Subgroups — #14284879

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#66059	C₂	1	1	yes	maximal, characteristic
2	#11747007	C₂	1	1	yes	maximal, characteristic, complemented
3	D₅×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746368	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4547	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745852	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745852	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599214	—	3	1	no	maximal, Hall
9	C₂×C₃×#713	—	2	1	no	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃×#4093	—	2	1	no	—
12	C₃×#4041	—	2	1	no	—
13	C₃×#4041	C₄	1	2	yes	—
14	C₃×#4039	—	2	1	no	—
15	C₃×#4039	C₄	1	2	yes	—
16	C₃×#4020	C₂²	1	1	yes	characteristic
17	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
18	C₂×#6649	—	2	1	no	—
19	C₂×D₅×C₃⋊C₈	—	2	1	no	—
20	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
24	#90982	—	2	1	no	—
25	#90982	—	2	1	no	—
26	#71806	—	5	1	no	maximal, Hall
27	#66059	D₃	1	1	yes	characteristic
28	#65126	—	3	1	no	—
29	D₅×C₄⋊C₈	—	3	1	no	—
30	C₅×#332	—	3	1	no	—
31	#64767	—	3	1	no	—
32	#64705	—	3	1	no	—
33	#64705	—	3	1	no	—
34	C₂×C₃×C₅×Q₈	—	2	1	no	—
35	C₃×#713	—	4	2	no	—
36	C₃×#713	—	4	2	no	—
37	C₂×C₃×D₂₀	—	2	2	no	—
38	C₂×C₃×D₂₀	—	2	1	no	—
39	C₂×C₃×D₂₀	—	2	1	no	—
40	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
41	C₂×C₃×C₄×D₅	—	2	2	no	—
42	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
44	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
45	C₃×C₅×C₄⋊C₄	—	2	1	no	—
46	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	C₂×C₃⋊#168	—	2	1	no	—
49	C₂×C₃⋊#168	—	2	1	no	—
50	C₂×C₅×C₃⋊C₈	—	2	1	no	—
51	C₂×C₅×C₃⋊C₈	—	2	1	no	—
52	C₃×#685	—	2	2	no	—
53	C₃×#685	—	2	2	no	—
54	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
55	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
56	C₃×C₄×Dic₅	D₄	1	1	yes	characteristic
57	#6649	—	4	2	no	—
58	D₅×C₃⋊C₈	—	4	2	no	—
59	C₃×#532	—	5	1	no	—
60	#4924	—	5	1	no	—
61	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
62	#4547	—	5	1	no	—
63	#4547	D₅	1	1	yes	characteristic, complemented
64	#4544	—	5	1	no	—
65	#4544	—	5	1	no	—
66	C₂×#713	—	2	1	no	—
67	C₅×C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
68	#4093	—	2	1	no	—
69	C₂×#676	—	6	1	no	—
70	C₂×C₈×D₅	—	6	1	no	—
71	#4041	—	2	1	no	—
72	#4041	Dic₃	1	2	yes	—
73	#4039	—	2	1	no	—
74	#4039	Dic₃	1	2	yes	—
75	#4020	C₂×D₃	1	1	yes	characteristic
76	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
77	C₅×C₄⋊C₈	—	3	1	no	—
78	C₅×C₄⋊C₈	—	3	1	no	—
79	#3946	—	6	1	no	—
80	#3946	—	6	1	no	—
81	C₄⋊#168	—	3	1	no	—
82	C₄⋊#168	—	3	1	no	—
83	#1547	—	15	1	no	Sylow
84	C₂²×C₃×D₅	—	2	2	no	—
85	C₂²×C₃×D₅	C₂²⋊C₄	1	1	yes	characteristic
86	C₃×C₅×Q₈	—	4	1	no	—
87	C₃×C₅×Q₈	—	4	1	no	—
88	C₂×C₃×C₄×C₅	—	2	2	no	—
89	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₃×Dic₅	C₂²⋊C₄	1	1	yes	characteristic
95	C₃×D₂₀	—	4	2	no	—
96	C₃×D₂₀	—	4	4	no	—
97	C₃×D₂₀	—	4	1	no	—
98	C₃×D₂₀	—	4	1	no	—
99	C₃×D₂₀	—	4	1	no	—
100	C₃×D₂₀	—	4	1	no	—
