Subgroups — C₅×#68408 #14291346

Order 1920.

344 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5076	C₂	1	1	yes	maximal, characteristic
2	C₅×#5075	C₂	1	1	yes	maximal, characteristic
3	C₅×#4851	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4830	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#331	C₂	1	1	yes	maximal, characteristic
6	C₅×#4482	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4480	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#1666	—	3	1	no	maximal, Hall
9	C₃×C₅×#127	C₂²	1	1	yes	characteristic
10	C₂×C₅×#214	—	2	1	no	—
11	C₂×C₅×#214	—	2	1	no	—
12	C₂×C₅×#214	—	2	1	no	—
13	C₅×#869	C₂²	1	1	yes	characteristic
14	C₄×C₅×Dic₆	C₂²	1	1	yes	characteristic
15	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₅×#832	—	2	1	no	—
18	C₅×#817	—	2	1	no	—
19	C₅×#816	—	2	1	no	—
20	C₅×Dic₃⋊C₈	—	2	1	no	—
21	C₅×C₈×Dic₃	—	2	1	no	—
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#68408	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#492	—	3	1	no	—
26	C₅×#475	—	3	1	no	—
27	C₅×#460	—	3	1	no	—
28	C₅×C₈×Q₈	—	3	1	no	—
29	C₅×#334	—	3	1	no	—
30	C₅×#332	—	3	1	no	—
31	C₅×#331	D₃	1	1	yes	characteristic
32	C₂×C₃×C₅×D₄	—	2	1	no	—
33	C₂×C₃×C₅×D₄	—	2	1	no	—
34	C₂×C₃×C₅×D₄	—	2	1	no	—
35	C₂×C₅×Dic₆	—	2	1	no	—
36	C₂×C₅×Dic₆	D₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	—	2	1	no	—
38	C₂×C₃×C₅×C₈	—	2	1	no	—
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₅×#214	—	4	2	no	—
41	C₅×#214	—	4	2	no	—
42	C₅×#214	—	4	2	no	—
43	C₅×C₄⋊Dic₃	—	2	1	no	—
44	C₅×C₄⋊Dic₃	—	2	1	no	—
45	C₅×C₄⋊Dic₃	D₄	1	1	yes	characteristic
46	C₅×Dic₃⋊C₄	—	2	1	no	—
47	C₄×C₅×Dic₃	—	2	1	no	—
48	C₂×C₅×C₃⋊C₈	—	2	1	no	—
49	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
50	C₂×C₅×C₃⋊C₈	D₄	1	1	yes	characteristic
51	#5076	C₂×C₅	1	1	yes	characteristic
52	#5075	C₂×C₅	1	1	yes	characteristic
53	#4851	C₂×C₅	1	1	yes	characteristic, complemented
54	#4830	C₂×C₅	1	1	yes	characteristic, complemented
55	C₃×#331	C₂×C₅	1	1	yes	characteristic
56	#4482	C₂×C₅	1	1	yes	characteristic, complemented
57	#4480	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂×C₅×SD₁₆	—	6	1	no	—
59	C₂×C₅×SD₁₆	—	6	1	no	—
60	C₂×C₅×SD₁₆	—	6	1	no	—
61	C₅×C₄⋊Q₈	—	3	1	no	—
62	C₅×#127	C₂×D₃	1	1	yes	characteristic
63	C₄×C₅×Q₈	—	3	1	no	—
64	C₅×#106	—	6	1	no	—
65	C₅×C₄⋊C₈	—	3	1	no	—
66	C₅×C₄⋊C₈	—	6	1	no	—
67	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
68	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
69	C₅×#103	—	6	1	no	—
70	C₅×#102	—	6	1	no	—
71	C₄×C₅×C₈	—	3	1	no	—
72	C₄×C₅×C₈	—	6	1	no	—
73	#1666	—	3	1	no	Sylow
74	C₂³×C₃×C₅	—	4	1	no	—
