Subgroups — C₃×#10600579 #14304929

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3946	C₂	1	2	yes	maximal
3	C₃×C₄×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
5	C₃×#64770	C₂	1	2	yes	maximal
6	#10600579	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄⋊Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₅	C₄	1	1	yes	characteristic
12	C₂×C₃×C₄×Dic₅	C₂²	1	2	yes	—
13	C₂²×C₃×#168	C₂²	1	2	yes	—
14	C₃×C₄×C₅×C₈	—	2	2	no	—
15	C₃×#3963	C₄	1	2	yes	—
16	C₃×#3946	—	2	4	no	—
17	C₃×C₄⋊#168	—	2	2	no	—
18	C₃×#1939	—	5	1	no	maximal, Hall
19	C₂×C₄×C₅×C₈	C₂×C₃	1	1	yes	characteristic
20	C₂×#3946	C₂×C₃	1	2	yes	—
21	C₄×C₄⋊Dic₅	C₂×C₃	1	1	yes	characteristic
22	C₂×C₄⋊#168	C₂×C₃	1	1	yes	characteristic
23	#64770	C₂×C₃	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
28	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	—	2	1	no	—
30	C₂×C₃×C₅×C₈	—	2	1	no	—
31	C₂×C₃×C₅×C₈	—	2	1	no	—
32	C₂×C₃×C₅×C₈	—	2	1	no	—
33	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
35	C₃×C₄²×C₅	D₄	1	2	yes	—
36	C₃×C₄²×C₅	Q₈	1	2	yes	—
37	C₃×C₄⋊Dic₅	—	2	2	no	—
38	C₃×C₄⋊Dic₅	C₈	1	4	yes	complemented
39	C₃×C₄×Dic₅	—	2	2	no	—
40	C₃×C₄×Dic₅	—	2	2	no	—
41	C₂×C₃×#168	—	2	2	no	—
42	C₂×C₃×#168	—	2	2	no	—
43	C₂×C₃×#168	—	2	2	no	—
44	C₂×C₃×C₄⋊C₈	—	5	2	no	—
45	C₂×C₃×C₄⋊C₈	—	5	1	no	—
46	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
47	C₃×C₄×C₄⋊C₄	—	5	1	no	—
48	C₃×#336	—	5	2	no	—
49	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₄⋊Dic₅	C₃×C₄	1	1	yes	characteristic
53	C₂×C₄⋊Dic₅	C₃×C₄	1	1	yes	characteristic
54	C₂×C₄×Dic₅	C₂²×C₃	1	2	yes	—
55	C₂²×#168	C₂²×C₃	1	2	yes	—
56	C₄×C₅×C₈	—	2	2	no	—
57	#3963	C₃×C₄	1	2	yes	—
58	#3946	—	2	4	no	—
59	C₄⋊#168	—	2	2	no	—
60	#1939	—	5	1	no	Sylow
61	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	—	2	1	no	—
63	C₂×C₃×C₄×C₅	—	2	1	no	—
64	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
67	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
74	C₂×C₃×Dic₅	—	2	2	no	—
75	C₂×C₃×Dic₅	—	2	2	no	—
76	C₂×C₃×Dic₅	—	2	2	no	—
77	C₂×C₃×Dic₅	—	2	2	no	—
78	C₂×C₃×Dic₅	—	2	2	no	—
79	C₂×C₃×Dic₅	C₂×C₈	1	4	yes	—
80	C₃×C₅×C₈	—	2	2	no	—
81	C₃×C₅×C₈	—	2	2	no	—
82	C₃×#168	—	4	2	no	—
83	C₂²×C₃×C₈	—	5	2	no	—
84	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
85	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
86	C₂×C₃×C₄⋊C₄	—	5	1	no	—
87	C₂×C₃×C₄⋊C₄	—	5	1	no	—
88	C₂×C₃×C₄²	—	5	2	no	—
89	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
90	C₃×C₄⋊C₈	—	10	4	no	—
91	C₃×C₄⋊C₈	—	10	2	no	—
92	C₃×C₄×C₈	—	2	2	no	—
93	C₃×#95	—	5	2	no	—
94	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
97	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
98	C₂²×Dic₅	C₂×C₃×C₄	1	2	yes	—
99	C₂×C₅×C₈	—	2	1	no	—
100	C₂×C₅×C₈	—	2	1	no	—
101	C₂×C₅×C₈	—	2	1	no	—
102	C₂×C₅×C₈	—	2	1	no	—
103	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
