Subgroups — C₃×#10599731 #14294653

Order 1920.

744 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₅×M₁₆	C₂	1	2	yes	maximal
2	C₂²×C₃×#676	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#341	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64778	C₂	1	2	yes	maximal, complemented
5	C₃×#64772	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599731	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×#681	C₂²	1	2	yes	—
11	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
12	C₃×D₅×M₁₆	—	2	4	no	—
13	C₃×D₅×M₁₆	C₄	1	8	yes	—
14	C₂×C₃×#676	—	2	4	no	—
15	C₂×C₃×#676	—	2	2	no	—
16	C₂×C₃×#676	C₄	1	4	yes	—
17	C₂×C₃×C₈×D₅	—	2	2	no	—
18	C₃×#1738	—	5	1	no	maximal, Hall
19	C₂×D₅×M₁₆	C₂×C₃	1	2	yes	—
20	C₂²×#676	C₂×C₃	1	1	yes	characteristic
21	C₅×#341	C₂×C₃	1	1	yes	characteristic, complemented
22	#64778	C₂×C₃	1	2	yes	complemented
23	#64772	C₂×C₃	1	1	yes	characteristic, complemented
24	C₂³×C₃×D₅	Q₈	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×C₃×Dic₅	Q₈	1	1	yes	characteristic
27	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×D₅	C₂×C₄	1	4	yes	—
29	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
30	C₂×C₃×C₄×D₅	D₄	1	4	yes	—
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	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₃×#681	—	2	2	no	—
37	C₃×#681	C₂×C₄	1	4	yes	—
38	C₂×C₃×#168	—	2	2	no	—
39	C₂×C₃×#168	—	2	2	no	—
40	C₂×C₃×#168	C₂×C₄	1	2	yes	—
41	C₃×#676	—	2	8	no	—
42	C₃×#676	—	2	8	no	—
43	C₃×C₈×D₅	—	2	8	no	—
44	C₂²×C₃×M₁₆	—	5	1	no	—
45	C₂²×C₃×M₁₆	—	5	2	no	—
46	C₃×#341	—	5	2	no	—
47	C₃×#341	—	5	1	no	—
48	C₃×#341	D₅	1	1	yes	characteristic, complemented
49	C₂²×C₄×D₅	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
51	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
52	C₂×#681	C₂²×C₃	1	2	yes	—
53	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
54	D₅×M₁₆	—	2	4	no	—
55	D₅×M₁₆	C₃×C₄	1	8	yes	—
56	C₂×#676	—	2	4	no	—
57	C₂×#676	—	2	2	no	—
58	C₂×#676	C₃×C₄	1	4	yes	—
59	C₂×C₈×D₅	—	2	2	no	—
60	#1738	—	5	1	no	Sylow
61	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₃×D₅	—	4	2	no	—
63	C₂²×C₃×D₅	C₄⋊C₄	1	2	yes	—
64	C₂²×C₃×D₅	C₄⋊C₄	1	4	yes	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×Dic₅	C₄⋊C₄	1	4	yes	—
70	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
71	C₃×C₄×D₅	C₄²	1	4	yes	—
72	C₃×C₄×D₅	C₂²⋊C₄	1	4	yes	—
73	C₃×C₄×D₅	C₂²⋊C₄	1	4	yes	—
74	C₃×C₄×D₅	C₂²⋊C₄	1	4	yes	—
75	C₃×C₅×C₈	—	2	4	no	—
76	C₃×C₅×C₈	—	2	2	no	—
77	C₃×#168	—	2	4	no	—
78	C₃×#168	—	2	2	no	—
79	C₂³×C₃×C₄	—	5	1	no	—
80	C₂×C₃×M₁₆	—	5	4	no	—
81	C₂×C₃×M₁₆	—	5	2	no	—
82	C₂×C₃×M₁₆	—	5	8	no	—
83	C₂×C₃×M₁₆	—	10	4	no	—
84	C₂×C₃×M₁₆	—	10	2	no	—
85	C₂×C₃×M₁₆	—	10	4	no	—
86	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
87	C₂²×C₃×C₈	—	5	1	no	—
88	C₂²×C₃×C₈	—	10	2	no	—
89	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
90	C₂³×D₅	C₃×Q₈	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
92	C₂²×Dic₅	C₃×Q₈	1	1	yes	characteristic
93	C₂×C₄×D₅	C₂×C₃×C₄	1	2	yes	—
94	C₂×C₄×D₅	C₂×C₃×C₄	1	4	yes	—
95	C₂×C₄×D₅	C₃×D₄	1	2	yes	—
96	C₂×C₄×D₅	C₃×D₄	1	4	yes	—
97	C₅×M₁₆	—	2	2	no	—
98	C₅×M₁₆	C₂×C₃×C₄	1	4	yes	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