75	C₃×C₅×D₄	—	4	1	no	—
76	C₃×C₅×D₄	—	4	1	no	—
77	C₃×C₅×D₄	—	4	1	no	—
78	C₃×C₅×D₄	—	4	1	no	—
79	C₃×C₅×D₄	—	8	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	C₂×C₅×Dic₃	—	2	1	no	—
85	C₂×C₅×Dic₃	—	4	1	no	—
86	C₅×Dic₆	—	2	1	no	—
87	C₅×Dic₆	—	4	1	no	—
88	C₅×Dic₆	—	4	1	no	—
89	C₅×Dic₆	SD₁₆	1	2	yes	complemented
90	C₃×C₅×C₈	—	4	1	no	—
91	C₃×C₅×C₈	—	4	1	no	—
92	C₅×C₃⋊C₈	—	2	1	no	—
93	C₅×C₃⋊C₈	—	4	1	no	—
94	C₅×C₃⋊C₈	SD₁₆	1	2	yes	complemented
95	C₃×#127	C₂²×C₅	1	1	yes	characteristic
96	C₂×#214	—	2	1	no	—
97	C₂×#214	—	2	1	no	—
98	C₂×#214	—	2	1	no	—
99	#869	C₂²×C₅	1	1	yes	characteristic
100	C₄×Dic₆	C₂²×C₅	1	1	yes	characteristic
101	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
102	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
103	#832	—	2	1	no	—
104	#817	—	2	1	no	—
105	#816	—	2	1	no	—
106	Dic₃⋊C₈	—	2	1	no	—
107	C₈×Dic₃	—	2	1	no	—
108	C₄⋊(C₃⋊C₈)	C₂²×C₅	1	1	yes	characteristic
109	C₄×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
110	C₂×C₅×Q₈	—	3	1	no	—
111	C₂×C₅×Q₈	—	6	1	no	—
112	C₂×C₅×D₄	—	2	1	no	—
113	C₂×C₅×D₄	—	2	1	no	—
114	C₂×C₅×D₄	—	2	1	no	—
115	C₅×SD₁₆	—	12	2	no	—
116	C₅×SD₁₆	—	12	2	no	—
117	C₅×SD₁₆	—	12	2	no	—
118	C₂×C₅×C₈	—	2	1	no	—
119	C₂×C₅×C₈	—	2	1	no	—
120	C₂×C₅×C₈	—	3	1	no	—
121	C₂×C₅×C₈	—	3	1	no	—
122	C₂×C₅×C₈	—	6	1	no	—
123	C₅×C₄⋊C₄	—	3	1	no	—
124	C₅×C₄⋊C₄	—	6	1	no	—
125	C₅×C₄⋊C₄	—	6	1	no	—
126	C₅×C₄⋊C₄	—	6	1	no	—
127	C₄²×C₅	—	6	1	no	—
128	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
129	#492	—	3	1	no	—
130	#475	—	3	1	no	—
131	#460	—	3	1	no	—
132	C₈×Q₈	—	3	1	no	—
133	#334	—	3	1	no	—
134	#332	—	3	1	no	—
135	#331	C₅×D₃	1	1	yes	characteristic
136	C₂²×C₃×C₅	—	8	1	no	—
137	C₂²×C₃×C₅	—	8	1	no	—
138	C₂²×C₃×C₅	—	8	1	no	—
139	C₂²×C₃×C₅	#121	1	1	yes	characteristic
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
143	C₃×C₄×C₅	#135	1	1	yes	characteristic
144	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₅×Dic₃	—	2	2	no	—
146	C₅×Dic₃	—	4	1	no	—
147	C₅×Dic₃	—	8	1	no	—
148	C₂×C₃×D₄	—	2	1	no	—
149	C₂×C₃×D₄	—	2	1	no	—
150	C₂×C₃×D₄	—	2	1	no	—
151	C₂×Dic₆	—	2	1	no	—
152	C₂×Dic₆	C₅×D₄	1	1	yes	characteristic
153	C₂×C₃×C₈	—	2	1	no	—
154	C₂×C₃×C₈	—	2	1	no	—
155	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
156	#214	—	4	2	no	—
157	#214	—	4	2	no	—
158	#214	—	4	2	no	—
159	C₄⋊Dic₃	—	2	1	no	—
160	C₄⋊Dic₃	—	2	1	no	—
161	C₄⋊Dic₃	C₅×D₄	1	1	yes	characteristic
162	Dic₃⋊C₄	—	2	1	no	—
163	C₄×Dic₃	—	2	1	no	—
164	C₂×C₃⋊C₈	—	2	1	no	—
165	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
166	C₂×C₃⋊C₈	C₅×D₄	1	1	yes	characteristic
167	C₂³×C₅	—	4	1	no	—
