Subgroups — C₄×#91121 #14295573

Order 1920.

720 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	C₃×C₄×C₅×#41	C₂	1	1	yes	maximal, characteristic
2	C₂×#91121	C₂	1	2	yes	maximal
3	C₄²×Dic₁₅	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746353	C₂	1	2	yes	maximal, complemented
6	C₄×#3969	—	3	1	no	maximal, Hall
7	C₂×C₃×C₅×#41	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₅×Q₈	C₄	1	2	yes	complemented
9	C₃×C₄×C₅×D₄	—	2	2	no	—
10	C₃×C₄×C₅×D₄	C₄	1	2	yes	complemented
11	C₃×C₅×#117	—	2	1	no	—
12	C₃×C₅×#117	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²×C₅	—	2	1	no	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₄×Dic₁₅	—	2	2	no	—
16	C₂×C₄×Dic₁₅	C₂²	1	2	yes	—
17	C₂×#6710	C₂²	1	2	yes	—
18	#91121	C₄	1	8	yes	direct product factor
19	#91087	—	2	1	no	—
20	C₄×C₃⋊#168	—	2	1	no	—
21	C₄×#837	—	5	1	no	maximal, Hall
22	C₄×C₅×#41	D₃	1	1	yes	characteristic
23	C₂×#3969	—	3	2	no	—
24	C₄²×Dic₅	—	3	1	no	—
25	C₄×#681	—	3	1	no	—
26	#64752	—	3	2	no	—
27	C₃×C₅×#41	—	2	2	no	—
28	C₃×C₅×#41	C₂×C₄	1	4	yes	—
29	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×D₄	—	2	1	no	—
31	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	—	2	1	no	—
33	C₂²×C₃×C₄×C₅	—	2	2	no	—
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂²×Dic₁₅	—	2	2	no	—
37	C₃×C₅×C₄⋊C₄	—	2	2	no	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
40	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
41	C₃×C₄²×C₅	—	2	2	no	—
42	C₃×C₄²×C₅	—	2	1	no	—
43	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
44	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₄×Dic₁₅	—	2	8	no	—
47	C₄×Dic₁₅	—	2	2	no	—
48	C₄×Dic₁₅	C₂×C₄	1	4	yes	complemented
49	#6710	—	2	2	no	—
50	#6710	C₂×C₄	1	4	yes	complemented
51	C₂×C₃⋊#168	—	2	2	no	—
52	C₃×C₄×#41	D₅	1	1	yes	characteristic
53	C₂×#837	—	5	2	no	—
54	C₄²×Dic₃	—	5	1	no	—
55	C₄×#208	—	5	1	no	—
56	#4532	—	5	2	no	—
57	C₂×C₅×#41	C₂×D₃	1	1	yes	characteristic
58	C₄×C₅×Q₈	Dic₃	1	2	yes	complemented
59	C₄×C₅×D₄	—	2	2	no	—
60	C₄×C₅×D₄	Dic₃	1	2	yes	complemented
61	C₅×#117	—	2	1	no	—
62	C₅×#117	C₂×D₃	1	1	yes	characteristic
63	C₂×C₄²×C₅	—	2	1	no	—
64	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
65	C₂×C₄×Dic₅	—	3	2	no	—
66	C₂×C₄×Dic₅	—	6	2	no	—
67	C₂×#681	—	3	2	no	—
68	#3969	—	3	8	no	—
69	#3935	—	6	1	no	—
70	C₄×#168	—	6	1	no	—
71	C₄×#104	—	15	1	no	Sylow
72	C₂³×C₃×C₅	—	2	1	no	—
73	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
74	C₃×C₅×Q₈	—	2	1	no	—
75	C₃×C₅×Q₈	C₄²	1	2	yes	complemented
76	C₃×C₅×D₄	—	2	4	no	—
77	C₃×C₅×D₄	—	2	1	no	—
78	C₃×C₅×D₄	C₄²	1	2	yes	complemented
79	C₂×C₃×C₄×C₅	—	2	1	no	—
80	C₂×C₃×C₄×C₅	—	2	4	no	—
81	C₂×C₃×C₄×C₅	—	2	4	no	—
82	C₂×C₃×C₄×C₅	—	2	2	no	—
83	C₂×C₃×C₄×C₅	—	4	4	no	—
84	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
85	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	#41	1	2	yes	—
90	C₂×Dic₁₅	—	2	8	no	—
91	C₂×Dic₁₅	—	2	4	no	—
92	C₃⋊#168	—	2	4	no	—
93	C₂×C₃×#41	C₂×D₅	1	1	yes	characteristic
