Subgroups — C₅×#69288 #14317194

Order 1920.

340 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#800	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×D₄₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#4696	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×D₂₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4509	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4503	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2210	—	3	1	no	maximal, Hall
9	C₂×C₅×Dic₁₂	C₂²	1	1	yes	characteristic
10	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
11	C₂×C₅×D₂₄	C₂²	1	1	yes	characteristic
12	C₂×C₅×#204	—	2	1	no	—
13	C₅×#875	—	2	1	no	—
14	C₄×C₅×D₁₂	—	2	1	no	—
15	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₅×#821	—	2	1	no	—
19	C₅×#818	C₂²	1	1	yes	characteristic
20	C₅×#800	—	2	2	no	—
21	C₅×D₄₈	—	2	2	no	—
22	#69288	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂×C₅×SD₃₂	—	3	1	no	—
24	C₂×C₅×D₁₆	—	3	1	no	—
25	C₅×#495	—	3	1	no	—
26	C₄×C₅×D₈	—	3	1	no	—
27	C₅×C₄⋊C₁₆	D₃	1	1	yes	characteristic, complemented
28	C₅×#358	—	3	1	no	—
29	C₅×#357	—	3	1	no	—
30	C₂×C₅×D₁₂	—	2	1	no	—
31	C₂×C₅×D₁₂	—	2	1	no	—
32	C₂×C₄×C₅×D₃	—	4	1	no	—
33	C₂×C₅×Dic₆	—	2	1	no	—
34	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₅×(C₂×D₃)⋊C₄	—	4	1	no	—
38	C₅×(C₂×D₃)⋊C₄	—	4	1	no	—
39	C₅×(C₂×D₃)⋊C₄	—	4	1	no	—
40	C₅×C₄⋊Dic₃	—	2	1	no	—
41	C₅×Dic₁₂	—	2	1	no	—
42	C₅×Dic₁₂	—	2	1	no	—
43	C₅×D₂₄	—	2	1	no	—
44	C₅×D₂₄	—	2	1	no	—
45	C₅×D₂₄	—	2	1	no	—
46	C₅×D₂₄	D₄	1	2	yes	—
47	C₅×#204	—	4	2	no	—
48	C₃×C₅×C₁₆	—	2	1	no	—
49	C₃×C₅×C₁₆	—	2	1	no	—
50	C₂×#800	C₂×C₅	1	1	yes	characteristic, complemented
51	C₂×D₄₈	C₂×C₅	1	1	yes	characteristic
52	#4696	C₂×C₅	1	1	yes	characteristic, complemented
53	C₄×D₂₄	C₂×C₅	1	1	yes	characteristic, complemented
54	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
55	#4509	C₂×C₅	1	1	yes	characteristic, complemented
56	#4503	C₂×C₅	1	1	yes	characteristic, complemented
57	C₂×C₅×Q₁₆	—	3	1	no	—
58	C₂×C₅×SD₁₆	—	6	1	no	—
59	C₂×C₅×D₈	—	3	1	no	—
60	C₂×C₅×D₈	—	3	1	no	—
61	C₅×#124	—	6	1	no	—
62	C₄×C₅×D₄	—	6	1	no	—
63	C₅×SD₃₂	—	6	2	no	—
64	C₅×D₁₆	—	6	2	no	—
65	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂×C₅×C₁₆	C₂×D₃	1	1	yes	characteristic
67	C₅×#107	—	3	1	no	—
68	C₅×#102	—	6	1	no	—
69	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
70	#2210	—	3	1	no	Sylow
71	C₂²×C₅×D₃	—	4	1	no	—
72	C₂²×C₅×D₃	—	4	1	no	—
73	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₅×Dic₃	—	4	1	no	—
77	C₂×C₅×Dic₃	—	4	1	no	—
78	C₅×D₁₂	—	2	2	no	—
79	C₅×D₁₂	—	4	1	no	—