104	C₂×C₅×C₈	C₃×Q₈	1	2	yes	—
105	C₄²×C₅	C₃×D₄	1	2	yes	—
106	C₄²×C₅	C₃×Q₈	1	2	yes	—
107	C₄⋊Dic₅	—	2	2	no	—
108	C₄⋊Dic₅	C₃×C₈	1	4	yes	complemented
109	C₄×Dic₅	—	2	2	no	—
110	C₄×Dic₅	—	2	2	no	—
111	C₂×#168	—	2	2	no	—
112	C₂×#168	—	2	2	no	—
113	C₂×#168	—	2	2	no	—
114	C₂×C₄⋊C₈	—	5	2	no	—
115	C₂×C₄⋊C₈	—	5	1	no	—
116	C₂×C₄×C₈	C₃×D₅	1	1	yes	characteristic
117	C₄×C₄⋊C₄	—	5	1	no	—
118	#336	—	5	2	no	—
119	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
121	C₂²×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
122	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
123	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
124	C₂²×C₃×C₅	#131	1	1	yes	characteristic
125	C₂²×C₃×C₅	#131	1	1	yes	characteristic
126	C₃×C₄×C₅	—	2	2	no	—
127	C₃×C₄×C₅	C₄⋊C₈	1	4	yes	complemented
128	C₃×C₄×C₅	#121	1	1	yes	characteristic
129	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
130	C₃×C₄×C₅	#125	1	1	yes	characteristic
131	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
132	C₃×Dic₅	—	2	4	no	—
133	C₃×Dic₅	—	4	2	no	—
134	C₂²×C₃×C₄	—	5	2	no	—
135	C₂²×C₃×C₄	—	5	2	no	—
136	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
137	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
138	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
139	C₂×C₃×C₈	—	2	1	no	—
140	C₂×C₃×C₈	—	2	1	no	—
141	C₂×C₃×C₈	—	2	1	no	—
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	10	2	no	—
144	C₂×C₃×C₈	—	10	2	no	—
145	C₂×C₃×C₈	—	10	2	no	—
146	C₂×C₃×C₈	Dic₁₀	1	2	yes	—
147	C₂×C₃×C₈	D₂₀	1	2	yes	—
148	C₃×C₄⋊C₄	—	5	4	no	—
149	C₃×C₄⋊C₄	—	10	2	no	—
150	C₃×C₄²	—	10	2	no	—
151	C₃×C₄²	—	10	2	no	—
152	C₃×C₄²	Dic₁₀	1	2	yes	—
153	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
154	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
155	C₂×C₄×C₅	—	2	1	no	—
156	C₂×C₄×C₅	—	2	1	no	—
157	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
158	C₂×C₄×C₅	C₃×C₄⋊C₄	1	2	yes	—
159	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
160	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
161	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
165	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
166	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
167	C₂×Dic₅	—	2	2	no	—
168	C₂×Dic₅	—	2	2	no	—
169	C₂×Dic₅	—	2	2	no	—
170	C₂×Dic₅	—	2	2	no	—
171	C₂×Dic₅	—	2	2	no	—
172	C₂×Dic₅	C₂×C₃×C₈	1	4	yes	—
173	C₅×C₈	—	2	2	no	—
174	C₅×C₈	—	2	2	no	—
175	#168	—	4	2	no	—
176	C₂²×C₈	—	5	2	no	—
177	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
178	C₂²×C₈	C₂×C₃×D₅	1	1	yes	characteristic
179	C₂×C₄⋊C₄	—	5	1	no	—
180	C₂×C₄⋊C₄	—	5	1	no	—
181	C₂×C₄²	—	5	2	no	—
182	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
183	C₄⋊C₈	—	10	4	no	—
184	C₄⋊C₈	—	10	2	no	—
185	C₄×C₈	—	2	2	no	—
186	#95	—	5	2	no	—
187	C₂×C₃×C₅	#389	1	1	yes	characteristic
