Subgroups — C₅×#72004 #14312857

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#812	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4549	C₂	1	4	yes	maximal, complemented
5	C₅×#2113	—	3	1	no	maximal, Hall
6	C₂×C₃×C₅×#41	C₄	1	1	yes	characteristic
7	C₃×C₅×#131	—	2	2	no	—
8	C₃×C₅×#131	—	2	2	no	—
9	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₄	1	2	yes	—
11	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₅×#812	—	2	2	no	—
14	C₂×C₅×C₃⋊C₁₆	—	2	2	no	—
15	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	—
16	C₂×C₅×C₃⋊C₁₆	C₂²	1	2	yes	—
17	#72004	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂×C₅×#131	D₃	1	1	yes	characteristic
19	C₂×C₅×M₃₂	—	3	1	no	—
20	C₂²×C₅×C₁₆	—	3	1	no	—
21	C₅×C₂²⋊C₁₆	—	3	4	no	—
22	C₃×C₅×#41	—	2	2	no	—
23	C₃×C₅×#41	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	C₈	1	1	yes	characteristic
25	C₂×C₃×C₅×D₄	C₈	1	2	yes	—
26	C₂×C₃×C₅×D₄	C₈	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₃×C₅×M₁₆	—	2	2	no	—
30	C₃×C₅×M₁₆	—	2	4	no	—
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
36	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
37	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₅×C₃⋊C₁₆	—	2	2	no	—
39	C₅×C₃⋊C₁₆	—	2	2	no	—
40	C₂×C₃×#131	C₂×C₅	1	1	yes	characteristic
41	C₂×#812	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂²×C₃⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
43	#4549	C₂×C₅	1	4	yes	complemented
44	C₂×C₅×#41	Dic₃	1	1	yes	characteristic
45	C₅×#131	—	2	2	no	—
46	C₅×#131	—	2	2	no	—
47	C₂×C₅×M₁₆	Dic₃	1	1	yes	characteristic
48	C₂×C₅×M₁₆	Dic₃	1	2	yes	—
49	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
50	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
51	C₅×M₃₂	—	6	2	no	—
52	C₂×C₅×C₁₆	—	3	2	no	—
53	C₂×C₅×C₁₆	—	3	2	no	—
54	C₂×C₅×C₁₆	—	6	2	no	—
55	#2113	—	3	1	no	Sylow
56	C₂³×C₃×C₅	C₂×C₈	1	2	yes	—
57	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
58	C₃×C₅×Q₈	—	2	2	no	—
59	C₃×C₅×D₄	—	2	4	no	—
60	C₃×C₅×D₄	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₃×C₅×C₈	—	2	2	no	—
69	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
70	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
71	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×#41	C₄×C₅	1	1	yes	characteristic
73	C₃×#131	—	2	2	no	—
74	C₃×#131	—	2	2	no	—
75	C₂×C₃×M₁₆	C₄×C₅	1	1	yes	characteristic
76	C₂×C₃×M₁₆	C₄×C₅	1	2	yes	—
77	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
78	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
79	#812	—	2	2	no	—
80	C₂×C₃⋊C₁₆	—	2	2	no	—
81	C₂×C₃⋊C₁₆	C₂²×C₅	1	2	yes	—
82	C₂×C₃⋊C₁₆	C₂²×C₅	1	2	yes	—
83	C₅×#41	—	2	2	no	—
84	C₅×#41	—	2	2	no	—
85	C₂×C₅×Q₈	C₃⋊C₈	1	1	yes	characteristic
86	C₂×C₅×D₄	C₃⋊C₈	1	2	yes	—
87	C₂×C₅×D₄	C₃⋊C₈	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
89	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
90	C₅×M₁₆	—	2	2	no	—
91	C₅×M₁₆	—	2	4	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
95	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
96	C₂×C₅×C₈	#67	1	2	yes	—
