Subgroups — C₃×#10602870 #14313850

Order 1920.

456 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×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	C₃×#65395	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃×#672	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64784	C₂	1	2	yes	maximal, complemented
5	C₃×#64782	C₂	1	2	yes	maximal, complemented
6	#10602870	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×#41	—	2	1	no	—
8	C₃×C₅×#135	—	2	2	no	—
9	C₃×C₅×#135	C₄	1	4	yes	complemented
10	C₂×C₃×C₅×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×SD₁₆	—	2	1	no	—
12	C₂×C₃×C₅×D₈	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₃×#4072	—	2	1	no	—
15	C₃×#3950	C₂²	1	2	yes	—
16	C₃×#3949	—	2	1	no	—
17	C₂×C₃×#672	—	2	2	no	—
18	C₂×C₃×#672	C₂²	1	2	yes	—
19	C₃×#2138	—	5	1	no	maximal, Hall
20	C₂×C₅×#135	C₂×C₃	1	1	yes	characteristic
21	#65395	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂²×#672	C₂×C₃	1	1	yes	characteristic, complemented
23	#64784	C₂×C₃	1	2	yes	complemented
24	#64782	C₂×C₃	1	2	yes	complemented
25	C₃×C₅×#41	—	2	4	no	—
26	C₃×C₅×#41	—	4	2	no	—
27	C₂×C₃×C₅×Q₈	—	2	1	no	—
28	C₂×C₃×C₅×D₄	—	2	1	no	—
29	C₂×C₃×C₅×D₄	—	4	1	no	—
30	C₂²×C₃×C₄×C₅	—	4	1	no	—
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₃×C₅×Q₁₆	—	2	1	no	—
33	C₃×C₅×Q₁₆	C₂×C₄	1	2	yes	—
34	C₃×C₅×SD₁₆	—	2	4	no	—
35	C₃×C₅×D₈	—	2	1	no	—
36	C₃×C₅×D₈	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
40	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
41	C₃×C₂²⋊Dic₅	—	4	1	no	—
42	C₃×C₄⋊Dic₅	—	2	2	no	—
43	C₃×C₄×Dic₅	—	4	1	no	—
44	C₃×#672	—	2	2	no	—
45	C₂×C₃×#135	D₅	1	1	yes	characteristic
46	C₂²×C₃×C₁₆	—	5	1	no	—
47	C₃×#427	—	5	1	no	—
48	C₃×#358	—	5	2	no	—
49	C₃×#357	—	5	2	no	—
50	C₂×C₅×#41	—	2	1	no	—
51	C₅×#135	—	2	2	no	—
52	C₅×#135	C₃×C₄	1	4	yes	complemented
53	C₂×C₅×Q₁₆	C₂²×C₃	1	1	yes	characteristic
54	C₂×C₅×SD₁₆	—	2	1	no	—
55	C₂×C₅×D₈	C₂²×C₃	1	1	yes	characteristic
56	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
57	#4072	—	2	1	no	—
58	#3950	C₂²×C₃	1	2	yes	—
59	#3949	—	2	1	no	—
60	C₂×#672	—	2	2	no	—
61	C₂×#672	C₂²×C₃	1	2	yes	—
62	#2138	—	5	1	no	Sylow
63	C₂³×C₃×C₅	—	4	1	no	—
64	C₂³×C₃×C₅	D₈	1	1	yes	characteristic
65	C₃×C₅×Q₈	—	2	2	no	—
66	C₃×C₅×Q₈	—	4	1	no	—
67	C₃×C₅×D₄	—	2	2	no	—
68	C₃×C₅×D₄	—	4	4	no	—
69	C₃×C₅×D₄	—	4	1	no	—
70	C₂×C₃×C₄×C₅	—	4	1	no	—
71	C₂×C₃×C₄×C₅	—	4	4	no	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
73	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×Dic₅	—	4	2	no	—
77	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
78	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×#41	—	2	1	no	—
81	C₃×#135	—	2	2	no	—
82	C₃×#135	Dic₅	1	4	yes	complemented
83	C₂×C₃×Q₁₆	C₂×D₅	1	1	yes	characteristic
84	C₂×C₃×SD₁₆	—	2	1	no	—
85	C₂×C₃×D₈	C₂×D₅	1	1	yes	characteristic
86	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
87	C₃×#117	—	10	1	no	—
88	C₂×C₃×C₁₆	—	5	2	no	—
89	C₂×C₃×C₁₆	—	10	2	no	—
90	C₃×#107	—	5	2	no	—
91	C₃×#106	—	10	1	no	—
92	C₅×#41	—	2	4	no	—
93	C₅×#41	—	4	2	no	—
94	C₂×C₅×Q₈	—	2	1	no	—
95	C₂×C₅×D₄	—	2	1	no	—
96	C₂×C₅×D₄	—	4	1	no	—
97	C₂²×C₄×C₅	—	4	1	no	—
98	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
99	C₅×Q₁₆	—	2	1	no	—
100	C₅×Q₁₆	C₂×C₃×C₄	1	2	yes	—
101	C₅×SD₁₆	—	2	4	no	—
102	C₅×D₈	—	2	1	no	—
103	C₅×D₈	C₂×C₃×C₄	1	2	yes	—
