Subgroups — C₅×#68531 #14285753

Order 1920.

332 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4837	C₂	1	1	yes	maximal, characteristic
2	C₅×#4835	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4480	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4479	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4458	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1569	—	3	1	no	maximal, Hall
9	C₂×C₅×ΓL(1,𝔽₂₅)	—	2	1	no	—
10	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
11	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
12	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
17	C₅×#815	—	2	1	no	—
18	C₅×#815	C₄	1	1	yes	characteristic
19	C₅×#815	C₄	1	1	yes	characteristic
20	C₅×Dic₃⋊C₈	—	2	1	no	—
21	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#68531	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#432	—	3	1	no	—
26	C₅×#423	—	3	1	no	—
27	C₄×C₅×M₁₆	—	3	1	no	—
28	C₅×#335	—	3	1	no	—
29	C₅×#335	—	3	1	no	—
30	C₅×#334	—	3	1	no	—
31	C₅×#334	D₃	1	1	yes	characteristic, complemented
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
34	C₂×C₄×C₅×D₃	Q₈	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
41	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
42	C₄×C₅×Dic₃	Q₈	1	1	yes	characteristic
43	C₂×C₅×C₃⋊C₈	—	2	1	no	—
44	C₂×C₅×C₃⋊C₈	—	2	1	no	—
45	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
48	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
49	C₅×ΓL(1,𝔽₂₅)	—	4	2	no	—
50	#4837	C₂×C₅	1	1	yes	characteristic
51	#4835	C₂×C₅	1	1	yes	characteristic
52	C₄×ΓL(1,𝔽₂₅)	C₂×C₅	1	1	yes	characteristic
53	C₃×#334	C₂×C₅	1	1	yes	characteristic, complemented
54	#4480	C₂×C₅	1	1	yes	characteristic, complemented
55	#4479	C₂×C₅	1	1	yes	characteristic, complemented
56	#4458	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₅×M₁₆	—	3	1	no	—
58	C₂×C₅×M₁₆	—	3	1	no	—
59	C₂×C₅×M₁₆	—	6	1	no	—
60	C₂×C₄²×C₅	—	3	1	no	—
61	C₅×C₄⋊C₈	—	3	1	no	—
62	C₅×C₄⋊C₈	—	3	1	no	—
63	C₅×C₄⋊C₈	—	6	1	no	—
64	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
65	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
66	C₅×C₂²⋊C₈	—	6	1	no	—
67	C₅×#97	—	3	1	no	—
68	C₅×#97	—	3	1	no	—
69	C₅×#97	—	6	1	no	—
70	C₄×C₅×C₈	—	3	1	no	—
71	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
72	#1569	—	3	1	no	Sylow
73	C₂²×C₅×D₃	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₅×Dic₃	C₄⋊C₄	1	1	yes	characteristic
80	C₄×C₅×D₃	—	2	1	no	—
81	C₄×C₅×D₃	—	2	1	no	—
82	C₄×C₅×D₃	—	2	1	no	—
83	C₄×C₅×D₃	—	2	1	no	—
84	C₄×C₅×D₃	—	4	1	no	—
85	C₃×C₅×C₈	—	2	1	no	—
86	C₃×C₅×C₈	—	4	1	no	—
87	C₃×C₅×C₈	—	4	1	no	—
88	C₃×C₅×C₈	M₁₆	1	2	yes	complemented
89	C₅×C₃⋊C₈	—	2	1	no	—
90	C₅×C₃⋊C₈	—	4	1	no	—
91	C₅×C₃⋊C₈	—	4	1	no	—
92	C₅×C₃⋊C₈	M₁₆	1	2	yes	complemented
93	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
94	C₂×ΓL(1,𝔽₂₅)	C₄×C₅	1	1	yes	characteristic
95	C₂×ΓL(1,𝔽₂₅)	C₄×C₅	1	1	yes	characteristic
96	C₄²×D₃	C₂²×C₅	1	1	yes	characteristic
97	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
98	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
99	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
100	(C₂×D₃)⋊C₈	—	2	1	no	—
101	#815	—	2	1	no	—
102	#815	C₄×C₅	1	1	yes	characteristic
103	#815	C₄×C₅	1	1	yes	characteristic
104	Dic₃⋊C₈	—	2	1	no	—
105	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
106	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
107	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
108	C₂²×C₄×C₅	—	3	1	no	—
109	C₂²×C₄×C₅	—	3	1	no	—
110	C₂²×C₄×C₅	—	3	1	no	—
111	C₅×M₁₆	—	6	2	no	—
112	C₅×M₁₆	—	6	2	no	—
113	C₅×M₁₆	—	12	2	no	—
114	C₂×C₅×C₈	—	2	1	no	—
115	C₂×C₅×C₈	—	2	1	no	—
116	C₂×C₅×C₈	—	3	1	no	—
117	C₂×C₅×C₈	—	3	1	no	—
118	C₂×C₅×C₈	—	6	1	no	—
119	C₂×C₅×C₈	—	6	1	no	—
120	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
121	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
122	C₄²×C₅	—	3	1	no	—
123	C₄²×C₅	—	3	1	no	—
124	C₄²×C₅	—	3	1	no	—
125	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
126	#432	—	3	1	no	—
127	#423	—	3	1	no	—
128	C₄×M₁₆	—	3	1	no	—
129	#335	—	3	1	no	—
130	#335	—	3	1	no	—
131	#334	—	3	1	no	—
132	#334	C₅×D₃	1	1	yes	characteristic, complemented
133	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₅×D₃	—	2	1	no	—
135	C₂×C₅×D₃	—	2	1	no	—
136	C₂×C₅×D₃	—	2	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
139	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
140	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₃×C₄×C₅	#137	1	1	yes	characteristic
142	C₅×Dic₃	—	2	1	no	—
143	C₅×Dic₃	—	2	1	no	—
144	C₅×Dic₃	—	2	1	no	—
145	C₂×C₄×D₃	C₂×C₄×C₅	1	1	yes	characteristic
146	C₂×C₄×D₃	C₅×D₄	1	1	yes	characteristic
147	C₂×C₄×D₃	C₅×Q₈	1	1	yes	characteristic
148	C₂×C₃×C₈	—	2	1	no	—
149	C₂×C₃×C₈	—	2	1	no	—
150	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
151	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
152	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
153	C₄×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
