Subgroups — C₅×#70135 #14299668

Order 1920.

504 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#5819	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#413	C₂	1	1	yes	maximal, characteristic
3	C₅×#5234	C₂	1	1	yes	maximal, characteristic
4	C₅×#5125	C₂	1	2	yes	maximal, complemented
5	C₅×#4558	C₂	1	2	yes	maximal, complemented
6	C₅×#1858	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₅×#948	—	2	4	no	—
10	C₂×C₅×#208	—	2	2	no	—
11	C₂×C₅×#208	C₂²	1	1	yes	characteristic
12	C₂×C₅×#208	C₂²	1	2	yes	—
13	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×#101	C₄	1	2	yes	—
15	C₃×C₅×#100	C₄	1	2	yes	—
16	C₅×#830	—	2	2	no	—
17	C₅×#819	—	2	4	no	—
18	#70135	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#568	—	3	1	no	—
20	C₅×#430	—	3	2	no	—
21	C₅×#413	D₃	1	1	yes	characteristic
22	C₅×#408	—	3	1	no	—
23	C₅×#341	—	3	2	no	—
24	C₃×C₅×#41	—	2	2	no	—
25	C₃×C₅×#41	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×D₄	Q₈	1	2	yes	—
29	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₃×C₅×M₁₆	—	2	2	no	—
32	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₅×#208	—	2	4	no	—
35	C₅×#208	—	2	2	no	—
36	C₅×#208	—	2	2	no	—
37	C₅×#208	D₄	1	4	yes	—
38	C₂×C₅×C₃⋊C₈	—	2	4	no	—
39	C₂×C₅×C₃⋊C₈	—	2	2	no	—
40	C₂×C₅×C₃⋊C₈	—	2	1	no	—
41	C₂×C₅×C₃⋊C₈	—	2	1	no	—
42	C₂×C₅×C₃⋊C₈	—	2	2	no	—
43	#5819	C₂×C₅	1	1	yes	characteristic
44	C₃×#413	C₂×C₅	1	1	yes	characteristic
45	#5234	C₂×C₅	1	1	yes	characteristic
46	#5125	C₂×C₅	1	2	yes	complemented
47	#4558	C₂×C₅	1	2	yes	complemented
48	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
49	C₅×#131	—	6	4	no	—
50	C₂×C₅×M₁₆	—	3	2	no	—
51	C₂×C₅×M₁₆	—	3	1	no	—
52	C₂×C₅×M₁₆	—	6	2	no	—
53	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
54	C₂²×C₅×C₈	—	3	1	no	—
55	C₅×#108	—	6	4	no	—
56	C₅×#101	Dic₃	1	2	yes	—
57	C₅×#100	Dic₃	1	2	yes	—
58	C₅×C₂²⋊C₈	—	6	2	no	—
59	#1858	—	3	1	no	Sylow
60	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
61	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₃×C₅×Q₈	—	2	2	no	—
63	C₃×C₅×D₄	—	2	4	no	—
64	C₃×C₅×D₄	—	2	2	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	#41	1	2	yes	—
72	C₃×C₅×C₈	—	2	4	no	—
73	C₅×C₃⋊C₈	—	2	4	no	—
74	C₅×C₃⋊C₈	—	4	1	no	—
75	C₅×C₃⋊C₈	—	4	1	no	—
76	C₂×C₃×#41	C₂²×C₅	1	1	yes	characteristic
77	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
78	#948	—	2	4	no	—
79	C₂×#208	—	2	2	no	—
80	C₂×#208	C₂²×C₅	1	1	yes	characteristic
81	C₂×#208	C₂²×C₅	1	2	yes	—
82	C₂²×C₃⋊C₈	C₂²×C₅	1	1	yes	characteristic
83	C₃×#101	C₄×C₅	1	2	yes	—
84	C₃×#100	C₄×C₅	1	2	yes	—
85	#830	—	2	2	no	—
86	#819	—	2	4	no	—
87	C₅×#41	—	2	2	no	—
88	C₅×#41	—	2	2	no	—
89	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic
90	C₂×C₅×D₄	Dic₆	1	2	yes	—
91	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
92	C₂²×C₄×C₅	D₁₂	1	2	yes	—
93	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
94	C₅×M₁₆	—	2	2	no	—
