Subgroups — C₅×#68991 #14298400

Order 1920.

544 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×#892	C₂	1	1	yes	maximal, characteristic
2	C₅×#5126	C₂	1	2	yes	maximal
3	C₃×C₅×#343	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4554	C₂	1	2	yes	maximal, complemented
5	C₅×#4530	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1822	—	3	1	no	maximal, Hall
7	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×C₅×(C₂×D₃)⋊C₄	C₄	1	2	yes	—
11	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×Dic₃	C₄	1	2	yes	—
13	C₂×C₅×#208	C₂²	1	2	yes	—
14	C₅×#892	—	2	4	no	—
15	C₅×(C₂×D₃)⋊C₈	—	2	4	no	—
16	#68991	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₅×#117	—	3	1	no	—
18	C₅×#407	—	3	2	no	—
19	C₅×#343	—	3	2	no	—
20	C₅×#343	—	3	1	no	—
21	C₅×#343	D₃	1	1	yes	characteristic, complemented
22	C₂³×C₅×D₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₅×D₃	—	2	2	no	—
28	C₂×C₄×C₅×D₃	—	2	1	no	—
29	C₂×C₄×C₅×D₃	—	2	1	no	—
30	C₂×C₄×C₅×D₃	D₄	1	4	yes	—
31	C₃×C₅×M₁₆	—	2	2	no	—
32	C₃×C₅×M₁₆	—	2	2	no	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₅×(C₂×D₃)⋊C₄	—	2	4	no	—
36	C₅×C₄⋊Dic₃	—	2	2	no	—
37	C₄×C₅×Dic₃	—	2	4	no	—
38	C₅×#208	—	2	2	no	—
39	C₅×#208	—	2	2	no	—
40	C₂×C₅×C₃⋊C₈	—	2	2	no	—
41	C₂×#892	C₂×C₅	1	1	yes	characteristic
42	#5126	C₂×C₅	1	2	yes	—
43	C₃×#343	C₂×C₅	1	1	yes	characteristic, complemented
44	#4554	C₂×C₅	1	2	yes	complemented
45	#4530	C₂×C₅	1	1	yes	characteristic, complemented
46	C₂³×C₄×C₅	—	3	1	no	—
47	C₂×C₅×M₁₆	—	3	2	no	—
48	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
49	C₅×#117	—	6	4	no	—
50	C₂×C₅×C₄⋊C₄	—	3	1	no	—
51	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
52	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
53	C₂×C₄²×C₅	—	3	2	no	—
54	C₅×C₂²⋊C₈	—	6	4	no	—
55	#1822	—	3	1	no	Sylow
56	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₅×D₃	—	2	1	no	—
58	C₂²×C₅×D₃	—	2	1	no	—
59	C₂²×C₅×D₃	—	4	1	no	—
60	C₂²×C₅×D₃	—	4	1	no	—
61	C₂²×C₅×D₃	C₂²⋊C₄	1	2	yes	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	4	2	no	—
64	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	#41	1	2	yes	—
66	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
68	C₂×C₅×Dic₃	—	2	1	no	—
69	C₂×C₅×Dic₃	—	2	1	no	—
70	C₂×C₅×Dic₃	—	2	2	no	—
71	C₂×C₅×Dic₃	—	4	2	no	—
72	C₂×C₅×Dic₃	C₂²⋊C₄	1	2	yes	—
73	C₄×C₅×D₃	—	2	4	no	—
74	C₄×C₅×D₃	—	2	4	no	—
75	C₃×C₅×C₈	—	4	2	no	—
76	C₅×C₃⋊C₈	—	4	2	no	—
77	C₂²×C₄×D₃	C₂²×C₅	1	1	yes	characteristic
78	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