104	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
105	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
106	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
107	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
108	C₂²⋊Dic₅	—	4	1	no	—
109	C₄⋊Dic₅	—	2	2	no	—
110	C₄×Dic₅	—	4	1	no	—
111	#672	—	2	2	no	—
112	C₂×#135	C₃×D₅	1	1	yes	characteristic
113	C₂²×C₁₆	—	5	1	no	—
114	#427	—	5	1	no	—
115	#358	—	5	2	no	—
116	#357	—	5	2	no	—
117	C₂²×C₃×C₅	—	2	2	no	—
118	C₂²×C₃×C₅	—	4	2	no	—
119	C₂²×C₃×C₅	—	8	2	no	—
120	C₂²×C₃×C₅	#102	1	2	yes	—
121	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
122	C₃×C₄×C₅	—	4	2	no	—
123	C₃×C₄×C₅	#102	1	2	yes	—
124	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
126	C₃×Dic₅	—	8	2	no	—
127	C₃×#41	—	2	4	no	—
128	C₃×#41	—	4	2	no	—
129	C₂×C₃×Q₈	—	2	1	no	—
130	C₂×C₃×D₄	—	2	1	no	—
131	C₂×C₃×D₄	—	4	1	no	—
132	C₂²×C₃×C₄	—	4	1	no	—
133	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
134	C₃×Q₁₆	—	2	1	no	—
135	C₃×Q₁₆	C₂×Dic₅	1	2	yes	—
136	C₃×SD₁₆	—	2	4	no	—
137	C₃×D₈	—	2	1	no	—
138	C₃×D₈	C₂×Dic₅	1	2	yes	—
139	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
140	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
141	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
142	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
143	C₃×C₄⋊C₄	—	10	2	no	—
144	C₃×C₂²⋊C₄	—	20	1	no	—
145	C₃×C₄²	—	20	1	no	—
146	C₃×C₁₆	—	10	2	no	—
147	C₂³×C₅	—	4	1	no	—
148	C₂³×C₅	C₃×D₈	1	1	yes	characteristic
149	C₅×Q₈	—	2	2	no	—
150	C₅×Q₈	—	4	1	no	—
151	C₅×D₄	—	2	2	no	—
152	C₅×D₄	—	4	4	no	—
153	C₅×D₄	—	4	1	no	—
154	C₂×C₄×C₅	—	4	1	no	—
155	C₂×C₄×C₅	—	4	4	no	—
156	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
157	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
158	C₂×C₄×C₅	C₃×SD₁₆	1	2	yes	—
159	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
160	C₂×Dic₅	—	4	2	no	—
161	C₅×C₈	C₃×C₂²⋊C₄	1	2	yes	—
162	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
163	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
164	C₂×#41	—	2	1	no	—
165	#135	—	2	2	no	—
166	#135	C₃×Dic₅	1	4	yes	complemented
167	C₂×Q₁₆	C₂×C₃×D₅	1	1	yes	characteristic
168	C₂×SD₁₆	—	2	1	no	—
169	C₂×D₈	C₂×C₃×D₅	1	1	yes	characteristic
170	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
171	#117	—	10	1	no	—
172	C₂×C₁₆	—	5	2	no	—
173	C₂×C₁₆	—	10	2	no	—
174	#107	—	5	2	no	—
175	#106	—	10	1	no	—
176	C₂×C₃×C₅	—	2	2	no	—
177	C₂×C₃×C₅	—	8	2	no	—
178	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
179	C₂×C₃×C₅	#508	1	2	yes	—
180	C₂³×C₃	—	4	1	no	—
181	C₂³×C₃	#686	1	1	yes	characteristic
182	C₃×Q₈	—	2	2	no	—
183	C₃×Q₈	—	4	1	no	—
184	C₃×D₄	—	2	2	no	—
185	C₃×D₄	—	4	4	no	—
186	C₃×D₄	—	4	1	no	—
187	C₂×C₃×C₄	—	4	1	no	—
188	C₂×C₃×C₄	—	4	4	no	—
189	C₂×C₃×C₄	—	20	2	no	—
190	C₂×C₃×C₄	#686	1	1	yes	characteristic
191	C₂×C₃×C₄	#687	1	2	yes	—
192	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
193	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
194	C₃×C₈	C₂²⋊Dic₅	1	2	yes	—
195	C₃×C₈	C₂²×Dic₅	1	1	yes	characteristic
196	C₃×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	4	2	no	—
199	C₂²×C₅	—	8	2	no	—
200	C₂²×C₅	C₃×#102	1	2	yes	—
201	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
202	C₄×C₅	—	4	2	no	—
203	C₄×C₅	C₃×#102	1	2	yes	—
204	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