97	C₂×C₅×C₈	#67	1	2	yes	—
98	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
99	C₅×C₁₆	—	6	2	no	—
100	C₅×C₁₆	—	6	2	no	—
101	C₂×#131	C₅×D₃	1	1	yes	characteristic
102	C₂×M₃₂	—	3	1	no	—
103	C₂²×C₁₆	—	3	1	no	—
104	C₂²⋊C₁₆	—	3	4	no	—
105	C₂²×C₃×C₅	—	2	2	no	—
106	C₂²×C₃×C₅	—	2	2	no	—
107	C₂²×C₃×C₅	—	2	4	no	—
108	C₂²×C₃×C₅	C₂²⋊C₈	1	2	yes	—
109	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	C₂²⋊C₈	1	2	yes	—
112	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
114	C₃×#41	—	2	2	no	—
115	C₃×#41	—	2	2	no	—
116	C₂×C₃×Q₈	C₅×C₈	1	1	yes	characteristic
117	C₂×C₃×D₄	C₅×C₈	1	2	yes	—
118	C₂×C₃×D₄	C₅×C₈	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
120	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
121	C₃×M₁₆	—	2	2	no	—
122	C₃×M₁₆	—	2	4	no	—
123	C₂×C₃×C₈	—	2	2	no	—
124	C₂×C₃×C₈	—	2	2	no	—
125	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
126	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
127	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
128	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
129	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
130	C₃⋊C₁₆	—	2	2	no	—
131	C₃⋊C₁₆	—	2	2	no	—
132	C₂³×C₅	C₂×C₃⋊C₈	1	2	yes	—
133	C₂³×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
134	C₅×Q₈	—	2	2	no	—
135	C₅×D₄	—	2	4	no	—
136	C₅×D₄	—	2	2	no	—
137	C₂×C₄×C₅	—	2	2	no	—
138	C₂×C₄×C₅	—	2	2	no	—
139	C₂×C₄×C₅	C₂×C₃⋊C₈	1	2	yes	—
140	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
141	C₂×C₄×C₅	#208	1	2	yes	—
142	C₂×C₄×C₅	#217	1	2	yes	—
143	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
144	C₅×C₈	—	2	2	no	—
145	C₅×C₈	#217	1	2	yes	—
146	C₅×C₈	C₂×#67	1	1	yes	characteristic
147	C₅×C₈	C₂×#67	1	1	yes	characteristic
148	C₂×#41	C₅×Dic₃	1	1	yes	characteristic
149	#131	—	2	2	no	—
150	#131	—	2	2	no	—
151	C₂×M₁₆	C₅×Dic₃	1	1	yes	characteristic
152	C₂×M₁₆	C₅×Dic₃	1	2	yes	—
153	C₂²×C₈	C₂×C₅×D₃	1	2	yes	—
154	C₂²×C₈	C₂×C₅×D₃	1	1	yes	characteristic
155	M₃₂	—	6	2	no	—
156	C₂×C₁₆	—	3	2	no	—
157	C₂×C₁₆	—	3	2	no	—
158	C₂×C₁₆	—	6	2	no	—
159	C₂×C₃×C₅	—	2	2	no	—
160	C₂×C₃×C₅	—	4	2	no	—
161	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₂×C₃×C₅	#504	1	2	yes	—
163	C₂³×C₃	C₂×C₅×C₈	1	2	yes	—
164	C₂³×C₃	C₂×C₅×C₈	1	1	yes	characteristic
165	C₃×Q₈	—	2	2	no	—
166	C₃×D₄	—	2	4	no	—
167	C₃×D₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	2	2	no	—
170	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
171	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
172	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
173	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
174	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
175	C₃×C₈	—	2	2	no	—
176	C₃×C₈	C₅×C₂²⋊C₄	1	2	yes	—
177	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
178	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	2	4	no	—
181	C₂²×C₅	—	2	2	no	—
182	C₂²×C₅	#830	1	2	yes	—
183	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
184	C₄×C₅	—	2	2	no	—
185	C₄×C₅	#830	1	2	yes	—
186	C₄×C₅	C₂×#208	1	1	yes	characteristic
