Subgroups — C₅×#70703 #14296525

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₅×#5122	C₂	1	2	yes	maximal
3	C₅×#5106	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#815	C₂	1	1	yes	maximal, characteristic
5	C₅×#4555	C₂	1	2	yes	maximal
6	C₅×#1773	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₅×#926	C₂²	1	2	yes	—
10	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#108	C₄	1	4	yes	complemented
13	C₅×#818	C₄	1	2	yes	complemented
14	C₅×#817	C₄	1	2	yes	complemented
15	C₅×#815	—	2	2	no	—
16	C₅×#815	C₄	1	4	yes	complemented
17	C₅×C₈×Dic₃	—	2	2	no	—
18	#70703	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₂×C₅×#108	D₃	1	1	yes	characteristic
20	C₅×#427	—	3	1	no	—
21	C₅×#405	—	3	2	no	—
22	C₂×C₅×#97	—	3	1	no	—
23	C₅×#339	—	3	2	no	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₅×Dic₃	—	2	1	no	—
26	C₃×C₅×M₁₆	—	2	2	no	—
27	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
33	C₅×#217	—	2	2	no	—
34	C₅×C₄⋊Dic₃	C₂×C₄	1	4	yes	—
35	C₄×C₅×Dic₃	—	2	2	no	—
36	C₄×C₅×Dic₃	—	2	1	no	—
37	C₄×C₅×Dic₃	C₂×C₄	1	2	yes	—
38	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
40	C₂×C₅×C₃⋊C₈	Q₈	1	2	yes	—
41	C₂×C₃×#108	C₂×C₅	1	1	yes	characteristic
42	#5122	C₂×C₅	1	2	yes	—
43	#5106	C₂×C₅	1	1	yes	characteristic
44	C₂×#815	C₂×C₅	1	1	yes	characteristic
45	#4555	C₂×C₅	1	2	yes	—
46	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
47	C₂²×C₅×C₈	—	3	1	no	—
48	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	C₅×#117	—	3	2	no	—
50	C₂×C₄²×C₅	—	3	1	no	—
51	C₅×#108	Dic₃	1	4	yes	complemented
52	C₅×#107	—	3	2	no	—
53	C₅×#106	—	3	2	no	—
54	C₅×#97	—	3	4	no	—
55	C₅×#97	—	6	2	no	—
56	C₄×C₅×C₈	—	6	2	no	—
57	#1773	—	3	1	no	Sylow
58	C₂³×C₃×C₅	#41	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
65	C₂×C₅×Dic₃	—	2	2	no	—
66	C₂×C₅×Dic₃	—	2	4	no	—
67	C₂×C₅×Dic₃	—	4	2	no	—
68	C₃×C₅×C₈	—	2	4	no	—
69	C₃×C₅×C₈	C₄²	1	2	yes	—
70	C₃×C₅×C₈	C₄²	1	2	yes	—
71	C₅×C₃⋊C₈	C₄⋊C₄	1	4	yes	—
72	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
73	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
74	#926	C₂²×C₅	1	2	yes	—
75	C₂×C₄×Dic₃	C₂²×C₅	1	1	yes	characteristic
76	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
77	C₃×#108	C₄×C₅	1	4	yes	complemented
78	#818	C₄×C₅	1	2	yes	complemented
79	#817	C₄×C₅	1	2	yes	complemented
80	#815	—	2	2	no	—
81	#815	C₄×C₅	1	4	yes	complemented
82	C₈×Dic₃	—	2	2	no	—
83	C₂²×C₄×C₅	—	6	1	no	—
84	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
85	C₅×M₁₆	—	2	2	no	—
86	C₅×M₁₆	C₂×Dic₃	1	4	yes	—
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	3	2	no	—
89	C₂×C₅×C₈	—	3	2	no	—
90	C₂×C₅×C₈	—	3	2	no	—
91	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
92	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
93	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
94	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
95	C₅×C₄⋊C₄	—	3	4	no	—
96	C₅×C₂²⋊C₄	—	6	2	no	—
97	C₄²×C₅	—	3	2	no	—
98	C₄²×C₅	—	6	2	no	—
99	C₄²×C₅	—	6	1	no	—
100	C₂×#108	C₅×D₃	1	1	yes	characteristic
101	#427	—	3	1	no	—
102	#405	—	3	2	no	—
103	C₂×#97	—	3	1	no	—
