Subgroups — C₅×#68448 #14292363

Order 1920.

364 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5087	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5085	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4843	C₂	1	1	yes	maximal, characteristic
4	C₅×#4838	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#332	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4485	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1685	—	3	1	no	maximal, Hall
9	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#215	—	2	1	no	—
11	C₂×C₅×#215	—	2	1	no	—
12	C₂×C₅×D₂₄	—	2	1	no	—
13	C₂×C₅×ΓL(1,𝔽₂₅)	—	2	1	no	—
14	C₅×#874	C₂²	1	1	yes	characteristic
15	C₄×C₅×D₁₂	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₅×#821	—	2	1	no	—
19	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
20	C₅×#809	—	2	1	no	—
21	C₅×#807	—	2	1	no	—
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#68448	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#492	—	3	1	no	—
26	C₅×#474	—	3	1	no	—
27	C₅×#459	—	3	1	no	—
28	C₅×#436	—	3	1	no	—
29	C₅×#335	—	3	1	no	—
30	C₅×#332	D₃	1	1	yes	characteristic, complemented
31	C₅×#331	—	3	1	no	—
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₅×D₁₂	—	2	1	no	—
34	C₂×C₅×D₁₂	—	2	1	no	—
35	C₂×C₅×D₁₂	—	2	1	no	—
36	C₂×C₅×D₁₂	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	—	2	1	no	—
38	C₂×C₃×C₅×C₈	—	2	1	no	—
39	C₂×C₃×C₅×C₈	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	—	2	1	no	—
41	C₃×C₅×C₄⋊C₄	—	2	1	no	—
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×#215	—	4	2	no	—
44	C₅×#215	—	4	2	no	—
45	C₅×(C₂×D₃)⋊C₄	—	2	1	no	—
46	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
47	C₂×C₅×C₃⋊C₈	—	2	1	no	—
48	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
49	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
50	C₅×D₂₄	—	4	2	no	—
51	C₅×ΓL(1,𝔽₂₅)	—	4	2	no	—
52	#5087	C₂×C₅	1	1	yes	characteristic, complemented
53	#5085	C₂×C₅	1	1	yes	characteristic, complemented
54	#4843	C₂×C₅	1	1	yes	characteristic
55	#4838	C₂×C₅	1	1	yes	characteristic, complemented
56	C₃×#332	C₂×C₅	1	1	yes	characteristic, complemented
57	#4485	C₂×C₅	1	1	yes	characteristic, complemented
58	#4479	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂×C₅×SD₁₆	—	6	1	no	—
60	C₂×C₅×SD₁₆	—	6	1	no	—
61	C₂×C₅×D₈	—	6	1	no	—
62	C₂×C₅×M₁₆	—	6	1	no	—
63	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
64	C₅×#127	—	3	1	no	—
65	C₄×C₅×D₄	—	3	1	no	—
66	C₅×#107	—	6	1	no	—
67	C₅×C₄⋊C₈	—	3	1	no	—
68	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
69	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
70	C₅×#102	—	6	1	no	—
71	C₅×#102	—	6	1	no	—
72	C₅×C₂²⋊C₈	—	6	1	no	—
73	C₄×C₅×C₈	—	3	1	no	—
74	#1685	—	3	1	no	Sylow
75	C₂²×C₅×D₃	—	2	1	no	—
76	C₂²×C₅×D₃	—	4	1	no	—
77	C₃×C₅×Q₈	—	4	1	no	—