187	C₄×C₅	C₂×#217	1	1	yes	characteristic
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	C₂×Q₈	C₅×C₃⋊C₈	1	1	yes	characteristic
191	C₂×D₄	C₅×C₃⋊C₈	1	2	yes	—
192	C₂×D₄	C₅×C₃⋊C₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×C₅×Dic₃	1	2	yes	—
194	C₂²×C₄	C₂×C₅×Dic₃	1	1	yes	characteristic
195	M₁₆	—	2	2	no	—
196	M₁₆	—	2	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂×C₅×Dic₃	1	2	yes	—
200	C₂×C₈	C₂×C₅×Dic₃	1	1	yes	characteristic
201	C₂×C₈	C₅×#67	1	2	yes	—
202	C₂×C₈	C₅×#67	1	2	yes	—
203	C₂×C₈	C₂²×C₅×D₃	1	1	yes	characteristic
204	C₁₆	—	6	2	no	—
205	C₁₆	—	6	2	no	—
206	C₃×C₅	#2113	1	1	yes	characteristic, complemented, Hall
207	C₂²×C₃	—	2	2	no	—
208	C₂²×C₃	—	2	2	no	—
209	C₂²×C₃	—	2	4	no	—
210	C₂²×C₃	C₅×C₂²⋊C₈	1	2	yes	—
211	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
212	C₃×C₄	—	2	2	no	—
213	C₃×C₄	C₅×C₂²⋊C₈	1	2	yes	—
214	C₃×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
215	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	—	4	2	no	—
218	C₂×C₅	#5140	1	2	yes	—
219	C₂×C₅	C₂×#830	1	1	yes	characteristic
220	C₂³	C₂×C₅×C₃⋊C₈	1	2	yes	—
221	C₂³	C₂×C₅×C₃⋊C₈	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	C₂×C₅×C₃⋊C₈	1	2	yes	—
228	C₂×C₄	C₂×C₅×C₃⋊C₈	1	1	yes	characteristic
229	C₂×C₄	C₅×#208	1	2	yes	—
230	C₂×C₄	C₅×#217	1	2	yes	—
231	C₂×C₄	C₂²×C₅×Dic₃	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	C₅×#217	1	2	yes	—
234	C₈	C₂×C₅×#67	1	1	yes	characteristic
235	C₈	C₂×C₅×#67	1	1	yes	characteristic
236	C₂×C₃	—	2	2	no	—
237	C₂×C₃	—	4	2	no	—
238	C₂×C₃	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
239	C₂×C₃	C₅×#504	1	2	yes	—
240	C₅	#72004	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	4	no	—
244	C₂²	C₅×#830	1	2	yes	—
245	C₂²	C₂²×C₅×C₃⋊C₈	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	C₅×#830	1	2	yes	—
248	C₄	C₂×C₅×#208	1	1	yes	characteristic
249	C₄	C₂×C₅×#217	1	1	yes	characteristic
250	C₃	C₅×#2113	1	1	yes	characteristic, complemented, Sylow
251	C₂	—	2	2	no	—
252	C₂	—	4	2	no	—
253	C₂	C₅×#5140	1	2	yes	—
254	C₂	C₂×C₅×#830	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅×C₈	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₅×C₈
Frattini	C₂×C₈	C₂²×C₅×D₃
Fitting	C₂×C₃×C₅×#131	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂×C₃×C₅×#41	C₄
cosocle	C₂×C₃×C₈	C₂³×C₅
hypercenter	C₂×C₅×#131	D₃
hypocenter	C₃	C₅×#2113
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₅×C₈ ◁ C₂×C₅×#131

Complements

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

N	H ≅ G/N	type
2. C₂×C₅×#812	C₂	semidirect
3. C₂²×C₅×C₃⋊C₁₆	C₂	semidirect
4. C₅×#4549	C₂	semidirect
17. #72004	C₅	direct (N × H)
41. C₂×#812	C₂×C₅	semidirect
42. C₂²×C₃⋊C₁₆	C₂×C₅	semidirect
43. #4549	C₂×C₅	semidirect
206. C₃×C₅	#2113	semidirect
240. C₅	#72004	direct (N × H)
250. C₃	C₅×#2113	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×#131	2
C₂×C₅×#812	2
C₂²×C₅×C₃⋊C₁₆	2
C₅×#4549	2
C₅×#2113	3
#72004	5

Subgroups — C5×#72004 #14312857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#72004 #14312857