Subgroups — C₅×#67696 #14279073

Order 1920.

332 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#410	C₂	1	1	yes	maximal, characteristic
2	C₅×#5240	C₂	1	1	yes	maximal, characteristic
3	C₅×#5234	C₂	1	1	yes	maximal, characteristic
4	C₅×#1391	—	3	1	no	maximal, Hall
5	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
6	C₃×C₅×#117	—	2	1	no	—
7	C₅×#926	—	2	1	no	—
8	C₂×C₅×#208	C₄	1	1	yes	characteristic, complemented
9	C₂²×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic, complemented
10	C₃×C₅×#99	—	2	1	no	—
11	C₃×C₅×#99	C₄	1	2	yes	complemented
12	C₅×#834	—	2	1	no	—
13	C₅×#834	C₄	1	2	yes	complemented
14	C₅×#830	—	2	1	no	—
15	C₅×#830	—	2	1	no	—
16	#67696	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#410	—	3	1	no	—
18	C₅×#410	D₃	1	1	yes	characteristic
19	C₅×#408	—	3	1	no	—
20	C₃×C₅×#41	—	4	1	no	—
21	C₃×C₅×#41	—	4	1	no	—
22	C₂×C₃×C₅×Q₈	Q₈	1	1	yes	characteristic
23	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
24	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	—	4	1	no	—
30	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
31	C₃×C₄²×C₅	—	4	1	no	—
32	C₅×#217	—	2	2	no	—
33	C₅×C₄⋊Dic₃	—	4	1	no	—
34	C₄×C₅×Dic₃	—	4	1	no	—
35	C₅×#208	—	4	1	no	—
36	C₂×C₅×C₃⋊C₈	—	2	1	no	—
37	C₂×C₅×C₃⋊C₈	—	2	1	no	—
38	C₂×C₅×C₃⋊C₈	—	4	1	no	—
39	C₃×#410	C₂×C₅	1	1	yes	characteristic
40	#5240	C₂×C₅	1	1	yes	characteristic
41	#5234	C₂×C₅	1	1	yes	characteristic
42	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
43	C₂×C₅×M₁₆	—	3	1	no	—
44	C₂²×C₅×C₈	—	3	1	no	—
45	C₅×#117	—	2	1	no	—
46	C₅×#117	—	6	1	no	—
47	C₅×#99	—	2	1	no	—
48	C₅×#99	—	3	2	no	—
49	C₅×#99	—	6	1	no	—
50	C₅×#99	Dic₃	1	2	yes	complemented
51	C₅×C₂²⋊C₈	—	6	1	no	—
52	C₅×C₂²⋊C₈	—	6	1	no	—
53	#1391	—	3	1	no	Sylow
54	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
55	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
56	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₃×C₅×Q₈	—	4	1	no	—
58	C₃×C₅×D₄	—	4	1	no	—
59	C₃×C₅×D₄	—	4	1	no	—
60	C₃×C₅×D₄	—	4	1	no	—
61	C₂×C₃×C₄×C₅	—	2	1	no	—
62	C₂×C₃×C₄×C₅	—	2	1	no	—
63	C₂×C₃×C₄×C₅	—	2	1	no	—
64	C₂×C₃×C₄×C₅	—	2	1	no	—
65	C₂×C₃×C₄×C₅	—	2	1	no	—
66	C₂×C₃×C₄×C₅	—	2	1	no	—
67	C₂×C₃×C₄×C₅	—	4	2	no	—
68	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₅×Dic₃	—	4	2	no	—
73	C₅×C₃⋊C₈	—	4	1	no	—
74	C₅×C₃⋊C₈	—	4	1	no	—
75	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
76	C₃×#117	—	2	1	no	—
77	#926	—	2	1	no	—
78	C₂×#208	C₄×C₅	1	1	yes	characteristic, complemented
79	C₂²×C₃⋊C₈	C₄×C₅	1	1	yes	characteristic, complemented
80	C₃×#99	—	2	1	no	—
81	C₃×#99	C₄×C₅	1	2	yes	complemented
82	#834	—	2	1	no	—
83	#834	C₄×C₅	1	2	yes	complemented
84	#830	—	2	1	no	—
85	#830	—	2	1	no	—
86	C₅×#41	—	4	1	no	—
87	C₅×#41	—	4	1	no	—
88	C₂×C₅×Q₈	Dic₆	1	1	yes	characteristic
89	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
90	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
91	C₂×C₅×D₄	#67	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
93	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
94	C₂²×C₄×C₅	#67	1	1	yes	characteristic
95	C₅×M₁₆	—	12	1	no	—
96	C₂×C₅×C₈	—	6	1	no	—
97	C₂×C₅×C₈	—	6	1	no	—
98	C₂×C₅×C₈	—	12	1	no	—
99	C₅×C₄⋊C₄	—	4	1	no	—
