Subgroups — C₄×C₅×#344 #11933430

Order 1280.

906 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#117	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#344	C₂	1	4	yes	maximal
5	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
7	C₂×C₅×#117	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₄×C₅×M₁₆	—	2	1	no	—
10	C₄×C₅×M₁₆	—	2	1	no	—
11	C₄×C₅×M₁₆	—	2	1	no	—
12	C₄×C₅×M₁₆	—	2	2	no	—
13	C₄×C₅×M₁₆	—	2	1	no	—
14	C₄×C₅×M₁₆	C₄	1	4	yes	complemented
15	C₂×C₅×#97	—	2	1	no	—
16	C₂×C₅×#97	C₄	1	2	yes	complemented
17	C₂×C₄×C₅×C₈	—	2	1	no	—
18	C₂×C₄×C₅×C₈	C₄	1	2	yes	complemented
19	C₄×C₅×C₄⋊C₄	—	2	2	no	—
20	C₄×C₅×C₂²⋊C₄	C₄	1	4	yes	—
21	C₅×#376	—	2	1	no	—
22	C₄³×C₅	—	2	1	no	—
23	C₅×#344	C₄	1	16	yes	direct product factor
24	C₄×#344	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	2	no	—
30	C₂×C₅×M₁₆	—	2	4	no	—
31	C₂×C₅×M₁₆	—	2	4	no	—
32	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
33	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
34	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
35	C₂²×C₅×C₈	—	2	2	no	—
36	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
37	C₅×#117	—	2	4	no	—
38	C₅×#117	C₂×C₄	1	8	yes	—
39	C₂×C₅×C₄⋊C₄	—	2	2	no	—
40	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₅	—	2	2	no	—
42	C₂×C₄²×C₅	—	2	4	no	—
43	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₅	D₄	1	2	yes	—
50	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₅	Q₈	1	2	yes	—
52	C₅×#97	—	2	4	no	—
53	C₅×#97	—	2	2	no	—
54	C₅×#97	—	2	2	no	—
55	C₄×C₅×C₈	—	2	4	no	—
56	C₄×C₅×C₈	—	2	2	no	—
57	C₄×C₅×C₈	—	2	2	no	—
58	C₅×#95	—	2	4	no	—
59	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
60	C₂×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
61	C₄×#117	C₂×C₅	1	1	yes	characteristic
62	C₂×#344	C₂×C₅	1	4	yes	—
63	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	—	2	4	no	—
65	C₂²×C₄×C₅	—	4	4	no	—
66	C₂²×C₄×C₅	C₄²	1	2	yes	—
67	C₂²×C₄×C₅	C₄²	1	2	yes	—
68	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
71	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₅	#41	1	4	yes	—
80	C₅×M₁₆	—	2	4	no	—
81	C₅×M₁₆	—	2	8	no	—
82	C₅×M₁₆	—	2	8	no	—
83	C₅×M₁₆	—	2	8	no	—
84	C₅×M₁₆	C₄²	1	8	yes	complemented
85	C₂×C₅×C₈	—	2	4	no	—
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	—	2	4	no	—
88	C₂×C₅×C₈	—	2	8	no	—
89	C₂×C₅×C₈	C₄²	1	8	yes	complemented
90	C₅×C₄⋊C₄	—	2	8	no	—
91	C₅×C₂²⋊C₄	—	2	4	no	—
92	C₅×C₂²⋊C₄	C₄²	1	8	yes	—
93	C₄²×C₅	—	2	8	no	—
94	C₄²×C₅	—	4	8	no	—
95	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
97	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
98	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
100	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
101	C₄²×C₅	C₄⋊C₄	1	2	yes	—
102	C₄²×C₅	C₄⋊C₄	1	2	yes	—
103	C₄²×C₅	C₄⋊C₄	1	2	yes	—
104	C₄²×C₅	C₄⋊C₄	1	2	yes	—
105	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
106	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
107	C₂×#117	C₂²×C₅	1	2	yes	—
108	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
109	C₄×M₁₆	—	2	2	no	—
110	C₄×M₁₆	—	2	1	no	—
111	C₄×M₁₆	—	2	1	no	—
112	C₄×M₁₆	—	2	1	no	—
113	C₄×M₁₆	—	2	1	no	—
114	C₄×M₁₆	C₄×C₅	1	4	yes	complemented
115	C₂×#97	—	2	1	no	—
116	C₂×#97	C₄×C₅	1	2	yes	complemented
117	C₂×C₄×C₈	—	2	1	no	—
118	C₂×C₄×C₈	C₄×C₅	1	2	yes	complemented
119	C₄×C₄⋊C₄	—	2	2	no	—
120	C₄×C₂²⋊C₄	C₄×C₅	1	4	yes	—
121	#376	—	2	1	no	—
122	C₄³	—	2	1	no	—
123	#344	C₄×C₅	1	16	yes	direct product factor
124	C₂³×C₅	—	4	2	no	—
125	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
126	C₂³×C₅	C₂×C₄²	1	2	yes	—
127	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂³×C₅	#117	1	2	yes	—
130	C₂×C₄×C₅	—	2	4	no	—
131	C₂×C₄×C₅	—	2	4	no	—
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	8	no	—
134	C₂×C₄×C₅	—	4	2	no	—
135	C₂×C₄×C₅	—	4	2	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	8	no	—
138	C₂×C₄×C₅	#95	1	2	yes	—
139	C₂×C₄×C₅	#95	1	2	yes	—
140	C₂×C₄×C₅	#95	1	2	yes	—
141	C₂×C₄×C₅	#95	1	2	yes	—
142	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
143	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
144	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
148	C₂×C₄×C₅	#117	1	2	yes	—
149	C₂×C₄×C₅	#117	1	2	yes	—
150	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
151	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
152	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
153	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
154	C₅×C₈	—	2	8	no	—
155	C₅×C₈	—	2	8	no	—
156	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
157	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	2	4	no	—
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