80	C₅×D₁₂	—	4	1	no	—
81	C₅×D₁₂	—	4	1	no	—
82	C₄×C₅×D₃	—	8	2	no	—
83	C₅×Dic₆	—	4	1	no	—
84	C₅×Dic₆	—	4	1	no	—
85	C₃×C₅×C₈	—	2	1	no	—
86	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
87	C₃×C₅×C₈	#41	1	1	yes	characteristic
88	C₂×Dic₁₂	C₂²×C₅	1	1	yes	characteristic
89	C₂×D₂₄	C₂²×C₅	1	1	yes	characteristic
90	C₂×D₂₄	C₂²×C₅	1	1	yes	characteristic
91	C₂×#204	—	2	1	no	—
92	#875	—	2	1	no	—
93	C₄×D₁₂	—	2	1	no	—
94	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
95	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
96	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
97	#821	—	2	1	no	—
98	#818	C₂²×C₅	1	1	yes	characteristic
99	#800	—	2	2	no	—
100	D₄₈	—	2	2	no	—
101	C₂×C₅×Q₈	—	6	1	no	—
102	C₂×C₅×D₄	—	6	1	no	—
103	C₂×C₅×D₄	—	6	1	no	—
104	C₂²×C₄×C₅	—	12	1	no	—
105	C₅×Q₁₆	—	6	1	no	—
106	C₅×Q₁₆	—	6	1	no	—
107	C₅×SD₁₆	—	12	2	no	—
108	C₅×D₈	—	3	2	no	—
109	C₅×D₈	—	6	1	no	—
110	C₅×D₈	—	6	1	no	—
111	C₅×D₈	—	6	1	no	—
112	C₂×C₅×C₈	D₁₂	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
114	C₅×C₄⋊C₄	—	6	1	no	—
115	C₅×C₂²⋊C₄	—	12	1	no	—
116	C₅×C₂²⋊C₄	—	12	1	no	—
117	C₅×C₂²⋊C₄	—	12	1	no	—
118	C₄²×C₅	D₁₂	1	1	yes	characteristic
119	C₅×C₁₆	—	2	1	no	—
120	C₅×C₁₆	—	2	1	no	—
121	C₂×SD₃₂	—	3	1	no	—
122	C₂×D₁₆	—	3	1	no	—
123	#495	—	3	1	no	—
124	C₄×D₈	—	3	1	no	—
125	C₄⋊C₁₆	C₅×D₃	1	1	yes	characteristic, complemented
126	#358	—	3	1	no	—
127	#357	—	3	1	no	—
128	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
129	C₂×C₅×D₃	—	4	2	no	—
130	C₂×C₅×D₃	—	8	1	no	—
131	C₂×C₅×D₃	—	8	1	no	—
132	C₂×C₅×D₃	—	8	1	no	—
133	C₂×C₅×D₃	—	8	1	no	—
134	C₂×C₅×D₃	—	8	1	no	—
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	#121	1	1	yes	characteristic
138	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₅×Dic₃	—	8	1	no	—
140	C₅×Dic₃	—	8	1	no	—
141	C₂×D₁₂	—	2	1	no	—
142	C₂×D₁₂	—	2	1	no	—
143	C₂×C₄×D₃	—	4	1	no	—
144	C₂×Dic₆	—	2	1	no	—
145	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
146	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
147	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
148	(C₂×D₃)⋊C₄	—	4	1	no	—
149	(C₂×D₃)⋊C₄	—	4	1	no	—
150	(C₂×D₃)⋊C₄	—	4	1	no	—
151	C₄⋊Dic₃	—	2	1	no	—
152	Dic₁₂	—	2	1	no	—
153	Dic₁₂	—	2	1	no	—
154	D₂₄	—	2	1	no	—
155	D₂₄	—	2	1	no	—
156	D₂₄	—	2	1	no	—
157	D₂₄	C₅×D₄	1	2	yes	—
158	#204	—	4	2	no	—
159	C₃×C₁₆	—	2	1	no	—
160	C₃×C₁₆	—	2	1	no	—
161	C₂³×C₅	—	12	1	no	—
162	C₂³×C₅	—	12	1	no	—
163	C₅×Q₈	—	12	1	no	—
164	C₅×Q₈	—	12	1	no	—
165	C₅×D₄	—	6	2	no	—
166	C₅×D₄	—	12	1	no	—
167	C₅×D₄	—	12	1	no	—
168	C₅×D₄	—	12	1	no	—
169	C₂×C₄×C₅	—	12	1	no	—