100	C₅×C₄⋊C₄	—	12	1	no	—
101	C₅×C₂²⋊C₄	—	2	2	no	—
102	C₅×C₂²⋊C₄	—	6	2	no	—
103	C₄²×C₅	—	4	1	no	—
104	C₄²×C₅	—	12	1	no	—
105	#410	—	3	1	no	—
106	#410	C₅×D₃	1	1	yes	characteristic
107	#408	—	3	1	no	—
108	C₂²×C₃×C₅	—	2	1	no	—
109	C₂²×C₃×C₅	—	2	1	no	—
110	C₂²×C₃×C₅	—	2	1	no	—
111	C₂²×C₃×C₅	—	4	1	no	—
112	C₂²×C₃×C₅	—	4	1	no	—
113	C₂²×C₃×C₅	—	4	1	no	—
114	C₂²×C₃×C₅	#95	1	1	yes	characteristic
115	C₃×C₄×C₅	—	2	1	no	—
116	C₃×C₄×C₅	—	2	1	no	—
117	C₃×C₄×C₅	—	2	1	no	—
118	C₃×C₄×C₅	—	8	2	no	—
119	C₃×C₄×C₅	#99	1	1	yes	characteristic
120	C₃×C₄×C₅	#99	1	1	yes	characteristic
121	C₅×Dic₃	—	8	2	no	—
122	C₃×#41	—	4	1	no	—
123	C₃×#41	—	4	1	no	—
124	C₂×C₃×Q₈	C₅×Q₈	1	1	yes	characteristic
125	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
126	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
127	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
131	C₃×C₄⋊C₄	—	4	1	no	—
132	C₃×C₂²⋊C₄	—	2	2	no	—
133	C₃×C₄²	—	4	1	no	—
134	#217	—	2	2	no	—
135	C₄⋊Dic₃	—	4	1	no	—
136	C₄×Dic₃	—	4	1	no	—
137	#208	—	4	1	no	—
138	C₂×C₃⋊C₈	—	2	1	no	—
139	C₂×C₃⋊C₈	—	2	1	no	—
140	C₂×C₃⋊C₈	—	4	1	no	—
141	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
142	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
143	C₂³×C₅	#217	1	1	yes	characteristic
144	C₅×Q₈	—	4	1	no	—
145	C₅×D₄	—	4	1	no	—
146	C₅×D₄	—	4	1	no	—
147	C₅×D₄	—	4	1	no	—
148	C₂×C₄×C₅	—	2	1	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	—	12	2	no	—
156	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
157	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
158	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
159	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
160	C₅×C₈	—	12	1	no	—
161	C₅×C₈	—	12	1	no	—
162	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
163	C₂×M₁₆	—	3	1	no	—
164	C₂²×C₈	—	3	1	no	—
165	#117	—	2	1	no	—
166	#117	—	6	1	no	—
167	#99	—	2	1	no	—
168	#99	—	3	2	no	—
169	#99	—	6	1	no	—
170	#99	C₅×Dic₃	1	2	yes	complemented
171	C₂²⋊C₈	—	6	1	no	—
172	C₂²⋊C₈	—	6	1	no	—
173	C₂×C₃×C₅	—	2	1	no	—
174	C₂×C₃×C₅	—	4	1	no	—
175	C₂×C₃×C₅	—	4	1	no	—
176	C₂×C₃×C₅	—	4	1	no	—
177	C₂×C₃×C₅	#342	1	1	yes	characteristic
178	C₂³×C₃	C₄²×C₅	1	1	yes	characteristic
179	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
180	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₃×Q₈	—	4	1	no	—
182	C₃×D₄	—	4	1	no	—
183	C₃×D₄	—	4	1	no	—
184	C₃×D₄	—	4	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	2	1	no	—
191	C₂×C₃×C₄	—	4	2	no	—
192	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
193	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
196	C₂×Dic₃	—	4	2	no	—
197	C₃⋊C₈	—	4	1	no	—
198	C₃⋊C₈	—	4	1	no	—
199	C₂²×C₅	—	2	1	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	2	1	no	—
202	C₂²×C₅	—	4	1	no	—
203	C₂²×C₅	—	4	1	no	—
204	C₂²×C₅	—	4	1	no	—
205	C₂²×C₅	C₃⋊#95	1	1	yes	characteristic
206	C₄×C₅	—	2	1	no	—
207	C₄×C₅	—	2	1	no	—
208	C₄×C₅	—	2	1	no	—
209	C₄×C₅	—	8	2	no	—
210	C₄×C₅	—	24	2	no	—
211	C₄×C₅	#806	1	1	yes	characteristic
212	C₄×C₅	#806	1	1	yes	characteristic
213	#41	—	4	1	no	—
214	#41	—	4	1	no	—
215	C₂×Q₈	C₅×Dic₆	1	1	yes	characteristic
216	C₂×D₄	C₄×C₅×D₃	1	1	yes	characteristic
