Subgroups — #14281831

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756337	C₂	1	1	yes	maximal, characteristic
2	#11747000	C₂	1	1	yes	maximal, characteristic
3	#11746937	C₂	1	1	yes	maximal, characteristic
4	C₅×#4535	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×Q₈⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745845	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745844	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599025	—	3	1	no	maximal, Hall
9	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	#91835	C₄	1	1	yes	characteristic
11	#91833	C₄	1	1	yes	characteristic
12	#91786	C₄	1	1	yes	characteristic
13	#91785	—	2	1	no	—
14	#91778	—	2	1	no	—
15	#91777	C₄	1	1	yes	characteristic
16	#91761	—	2	1	no	—
17	C₄²×D₁₅	C₂²	1	1	yes	characteristic
18	C₄×Dic₃₀	—	2	1	no	—
19	C₄×Dic₃₀	C₂²	1	1	yes	characteristic
20	C₂×#6648	—	2	1	no	—
21	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
22	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
24	C₃×C₄×#168	C₂²	1	1	yes	characteristic
25	#90983	—	2	1	no	—
26	#90983	—	2	1	no	—
27	#90957	—	2	1	no	—
28	#68232	—	5	1	no	maximal, Hall
29	#65912	—	3	1	no	—
30	D₅×C₄⋊C₈	—	3	1	no	—
31	#64974	—	3	1	no	—
32	C₅×Q₈⋊C₈	—	3	1	no	—
33	Q₈⋊#168	D₃	1	1	yes	characteristic, complemented
34	#64704	—	3	1	no	—
35	#64676	—	3	1	no	—
36	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
39	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
40	C₂×Dic₃₀	—	2	1	no	—
41	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
42	C₃×C₅×C₄⋊C₄	—	2	1	no	—
43	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	—	2	1	no	—
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	#6714	—	2	1	no	—
47	#6714	—	2	1	no	—
48	C₄⋊Dic₁₅	—	2	1	no	—
49	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
50	#6712	—	2	1	no	—
51	#6712	—	2	1	no	—
52	#6712	—	2	1	no	—
53	#6712	—	2	1	no	—
54	#6712	—	2	1	no	—
55	C₄×Dic₁₅	—	2	1	no	—
56	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₁₅	D₄	1	1	yes	characteristic
58	C₄×Dic₁₅	D₄	1	1	yes	characteristic
59	C₂×C₅×C₃⋊C₈	—	2	1	no	—
60	C₂×C₅×C₃⋊C₈	—	2	1	no	—
61	C₂×C₃×#168	—	2	1	no	—
62	C₂×C₃×#168	—	2	1	no	—
63	#6648	—	4	2	no	—
64	#5578	—	5	1	no	—
65	D₃×C₄⋊C₈	—	5	1	no	—
66	#4685	—	5	1	no	—
67	C₃×Q₈⋊C₈	—	5	1	no	—
68	#4535	D₅	1	1	yes	characteristic, complemented
69	#4484	—	5	1	no	—
70	#4456	—	5	1	no	—
71	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
72	#4092	—	3	1	no	—
73	#4090	—	3	1	no	—
