Subgroups — C₅×C₄⋊C₄×Q₁₆ #11945404

Order 1280.

926 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×Q₁₆	C₂	1	2	yes	maximal
2	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×Q₈×C₄⋊C₄	C₂	1	2	yes	maximal
4	C₅×#1942	C₂	1	2	yes	maximal
5	C₅×#1863	C₂	1	4	yes	maximal
6	C₅×#1845	C₂	1	1	yes	maximal, characteristic
7	C₅×#1803	C₂	1	2	yes	maximal
8	C₅×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	—
11	C₂×C₄×C₅×Q₈	C₂²	1	2	yes	—
12	C₄×C₅×Q₁₆	—	2	8	no	—
13	C₄×C₅×Q₁₆	C₄	1	8	yes	complemented
14	C₂×C₅×#107	C₂²	1	2	yes	—
15	C₂×C₅×#107	C₂²	1	2	yes	—
16	C₂×C₅×#107	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₅×#103	C₂²	1	4	yes	—
19	C₂×C₅×#103	C₂²	1	2	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
21	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
22	C₅×#389	—	2	4	no	—
23	C₅×#389	C₂²	1	4	yes	—
24	C₅×#384	—	2	2	no	—
25	C₅×#384	C₂²	1	2	yes	—
26	C₄×C₅×C₄⋊C₄	—	2	2	no	—
27	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₅×#340	C₂²	1	4	yes	—
29	C₅×#336	C₂²	1	2	yes	—
30	C₄⋊C₄×Q₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
31	C₂²×C₅×Q₈	C₂³	1	2	yes	—
32	C₂×C₅×Q₁₆	C₂×C₄	1	4	yes	—
33	C₂×C₅×Q₁₆	D₄	1	4	yes	—
34	C₂×C₅×Q₁₆	Q₈	1	4	yes	—
35	C₂²×C₅×C₈	C₂³	1	2	yes	—
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₄×C₅×Q₈	—	2	4	no	—
39	C₄×C₅×Q₈	—	2	8	no	—
40	C₄×C₅×Q₈	—	4	4	no	—
41	C₄×C₅×Q₈	C₂×C₄	1	8	yes	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	—	2	4	no	—
45	C₂×C₅×C₄⋊C₄	—	2	4	no	—
46	C₂×C₅×C₄⋊C₄	—	2	2	no	—
47	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
48	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
49	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
50	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₄²×C₅	—	2	2	no	—
52	C₂×C₄²×C₅	—	2	4	no	—
53	C₂×C₄²×C₅	C₂³	1	2	yes	—
54	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₅×#107	—	2	4	no	—
56	C₅×#107	—	4	2	no	—
57	C₅×#107	C₂×C₄	1	4	yes	—
58	C₅×C₄⋊C₈	—	2	2	no	—
59	C₅×#103	—	2	8	no	—
60	C₅×#103	C₂×C₄	1	8	yes	—
61	C₄×C₅×C₈	—	2	4	no	—
62	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
63	C₅×#95	—	2	4	no	—
64	C₅×#95	D₄	1	2	yes	—
65	C₂×C₄×Q₁₆	C₂×C₅	1	2	yes	—
66	C₂×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
67	Q₈×C₄⋊C₄	C₂×C₅	1	2	yes	—
68	#1942	C₂×C₅	1	2	yes	—
69	#1863	C₂×C₅	1	4	yes	—
70	#1845	C₂×C₅	1	1	yes	characteristic
71	#1803	C₂×C₅	1	2	yes	—
72	C₈×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
73	C₂×C₅×Q₈	—	2	2	no	—
74	C₂×C₅×Q₈	—	2	2	no	—
75	C₂×C₅×Q₈	—	2	2	no	—
76	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
77	C₂×C₅×Q₈	C₂×D₄	1	4	yes	—
78	C₂×C₅×Q₈	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₅	—	2	2	no	—
80	C₂²×C₄×C₅	—	2	2	no	—
81	C₂²×C₄×C₅	—	2	4	no	—
82	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
87	C₅×Q₁₆	C₄⋊C₄	1	16	yes	direct product factor
88	C₂×C₅×C₈	—	2	1	no	—
89	C₂×C₅×C₈	—	2	1	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
96	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
97	C₅×C₄⋊C₄	—	2	8	no	—
98	C₅×C₄⋊C₄	—	2	2	no	—
99	C₅×C₄⋊C₄	—	2	2	no	—
100	C₅×C₄⋊C₄	—	2	4	no	—
101	C₅×C₄⋊C₄	—	2	4	no	—
102	C₅×C₄⋊C₄	—	4	8	no	—
103	C₅×C₄⋊C₄	—	4	4	no	—
104	C₅×C₄⋊C₄	—	8	2	no	—
105	C₅×C₄⋊C₄	—	8	4	no	—
106	C₅×C₄⋊C₄	Q₁₆	1	4	yes	direct product factor
107	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
108	C₄²×C₅	—	2	8	no	—
109	C₄²×C₅	—	2	2	no	—
110	C₄²×C₅	—	2	2	no	—
111	C₄²×C₅	—	4	8	no	—