104	#339	—	3	2	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	#117	1	1	yes	characteristic
107	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
109	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
110	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
112	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
113	C₅×Dic₃	—	4	2	no	—
114	C₅×Dic₃	—	4	4	no	—
115	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
116	C₂²×Dic₃	—	2	1	no	—
117	C₃×M₁₆	—	2	2	no	—
118	C₃×M₁₆	C₂×C₄×C₅	1	4	yes	—
119	C₂×C₃×C₈	—	2	2	no	—
120	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
121	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
122	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
124	#217	—	2	2	no	—
125	C₄⋊Dic₃	C₂×C₄×C₅	1	4	yes	—
126	C₄×Dic₃	—	2	2	no	—
127	C₄×Dic₃	—	2	1	no	—
128	C₄×Dic₃	C₂×C₄×C₅	1	2	yes	—
129	C₂×C₃⋊C₈	C₂×C₄×C₅	1	2	yes	—
130	C₂×C₃⋊C₈	C₅×D₄	1	2	yes	—
131	C₂×C₃⋊C₈	C₅×Q₈	1	2	yes	—
132	C₂³×C₅	#239	1	1	yes	characteristic
133	C₂×C₄×C₅	—	6	2	no	—
134	C₂×C₄×C₅	—	6	4	no	—
135	C₂×C₄×C₅	—	12	2	no	—
136	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
138	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	#237	1	1	yes	characteristic
140	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
142	C₅×C₈	—	2	4	no	—
143	C₅×C₈	—	3	4	no	—
144	C₅×C₈	C₄×Dic₃	1	2	yes	—
145	C₅×C₈	C₄×Dic₃	1	2	yes	—
146	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
147	C₂²×C₈	—	3	1	no	—
148	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
149	#117	—	3	2	no	—
150	C₂×C₄²	—	3	1	no	—
151	#108	C₅×Dic₃	1	4	yes	complemented
152	#107	—	3	2	no	—
153	#106	—	3	2	no	—
154	#97	—	3	4	no	—
155	#97	—	6	2	no	—
156	C₄×C₈	—	6	2	no	—
157	C₂×C₃×C₅	—	2	1	no	—
158	C₂×C₃×C₅	—	2	1	no	—
159	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
160	C₂×C₃×C₅	#444	1	2	yes	—
161	C₂³×C₃	C₅×#41	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
163	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
164	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
165	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
166	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
167	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
168	C₂×Dic₃	—	2	2	no	—
169	C₂×Dic₃	—	2	4	no	—
170	C₂×Dic₃	—	4	2	no	—
171	C₃×C₈	—	2	4	no	—
172	C₃×C₈	C₄²×C₅	1	2	yes	—
173	C₃×C₈	C₄²×C₅	1	2	yes	—
174	C₃⋊C₈	C₅×C₄⋊C₄	1	4	yes	—
175	C₂²×C₅	—	2	2	no	—
176	C₂²×C₅	#892	1	1	yes	characteristic
177	C₂²×C₅	#893	1	1	yes	characteristic
178	C₂²×C₅	Q₈×Dic₃	1	1	yes	characteristic
179	C₄×C₅	—	12	2	no	—
180	C₄×C₅	—	12	4	no	—
181	C₄×C₅	#887	1	1	yes	characteristic
182	C₄×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
183	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
184	C₄×C₅	D₄×Dic₃	1	1	yes	characteristic
185	C₂²×C₄	—	6	1	no	—
186	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
187	M₁₆	—	2	2	no	—
188	M₁₆	C₂×C₅×Dic₃	1	4	yes	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	3	2	no	—
191	C₂×C₈	—	3	2	no	—
192	C₂×C₈	—	3	2	no	—
193	C₂×C₈	C₄×C₅×D₃	1	2	yes	—
194	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
