Subgroups — C₃×#10600078 #14314354

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3930	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#64978	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64721	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#64721	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₄⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600078	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×#676	C₄	1	1	yes	characteristic
10	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₃×#4018	C₄	1	1	yes	characteristic
12	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×#3947	C₄	1	1	yes	characteristic
16	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
17	C₂×C₃×#672	C₂²	1	1	yes	characteristic
18	C₂×C₃×#672	C₂²	1	1	yes	characteristic
19	C₃×#3935	C₄	1	1	yes	characteristic
20	C₃×#3930	—	2	2	no	—
21	C₃×D₅×C₁₆	—	2	2	no	—
22	C₃×#2151	—	5	1	no	maximal, Hall
23	C₂×#3930	C₂×C₃	1	1	yes	characteristic
24	C₂×D₅×C₁₆	C₂×C₃	1	1	yes	characteristic
25	#64978	C₂×C₃	1	1	yes	characteristic
26	C₅×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
27	#64721	C₂×C₃	1	1	yes	characteristic, complemented
28	#64721	C₂×C₃	1	1	yes	characteristic, complemented
29	C₄⋊#672	C₂×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃×#685	C₈	1	1	yes	characteristic
35	C₃×#685	C₈	1	1	yes	characteristic
36	C₃×#683	C₈	1	1	yes	characteristic
37	C₃×#683	C₈	1	1	yes	characteristic
38	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
41	C₃×#676	—	2	2	no	—
42	C₃×C₈×D₅	D₄	1	2	yes	—
43	C₃×C₈×D₅	Q₈	1	2	yes	—
44	C₃×C₅×C₁₆	—	2	1	no	—
45	C₃×C₅×C₁₆	—	2	1	no	—
46	C₃×#672	—	2	1	no	—
47	C₃×#672	—	2	1	no	—
48	C₂×C₃×M₃₂	—	5	1	no	—
49	C₂²×C₃×C₁₆	—	5	1	no	—
50	C₃×#405	—	5	1	no	—
51	C₃×C₄⋊C₁₆	—	5	1	no	—
52	C₃×C₄⋊C₁₆	—	5	1	no	—
53	C₃×C₄⋊C₁₆	—	5	1	no	—
54	C₃×C₄⋊C₁₆	D₅	1	1	yes	characteristic, complemented
55	C₂×#676	C₃×C₄	1	1	yes	characteristic
56	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
57	#4018	C₃×C₄	1	1	yes	characteristic
58	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
59	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
60	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
61	#3947	C₃×C₄	1	1	yes	characteristic
62	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
63	C₂×#672	C₂²×C₃	1	1	yes	characteristic
64	C₂×#672	C₂²×C₃	1	1	yes	characteristic
65	#3935	C₃×C₄	1	1	yes	characteristic
66	#3930	—	2	2	no	—
67	D₅×C₁₆	—	2	2	no	—
68	#2151	—	5	1	no	Sylow
69	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
74	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
76	C₃×C₄×D₅	C₄⋊C₄	1	2	yes	—
77	C₃×C₄×D₅	M₁₆	1	2	yes	—
78	C₃×C₅×C₈	—	2	1	no	—
79	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
80	C₃×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
81	C₃×#168	—	2	1	no	—
82	C₃×#168	C₄⋊C₄	1	2	yes	—
83	C₂×C₃×M₁₆	—	5	1	no	—
84	C₂²×C₃×C₈	—	5	1	no	—
85	C₃×#117	—	5	1	no	—
86	C₃×M₃₂	—	10	2	no	—
87	C₂×C₃×C₁₆	—	5	1	no	—
88	C₂×C₃×C₁₆	—	5	1	no	—
89	C₂×C₃×C₁₆	—	10	2	no	—
90	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
91	C₂×C₃×C₁₆	C₂×D₅	1	1	yes	characteristic
92	C₃×#97	—	5	1	no	—
93	C₃×#97	—	5	1	no	—
94	C₃×C₄×C₈	—	5	1	no	—
95	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
96	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