94	C₃×C₄×Q₈	Dic₅	1	2	yes	complemented
95	C₃×C₄×D₄	—	2	2	no	—
96	C₃×C₄×D₄	Dic₅	1	2	yes	complemented
97	C₃×#117	—	2	1	no	—
98	C₃×#117	C₂×D₅	1	1	yes	characteristic
99	C₂×C₃×C₄²	—	2	1	no	—
100	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
101	C₂×C₄×Dic₃	—	5	2	no	—
102	C₂×C₄×Dic₃	—	10	2	no	—
103	C₂×#208	—	5	2	no	—
104	#837	—	5	8	no	—
105	#803	—	10	1	no	—
106	C₄×C₃⋊C₈	—	10	1	no	—
107	C₅×#41	—	2	2	no	—
108	C₅×#41	C₄×D₃	1	4	yes	—
109	C₂×C₅×Q₈	C₂×Dic₃	1	1	yes	characteristic
110	C₂×C₅×D₄	—	2	1	no	—
111	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₂²×C₄×C₅	—	2	2	no	—
114	C₂²×C₄×C₅	#67	1	2	yes	—
115	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
116	C₂²×Dic₅	—	6	2	no	—
117	C₅×C₄⋊C₄	—	2	2	no	—
118	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
119	C₅×C₂²⋊C₄	—	2	1	no	—
120	C₅×C₂²⋊C₄	—	2	2	no	—
121	C₄²×C₅	—	2	2	no	—
122	C₄²×C₅	—	2	1	no	—
123	C₄²×C₅	C₂×Dic₃	1	2	yes	—
124	C₄²×C₅	#67	1	1	yes	characteristic
125	C₄²×C₅	#67	1	1	yes	characteristic
126	C₄×Dic₅	—	3	4	no	—
127	C₄×Dic₅	—	6	8	no	—
128	C₄×Dic₅	—	6	2	no	—
129	#681	—	3	4	no	—
130	#681	—	6	2	no	—
131	C₂×#168	—	6	2	no	—
132	C₄×#41	D₁₅	1	1	yes	characteristic
133	C₂×#104	—	15	2	no	—
134	C₄×M₁₆	—	15	1	no	—
135	C₄³	—	15	1	no	—
136	#339	—	15	2	no	—
137	C₂²×C₃×C₅	—	2	2	no	—
138	C₂²×C₃×C₅	—	2	2	no	—
139	C₂²×C₃×C₅	—	4	2	no	—
140	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
142	C₃×C₄×C₅	—	2	2	no	—
143	C₃×C₄×C₅	—	2	2	no	—
144	C₃×C₄×C₅	—	4	2	no	—
145	C₃×C₄×C₅	#104	1	4	yes	complemented
146	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
147	C₃×C₄×C₅	#117	1	1	yes	characteristic
148	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
149	Dic₁₅	—	2	8	no	—
150	C₃×#41	—	2	2	no	—
151	C₃×#41	C₄×D₅	1	4	yes	—
152	C₂×C₃×Q₈	C₂×Dic₅	1	1	yes	characteristic
153	C₂×C₃×D₄	—	2	1	no	—
154	C₂×C₃×D₄	C₂×Dic₅	1	1	yes	characteristic
155	C₂²×C₃×C₄	—	2	1	no	—
156	C₂²×C₃×C₄	—	2	2	no	—
157	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
158	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
159	C₂²×Dic₃	—	10	2	no	—
160	C₃×C₄⋊C₄	—	2	2	no	—
161	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
162	C₃×C₂²⋊C₄	—	2	1	no	—
163	C₃×C₂²⋊C₄	—	2	2	no	—
164	C₃×C₄²	—	2	2	no	—
165	C₃×C₄²	—	2	1	no	—
166	C₃×C₄²	C₂×Dic₅	1	2	yes	—
167	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
168	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
169	C₄×Dic₃	—	5	4	no	—
170	C₄×Dic₃	—	10	8	no	—
171	C₄×Dic₃	—	10	2	no	—
172	#208	—	5	4	no	—
173	#208	—	10	2	no	—
174	C₂×C₃⋊C₈	—	10	2	no	—
175	C₂³×C₅	—	2	1	no	—
176	C₂³×C₅	C₂×#67	1	1	yes	characteristic
177	C₅×Q₈	—	2	1	no	—
178	C₅×Q₈	C₄×Dic₃	1	2	yes	complemented
179	C₅×D₄	—	2	4	no	—
180	C₅×D₄	—	2	1	no	—
181	C₅×D₄	C₄×Dic₃	1	2	yes	complemented
182	C₂×C₄×C₅	—	2	1	no	—
183	C₂×C₄×C₅	—	2	4	no	—
184	C₂×C₄×C₅	—	2	4	no	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	—	4	4	no	—
187	C₂×C₄×C₅	#217	1	2	yes	—
188	C₂×C₄×C₅	#217	1	2	yes	—
