Subgroups — C₅×#70922 #14315682

Order 1920.

288 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#443	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#5194	C₂	1	1	yes	maximal, characteristic
3	C₅×#4700	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4497	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#4493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#4461	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2179	—	3	1	no	maximal, Hall
9	C₃×C₅×#135	C₂²	1	1	yes	characteristic
10	C₃×C₅×#131	—	2	1	no	—
11	C₅×#904	C₂²	1	1	yes	characteristic
12	C₅×#901	—	2	1	no	—
13	C₃×C₅×#108	C₂²	1	1	yes	characteristic
14	C₃×C₅×#104	—	2	1	no	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×#829	C₄	1	1	yes	characteristic, complemented
17	C₅×#828	C₄	1	1	yes	characteristic, complemented
18	C₅×#827	C₄	1	1	yes	characteristic, complemented
19	C₅×#826	C₄	1	1	yes	characteristic, complemented
20	C₅×#819	C₂²	1	1	yes	characteristic
21	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
23	C₅×#805	—	2	1	no	—
24	#70922	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#508	—	3	1	no	—
26	C₅×#443	—	3	1	no	—
27	C₅×#443	D₃	1	1	yes	characteristic, complemented
28	C₅×#368	—	3	1	no	—
29	C₅×#359	—	3	1	no	—
30	C₅×#359	—	3	1	no	—
31	C₅×#346	—	3	1	no	—
32	C₃×C₅×#41	—	2	1	no	—
33	C₅×#235	—	2	1	no	—
34	C₃×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic
35	C₃×C₅×SD₁₆	—	2	1	no	—
36	C₃×C₅×D₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₅×M₁₆	—	2	1	no	—
38	C₃×C₅×M₁₆	—	4	1	no	—
39	C₂×C₃×C₅×C₈	—	4	1	no	—
40	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₅×#208	—	2	1	no	—
44	C₅×Dic₁₂	C₂×C₄	1	1	yes	characteristic
45	C₅×D₂₄	C₂×C₄	1	1	yes	characteristic
46	C₅×#204	—	2	1	no	—
47	C₅×ΓL(1,𝔽₂₅)	—	4	1	no	—
48	C₅×D₃×C₈	—	4	1	no	—
49	C₅×C₃⋊C₁₆	—	2	1	no	—
50	C₅×C₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
51	C₅×C₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
52	C₃×#443	C₂×C₅	1	1	yes	characteristic, complemented
53	#5194	C₂×C₅	1	1	yes	characteristic
54	#4700	C₂×C₅	1	1	yes	characteristic, complemented
55	#4567	C₂×C₅	1	1	yes	characteristic, complemented
56	#4497	C₂×C₅	1	1	yes	characteristic, complemented
57	#4493	C₂×C₅	1	1	yes	characteristic, complemented
58	#4461	C₂×C₅	1	1	yes	characteristic, complemented
59	C₅×#135	—	3	1	no	—
60	C₅×#135	C₂×D₃	1	1	yes	characteristic
61	C₅×#131	—	2	1	no	—
62	C₅×#131	—	6	1	no	—
63	C₅×Q₃₂	—	3	1	no	—
64	C₅×SD₃₂	—	3	1	no	—
65	C₅×SD₃₂	—	3	1	no	—
66	C₅×D₁₆	—	3	1	no	—
67	C₂×C₅×C₁₆	—	3	1	no	—
68	C₂×C₅×C₁₆	—	3	1	no	—
69	C₅×#108	—	3	1	no	—
70	C₅×#108	C₂×D₃	1	1	yes	characteristic
71	C₅×#104	—	2	1	no	—
72	C₅×#104	—	6	1	no	—
