Subgroups — C₄×#91160 #14296646

Order 1920.

560 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	C₄×C₄⋊ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊AGL(1,𝔽₅)	C₂	1	1	yes	maximal, characteristic
3	C₂×#91160	C₂	1	2	yes	maximal
4	C₄×D₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#11746529	C₂	1	2	yes	maximal
6	C₄×#3994	—	3	1	no	maximal, Hall
7	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
8	C₂×C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
9	C₂×C₃×C₄⋊AGL(1,𝔽₅)	C₂²	1	2	yes	—
10	C₂×C₃×C₄×AGL(1,𝔽₅)	—	2	1	no	—
11	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
12	C₂×D₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×D₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
14	#91235	—	2	1	no	—
15	C₃×(C₂×AGL(1,𝔽₅))⋊C₄	—	2	1	no	—
16	#91160	C₄	1	8	yes	direct product factor
17	C₄×C₃⋊#168	C₄	1	1	yes	characteristic, complemented
18	C₄×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic, complemented
19	Dic₅×C₃⋊C₈	C₄	1	1	yes	characteristic, complemented
20	Dic₅×C₃⋊C₈	C₄	1	1	yes	characteristic, complemented
21	C₄×#807	—	5	1	no	maximal, Hall
22	C₂×#3994	—	3	2	no	—
23	C₄×C₄⋊AGL(1,𝔽₅)	—	3	1	no	—
24	C₄×C₄⋊AGL(1,𝔽₅)	D₃	1	1	yes	characteristic
25	C₄×C₈×D₅	—	3	1	no	—
26	#64893	—	3	2	no	—
27	C₂²×ΓL(1,𝔽₁₆)	—	2	2	no	—
28	C₂²×C₃×AGL(1,𝔽₅)	—	2	2	no	—
29	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
31	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
32	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
33	C₄⋊ΓL(1,𝔽₁₆)	C₂×C₄	1	4	yes	—
34	C₄×ΓL(1,𝔽₁₆)	—	4	2	no	—
35	C₃×C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
36	C₃×C₄⋊AGL(1,𝔽₅)	C₂×C₄	1	4	yes	—
37	C₃×C₄×AGL(1,𝔽₅)	—	4	2	no	—
38	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
45	C₃×C₄×Dic₅	Q₈	1	1	yes	characteristic
46	D₅×C₃⋊C₈	—	2	2	no	—
47	D₅×C₃⋊C₈	C₂×C₄	1	4	yes	—
48	C₃×C₄×C₄⋊C₄	—	5	1	no	—
49	C₂×#807	—	5	2	no	—
50	C₄×C₄⋊Dic₃	—	5	1	no	—
51	C₂×C₄×C₃⋊C₈	—	5	1	no	—
52	#4529	—	5	2	no	—
53	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
54	C₂×C₄⋊AGL(1,𝔽₅)	C₂×D₃	1	2	yes	—
55	C₂×C₄×AGL(1,𝔽₅)	—	2	1	no	—
56	C₂×C₄×AGL(1,𝔽₅)	—	6	1	no	—
57	C₂×C₈×D₅	—	3	1	no	—
58	C₂×C₈×D₅	—	3	1	no	—
59	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
60	(C₂×AGL(1,𝔽₅))⋊C₄	—	2	1	no	—
61	(C₂×AGL(1,𝔽₅))⋊C₄	—	6	1	no	—
62	#3994	—	3	8	no	—
63	C₄×C₅×C₈	—	3	1	no	—
64	C₈×Dic₅	—	3	1	no	—
65	C₈×Dic₅	—	3	1	no	—
66	C₄×#168	—	3	1	no	—
67	C₄×#107	—	15	1	no	Sylow
68	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
69	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
70	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
71	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
72	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
73	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
74	C₂×C₃×AGL(1,𝔽₅)	—	4	2	no	—
75	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
81	C₃×C₄×D₅	—	2	2	no	—
