Subgroups — #14285348

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃×#4015	C₂	1	1	yes	maximal, characteristic
2	C₅×#4835	C₂	1	1	yes	maximal, characteristic
3	#11746875	C₂	1	1	yes	maximal, characteristic
4	#11746302	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64701	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745863	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745861	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601617	—	3	1	no	maximal, Hall
9	C₂×D₃×Dic₁₀	C₄	1	2	yes	—
10	#91758	C₂²	1	1	yes	characteristic
11	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
12	C₃×#4015	C₂²	1	1	yes	characteristic
13	#91469	—	2	1	no	—
14	#91469	—	2	1	no	—
15	D₃×C₄⋊Dic₅	—	2	1	no	—
16	D₃×C₄⋊Dic₅	—	2	1	no	—
17	#91389	C₄	1	2	yes	—
18	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
20	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
21	C₅×#815	—	2	1	no	—
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
24	(C₂×D₃)⋊#168	—	2	1	no	—
25	#90953	—	2	1	no	—
26	#68487	—	5	1	no	maximal, Hall
27	C₂×#4015	—	3	1	no	—
28	C₅×#432	—	3	1	no	—
29	#65215	—	3	1	no	—
30	#64701	—	3	1	no	—
31	#64701	—	3	1	no	—
32	#64701	—	3	1	no	—
33	#64701	D₃	1	1	yes	characteristic, complemented
34	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
38	C₂×C₃×Dic₁₀	C₂×C₄	1	2	yes	—
39	C₂×#1204	—	2	2	no	—
40	C₂×D₃×Dic₅	—	2	2	no	—
41	D₃×Dic₁₀	—	4	4	no	—
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₄⋊Dic₁₅	—	2	1	no	—
45	C₄⋊Dic₁₅	—	2	1	no	—
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
48	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
49	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
50	C₂×C₅×C₃⋊C₈	—	2	1	no	—
51	C₃×C₄⋊Dic₅	—	2	1	no	—
52	C₃×C₄⋊Dic₅	—	2	1	no	—
53	C₂×C₃×#168	—	2	1	no	—
54	Dic₃⋊Dic₅	—	2	2	no	—
55	Dic₃⋊Dic₅	—	2	2	no	—
56	D₃×C₄⋊Q₈	—	5	1	no	—
57	#4835	—	5	1	no	—
58	#4835	D₅	1	1	yes	characteristic
59	C₃×#333	—	5	1	no	—
60	#4546	—	5	1	no	—
61	#4481	—	5	1	no	—
62	#4481	—	5	1	no	—
63	C₂²×Dic₁₀	—	3	2	no	—
64	C₂×C₄²×C₅	—	3	1	no	—
65	C₂×C₄⋊Dic₅	—	6	1	no	—
66	C₂×C₄⋊Dic₅	—	6	1	no	—
67	#4015	—	3	2	no	—
68	#4015	—	3	1	no	—
69	#4015	—	6	1	no	—
70	#4015	—	6	1	no	—
71	#4015	C₂×D₃	1	1	yes	characteristic
72	C₅×C₄⋊C₈	—	3	1	no	—
73	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
74	C₅×C₂²⋊C₈	—	6	1	no	—
75	C₅×#97	—	6	1	no	—
76	C₂²⋊#168	—	6	1	no	—
77	C₄⋊#168	—	3	1	no	—
78	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
79	#3935	—	6	1	no	—
80	#1558	—	15	1	no	Sylow
81	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×Dic₁₅	—	2	2	no	—
86	Dic₃₀	—	4	2	no	—
87	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₄×C₅×D₃	—	2	2	no	—
91	C₄×C₅×D₃	—	2	1	no	—