73	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
74	#2179	—	3	1	no	Sylow
75	C₃×C₅×Q₈	—	2	1	no	—
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	—	4	1	no	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	4	1	no	—
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₅×#67	—	4	1	no	—
82	C₅×D₁₂	—	2	1	no	—
83	C₄×C₅×D₃	—	4	1	no	—
84	C₅×Dic₆	—	2	1	no	—
85	C₃×C₅×C₈	—	4	1	no	—
86	C₃×C₅×C₈	D₈	1	1	yes	characteristic
87	C₃×C₅×C₈	D₈	1	1	yes	characteristic
88	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₃×C₅×C₈	#41	1	1	yes	characteristic
90	C₅×C₃⋊C₈	—	4	1	no	—
91	C₃×#135	C₂²×C₅	1	1	yes	characteristic
92	C₃×#131	—	2	1	no	—
93	#904	C₂²×C₅	1	1	yes	characteristic
94	#901	—	2	1	no	—
95	C₃×#108	C₂²×C₅	1	1	yes	characteristic
96	C₃×#104	—	2	1	no	—
97	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
98	#829	C₄×C₅	1	1	yes	characteristic, complemented
99	#828	C₄×C₅	1	1	yes	characteristic, complemented
100	#827	C₄×C₅	1	1	yes	characteristic, complemented
101	#826	C₄×C₅	1	1	yes	characteristic, complemented
102	#819	C₂²×C₅	1	1	yes	characteristic
103	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
104	C₂×C₃⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
105	#805	—	2	1	no	—
106	C₅×#41	—	2	1	no	—
107	C₅×#41	—	6	1	no	—
108	C₅×Q₁₆	—	3	1	no	—
109	C₅×Q₁₆	C₄×D₃	1	1	yes	characteristic
110	C₅×SD₁₆	—	2	1	no	—
111	C₅×SD₁₆	—	6	1	no	—
112	C₅×D₈	—	3	1	no	—
113	C₅×D₈	C₄×D₃	1	1	yes	characteristic
114	C₅×M₁₆	—	2	1	no	—
115	C₅×M₁₆	—	4	1	no	—
116	C₅×M₁₆	—	6	1	no	—
117	C₅×M₁₆	—	12	1	no	—
118	C₂×C₅×C₈	—	4	1	no	—
119	C₂×C₅×C₈	—	12	1	no	—
120	C₂×C₅×C₈	#67	1	1	yes	characteristic
121	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
122	C₄²×C₅	#67	1	1	yes	characteristic
123	C₅×C₁₆	—	3	1	no	—
124	C₅×C₁₆	—	3	1	no	—
125	C₅×C₁₆	—	6	1	no	—
126	#508	—	3	1	no	—
127	#443	—	3	1	no	—
128	#443	C₅×D₃	1	1	yes	characteristic, complemented
129	#368	—	3	1	no	—
130	#359	—	3	1	no	—
131	#359	—	3	1	no	—
132	#346	—	3	1	no	—
133	C₂²×C₃×C₅	—	4	1	no	—
134	C₂²×C₃×C₅	#135	1	1	yes	characteristic
135	C₂×C₅×D₃	—	4	1	no	—
136	C₃×C₄×C₅	—	4	1	no	—
137	C₃×C₄×C₅	—	4	1	no	—
138	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
139	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
140	C₅×Dic₃	—	4	1	no	—
141	C₃×#41	—	2	1	no	—
142	#235	—	2	1	no	—
143	C₃×Q₁₆	C₂×C₄×C₅	1	1	yes	characteristic
144	C₃×SD₁₆	—	2	1	no	—
145	C₃×D₈	C₂×C₄×C₅	1	1	yes	characteristic
146	C₃×M₁₆	—	2	1	no	—
147	C₃×M₁₆	—	4	1	no	—
148	C₂×C₃×C₈	—	4	1	no	—
149	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
150	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
151	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