189	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
190	C₂×C₄×C₅	#235	1	2	yes	—
191	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
193	C₂×Dic₅	—	6	8	no	—
194	C₂×Dic₅	—	6	4	no	—
195	#168	—	6	4	no	—
196	C₂×#41	C₂×D₁₅	1	1	yes	characteristic
197	C₂×M₁₆	—	15	2	no	—
198	C₄×Q₈	Dic₁₅	1	2	yes	complemented
199	C₄×D₄	—	2	2	no	—
200	C₄×D₄	Dic₁₅	1	2	yes	complemented
201	#117	—	2	1	no	—
202	#117	C₂×D₁₅	1	1	yes	characteristic
203	C₂×C₄²	—	2	1	no	—
204	C₂×C₄²	—	15	2	no	—
205	C₂×C₄²	—	30	2	no	—
206	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
207	#104	—	15	8	no	—
208	#97	—	30	1	no	—
209	C₄×C₈	—	30	1	no	—
210	C₂×C₃×C₅	—	2	2	no	—
211	C₂×C₃×C₅	—	4	2	no	—
212	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
213	C₂×C₃×C₅	C₂×#104	1	2	yes	—
214	C₂³×C₃	—	2	1	no	—
215	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
216	C₃×Q₈	—	2	1	no	—
217	C₃×Q₈	C₄×Dic₅	1	2	yes	complemented
218	C₃×D₄	—	2	4	no	—
219	C₃×D₄	—	2	1	no	—
220	C₃×D₄	C₄×Dic₅	1	2	yes	complemented
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	2	4	no	—
223	C₂×C₃×C₄	—	2	4	no	—
224	C₂×C₃×C₄	—	2	2	no	—
225	C₂×C₃×C₄	—	4	4	no	—
226	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
227	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
228	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
229	C₂×C₃×C₄	#709	1	2	yes	—
230	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
231	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
232	C₂×Dic₃	—	10	8	no	—
233	C₂×Dic₃	—	10	4	no	—
234	C₃⋊C₈	—	10	4	no	—
235	C₂²×C₅	—	2	2	no	—
236	C₂²×C₅	—	2	2	no	—
237	C₂²×C₅	—	4	2	no	—
238	C₂²×C₅	C₄×#67	1	2	yes	—
239	C₂²×C₅	C₂×#217	1	1	yes	characteristic
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	—	2	2	no	—
242	C₄×C₅	—	4	2	no	—
243	C₄×C₅	#837	1	4	yes	complemented
244	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
245	C₄×C₅	#926	1	1	yes	characteristic
246	C₄×C₅	C₄×#67	1	2	yes	—
247	Dic₅	—	6	8	no	—
248	#41	—	2	2	no	—
249	#41	C₄×D₁₅	1	4	yes	—
250	C₂×Q₈	C₂×Dic₁₅	1	1	yes	characteristic
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	C₂×Dic₁₅	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	30	2	no	—
256	C₂²×C₄	#1220	1	2	yes	—
257	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
258	M₁₆	—	15	4	no	—
259	M₁₆	—	30	2	no	—
260	C₂×C₈	—	30	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	C₂×Dic₁₅	1	2	yes	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	15	4	no	—
268	C₄²	—	30	8	no	—
269	C₄²	—	30	2	no	—
270	C₄²	C₂×Dic₁₅	1	2	yes	—
271	C₄²	#1220	1	1	yes	characteristic
272	C₄²	#1220	1	1	yes	characteristic
273	C₃×C₅	C₄×#104	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	—	2	2	no	—
275	C₂²×C₃	—	2	2	no	—
276	C₂²×C₃	—	4	2	no	—
277	C₂²×C₃	C₄×C₂²⋊D₅	1	2	yes	—
278	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
279	C₃×C₄	—	2	2	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	—	4	2	no	—
282	C₃×C₄	#3969	1	4	yes	complemented