205	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
206	Dic₅	—	8	2	no	—
207	#41	—	2	4	no	—
208	#41	—	4	2	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	C₃×C₂²⋊D₅	1	1	yes	characteristic
214	Q₁₆	—	2	1	no	—
215	Q₁₆	C₂×C₃×Dic₅	1	2	yes	—
216	SD₁₆	—	2	4	no	—
217	D₈	—	2	1	no	—
218	D₈	C₂×C₃×Dic₅	1	2	yes	—
219	C₂×C₈	C₂×C₃×Dic₅	1	2	yes	—
220	C₂×C₈	C₃×C₂²⋊D₅	1	1	yes	characteristic
221	C₂×C₈	C₃×C₂²⋊D₅	1	2	yes	—
222	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
223	C₄⋊C₄	—	10	2	no	—
224	C₂²⋊C₄	—	20	1	no	—
225	C₄²	—	20	1	no	—
226	C₁₆	—	10	2	no	—
227	C₃×C₅	#2138	1	1	yes	characteristic, complemented, Hall
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	4	2	no	—
230	C₂²×C₃	—	8	2	no	—
231	C₂²×C₃	#3964	1	2	yes	—
232	C₂²×C₃	C₂×#686	1	1	yes	characteristic
233	C₃×C₄	—	4	2	no	—
234	C₃×C₄	—	40	2	no	—
235	C₃×C₄	#3964	1	2	yes	—
236	C₃×C₄	C₂×#687	1	1	yes	characteristic
237	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
238	C₂×C₅	—	2	2	no	—
239	C₂×C₅	—	8	2	no	—
240	C₂×C₅	C₂×C₃×#102	1	1	yes	characteristic
241	C₂×C₅	C₃×#508	1	2	yes	—
242	C₂³	—	4	1	no	—
243	C₂³	C₃×#686	1	1	yes	characteristic
244	Q₈	—	2	2	no	—
245	Q₈	—	4	1	no	—
246	D₄	—	2	2	no	—
247	D₄	—	4	4	no	—
248	D₄	—	4	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	20	2	no	—
252	C₂×C₄	C₃×#686	1	1	yes	characteristic
253	C₂×C₄	C₃×#687	1	2	yes	—
254	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
255	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
256	C₈	C₃×C₂²⋊Dic₅	1	2	yes	—
257	C₈	C₂²×C₃×Dic₅	1	1	yes	characteristic
258	C₈	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	—	8	2	no	—
261	C₂×C₃	#65483	1	2	yes	—
262	C₂×C₃	C₂×#3964	1	1	yes	characteristic
263	C₅	C₃×#2138	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	C₃×#3964	1	2	yes	—
268	C₂²	C₂×C₃×#686	1	1	yes	characteristic
269	C₄	—	4	2	no	—
270	C₄	—	40	2	no	—
271	C₄	C₃×#3964	1	2	yes	—
272	C₄	C₂×C₃×#687	1	1	yes	characteristic
273	C₄	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
274	C₃	#10602870	1	1	yes	characteristic, direct product factor, Sylow
275	C₂	—	2	2	no	—
276	C₂	—	8	2	no	—
277	C₂	C₃×#65483	1	2	yes	—
278	C₂	C₂×C₃×#3964	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	#686
commutator	C₅×C₈	C₂²×C₃×C₄
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting, prime radical	C₂×C₃×C₅×#135	C₂
socle	C₂²×C₃×C₅	C₂×D₈
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₂×C₃×#135	D₅
hypocenter	C₅	C₃×#2138
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×C₈ ◁ C₂×C₃×#135

Complements

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

N	H ≅ G/N	type
2. C₃×#65395	C₂	semidirect
3. C₂²×C₃×#672	C₂	semidirect
4. C₃×#64784	C₂	semidirect
5. C₃×#64782	C₂	semidirect
6. #10602870	C₃	direct (N × H)
9. C₃×C₅×#135	C₄	semidirect
21. #65395	C₂×C₃	semidirect
22. C₂²×#672	C₂×C₃	semidirect
23. #64784	C₂×C₃	semidirect
24. #64782	C₂×C₃	semidirect
52. C₅×#135	C₃×C₄	semidirect
82. C₃×#135	Dic₅	semidirect
166. #135	C₃×Dic₅	semidirect
227. C₃×C₅	#2138	semidirect
263. C₅	C₃×#2138	semidirect
274. C₃	#10602870	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×C₅×#135	2
C₃×#65395	2
C₂²×C₃×#672	2
C₃×#64784	2
C₃×#64782	2
#10602870	3
C₃×#2138	5

Subgroups — C3×#10602870 #14313850

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602870 #14313850