Subgroups — C₅×#71037 #14291688

Order 1920.

344 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#461	C₂	1	1	yes	maximal, characteristic
2	C₅×#5078	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#5019	C₂	1	1	yes	maximal, characteristic
4	C₅×D₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₅×#4547	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4482	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4479	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1673	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₅×#214	—	2	1	no	—
13	C₂²×C₅×C₃⋊C₈	—	2	1	no	—
14	C₅×#869	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×#103	—	2	1	no	—
17	C₅×#830	—	2	1	no	—
18	C₅×#810	—	2	1	no	—
19	C₅×#807	—	2	1	no	—
20	C₅×#807	—	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	#71037	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#500	—	3	1	no	—
26	C₅×#461	—	3	1	no	—
27	C₅×#461	D₃	1	1	yes	characteristic
28	C₅×C₈×D₄	—	3	1	no	—
29	C₅×#335	—	3	1	no	—
30	C₅×#332	—	3	1	no	—
31	C₅×#332	—	3	1	no	—
32	C₂×C₃×C₅×Q₈	—	2	1	no	—
33	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	—	2	1	no	—
35	C₂×C₅×Dic₆	—	2	1	no	—
36	C₃×C₅×SD₁₆	—	4	2	no	—
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	—	2	1	no	—
40	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
41	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₅×#214	—	4	2	no	—
44	C₅×C₄⋊Dic₃	—	2	1	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₂×C₅×C₃⋊C₈	—	2	1	no	—
47	C₂×C₅×C₃⋊C₈	—	2	1	no	—
48	C₂×C₅×C₃⋊C₈	—	4	2	no	—
49	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
50	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
51	C₃×#461	C₂×C₅	1	1	yes	characteristic
52	#5078	C₂×C₅	1	1	yes	characteristic, complemented
53	#5019	C₂×C₅	1	1	yes	characteristic
54	D₄×C₃⋊C₈	C₂×C₅	1	1	yes	characteristic
55	#4547	C₂×C₅	1	1	yes	characteristic, complemented
56	#4482	C₂×C₅	1	1	yes	characteristic, complemented
57	#4479	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×C₅×SD₁₆	—	2	1	no	—
59	C₂×C₅×SD₁₆	—	6	1	no	—
60	C₂²×C₅×C₈	—	6	1	no	—
61	C₅×C₄⋊Q₈	—	3	1	no	—
62	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
63	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
64	C₅×#107	—	6	1	no	—
65	C₅×#107	—	6	1	no	—
66	C₅×C₄⋊C₈	—	3	1	no	—
67	C₅×C₄⋊C₈	—	3	1	no	—
68	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
69	C₅×#103	—	2	1	no	—
70	C₅×#103	—	6	1	no	—
71	C₅×C₂²⋊C₈	—	6	1	no	—
72	C₄×C₅×C₈	—	3	1	no	—
73	#1673	—	3	1	no	Sylow
74	C₂³×C₃×C₅	—	2	1	no	—
75	C₃×C₅×Q₈	—	4	1	no	—
76	C₃×C₅×Q₈	—	4	1	no	—
77	C₃×C₅×D₄	—	2	1	no	—
78	C₃×C₅×D₄	D₈	1	2	yes	—
79	C₂×C₃×C₄×C₅	—	2	1	no	—
80	C₂×C₃×C₄×C₅	—	4	1	no	—
81	C₂×C₃×C₄×C₅	—	4	1	no	—
82	C₂×C₃×C₄×C₅	—	4	1	no	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
86	C₂×C₅×Dic₃	—	4	1	no	—
87	C₅×Dic₆	—	4	1	no	—
88	C₅×Dic₆	—	4	1	no	—
89	C₃×C₅×C₈	—	4	1	no	—
90	C₅×C₃⋊C₈	—	2	1	no	—
91	C₅×C₃⋊C₈	—	4	1	no	—
92	C₅×C₃⋊C₈	—	4	1	no	—
93	C₅×C₃⋊C₈	SD₁₆	1	2	yes	complemented
94	C₂×C₃×SD₁₆	—	2	1	no	—
95	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
96	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
97	C₂×#214	—	2	1	no	—
98	C₂²×C₃⋊C₈	—	2	1	no	—
99	#869	C₂²×C₅	1	1	yes	characteristic
100	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
101	C₃×#103	—	2	1	no	—
102	#830	—	2	1	no	—
103	#810	—	2	1	no	—
104	#807	—	2	1	no	—
105	#807	—	2	1	no	—
106	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
107	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