154	C₄×Dic₃	C₅×D₄	1	1	yes	characteristic
155	C₄×Dic₃	C₅×Q₈	1	1	yes	characteristic
156	C₂×C₃⋊C₈	—	2	1	no	—
157	C₂×C₃⋊C₈	—	2	1	no	—
158	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
159	C₂×C₃⋊C₈	C₂×C₄×C₅	1	1	yes	characteristic
160	ΓL(1,𝔽₂₅)	—	2	2	no	—
161	ΓL(1,𝔽₂₅)	—	2	2	no	—
162	ΓL(1,𝔽₂₅)	—	4	2	no	—
163	C₂³×C₅	—	3	1	no	—
164	C₂×C₄×C₅	—	3	1	no	—
165	C₂×C₄×C₅	—	3	1	no	—
166	C₂×C₄×C₅	—	3	1	no	—
167	C₂×C₄×C₅	—	6	1	no	—
168	C₂×C₄×C₅	—	6	1	no	—
169	C₂×C₄×C₅	—	6	1	no	—
170	C₂×C₄×C₅	—	6	1	no	—
171	C₂×C₄×C₅	—	12	1	no	—
172	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
173	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	—	3	2	no	—
177	C₅×C₈	—	4	1	no	—
178	C₅×C₈	—	4	1	no	—
179	C₅×C₈	—	6	1	no	—
180	C₅×C₈	—	12	1	no	—
181	C₅×C₈	—	12	1	no	—
182	C₅×C₈	ΓL(1,𝔽₂₅)	1	2	yes	complemented
183	C₂×M₁₆	—	3	1	no	—
184	C₂×M₁₆	—	3	1	no	—
185	C₂×M₁₆	—	6	1	no	—
186	C₂×C₄²	—	3	1	no	—
187	C₄⋊C₈	—	3	1	no	—
188	C₄⋊C₈	—	3	1	no	—
189	C₄⋊C₈	—	6	1	no	—
190	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
191	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
192	C₂²⋊C₈	—	6	1	no	—
193	#97	—	3	1	no	—
194	#97	—	3	1	no	—
195	#97	—	6	1	no	—
196	C₄×C₈	—	3	1	no	—
197	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
198	C₂×C₃×C₅	#423	1	1	yes	characteristic
199	C₂×C₃×C₅	#428	1	1	yes	characteristic
200	C₂×C₃×C₅	#430	1	1	yes	characteristic
201	C₅×D₃	—	4	1	no	—
202	C₂²×D₃	C₅×C₄⋊C₄	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
206	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
207	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
208	C₂×Dic₃	C₅×C₄⋊C₄	1	1	yes	characteristic
209	C₄×D₃	—	2	1	no	—
210	C₄×D₃	—	2	1	no	—
211	C₄×D₃	—	2	1	no	—
212	C₄×D₃	—	2	1	no	—
213	C₄×D₃	—	4	1	no	—
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	4	1	no	—
216	C₃×C₈	—	4	1	no	—
217	C₃×C₈	C₅×M₁₆	1	2	yes	complemented
218	C₃⋊C₈	—	2	1	no	—
219	C₃⋊C₈	—	4	1	no	—
220	C₃⋊C₈	—	4	1	no	—
221	C₃⋊C₈	C₅×M₁₆	1	2	yes	complemented
222	C₂²×C₅	—	6	1	no	—
223	C₂²×C₅	—	6	1	no	—
224	C₂²×C₅	—	6	1	no	—
225	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	6	1	no	—
228	C₄×C₅	—	6	1	no	—
229	C₄×C₅	—	6	1	no	—
230	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
231	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
232	C₄×C₅	#914	1	1	yes	characteristic
233	C₄×C₅	#915	1	1	yes	characteristic
234	C₂²×C₄	—	3	1	no	—
235	C₂²×C₄	—	3	1	no	—
236	C₂²×C₄	—	3	1	no	—
237	M₁₆	—	6	2	no	—
238	M₁₆	—	6	2	no	—
239	M₁₆	—	12	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	3	1	no	—
243	C₂×C₈	—	3	1	no	—
244	C₂×C₈	—	6	1	no	—
245	C₂×C₈	—	6	1	no	—
246	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
247	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
248	C₄²	—	3	1	no	—
249	C₄²	—	3	1	no	—
250	C₄²	—	3	1	no	—
251	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
252	C₃×C₅	#1569	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
254	C₂×D₃	—	2	1	no	—
255	C₂×D₃	—	2	1	no	—
256	C₂×D₃	—	2	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
259	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
260	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₃×C₄	C₅×#137	1	1	yes	characteristic
262	Dic₃	—	2	1	no	—
263	Dic₃	—	2	1	no	—
264	Dic₃	—	2	1	no	—
265	C₂×C₅	—	12	1	no	—
266	C₂×C₅	#4837	1	1	yes	characteristic
267	C₂×C₅	#4861	1	1	yes	characteristic
268	C₂×C₅	#4893	1	1	yes	characteristic
269	C₂³	—	3	1	no	—
270	C₂×C₄	—	3	1	no	—
271	C₂×C₄	—	3	1	no	—
272	C₂×C₄	—	3	1	no	—
273	C₂×C₄	—	6	1	no	—
274	C₂×C₄	—	6	1	no	—
275	C₂×C₄	—	6	1	no	—
276	C₂×C₄	—	6	1	no	—
277	C₂×C₄	—	12	1	no	—
278	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
279	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
280	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
281	C₈	—	2	1	no	—
282	C₈	—	3	2	no	—
283	C₈	—	4	1	no	—
284	C₈	—	4	1	no	—
285	C₈	—	6	1	no	—
286	C₈	—	12	1	no	—
287	C₈	—	12	1	no	—
288	C₈	C₅×ΓL(1,𝔽₂₅)	1	2	yes	complemented
289	C₂×C₃	C₅×#423	1	1	yes	characteristic
290	C₂×C₃	C₅×#428	1	1	yes	characteristic
291	C₂×C₃	C₅×#430	1	1	yes	characteristic
292	D₃	—	4	1	no	—
293	C₅	#68531	1	1	yes	characteristic, direct product factor, Sylow
294	C₂²	—	6	1	no	—
295	C₂²	—	6	1	no	—
296	C₂²	—	6	1	no	—
297	C₂²	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	6	1	no	—
300	C₄	—	6	1	no	—
301	C₄	—	6	1	no	—
302	C₄	C₂×C₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
303	C₄	C₅×D₃×M₁₆	1	1	yes	characteristic
304	C₄	C₅×#914	1	1	yes	characteristic
305	C₄	C₅×#915	1	1	yes	characteristic
306	C₃	C₅×#1569	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	12	1	no	—
308	C₂	C₅×#4837	1	1	yes	characteristic
309	C₂	C₅×#4861	1	1	yes	characteristic
310	C₂	C₅×#4893	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#334	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₅×D₃	C₄⋊C₄
cosocle	C₃×C₄²	C₂³×C₅
hypercenter	C₅×#334	D₃
hypocenter	C₃	C₅×#1569
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₄²×C₅ ◁ C₅×#334