78	C₃×C₅×Q₈	—	4	1	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
83	C₂×C₅×Dic₃	—	2	1	no	—
84	C₅×D₁₂	—	2	1	no	—
85	C₅×D₁₂	—	2	1	no	—
86	C₅×D₁₂	—	4	1	no	—
87	C₅×D₁₂	—	4	1	no	—
88	C₅×D₁₂	—	4	1	no	—
89	C₅×D₁₂	—	4	1	no	—
90	C₅×D₁₂	—	8	1	no	—
91	C₄×C₅×D₃	—	4	1	no	—
92	C₄×C₅×D₃	—	4	1	no	—
93	C₃×C₅×C₈	—	4	1	no	—
94	C₃×C₅×C₈	—	4	1	no	—
95	C₅×C₃⋊C₈	—	2	1	no	—
96	C₅×C₃⋊C₈	—	4	1	no	—
97	C₅×C₃⋊C₈	D₈	1	2	yes	complemented
98	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
99	C₂×#215	—	2	1	no	—
100	C₂×#215	—	2	1	no	—
101	C₂×D₂₄	—	2	1	no	—
102	C₂×ΓL(1,𝔽₂₅)	—	2	1	no	—
103	#874	C₂²×C₅	1	1	yes	characteristic
104	C₄×D₁₂	C₂²×C₅	1	1	yes	characteristic
105	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
106	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
107	#821	—	2	1	no	—
108	(C₂×D₃)⋊C₈	—	2	1	no	—
109	#809	—	2	1	no	—
110	#807	—	2	1	no	—
111	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
112	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
113	C₂×C₅×Q₈	—	2	1	no	—
114	C₂×C₅×D₄	—	3	1	no	—
115	C₂×C₅×D₄	—	6	1	no	—
116	C₂×C₅×D₄	—	6	1	no	—
117	C₂×C₅×D₄	—	6	1	no	—
118	C₂²×C₄×C₅	—	6	1	no	—
119	C₅×SD₁₆	—	12	2	no	—
120	C₅×SD₁₆	—	12	2	no	—
121	C₅×D₈	—	12	2	no	—
122	C₅×M₁₆	—	12	2	no	—
123	C₂×C₅×C₈	—	2	1	no	—
124	C₂×C₅×C₈	—	2	1	no	—
125	C₂×C₅×C₈	—	3	1	no	—
126	C₂×C₅×C₈	—	3	1	no	—
127	C₂×C₅×C₈	—	6	1	no	—
128	C₅×C₄⋊C₄	—	2	1	no	—
129	C₅×C₄⋊C₄	—	2	1	no	—
130	C₅×C₄⋊C₄	—	3	1	no	—
131	C₅×C₂²⋊C₄	—	6	1	no	—
132	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
133	#492	—	3	1	no	—
134	#474	—	3	1	no	—
135	#459	—	3	1	no	—
136	#436	—	3	1	no	—
137	#335	—	3	1	no	—
138	#332	C₅×D₃	1	1	yes	characteristic, complemented
139	#331	—	3	1	no	—
140	C₂²×C₃×C₅	#121	1	1	yes	characteristic
141	C₂×C₅×D₃	—	4	1	no	—
142	C₂×C₅×D₃	—	4	1	no	—
143	C₂×C₅×D₃	—	4	1	no	—
144	C₂×C₅×D₃	—	8	1	no	—
145	C₂×C₅×D₃	—	8	1	no	—
146	C₂×C₅×D₃	—	8	1	no	—
147	C₃×C₄×C₅	—	2	1	no	—
148	C₃×C₄×C₅	—	8	1	no	—
149	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
150	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₃×C₄×C₅	#137	1	1	yes	characteristic
153	C₅×Dic₃	—	4	1	no	—
154	C₂×C₃×Q₈	—	2	1	no	—
155	C₂×D₁₂	—	2	1	no	—
156	C₂×D₁₂	—	2	1	no	—
157	C₂×D₁₂	—	2	1	no	—
158	C₂×D₁₂	C₅×D₄	1	1	yes	characteristic
159	C₂×C₄×D₃	—	2	1	no	—
160	C₂×C₃×C₈	—	2	1	no	—
161	C₂×C₃×C₈	—	2	1	no	—
162	C₃×C₄⋊C₄	—	2	1	no	—
163	C₃×C₄⋊C₄	—	2	1	no	—
164	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
165	#215	—	4	2	no	—
166	#215	—	4	2	no	—
167	(C₂×D₃)⋊C₄	—	2	1	no	—
168	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
169	C₂×C₃⋊C₈	—	2	1	no	—
170	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
171	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
172	D₂₄	—	4	2	no	—
173	ΓL(1,𝔽₂₅)	—	4	2	no	—
174	C₂³×C₅	—	6	1	no	—
175	C₂³×C₅	—	12	1	no	—
176	C₅×Q₈	—	4	1	no	—
177	C₅×Q₈	—	4	1	no	—
178	C₅×D₄	—	6	1	no	—