92	C₄×C₅×D₃	—	2	1	no	—
93	C₄×C₅×D₃	—	2	1	no	—
94	C₄×C₅×D₃	—	2	1	no	—
95	C₂×C₃×Dic₅	—	2	2	no	—
96	C₃×Dic₁₀	—	4	2	no	—
97	#1204	—	4	4	no	—
98	D₃×Dic₅	—	4	4	no	—
99	C₃×C₅×C₈	—	4	1	no	—
100	C₃⋊#168	—	4	1	no	—
101	C₃×#168	—	4	1	no	—
102	C₅×C₃⋊C₈	—	4	1	no	—
103	C₂×D₃×Q₈	—	5	2	no	—
104	C₃×C₄⋊Q₈	—	5	1	no	—
105	#944	—	5	2	no	—
106	D₃×C₄⋊C₄	—	10	1	no	—
107	D₃×C₄⋊C₄	—	10	1	no	—
108	#888	—	10	1	no	—
109	#888	—	10	1	no	—
110	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
111	#869	—	5	1	no	—
112	C₃×C₄⋊C₈	—	5	1	no	—
113	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
114	(C₂×D₃)⋊C₈	—	2	1	no	—
115	(C₂×D₃)⋊C₈	—	10	1	no	—
116	#815	—	2	1	no	—
117	#815	—	10	1	no	—
118	C₄⋊(C₃⋊C₈)	—	5	1	no	—
119	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
120	C₂²×C₄×C₅	—	3	1	no	—
121	C₂²×C₄×C₅	—	3	1	no	—
122	C₂²×C₄×C₅	—	3	1	no	—
123	C₂²×Dic₅	—	6	2	no	—
124	C₂×Dic₁₀	—	3	2	no	—
125	C₂×Dic₁₀	—	6	2	no	—
126	C₂×Dic₁₀	—	12	4	no	—
127	C₂×Dic₁₀	C₄×D₃	1	2	yes	—
128	C₂×C₅×C₈	—	2	1	no	—
129	C₂×C₅×C₈	—	6	1	no	—
130	C₄²×C₅	—	3	1	no	—
131	C₄²×C₅	—	3	1	no	—
132	C₄²×C₅	—	3	1	no	—
133	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
134	C₄⋊Dic₅	—	2	1	no	—
135	C₄⋊Dic₅	—	2	1	no	—
136	C₄⋊Dic₅	—	6	2	no	—
137	C₄⋊Dic₅	—	6	2	no	—
138	C₄⋊Dic₅	—	6	1	no	—
139	C₄⋊Dic₅	—	6	1	no	—
140	C₂×#168	—	2	1	no	—
141	C₂×#168	—	6	1	no	—
142	C₂×C₄⋊Q₈	—	15	1	no	—
143	#432	—	3	1	no	—
144	#432	—	15	1	no	—
145	#333	—	5	1	no	—
146	#333	—	15	1	no	—
147	#333	—	15	1	no	—
148	#333	—	15	1	no	—
149	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₂×C₅×D₃	—	2	1	no	—
151	C₂×C₅×D₃	—	2	1	no	—
152	C₂×C₅×D₃	#100	1	2	yes	—
153	C₃×C₄×C₅	—	2	1	no	—
154	C₃×C₄×C₅	#137	1	1	yes	characteristic
155	C₃×C₄×C₅	#137	1	1	yes	characteristic
156	C₃×C₄×C₅	#137	1	1	yes	characteristic
157	C₃×C₄×C₅	#137	1	1	yes	characteristic
158	Dic₁₅	—	4	2	no	—
159	C₃×Dic₅	—	4	2	no	—
160	C₅×Dic₃	—	2	1	no	—
161	C₅×Dic₃	—	2	1	no	—
162	C₅×Dic₃	—	2	1	no	—
163	C₂×C₃×Q₈	—	5	2	no	—
164	D₃×Q₈	—	20	4	no	—
165	C₂×C₄×D₃	—	10	2	no	—
166	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
167	C₂×C₄×D₃	D₂₀	1	1	yes	characteristic
168	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
169	C₂×Dic₆	—	5	2	no	—
170	C₂×Dic₆	—	10	2	no	—
171	C₂×C₃×C₈	—	2	1	no	—
172	C₂×C₃×C₈	—	10	1	no	—
173	C₃×C₄⋊C₄	—	10	1	no	—
174	C₃×C₄⋊C₄	—	10	1	no	—
175	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
176	C₄⋊Dic₃	—	10	1	no	—
177	C₄⋊Dic₃	—	10	1	no	—
178	Dic₃⋊C₄	—	10	2	no	—
179	Dic₃⋊C₄	—	10	2	no	—
180	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
181	C₄×Dic₃	D₂₀	1	1	yes	characteristic
182	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
183	C₂×C₃⋊C₈	—	2	1	no	—
