	no	—
84	C₄×C₅×D₃	—	2	2	no	—
85	C₅×Dic₆	—	2	1	no	—
86	C₅×Dic₆	C₂×C₈	1	2	yes	—
87	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₃×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
90	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
91	C₅×C₃⋊C₈	—	2	2	no	—
92	C₂×#235	C₄×C₅	1	1	yes	characteristic
93	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
94	C₂×#208	C₄×C₅	1	1	yes	characteristic
95	#901	—	2	1	no	—
96	#901	—	2	1	no	—
97	#901	C₄×C₅	1	2	yes	—
98	#901	C₂²×C₅	1	2	yes	—
99	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
100	C₂×D₃×C₈	—	2	1	no	—
101	C₃×M₃₂	—	2	1	no	—
102	C₃×M₃₂	—	2	1	no	—
103	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	complemented
104	#812	—	2	1	no	—
105	#812	C₂²×C₅	1	2	yes	complemented
106	C₂×C₃⋊C₁₆	—	2	1	no	—
107	C₅×#41	—	3	2	no	—
108	C₅×#41	—	3	2	no	—
109	C₅×#41	—	6	1	no	—
110	C₅×#41	—	6	1	no	—
111	C₂×C₅×Q₈	—	3	1	no	—
112	C₂×C₅×D₄	—	3	1	no	—
113	C₂×C₅×D₄	—	6	1	no	—
114	C₂²×C₄×C₅	—	6	1	no	—
115	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
116	C₅×M₁₆	—	3	2	no	—
117	C₅×M₁₆	—	6	1	no	—
118	C₅×M₁₆	—	6	2	no	—
119	C₅×M₁₆	—	6	2	no	—
120	C₂×C₅×C₈	—	6	1	no	—
121	C₂×C₅×C₈	—	6	2	no	—
122	C₂×C₅×C₈	—	6	2	no	—
123	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
124	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
125	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
126	C₂×C₅×C₈	#67	1	1	yes	characteristic
127	C₂×C₅×C₈	#67	1	1	yes	characteristic
128	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
129	C₅×C₁₆	—	2	2	no	—
130	C₅×C₁₆	—	6	2	no	—
131	C₂×#131	—	3	1	no	—
132	C₂×M₃₂	—	3	1	no	—
133	C₂×M₃₂	C₅×D₃	1	1	yes	characteristic, complemented
134	#350	—	3	4	no	—
135	C₂²×C₃×C₅	—	4	1	no	—
136	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
137	C₂²×C₃×C₅	C₂²⋊C₈	1	1	yes	characteristic
138	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
139	C₂×C₅×D₃	—	2	2	no	—
140	C₂×C₅×D₃	—	4	2	no	—
141	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
143	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
145	C₅×Dic₃	—	2	2	no	—
146	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
147	C₂×#67	—	2	1	no	—
148	#235	—	2	1	no	—
149	#235	—	2	1	no	—
150	#235	C₅×C₈	1	2	yes	—
151	#235	C₂×C₄×C₅	1	2	yes	—
152	C₂×D₁₂	C₅×C₈	1	1	yes	characteristic
153	C₂×C₄×D₃	—	2	1	no	—
154	C₂×Dic₆	C₅×C₈	1	1	yes	characteristic
155	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
156	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
157	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
158	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
159	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
160	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
161	#208	—	2	1	no	—
162	#208	C₂×C₄×C₅	1	2	yes	—
163	C₂×C₃⋊C₈	—	2	1	no	—
164	ΓL(1,𝔽₂₅)	—	2	2	no	—
165	ΓL(1,𝔽₂₅)	—	2	2	no	—
166	D₃×C₈	—	2	2	no	—
167	D₃×C₈	—	2	2	no	—
168	C₃×C₁₆	—	2	2	no	—
169	C₃⋊C₁₆	—	2	2	no	—
170	C₂³×C₅	—	6	1	no	—
171	C₂³×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
172	C₅×Q₈	—	3	2	no	—
173	C₅×Q₈	—	6	1	no	—
174	C₅×D₄	—	3	2	no	—
175	C₅×D₄	—	6	2	no	—
176	C₅×D₄	—	6	2	no	—
177	C₅×D₄	—	6	1	no	—
178	C₂×C₄×C₅	—	6	1	no	—
179	C₂×C₄×C₅	—	6	2	no	—