179	C₅×D₄	—	6	1	no	—
180	C₅×D₄	—	12	1	no	—
181	C₅×D₄	—	12	1	no	—
182	C₅×D₄	—	12	1	no	—
183	C₅×D₄	—	12	1	no	—
184	C₅×D₄	—	24	1	no	—
185	C₂×C₄×C₅	—	4	1	no	—
186	C₂×C₄×C₅	—	6	1	no	—
187	C₂×C₄×C₅	—	12	1	no	—
188	C₂×C₄×C₅	—	12	1	no	—
189	C₂×C₄×C₅	#235	1	1	yes	characteristic
190	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
192	C₅×C₈	—	3	2	no	—
193	C₅×C₈	—	4	1	no	—
194	C₅×C₈	—	4	1	no	—
195	C₅×C₈	—	6	1	no	—
196	C₅×C₈	—	12	1	no	—
197	C₂×SD₁₆	—	6	1	no	—
198	C₂×SD₁₆	—	6	1	no	—
199	C₂×D₈	—	6	1	no	—
200	C₂×M₁₆	—	6	1	no	—
201	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
202	#127	—	3	1	no	—
203	C₄×D₄	—	3	1	no	—
204	#107	—	6	1	no	—
205	C₄⋊C₈	—	3	1	no	—
206	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
207	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
208	#102	—	6	1	no	—
209	#102	—	6	1	no	—
210	C₂²⋊C₈	—	6	1	no	—
211	C₄×C₈	—	3	1	no	—
212	C₂×C₃×C₅	#459	1	1	yes	characteristic
213	C₂×C₃×C₅	#468	1	1	yes	characteristic
214	C₂×C₃×C₅	#471	1	1	yes	characteristic
215	C₅×D₃	—	8	1	no	—
216	C₅×D₃	—	16	1	no	—
217	C₂²×D₃	—	2	1	no	—
218	C₂²×D₃	—	4	1	no	—
219	C₃×Q₈	—	4	1	no	—
220	C₃×Q₈	—	4	1	no	—
221	C₂×C₃×C₄	—	4	1	no	—
222	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
225	C₂×Dic₃	—	2	1	no	—
226	D₁₂	—	2	1	no	—
227	D₁₂	—	2	1	no	—
228	D₁₂	—	4	1	no	—
229	D₁₂	—	4	1	no	—
230	D₁₂	—	4	1	no	—
231	D₁₂	—	4	1	no	—
232	D₁₂	—	8	1	no	—
233	C₄×D₃	—	4	1	no	—
234	C₄×D₃	—	4	1	no	—
235	C₃×C₈	—	4	1	no	—
236	C₃×C₈	—	4	1	no	—
237	C₃⋊C₈	—	2	1	no	—
238	C₃⋊C₈	—	4	1	no	—
239	C₃⋊C₈	C₅×D₈	1	2	yes	complemented
240	C₂²×C₅	—	12	1	no	—
241	C₂²×C₅	—	12	1	no	—
242	C₂²×C₅	—	12	1	no	—
243	C₂²×C₅	—	24	1	no	—
244	C₂²×C₅	—	24	1	no	—
245	C₂²×C₅	—	24	1	no	—
246	C₂²×C₅	#884	1	1	yes	characteristic
247	C₄×C₅	—	2	1	no	—
248	C₄×C₅	—	8	1	no	—
249	C₄×C₅	—	12	1	no	—
250	C₄×C₅	D₃×D₈	1	1	yes	characteristic
251	C₄×C₅	#914	1	1	yes	characteristic
252	C₄×C₅	#914	1	1	yes	characteristic
253	C₄×C₅	#918	1	1	yes	characteristic
254	C₂×Q₈	—	2	1	no	—
255	C₂×D₄	—	3	1	no	—
256	C₂×D₄	—	6	1	no	—
257	C₂×D₄	—	6	1	no	—
258	C₂×D₄	—	6	1	no	—
259	C₂²×C₄	—	6	1	no	—
260	SD₁₆	—	12	2	no	—
261	SD₁₆	—	12	2	no	—
262	D₈	—	12	2	no	—
263	M₁₆	—	12	2	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	3	1	no	—
267	C₂×C₈	—	3	1	no	—
268	C₂×C₈	—	6	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	3	1	no	—
272	C₂²⋊C₄	—	6	1	no	—
273	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
274	C₃×C₅	#1685	1	1	yes	characteristic, complemented, Hall
275	C₂²×C₃	C₅×#121	1	1	yes	characteristic
276	C₂×D₃	—	4	1	no	—
277	C₂×D₃	—	4	1	no	—
278	C₂×D₃	—	4	1	no	—
279	C₂×D₃	—	8	1	no	—
280	C₂×D₃	—	8	1	no	—
281	C₂×D₃	—	8	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	8	1	no	—
284	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
285	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