99	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
100	#685	C₃×C₈	1	1	yes	characteristic
101	#685	C₃×C₈	1	1	yes	characteristic
102	#683	C₃×C₈	1	1	yes	characteristic
103	#683	C₃×C₈	1	1	yes	characteristic
104	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
105	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
106	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
107	#676	—	2	2	no	—
108	C₈×D₅	C₃×D₄	1	2	yes	—
109	C₈×D₅	C₃×Q₈	1	2	yes	—
110	C₅×C₁₆	—	2	1	no	—
111	C₅×C₁₆	—	2	1	no	—
112	#672	—	2	1	no	—
113	#672	—	2	1	no	—
114	C₂×M₃₂	—	5	1	no	—
115	C₂²×C₁₆	—	5	1	no	—
116	#405	—	5	1	no	—
117	C₄⋊C₁₆	—	5	1	no	—
118	C₄⋊C₁₆	—	5	1	no	—
119	C₄⋊C₁₆	—	5	1	no	—
120	C₄⋊C₁₆	C₃×D₅	1	1	yes	characteristic, complemented
121	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₃×D₅	—	2	1	no	—
123	C₂×C₃×D₅	—	2	1	no	—
124	C₂×C₃×D₅	C₄⋊C₈	1	2	yes	—
125	C₃×C₄×C₅	—	2	1	no	—
126	C₃×C₄×C₅	—	2	1	no	—
127	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
129	C₃×Dic₅	—	2	1	no	—
130	C₃×Dic₅	—	2	1	no	—
131	C₃×Dic₅	C₄⋊C₈	1	2	yes	—
132	C₂²×C₃×C₄	—	5	1	no	—
133	C₃×M₁₆	—	10	2	no	—
134	C₂×C₃×C₈	—	5	1	no	—
135	C₂×C₃×C₈	—	5	2	no	—
136	C₂×C₃×C₈	—	5	2	no	—
137	C₂×C₃×C₈	—	5	1	no	—
138	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
139	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
140	C₃×C₄⋊C₄	—	5	1	no	—
141	C₃×C₄⋊C₄	—	5	1	no	—
142	C₃×C₂²⋊C₄	—	5	1	no	—
143	C₃×C₂²⋊C₄	—	5	1	no	—
144	C₃×C₄²	—	5	1	no	—
145	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
146	C₃×C₁₆	—	2	1	no	—
147	C₃×C₁₆	—	2	1	no	—
148	C₃×C₁₆	—	10	1	no	—
149	C₃×C₁₆	—	10	1	no	—
150	C₂²×D₅	C₂×C₃×C₈	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
154	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
155	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
156	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
157	C₄×D₅	C₃×C₄⋊C₄	1	2	yes	—
158	C₄×D₅	C₃×M₁₆	1	2	yes	—
159	C₅×C₈	—	2	1	no	—
160	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
161	C₅×C₈	C₂×C₃×Q₈	1	1	yes	characteristic
162	#168	—	2	1	no	—
163	#168	C₃×C₄⋊C₄	1	2	yes	—
164	C₂×M₁₆	—	5	1	no	—
165	C₂²×C₈	—	5	1	no	—
166	#117	—	5	1	no	—
167	M₃₂	—	10	2	no	—
168	C₂×C₁₆	—	5	1	no	—
169	C₂×C₁₆	—	5	1	no	—
170	C₂×C₁₆	—	10	2	no	—
171	C₂×C₁₆	C₂×C₃×D₅	1	1	yes	characteristic
172	C₂×C₁₆	C₂×C₃×D₅	1	1	yes	characteristic
173	#97	—	5	1	no	—
174	#97	—	5	1	no	—
175	C₄×C₈	—	5	1	no	—
176	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
177	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
178	C₂×C₃×C₅	#504	1	1	yes	characteristic
179	C₂×C₃×C₅	#504	1	1	yes	characteristic
180	C₃×D₅	—	2	2	no	—
181	C₂³×C₃	—	5	1	no	—
182	C₂×C₃×C₄	—	5	1	no	—
183	C₂×C₃×C₄	—	5	2	no	—
184	C₂×C₃×C₄	—	5	2	no	—
185	C₂×C₃×C₄	—	5	1	no	—
186	C₂×C₃×C₄	—	5	1	no	—
187	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
188	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
190	C₃×C₈	—	2	1	no	—
191	C₃×C₈	—	5	2	no	—
192	C₃×C₈	—	10	1	no	—
193	C₃×C₈	D₄×D₅	1	1	yes	characteristic
194	C₃×C₈	Q₈×D₅	1	1	yes	characteristic
195	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
196	C₂×D₅	—	2	1	no	—
197	C₂×D₅	—	2	1	no	—
198	C₂×D₅	C₃×C₄⋊C₈	1	2	yes	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	2	1	no	—
201	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