195	C₂×C₈	C₄×C₅×D₃	1	1	yes	characteristic
196	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
197	C₄⋊C₄	—	3	4	no	—
198	C₂²⋊C₄	—	6	2	no	—
199	C₄²	—	3	2	no	—
200	C₄²	—	6	2	no	—
201	C₄²	—	6	1	no	—
202	C₃×C₅	#1773	1	1	yes	characteristic, complemented, Hall
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	C₅×#117	1	1	yes	characteristic
205	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
206	C₂²×C₃	C₄×C₅×Q₈	1	1	yes	characteristic
207	C₃×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
208	C₃×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
209	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
210	C₃×C₄	C₄×C₅×Q₈	1	1	yes	characteristic
211	Dic₃	—	4	2	no	—
212	Dic₃	—	4	4	no	—
213	C₂×C₅	—	2	1	no	—
214	C₂×C₅	—	2	1	no	—
215	C₂×C₅	#4894	1	2	yes	—
216	C₂×C₅	Dic₃×C₄⋊C₄	1	1	yes	characteristic
217	C₂³	C₅×#239	1	1	yes	characteristic
218	C₂×C₄	—	6	2	no	—
219	C₂×C₄	—	6	4	no	—
220	C₂×C₄	—	12	2	no	—
221	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
222	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
223	C₂×C₄	C₅×D₃×D₄	1	1	yes	characteristic
224	C₂×C₄	C₅×#237	1	1	yes	characteristic
225	C₂×C₄	C₅×D₃×Q₈	1	1	yes	characteristic
226	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
227	C₈	—	2	4	no	—
228	C₈	—	3	4	no	—
229	C₈	C₄×C₅×Dic₃	1	2	yes	—
230	C₈	C₄×C₅×Dic₃	1	2	yes	—
231	C₂×C₃	—	2	1	no	—
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
234	C₂×C₃	C₅×#444	1	2	yes	—
235	C₅	#70703	1	1	yes	characteristic, direct product factor, Sylow
236	C₂²	—	2	2	no	—
237	C₂²	C₅×#892	1	1	yes	characteristic
238	C₂²	C₅×#893	1	1	yes	characteristic
239	C₂²	C₅×Q₈×Dic₃	1	1	yes	characteristic
240	C₄	—	12	2	no	—
241	C₄	—	12	4	no	—
242	C₄	C₅×#887	1	1	yes	characteristic
243	C₄	C₅×D₃×C₄⋊C₄	1	1	yes	characteristic
244	C₄	C₂×C₄×C₅×Dic₃	1	1	yes	characteristic
245	C₄	C₅×D₄×Dic₃	1	1	yes	characteristic
246	C₃	C₅×#1773	1	1	yes	characteristic, complemented, Sylow
247	C₂	—	2	1	no	—
248	C₂	—	2	1	no	—
249	C₂	C₅×#4894	1	2	yes	—
250	C₂	C₅×Dic₃×C₄⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	D₃×D₄
commutator	C₃×C₄	C₂×C₄²×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#108	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂³×C₃×C₅	#41
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₂×C₅×#108	D₃
hypocenter	C₃	C₅×#1773
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₂×C₅×#108

Complements

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

N	H ≅ G/N	type
12. C₃×C₅×#108	C₄	semidirect
13. C₅×#818	C₄	semidirect
14. C₅×#817	C₄	semidirect
16. C₅×#815	C₄	semidirect
18. #70703	C₅	direct (N × H)
51. C₅×#108	Dic₃	semidirect
77. C₃×#108	C₄×C₅	semidirect
78. #818	C₄×C₅	semidirect
79. #817	C₄×C₅	semidirect
81. #815	C₄×C₅	semidirect
151. #108	C₅×Dic₃	semidirect
202. C₃×C₅	#1773	semidirect
235. C₅	#70703	direct (N × H)
246. C₃	C₅×#1773	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#108	2
C₅×#5122	2
C₅×#5106	2
C₂×C₅×#815	2
C₅×#4555	2
C₅×#1773	3
#70703	5

Subgroups — C5×#70703 #14296525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70703 #14296525