74	C₂×C₈×D₅	—	6	1	no	—
75	#4043	—	3	1	no	—
76	#4042	—	6	1	no	—
77	#4035	—	6	1	no	—
78	#4034	—	3	1	no	—
79	#4018	—	6	1	no	—
80	C₄²×D₅	—	3	1	no	—
81	C₄×Dic₁₀	—	3	1	no	—
82	C₄×Dic₁₀	—	6	1	no	—
83	C₅×C₄⋊C₈	—	3	1	no	—
84	C₄×C₅×C₈	—	3	1	no	—
85	#3952	—	6	1	no	—
86	#3946	—	6	1	no	—
87	#3946	—	6	1	no	—
88	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
89	C₄×#168	C₂×D₃	1	1	yes	characteristic
90	#1481	—	15	1	no	Sylow
91	C₂²×D₁₅	C₂²⋊C₄	1	1	yes	characteristic
92	C₃×C₅×Q₈	—	2	1	no	—
93	C₃×C₅×Q₈	—	2	1	no	—
94	C₂×C₃×C₄×C₅	—	2	1	no	—
95	C₂×C₃×C₄×C₅	—	2	1	no	—
96	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₁₅	—	2	1	no	—
100	C₂×Dic₁₅	—	2	1	no	—
101	C₂×Dic₁₅	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×Dic₁₅	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×Dic₁₅	C₂²⋊C₄	1	1	yes	characteristic
104	C₄×D₁₅	—	2	1	no	—
105	C₄×D₁₅	—	2	1	no	—
106	C₄×D₁₅	—	4	1	no	—
107	C₄×D₁₅	SD₁₆	1	2	yes	—
108	C₄×D₁₅	Q₁₆	1	2	yes	—
109	Dic₃₀	—	2	1	no	—
110	Dic₃₀	—	2	1	no	—
111	Dic₃₀	—	4	2	no	—
112	C₃×#168	—	4	1	no	—
113	C₃×#168	—	4	1	no	—
114	C₅×C₃⋊C₈	—	4	1	no	—
115	C₅×C₃⋊C₈	—	4	1	no	—
116	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
117	#946	—	5	1	no	—
118	#944	—	5	1	no	—
119	C₂×D₃×C₈	—	10	1	no	—
120	#897	—	5	1	no	—
121	#896	—	10	1	no	—
122	#889	—	10	1	no	—
123	#888	—	5	1	no	—
124	#872	—	10	1	no	—
125	C₄²×D₃	—	5	1	no	—
126	C₄×Dic₆	—	5	1	no	—
127	C₄×Dic₆	—	10	1	no	—
128	C₃×C₄⋊C₈	—	5	1	no	—
129	C₃×C₄×C₈	—	5	1	no	—
130	(C₂×D₃)⋊C₈	—	10	1	no	—
131	Dic₃⋊C₈	—	10	1	no	—
132	Dic₃⋊C₈	—	10	1	no	—
133	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
134	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
135	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
136	C₂×C₄×D₅	—	3	1	no	—
137	C₂×C₄×D₅	—	3	1	no	—
138	C₂×C₄×D₅	—	3	1	no	—
139	C₂×Dic₁₀	—	3	1	no	—
140	C₂×Dic₁₀	—	6	1	no	—
141	C₂×C₅×C₈	—	6	1	no	—
142	C₂×C₅×C₈	—	6	1	no	—
143	C₅×C₄⋊C₄	—	2	1	no	—
144	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
145	C₄²×C₅	—	2	1	no	—
146	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
147	#685	—	6	1	no	—
148	#685	—	6	1	no	—
149	C₄⋊Dic₅	—	3	1	no	—
150	C₄⋊Dic₅	—	6	1	no	—
151	#683	—	6	1	no	—
152	#683	—	6	1	no	—
153	#683	—	6	1	no	—
154	#683	—	6	1	no	—
155	#683	—	6	1	no	—
156	C₄×Dic₅	—	3	1	no	—
157	C₄×Dic₅	—	3	1	no	—
158	C₄×Dic₅	—	3	1	no	—
159	C₄×Dic₅	—	6	1	no	—
160	C₂×#168	—	2	1	no	—
161	C₂×#168	—	2	1	no	—
162	C₈×D₅	—	12	2	no	—
163	#533	—	15	1	no	—
164	#431	—	15	1	no	—