82	C₃×C₄×D₅	D₈	1	2	yes	—
83	C₃×C₄×D₅	Q₁₆	1	2	yes	—
84	C₃×C₄×D₅	C₂²×C₄	1	2	yes	—
85	C₃×C₄×D₅	#41	1	2	yes	—
86	C₃⋊#168	—	2	1	no	—
87	C₃⋊#168	C₄²	1	2	yes	complemented
88	C₅×C₃⋊C₈	—	2	1	no	—
89	C₅×C₃⋊C₈	C₄²	1	2	yes	complemented
90	C₂×C₃×C₄⋊C₄	—	5	2	no	—
91	C₂×C₃×C₄²	—	5	1	no	—
92	C₂×C₃×C₄²	—	10	1	no	—
93	C₂×C₄⋊Dic₃	—	5	2	no	—
94	C₂×C₄×Dic₃	—	10	1	no	—
95	C₂²×C₃⋊C₈	—	5	1	no	—
96	C₂²×C₃⋊C₈	—	5	1	no	—
97	C₃×#95	—	10	1	no	—
98	C₃⋊#95	—	10	1	no	—
99	#807	—	5	8	no	—
100	C₄×C₃⋊C₈	—	5	1	no	—
101	C₄×C₃⋊C₈	—	5	1	no	—
102	C₄×C₃⋊C₈	—	5	1	no	—
103	C₄×C₃⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
104	C₂²×AGL(1,𝔽₅)	—	2	2	no	—
105	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
106	C₂×C₄×D₅	#67	1	1	yes	characteristic
107	C₂×C₄×D₅	#67	1	1	yes	characteristic
108	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
109	C₄⋊AGL(1,𝔽₅)	—	2	2	no	—
110	C₄⋊AGL(1,𝔽₅)	—	3	4	no	—
111	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
112	C₄⋊AGL(1,𝔽₅)	C₄×D₃	1	4	yes	—
113	C₄×AGL(1,𝔽₅)	—	4	2	no	—
114	C₄×AGL(1,𝔽₅)	—	12	2	no	—
115	C₂×C₅×C₈	—	3	1	no	—
116	C₂×C₅×C₈	—	3	1	no	—
117	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
118	C₄×Dic₅	Dic₆	1	1	yes	characteristic
119	C₄×Dic₅	Dic₆	1	1	yes	characteristic
120	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
121	C₂×#168	—	3	1	no	—
122	C₂×#168	—	3	1	no	—
123	C₈×D₅	—	3	4	no	—
124	C₈×D₅	—	6	2	no	—
125	C₂×#107	—	15	2	no	—
126	C₂×C₄×C₈	—	15	1	no	—
127	C₄×C₄⋊C₄	—	5	1	no	—
128	C₄×C₄⋊C₄	—	15	1	no	—
129	#340	—	15	2	no	—
130	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₃×D₅	C₄×D₄	1	2	yes	—
132	C₂×C₃×D₅	C₂×D₈	1	1	yes	characteristic
133	C₂×C₃×D₅	C₂×Q₁₆	1	1	yes	characteristic
134	C₂×C₃×D₅	#135	1	2	yes	—
135	ΓL(1,𝔽₁₆)	—	4	4	no	—
136	C₃×AGL(1,𝔽₅)	—	4	4	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	#107	1	2	yes	complemented
139	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
140	C₃×C₄×C₅	#117	1	1	yes	characteristic
141	C₃×Dic₅	—	2	1	no	—
142	C₃×Dic₅	#107	1	2	yes	complemented
143	C₃×Dic₅	C₄×Q₈	1	2	yes	—
144	C₂²×C₃×C₄	—	5	1	no	—
145	C₂²×C₃×C₄	—	5	1	no	—
146	C₂²×C₃×C₄	—	5	1	no	—
147	C₂²×C₃×C₄	—	10	2	no	—
148	C₂²×Dic₃	—	10	2	no	—
149	C₃×C₄⋊C₄	—	5	4	no	—
150	C₃×C₄⋊C₄	—	10	2	no	—
151	C₃×C₄²	—	5	1	no	—
152	C₃×C₄²	—	5	1	no	—
153	C₃×C₄²	—	5	1	no	—
154	C₃×C₄²	—	20	2	no	—
155	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
156	C₄⋊Dic₃	—	5	4	no	—
157	C₄⋊Dic₃	—	10	2	no	—
158	C₄×Dic₃	—	20	2	no	—
159	C₂×C₃⋊C₈	—	5	1	no	—
160	C₂×C₃⋊C₈	—	5	4	no	—
161	C₂×C₃⋊C₈	—	5	1	no	—
162	C₂×C₃⋊C₈	—	10	2	no	—
163	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
164	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
165	C₂²×D₅	C₂×#67	1	1	yes	characteristic
166	C₂×AGL(1,𝔽₅)	—	2	4	no	—
167	C₂×AGL(1,𝔽₅)	—	4	2	no	—
168	C₂×AGL(1,𝔽₅)	—	4	2	no	—
169	C₂×AGL(1,𝔽₅)	—	6	4	no	—
170	C₂×AGL(1,𝔽₅)	—	12	2	no	—
171	C₂×AGL(1,𝔽₅)	—	12	2	no	—
