Subgroups — #14296342

Order 1920.

544 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×D₅⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×#91150	C₂	1	2	yes	maximal
4	C₄×D₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#11746517	C₂	1	2	yes	maximal
6	#10603842	—	3	1	no	maximal, Hall
7	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
8	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
9	C₂×C₃×D₅⋊C₈	C₂²	1	2	yes	—
10	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
11	C₂×D₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×D₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	#91235	C₄	1	2	yes	—
14	C₃×#4004	C₄	1	2	yes	complemented
15	C₃×C₄⋊#170	C₄	1	2	yes	complemented
16	#91150	—	2	4	no	—
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	#70466	—	5	1	no	maximal, Hall
22	C₂×#3992	—	3	2	no	—
23	C₄×C₄⋊AGL(1,𝔽₅)	—	3	1	no	—
24	D₅⋊(C₄⋊C₈)	D₃	1	1	yes	characteristic
25	C₄×C₈×D₅	—	3	1	no	—
26	(C₂×AGL(1,𝔽₅))⋊C₈	—	3	2	no	—
27	C₂²×ΓL(1,𝔽₁₆)	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
31	C₄⋊ΓL(1,𝔽₁₆)	—	2	4	no	—
32	C₄×ΓL(1,𝔽₁₆)	—	2	2	no	—
33	C₄×ΓL(1,𝔽₁₆)	—	2	2	no	—
34	C₂×C₃×#170	C₂×C₄	1	4	yes	—
35	C₃×D₅⋊C₈	—	2	2	no	—
36	C₃×D₅⋊C₈	—	2	2	no	—
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
43	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
45	D₅×C₃⋊C₈	—	2	1	no	—
46	D₅×C₃⋊C₈	—	2	1	no	—
47	D₅×C₃⋊C₈	D₄	1	2	yes	—
48	D₅×C₃⋊C₈	Q₈	1	2	yes	—
49	C₂×C₃×C₄⋊C₈	—	5	1	no	—
50	C₂×#815	—	5	2	no	—
51	C₄×C₄⋊Dic₃	—	5	1	no	—
52	C₂×C₄×C₃⋊C₈	—	5	1	no	—
53	#4550	—	5	2	no	—
54	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
55	C₂×C₄×AGL(1,𝔽₅)	—	3	2	no	—
56	C₂×D₅⋊C₈	C₂×D₃	1	2	yes	—
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	—	3	2	no	—
61	#4004	Dic₃	1	2	yes	complemented
62	C₄⋊#170	Dic₃	1	2	yes	complemented
63	#3992	—	6	4	no	—
64	C₄×C₅×C₈	—	3	1	no	—
65	C₈×Dic₅	—	3	1	no	—
66	C₈×Dic₅	—	3	1	no	—
67	C₄×#168	—	3	1	no	—
68	#1770	—	15	1	no	Sylow
69	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
71	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
72	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
73	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
77	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
78	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₄×D₅	—	2	2	no	—
80	C₃×C₄×D₅	M₁₆	1	4	yes	—
81	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
82	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
83	C₃×C₄×D₅	#41	1	1	yes	characteristic
84	C₃×C₄×D₅	#41	1	1	yes	characteristic
85	C₃×#170	—	2	4	no	—
86	C₃⋊#168	—	2	1	no	—
87	C₃⋊#168	C₄⋊C₄	1	2	yes	complemented
88	C₅×C₃⋊C₈	—	2	1	no	—
89	C₅×C₃⋊C₈	C₄⋊C₄	1	2	yes	complemented
