Subgroups — C₅×#72027 #14299662

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	C₃×C₅×#568	C₂	1	1	yes	maximal, characteristic
2	C₅×#5237	C₂	1	1	yes	maximal, characteristic
3	C₅×#5234	C₂	1	1	yes	maximal, characteristic
4	C₅×#5124	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	C₃×C₅×#131	—	2	4	no	—
9	C₂×C₃×C₅×M₁₆	—	2	2	no	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
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	C₅×#836	C₄	1	2	yes	—
15	C₅×#833	C₄	1	2	yes	—
16	C₅×#830	—	2	2	no	—
17	C₅×#822	—	2	4	no	—
18	#72027	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#568	D₃	1	1	yes	characteristic
20	C₅×#430	—	3	2	no	—
21	C₅×#413	—	3	1	no	—
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	4	no	—
32	C₃×C₅×M₁₆	—	2	2	no	—
33	C₃×C₅×M₁₆	D₄	1	4	yes	—
34	C₂×C₃×C₅×C₈	—	2	4	no	—
35	C₂×C₃×C₅×C₈	—	2	2	no	—
36	C₅×#208	—	2	2	no	—
37	C₅×#208	—	2	2	no	—
38	C₅×#208	C₂×C₄	1	4	yes	—
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	C₃×#568	C₂×C₅	1	1	yes	characteristic
44	#5237	C₂×C₅	1	1	yes	characteristic
45	#5234	C₂×C₅	1	1	yes	characteristic
46	#5124	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	—	2	4	no	—
50	C₂×C₅×M₁₆	—	2	2	no	—
51	C₂×C₅×M₁₆	—	3	2	no	—
52	C₂×C₅×M₁₆	—	3	1	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	—	3	2	no	—
57	C₅×#100	—	3	2	no	—
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	C₃×#131	—	2	4	no	—
78	C₂×C₃×M₁₆	—	2	2	no	—
79	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
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	#836	C₄×C₅	1	2	yes	—
84	#833	C₄×C₅	1	2	yes	—
85	#830	—	2	2	no	—
86	#822	—	2	4	no	—
87	C₅×#41	—	2	2	no	—
88	C₅×#41	—	2	2	no	—
89	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
90	C₂×C₅×D₄	Dic₆	1	2	yes	—
91	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
92	C₂²×C₄×C₅	#67	1	2	yes	—
93	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
94	C₅×M₁₆	—	2	4	no	—
95	C₅×M₁₆	—	2	2	no	—
96	C₅×M₁₆	—	3	4	no	—
97	C₅×M₁₆	—	6	2	no	—
98	C₅×M₁₆	—	6	2	no	—
99	C₅×M₁₆	#67	1	4	yes	—
100	C₂×C₅×C₈	—	2	4	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	6	2	no	—
103	C₂×C₅×C₈	—	6	2	no	—
104	C₂×C₅×C₈	—	6	1	no	—
105	C₂×C₅×C₈	—	6	1	no	—
106	#568	C₅×D₃	1	1	yes	characteristic
107	#430	—	3	2	no	—
108	#413	—	3	1	no	—
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	4	no	—
128	C₃×M₁₆	—	2	2	no	—
129	C₃×M₁₆	C₅×D₄	1	4	yes	—
130	C₂×C₃×C₈	—	2	4	no	—
131	C₂×C₃×C₈	—	2	2	no	—
132	#208	—	2	2	no	—
133	#208	—	2	2	no	—
134	#208	C₂×C₄×C₅	1	4	yes	—
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	Dic₃⋊C₄	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	Dic₃⋊C₄	1	2	yes	—
147	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
148	C₂×C₄×C₅	#235	1	2	yes	—
149	C₂×C₄×C₅	C₂×#67	1	2	yes	—
150	C₂×C₄×C₅	C₂×#67	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	—	2	4	no	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂×M₁₆	—	3	2	no	—
159	C₂×M₁₆	—	3	1	no	—
160	C₂×M₁₆	C₂×C₅×D₃	1	2	yes	—
161	C₂²×C₈	—	3	1	no	—
162	#108	—	6	4	no	—
163	#101	—	3	2	no	—
164	#100	—	3	2	no	—
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	2	yes	—
180	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
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	C₂×Dic₃⋊C₄	1	1	yes	characteristic
190	C₂²×C₅	#924	1	2	yes	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	C₄×#67	1	2	yes	—
193	C₄×C₅	#929	1	1	yes	characteristic
194	C₄×C₅	#953	1	1	yes	characteristic
195	#41	—	2	2	no	—
196	#41	—	2	2	no	—
197	C₂×Q₈	C₄×C₅×D₃	1	1	yes	characteristic
198	C₂×D₄	C₅×Dic₆	1	2	yes	—
199	C₂×D₄	C₄×C₅×D₃	1	1	yes	characteristic
200	C₂²×C₄	C₅×#67	1	2	yes	—
201	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
202	M₁₆	—	2	4	no	—
203	M₁₆	—	2	2	no	—
204	M₁₆	—	3	4	no	—
205	M₁₆	—	6	2	no	—
206	M₁₆	—	6	2	no	—
207	M₁₆	C₅×#67	1	4	yes	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	—	6	2	no	—
211	C₂×C₈	—	6	2	no	—
212	C₂×C₈	—	6	1	no	—
213	C₂×C₈	—	6	1	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	#5210	1	1	yes	characteristic
228	C₂³	C₅×Dic₃⋊C₄	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×Dic₃⋊C₄	1	2	yes	—
236	C₂×C₄	C₂×C₄×C₅×D₃	1	1	yes	characteristic
237	C₂×C₄	C₅×#235	1	2	yes	—
238	C₂×C₄	C₂×C₅×#67	1	2	yes	—
239	C₂×C₄	C₂×C₅×#67	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	#72027	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	C₂×C₅×Dic₃⋊C₄	1	1	yes	characteristic
253	C₂²	C₅×#924	1	2	yes	—
254	C₄	—	2	2	no	—
255	C₄	C₄×C₅×#67	1	2	yes	—
256	C₄	C₅×#929	1	1	yes	characteristic
257	C₄	C₅×#953	1	1	yes	characteristic
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×#5210	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₅	#953
commutator	C₂×C₃×C₄	C₂²×C₄×C₅
Frattini	C₂²×C₄	C₂²×C₅×D₃
Fitting	C₃×C₅×#568	C₂
socle	C₂×C₃×C₅	#385
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₃×C₄	C₂³×C₅
hypercenter	C₅×#568	D₃
hypocenter	C₃	C₅×#1858
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#568

Complements

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

N	H ≅ G/N	type
4. C₅×#5124	C₂	semidirect
5. C₅×#4558	C₂	semidirect
18. #72027	C₅	direct (N × H)
46. #5124	C₂×C₅	semidirect
47. #4558	C₂×C₅	semidirect
214. C₃×C₅	#1858	semidirect
248. C₅	#72027	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
C₃×C₅×#568	2
C₅×#5237	2
C₅×#5234	2
C₅×#5124	2
C₅×#4558	2
C₅×#1858	3
#72027	5

Subgroups — C5×#72027 #14299662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#72027 #14299662