108	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₅×Q₈	—	2	1	no	—
110	C₂×C₅×Q₈	—	6	1	no	—
111	C₂×C₅×D₄	#67	1	1	yes	characteristic
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₅×SD₁₆	—	4	2	no	—
114	C₅×SD₁₆	—	12	2	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	—	12	2	no	—
121	C₅×C₄⋊C₄	—	2	1	no	—
122	C₅×C₄⋊C₄	—	2	1	no	—
123	C₅×C₄⋊C₄	—	6	1	no	—
124	C₅×C₄⋊C₄	—	6	1	no	—
125	C₅×C₄⋊C₄	#67	1	1	yes	characteristic
126	C₅×C₂²⋊C₄	—	2	1	no	—
127	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
128	#500	—	3	1	no	—
129	#461	—	3	1	no	—
130	#461	C₅×D₃	1	1	yes	characteristic
131	C₈×D₄	—	3	1	no	—
132	#335	—	3	1	no	—
133	#332	—	3	1	no	—
134	#332	—	3	1	no	—
135	C₂²×C₃×C₅	—	2	2	no	—
136	C₂²×C₃×C₅	—	4	1	no	—
137	C₂²×C₃×C₅	—	4	1	no	—
138	C₂²×C₃×C₅	#121	1	1	yes	characteristic
139	C₃×C₄×C₅	—	2	1	no	—
140	C₃×C₄×C₅	—	4	1	no	—
141	C₃×C₄×C₅	—	8	1	no	—
142	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
143	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
144	C₃×C₄×C₅	#135	1	1	yes	characteristic
145	C₃×C₄×C₅	#137	1	1	yes	characteristic
146	C₅×Dic₃	—	8	1	no	—
147	C₂×C₃×Q₈	—	2	1	no	—
148	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
149	C₂²×C₃×C₄	—	2	1	no	—
150	C₂×Dic₆	—	2	1	no	—
151	C₃×SD₁₆	—	4	2	no	—
152	C₂×C₃×C₈	—	2	1	no	—
153	C₃×C₄⋊C₄	—	2	1	no	—
154	C₃×C₄⋊C₄	—	2	1	no	—
155	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
156	C₃×C₂²⋊C₄	—	2	1	no	—
157	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
158	#214	—	4	2	no	—
159	C₄⋊Dic₃	—	2	1	no	—
160	C₄⋊Dic₃	—	2	1	no	—
161	C₂×C₃⋊C₈	—	2	1	no	—
162	C₂×C₃⋊C₈	—	2	1	no	—
163	C₂×C₃⋊C₈	—	4	2	no	—
164	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
165	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
166	C₂³×C₅	—	2	1	no	—
167	C₅×Q₈	—	4	1	no	—
168	C₅×Q₈	—	4	1	no	—
169	C₅×Q₈	—	12	1	no	—
170	C₅×Q₈	—	12	1	no	—
171	C₅×D₄	—	2	1	no	—
172	C₅×D₄	#213	1	2	yes	—
173	C₂×C₄×C₅	—	2	1	no	—
174	C₂×C₄×C₅	—	4	1	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	4	1	no	—
177	C₂×C₄×C₅	—	12	1	no	—
178	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	#237	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
181	C₅×C₈	—	3	2	no	—
182	C₅×C₈	—	4	1	no	—
183	C₅×C₈	—	6	1	no	—
184	C₅×C₈	—	12	1	no	—
185	C₅×C₈	—	12	1	no	—
186	C₂×SD₁₆	—	2	1	no	—
187	C₂×SD₁₆	—	6	1	no	—
188	C₂²×C₈	—	6	1	no	—
189	C₄⋊Q₈	—	3	1	no	—
190	C₄⋊Q₈	C₂×C₅×D₃	1	1	yes	characteristic
191	C₄×D₄	C₂×C₅×D₃	1	1	yes	characteristic
192	#107	—	6	1	no	—
193	#107	—	6	1	no	—
194	C₄⋊C₈	—	3	1	no	—
195	C₄⋊C₈	—	3	1	no	—
196	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
197	#103	—	2	1	no	—
198	#103	—	6	1	no	—
199	C₂²⋊C₈	—	6	1	no	—
200	C₄×C₈	—	3	1	no	—
201	C₂×C₃×C₅	—	4	2	no	—
202	C₂×C₃×C₅	#461	1	1	yes	characteristic
203	C₂×C₃×C₅	#466	1	1	yes	characteristic
204	C₂×C₃×C₅	#473	1	1	yes	characteristic
205	C₂³×C₃	—	2	1	no	—
206	C₃×Q₈	—	4	1	no	—
207	C₃×Q₈	—	4	1	no	—
208	C₃×D₄	—	2	1	no	—
209	C₃×D₄	C₅×D₈	1	2	yes	—
210	C₂×C₃×C₄	—	2	1	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	—	4	1	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
216	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
217	C₂×Dic₃	—	4	1	no	—
218	Dic₆	—	4	1	no	—
219	Dic₆	—	4	1	no	—
220	C₃×C₈	—	4	1	no	—
221	C₃⋊C₈	—	2	1	no	—
222	C₃⋊C₈	—	4	1	no	—
223	C₃⋊C₈	—	4	1	no	—
224	C₃⋊C₈	C₅×SD₁₆	1	2	yes	complemented
225	C₂²×C₅	—	2	2	no	—
226	C₂²×C₅	—	4	1	no	—
227	C₂²×C₅	—	4	1	no	—
228	C₂²×C₅	#939	1	1	yes	characteristic
229	C₄×C₅	—	2	1	no	—
230	C₄×C₅	—	4	1	no	—
231	C₄×C₅	—	8	1	no	—
232	C₄×C₅	—	24	1	no	—
233	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