286	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
287	C₃×C₄	C₅×#137	1	1	yes	characteristic
288	Dic₃	—	4	1	no	—
289	C₂×C₅	—	24	1	no	—
290	C₂×C₅	—	48	1	no	—
291	C₂×C₅	#4754	1	1	yes	characteristic
292	C₂×C₅	#4786	1	1	yes	characteristic
293	C₂×C₅	#4812	1	1	yes	characteristic
294	C₂³	—	6	1	no	—
295	C₂³	—	12	1	no	—
296	Q₈	—	4	1	no	—
297	Q₈	—	4	1	no	—
298	D₄	—	6	1	no	—
299	D₄	—	6	1	no	—
300	D₄	—	12	1	no	—
301	D₄	—	12	1	no	—
302	D₄	—	12	1	no	—
303	D₄	—	12	1	no	—
304	D₄	—	24	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	6	1	no	—
307	C₂×C₄	—	12	1	no	—
308	C₂×C₄	—	12	1	no	—
309	C₂×C₄	C₅×#235	1	1	yes	characteristic
310	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
311	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
312	C₈	—	3	2	no	—
313	C₈	—	4	1	no	—
314	C₈	—	4	1	no	—
315	C₈	—	6	1	no	—
316	C₈	—	12	1	no	—
317	C₂×C₃	C₅×#459	1	1	yes	characteristic
318	C₂×C₃	C₅×#468	1	1	yes	characteristic
319	C₂×C₃	C₅×#471	1	1	yes	characteristic
320	D₃	—	8	1	no	—
321	D₃	—	16	1	no	—
322	C₅	#68448	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²	—	12	1	no	—
324	C₂²	—	12	1	no	—
325	C₂²	—	12	1	no	—
326	C₂²	—	24	1	no	—
327	C₂²	—	24	1	no	—
328	C₂²	—	24	1	no	—
329	C₂²	C₅×#884	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	8	1	no	—
332	C₄	—	12	1	no	—
333	C₄	C₅×D₃×D₈	1	1	yes	characteristic
334	C₄	C₅×#914	1	1	yes	characteristic
335	C₄	C₅×#914	1	1	yes	characteristic
336	C₄	C₅×#918	1	1	yes	characteristic
337	C₃	C₅×#1685	1	1	yes	characteristic, complemented, Sylow
338	C₂	—	24	1	no	—
339	C₂	—	48	1	no	—
340	C₂	C₅×#4754	1	1	yes	characteristic
341	C₂	C₅×#4786	1	1	yes	characteristic
342	C₂	C₅×#4812	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#884
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#332	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₅×#4843	C₂
hypercenter	C₅×#332	D₃
hypocenter	C₃	C₅×#1685
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#332

Complements

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

N	H ≅ G/N	type
1. C₅×#5087	C₂	semidirect
2. C₅×#5085	C₂	semidirect
4. C₅×#4838	C₂	semidirect
5. C₃×C₅×#332	C₂	semidirect
6. C₅×#4485	C₂	semidirect
7. C₅×#4479	C₂	semidirect
24. #68448	C₅	direct (N × H)
30. C₅×#332	D₃	semidirect
52. #5087	C₂×C₅	semidirect
53. #5085	C₂×C₅	semidirect
55. #4838	C₂×C₅	semidirect
56. C₃×#332	C₂×C₅	semidirect
57. #4485	C₂×C₅	semidirect
58. #4479	C₂×C₅	semidirect
97. C₅×C₃⋊C₈	D₈	semidirect
138. #332	C₅×D₃	semidirect
239. C₃⋊C₈	C₅×D₈	semidirect
274. C₃×C₅	#1685	semidirect
322. C₅	#68448	direct (N × H)
337. C₃	C₅×#1685	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5087	2
C₅×#5085	2
C₅×#4843	2
C₅×#4838	2
C₃×C₅×#332	2
C₅×#4485	2
C₅×#4479	2
C₅×#1685	3
#68448	5

Subgroups — C5×#68448 #14292363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68448 #14292363