164	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
165	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
168	#117	—	2	4	no	—
169	#117	C₂×C₄×C₅	1	8	yes	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	—	2	4	no	—
174	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
176	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
177	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
178	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
179	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
180	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
181	C₂×C₄²	C₅×D₄	1	2	yes	—
182	C₂×C₄²	C₅×Q₈	1	2	yes	—
183	#97	—	2	4	no	—
184	#97	—	2	2	no	—
185	#97	—	2	2	no	—
186	C₄×C₈	—	2	4	no	—
187	C₄×C₈	—	2	2	no	—
188	C₄×C₈	—	2	2	no	—
189	#95	—	2	4	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	2	no	—
193	C₂²×C₅	—	4	2	no	—
194	C₂²×C₅	—	4	2	no	—
195	C₂²×C₅	C₄³	1	1	yes	characteristic
196	C₂²×C₅	C₂×#95	1	1	yes	characteristic
197	C₂²×C₅	#376	1	1	yes	characteristic
198	C₂²×C₅	C₄×C₂²⋊C₄	1	2	yes	—
199	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
200	C₄×C₅	—	2	2	no	—
201	C₄×C₅	—	2	2	no	—
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	—	4	8	no	—
204	C₄×C₅	#344	1	4	yes	direct product factor
205	C₄×C₅	C₄×C₂²⋊C₄	1	2	yes	—
206	C₄×C₅	C₄×C₂²⋊C₄	1	2	yes	—
207	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
208	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
209	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	C₄²×C₅	1	2	yes	—
213	C₂²×C₄	C₄²×C₅	1	2	yes	—
214	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
215	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
216	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
217	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
219	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
221	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
224	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
225	C₂²×C₄	C₅×#41	1	4	yes	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	8	no	—
228	M₁₆	—	2	8	no	—
229	M₁₆	—	2	8	no	—
230	M₁₆	C₄²×C₅	1	8	yes	complemented
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	8	no	—
235	C₂×C₈	C₄²×C₅	1	8	yes	complemented
236	C₄⋊C₄	—	2	8	no	—
237	C₂²⋊C₄	—	2	4	no	—
238	C₂²⋊C₄	C₄²×C₅	1	8	yes	—
239	C₄²	—	2	8	no	—
240	C₄²	—	4	8	no	—
241	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
242	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
243	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
244	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
245	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
246	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
247	C₄²	C₅×C₄⋊C₄	1	2	yes	—
248	C₄²	C₅×C₄⋊C₄	1	2	yes	—
249	C₄²	C₅×C₄⋊C₄	1	2	yes	—
250	C₄²	C₅×C₄⋊C₄	1	2	yes	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	—	2	1	no	—
253	C₂×C₅	—	2	1	no	—
254	C₂×C₅	—	2	2	no	—
255	C₂×C₅	C₄×#95	1	1	yes	characteristic
256	C₂×C₅	C₂×#344	1	2	yes	—
257	C₂³	—	4	2	no	—
258	C₂³	C₂×C₄²×C₅	1	1	yes	characteristic
259	C₂³	C₂×C₄²×C₅	1	2	yes	—
260	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
261	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
262	C₂³	C₅×#117	1	2	yes	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	C₅×#95	1	2	yes	—
272	C₂×C₄	C₅×#95	1	2	yes	—
273	C₂×C₄	C₅×#95	1	2	yes	—
274	C₂×C₄	C₅×#95	1	2	yes	—
275	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
276	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
277	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
278	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
279	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
280	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
281	C₂×C₄	C₅×#117	1	2	yes	—
282	C₂×C₄	C₅×#117	1	2	yes	—
283	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
284	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
285	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
286	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
287	C₈	—	2	8	no	—
288	C₈	—	2	8	no	—
289	C₅	C₄×#344	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	C₄³×C₅	1	1	yes	characteristic
296	C₂²	C₂×C₅×#95	1	1	yes	characteristic
297	C₂²	C₅×#376	1	1	yes	characteristic
298	C₂²	C₄×C₅×C₂²⋊C₄	1	2	yes	—
299	C₂²	C₄×C₅×C₄⋊C₄	1	2	yes	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	2	2	no	—
303	C₄	—	4	8	no	—
304	C₄	C₅×#344	1	4	yes	direct product factor
305	C₄	C₄×C₅×C₂²⋊C₄	1	2	yes	—
306	C₄	C₄×C₅×C₂²⋊C₄	1	2	yes	—
307	C₄	C₄×C₅×C₄⋊C₄	1	2	yes	—
308	C₄	C₄×C₅×C₄⋊C₄	1	2	yes	—
309	C₂	—	2	2	no	—
310	C₂	—	2	1	no	—
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	C₄×C₅×#95	1	1	yes	characteristic
314	C₂	C₂×C₅×#344	1	2	yes	—