165	C₂×C₄⋊C₈	—	15	1	no	—
166	Q₈⋊C₈	—	3	1	no	—
167	Q₈⋊C₈	—	5	1	no	—
168	Q₈⋊C₈	—	15	1	no	—
169	Q₈⋊C₈	—	15	1	no	—
170	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₂×D₁₅	—	2	1	no	—
172	C₂×D₁₅	—	2	1	no	—
173	C₂×D₁₅	#103	1	2	yes	—
174	C₃×C₄×C₅	—	2	1	no	—
175	C₃×C₄×C₅	—	4	1	no	—
176	C₃×C₄×C₅	—	4	1	no	—
177	C₃×C₄×C₅	#131	1	1	yes	characteristic
178	C₃×C₄×C₅	#131	1	1	yes	characteristic
179	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
180	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
181	Dic₁₅	—	2	1	no	—
182	Dic₁₅	—	2	1	no	—
183	Dic₁₅	—	4	1	no	—
184	Dic₁₅	—	4	1	no	—
185	Dic₁₅	#103	1	2	yes	—
186	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
187	C₂×C₄×D₃	—	5	1	no	—
188	C₂×C₄×D₃	—	5	1	no	—
189	C₂×C₄×D₃	—	5	1	no	—
190	C₂×Dic₆	—	5	1	no	—
191	C₂×Dic₆	—	10	1	no	—
192	C₂×C₃×C₈	—	10	1	no	—
193	C₂×C₃×C₈	—	10	1	no	—
194	C₃×C₄⋊C₄	—	2	1	no	—
195	C₃×C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
196	C₃×C₄²	—	2	1	no	—
197	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
198	(C₂×D₃)⋊C₄	—	10	1	no	—
199	(C₂×D₃)⋊C₄	—	10	1	no	—
200	C₄⋊Dic₃	—	5	1	no	—
201	C₄⋊Dic₃	—	10	1	no	—
202	Dic₃⋊C₄	—	10	1	no	—
203	Dic₃⋊C₄	—	10	1	no	—
204	Dic₃⋊C₄	—	10	1	no	—
205	Dic₃⋊C₄	—	10	1	no	—
206	Dic₃⋊C₄	—	10	1	no	—
207	C₄×Dic₃	—	5	1	no	—
208	C₄×Dic₃	—	5	1	no	—
209	C₄×Dic₃	—	5	1	no	—
210	C₄×Dic₃	—	10	1	no	—
211	C₂×C₃⋊C₈	—	2	1	no	—
212	C₂×C₃⋊C₈	—	2	1	no	—
213	D₃×C₈	—	20	2	no	—
214	C₂²×D₅	—	3	1	no	—
215	C₅×Q₈	—	2	1	no	—
216	C₅×Q₈	—	2	1	no	—
217	C₂×C₄×C₅	—	2	1	no	—
218	C₂×C₄×C₅	—	2	1	no	—
219	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
220	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
221	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
222	C₂×Dic₅	—	3	1	no	—
223	C₂×Dic₅	—	3	1	no	—
224	C₂×Dic₅	—	3	1	no	—
225	C₂×Dic₅	—	6	1	no	—
226	C₂×Dic₅	—	6	1	no	—
227	C₄×D₅	—	3	2	no	—
228	C₄×D₅	—	3	2	no	—
229	C₄×D₅	—	6	1	no	—
230	C₄×D₅	—	6	1	no	—
231	C₄×D₅	—	12	1	no	—
232	Dic₁₀	—	6	1	no	—
233	Dic₁₀	—	6	1	no	—
234	Dic₁₀	—	12	2	no	—
235	C₅×C₈	—	12	1	no	—
236	C₅×C₈	—	12	1	no	—
237	#168	—	4	1	no	—
238	#168	—	4	1	no	—
239	C₂²×C₈	—	30	1	no	—
240	C₄⋊Q₈	—	15	1	no	—
241	C₄⋊Q₈	—	15	1	no	—
242	#125	—	30	1	no	—
243	C₂²⋊Q₈	—	15	1	no	—
244	C₂²⋊Q₈	—	15	1	no	—
245	C₂²⋊Q₈	—	30	1	no	—
246	C₄×Q₈	—	15	1	no	—
247	C₄×Q₈	—	30	1	no	—
248	C₄×Q₈	D₃×D₅	1	1	yes	characteristic
249	#117	—	30	1	no	—
250	C₂×C₄²	—	15	1	no	—
251	C₄⋊C₈	—	3	1	no	—
252	C₄⋊C₈	—	5	1	no	—
253	C₄⋊C₈	—	30	1	no	—