79	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
80	C₂×(C₂×D₃)⋊C₄	C₄×C₅	1	2	yes	—
81	C₂×C₄⋊Dic₃	C₂²×C₅	1	1	yes	characteristic
82	C₂×C₄×Dic₃	C₄×C₅	1	2	yes	—
83	C₂×#208	C₂²×C₅	1	2	yes	—
84	#892	—	2	4	no	—
85	(C₂×D₃)⋊C₈	—	2	4	no	—
86	C₂⁴×C₅	—	3	1	no	—
87	C₂²×C₄×C₅	—	3	1	no	—
88	C₂²×C₄×C₅	—	3	4	no	—
89	C₂²×C₄×C₅	—	3	2	no	—
90	C₂²×C₄×C₅	—	6	2	no	—
91	C₂²×C₄×C₅	—	6	1	no	—
92	C₂²×C₄×C₅	—	6	1	no	—
93	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
94	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
95	C₅×M₁₆	—	2	2	no	—
96	C₅×M₁₆	—	2	2	no	—
97	C₅×M₁₆	—	6	2	no	—
98	C₅×M₁₆	—	6	2	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	6	2	no	—
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	6	2	no	—
103	C₅×C₂²⋊C₄	—	6	4	no	—
104	C₄²×C₅	—	6	4	no	—
105	C₂×#117	—	3	1	no	—
106	#407	—	3	2	no	—
107	#343	—	3	2	no	—
108	#343	—	3	1	no	—
109	#343	C₅×D₃	1	1	yes	characteristic, complemented
110	C₂²×C₃×C₅	—	2	1	no	—
111	C₂²×C₃×C₅	—	2	1	no	—
112	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂²×C₃×C₅	#117	1	1	yes	characteristic
114	C₂²×C₃×C₅	#117	1	1	yes	characteristic
115	C₂×C₅×D₃	—	2	2	no	—
116	C₂×C₅×D₃	—	2	2	no	—
117	C₂×C₅×D₃	—	2	2	no	—
118	C₂×C₅×D₃	—	4	2	no	—
119	C₂×C₅×D₃	—	4	2	no	—
120	C₃×C₄×C₅	—	4	2	no	—
121	C₃×C₄×C₅	#123	1	2	yes	—
122	C₃×C₄×C₅	#123	1	2	yes	—
123	C₅×Dic₃	—	2	2	no	—
124	C₅×Dic₃	—	4	2	no	—
125	C₂³×D₃	C₂×C₄×C₅	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
127	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
128	C₂²×Dic₃	C₂×C₄×C₅	1	1	yes	characteristic
129	C₂²×Dic₃	C₂×C₄×C₅	1	2	yes	—
130	C₂×C₄×D₃	—	2	2	no	—
131	C₂×C₄×D₃	—	2	1	no	—
132	C₂×C₄×D₃	—	2	1	no	—
133	C₂×C₄×D₃	C₅×D₄	1	4	yes	—
134	C₃×M₁₆	—	2	2	no	—
135	C₃×M₁₆	—	2	2	no	—
136	C₂×C₃×C₈	—	2	2	no	—
137	C₃×C₄⋊C₄	—	2	2	no	—
138	(C₂×D₃)⋊C₄	—	2	4	no	—
139	C₄⋊Dic₃	—	2	2	no	—
140	C₄×Dic₃	—	2	4	no	—
141	#208	—	2	2	no	—
142	#208	—	2	2	no	—
143	C₂×C₃⋊C₈	—	2	2	no	—
144	C₂³×C₅	—	3	2	no	—
145	C₂³×C₅	—	6	1	no	—
146	C₂³×C₅	—	6	1	no	—
147	C₂³×C₅	—	12	1	no	—
148	C₂³×C₅	—	12	1	no	—
149	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	3	2	no	—
152	C₂×C₄×C₅	—	4	2	no	—
153	C₂×C₄×C₅	—	6	1	no	—
154	C₂×C₄×C₅	—	6	1	no	—
155	C₂×C₄×C₅	—	6	4	no	—
156	C₂×C₄×C₅	—	6	4	no	—
157	C₂×C₄×C₅	—	6	2	no	—
158	C₂×C₄×C₅	—	12	2	no	—
159	C₂×C₄×C₅	#235	1	2	yes	—
160	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
161	C₂×C₄×C₅	#237	1	1	yes	characteristic
162	C₂×C₄×C₅	#239	1	1	yes	characteristic
163	C₅×C₈	—	4	2	no	—
164	C₅×C₈	—	12	2	no	—
165	C₂³×C₄	—	3	1	no	—
166	C₂×M₁₆	—	3	2	no	—
