Subgroups — #14411389

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₁₅	C₂	1	2	yes	maximal
2	C₂×#91761	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	M₁₆×Dic₁₅	C₂	1	2	yes	maximal, complemented
5	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
6	#11750809	C₂	1	4	yes	maximal, complemented
7	#11750790	C₂	1	4	yes	maximal, complemented
8	#10610245	—	3	1	no	maximal, Hall
9	C₂²×C₄×D₁₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×#6714	C₄	1	2	yes	—
13	C₂×#6712	C₄	1	2	yes	—
14	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
15	C₂×#6710	C₂²	1	2	yes	—
16	M₁₆×D₁₅	C₄	1	16	yes	complemented
17	C₂×#6705	C₂²	1	4	yes	—
18	C₂×C₈×D₁₅	C₂²	1	4	yes	—
19	#91761	C₄	1	4	yes	—
20	#91761	C₂²	1	2	yes	—
21	#91761	C₂²	1	2	yes	—
22	C₃×C₅×#97	C₂²	1	2	yes	complemented
23	C₃×C₄×C₅×C₈	C₂²	1	2	yes	complemented
24	#91099	C₂²	1	4	yes	complemented
25	C₈×Dic₁₅	C₂²	1	4	yes	complemented
26	#91087	C₂²	1	2	yes	complemented
27	C₄×C₃⋊#168	C₂²	1	2	yes	complemented
28	#78514	—	5	1	no	maximal, Hall
29	C₂×D₅×M₁₆	—	3	2	no	—
30	C₂×#4018	—	3	1	no	—
31	C₄×C₅×M₁₆	D₃	1	1	yes	characteristic, complemented
32	M₁₆×Dic₅	—	3	2	no	—
33	C₄×#681	—	3	1	no	—
34	#64997	—	3	4	no	—
35	#64978	—	3	4	no	—
36	C₂³×D₁₅	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×Dic₁₅	C₂×C₄	1	2	yes	—
40	C₂²×Dic₁₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₁₅	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₁₅	C₂×C₄	1	8	yes	—
43	C₂×C₄×D₁₅	C₂³	1	2	yes	—
44	C₃×C₅×M₁₆	C₂×C₄	1	8	yes	—
45	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
46	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	#6714	C₂×C₄	1	4	yes	—
50	#6714	C₂×C₄	1	4	yes	—
51	#6712	C₂×C₄	1	4	yes	—
52	#6712	C₂×C₄	1	4	yes	—
53	C₄×Dic₁₅	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₅	C₂³	1	2	yes	—
55	#6710	C₂×C₄	1	8	yes	—
56	C₂×C₃⋊#168	C₂³	1	4	yes	—
57	#6705	C₂×C₄	1	16	yes	complemented
58	C₈×D₁₅	C₂×C₄	1	16	yes	complemented
59	C₂×D₃×M₁₆	—	5	2	no	—
60	C₂×#872	—	5	1	no	—
61	C₃×C₄×M₁₆	D₅	1	1	yes	characteristic, complemented
62	Dic₃×M₁₆	—	5	2	no	—
63	C₄×#208	—	5	1	no	—
64	#4708	—	5	4	no	—
65	#4689	—	5	4	no	—
66	C₂²×C₄×D₅	—	3	1	no	—
67	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
68	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
69	C₂×#685	—	3	2	no	—
70	C₂×#683	—	3	2	no	—
71	C₂×C₄×Dic₅	—	3	1	no	—
72	C₂×#681	—	3	2	no	—
73	D₅×M₁₆	—	3	16	no	—
74	C₂×#676	—	3	4	no	—
75	C₂×C₈×D₅	—	3	4	no	—
76	#4018	—	3	4	no	—
77	#4018	—	3	2	no	—
78	#4018	—	3	2	no	—
79	C₅×#97	C₂×D₃	1	2	yes	complemented
80	C₄×C₅×C₈	C₂×D₃	1	2	yes	complemented
81	#3947	—	3	4	no	—
82	C₈×Dic₅	—	3	4	no	—
83	#3935	—	3	2	no	—
84	C₄×#168	—	3	2	no	—
85	#2873	—	15	1	no	Sylow
86	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₁₅	—	4	2	no	—
88	C₂²×D₁₅	C₄²	1	4	yes	—
89	C₂²×D₁₅	C₂²×C₄	1	2	yes	—
90	C₂×C₃×C₄×C₅	—	4	2	no	—
91	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