188	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
189	C₂×C₃×C₅	#424	1	1	yes	characteristic
190	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
191	C₂×C₃×C₅	#436	1	1	yes	characteristic
192	C₂×C₃×C₅	C₈×Q₈	1	1	yes	characteristic
193	C₂×C₃×C₅	#446	1	1	yes	characteristic
194	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	5	4	no	—
198	C₂×C₃×C₄	—	10	2	no	—
199	C₂×C₃×C₄	—	10	2	no	—
200	C₂×C₃×C₄	—	10	2	no	—
201	C₂×C₃×C₄	—	10	2	no	—
202	C₂×C₃×C₄	—	10	2	no	—
203	C₂×C₃×C₄	C₈×D₅	1	2	yes	—
204	C₂×C₃×C₄	#676	1	2	yes	—
205	C₂×C₃×C₄	#683	1	2	yes	—
206	C₂×C₃×C₄	#683	1	2	yes	—
207	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
210	C₂×C₃×C₄	#709	1	1	yes	characteristic
211	C₂×C₃×C₄	#709	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
213	C₃×C₈	—	2	2	no	—
214	C₃×C₈	—	2	2	no	—
215	C₃×C₈	—	20	2	no	—
216	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
217	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
218	C₂²×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
219	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
220	C₂²×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
221	C₂²×C₅	C₃×#131	1	1	yes	characteristic
222	C₂²×C₅	C₃×#131	1	1	yes	characteristic
223	C₄×C₅	—	2	2	no	—
224	C₄×C₅	C₃×C₄⋊C₈	1	4	yes	complemented
225	C₄×C₅	C₃×#121	1	1	yes	characteristic
226	C₄×C₅	C₃×C₂²⋊Q₈	1	1	yes	characteristic
227	C₄×C₅	C₃×#125	1	1	yes	characteristic
228	C₄×C₅	C₃×C₄⋊Q₈	1	1	yes	characteristic
229	Dic₅	—	2	4	no	—
230	Dic₅	—	4	2	no	—
231	C₂²×C₄	—	5	2	no	—
232	C₂²×C₄	—	5	2	no	—
233	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
234	C₂²×C₄	C₃×C₄×D₅	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	10	2	no	—
241	C₂×C₈	—	10	2	no	—
242	C₂×C₈	—	10	2	no	—
243	C₂×C₈	C₃×Dic₁₀	1	2	yes	—
244	C₂×C₈	C₃×D₂₀	1	2	yes	—
245	C₄⋊C₄	—	5	4	no	—
246	C₄⋊C₄	—	10	2	no	—
247	C₄²	—	10	2	no	—
248	C₄²	—	10	2	no	—
249	C₄²	C₃×Dic₁₀	1	2	yes	—
250	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
251	C₃×C₅	#1939	1	1	yes	characteristic, complemented, Hall
252	C₂²×C₃	C₄×Dic₁₀	1	1	yes	characteristic
253	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
254	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
255	C₂²×C₃	C₂×#676	1	1	yes	characteristic
256	C₂²×C₃	#4047	1	1	yes	characteristic
257	C₂²×C₃	#4047	1	1	yes	characteristic
258	C₂²×C₃	C₂×#683	1	1	yes	characteristic
259	C₃×C₄	—	2	2	no	—
260	C₃×C₄	—	10	4	no	—
261	C₃×C₄	—	20	2	no	—
262	C₃×C₄	#3946	1	4	yes	complemented
263	C₃×C₄	#4015	1	1	yes	characteristic
264	C₃×C₄	#4016	1	1	yes	characteristic
265	C₃×C₄	#4070	1	1	yes	characteristic
266	C₃×C₄	#4076	1	1	yes	characteristic
267	C₂×C₅	C₃×#389	1	1	yes	characteristic
268	C₂×C₅	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
269	C₂×C₅	C₃×#424	1	1	yes	characteristic
270	C₂×C₅	C₃×C₈×D₄	1	1	yes	characteristic