254	C₄⋊C₈	—	30	1	no	—
255	C₂²⋊C₈	—	30	1	no	—
256	C₄×C₈	—	3	1	no	—
257	C₄×C₈	—	5	1	no	—
258	C₂×C₃×C₅	#408	1	1	yes	characteristic
259	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
260	C₂×C₃×C₅	#421	1	1	yes	characteristic
261	D₁₅	—	2	2	no	—
262	C₂²×D₃	—	5	1	no	—
263	C₃×Q₈	—	2	1	no	—
264	C₃×Q₈	—	2	1	no	—
265	C₂×C₃×C₄	—	2	1	no	—
266	C₂×C₃×C₄	—	2	1	no	—
267	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
268	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
269	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
270	C₂×Dic₃	—	5	1	no	—
271	C₂×Dic₃	—	5	1	no	—
272	C₂×Dic₃	—	5	1	no	—
273	C₂×Dic₃	—	10	1	no	—
274	C₂×Dic₃	—	10	1	no	—
275	C₄×D₃	—	5	2	no	—
276	C₄×D₃	—	5	2	no	—
277	C₄×D₃	—	10	1	no	—
278	C₄×D₃	—	10	1	no	—
279	C₄×D₃	—	20	1	no	—
280	Dic₆	—	10	1	no	—
281	Dic₆	—	10	1	no	—
282	Dic₆	—	20	2	no	—
283	C₃×C₈	—	20	1	no	—
284	C₃×C₈	—	20	1	no	—
285	C₃⋊C₈	—	4	1	no	—
286	C₃⋊C₈	—	4	1	no	—
287	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
288	C₂×D₅	—	3	2	no	—
289	C₂×D₅	—	6	1	no	—
290	C₂×D₅	—	6	1	no	—
291	C₄×C₅	—	2	1	no	—
292	C₄×C₅	—	4	1	no	—
293	C₄×C₅	—	4	1	no	—
294	C₄×C₅	#901	1	1	yes	characteristic
295	C₄×C₅	#907	1	1	yes	characteristic
296	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
297	C₄×C₅	D₃×Q₁₆	1	1	yes	characteristic
298	Dic₅	—	3	2	no	—
299	Dic₅	—	6	1	no	—
300	Dic₅	—	6	1	no	—
301	Dic₅	—	12	1	no	—
302	Dic₅	—	12	1	no	—
303	C₂×Q₈	—	15	1	no	—
304	C₂×Q₈	—	30	1	no	—
305	C₂×Q₈	#1200	1	1	yes	characteristic
306	C₂²×C₄	—	15	1	no	—
307	C₂²×C₄	—	15	1	no	—
308	C₂²×C₄	—	15	1	no	—
309	C₂×C₈	—	6	1	no	—
310	C₂×C₈	—	6	1	no	—
311	C₂×C₈	—	10	1	no	—
312	C₂×C₈	—	10	1	no	—
313	C₂×C₈	—	60	2	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	15	1	no	—
316	C₄⋊C₄	—	30	1	no	—
317	C₄⋊C₄	—	30	1	no	—
318	C₄⋊C₄	—	30	1	no	—
319	C₄⋊C₄	—	30	1	no	—
320	C₄⋊C₄	—	30	1	no	—
321	C₄⋊C₄	—	30	1	no	—
322	C₄⋊C₄	#1200	1	1	yes	characteristic
323	C₂²⋊C₄	—	30	1	no	—
324	C₂²⋊C₄	—	30	1	no	—
325	C₄²	—	2	1	no	—
326	C₄²	—	15	1	no	—
327	C₄²	—	15	1	no	—
328	C₄²	—	15	1	no	—
329	C₄²	—	30	1	no	—
330	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
331	C₃×C₅	#1481	1	1	yes	characteristic, complemented, Hall
332	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
333	C₂×D₃	—	5	2	no	—
334	C₂×D₃	—	10	1	no	—
335	C₂×D₃	—	10	1	no	—
336	C₃×C₄	—	2	1	no	—
337	C₃×C₄	—	4	1	no	—
338	C₃×C₄	—	4	1	no	—
339	C₃×C₄	#4047	1	1	yes	characteristic
340	C₃×C₄	#4053	1	1	yes	characteristic