95	C₅×M₁₆	—	3	4	no	—
96	C₅×M₁₆	—	6	4	no	—
97	C₅×M₁₆	—	6	2	no	—
98	C₅×M₁₆	—	6	2	no	—
99	C₅×M₁₆	C₂×Dic₃	1	4	yes	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	—	6	4	no	—
102	C₂×C₅×C₈	—	6	2	no	—
103	C₂×C₅×C₈	—	6	1	no	—
104	C₂×C₅×C₈	—	6	1	no	—
105	C₂×C₅×C₈	—	6	2	no	—
106	#568	—	3	1	no	—
107	#430	—	3	2	no	—
108	#413	C₅×D₃	1	1	yes	characteristic
109	#408	—	3	1	no	—
110	#341	—	3	2	no	—
111	C₂²×C₃×C₅	—	2	2	no	—
112	C₂²×C₃×C₅	—	4	1	no	—
113	C₂²×C₃×C₅	—	4	2	no	—
114	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
116	C₃×C₄×C₅	—	2	2	no	—
117	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
118	C₃×C₄×C₅	#120	1	1	yes	characteristic
119	C₃×C₄×C₅	#123	1	1	yes	characteristic
120	C₃×#41	—	2	2	no	—
121	C₃×#41	—	2	2	no	—
122	C₂×C₃×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
123	C₂×C₃×D₄	C₂×C₄×C₅	1	1	yes	characteristic
124	C₂×C₃×D₄	C₅×Q₈	1	2	yes	—
125	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
126	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
127	C₃×M₁₆	—	2	2	no	—
128	C₃×M₁₆	C₂×C₄×C₅	1	4	yes	—
129	C₂×C₃×C₈	—	2	2	no	—
130	#208	—	2	4	no	—
131	#208	—	2	2	no	—
132	#208	—	2	2	no	—
133	#208	C₅×D₄	1	4	yes	—
134	C₂×C₃⋊C₈	—	2	4	no	—
135	C₂×C₃⋊C₈	—	2	2	no	—
136	C₂×C₃⋊C₈	—	2	1	no	—
137	C₂×C₃⋊C₈	—	2	1	no	—
138	C₂×C₃⋊C₈	—	2	2	no	—
139	C₂³×C₅	C₄⋊Dic₃	1	2	yes	—
140	C₂³×C₅	C₂×Dic₆	1	1	yes	characteristic
141	C₅×Q₈	—	2	2	no	—
142	C₅×D₄	—	2	4	no	—
143	C₅×D₄	—	2	2	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	2	no	—
146	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
147	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
148	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
149	C₂×C₄×C₅	#237	1	2	yes	—
150	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
151	C₅×C₈	—	2	4	no	—
152	C₅×C₈	—	6	4	no	—
153	C₅×C₈	—	12	1	no	—
154	C₅×C₈	—	12	1	no	—
155	C₂×#41	C₂×C₅×D₃	1	1	yes	characteristic
156	#131	—	6	4	no	—
157	C₂×M₁₆	—	3	2	no	—
158	C₂×M₁₆	—	3	1	no	—
159	C₂×M₁₆	—	6	2	no	—
160	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
161	C₂²×C₈	—	3	1	no	—
162	#108	—	6	4	no	—
163	#101	C₅×Dic₃	1	2	yes	—
164	#100	C₅×Dic₃	1	2	yes	—
165	C₂²⋊C₈	—	6	2	no	—
166	C₂×C₃×C₅	—	2	1	no	—
167	C₂×C₃×C₅	—	2	2	no	—
168	C₂×C₃×C₅	—	4	2	no	—
169	C₂×C₃×C₅	#385	1	1	yes	characteristic
170	C₂³×C₃	C₅×C₄⋊C₄	1	2	yes	—
171	C₂³×C₃	C₂×C₅×Q₈	1	1	yes	characteristic
172	C₃×Q₈	—	2	2	no	—
173	C₃×D₄	—	2	4	no	—
174	C₃×D₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	C₅×C₄⋊C₄	1	2	yes	—
178	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
179	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
181	C₂×C₃×C₄	C₅×#41	1	2	yes	—
182	C₃×C₈	—	2	4	no	—
183	C₃⋊C₈	—	2	4	no	—
184	C₃⋊C₈	—	4	1	no	—
185	C₃⋊C₈	—	4	1	no	—
186	C₂²×C₅	—	2	2	no	—
187	C₂²×C₅	—	4	1	no	—
188	C₂²×C₅	—	4	2	no	—
189	C₂²×C₅	#878	1	2	yes	—
190	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	#882	1	1	yes	characteristic