	subgroup	G/H
center	C₄²×C₅	C₂²⋊C₄
commutator	C₂²	C₄³×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂²×C₅	C₂×#95
prime radical	C₂⁴×C₅	C₂²×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₄×C₅×M₁₆	C₄	semidirect
16. C₂×C₅×#97	C₄	semidirect
18. C₂×C₄×C₅×C₈	C₄	semidirect
23. C₅×#344	C₄	direct (N × H)
24. C₄×#344	C₅	direct (N × H)
84. C₅×M₁₆	C₄²	semidirect
89. C₂×C₅×C₈	C₄²	semidirect
114. C₄×M₁₆	C₄×C₅	semidirect
116. C₂×#97	C₄×C₅	semidirect
118. C₂×C₄×C₈	C₄×C₅	semidirect
123. #344	C₄×C₅	direct (N × H)
204. C₄×C₅	#344	direct (N × H)
230. M₁₆	C₄²×C₅	semidirect
235. C₂×C₈	C₄²×C₅	semidirect
289. C₅	C₄×#344	direct (N × H)
304. C₄	C₅×#344	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×C₄×C₅×M₁₆	2
C₄×C₅×#117	2
C₂×C₅×#344	2
C₄×#344	5

Subgroups — C4×C5×#344 #11933430

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×#344 #11933430