172	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅	Dic₃⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
175	C₂×Dic₅	Dic₃⋊C₄	1	1	yes	characteristic
176	C₂×Dic₅	Dic₃⋊C₄	1	1	yes	characteristic
177	C₂×Dic₅	C₂×Dic₆	1	1	yes	characteristic
178	C₄×D₅	—	2	2	no	—
179	C₄×D₅	#213	1	2	yes	—
180	C₄×D₅	#216	1	2	yes	—
181	C₄×D₅	C₂×C₄×D₃	1	2	yes	—
182	C₄×D₅	#235	1	2	yes	—
183	C₅×C₈	—	3	2	no	—
184	C₅×C₈	—	6	1	no	—
185	#168	—	3	2	no	—
186	#168	—	6	1	no	—
187	C₂²×C₈	—	15	1	no	—
188	C₂²×C₈	—	15	1	no	—
189	C₂×C₄⋊C₄	—	5	2	no	—
190	C₂×C₄⋊C₄	—	15	2	no	—
191	C₂×C₄²	—	5	1	no	—
192	C₂×C₄²	—	10	1	no	—
193	C₂×C₄²	—	30	1	no	—
194	#107	—	15	8	no	—
195	C₄×C₈	—	3	1	no	—
196	C₄×C₈	—	15	1	no	—
197	C₄×C₈	—	15	1	no	—
198	C₄×C₈	—	15	1	no	—
199	#95	—	10	1	no	—
200	#95	—	30	1	no	—
201	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
202	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
203	C₂×C₃×C₅	#427	1	1	yes	characteristic
204	C₃×D₅	C₄×D₈	1	2	yes	—
205	C₃×D₅	C₄×Q₁₆	1	2	yes	—
206	C₂³×C₃	—	5	1	no	—
207	C₂×C₃×C₄	—	5	1	no	—
208	C₂×C₃×C₄	—	5	1	no	—
209	C₂×C₃×C₄	—	5	2	no	—
210	C₂×C₃×C₄	—	5	2	no	—
211	C₂×C₃×C₄	—	5	2	no	—
212	C₂×C₃×C₄	—	5	2	no	—
213	C₂×C₃×C₄	—	5	1	no	—
214	C₂×C₃×C₄	—	10	2	no	—
215	C₂×C₃×C₄	—	10	4	no	—
216	C₂×C₃×C₄	—	20	2	no	—
217	C₂×C₃×C₄	—	20	2	no	—
218	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
219	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
221	C₂×Dic₃	—	10	4	no	—
222	C₂×Dic₃	—	20	2	no	—
223	C₂×Dic₃	—	20	2	no	—
224	C₃⋊C₈	—	2	1	no	—
225	C₃⋊C₈	—	5	2	no	—
226	C₃⋊C₈	—	10	1	no	—
227	C₃⋊C₈	C₄×AGL(1,𝔽₅)	1	2	yes	complemented
228	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
229	C₂×D₅	C₄×#67	1	2	yes	—
230	C₂×D₅	C₂×#213	1	1	yes	characteristic
231	C₂×D₅	C₂×#216	1	1	yes	characteristic
232	C₂×D₅	#950	1	2	yes	—
233	AGL(1,𝔽₅)	—	4	4	no	—
234	AGL(1,𝔽₅)	—	12	4	no	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	#807	1	2	yes	complemented
237	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
238	C₄×C₅	#872	1	1	yes	characteristic
239	Dic₅	—	2	1	no	—
240	Dic₅	#807	1	2	yes	complemented
241	Dic₅	C₄×Dic₆	1	2	yes	—
242	C₂²×C₄	—	5	1	no	—
243	C₂²×C₄	—	5	1	no	—
244	C₂²×C₄	—	5	1	no	—
245	C₂²×C₄	—	10	2	no	—
246	C₂²×C₄	—	30	2	no	—
247	C₂×C₈	—	3	1	no	—
248	C₂×C₈	—	3	1	no	—
249	C₂×C₈	—	15	4	no	—
250	C₂×C₈	—	15	1	no	—
251	C₂×C₈	—	15	1	no	—
252	C₂×C₈	—	30	2	no	—
253	C₄⋊C₄	—	5	4	no	—
254	C₄⋊C₄	—	10	2	no	—
255	C₄⋊C₄	—	15	4	no	—
256	C₄⋊C₄	—	30	2	no	—
257	C₄²	—	5	1	no	—
258	C₄²	—	5	1	no	—
259	C₄²	—	5	1	no	—
260	C₄²	—	20	2	no	—
261	C₄²	—	60	2	no	—
262	C₄²	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
263	C₃×C₅	C₄×#107	1	1	yes	characteristic, complemented, Hall
264	C₂²×C₃	—	5	2	no	—
265	C₂²×C₃	—	5	1	no	—
266	C₂²×C₃	—	5	1	no	—
267	C₂²×C₃	—	5	2	no	—
268	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	5	2	no	—
271	C₃×C₄	—	5	2	no	—
272	C₃×C₄	—	10	1	no	—
273	C₃×C₄	—	20	4	no	—
274	C₃×C₄	#3994	1	2	yes	complemented