202	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
203	Dic₅	—	2	1	no	—
204	Dic₅	—	2	1	no	—
205	Dic₅	C₃×C₄⋊C₈	1	2	yes	—
206	C₂²×C₄	—	5	1	no	—
207	M₁₆	—	10	2	no	—
208	C₂×C₈	—	5	1	no	—
209	C₂×C₈	—	5	2	no	—
210	C₂×C₈	—	5	2	no	—
211	C₂×C₈	—	5	1	no	—
212	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
213	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
214	C₄⋊C₄	—	5	1	no	—
215	C₄⋊C₄	—	5	1	no	—
216	C₂²⋊C₄	—	5	1	no	—
217	C₂²⋊C₄	—	5	1	no	—
218	C₄²	—	5	1	no	—
219	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
220	C₁₆	—	2	1	no	—
221	C₁₆	—	2	1	no	—
222	C₁₆	—	10	1	no	—
223	C₁₆	—	10	1	no	—
224	C₃×C₅	#2151	1	1	yes	characteristic, complemented, Hall
225	C₂²×C₃	—	5	2	no	—
226	C₂²×C₃	—	10	1	no	—
227	C₂²×C₃	—	10	1	no	—
228	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
229	C₃×C₄	—	2	1	no	—
230	C₃×C₄	—	2	1	no	—
231	C₃×C₄	—	5	2	no	—
232	C₃×C₄	—	10	1	no	—
233	C₃×C₄	—	10	1	no	—
234	C₃×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
235	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
236	C₂×C₅	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
237	C₂×C₅	C₃×#504	1	1	yes	characteristic
238	C₂×C₅	C₃×#504	1	1	yes	characteristic
239	D₅	—	2	2	no	—
240	C₂³	—	5	1	no	—
241	C₂×C₄	—	5	1	no	—
242	C₂×C₄	—	5	2	no	—
243	C₂×C₄	—	5	2	no	—
244	C₂×C₄	—	5	1	no	—
245	C₂×C₄	—	5	1	no	—
246	C₂×C₄	C₃×C₈×D₅	1	1	yes	characteristic
247	C₂×C₄	C₃×C₈×D₅	1	1	yes	characteristic
248	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
249	C₈	—	2	1	no	—
250	C₈	—	5	2	no	—
251	C₈	—	10	1	no	—
252	C₈	C₃×D₄×D₅	1	1	yes	characteristic
253	C₈	C₃×Q₈×D₅	1	1	yes	characteristic
254	C₂×C₃	—	10	2	no	—
255	C₂×C₃	D₅×C₄⋊C₈	1	1	yes	characteristic
256	C₂×C₃	#65190	1	1	yes	characteristic
257	C₂×C₃	#65196	1	1	yes	characteristic
258	C₅	C₃×#2151	1	1	yes	characteristic, complemented, Sylow
259	C₂²	—	5	2	no	—
260	C₂²	—	10	1	no	—
261	C₂²	—	10	1	no	—
262	C₂²	C₂×C₃×C₈×D₅	1	1	yes	characteristic
263	C₄	—	2	1	no	—
264	C₄	—	2	1	no	—
265	C₄	—	5	2	no	—
266	C₄	—	10	1	no	—
267	C₄	—	10	1	no	—
268	C₄	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
269	C₄	C₃×D₅×M₁₆	1	1	yes	characteristic
270	C₃	#10600078	1	1	yes	characteristic, direct product factor, Sylow
271	C₂	—	10	2	no	—
272	C₂	C₃×D₅×C₄⋊C₈	1	1	yes	characteristic
273	C₂	C₃×#65190	1	1	yes	characteristic
274	C₂	C₃×#65196	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₃×C₈
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₃×C₅×C₄⋊C₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂³×C₃
hypercenter	C₃×C₄⋊C₁₆	D₅
hypocenter	C₅	C₃×#2151
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₃×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
5. C₃×#64721	C₂	semidirect
6. C₃×#64721	C₂	semidirect
7. C₃×C₄⋊#672	C₂	semidirect
8. #10600078	C₃	direct (N × H)
26. C₅×C₄⋊C₁₆	C₂×C₃	semidirect
27. #64721	C₂×C₃	semidirect
28. #64721	C₂×C₃	semidirect
29. C₄⋊#672	C₂×C₃	semidirect
54. C₃×C₄⋊C₁₆	D₅	semidirect
120. C₄⋊C₁₆	C₃×D₅	semidirect
224. C₃×C₅	#2151	semidirect
258. C₅	C₃×#2151	semidirect
270. C₃	#10600078	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×#3930	2
C₂×C₃×D₅×C₁₆	2
C₃×#64978	2
C₃×C₅×C₄⋊C₁₆	2
C₃×#64721	2
C₃×C₄⋊#672	2
#10600078	3
C₃×#2151	5

Subgroups — C3×#10600078 #14314354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600078 #14314354