184	C₂×C₃⋊C₈	—	10	1	no	—
185	C₂³×C₅	—	3	1	no	—
186	C₂×C₄×C₅	—	3	1	no	—
187	C₂×C₄×C₅	—	3	1	no	—
188	C₂×C₄×C₅	—	3	1	no	—
189	C₂×C₄×C₅	—	6	2	no	—
190	C₂×C₄×C₅	—	6	1	no	—
191	C₂×C₄×C₅	—	6	1	no	—
192	C₂×C₄×C₅	—	6	1	no	—
193	C₂×C₄×C₅	—	6	1	no	—
194	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
195	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
196	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
197	C₂×Dic₅	—	2	2	no	—
198	C₂×Dic₅	—	6	2	no	—
199	C₂×Dic₅	—	12	4	no	—
200	Dic₁₀	—	4	2	no	—
201	Dic₁₀	—	12	4	no	—
202	Dic₁₀	—	12	2	no	—
203	C₅×C₈	—	4	1	no	—
204	C₅×C₈	—	12	1	no	—
205	#168	—	4	1	no	—
206	#168	—	12	1	no	—
207	C₂²×Q₈	—	15	2	no	—
208	C₄⋊Q₈	—	5	1	no	—
209	C₄⋊Q₈	—	15	2	no	—
210	C₄⋊Q₈	—	15	1	no	—
211	C₄⋊Q₈	—	30	1	no	—
212	C₄⋊Q₈	—	30	1	no	—
213	C₂×C₄⋊C₄	—	30	1	no	—
214	C₂×C₄⋊C₄	—	30	1	no	—
215	C₂×C₄²	—	3	1	no	—
216	C₄⋊C₈	—	3	1	no	—
217	C₄⋊C₈	—	5	1	no	—
218	C₄⋊C₈	—	15	1	no	—
219	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
220	C₂²⋊C₈	—	6	1	no	—
221	C₂²⋊C₈	—	30	1	no	—
222	#97	—	6	1	no	—
223	#97	—	30	1	no	—
224	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
225	C₂×C₃×C₅	#419	1	1	yes	characteristic
226	C₂×C₃×C₅	#419	1	1	yes	characteristic
227	C₅×D₃	—	2	2	no	—
228	C₂²×D₃	#685	1	1	yes	characteristic
229	C₃×Q₈	—	20	2	no	—
230	C₂×C₃×C₄	—	10	2	no	—
231	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
234	C₂×Dic₃	—	10	2	no	—
235	C₂×Dic₃	#685	1	1	yes	characteristic
236	C₂×Dic₃	#685	1	1	yes	characteristic
237	C₂×Dic₃	#685	1	1	yes	characteristic
238	C₄×D₃	—	2	2	no	—
239	C₄×D₃	—	2	1	no	—
240	C₄×D₃	—	2	1	no	—
241	C₄×D₃	—	2	1	no	—
242	C₄×D₃	—	2	1	no	—
243	C₄×D₃	—	20	4	no	—
244	Dic₆	—	20	4	no	—
245	Dic₆	—	20	2	no	—
246	C₃×C₈	—	4	1	no	—
247	C₃×C₈	—	20	1	no	—
248	C₃⋊C₈	—	4	1	no	—
249	C₃⋊C₈	—	20	1	no	—
250	C₂²×C₅	—	3	2	no	—
251	C₂²×C₅	—	6	1	no	—
252	C₂²×C₅	—	6	1	no	—
253	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
254	C₄×C₅	—	2	1	no	—
255	C₄×C₅	—	6	1	no	—
256	C₄×C₅	—	6	1	no	—
257	C₄×C₅	—	6	1	no	—
258	C₄×C₅	#915	1	1	yes	characteristic
259	C₄×C₅	#915	1	1	yes	characteristic
260	C₄×C₅	#918	1	1	yes	characteristic
261	C₄×C₅	#918	1	1	yes	characteristic
262	Dic₅	—	4	2	no	—
263	Dic₅	—	12	2	no	—
264	C₂×Q₈	—	5	2	no	—
265	C₂×Q₈	—	15	2	no	—
266	C₂×Q₈	—	30	2	no	—
267	C₂×Q₈	—	60	4	no	—
268	C₂²×C₄	—	3	1	no	—
269	C₂²×C₄	—	3	1	no	—
270	C₂²×C₄	—	3	1	no	—
271	C₂²×C₄	—	30	2	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	6	1	no	—
274	C₂×C₈	—	10	1	no	—
275	C₂×C₈	—	30	1	no	—
276	C₄⋊C₄	—	10	1	no	—
277	C₄⋊C₄	—	10	1	no	—
278	C₄⋊C₄	—	30	2	no	—
279	C₄⋊C₄	—	30	2	no	—
280	C₄⋊C₄	—	30	1	no	—
281	C₄⋊C₄	—	30	1	no	—
282	C₄²	—	3	1	no	—
283	C₄²	—	3	1	no	—
284	C₄²	—	3	1	no	—