275	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
276	C₃×C₄	#4130	1	1	yes	characteristic
277	Dic₃	—	20	4	no	—
278	C₂×C₅	C₄×Dic₃⋊C₄	1	1	yes	characteristic
279	C₂×C₅	C₂×#807	1	1	yes	characteristic
280	C₂×C₅	#4998	1	1	yes	characteristic
281	D₅	C₄×#213	1	2	yes	—
282	D₅	C₄×#216	1	2	yes	—
283	C₂³	—	5	1	no	—
284	C₂×C₄	—	5	1	no	—
285	C₂×C₄	—	5	1	no	—
286	C₂×C₄	—	5	2	no	—
287	C₂×C₄	—	5	2	no	—
288	C₂×C₄	—	5	2	no	—
289	C₂×C₄	—	5	2	no	—
290	C₂×C₄	—	5	1	no	—
291	C₂×C₄	—	10	2	no	—
292	C₂×C₄	—	10	4	no	—
293	C₂×C₄	—	20	2	no	—
294	C₂×C₄	—	20	2	no	—
295	C₂×C₄	—	30	4	no	—
296	C₂×C₄	—	60	2	no	—
297	C₂×C₄	—	60	2	no	—
298	C₂×C₄	Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic
299	C₂×C₄	Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic
300	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
301	C₈	—	3	2	no	—
302	C₈	—	6	1	no	—
303	C₈	—	15	2	no	—
304	C₈	—	30	1	no	—
305	C₂×C₃	—	5	2	no	—
306	C₂×C₃	—	5	2	no	—
307	C₂×C₃	C₄×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
308	C₂×C₃	C₂×#3994	1	1	yes	characteristic
309	C₂×C₃	#65721	1	1	yes	characteristic
310	C₅	C₄×#807	1	1	yes	characteristic, complemented, Sylow
311	C₂²	—	5	2	no	—
312	C₂²	—	5	1	no	—
313	C₂²	—	5	1	no	—
314	C₂²	—	5	2	no	—
315	C₂²	C₂×Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	5	2	no	—
318	C₄	—	5	2	no	—
319	C₄	—	10	1	no	—
320	C₄	—	20	4	no	—
321	C₄	—	60	4	no	—
322	C₄	#91160	1	2	yes	direct product factor
323	C₄	#91909	1	1	yes	characteristic
324	C₄	C₄×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
325	C₃	C₄×#3994	1	1	yes	characteristic, complemented, Sylow
326	C₂	—	5	2	no	—
327	C₂	—	5	2	no	—
328	C₂	#11751726	1	1	yes	characteristic
329	C₂	C₂×#91160	1	1	yes	characteristic
330	C₂	C₄×Dic₃⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	Dic₃⋊AGL(1,𝔽₅)
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
Fitting	C₃×C₄²×C₅	C₂×C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₂×D₄
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	C₄²	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	C₄×#107
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
16. #91160	C₄	direct (N × H)
17. C₄×C₃⋊#168	C₄	semidirect
18. C₄×C₅×C₃⋊C₈	C₄	semidirect
19. Dic₅×C₃⋊C₈	C₄	semidirect
20. Dic₅×C₃⋊C₈	C₄	semidirect
87. C₃⋊#168	C₄²	semidirect
89. C₅×C₃⋊C₈	C₄²	semidirect
103. C₄×C₃⋊C₈	AGL(1,𝔽₅)	semidirect
138. C₃×C₄×C₅	#107	semidirect
142. C₃×Dic₅	#107	semidirect
227. C₃⋊C₈	C₄×AGL(1,𝔽₅)	semidirect
236. C₄×C₅	#807	semidirect
240. Dic₅	#807	semidirect
263. C₃×C₅	C₄×#107	semidirect
274. C₃×C₄	#3994	semidirect
310. C₅	C₄×#807	semidirect
322. C₄	#91160	direct (N × H)
325. C₃	C₄×#3994	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊ΓL(1,𝔽₁₆)	2
C₃×C₄×C₄⋊AGL(1,𝔽₅)	2
C₂×#91160	2
C₄×D₅×C₃⋊C₈	2
#11746529	2
C₄×#3994	3
C₄×#807	5

Subgroups — C4×#91160 #14296646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#91160 #14296646