180	C₂×C₄×C₅	—	6	2	no	—
181	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
182	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
183	C₂×C₄×C₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
184	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
185	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
186	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
187	C₅×C₈	—	6	2	no	—
188	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
189	C₅×C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
190	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
191	C₅×C₈	C₂×#67	1	1	yes	characteristic
192	C₂×#41	—	3	1	no	—
193	#131	—	3	2	no	—
194	#131	—	3	2	no	—
195	#131	—	6	1	no	—
196	#131	—	6	1	no	—
197	C₂×M₁₆	—	3	1	no	—
198	C₂×M₁₆	—	6	1	no	—
199	C₂²×C₈	—	6	1	no	—
200	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
201	M₃₂	—	2	1	no	—
202	M₃₂	—	2	1	no	—
203	M₃₂	—	3	2	no	—
204	M₃₂	—	6	1	no	—
205	C₂×C₁₆	—	6	1	no	—
206	C₂×C₁₆	C₂×C₅×D₃	1	2	yes	complemented
207	C₂×C₃×C₅	—	2	1	no	—
208	C₂×C₃×C₅	—	2	1	no	—
209	C₂×C₃×C₅	—	2	1	no	—
210	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
211	C₅×D₃	—	4	2	no	—
212	C₂³×C₃	C₅×M₁₆	1	1	yes	characteristic
213	C₂²×D₃	—	2	1	no	—
214	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
218	C₂×C₃×C₄	C₅×M₁₆	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
220	#67	—	2	2	no	—
221	#67	—	2	2	no	—
222	C₂×Dic₃	—	2	1	no	—
223	D₁₂	—	2	1	no	—
224	D₁₂	C₂×C₅×C₈	1	2	yes	—
225	C₄×D₃	—	2	2	no	—
226	C₄×D₃	—	2	2	no	—
227	Dic₆	—	2	1	no	—
228	Dic₆	C₂×C₅×C₈	1	2	yes	—
229	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
230	C₃×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
231	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
232	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
233	C₃⋊C₈	—	2	2	no	—
234	C₂²×C₅	—	4	1	no	—
235	C₂²×C₅	—	6	2	no	—
236	C₂²×C₅	—	12	2	no	—
237	C₂²×C₅	(C₂×D₃)⋊C₈	1	1	yes	characteristic
238	C₂²×C₅	(C₂×D₃)⋊C₈	1	1	yes	characteristic
239	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
240	C₄×C₅	—	6	2	no	—
241	C₄×C₅	(C₂×D₃)⋊C₈	1	1	yes	characteristic
242	C₄×C₅	(C₂×D₃)⋊C₈	1	1	yes	characteristic
243	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
244	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
245	#41	—	3	2	no	—
246	#41	—	3	2	no	—
247	#41	—	6	1	no	—
248	#41	—	6	1	no	—
249	C₂×Q₈	—	3	1	no	—
250	C₂×D₄	—	3	1	no	—
251	C₂×D₄	—	6	1	no	—
252	C₂²×C₄	—	6	1	no	—
253	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
254	M₁₆	—	3	2	no	—
255	M₁₆	—	6	1	no	—
256	M₁₆	—	6	2	no	—
257	M₁₆	—	6	2	no	—
258	C₂×C₈	—	6	1	no	—
259	C₂×C₈	—	6	2	no	—
260	C₂×C₈	—	6	2	no	—
261	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
262	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
263	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
264	C₂×C₈	C₅×#67	1	1	yes	characteristic
265	C₂×C₈	C₅×#67	1	1	yes	characteristic
266	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
267	C₁₆	—	2	2	no	—
268	C₁₆	—	6	2	no	—
269	C₃×C₅	#2117	1	1	yes	characteristic, complemented, Hall
270	C₂²×C₃	—	4	1	no	—
271	C₂²×C₃	C₅×C₂²⋊C₈	1	1	yes	characteristic
272	C₂²×C₃	C₅×C₂²⋊C₈	1	1	yes	characteristic