92	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
93	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
95	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	C₂×Dic₁₅	—	4	2	no	—
97	C₂×Dic₁₅	C₄²	1	4	yes	—
98	C₂×Dic₁₅	C₂²×C₄	1	4	yes	—
99	C₂×Dic₁₅	C₂²×C₄	1	2	yes	—
100	C₄×D₁₅	C₄²	1	8	yes	—
101	C₄×D₁₅	C₂²×C₄	1	4	yes	—
102	C₄×D₁₅	C₂²×C₄	1	4	yes	—
103	C₃×C₅×C₈	C₂²×C₄	1	8	yes	—
104	C₃⋊#168	C₂²×C₄	1	8	yes	—
105	C₂²×C₄×D₃	—	5	1	no	—
106	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
107	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
108	C₂×(C₂×D₃)⋊C₄	—	5	2	no	—
109	C₂×Dic₃⋊C₄	—	5	2	no	—
110	C₂×C₄×Dic₃	—	5	1	no	—
111	C₂×#208	—	5	2	no	—
112	D₃×M₁₆	—	5	16	no	—
113	C₂×ΓL(1,𝔽₂₅)	—	5	4	no	—
114	C₂×D₃×C₈	—	5	4	no	—
115	#872	—	5	4	no	—
116	#872	—	5	2	no	—
117	#872	—	5	2	no	—
118	C₃×#97	C₂×D₅	1	2	yes	complemented
119	C₃×C₄×C₈	C₂×D₅	1	2	yes	complemented
120	#815	—	5	4	no	—
121	C₈×Dic₃	—	5	4	no	—
122	#803	—	5	2	no	—
123	C₄×C₃⋊C₈	—	5	2	no	—
124	C₂³×D₅	—	3	1	no	—
125	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
126	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
127	C₂²×Dic₅	—	3	1	no	—
128	C₂²×Dic₅	—	3	2	no	—
129	C₂×C₄×D₅	—	3	2	no	—
130	C₂×C₄×D₅	—	3	8	no	—
131	C₂×C₄×D₅	—	3	2	no	—
132	C₅×M₁₆	C₄×D₃	1	8	yes	—
133	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
134	C₄²×C₅	C₄×D₃	1	2	yes	—
135	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
136	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
137	#685	—	3	4	no	—
138	#685	—	3	4	no	—
139	#683	—	3	4	no	—
140	#683	—	3	4	no	—
141	C₄×Dic₅	—	3	2	no	—
142	C₄×Dic₅	—	3	2	no	—
143	#681	—	3	8	no	—
144	C₂×#168	—	3	4	no	—
145	#676	—	3	16	no	—
146	C₈×D₅	—	3	16	no	—
147	C₂²×M₁₆	—	15	2	no	—
148	C₂×#117	—	15	1	no	—
149	#405	—	15	4	no	—
150	#405	—	15	4	no	—
151	C₄×M₁₆	—	15	2	no	—
152	C₄×M₁₆	—	15	1	no	—
153	C₄×M₁₆	D₁₅	1	1	yes	characteristic, complemented
154	C₂²×C₃×C₅	—	2	2	no	—
155	C₂²×C₃×C₅	C₂×C₄²	1	2	yes	—
156	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
157	C₂×D₁₅	—	2	4	no	—
158	C₂×D₁₅	—	4	4	no	—
159	C₂×D₁₅	C₂×C₄²	1	4	yes	—
160	C₃×C₄×C₅	—	2	2	no	—
161	C₃×C₄×C₅	—	2	2	no	—
162	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
163	C₃×C₄×C₅	C₂³×C₄	1	2	yes	—
164	Dic₁₅	—	2	4	no	—
165	Dic₁₅	C₂×C₄²	1	4	yes	—
166	C₂³×D₃	—	5	1	no	—
167	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
168	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
169	C₂²×Dic₃	—	5	1	no	—
170	C₂²×Dic₃	—	5	2	no	—
171	C₂×C₄×D₃	—	5	2	no	—
172	C₂×C₄×D₃	—	5	8	no	—
173	C₂×C₄×D₃	—	5	2	no	—
174	C₃×M₁₆	C₄×D₅	1	8	yes	—
175	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
176	C₃×C₄²	C₄×D₅	1	2	yes	—
177	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
178	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
179	(C₂×D₃)⋊C₄	—	5	4	no	—
180	(C₂×D₃)⋊C₄	—	5	4	no	—
181	Dic₃⋊C₄	—	5	4	no	—
182	Dic₃⋊C₄	—	5	4	no	—
183	C₄×Dic₃	—	5	2	no	—
