Subgroups — C₃×#10614934 #14439208

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×#4088	C₂	1	4	yes	maximal, complemented
3	C₃×#65811	C₂	1	1	yes	maximal, characteristic
4	C₃×#65381	C₂	1	2	yes	maximal, complemented
5	C₃×#65375	C₂	1	4	yes	maximal, complemented
6	C₃×#65374	C₂	1	2	yes	maximal, complemented
7	C₃×C₄×#681	C₂	1	1	yes	maximal, characteristic, complemented
8	#10614934	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
10	C₂×C₃×#709	C₂²	1	2	yes	—
11	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
15	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
16	C₂×C₃×#689	C₂²	1	8	yes	—
17	C₃×#4088	—	2	16	no	—
18	C₂×C₃×#688	C₂²	1	8	yes	—
19	C₃×#4076	—	2	2	no	—
20	C₂×C₃×#681	C₂²	1	2	yes	—
21	C₃×#4021	C₂²	1	2	yes	—
22	C₃×#4020	C₂²	1	1	yes	characteristic
23	C₃×C₄×D₂₀	—	2	2	no	—
24	C₃×#4015	C₂²	1	1	yes	characteristic
25	C₃×#3935	C₂²	1	2	yes	—
26	C₃×C₄×#168	C₂²	1	2	yes	—
27	C₃×#3152	—	5	1	no	maximal, Hall
28	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic, complemented
29	C₂×#4088	C₂×C₃	1	4	yes	complemented
30	#65811	C₂×C₃	1	1	yes	characteristic
31	#65381	C₂×C₃	1	2	yes	complemented
32	#65375	C₂×C₃	1	4	yes	complemented
33	#65374	C₂×C₃	1	2	yes	complemented
34	C₄×#681	C₂×C₃	1	1	yes	characteristic, complemented
35	C₂×C₃×C₅×Q₈	—	2	2	no	—
36	C₂×C₃×C₅×Q₈	—	2	8	no	—
37	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
38	C₂²×C₃×C₄×C₅	—	2	2	no	—
39	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
42	C₃×#709	—	2	8	no	—
43	C₂×C₃×D₂₀	—	2	2	no	—
44	C₂×C₃×D₂₀	C₂³	1	2	yes	—
45	C₂×C₃×C₄×D₅	—	2	2	no	—
46	C₂×C₃×Dic₁₀	C₂³	1	2	yes	—
47	C₃×C₅×C₄⋊C₄	—	2	4	no	—
48	C₃×C₅×C₄⋊C₄	—	2	4	no	—
49	C₃×C₄²×C₅	D₄	1	2	yes	—
50	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₃×#689	—	2	16	no	—
53	C₃×#688	—	2	16	no	—
54	C₃×#685	—	2	4	no	—
55	C₃×C₄⋊Dic₅	—	2	2	no	—
56	C₃×#681	D₄	1	8	yes	complemented
57	C₂×C₃×#168	C₂³	1	4	yes	—
58	C₂×C₃×#137	—	5	4	no	—
59	C₃×#532	—	5	1	no	—
60	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic, complemented
61	C₃×#497	—	5	4	no	—
62	C₃×#494	—	5	2	no	—
63	C₃×#492	—	5	2	no	—
64	C₃×C₄×M₁₆	—	5	1	no	—
65	C₂²×C₅×Q₈	C₂²×C₃	1	2	yes	—
66	C₂×#709	C₂²×C₃	1	2	yes	—
67	C₅×C₄⋊Q₈	—	2	2	no	—
68	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
69	C₅×C₄⋊Q₈	C₂²×C₃	1	2	yes	—
70	C₂×C₅×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
72	C₂×#689	C₂²×C₃	1	8	yes	—
73	#4088	—	2	16	no	—
74	C₂×#688	C₂²×C₃	1	8	yes	—
75	#4076	—	2	2	no	—
76	C₂×#681	C₂²×C₃	1	2	yes	—
77	#4021	C₂²×C₃	1	2	yes	—
78	#4020	C₂²×C₃	1	1	yes	characteristic
79	C₄×D₂₀	—	2	2	no	—
80	#4015	C₂²×C₃	1	1	yes	characteristic
81	#3935	C₂²×C₃	1	2	yes	—
82	C₄×#168	C₂²×C₃	1	2	yes	—
83	#3152	—	5	1	no	Sylow
84	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₃×D₅	—	2	2	no	—
86	C₃×C₅×Q₈	—	2	8	no	—
87	C₃×C₅×Q₈	—	4	4	no	—
88	C₂×C₃×C₄×C₅	—	2	4	no	—
89	C₂×C₃×C₄×C₅	—	2	4	no	—
90	C₂×C₃×C₄×C₅	—	4	4	no	—
91	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