273	C₂²×C₃	C₂×C₅×M₁₆	1	1	yes	characteristic
274	C₂×D₃	—	2	2	no	—
275	C₂×D₃	—	4	2	no	—
276	C₃×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
277	C₃×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
278	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
279	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
280	Dic₃	—	2	2	no	—
281	C₂×C₅	—	2	1	no	—
282	C₂×C₅	—	2	1	no	—
283	C₂×C₅	—	2	1	no	—
284	C₂×C₅	—	12	2	no	—
285	C₂×C₅	C₂×(C₂×D₃)⋊C₈	1	1	yes	characteristic
286	C₂³	—	6	1	no	—
287	C₂³	C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
288	Q₈	—	3	2	no	—
289	Q₈	—	6	1	no	—
290	D₄	—	3	2	no	—
291	D₄	—	6	2	no	—
292	D₄	—	6	2	no	—
293	D₄	—	6	1	no	—
294	C₂×C₄	—	6	1	no	—
295	C₂×C₄	—	6	2	no	—
296	C₂×C₄	—	6	2	no	—
297	C₂×C₄	C₅×D₃×C₈	1	1	yes	characteristic
298	C₂×C₄	C₅×D₃×C₈	1	1	yes	characteristic
299	C₂×C₄	C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
300	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
301	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
302	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
303	C₈	—	6	2	no	—
304	C₈	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
305	C₈	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
306	C₈	C₂×C₅×D₁₂	1	1	yes	characteristic
307	C₈	C₂×C₅×#67	1	1	yes	characteristic
308	C₂×C₃	—	2	1	no	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
312	D₃	—	4	2	no	—
313	C₅	#71437	1	1	yes	characteristic, direct product factor, Sylow
314	C₂²	—	4	1	no	—
315	C₂²	—	6	2	no	—
316	C₂²	—	12	2	no	—
317	C₂²	C₅×(C₂×D₃)⋊C₈	1	1	yes	characteristic
318	C₂²	C₅×(C₂×D₃)⋊C₈	1	1	yes	characteristic
319	C₂²	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
320	C₄	—	6	2	no	—
321	C₄	C₅×(C₂×D₃)⋊C₈	1	1	yes	characteristic
322	C₄	C₅×(C₂×D₃)⋊C₈	1	1	yes	characteristic
323	C₄	C₂×C₅×D₃×C₈	1	1	yes	characteristic
324	C₄	C₂×C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
325	C₃	C₅×#2117	1	1	yes	characteristic, complemented, Sylow
326	C₂	—	2	1	no	—
327	C₂	—	2	1	no	—
328	C₂	—	2	1	no	—
329	C₂	—	12	2	no	—
330	C₂	C₂×C₅×(C₂×D₃)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₅×C₈	C₂×D₁₂
commutator	C₃×C₄	C₂²×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂×C₃×C₅	C₂×C₂²⋊C₈
prime radical	C₂×C₅×#235	C₄
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂×C₅×M₃₂	D₃
hypocenter	C₃	C₅×#2117
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₅×C₈ ◁ C₂²×C₅×C₈ ◁ C₂×C₅×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₂×C₅×#812	C₂	semidirect
4. C₅×#4508	C₂	semidirect
17. C₂×C₃×C₅×C₁₆	C₂²	semidirect
19. C₅×#812	C₂²	semidirect
21. #71437	C₅	direct (N × H)
24. C₂×C₅×M₃₂	D₃	semidirect
51. C₂×C₃×M₃₂	C₂×C₅	semidirect
52. C₂×#812	C₂×C₅	semidirect
53. #4508	C₂×C₅	semidirect
68. C₂×C₅×C₁₆	C₂×D₃	semidirect
103. C₂×C₃×C₁₆	C₂²×C₅	semidirect
105. #812	C₂²×C₅	semidirect
133. C₂×M₃₂	C₅×D₃	semidirect
206. C₂×C₁₆	C₂×C₅×D₃	semidirect
269. C₃×C₅	#2117	semidirect
313. C₅	#71437	direct (N × H)
325. C₃	C₅×#2117	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#901	2
C₂×C₃×C₅×M₃₂	2
C₂×C₅×#812	2
C₅×#4508	2
C₅×#2117	3
#71437	5

Subgroups — C5×#71437 #14313002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71437 #14313002