112	C₄²×C₅	C₂²×C₄	1	2	yes	—
113	C₄²×C₅	#41	1	2	yes	—
114	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
115	C₂×C₄×Q₈	C₂²×C₅	1	4	yes	—
116	C₂×C₄×Q₈	C₂²×C₅	1	2	yes	—
117	C₄×Q₁₆	—	2	8	no	—
118	C₄×Q₁₆	C₄×C₅	1	8	yes	complemented
119	C₂×#107	C₂²×C₅	1	2	yes	—
120	C₂×#107	C₂²×C₅	1	2	yes	—
121	C₂×#107	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
123	C₂×#103	C₂²×C₅	1	4	yes	—
124	C₂×#103	C₂²×C₅	1	2	yes	—
125	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
126	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
127	#389	—	2	4	no	—
128	#389	C₂²×C₅	1	4	yes	—
129	#384	—	2	2	no	—
130	#384	C₂²×C₅	1	2	yes	—
131	C₄×C₄⋊C₄	—	2	2	no	—
132	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
133	#340	C₂²×C₅	1	4	yes	—
134	#336	C₂²×C₅	1	2	yes	—
135	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
136	C₅×Q₈	—	2	4	no	—
137	C₅×Q₈	—	2	4	no	—
138	C₅×Q₈	—	2	4	no	—
139	C₅×Q₈	C₂×C₄⋊C₄	1	8	yes	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	4	no	—
143	C₂×C₄×C₅	—	2	4	no	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	2	1	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	4	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	4	2	no	—
152	C₂×C₄×C₅	—	4	2	no	—
153	C₂×C₄×C₅	—	4	4	no	—
154	C₂×C₄×C₅	—	4	4	no	—
155	C₂×C₄×C₅	—	4	4	no	—
156	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
157	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
158	C₂×C₄×C₅	C₂×Q₁₆	1	4	yes	—
159	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
160	C₂×C₄×C₅	#135	1	2	yes	—
161	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
167	C₅×C₈	—	2	2	no	—
168	C₅×C₈	—	4	2	no	—
169	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
170	C₂²×Q₈	C₂³×C₅	1	2	yes	—
171	C₂×Q₁₆	C₂×C₄×C₅	1	4	yes	—
172	C₂×Q₁₆	C₅×D₄	1	4	yes	—
173	C₂×Q₁₆	C₅×Q₈	1	4	yes	—
174	C₂²×C₈	C₂³×C₅	1	2	yes	—
175	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
176	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
177	C₄×Q₈	—	2	4	no	—
178	C₄×Q₈	—	2	8	no	—
179	C₄×Q₈	—	4	4	no	—
180	C₄×Q₈	C₂×C₄×C₅	1	8	yes	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
188	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
189	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	—	2	4	no	—
192	C₂×C₄²	C₂³×C₅	1	2	yes	—
193	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
194	#107	—	2	4	no	—
195	#107	—	4	2	no	—
196	#107	C₂×C₄×C₅	1	4	yes	—
197	C₄⋊C₈	—	2	2	no	—
198	#103	—	2	8	no	—
199	#103	C₂×C₄×C₅	1	8	yes	—
200	C₄×C₈	—	2	4	no	—
201	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
202	#95	—	2	4	no	—
203	#95	C₅×D₄	1	2	yes	—
204	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
205	C₂²×C₅	D₄²	1	1	yes	characteristic
206	C₂²×C₅	D₄×Q₈	1	1	yes	characteristic
207	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
208	C₂²×C₅	C₂×#135	1	1	yes	characteristic
209	C₂²×C₅	#576	1	1	yes	characteristic
210	C₂²×C₅	#578	1	1	yes	characteristic
211	C₄×C₅	—	2	8	no	—
212	C₄×C₅	—	2	2	no	—
213	C₄×C₅	—	2	4	no	—
214	C₄×C₅	—	4	4	no	—
215	C₄×C₅	—	4	2	no	—
216	C₄×C₅	—	8	4	no	—
217	C₄×C₅	C₄×Q₁₆	1	4	yes	complemented
218	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
219	C₄×C₅	#520	1	1	yes	characteristic
220	C₄×C₅	#547	1	1	yes	characteristic
221	C₄×C₅	#554	1	1	yes	characteristic
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
226	C₂×Q₈	C₂×C₅×D₄	1	4	yes	—
227	C₂×Q₈	C₂×C₅×Q₈	1	4	yes	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