92	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
94	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
95	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	C₃×C₂²⋊D₅	—	4	4	no	—
97	C₂×C₃×Dic₅	—	2	2	no	—
98	C₃×D₂₀	—	2	4	no	—
99	C₃×D₂₀	—	4	4	no	—
100	C₃×C₄×D₅	—	4	4	no	—
101	C₃×Dic₁₀	—	2	4	no	—
102	C₃×#168	C₂×D₄	1	8	yes	—
103	C₂×C₃×#41	—	5	2	no	—
104	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
105	C₃×#137	—	10	16	no	—
106	C₂×C₃×Q₁₆	—	5	8	no	—
107	C₂×C₃×SD₁₆	—	5	8	no	—
108	C₂×C₃×M₁₆	—	5	2	no	—
109	C₃×C₄⋊Q₈	—	2	2	no	—
110	C₃×C₄⋊Q₈	—	5	1	no	—
111	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
112	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
113	C₃×#127	—	5	1	no	—
114	C₃×#124	—	5	2	no	—
115	C₃×#121	—	10	2	no	—
116	C₃×C₄×D₄	—	10	2	no	—
117	C₂×C₃×C₄⋊C₄	—	2	2	no	—
118	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
119	C₃×#97	—	5	2	no	—
120	C₃×C₄×C₈	—	5	2	no	—
121	C₂×C₅×Q₈	—	2	2	no	—
122	C₂×C₅×Q₈	—	2	8	no	—
123	C₂×C₅×Q₈	C₂³×C₃	1	4	yes	—
124	C₂²×C₄×C₅	—	2	2	no	—
125	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
126	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
127	C₂×C₂²⋊D₅	—	2	2	no	—
128	#709	—	2	8	no	—
129	C₂×D₂₀	—	2	2	no	—
130	C₂×D₂₀	C₂³×C₃	1	2	yes	—
131	C₂×C₄×D₅	—	2	2	no	—
132	C₂×Dic₁₀	C₂³×C₃	1	2	yes	—
133	C₅×C₄⋊C₄	—	2	4	no	—
134	C₅×C₄⋊C₄	—	2	4	no	—
135	C₄²×C₅	C₃×D₄	1	2	yes	—
136	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
137	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
138	#689	—	2	16	no	—
139	#688	—	2	16	no	—
140	#685	—	2	4	no	—
141	C₄⋊Dic₅	—	2	2	no	—
142	#681	C₃×D₄	1	8	yes	complemented
143	C₂×#168	C₂³×C₃	1	4	yes	—
144	C₂×#137	—	5	4	no	—
145	#532	—	5	1	no	—
146	C₂×C₄⋊Q₈	C₃×D₅	1	1	yes	characteristic, complemented
147	#497	—	5	4	no	—
148	#494	—	5	2	no	—
149	#492	—	5	2	no	—
150	C₄×M₁₆	—	5	1	no	—
151	C₂²×C₃×C₅	—	2	2	no	—
152	C₂²×C₃×C₅	#127	1	2	yes	—
153	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₃×D₅	—	4	2	no	—
155	C₂×C₃×D₅	—	4	4	no	—
156	C₃×C₄×C₅	—	2	2	no	—
157	C₃×C₄×C₅	—	4	4	no	—
158	C₃×C₄×C₅	#127	1	2	yes	—
159	C₃×C₄×C₅	#137	1	4	yes	complemented
160	C₃×C₄×C₅	C₂²×D₄	1	2	yes	—
161	C₃×Dic₅	—	4	2	no	—
162	C₃×#41	—	10	8	no	—
163	C₂×C₃×Q₈	—	2	2	no	—
164	C₂×C₃×Q₈	—	2	8	no	—
165	C₂×C₃×Q₈	—	5	2	no	—
166	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	—
167	C₂×C₃×D₄	—	5	2	no	—
168	C₂×C₃×D₄	—	10	2	no	—
169	C₂×C₃×D₄	—	10	2	no	—
170	C₂²×C₃×C₄	—	2	2	no	—
171	C₂²×C₃×C₄	—	10	2	no	—
172	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
173	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₃×Q₁₆	—	10	16	no	—
175	C₃×SD₁₆	—	10	16	no	—
176	C₃×M₁₆	—	5	8	no	—
177	C₂×C₃×C₈	—	5	4	no	—
178	C₃×C₄⋊C₄	—	2	4	no	—
179	C₃×C₄⋊C₄	—	2	4	no	—
180	C₃×C₄⋊C₄	—	10	2	no	—
181	C₃×C₂²⋊C₄	—	10	4	no	—
182	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
183	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