217	C₂×D₄	C₂×C₅×Dic₃	1	1	yes	characteristic
218	C₂×D₄	C₅×#67	1	1	yes	characteristic
219	C₂²×C₄	C₄×C₅×D₃	1	1	yes	characteristic
220	C₂²×C₄	C₅×D₁₂	1	1	yes	characteristic
221	C₂²×C₄	C₅×#67	1	1	yes	characteristic
222	M₁₆	—	12	1	no	—
223	C₂×C₈	—	6	1	no	—
224	C₂×C₈	—	6	1	no	—
225	C₂×C₈	—	12	1	no	—
226	C₄⋊C₄	—	4	1	no	—
227	C₄⋊C₄	—	12	1	no	—
228	C₂²⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	6	2	no	—
230	C₄²	—	4	1	no	—
231	C₄²	—	12	1	no	—
232	C₃×C₅	#1391	1	1	yes	characteristic, complemented, Hall
233	C₂²×C₃	—	2	1	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	1	no	—
236	C₂²×C₃	—	4	1	no	—
237	C₂²×C₃	—	4	1	no	—
238	C₂²×C₃	—	4	1	no	—
239	C₂²×C₃	C₅×#95	1	1	yes	characteristic
240	C₃×C₄	—	2	1	no	—
241	C₃×C₄	—	2	1	no	—
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	8	2	no	—
244	C₃×C₄	C₅×#99	1	1	yes	characteristic
245	C₃×C₄	C₅×#99	1	1	yes	characteristic
246	Dic₃	—	8	2	no	—
247	C₂×C₅	—	2	1	no	—
248	C₂×C₅	—	4	1	no	—
249	C₂×C₅	—	4	1	no	—
250	C₂×C₅	—	4	1	no	—
251	C₂×C₅	#4527	1	1	yes	characteristic
252	C₂³	C₄×C₅×Dic₃	1	1	yes	characteristic
253	C₂³	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
254	C₂³	C₅×#217	1	1	yes	characteristic
255	Q₈	—	4	1	no	—
256	D₄	—	4	1	no	—
257	D₄	—	4	1	no	—
258	D₄	—	4	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	12	2	no	—
267	C₂×C₄	C₅×Dic₃⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₅×Dic₃⋊C₄	1	1	yes	characteristic
269	C₂×C₄	C₅×C₄⋊Dic₃	1	1	yes	characteristic
270	C₂×C₄	C₅×(C₂×D₃)⋊C₄	1	1	yes	characteristic
271	C₈	—	12	1	no	—
272	C₈	—	12	1	no	—
273	C₂×C₃	—	2	1	no	—
274	C₂×C₃	—	4	1	no	—
275	C₂×C₃	—	4	1	no	—
276	C₂×C₃	—	4	1	no	—
277	C₂×C₃	C₅×#342	1	1	yes	characteristic
278	C₅	#67696	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	C₅×C₃⋊#95	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	2	1	no	—
289	C₄	—	8	2	no	—
290	C₄	—	24	2	no	—
291	C₄	C₅×#806	1	1	yes	characteristic
292	C₄	C₅×#806	1	1	yes	characteristic
293	C₃	C₅×#1391	1	1	yes	characteristic, complemented, Sylow
294	C₂	—	2	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	4	1	no	—
297	C₂	—	4	1	no	—
298	C₂	C₅×#4527	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#806
commutator	C₂³×C₃	C₄²×C₅
Frattini	C₂×#41	C₂×C₅×D₃
Fitting	C₃×C₅×#410	C₂
socle	C₂×C₃×C₅	#342
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂×C₃×#41	C₂²×C₅
hypercenter	C₅×#410	D₃
hypocenter	C₃	C₅×#1391
LCS	G ▷ C₂³×C₃ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41 ◁ C₅×#410

Complements

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

N	H ≅ G/N	type
8. C₂×C₅×#208	C₄	semidirect
9. C₂²×C₅×C₃⋊C₈	C₄	semidirect
11. C₃×C₅×#99	C₄	semidirect
13. C₅×#834	C₄	semidirect
16. #67696	C₅	direct (N × H)
50. C₅×#99	Dic₃	semidirect
78. C₂×#208	C₄×C₅	semidirect
79. C₂²×C₃⋊C₈	C₄×C₅	semidirect
81. C₃×#99	C₄×C₅	semidirect
83. #834	C₄×C₅	semidirect
170. #99	C₅×Dic₃	semidirect
232. C₃×C₅	#1391	semidirect
278. C₅	#67696	direct (N × H)
293. C₃	C₅×#1391	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#410	2
C₅×#5240	2
C₅×#5234	2
C₅×#1391	3
#67696	5

Subgroups — C5×#67696 #14279073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#67696 #14279073