285	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
286	C₃×C₅	#1558	1	1	yes	characteristic, complemented, Hall
287	C₂²×C₃	C₂×#685	1	1	yes	characteristic
288	C₂×D₃	—	2	1	no	—
289	C₂×D₃	—	2	1	no	—
290	C₂×D₃	#3955	1	2	yes	—
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	20	2	no	—
293	C₃×C₄	#4055	1	1	yes	characteristic
294	C₃×C₄	#4055	1	1	yes	characteristic
295	C₃×C₄	#4097	1	1	yes	characteristic
296	C₃×C₄	#4097	1	1	yes	characteristic
297	Dic₃	—	2	1	no	—
298	Dic₃	—	2	1	no	—
299	Dic₃	—	2	1	no	—
300	Dic₃	—	20	2	no	—
301	C₂×C₅	—	6	2	no	—
302	C₂×C₅	D₃×#100	1	1	yes	characteristic
303	C₂×C₅	#4802	1	1	yes	characteristic
304	C₂×C₅	#4802	1	1	yes	characteristic
305	C₂³	—	3	1	no	—
306	Q₈	—	20	2	no	—
307	Q₈	—	60	2	no	—
308	Q₈	—	60	4	no	—
309	C₂×C₄	—	3	1	no	—
310	C₂×C₄	—	3	1	no	—
311	C₂×C₄	—	3	1	no	—
312	C₂×C₄	—	6	2	no	—
313	C₂×C₄	—	6	1	no	—
314	C₂×C₄	—	6	1	no	—
315	C₂×C₄	—	6	1	no	—
316	C₂×C₄	—	6	1	no	—
317	C₂×C₄	—	10	2	no	—
318	C₂×C₄	—	30	2	no	—
319	C₂×C₄	—	60	4	no	—
320	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
321	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
322	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
323	C₈	—	4	1	no	—
324	C₈	—	12	1	no	—
325	C₈	—	20	1	no	—
326	C₈	—	60	1	no	—
327	C₂×C₃	#65293	1	1	yes	characteristic
328	C₂×C₃	#65409	1	1	yes	characteristic
329	C₂×C₃	C₂×#3955	1	1	yes	characteristic
330	D₃	—	2	2	no	—
331	C₅	#68487	1	1	yes	characteristic, complemented, Sylow
332	C₂²	—	3	2	no	—
333	C₂²	—	6	1	no	—
334	C₂²	—	6	1	no	—
335	C₂²	D₃×#685	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	6	1	no	—
338	C₄	—	6	1	no	—
339	C₄	—	6	1	no	—
340	C₄	—	20	2	no	—
341	C₄	—	60	2	no	—
342	C₄	#91263	1	1	yes	characteristic
343	C₄	#91274	1	1	yes	characteristic
344	C₄	#91486	1	1	yes	characteristic
345	C₄	#91510	1	1	yes	characteristic
346	C₃	#10601617	1	1	yes	characteristic, complemented, Sylow
347	C₂	—	6	2	no	—
348	C₂	#11747489	1	1	yes	characteristic
349	C₂	#11748123	1	1	yes	characteristic
350	C₂	D₃×#3955	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×#685
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₃	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊C₈	D₃×D₅
hypocenter	C₃×C₅	#1558
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #11746302	C₂	semidirect
5. C₃×#64701	C₂	semidirect
6. #11745863	C₂	semidirect
7. #11745861	C₂	semidirect
33. #64701	D₃	semidirect
286. C₃×C₅	#1558	semidirect
331. C₅	#68487	semidirect
346. C₃	#10601617	semidirect

Maximal subgroups

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

subgroup	index
D₃×#4015	2
C₅×#4835	2
#11746875	2
#11746302	2
C₃×#64701	2
#11745863	2
#11745861	2
#10601617	3
#68487	5