90	C₂²×C₃×C₈	—	5	2	no	—
91	C₂×C₃×C₄²	—	5	1	no	—
92	C₂×C₄⋊Dic₃	—	5	2	no	—
93	C₂×C₄×Dic₃	—	5	2	no	—
94	C₂²×C₃⋊C₈	—	5	1	no	—
95	C₂²×C₃⋊C₈	—	5	1	no	—
96	C₃×C₄⋊C₈	—	5	2	no	—
97	C₃×C₄⋊C₈	—	5	2	no	—
98	C₃⋊#95	—	5	2	no	—
99	#815	—	10	4	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)	—	3	4	no	—
105	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
106	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
107	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
108	C₂×#170	C₂×Dic₃	1	4	yes	—
109	C₄⋊AGL(1,𝔽₅)	—	6	4	no	—
110	C₄×AGL(1,𝔽₅)	—	6	2	no	—
111	C₄×AGL(1,𝔽₅)	—	6	2	no	—
112	D₅⋊C₈	—	2	2	no	—
113	D₅⋊C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	3	1	no	—
115	C₂×C₅×C₈	—	3	1	no	—
116	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
117	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
118	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
119	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
120	C₂×#168	—	3	1	no	—
121	C₂×#168	—	3	1	no	—
122	C₈×D₅	—	3	2	no	—
123	C₈×D₅	—	3	2	no	—
124	C₈×D₅	—	6	1	no	—
125	C₈×D₅	—	6	1	no	—
126	C₂×C₄⋊C₈	—	5	1	no	—
127	C₂×#97	—	15	2	no	—
128	C₂×C₄×C₈	—	15	1	no	—
129	C₄×C₄⋊C₄	—	15	1	no	—
130	#336	—	15	2	no	—
131	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₃×D₅	C₄×D₄	1	1	yes	characteristic
133	C₂×C₃×D₅	C₄×Q₈	1	1	yes	characteristic
134	C₂×C₃×D₅	C₂×M₁₆	1	2	yes	—
135	C₂×C₃×D₅	#131	1	2	yes	—
136	ΓL(1,𝔽₁₆)	—	4	4	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	#97	1	2	yes	complemented
139	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₃×C₄×C₅	#117	1	1	yes	characteristic
141	C₃×Dic₅	—	2	1	no	—
142	C₃×Dic₅	#97	1	2	yes	complemented
143	C₃×Dic₅	C₄×D₄	1	1	yes	characteristic
144	C₃×Dic₅	C₄×Q₈	1	1	yes	characteristic
145	C₂²×C₃×C₄	—	5	1	no	—
146	C₂²×C₃×C₄	—	5	1	no	—
147	C₂²×C₃×C₄	—	5	1	no	—
148	C₂²×Dic₃	—	5	4	no	—
149	C₂×C₃×C₈	—	5	4	no	—
150	C₂×C₃×C₈	—	10	2	no	—
151	C₂×C₃×C₈	—	10	2	no	—
152	C₃×C₄²	—	5	1	no	—
153	C₃×C₄²	—	5	1	no	—
154	C₃×C₄²	—	5	1	no	—
155	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
156	C₄⋊Dic₃	—	10	4	no	—
157	C₄×Dic₃	—	10	2	no	—
158	C₄×Dic₃	—	10	2	no	—
159	C₂×C₃⋊C₈	—	5	1	no	—
160	C₂×C₃⋊C₈	—	5	2	no	—
161	C₂×C₃⋊C₈	—	5	2	no	—
162	C₂×C₃⋊C₈	—	5	1	no	—
163	C₂×C₃⋊C₈	—	10	1	no	—
164	C₂×C₃⋊C₈	—	10	1	no	—
165	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
166	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
167	C₂²×D₅	C₂×C₄×D₃	1	1	yes	characteristic
168	C₂×AGL(1,𝔽₅)	—	6	4	no	—
169	C₂×AGL(1,𝔽₅)	—	6	4	no	—
170	C₂×AGL(1,𝔽₅)	—	6	4	no	—
171	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
172	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
175	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