184	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
185	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
186	C₂²×D₅	—	2	2	no	—
187	C₅×Q₈	—	2	8	no	—
188	C₅×Q₈	—	4	4	no	—
189	C₂×C₄×C₅	—	2	4	no	—
190	C₂×C₄×C₅	—	2	4	no	—
191	C₂×C₄×C₅	—	4	4	no	—
192	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
193	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
194	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
196	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
197	C₂²⋊D₅	—	4	4	no	—
198	C₂×Dic₅	—	2	2	no	—
199	D₂₀	—	2	4	no	—
200	D₂₀	—	4	4	no	—
201	C₄×D₅	—	4	4	no	—
202	Dic₁₀	—	2	4	no	—
203	#168	C₂×C₃×D₄	1	8	yes	—
204	C₂×#41	—	5	2	no	—
205	C₂²×Q₈	C₂×C₃×D₅	1	2	yes	—
206	#137	—	10	16	no	—
207	C₂×Q₁₆	—	5	8	no	—
208	C₂×SD₁₆	—	5	8	no	—
209	C₂×M₁₆	—	5	2	no	—
210	C₄⋊Q₈	—	2	2	no	—
211	C₄⋊Q₈	—	5	1	no	—
212	C₄⋊Q₈	C₂×C₃×D₅	1	2	yes	—
213	C₄⋊Q₈	C₂×C₃×D₅	1	2	yes	—
214	#127	—	5	1	no	—
215	#124	—	5	2	no	—
216	#121	—	10	2	no	—
217	C₄×D₄	—	10	2	no	—
218	C₂×C₄⋊C₄	—	2	2	no	—
219	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
220	#97	—	5	2	no	—
221	C₄×C₈	—	5	2	no	—
222	C₂×C₃×C₅	—	2	2	no	—
223	C₂×C₃×C₅	C₂×#127	1	1	yes	characteristic
224	C₂×C₃×C₅	C₂×#137	1	2	yes	—
225	C₃×D₅	—	8	2	no	—
226	C₂³×C₃	—	10	2	no	—
227	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
228	C₃×Q₈	—	2	8	no	—
229	C₃×Q₈	—	4	4	no	—
230	C₃×Q₈	—	10	4	no	—
231	C₃×D₄	—	10	4	no	—
232	C₃×D₄	—	20	4	no	—
233	C₃×D₄	—	20	4	no	—
234	C₂×C₃×C₄	—	2	4	no	—
235	C₂×C₃×C₄	—	2	4	no	—
236	C₂×C₃×C₄	—	4	4	no	—
237	C₂×C₃×C₄	—	10	2	no	—
238	C₂×C₃×C₄	—	20	4	no	—
239	C₂×C₃×C₄	D₄×D₅	1	4	yes	—
240	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
241	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
242	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
243	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
244	C₃×C₈	—	5	8	no	—
245	C₂²×C₅	—	2	2	no	—
246	C₂²×C₅	C₃×#127	1	2	yes	—
247	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
248	C₂×D₅	—	4	4	no	—
249	C₂×D₅	—	4	2	no	—
250	C₄×C₅	—	2	2	no	—
251	C₄×C₅	—	4	4	no	—
252	C₄×C₅	C₃×#127	1	2	yes	—
253	C₄×C₅	C₃×#137	1	4	yes	complemented
254	C₄×C₅	C₂²×C₃×D₄	1	2	yes	—
255	Dic₅	—	4	2	no	—
256	#41	—	10	8	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	2	8	no	—
259	C₂×Q₈	—	5	2	no	—
260	C₂×Q₈	C₂²×C₃×D₅	1	4	yes	—
261	C₂×D₄	—	5	2	no	—
262	C₂×D₄	—	10	2	no	—
263	C₂×D₄	—	10	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	10	2	no	—
266	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
267	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
268	Q₁₆	—	10	16	no	—
269	SD₁₆	—	10	16	no	—
270	M₁₆	—	5	8	no	—
271	C₂×C₈	—	5	4	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	2	4	no	—
274	C₄⋊C₄	—	10	2	no	—
275	C₂²⋊C₄	—	10	4	no	—
276	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
277	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
278	C₄²	C₂²×C₃×D₅	1	1	yes	characteristic
279	C₃×C₅	#3152	1	1	yes	characteristic, complemented, Hall