170	C₂×C₄×C₅	—	12	1	no	—
171	C₂×C₄×C₅	—	24	2	no	—
172	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
173	C₂×C₄×C₅	D₂₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
175	C₅×C₈	—	2	1	no	—
176	C₅×C₈	D₃×D₄	1	1	yes	characteristic
177	C₅×C₈	#239	1	1	yes	characteristic
178	C₂×Q₁₆	—	3	1	no	—
179	C₂×SD₁₆	—	6	1	no	—
180	C₂×D₈	—	3	1	no	—
181	C₂×D₈	—	3	1	no	—
182	#124	—	6	1	no	—
183	C₄×D₄	—	6	1	no	—
184	SD₃₂	—	6	2	no	—
185	D₁₆	—	6	2	no	—
186	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
187	C₂×C₁₆	C₂×C₅×D₃	1	1	yes	characteristic
188	#107	—	3	1	no	—
189	#102	—	6	1	no	—
190	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
191	C₂×C₃×C₅	#459	1	1	yes	characteristic
192	C₂×C₃×C₅	#508	1	1	yes	characteristic
193	C₂×C₃×C₅	#509	1	1	yes	characteristic
194	C₅×D₃	—	8	2	no	—
195	C₅×D₃	—	16	1	no	—
196	C₂²×D₃	—	4	1	no	—
197	C₂²×D₃	—	4	1	no	—
198	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
199	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
201	C₂×Dic₃	—	4	1	no	—
202	C₂×Dic₃	—	4	1	no	—
203	D₁₂	—	2	2	no	—
204	D₁₂	—	4	1	no	—
205	D₁₂	—	4	1	no	—
206	D₁₂	—	4	1	no	—
207	C₄×D₃	—	8	2	no	—
208	Dic₆	—	4	1	no	—
209	Dic₆	—	4	1	no	—
210	C₃×C₈	—	2	1	no	—
211	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
212	C₃×C₈	C₅×#41	1	1	yes	characteristic
213	C₂²×C₅	—	12	2	no	—
214	C₂²×C₅	—	24	1	no	—
215	C₂²×C₅	—	24	1	no	—
216	C₂²×C₅	—	24	1	no	—
217	C₂²×C₅	—	24	1	no	—
218	C₂²×C₅	—	24	1	no	—
219	C₂²×C₅	C₂×D₂₄	1	1	yes	characteristic
220	C₄×C₅	—	2	1	no	—
221	C₄×C₅	—	2	1	no	—
222	C₄×C₅	—	24	1	no	—
223	C₄×C₅	—	24	1	no	—
224	C₄×C₅	#895	1	1	yes	characteristic
225	C₄×C₅	#908	1	1	yes	characteristic
226	C₂×Q₈	—	6	1	no	—
227	C₂×D₄	—	6	1	no	—
228	C₂×D₄	—	6	1	no	—
229	C₂²×C₄	—	12	1	no	—
230	Q₁₆	—	6	1	no	—
231	Q₁₆	—	6	1	no	—
232	SD₁₆	—	12	2	no	—
233	D₈	—	3	2	no	—
234	D₈	—	6	1	no	—
235	D₈	—	6	1	no	—
236	D₈	—	6	1	no	—
237	C₂×C₈	C₅×D₁₂	1	1	yes	characteristic
238	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
239	C₄⋊C₄	—	6	1	no	—
240	C₂²⋊C₄	—	12	1	no	—
241	C₂²⋊C₄	—	12	1	no	—
242	C₂²⋊C₄	—	12	1	no	—
243	C₄²	C₅×D₁₂	1	1	yes	characteristic
244	C₁₆	—	2	1	no	—
245	C₁₆	—	2	1	no	—
246	C₃×C₅	#2210	1	1	yes	characteristic, complemented, Hall
247	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
248	C₂×D₃	—	4	2	no	—
249	C₂×D₃	—	8	1	no	—
250	C₂×D₃	—	8	1	no	—
251	C₂×D₃	—	8	1	no	—
252	C₂×D₃	—	8	1	no	—
253	C₂×D₃	—	8	1	no	—
254	C₃×C₄	—	2	1	no	—
255	C₃×C₄	—	2	1	no	—
256	C₃×C₄	C₅×#121	1	1	yes	characteristic
257	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
258	Dic₃	—	8	1	no	—
259	Dic₃	—	8	1	no	—