184	C₄×Dic₃	—	5	2	no	—
185	#208	—	5	8	no	—
186	C₂×C₃⋊C₈	—	5	4	no	—
187	ΓL(1,𝔽₂₅)	—	5	16	no	—
188	D₃×C₈	—	5	16	no	—
189	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
190	C₂²×D₅	—	3	2	no	—
191	C₂²×D₅	—	3	4	no	—
192	C₂²×D₅	—	12	2	no	—
193	C₂×C₄×C₅	—	4	2	no	—
194	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
195	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
196	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
197	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
198	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
199	C₂×Dic₅	—	3	4	no	—
200	C₂×Dic₅	—	3	2	no	—
201	C₂×Dic₅	—	3	4	no	—
202	C₂×Dic₅	—	12	2	no	—
203	C₄×D₅	—	3	8	no	—
204	C₄×D₅	—	3	4	no	—
205	C₄×D₅	—	3	4	no	—
206	C₅×C₈	C₂×C₄×D₃	1	8	yes	—
207	#168	—	3	8	no	—
208	C₂³×C₄	—	15	1	no	—
209	C₂×M₁₆	—	15	2	no	—
210	C₂×M₁₆	—	15	16	no	—
211	C₂×M₁₆	—	15	4	no	—
212	C₂×M₁₆	C₂×D₁₅	1	2	yes	—
213	C₂²×C₈	—	15	4	no	—
214	#117	—	15	4	no	—
215	#117	—	15	2	no	—
216	#117	—	15	2	no	—
217	C₂×C₄⋊C₄	—	15	2	no	—
218	C₂×C₂²⋊C₄	—	15	2	no	—
219	C₂×C₄²	—	15	1	no	—
220	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
221	#97	—	15	4	no	—
222	#97	—	15	2	no	—
223	#97	C₂×D₁₅	1	2	yes	complemented
224	C₄×C₈	—	15	4	no	—
225	C₄×C₈	—	15	2	no	—
226	C₄×C₈	C₂×D₁₅	1	2	yes	complemented
227	C₂×C₃×C₅	—	2	2	no	—
228	C₂×C₃×C₅	C₂²×C₄²	1	1	yes	characteristic
229	C₂×C₃×C₅	#568	1	2	yes	—
230	D₁₅	—	2	4	no	—
231	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
232	C₂²×D₃	—	5	2	no	—
233	C₂²×D₃	—	5	4	no	—
234	C₂²×D₃	—	20	2	no	—
235	C₂×C₃×C₄	—	4	2	no	—
236	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
237	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
238	C₂×C₃×C₄	C₂×C₄×D₅	1	4	yes	—
239	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
241	C₂×Dic₃	—	5	2	no	—
242	C₂×Dic₃	—	5	4	no	—
243	C₂×Dic₃	—	5	4	no	—
244	C₂×Dic₃	—	20	2	no	—
245	C₄×D₃	—	5	4	no	—
246	C₄×D₃	—	5	4	no	—
247	C₄×D₃	—	5	8	no	—
248	C₃×C₈	C₂×C₄×D₅	1	8	yes	—
249	C₃⋊C₈	—	5	8	no	—
250	C₂²×C₅	—	2	2	no	—
251	C₂²×C₅	C₄²×D₃	1	2	yes	—
252	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
253	C₂×D₅	—	3	4	no	—
254	C₂×D₅	—	6	4	no	—
255	C₂×D₅	—	12	4	no	—
256	C₄×C₅	—	2	2	no	—
257	C₄×C₅	—	2	2	no	—
258	C₄×C₅	C₄²×D₃	1	2	yes	—
259	C₄×C₅	C₂²×C₄×D₃	1	2	yes	—
260	Dic₅	—	3	4	no	—
261	Dic₅	—	6	4	no	—
262	C₂⁴	—	15	1	no	—
263	C₂²×C₄	—	15	1	no	—
264	C₂²×C₄	—	15	2	no	—
265	C₂²×C₄	—	15	8	no	—
266	C₂²×C₄	—	15	2	no	—
267	C₂²×C₄	—	15	2	no	—
268	C₂²×C₄	C₄×D₁₅	1	2	yes	—
269	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
270	M₁₆	—	15	8	no	—
271	M₁₆	—	15	16	no	—
272	M₁₆	C₄×D₁₅	1	8	yes	—
273	C₂×C₈	—	15	4	no	—
274	C₂×C₈	—	15	16	no	—
275	C₂×C₈	C₂²×D₁₅	1	4	yes	—
276	C₄⋊C₄	—	15	4	no	—
277	C₄⋊C₄	—	15	4	no	—
278	C₂²⋊C₄	—	15	4	no	—
279	C₂²⋊C₄	—	15	4	no	—
280	C₄²	—	15	2	no	—
281	C₄²	—	15	2	no	—
282	C₄²	C₄×D₁₅	1	2	yes	—
283	C₄²	C₂²×D₁₅	1	1	yes	characteristic
284	C₄²	C₂²×D₁₅	1	1	yes	characteristic