234	C₄×C₅	#915	1	1	yes	characteristic
235	C₄×C₅	C₂×#213	1	1	yes	characteristic
236	C₄×C₅	#950	1	1	yes	characteristic
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	—	6	1	no	—
239	C₂×D₄	C₅×#67	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	SD₁₆	—	4	2	no	—
242	SD₁₆	—	12	2	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	3	1	no	—
245	C₂×C₈	—	3	1	no	—
246	C₂×C₈	—	6	1	no	—
247	C₂×C₈	—	6	1	no	—
248	C₂×C₈	—	12	2	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	6	1	no	—
252	C₄⋊C₄	—	6	1	no	—
253	C₄⋊C₄	C₅×#67	1	1	yes	characteristic
254	C₂²⋊C₄	—	2	1	no	—
255	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
256	C₃×C₅	#1673	1	1	yes	characteristic, complemented, Hall
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	4	1	no	—
259	C₂²×C₃	—	4	1	no	—
260	C₂²×C₃	C₅×#121	1	1	yes	characteristic
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	4	1	no	—
263	C₃×C₄	—	8	1	no	—
264	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
265	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
266	C₃×C₄	C₅×#135	1	1	yes	characteristic
267	C₃×C₄	C₅×#137	1	1	yes	characteristic
268	Dic₃	—	8	1	no	—
269	C₂×C₅	—	4	2	no	—
270	C₂×C₅	#5038	1	1	yes	characteristic
271	C₂×C₅	#5162	1	1	yes	characteristic
272	C₂×C₅	#5204	1	1	yes	characteristic
273	C₂³	—	2	1	no	—
274	Q₈	—	4	1	no	—
275	Q₈	—	4	1	no	—
276	Q₈	—	12	1	no	—
277	Q₈	—	12	1	no	—
278	D₄	—	2	1	no	—
279	D₄	C₅×#213	1	2	yes	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	12	1	no	—
285	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
286	C₂×C₄	C₅×#237	1	1	yes	characteristic
287	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
288	C₈	—	3	2	no	—
289	C₈	—	4	1	no	—
290	C₈	—	6	1	no	—
291	C₈	—	12	1	no	—
292	C₈	—	12	1	no	—
293	C₂×C₃	—	4	2	no	—
294	C₂×C₃	C₅×#461	1	1	yes	characteristic
295	C₂×C₃	C₅×#466	1	1	yes	characteristic
296	C₂×C₃	C₅×#473	1	1	yes	characteristic
297	C₅	#71037	1	1	yes	characteristic, direct product factor, Sylow
298	C₂²	—	2	2	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₅×#939	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	8	1	no	—
305	C₄	—	24	1	no	—
306	C₄	C₅×D₃×SD₁₆	1	1	yes	characteristic
307	C₄	C₅×#915	1	1	yes	characteristic
308	C₄	C₂×C₅×#213	1	1	yes	characteristic
309	C₄	C₅×#950	1	1	yes	characteristic
310	C₃	C₅×#1673	1	1	yes	characteristic, complemented, Sylow
311	C₂	—	4	2	no	—
312	C₂	C₅×#5038	1	1	yes	characteristic
313	C₂	C₅×#5162	1	1	yes	characteristic
314	C₂	C₅×#5204	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₅	#939
commutator, cosocle	C₃×C₄²	C₂³×C₅
Frattini	C₄²	C₂²×C₅×D₃
Fitting	C₃×C₅×#461	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₂×C₃×C₅×D₄	D₄
hypercenter	C₅×#461	D₃
hypocenter	C₃	C₅×#1673
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#461

Complements

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

N	H ≅ G/N	type
2. C₅×#5078	C₂	semidirect
5. C₅×#4547	C₂	semidirect
6. C₅×#4482	C₂	semidirect
7. C₅×#4479	C₂	semidirect
24. #71037	C₅	direct (N × H)
52. #5078	C₂×C₅	semidirect
55. #4547	C₂×C₅	semidirect
56. #4482	C₂×C₅	semidirect
57. #4479	C₂×C₅	semidirect
93. C₅×C₃⋊C₈	SD₁₆	semidirect
224. C₃⋊C₈	C₅×SD₁₆	semidirect
256. C₃×C₅	#1673	semidirect
297. C₅	#71037	direct (N × H)
310. C₃	C₅×#1673	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#461	2
C₅×#5078	2
C₅×#5019	2
C₅×D₄×C₃⋊C₈	2
C₅×#4547	2
C₅×#4482	2
C₅×#4479	2
C₅×#1673	3
#71037	5

Subgroups — C5×#71037 #14291688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#71037 #14291688