260	C₂×C₅	—	24	2	no	—
261	C₂×C₅	—	48	1	no	—
262	C₂×C₅	#4843	1	1	yes	characteristic
263	C₂×C₅	#4904	1	1	yes	characteristic
264	C₂×C₅	#4908	1	1	yes	characteristic
265	C₂³	—	12	1	no	—
266	C₂³	—	12	1	no	—
267	Q₈	—	12	1	no	—
268	Q₈	—	12	1	no	—
269	D₄	—	6	2	no	—
270	D₄	—	12	1	no	—
271	D₄	—	12	1	no	—
272	D₄	—	12	1	no	—
273	C₂×C₄	—	12	1	no	—
274	C₂×C₄	—	12	1	no	—
275	C₂×C₄	—	24	2	no	—
276	C₂×C₄	C₅×D₂₄	1	1	yes	characteristic
277	C₂×C₄	C₅×D₂₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
279	C₈	—	2	1	no	—
280	C₈	C₅×D₃×D₄	1	1	yes	characteristic
281	C₈	C₅×#239	1	1	yes	characteristic
282	C₂×C₃	C₅×#459	1	1	yes	characteristic
283	C₂×C₃	C₅×#508	1	1	yes	characteristic
284	C₂×C₃	C₅×#509	1	1	yes	characteristic
285	D₃	—	8	2	no	—
286	D₃	—	16	1	no	—
287	C₅	#69288	1	1	yes	characteristic, direct product factor, Sylow
288	C₂²	—	12	2	no	—
289	C₂²	—	24	1	no	—
290	C₂²	—	24	1	no	—
291	C₂²	—	24	1	no	—
292	C₂²	—	24	1	no	—
293	C₂²	—	24	1	no	—
294	C₂²	C₂×C₅×D₂₄	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	24	1	no	—
298	C₄	—	24	1	no	—
299	C₄	C₅×#895	1	1	yes	characteristic
300	C₄	C₅×#908	1	1	yes	characteristic
301	C₃	C₅×#2210	1	1	yes	characteristic, complemented, Sylow
302	C₂	—	24	2	no	—
303	C₂	—	48	1	no	—
304	C₂	C₅×#4843	1	1	yes	characteristic
305	C₂	C₅×#4904	1	1	yes	characteristic
306	C₂	C₅×#4908	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	C₂×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂³×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×C₄⋊C₁₆	C₂
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₅×D₄₈	C₂
hypercenter	C₅×C₄⋊C₁₆	D₃
hypocenter	C₃	C₅×#2210
LCS	G ▷ C₂×C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₄×C₅×C₈ ◁ C₅×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#800	C₂	semidirect
3. C₅×#4696	C₂	semidirect
4. C₄×C₅×D₂₄	C₂	semidirect
5. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
6. C₅×#4509	C₂	semidirect
7. C₅×#4503	C₂	semidirect
22. #69288	C₅	direct (N × H)
27. C₅×C₄⋊C₁₆	D₃	semidirect
50. C₂×#800	C₂×C₅	semidirect
52. #4696	C₂×C₅	semidirect
53. C₄×D₂₄	C₂×C₅	semidirect
54. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
55. #4509	C₂×C₅	semidirect
56. #4503	C₂×C₅	semidirect
125. C₄⋊C₁₆	C₅×D₃	semidirect
246. C₃×C₅	#2210	semidirect
287. C₅	#69288	direct (N × H)
301. C₃	C₅×#2210	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#800	2
C₂×C₅×D₄₈	2
C₅×#4696	2
C₄×C₅×D₂₄	2
C₃×C₅×C₄⋊C₁₆	2
C₅×#4509	2
C₅×#4503	2
C₅×#2210	3
#69288	5

Subgroups — C5×#69288 #14317194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#69288 #14317194