280	C₂²×C₃	—	2	2	no	—
281	C₂²×C₃	—	20	2	no	—
282	C₂²×C₃	—	20	4	no	—
283	C₂²×C₃	#4086	1	2	yes	—
284	C₂²×C₃	C₂²×C₂²⋊D₅	1	1	yes	characteristic
285	C₃×C₄	—	2	2	no	—
286	C₃×C₄	—	4	4	no	—
287	C₃×C₄	—	20	2	no	—
288	C₃×C₄	#4086	1	2	yes	—
289	C₃×C₄	#4088	1	4	yes	complemented
290	C₃×C₄	C₂×D₄×D₅	1	2	yes	—
291	C₂×C₅	—	2	2	no	—
292	C₂×C₅	C₂×C₃×#127	1	1	yes	characteristic
293	C₂×C₅	C₂×C₃×#137	1	2	yes	—
294	D₅	—	8	2	no	—
295	C₂³	—	10	2	no	—
296	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
297	Q₈	—	2	8	no	—
298	Q₈	—	4	4	no	—
299	Q₈	—	10	4	no	—
300	D₄	—	10	4	no	—
301	D₄	—	20	4	no	—
302	D₄	—	20	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	10	2	no	—
307	C₂×C₄	—	20	4	no	—
308	C₂×C₄	C₃×D₄×D₅	1	4	yes	—
309	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
310	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
311	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
312	C₂×C₄	C₂³×C₃×D₅	1	1	yes	characteristic
313	C₈	—	5	8	no	—
314	C₂×C₃	—	2	2	no	—
315	C₂×C₃	—	40	2	no	—
316	C₂×C₃	C₂×#4086	1	1	yes	characteristic
317	C₂×C₃	C₂×#4088	1	2	yes	—
318	C₅	C₃×#3152	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	20	2	no	—
321	C₂²	—	20	4	no	—
322	C₂²	C₃×#4086	1	2	yes	—
323	C₂²	C₂²×C₃×C₂²⋊D₅	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	4	4	no	—
326	C₄	—	20	2	no	—
327	C₄	C₃×#4086	1	2	yes	—
328	C₄	C₃×#4088	1	4	yes	complemented
329	C₄	C₂×C₃×D₄×D₅	1	2	yes	—
330	C₃	#10614934	1	1	yes	characteristic, direct product factor, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	40	2	no	—
333	C₂	C₂×C₃×#4086	1	1	yes	characteristic
334	C₂	C₂×C₃×#4088	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂²×C₂²⋊D₅
commutator, cosocle	C₂×C₄×C₅	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×C₃×D₅
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₃×#65811	C₂
hypercenter	C₂×C₃×C₄⋊Q₈	D₅
hypocenter	C₅	C₃×#3152
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₄⋊Q₈	C₂	semidirect
2. C₂×C₃×#4088	C₂	semidirect
4. C₃×#65381	C₂	semidirect
5. C₃×#65375	C₂	semidirect
6. C₃×#65374	C₂	semidirect
7. C₃×C₄×#681	C₂	semidirect
8. #10614934	C₃	direct (N × H)
28. C₂×C₅×C₄⋊Q₈	C₂×C₃	semidirect
29. C₂×#4088	C₂×C₃	semidirect
31. #65381	C₂×C₃	semidirect
32. #65375	C₂×C₃	semidirect
33. #65374	C₂×C₃	semidirect
34. C₄×#681	C₂×C₃	semidirect
56. C₃×#681	D₄	semidirect
60. C₂×C₃×C₄⋊Q₈	D₅	semidirect
142. #681	C₃×D₄	semidirect
146. C₂×C₄⋊Q₈	C₃×D₅	semidirect
159. C₃×C₄×C₅	#137	semidirect
253. C₄×C₅	C₃×#137	semidirect
279. C₃×C₅	#3152	semidirect
289. C₃×C₄	#4088	semidirect
318. C₅	C₃×#3152	semidirect
328. C₄	C₃×#4088	semidirect
330. C₃	#10614934	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×C₄⋊Q₈	2
C₂×C₃×#4088	2
C₃×#65811	2
C₃×#65381	2
C₃×#65375	2
C₃×#65374	2
C₃×C₄×#681	2
#10614934	3
C₃×#3152	5

Subgroups — C3×#10614934 #14439208

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10614934 #14439208