176	C₂×Dic₅	C₂²×Dic₃	1	1	yes	characteristic
177	C₄×D₅	—	2	2	no	—
178	C₄×D₅	ΓL(1,𝔽₂₅)	1	4	yes	—
179	C₄×D₅	D₃×D₄	1	1	yes	characteristic
180	C₄×D₅	#237	1	1	yes	characteristic
181	C₄×D₅	D₃×Q₈	1	1	yes	characteristic
182	C₄×D₅	#239	1	1	yes	characteristic
183	#170	—	2	4	no	—
184	C₅×C₈	—	3	2	no	—
185	C₅×C₈	—	6	1	no	—
186	#168	—	3	2	no	—
187	#168	—	6	1	no	—
188	C₂²×C₈	—	5	2	no	—
189	C₂²×C₈	—	15	1	no	—
190	C₂²×C₈	—	15	1	no	—
191	C₂×C₄⋊C₄	—	15	2	no	—
192	C₂×C₄²	—	5	1	no	—
193	C₂×C₄²	—	15	2	no	—
194	C₄⋊C₈	—	5	2	no	—
195	C₄⋊C₈	—	5	2	no	—
196	#97	—	30	4	no	—
197	C₄×C₈	—	3	1	no	—
198	C₄×C₈	—	15	1	no	—
199	C₄×C₈	—	15	1	no	—
200	C₄×C₈	—	15	1	no	—
201	#95	—	15	2	no	—
202	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
203	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
204	C₂×C₃×C₅	#405	1	1	yes	characteristic
205	C₃×D₅	#436	1	2	yes	—
206	C₃×D₅	#446	1	2	yes	—
207	C₂³×C₃	—	5	1	no	—
208	C₂×C₃×C₄	—	5	1	no	—
209	C₂×C₃×C₄	—	5	1	no	—
210	C₂×C₃×C₄	—	5	1	no	—
211	C₂×C₃×C₄	—	5	1	no	—
212	C₂×C₃×C₄	—	5	1	no	—
213	C₂×C₃×C₄	—	5	1	no	—
214	C₂×C₃×C₄	—	5	4	no	—
215	C₂×C₃×C₄	—	5	1	no	—
216	C₂×C₃×C₄	—	10	2	no	—
217	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
218	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
220	C₂×Dic₃	—	10	4	no	—
221	C₂×Dic₃	—	10	4	no	—
222	C₂×Dic₃	—	10	4	no	—
223	C₃×C₈	—	10	4	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×C₄×Dic₃	1	1	yes	characteristic
229	C₂×D₅	#887	1	1	yes	characteristic
230	C₂×D₅	#893	1	1	yes	characteristic
231	C₂×D₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
232	C₂×D₅	#907	1	2	yes	—
233	AGL(1,𝔽₅)	—	12	4	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	#815	1	2	yes	complemented
236	C₄×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
237	C₄×C₅	#892	1	1	yes	characteristic
238	Dic₅	—	2	1	no	—
239	Dic₅	#815	1	2	yes	complemented
240	Dic₅	D₄×Dic₃	1	1	yes	characteristic
241	Dic₅	Q₈×Dic₃	1	1	yes	characteristic
242	C₂²×C₄	—	5	1	no	—
243	C₂²×C₄	—	5	1	no	—
244	C₂²×C₄	—	5	1	no	—
245	C₂²×C₄	—	15	4	no	—
246	C₂×C₈	—	3	1	no	—
247	C₂×C₈	—	3	1	no	—
248	C₂×C₈	—	5	4	no	—
249	C₂×C₈	—	10	2	no	—
250	C₂×C₈	—	10	2	no	—
251	C₂×C₈	—	15	2	no	—
252	C₂×C₈	—	15	2	no	—
253	C₂×C₈	—	15	1	no	—
254	C₂×C₈	—	15	1	no	—
255	C₂×C₈	—	30	1	no	—
256	C₂×C₈	—	30	1	no	—
257	C₄⋊C₄	—	30	4	no	—
258	C₄²	—	5	1	no	—
259	C₄²	—	5	1	no	—
260	C₄²	—	5	1	no	—
261	C₄²	—	30	2	no	—
262	C₄²	—	30	2	no	—
263	C₄²	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
264	C₃×C₅	#1770	1	1	yes	characteristic, complemented, Hall
265	C₂²×C₃	—	5	1	no	—
266	C₂²×C₃	—	5	1	no	—
267	C₂²×C₃	—	5	2	no	—
268	C₂²×C₃	—	5	2	no	—
269	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	5	1	no	—