168	C₅×Q₈	—	3	2	no	—
169	C₅×Q₈	—	6	1	no	—
170	C₅×Q₈	—	12	1	no	—
171	C₅×Q₈	—	12	1	no	—
172	C₅×D₄	—	4	1	no	—
173	C₅×D₄	—	4	1	no	—
174	C₅×D₄	—	4	1	no	—
175	C₅×D₄	—	4	1	no	—
176	C₅×D₄	—	8	1	no	—
177	C₂×C₄×C₅	—	6	1	no	—
178	C₂×C₄×C₅	—	6	1	no	—
179	C₂×C₄×C₅	—	12	1	no	—
180	C₂×C₄×C₅	#235	1	1	yes	characteristic
181	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
183	C₅×C₈	—	3	2	no	—
184	C₅×C₈	—	4	1	no	—
185	C₅×C₈	—	4	1	no	—
186	C₅×C₈	—	6	1	no	—
187	C₅×C₈	—	12	1	no	—
188	C₂×SD₁₆	—	6	1	no	—
189	C₂×SD₁₆	—	6	1	no	—
190	C₂×SD₁₆	—	6	1	no	—
191	C₄⋊Q₈	—	3	1	no	—
192	#127	C₂×C₅×D₃	1	1	yes	characteristic
193	C₄×Q₈	—	3	1	no	—
194	#106	—	6	1	no	—
195	C₄⋊C₈	—	3	1	no	—
196	C₄⋊C₈	—	6	1	no	—
197	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
198	C₄⋊C₈	C₂×C₅×D₃	1	1	yes	characteristic
199	#103	—	6	1	no	—
200	#102	—	6	1	no	—
201	C₄×C₈	—	3	1	no	—
202	C₄×C₈	—	6	1	no	—
203	C₂×C₃×C₅	—	16	1	no	—
204	C₂×C₃×C₅	#460	1	1	yes	characteristic
205	C₂×C₃×C₅	#465	1	1	yes	characteristic
206	C₂×C₃×C₅	#471	1	1	yes	characteristic
207	C₂³×C₃	—	4	1	no	—
208	C₃×D₄	—	4	1	no	—
209	C₃×D₄	—	4	1	no	—
210	C₃×D₄	—	4	1	no	—
211	C₃×D₄	—	4	1	no	—
212	C₃×D₄	—	8	1	no	—
213	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
216	C₂×Dic₃	—	2	1	no	—
217	C₂×Dic₃	—	2	1	no	—
218	C₂×Dic₃	—	4	1	no	—
219	Dic₆	—	2	1	no	—
220	Dic₆	—	4	1	no	—
221	Dic₆	—	4	1	no	—
222	Dic₆	C₅×SD₁₆	1	2	yes	complemented
223	C₃×C₈	—	4	1	no	—
224	C₃×C₈	—	4	1	no	—
225	C₃⋊C₈	—	2	1	no	—
226	C₃⋊C₈	—	4	1	no	—
227	C₃⋊C₈	C₅×SD₁₆	1	2	yes	complemented
228	C₂²×C₅	—	8	1	no	—
229	C₂²×C₅	—	8	1	no	—
230	C₂²×C₅	—	8	1	no	—
231	C₂²×C₅	#884	1	1	yes	characteristic
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	6	2	no	—
234	C₄×C₅	—	12	1	no	—
235	C₄×C₅	—	24	1	no	—
236	C₄×C₅	#911	1	1	yes	characteristic
237	C₄×C₅	#912	1	1	yes	characteristic
238	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
239	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
240	C₂×Q₈	—	3	1	no	—
241	C₂×Q₈	—	6	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	2	1	no	—
245	SD₁₆	—	12	2	no	—
246	SD₁₆	—	12	2	no	—
247	SD₁₆	—	12	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	3	1	no	—
251	C₂×C₈	—	3	1	no	—
252	C₂×C₈	—	6	1	no	—
253	C₄⋊C₄	—	3	1	no	—
254	C₄⋊C₄	—	6	1	no	—
255	C₄⋊C₄	—	6	1	no	—
256	C₄⋊C₄	—	6	1	no	—
257	C₄²	—	6	1	no	—
258	C₄²	C₂²×C₅×D₃	1	1	yes	characteristic
259	C₃×C₅	#1666	1	1	yes	characteristic, complemented, Hall
260	C₂²×C₃	—	8	1	no	—
261	C₂²×C₃	—	8	1	no	—
262	C₂²×C₃	—	8	1	no	—
263	C₂²×C₃	C₅×#121	1	1	yes	characteristic
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