283	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
284	C₃×C₄	#4072	1	1	yes	characteristic
285	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
286	Dic₃	—	10	8	no	—
287	C₂×C₅	—	2	2	no	—
288	C₂×C₅	—	4	2	no	—
289	C₂×C₅	C₄×#217	1	1	yes	characteristic
290	C₂×C₅	C₂×#837	1	2	yes	—
291	C₂³	—	2	1	no	—
292	C₂³	C₂×#1220	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	C₄×Dic₁₅	1	2	yes	complemented
295	D₄	—	2	4	no	—
296	D₄	—	2	1	no	—
297	D₄	C₄×Dic₁₅	1	2	yes	complemented
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	30	8	no	—
304	C₂×C₄	—	30	4	no	—
305	C₂×C₄	#6719	1	2	yes	—
306	C₂×C₄	#6719	1	2	yes	—
307	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
308	C₂×C₄	#6817	1	2	yes	—
309	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
310	C₂×C₄	C₂×#1220	1	1	yes	characteristic
311	C₈	—	30	4	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	—	4	2	no	—
314	C₂×C₃	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
315	C₂×C₃	C₂×#3969	1	2	yes	—
316	C₅	C₄×#837	1	1	yes	characteristic, complemented, Sylow
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	4	2	no	—
320	C₂²	C₄×#1220	1	2	yes	—
321	C₂²	C₂×#6719	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	30	8	no	—
326	C₄	#91121	1	4	yes	direct product factor
327	C₄	C₂×C₄×Dic₁₅	1	1	yes	characteristic
328	C₄	#91815	1	1	yes	characteristic
329	C₄	C₄×#1220	1	2	yes	—
330	C₃	C₄×#3969	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	4	2	no	—
333	C₂	C₄×#6719	1	1	yes	characteristic
334	C₂	C₂×#91121	1	2	yes	—

	subgroup	G/H
center	C₄²	#1220
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₄×C₅×#41	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂×C₃×C₅×#41	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₄×#41	D₁₅
hypocenter	C₃×C₅	C₄×#104
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
3. C₄²×Dic₁₅	C₂	semidirect
4. C₄×#6710	C₂	semidirect
5. #11746353	C₂	semidirect
8. C₃×C₄×C₅×Q₈	C₄	semidirect
10. C₃×C₄×C₅×D₄	C₄	semidirect
18. #91121	C₄	direct (N × H)
48. C₄×Dic₁₅	C₂×C₄	semidirect
50. #6710	C₂×C₄	semidirect
58. C₄×C₅×Q₈	Dic₃	semidirect
60. C₄×C₅×D₄	Dic₃	semidirect
75. C₃×C₅×Q₈	C₄²	semidirect
78. C₃×C₅×D₄	C₄²	semidirect
94. C₃×C₄×Q₈	Dic₅	semidirect
96. C₃×C₄×D₄	Dic₅	semidirect
145. C₃×C₄×C₅	#104	semidirect
178. C₅×Q₈	C₄×Dic₃	semidirect
181. C₅×D₄	C₄×Dic₃	semidirect
198. C₄×Q₈	Dic₁₅	semidirect
200. C₄×D₄	Dic₁₅	semidirect
217. C₃×Q₈	C₄×Dic₅	semidirect
220. C₃×D₄	C₄×Dic₅	semidirect
243. C₄×C₅	#837	semidirect
273. C₃×C₅	C₄×#104	semidirect
282. C₃×C₄	#3969	semidirect
294. Q₈	C₄×Dic₁₅	semidirect
297. D₄	C₄×Dic₁₅	semidirect
316. C₅	C₄×#837	semidirect
326. C₄	#91121	direct (N × H)
330. C₃	C₄×#3969	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×#41	2
C₂×#91121	2
C₄²×Dic₁₅	2
C₄×#6710	2
#11746353	2
C₄×#3969	3
C₄×#837	5

Subgroups — C4×#91121 #14295573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#91121 #14295573