271	C₂×C₅	C₃×#436	1	1	yes	characteristic
272	C₂×C₅	C₃×C₈×Q₈	1	1	yes	characteristic
273	C₂×C₅	C₃×#446	1	1	yes	characteristic
274	C₂³	C₂×C₃×C₄×D₅	1	1	yes	characteristic
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	5	4	no	—
278	C₂×C₄	—	10	2	no	—
279	C₂×C₄	—	10	2	no	—
280	C₂×C₄	—	10	2	no	—
281	C₂×C₄	—	10	2	no	—
282	C₂×C₄	—	10	2	no	—
283	C₂×C₄	C₃×C₈×D₅	1	2	yes	—
284	C₂×C₄	C₃×#676	1	2	yes	—
285	C₂×C₄	C₃×#683	1	2	yes	—
286	C₂×C₄	C₃×#683	1	2	yes	—
287	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
288	C₂×C₄	C₂×C₃×Dic₁₀	1	1	yes	characteristic
289	C₂×C₄	C₂×C₃×D₂₀	1	1	yes	characteristic
290	C₂×C₄	C₃×#709	1	1	yes	characteristic
291	C₂×C₄	C₃×#709	1	1	yes	characteristic
292	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	2	2	no	—
295	C₈	—	20	2	no	—
296	C₂×C₃	C₈×Dic₁₀	1	1	yes	characteristic
297	C₂×C₃	#64968	1	1	yes	characteristic
298	C₂×C₃	C₈×D₂₀	1	1	yes	characteristic
299	C₂×C₃	#64977	1	1	yes	characteristic
300	C₂×C₃	#65217	1	1	yes	characteristic
301	C₂×C₃	C₂×#3946	1	1	yes	characteristic
302	C₂×C₃	#65391	1	1	yes	characteristic
303	C₅	C₃×#1939	1	1	yes	characteristic, complemented, Sylow
304	C₂²	C₃×C₄×Dic₁₀	1	1	yes	characteristic
305	C₂²	C₃×C₄×D₂₀	1	1	yes	characteristic
306	C₂²	C₂×C₃×C₈×D₅	1	1	yes	characteristic
307	C₂²	C₂×C₃×#676	1	1	yes	characteristic
308	C₂²	C₃×#4047	1	1	yes	characteristic
309	C₂²	C₃×#4047	1	1	yes	characteristic
310	C₂²	C₂×C₃×#683	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	10	4	no	—
313	C₄	—	20	2	no	—
314	C₄	C₃×#3946	1	4	yes	complemented
315	C₄	C₃×#4015	1	1	yes	characteristic
316	C₄	C₃×#4016	1	1	yes	characteristic
317	C₄	C₃×#4070	1	1	yes	characteristic
318	C₄	C₃×#4076	1	1	yes	characteristic
319	C₃	#10600579	1	1	yes	characteristic, direct product factor, Sylow
320	C₂	C₃×C₈×Dic₁₀	1	1	yes	characteristic
321	C₂	C₃×#64968	1	1	yes	characteristic
322	C₂	C₃×C₈×D₂₀	1	1	yes	characteristic
323	C₂	C₃×#64977	1	1	yes	characteristic
324	C₂	C₃×#65217	1	1	yes	characteristic
325	C₂	C₂×C₃×#3946	1	1	yes	characteristic
326	C₂	C₃×#65391	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₅
commutator	C₂²×C₅	C₂²×C₃×C₈
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄×C₈	D₅
hypocenter	C₅	C₃×#1939
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
6. #10600579	C₃	direct (N × H)
38. C₃×C₄⋊Dic₅	C₈	semidirect
108. C₄⋊Dic₅	C₃×C₈	semidirect
127. C₃×C₄×C₅	C₄⋊C₈	semidirect
224. C₄×C₅	C₃×C₄⋊C₈	semidirect
251. C₃×C₅	#1939	semidirect
262. C₃×C₄	#3946	semidirect
303. C₅	C₃×#1939	semidirect
314. C₄	C₃×#3946	semidirect
319. C₃	#10600579	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×C₄×C₅×C₈	2
C₂×C₃×#3946	2
C₃×C₄×C₄⋊Dic₅	2
C₂×C₃×C₄⋊#168	2
C₃×#64770	2
#10600579	3
C₃×#1939	5

Subgroups — C3×#10600579 #14304929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10600579 #14304929