167	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
168	#117	—	6	4	no	—
169	C₂×C₄⋊C₄	—	3	1	no	—
170	C₂×C₄⋊C₄	C₂×C₅×D₃	1	1	yes	characteristic
171	C₂×C₂²⋊C₄	—	3	2	no	—
172	C₂×C₄²	—	3	2	no	—
173	C₂²⋊C₈	—	6	4	no	—
174	C₂×C₃×C₅	—	2	1	no	—
175	C₂×C₃×C₅	—	2	1	no	—
176	C₂×C₃×C₅	#381	1	1	yes	characteristic
177	C₂×C₃×C₅	#413	1	1	yes	characteristic
178	C₂×C₃×C₅	#413	1	1	yes	characteristic
179	C₅×D₃	—	4	2	no	—
180	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
181	C₂²×D₃	—	2	1	no	—
182	C₂²×D₃	—	2	1	no	—
183	C₂²×D₃	—	4	1	no	—
184	C₂²×D₃	—	4	1	no	—
185	C₂²×D₃	C₅×C₂²⋊C₄	1	2	yes	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	4	2	no	—
188	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
189	C₂×C₃×C₄	C₅×#41	1	2	yes	—
190	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
191	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
192	C₂×Dic₃	—	2	1	no	—
193	C₂×Dic₃	—	2	1	no	—
194	C₂×Dic₃	—	2	2	no	—
195	C₂×Dic₃	—	4	2	no	—
196	C₂×Dic₃	C₅×C₂²⋊C₄	1	2	yes	—
197	C₄×D₃	—	2	4	no	—
198	C₄×D₃	—	2	4	no	—
199	C₃×C₈	—	4	2	no	—
200	C₃⋊C₈	—	4	2	no	—
201	C₂²×C₅	—	2	1	no	—
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	—	6	2	no	—
204	C₂²×C₅	—	6	2	no	—
205	C₂²×C₅	—	6	2	no	—
206	C₂²×C₅	—	12	2	no	—
207	C₂²×C₅	—	12	2	no	—
208	C₂²×C₅	#872	1	1	yes	characteristic
209	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
210	C₂²×C₅	#892	1	1	yes	characteristic
211	C₄×C₅	—	4	2	no	—
212	C₄×C₅	—	6	2	no	—
213	C₄×C₅	—	12	2	no	—
214	C₄×C₅	#883	1	2	yes	—
215	C₄×C₅	#894	1	2	yes	—
216	C₂⁴	—	3	1	no	—
217	C₂²×C₄	—	3	1	no	—
218	C₂²×C₄	—	3	4	no	—
219	C₂²×C₄	—	3	2	no	—
220	C₂²×C₄	—	6	2	no	—
221	C₂²×C₄	—	6	1	no	—
222	C₂²×C₄	—	6	1	no	—
223	C₂²×C₄	C₄×C₅×D₃	1	2	yes	—
224	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
225	M₁₆	—	2	2	no	—
226	M₁₆	—	2	2	no	—
227	M₁₆	—	6	2	no	—
228	M₁₆	—	6	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	6	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	6	2	no	—
233	C₂²⋊C₄	—	6	4	no	—
234	C₄²	—	6	4	no	—
235	C₃×C₅	#1822	1	1	yes	characteristic, complemented, Hall
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
239	C₂²×C₃	C₅×#117	1	1	yes	characteristic
240	C₂²×C₃	C₅×#117	1	1	yes	characteristic
241	C₂×D₃	—	2	2	no	—
242	C₂×D₃	—	2	2	no	—
243	C₂×D₃	—	2	2	no	—
244	C₂×D₃	—	4	2	no	—
245	C₂×D₃	—	4	2	no	—
246	C₃×C₄	—	4	2	no	—
247	C₃×C₄	C₅×#123	1	2	yes	—
248	C₃×C₄	C₅×#123	1	2	yes	—
249	Dic₃	—	2	2	no	—
250	Dic₃	—	4	2	no	—
251	C₂×C₅	—	2	1	no	—
252	C₂×C₅	—	2	1	no	—
253	C₂×C₅	—	12	2	no	—