102	#681	—	2	2	no	—
103	#681	C₂×C₃×C₄	1	4	yes	—
104	C₂×#168	—	2	2	no	—
105	C₂×#168	—	2	2	no	—
106	C₂×#168	C₂×C₃×C₄	1	2	yes	—
107	#676	—	2	8	no	—
108	#676	—	2	8	no	—
109	C₈×D₅	—	2	8	no	—
110	C₂²×M₁₆	—	5	1	no	—
111	C₂²×M₁₆	—	5	2	no	—
112	#341	—	5	2	no	—
113	#341	—	5	1	no	—
114	#341	C₃×D₅	1	1	yes	characteristic, complemented
115	C₂²×C₃×C₅	—	2	2	no	—
116	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
117	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₃×D₅	—	2	4	no	—
119	C₂×C₃×D₅	—	4	4	no	—
120	C₂×C₃×D₅	#95	1	4	yes	—
121	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
122	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
124	C₃×Dic₅	#95	1	4	yes	—
125	C₂⁴×C₃	—	5	1	no	—
126	C₂²×C₃×C₄	—	5	1	no	—
127	C₂²×C₃×C₄	—	5	4	no	—
128	C₂²×C₃×C₄	—	5	2	no	—
129	C₂²×C₃×C₄	—	5	4	no	—
130	C₂²×C₃×C₄	—	5	2	no	—
131	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
132	C₃×M₁₆	—	2	2	no	—
133	C₃×M₁₆	—	5	4	no	—
134	C₃×M₁₆	—	10	8	no	—
135	C₃×M₁₆	—	10	8	no	—
136	C₃×M₁₆	—	10	2	no	—
137	C₃×M₁₆	C₄×D₅	1	4	yes	—
138	C₂×C₃×C₈	—	2	2	no	—
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	—	5	2	no	—
141	C₂×C₃×C₈	—	10	8	no	—
142	C₂×C₃×C₈	—	10	2	no	—
143	C₂×C₃×C₈	—	10	2	no	—
144	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
145	C₂³×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
146	C₂²×D₅	—	4	2	no	—
147	C₂²×D₅	C₃×C₄⋊C₄	1	2	yes	—
148	C₂²×D₅	C₃×C₄⋊C₄	1	4	yes	—
149	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
151	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
153	C₂×Dic₅	C₃×C₄⋊C₄	1	4	yes	—
154	C₂×Dic₅	C₃×C₄⋊C₄	1	2	yes	—
155	C₄×D₅	C₃×C₄²	1	4	yes	—
156	C₄×D₅	C₃×C₂²⋊C₄	1	4	yes	—
157	C₄×D₅	C₃×C₂²⋊C₄	1	4	yes	—
158	C₄×D₅	C₃×C₂²⋊C₄	1	4	yes	—
159	C₅×C₈	—	2	4	no	—
160	C₅×C₈	—	2	2	no	—
161	#168	—	2	4	no	—
162	#168	—	2	2	no	—
163	C₂³×C₄	—	5	1	no	—
164	C₂×M₁₆	—	5	4	no	—
165	C₂×M₁₆	—	5	2	no	—
166	C₂×M₁₆	—	5	8	no	—
167	C₂×M₁₆	—	10	2	no	—
168	C₂×M₁₆	—	10	4	no	—
169	C₂×M₁₆	—	10	4	no	—
170	C₂×M₁₆	C₂×C₃×D₅	1	2	yes	—
171	C₂²×C₈	—	5	1	no	—
172	C₂²×C₈	—	10	2	no	—
173	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
174	C₂×C₃×C₅	—	2	2	no	—
175	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
176	C₂×C₃×C₅	#430	1	2	yes	—
177	C₃×D₅	—	2	4	no	—
178	C₂³×C₃	—	5	2	no	—
179	C₂³×C₃	—	5	4	no	—
180	C₂³×C₃	—	20	2	no	—
181	C₂³×C₃	Q₈×D₅	1	1	yes	characteristic
182	C₂×C₃×C₄	—	5	4	no	—
183	C₂×C₃×C₄	—	5	2	no	—
184	C₂×C₃×C₄	—	5	4	no	—
185	C₂×C₃×C₄	—	5	4	no	—
186	C₂×C₃×C₄	—	5	4	no	—
187	C₂×C₃×C₄	—	5	4	no	—
188	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
190	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
191	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
192	C₃×C₈	—	2	2	no	—
193	C₃×C₈	—	2	4	no	—
194	C₃×C₈	—	10	2	no	—
195	C₃×C₈	—	10	4	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	2	yes	—
198	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
199	C₂×D₅	—	2	4	no	—
200	C₂×D₅	—	4	4	no	—
201	C₂×D₅	C₃×#95	1	4	yes	—
202	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
203	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	2	yes	—
204	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
205	Dic₅	C₃×#95	1	4	yes	—
206	C₂⁴	—	5	1	no	—