152	#208	—	2	1	no	—
153	Dic₁₂	C₂×C₄×C₅	1	1	yes	characteristic
154	D₂₄	C₂×C₄×C₅	1	1	yes	characteristic
155	#204	—	2	1	no	—
156	ΓL(1,𝔽₂₅)	—	4	1	no	—
157	D₃×C₈	—	4	1	no	—
158	C₃⋊C₁₆	—	2	1	no	—
159	C₃⋊C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
160	C₃⋊C₁₆	C₂×C₄×C₅	1	1	yes	characteristic
161	C₅×Q₈	—	2	1	no	—
162	C₅×Q₈	—	6	1	no	—
163	C₅×D₄	—	2	1	no	—
164	C₅×D₄	—	4	1	no	—
165	C₅×D₄	—	6	1	no	—
166	C₅×D₄	—	12	1	no	—
167	C₂×C₄×C₅	—	2	1	no	—
168	C₂×C₄×C₅	—	4	1	no	—
169	C₂×C₄×C₅	—	12	1	no	—
170	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
171	C₅×C₈	—	4	1	no	—
172	C₅×C₈	—	12	1	no	—
173	C₅×C₈	#213	1	1	yes	characteristic
174	C₅×C₈	#213	1	1	yes	characteristic
175	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
176	C₅×C₈	#235	1	1	yes	characteristic
177	#135	—	3	1	no	—
178	#135	C₂×C₅×D₃	1	1	yes	characteristic
179	#131	—	2	1	no	—
180	#131	—	6	1	no	—
181	Q₃₂	—	3	1	no	—
182	SD₃₂	—	3	1	no	—
183	SD₃₂	—	3	1	no	—
184	D₁₆	—	3	1	no	—
185	C₂×C₁₆	—	3	1	no	—
186	C₂×C₁₆	—	3	1	no	—
187	#108	—	3	1	no	—
188	#108	C₂×C₅×D₃	1	1	yes	characteristic
189	#104	—	2	1	no	—
190	#104	—	6	1	no	—
191	C₄×C₈	C₂×C₅×D₃	1	1	yes	characteristic
192	C₂×C₃×C₅	—	2	1	no	—
193	C₂×C₃×C₅	—	8	1	no	—
194	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
195	C₅×D₃	—	8	1	no	—
196	C₃×Q₈	—	2	1	no	—
197	C₃×D₄	—	2	1	no	—
198	C₃×D₄	—	4	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	4	1	no	—
201	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
202	#67	—	4	1	no	—
203	D₁₂	—	2	1	no	—
204	C₄×D₃	—	4	1	no	—
205	Dic₆	—	2	1	no	—
206	C₃×C₈	—	4	1	no	—
207	C₃×C₈	C₅×D₈	1	1	yes	characteristic
208	C₃×C₈	C₅×D₈	1	1	yes	characteristic
209	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
210	C₃×C₈	C₅×#41	1	1	yes	characteristic
211	C₃⋊C₈	—	4	1	no	—
212	C₂²×C₅	—	4	1	no	—
213	C₂²×C₅	—	12	1	no	—
214	C₂²×C₅	#950	1	1	yes	characteristic
215	C₄×C₅	—	4	1	no	—
216	C₄×C₅	—	4	1	no	—
217	C₄×C₅	—	12	1	no	—
218	C₄×C₅	C₄×#67	1	1	yes	characteristic
219	C₄×C₅	C₂×#213	1	1	yes	characteristic
220	#41	—	2	1	no	—
221	#41	—	6	1	no	—
222	Q₁₆	—	3	1	no	—
223	Q₁₆	C₄×C₅×D₃	1	1	yes	characteristic
224	SD₁₆	—	2	1	no	—
225	SD₁₆	—	6	1	no	—
226	D₈	—	3	1	no	—
227	D₈	C₄×C₅×D₃	1	1	yes	characteristic
228	M₁₆	—	2	1	no	—
229	M₁₆	—	4	1	no	—
230	M₁₆	—	6	1	no	—
231	M₁₆	—	12	1	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	12	1	no	—
234	C₂×C₈	C₅×#67	1	1	yes	characteristic
235	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
236	C₄²	C₅×#67	1	1	yes	characteristic
237	C₁₆	—	3	1	no	—
238	C₁₆	—	3	1	no	—
239	C₁₆	—	6	1	no	—
240	C₃×C₅	#2179	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	4	1	no	—
242	C₂²×C₃	C₅×#135	1	1	yes	characteristic