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#334	C₂	semidirect
5. C₅×#4480	C₂	semidirect
6. C₅×#4479	C₂	semidirect
7. C₅×#4458	C₂	semidirect
24. #68531	C₅	direct (N × H)
31. C₅×#334	D₃	semidirect
53. C₃×#334	C₂×C₅	semidirect
54. #4480	C₂×C₅	semidirect
55. #4479	C₂×C₅	semidirect
56. #4458	C₂×C₅	semidirect
88. C₃×C₅×C₈	M₁₆	semidirect
92. C₅×C₃⋊C₈	M₁₆	semidirect
132. #334	C₅×D₃	semidirect
182. C₅×C₈	ΓL(1,𝔽₂₅)	semidirect
217. C₃×C₈	C₅×M₁₆	semidirect
221. C₃⋊C₈	C₅×M₁₆	semidirect
252. C₃×C₅	#1569	semidirect
288. C₈	C₅×ΓL(1,𝔽₂₅)	semidirect
293. C₅	#68531	direct (N × H)
306. C₃	C₅×#1569	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4837	2
C₅×#4835	2
C₄×C₅×ΓL(1,𝔽₂₅)	2
C₃×C₅×#334	2
C₅×#4480	2
C₅×#4479	2
C₅×#4458	2
C₅×#1569	3
#68531	5

Subgroups — C5×#68531 #14285753

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68531 #14285753