254	C₂×C₅	#4664	1	1	yes	characteristic
255	C₂×C₅	#4745	1	1	yes	characteristic
256	C₂×C₅	#4751	1	1	yes	characteristic
257	C₂³	—	3	2	no	—
258	C₂³	—	6	1	no	—
259	C₂³	—	6	1	no	—
260	C₂³	—	12	1	no	—
261	C₂³	—	12	1	no	—
262	C₂³	C₂×C₄×C₅×D₃	1	1	yes	characteristic
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	3	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	6	1	no	—
267	C₂×C₄	—	6	1	no	—
268	C₂×C₄	—	6	4	no	—
269	C₂×C₄	—	6	4	no	—
270	C₂×C₄	—	6	2	no	—
271	C₂×C₄	—	12	2	no	—
272	C₂×C₄	C₅×#235	1	2	yes	—
273	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
274	C₂×C₄	C₅×#237	1	1	yes	characteristic
275	C₂×C₄	C₅×#239	1	1	yes	characteristic
276	C₈	—	4	2	no	—
277	C₈	—	12	2	no	—
278	C₂×C₃	—	2	1	no	—
279	C₂×C₃	—	2	1	no	—
280	C₂×C₃	C₅×#381	1	1	yes	characteristic
281	C₂×C₃	C₅×#413	1	1	yes	characteristic
282	C₂×C₃	C₅×#413	1	1	yes	characteristic
283	D₃	—	4	2	no	—
284	C₅	#68991	1	1	yes	characteristic, direct product factor, Sylow
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	6	2	no	—
288	C₂²	—	6	2	no	—
289	C₂²	—	6	2	no	—
290	C₂²	—	12	2	no	—
291	C₂²	—	12	2	no	—
292	C₂²	C₅×#872	1	1	yes	characteristic
293	C₂²	C₅×D₃×C₂²⋊C₄	1	1	yes	characteristic
294	C₂²	C₅×#892	1	1	yes	characteristic
295	C₄	—	4	2	no	—
296	C₄	—	6	2	no	—
297	C₄	—	12	2	no	—
298	C₄	C₅×#883	1	2	yes	—
299	C₄	C₅×#894	1	2	yes	—
300	C₃	C₅×#1822	1	1	yes	characteristic, complemented, Sylow
301	C₂	—	2	1	no	—
302	C₂	—	2	1	no	—
303	C₂	—	12	2	no	—
304	C₂	C₅×#4664	1	1	yes	characteristic
305	C₂	C₅×#4745	1	1	yes	characteristic
306	C₂	C₅×#4751	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	D₃×C₂²⋊C₄
commutator	C₂³×C₃	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#343	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₅×D₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#343	D₃
hypocenter	C₃	C₅×#1822
LCS	G ▷ C₂³×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#343

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#343	C₂	semidirect
4. C₅×#4554	C₂	semidirect
5. C₅×#4530	C₂	semidirect
16. #68991	C₅	direct (N × H)
21. C₅×#343	D₃	semidirect
43. C₃×#343	C₂×C₅	semidirect
44. #4554	C₂×C₅	semidirect
45. #4530	C₂×C₅	semidirect
109. #343	C₅×D₃	semidirect
235. C₃×C₅	#1822	semidirect
284. C₅	#68991	direct (N × H)
300. C₃	C₅×#1822	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#892	2
C₅×#5126	2
C₃×C₅×#343	2
C₅×#4554	2
C₅×#4530	2
C₅×#1822	3
#68991	5

Subgroups — C5×#68991 #14298400

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68991 #14298400