193	C₄×C₅	#886	1	1	yes	characteristic
194	C₄×C₅	D₄×Dic₃	1	2	yes	—
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	C₂×Q₈	C₂×C₅×Dic₃	1	1	yes	characteristic
198	C₂×D₄	C₅×Dic₆	1	2	yes	—
199	C₂×D₄	C₂×C₅×Dic₃	1	1	yes	characteristic
200	C₂²×C₄	C₅×D₁₂	1	2	yes	—
201	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
202	M₁₆	—	2	2	no	—
203	M₁₆	—	3	4	no	—
204	M₁₆	—	6	4	no	—
205	M₁₆	—	6	2	no	—
206	M₁₆	—	6	2	no	—
207	M₁₆	C₂×C₅×Dic₃	1	4	yes	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	6	4	no	—
210	C₂×C₈	—	6	2	no	—
211	C₂×C₈	—	6	1	no	—
212	C₂×C₈	—	6	1	no	—
213	C₂×C₈	—	6	2	no	—
214	C₃×C₅	#1858	1	1	yes	characteristic, complemented, Hall
215	C₂²×C₃	—	2	2	no	—
216	C₂²×C₃	—	4	1	no	—
217	C₂²×C₃	—	4	2	no	—
218	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
219	C₂²×C₃	C₅×C₂²⋊Q₈	1	2	yes	—
220	C₃×C₄	—	2	2	no	—
221	C₃×C₄	C₄×C₅×D₄	1	2	yes	—
222	C₃×C₄	C₅×#120	1	1	yes	characteristic
223	C₃×C₄	C₅×#123	1	1	yes	characteristic
224	C₂×C₅	—	2	1	no	—
225	C₂×C₅	—	2	2	no	—
226	C₂×C₅	—	4	2	no	—
227	C₂×C₅	#4950	1	1	yes	characteristic
228	C₂³	C₅×C₄⋊Dic₃	1	2	yes	—
229	C₂³	C₂×C₅×Dic₆	1	1	yes	characteristic
230	Q₈	—	2	2	no	—
231	D₄	—	2	4	no	—
232	D₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	C₅×C₄⋊Dic₃	1	2	yes	—
236	C₂×C₄	C₂×C₅×D₁₂	1	1	yes	characteristic
237	C₂×C₄	C₅×D₃×D₄	1	2	yes	—
238	C₂×C₄	C₅×#237	1	2	yes	—
239	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
240	C₈	—	2	4	no	—
241	C₈	—	6	4	no	—
242	C₈	—	12	1	no	—
243	C₈	—	12	1	no	—
244	C₂×C₃	—	2	1	no	—
245	C₂×C₃	—	2	2	no	—
246	C₂×C₃	—	4	2	no	—
247	C₂×C₃	C₅×#385	1	1	yes	characteristic
248	C₅	#70135	1	1	yes	characteristic, direct product factor, Sylow
249	C₂²	—	2	2	no	—
250	C₂²	—	4	1	no	—
251	C₂²	—	4	2	no	—
252	C₂²	C₅×#878	1	2	yes	—
253	C₂²	C₂×C₅×C₄⋊Dic₃	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	C₅×#882	1	1	yes	characteristic
256	C₄	C₅×#886	1	1	yes	characteristic
257	C₄	C₅×D₄×Dic₃	1	2	yes	—
258	C₃	C₅×#1858	1	1	yes	characteristic, complemented, Sylow
259	C₂	—	2	1	no	—
260	C₂	—	2	2	no	—
261	C₂	—	4	2	no	—
262	C₂	C₅×#4950	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#882
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#413	C₂
socle	C₂×C₃×C₅	#385
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#413	D₃
hypocenter	C₃	C₅×#1858
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#413

Complements

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

N	H ≅ G/N	type
4. C₅×#5125	C₂	semidirect
5. C₅×#4558	C₂	semidirect
18. #70135	C₅	direct (N × H)
46. #5125	C₂×C₅	semidirect
47. #4558	C₂×C₅	semidirect
214. C₃×C₅	#1858	semidirect
248. C₅	#70135	direct (N × H)
258. C₃	C₅×#1858	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5819	2
C₃×C₅×#413	2
C₅×#5234	2
C₅×#5125	2
C₅×#4558	2
C₅×#1858	3
#70135	5

Subgroups — C5×#70135 #14299668

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#70135 #14299668