266	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
267	C₃×C₄	C₅×#135	1	1	yes	characteristic
268	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
269	Dic₃	—	2	2	no	—
270	Dic₃	—	4	1	no	—
271	Dic₃	—	8	1	no	—
272	C₂×C₅	—	16	1	no	—
273	C₂×C₅	#4748	1	1	yes	characteristic
274	C₂×C₅	#4762	1	1	yes	characteristic
275	C₂×C₅	#4778	1	1	yes	characteristic
276	C₂³	—	4	1	no	—
277	Q₈	—	3	2	no	—
278	Q₈	—	6	1	no	—
279	Q₈	—	12	1	no	—
280	Q₈	—	12	1	no	—
281	D₄	—	4	1	no	—
282	D₄	—	4	1	no	—
283	D₄	—	4	1	no	—
284	D₄	—	4	1	no	—
285	D₄	—	8	1	no	—
286	C₂×C₄	—	6	1	no	—
287	C₂×C₄	—	6	1	no	—
288	C₂×C₄	—	12	1	no	—
289	C₂×C₄	C₅×#235	1	1	yes	characteristic
290	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
291	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
292	C₈	—	3	2	no	—
293	C₈	—	4	1	no	—
294	C₈	—	4	1	no	—
295	C₈	—	6	1	no	—
296	C₈	—	12	1	no	—
297	C₂×C₃	—	16	1	no	—
298	C₂×C₃	C₅×#460	1	1	yes	characteristic
299	C₂×C₃	C₅×#465	1	1	yes	characteristic
300	C₂×C₃	C₅×#471	1	1	yes	characteristic
301	C₅	#68408	1	1	yes	characteristic, direct product factor, Sylow
302	C₂²	—	8	1	no	—
303	C₂²	—	8	1	no	—
304	C₂²	—	8	1	no	—
305	C₂²	C₅×#884	1	1	yes	characteristic
306	C₄	—	2	1	no	—
307	C₄	—	6	2	no	—
308	C₄	—	12	1	no	—
309	C₄	—	24	1	no	—
310	C₄	C₅×#911	1	1	yes	characteristic
311	C₄	C₅×#912	1	1	yes	characteristic
312	C₄	C₅×D₃×SD₁₆	1	1	yes	characteristic
313	C₄	C₅×D₃×SD₁₆	1	1	yes	characteristic
314	C₃	C₅×#1666	1	1	yes	characteristic, complemented, Sylow
315	C₂	—	16	1	no	—
316	C₂	C₅×#4748	1	1	yes	characteristic
317	C₂	C₅×#4762	1	1	yes	characteristic
318	C₂	C₅×#4778	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×#331	C₂
socle	C₂²×C₃×C₅	#121
prime radical	C₃×C₅×#127	C₂²
hypercenter	C₅×#331	D₃
hypocenter	C₃	C₅×#1666
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ C₅×#331

Complements

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

N	H ≅ G/N	type
3. C₅×#4851	C₂	semidirect
4. C₅×#4830	C₂	semidirect
6. C₅×#4482	C₂	semidirect
7. C₅×#4480	C₂	semidirect
24. #68408	C₅	direct (N × H)
53. #4851	C₂×C₅	semidirect
54. #4830	C₂×C₅	semidirect
56. #4482	C₂×C₅	semidirect
57. #4480	C₂×C₅	semidirect
89. C₅×Dic₆	SD₁₆	semidirect
94. C₅×C₃⋊C₈	SD₁₆	semidirect
222. Dic₆	C₅×SD₁₆	semidirect
227. C₃⋊C₈	C₅×SD₁₆	semidirect
259. C₃×C₅	#1666	semidirect
301. C₅	#68408	direct (N × H)
314. C₃	C₅×#1666	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5076	2
C₅×#5075	2
C₅×#4851	2
C₅×#4830	2
C₃×C₅×#331	2
C₅×#4482	2
C₅×#4480	2
C₅×#1666	3
#68408	5

Subgroups — C5×#68408 #14291346

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#68408 #14291346