272	C₃×C₄	—	5	1	no	—
273	C₃×C₄	—	5	2	no	—
274	C₃×C₄	—	10	1	no	—
275	C₃×C₄	#3992	1	2	yes	complemented
276	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
277	C₃×C₄	#4130	1	1	yes	characteristic
278	Dic₃	—	20	4	no	—
279	C₂×C₅	Dic₃×C₄⋊C₄	1	1	yes	characteristic
280	C₂×C₅	C₂×#815	1	1	yes	characteristic
281	C₂×C₅	#5122	1	1	yes	characteristic
282	D₅	#4831	1	2	yes	—
283	D₅	#4838	1	2	yes	—
284	C₂³	—	5	1	no	—
285	C₂×C₄	—	5	1	no	—
286	C₂×C₄	—	5	1	no	—
287	C₂×C₄	—	5	1	no	—
288	C₂×C₄	—	5	1	no	—
289	C₂×C₄	—	5	4	no	—
290	C₂×C₄	—	5	1	no	—
291	C₂×C₄	—	5	1	no	—
292	C₂×C₄	—	5	1	no	—
293	C₂×C₄	—	10	2	no	—
294	C₂×C₄	—	30	4	no	—
295	C₂×C₄	—	30	4	no	—
296	C₂×C₄	—	30	4	no	—
297	C₂×C₄	Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
298	C₂×C₄	Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
299	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
300	C₈	—	3	2	no	—
301	C₈	—	6	1	no	—
302	C₈	—	10	4	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×#3992	1	1	yes	characteristic
309	C₂×C₃	#65718	1	1	yes	characteristic
310	C₅	#70466	1	1	yes	characteristic, complemented, Sylow
311	C₂²	—	5	2	no	—
312	C₂²	—	5	2	no	—
313	C₂²	—	5	1	no	—
314	C₂²	—	5	1	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	1	no	—
319	C₄	—	5	1	no	—
320	C₄	—	10	1	no	—
321	C₄	—	60	4	no	—
322	C₄	#91150	1	2	yes	complemented
323	C₄	#91907	1	1	yes	characteristic
324	C₄	D₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
325	C₃	#10603842	1	1	yes	characteristic, complemented, Sylow
326	C₂	—	5	2	no	—
327	C₂	—	5	2	no	—
328	C₂	#11751696	1	1	yes	characteristic
329	C₂	C₂×#91150	1	1	yes	characteristic
330	C₂	Dic₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator, socle	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₃×C₄²×C₅	C₂×C₄
prime radical	C₂²×C₃×D₅	C₂²×C₄
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	C₄²	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#1770
LCS	G ▷ C₂²×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
14. C₃×#4004	C₄	semidirect
15. C₃×C₄⋊#170	C₄	semidirect
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
61. #4004	Dic₃	semidirect
62. C₄⋊#170	Dic₃	semidirect
87. C₃⋊#168	C₄⋊C₄	semidirect
89. C₅×C₃⋊C₈	C₄⋊C₄	semidirect
103. C₄×C₃⋊C₈	AGL(1,𝔽₅)	semidirect
138. C₃×C₄×C₅	#97	semidirect
142. C₃×Dic₅	#97	semidirect
227. C₃⋊C₈	C₄⋊AGL(1,𝔽₅)	semidirect
235. C₄×C₅	#815	semidirect
239. Dic₅	#815	semidirect
264. C₃×C₅	#1770	semidirect
275. C₃×C₄	#3992	semidirect
310. C₅	#70466	semidirect
322. C₄	#91150	semidirect
325. C₃	#10603842	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×D₅⋊(C₄⋊C₈)	2
C₂×#91150	2
C₄×D₅×C₃⋊C₈	2
#11746517	2
#10603842	3
#70466	5