232	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
234	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
236	Q₁₆	C₅×C₄⋊C₄	1	16	yes	direct product factor
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
243	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
244	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
245	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	8	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	8	2	no	—
254	C₄⋊C₄	—	8	4	no	—
255	C₄⋊C₄	C₅×Q₁₆	1	4	yes	direct product factor
256	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
257	C₄²	—	2	8	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	4	8	no	—
261	C₄²	C₂²×C₄×C₅	1	2	yes	—
262	C₄²	C₅×#41	1	2	yes	—
263	C₂×C₅	D₄×C₄⋊C₄	1	1	yes	characteristic
264	C₂×C₅	C₂×C₄×Q₁₆	1	1	yes	characteristic
265	C₂×C₅	#2951	1	1	yes	characteristic
266	C₂×C₅	D₄×Q₁₆	1	1	yes	characteristic
267	C₂×C₅	#3287	1	1	yes	characteristic
268	C₂×C₅	Q₈×Q₁₆	1	1	yes	characteristic
269	C₂×C₅	#3383	1	1	yes	characteristic
270	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
271	Q₈	—	2	4	no	—
272	Q₈	—	2	4	no	—
273	Q₈	—	2	4	no	—
274	Q₈	C₂×C₅×C₄⋊C₄	1	8	yes	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
292	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
293	C₂×C₄	C₂×C₅×Q₁₆	1	4	yes	—
294	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
295	C₂×C₄	C₅×#135	1	2	yes	—
296	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
297	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
299	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
300	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
301	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	4	2	no	—
304	C₈	C₂×C₅×C₄⋊C₄	1	4	yes	—
305	C₅	C₄⋊C₄×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
306	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
307	C₂²	C₅×D₄²	1	1	yes	characteristic
308	C₂²	C₅×D₄×Q₈	1	1	yes	characteristic
309	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
310	C₂²	C₂×C₅×#135	1	1	yes	characteristic
311	C₂²	C₅×#576	1	1	yes	characteristic
312	C₂²	C₅×#578	1	1	yes	characteristic
313	C₄	—	2	4	no	—
314	C₄	—	2	8	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	4	no	—
319	C₄	C₄×C₅×Q₁₆	1	4	yes	complemented
320	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
321	C₄	C₅×#520	1	1	yes	characteristic
322	C₄	C₅×#547	1	1	yes	characteristic
323	C₄	C₅×#554	1	1	yes	characteristic
324	C₂	C₅×D₄×C₄⋊C₄	1	1	yes	characteristic
325	C₂	C₂×C₄×C₅×Q₁₆	1	1	yes	characteristic
326	C₂	C₅×#2951	1	1	yes	characteristic
327	C₂	C₅×D₄×Q₁₆	1	1	yes	characteristic
328	C₂	C₅×#3287	1	1	yes	characteristic
329	C₂	C₅×Q₈×Q₁₆	1	1	yes	characteristic
330	C₂	C₅×#3383	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄×C₅×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄×C₅×Q₁₆	C₄	semidirect
30. C₄⋊C₄×Q₁₆	C₅	direct (N × H)
87. C₅×Q₁₆	C₄⋊C₄	direct (N × H)
106. C₅×C₄⋊C₄	Q₁₆	direct (N × H)
118. C₄×Q₁₆	C₄×C₅	semidirect
217. C₄×C₅	C₄×Q₁₆	semidirect
236. Q₁₆	C₅×C₄⋊C₄	direct (N × H)
255. C₄⋊C₄	C₅×Q₁₆	direct (N × H)
305. C₅	C₄⋊C₄×Q₁₆	direct (N × H)
319. C₄	C₄×C₅×Q₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×Q₁₆	2
C₅×Q₈×C₄⋊C₄	2
C₅×#1942	2
C₅×#1863	2
C₅×#1845	2
C₅×#1803	2
C₅×C₈×C₄⋊C₄	2
C₄⋊C₄×Q₁₆	5

Subgroups — C5×C4⋊C4×Q16 #11945404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×C₄⋊C₄×Q₁₆ #11945404