285	C₃×C₅	#2873	1	1	yes	characteristic, complemented, Hall
286	C₂²×C₃	—	2	2	no	—
287	C₂²×C₃	C₄²×D₅	1	2	yes	—
288	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
289	C₂×D₃	—	5	4	no	—
290	C₂×D₃	—	10	4	no	—
291	C₂×D₃	—	20	4	no	—
292	C₃×C₄	—	2	2	no	—
293	C₃×C₄	—	2	2	no	—
294	C₃×C₄	C₄²×D₅	1	2	yes	—
295	C₃×C₄	C₂²×C₄×D₅	1	2	yes	—
296	Dic₃	—	5	4	no	—
297	Dic₃	—	10	4	no	—
298	C₂×C₅	—	2	2	no	—
299	C₂×C₅	C₂×C₄²×D₃	1	1	yes	characteristic
300	C₂×C₅	#5745	1	2	yes	—
301	D₅	—	6	4	no	—
302	C₂³	—	15	2	no	—
303	C₂³	—	15	4	no	—
304	C₂³	—	60	2	no	—
305	C₂³	C₂×C₄×D₁₅	1	1	yes	characteristic
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	15	4	no	—
308	C₂×C₄	—	15	2	no	—
309	C₂×C₄	—	15	8	no	—
310	C₂×C₄	—	15	4	no	—
311	C₂×C₄	—	15	4	no	—
312	C₂×C₄	—	15	4	no	—
313	C₂×C₄	—	60	2	no	—
314	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
315	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
316	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄×D₁₅	1	4	yes	—
318	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
319	C₈	—	15	8	no	—
320	C₈	C₂×C₄×D₁₅	1	8	yes	—
321	C₂×C₃	—	2	2	no	—
322	C₂×C₃	C₂×C₄²×D₅	1	1	yes	characteristic
323	C₂×C₃	#66079	1	2	yes	—
324	D₃	—	10	4	no	—
325	C₅	#78514	1	1	yes	characteristic, complemented, Sylow
326	C₂²	—	2	2	no	—
327	C₂²	—	15	4	no	—
328	C₂²	—	30	4	no	—
329	C₂²	—	60	4	no	—
330	C₂²	C₄²×D₁₅	1	2	yes	—
331	C₂²	C₂²×C₄×D₁₅	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	15	4	no	—
335	C₄	—	30	4	no	—
336	C₄	C₄²×D₁₅	1	2	yes	—
337	C₄	C₂²×C₄×D₁₅	1	2	yes	—
338	C₃	#10610245	1	1	yes	characteristic, complemented, Sylow
339	C₂	—	2	2	no	—
340	C₂	—	30	4	no	—
341	C₂	C₂×C₄²×D₁₅	1	1	yes	characteristic
342	C₂	#11756504	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅
commutator	C₂×C₃×C₅	C₂²×C₄²
Fitting	C₃×C₄×C₅×M₁₆	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂³×D₁₅	C₂×C₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₄×M₁₆	D₁₅
hypocenter	C₃×C₅	#2873
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₅×M₁₆	C₂	semidirect
4. M₁₆×Dic₁₅	C₂	semidirect
5. C₄×#6710	C₂	semidirect
6. #11750809	C₂	semidirect
7. #11750790	C₂	semidirect
16. M₁₆×D₁₅	C₄	semidirect
22. C₃×C₅×#97	C₂²	semidirect
23. C₃×C₄×C₅×C₈	C₂²	semidirect
24. #91099	C₂²	semidirect
25. C₈×Dic₁₅	C₂²	semidirect
26. #91087	C₂²	semidirect
27. C₄×C₃⋊#168	C₂²	semidirect
31. C₄×C₅×M₁₆	D₃	semidirect
57. #6705	C₂×C₄	semidirect
58. C₈×D₁₅	C₂×C₄	semidirect
61. C₃×C₄×M₁₆	D₅	semidirect
79. C₅×#97	C₂×D₃	semidirect
80. C₄×C₅×C₈	C₂×D₃	semidirect
118. C₃×#97	C₂×D₅	semidirect
119. C₃×C₄×C₈	C₂×D₅	semidirect
153. C₄×M₁₆	D₁₅	semidirect
223. #97	C₂×D₁₅	semidirect
226. C₄×C₈	C₂×D₁₅	semidirect
285. C₃×C₅	#2873	semidirect
325. C₅	#78514	semidirect
338. C₃	#10610245	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₁₅	2
C₂×#91761	2
C₃×C₄×C₅×M₁₆	2
M₁₆×Dic₁₅	2
C₄×#6710	2
#11750809	2
#11750790	2
#10610245	3
#78514	5