243	C₂×D₃	—	4	1	no	—
244	C₃×C₄	—	4	1	no	—
245	C₃×C₄	—	4	1	no	—
246	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
247	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
248	Dic₃	—	4	1	no	—
249	C₂×C₅	—	2	1	no	—
250	C₂×C₅	—	8	1	no	—
251	C₂×C₅	—	24	1	no	—
252	C₂×C₅	C₄×#213	1	1	yes	characteristic
253	Q₈	—	2	1	no	—
254	Q₈	—	6	1	no	—
255	D₄	—	2	1	no	—
256	D₄	—	4	1	no	—
257	D₄	—	6	1	no	—
258	D₄	—	12	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	12	1	no	—
262	C₂×C₄	C₂×C₅×#67	1	1	yes	characteristic
263	C₈	—	4	1	no	—
264	C₈	—	12	1	no	—
265	C₈	C₅×#213	1	1	yes	characteristic
266	C₈	C₅×#213	1	1	yes	characteristic
267	C₈	C₂×C₄×C₅×D₃	1	1	yes	characteristic
268	C₈	C₅×#235	1	1	yes	characteristic
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	8	1	no	—
271	C₂×C₃	C₄×C₅×D₈	1	1	yes	characteristic
272	D₃	—	8	1	no	—
273	C₅	#70922	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	—	4	1	no	—
275	C₂²	—	12	1	no	—
276	C₂²	C₅×#950	1	1	yes	characteristic
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	12	1	no	—
280	C₄	C₄×C₅×#67	1	1	yes	characteristic
281	C₄	C₂×C₅×#213	1	1	yes	characteristic
282	C₃	C₅×#2179	1	1	yes	characteristic, complemented, Sylow
283	C₂	—	2	1	no	—
284	C₂	—	8	1	no	—
285	C₂	—	24	1	no	—
286	C₂	C₄×C₅×#213	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	C₂×#213
commutator	C₃×C₈	C₂²×C₄×C₅
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₃×C₅×#443	C₂
socle	C₂×C₃×C₅	C₄×D₈
prime radical	C₅×#5194	C₂
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₅×#443	D₃
hypocenter	C₃	C₅×#2179
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂×C₅×C₈ ◁ C₄×C₅×C₈ ◁ C₅×#443

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#443	C₂	semidirect
3. C₅×#4700	C₂	semidirect
4. C₅×#4567	C₂	semidirect
5. C₅×#4497	C₂	semidirect
6. C₅×#4493	C₂	semidirect
7. C₅×#4461	C₂	semidirect
16. C₅×#829	C₄	semidirect
17. C₅×#828	C₄	semidirect
18. C₅×#827	C₄	semidirect
19. C₅×#826	C₄	semidirect
24. #70922	C₅	direct (N × H)
27. C₅×#443	D₃	semidirect
52. C₃×#443	C₂×C₅	semidirect
54. #4700	C₂×C₅	semidirect
55. #4567	C₂×C₅	semidirect
56. #4497	C₂×C₅	semidirect
57. #4493	C₂×C₅	semidirect
58. #4461	C₂×C₅	semidirect
98. #829	C₄×C₅	semidirect
99. #828	C₄×C₅	semidirect
100. #827	C₄×C₅	semidirect
101. #826	C₄×C₅	semidirect
128. #443	C₅×D₃	semidirect
240. C₃×C₅	#2179	semidirect
273. C₅	#70922	direct (N × H)
282. C₃	C₅×#2179	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#443	2
C₅×#5194	2
C₅×#4700	2
C₅×#4567	2
C₅×#4497	2
C₅×#4493	2
C₅×#4461	2
C₅×#2179	3
#70922	5

Subgroups — C5×#70922 #14315682

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70922 #14315682