341	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
342	C₃×C₄	D₅×Q₁₆	1	1	yes	characteristic
343	Dic₃	—	5	2	no	—
344	Dic₃	—	10	1	no	—
345	Dic₃	—	10	1	no	—
346	Dic₃	—	20	1	no	—
347	Dic₃	—	20	1	no	—
348	C₂×C₅	#4727	1	1	yes	characteristic
349	C₂×C₅	D₃×#103	1	1	yes	characteristic
350	C₂×C₅	#4821	1	1	yes	characteristic
351	D₅	—	6	2	no	—
352	C₂³	—	15	1	no	—
353	Q₈	—	2	1	no	—
354	Q₈	—	2	1	no	—
355	Q₈	—	30	1	no	—
356	Q₈	—	30	1	no	—
357	Q₈	—	60	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	15	1	no	—
361	C₂×C₄	—	15	2	no	—
362	C₂×C₄	—	15	2	no	—
363	C₂×C₄	—	15	1	no	—
364	C₂×C₄	—	15	1	no	—
365	C₂×C₄	—	30	1	no	—
366	C₂×C₄	—	30	1	no	—
367	C₂×C₄	—	30	1	no	—
368	C₂×C₄	—	30	1	no	—
369	C₂×C₄	—	60	1	no	—
370	C₂×C₄	C₂×#1200	1	1	yes	characteristic
371	C₂×C₄	#6790	1	1	yes	characteristic
372	C₂×C₄	#6790	1	1	yes	characteristic
373	C₈	—	12	1	no	—
374	C₈	—	12	1	no	—
375	C₈	—	20	1	no	—
376	C₈	—	20	1	no	—
377	C₂×C₃	#65016	1	1	yes	characteristic
378	C₂×C₃	D₅×#103	1	1	yes	characteristic
379	C₂×C₃	#65110	1	1	yes	characteristic
380	D₃	—	10	2	no	—
381	C₅	#68232	1	1	yes	characteristic, complemented, Sylow
382	C₂²	—	15	2	no	—
383	C₂²	—	30	1	no	—
384	C₂²	—	30	1	no	—
385	C₂²	#91561	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	4	1	no	—
389	C₄	—	15	2	no	—
390	C₄	—	30	1	no	—
391	C₄	—	30	1	no	—
392	C₄	—	60	1	no	—
393	C₄	—	60	1	no	—
394	C₄	#91290	1	1	yes	characteristic
395	C₄	#91291	1	1	yes	characteristic
396	C₄	#91505	1	1	yes	characteristic
397	C₄	#91511	1	1	yes	characteristic
398	C₃	#10599025	1	1	yes	characteristic, complemented, Sylow
399	C₂	—	30	2	no	—
400	C₂	#11748498	1	1	yes	characteristic
401	C₂	#11749203	1	1	yes	characteristic
402	C₂	#11749431	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6790
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₄×C₅×Q₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₅	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×Q₈	D₃×D₅
hypocenter	C₃×C₅	#1481
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
4. C₅×#4535	C₂	semidirect
5. C₃×Q₈⋊#168	C₂	semidirect
6. #11745845	C₂	semidirect
7. #11745844	C₂	semidirect
33. Q₈⋊#168	D₃	semidirect
68. #4535	D₅	semidirect
331. C₃×C₅	#1481	semidirect
381. C₅	#68232	semidirect
398. C₃	#10599025	semidirect

Maximal subgroups

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

subgroup	index
#11756337	2
#11747000	2
#11746937	2
C₅×#4535	2
C₃×Q₈⋊#168	2
#11745845	2
#11745844	2
#10599025	3
#68232	5