207	C₂²×C₄	—	5	1	no	—
208	C₂²×C₄	—	5	4	no	—
209	C₂²×C₄	—	5	2	no	—
210	C₂²×C₄	—	5	2	no	—
211	C₂²×C₄	—	5	4	no	—
212	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
213	M₁₆	—	2	2	no	—
214	M₁₆	—	5	4	no	—
215	M₁₆	—	10	8	no	—
216	M₁₆	—	10	8	no	—
217	M₁₆	—	10	2	no	—
218	M₁₆	C₃×C₄×D₅	1	4	yes	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	5	2	no	—
222	C₂×C₈	—	10	8	no	—
223	C₂×C₈	—	10	2	no	—
224	C₂×C₈	—	10	2	no	—
225	C₂×C₈	C₃×C₄×D₅	1	2	yes	—
226	C₃×C₅	#1738	1	1	yes	characteristic, complemented, Hall
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	—	5	4	no	—
229	C₂²×C₃	—	10	4	no	—
230	C₂²×C₃	—	20	4	no	—
231	C₂²×C₃	D₅×C₄⋊C₄	1	2	yes	—
232	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
233	C₃×C₄	—	5	4	no	—
234	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
235	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
236	C₃×C₄	D₅×C₂²⋊C₄	1	2	yes	—
237	C₂×C₅	—	2	2	no	—
238	C₂×C₅	C₂×C₃×#95	1	1	yes	characteristic
239	C₂×C₅	C₃×#430	1	2	yes	—
240	D₅	—	2	4	no	—
241	C₂³	—	5	2	no	—
242	C₂³	—	5	4	no	—
243	C₂³	—	20	2	no	—
244	C₂³	C₃×Q₈×D₅	1	1	yes	characteristic
245	C₂×C₄	—	5	4	no	—
246	C₂×C₄	—	5	2	no	—
247	C₂×C₄	—	5	4	no	—
248	C₂×C₄	—	5	4	no	—
249	C₂×C₄	—	5	4	no	—
250	C₂×C₄	—	5	4	no	—
251	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
252	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
253	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
254	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	2	4	no	—
257	C₈	—	10	2	no	—
258	C₈	—	10	4	no	—
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	—	10	4	no	—
261	C₂×C₃	D₅×#95	1	1	yes	characteristic
262	C₂×C₃	#65182	1	2	yes	—
263	C₅	C₃×#1738	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	5	4	no	—
266	C₂²	—	10	4	no	—
267	C₂²	—	20	4	no	—
268	C₂²	C₃×D₅×C₄⋊C₄	1	2	yes	—
269	C₂²	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
270	C₄	—	5	4	no	—
271	C₄	C₃×C₄²×D₅	1	1	yes	characteristic
272	C₄	C₃×D₅×C₂²⋊C₄	1	1	yes	characteristic
273	C₄	C₃×D₅×C₂²⋊C₄	1	2	yes	—
274	C₃	#10599731	1	1	yes	characteristic, direct product factor, Sylow
275	C₂	—	2	2	no	—
276	C₂	—	10	4	no	—
277	C₂	C₃×D₅×#95	1	1	yes	characteristic
278	C₂	C₃×#65182	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄	D₄×D₅
commutator	C₄×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₅×#341	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×D₅	Q₈
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×#341	D₅
hypocenter	C₅	C₃×#1738
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#341

Complements

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

N	H ≅ G/N	type
3. C₃×C₅×#341	C₂	semidirect
4. C₃×#64778	C₂	semidirect
5. C₃×#64772	C₂	semidirect
6. #10599731	C₃	direct (N × H)
21. C₅×#341	C₂×C₃	semidirect
22. #64778	C₂×C₃	semidirect
23. #64772	C₂×C₃	semidirect
48. C₃×#341	D₅	semidirect
114. #341	C₃×D₅	semidirect
226. C₃×C₅	#1738	semidirect
263. C₅	C₃×#1738	semidirect
274. C₃	#10599731	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×D₅×M₁₆	2
C₂²×C₃×#676	2
C₃×C₅×#341	2
C₃×#64778	2
C₃×#64772	2
#10599731	3
C₃×#1738	5

Subgroups — C3×#10599731 #14294653

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10599731 #14294653