101	C₃×C₄×D₅	—	2	1	no	—
102	C₃×C₄×D₅	—	2	1	no	—
103	C₃×C₄×D₅	—	4	1	no	—
104	C₃×C₄×D₅	—	4	4	no	—
105	C₃×C₄×D₅	D₈	1	2	yes	—
106	C₃×C₄×D₅	SD₁₆	1	2	yes	—
107	C₃⋊#168	—	4	1	no	—
108	C₃⋊#168	—	4	1	no	—
109	C₅×C₃⋊C₈	—	4	1	no	—
110	C₅×C₃⋊C₈	—	4	1	no	—
111	C₂×C₃×#41	—	10	1	no	—
112	C₃×C₄⋊Q₈	C₂×D₅	1	1	yes	characteristic
113	C₃×#127	—	5	1	no	—
114	C₃×#124	—	10	1	no	—
115	C₃×#121	—	5	2	no	—
116	C₃×#121	—	10	1	no	—
117	C₃×C₄×D₄	—	5	2	no	—
118	C₃×C₄×D₄	—	10	1	no	—
119	C₂×C₃×C₄²	—	5	1	no	—
120	C₂×#208	—	10	1	no	—
121	C₂²×C₃⋊C₈	—	10	1	no	—
122	C₄⋊(C₃⋊C₈)	—	5	1	no	—
123	C₄⋊(C₃⋊C₈)	—	5	1	no	—
124	C₄⋊(C₃⋊C₈)	—	10	1	no	—
125	C₄⋊(C₃⋊C₈)	—	10	1	no	—
126	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
127	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
128	C₂×C₅×Q₈	—	2	1	no	—
129	#713	—	4	2	no	—
130	#713	—	4	2	no	—
131	C₂×D₂₀	—	2	2	no	—
132	C₂×D₂₀	—	2	1	no	—
133	C₂×D₂₀	—	2	1	no	—
134	C₂×D₂₀	C₂×Dic₃	1	2	yes	—
135	C₂×C₄×D₅	—	2	2	no	—
136	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
137	C₂×C₄×D₅	#67	1	1	yes	characteristic
138	C₂×C₄×D₅	#67	1	1	yes	characteristic
139	C₂×C₅×C₈	—	6	1	no	—
140	C₂×C₅×C₈	—	6	1	no	—
141	C₅×C₄⋊C₄	—	2	1	no	—
142	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
143	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
144	#685	—	2	2	no	—
145	#685	—	2	2	no	—
146	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
147	C₄×Dic₅	#67	1	1	yes	characteristic
148	C₄×Dic₅	#67	1	1	yes	characteristic
149	C₂×#168	—	6	1	no	—
150	C₂×#168	—	6	1	no	—
151	#676	—	12	2	no	—
152	C₈×D₅	—	12	2	no	—
153	#532	—	5	1	no	—
154	#423	—	15	1	no	—
155	C₂×C₄⋊C₈	—	15	1	no	—
156	#332	—	3	1	no	—
157	#332	—	15	1	no	—
158	#331	—	15	1	no	—
159	#331	—	15	1	no	—
160	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₃×D₅	—	2	1	no	—
162	C₂×C₃×D₅	—	2	1	no	—
163	C₂×C₃×D₅	—	4	2	no	—
164	C₂×C₃×D₅	—	4	2	no	—
165	C₂×C₃×D₅	—	4	2	no	—
166	C₂×C₃×D₅	#102	1	2	yes	—
167	C₃×C₄×C₅	—	2	1	no	—
168	C₃×C₄×C₅	—	4	2	no	—
169	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
170	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
171	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
172	C₃×C₄×C₅	#137	1	1	yes	characteristic
173	C₃×Dic₅	—	2	1	no	—
174	C₃×Dic₅	—	2	1	no	—
175	C₃×Dic₅	#102	1	2	yes	—
176	C₃×#41	—	20	2	no	—
177	C₃×#41	—	20	2	no	—
178	C₂×C₃×Q₈	—	2	1	no	—
179	C₂×C₃×D₄	—	5	2	no	—
180	C₂×C₃×D₄	—	10	2	no	—
181	C₂×C₃×D₄	—	10	1	no	—
182	C₂×C₃×D₄	—	10	1	no	—
183	C₂²×C₃×C₄	—	5	1	no	—
184	C₂²×C₃×C₄	—	5	1	no	—
185	C₂²×C₃×C₄	—	5	1	no	—
186	C₂²×C₃×C₄	—	10	2	no	—
187	C₃×C₄⋊C₄	—	2	1	no	—
188	C₃×C₄⋊C₄	C₄×D₅	1	2	yes	—
189	C₃×C₂²⋊C₄	—	10	2	no	—
190	C₃×C₂²⋊C₄	—	10	2	no	—
191	C₃×C₄²	—	5	1	no	—
192	C₃×C₄²	—	5	1	no	—
193	C₃×C₄²	—	5	1	no	—
194	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
195	#208	—	20	2	no	—
196	C₂×C₃⋊C₈	—	2	1	no	—
197	C₂×C₃⋊C₈	—	2	1	no	—
198	C₂×C₃⋊C₈	—	10	1	no	—
199	C₂×C₃⋊C₈	—	10	1	no	—
200	C₂×C₃⋊C₈	—	20	2	no	—
201	C₂²×D₅	—	2	2	no	—
202	C₂²×D₅	#217	1	1	yes	characteristic
203	C₅×Q₈	—	4	1	no	—
204	C₅×Q₈	—	4	1	no	—
205	C₂×C₄×C₅	—	2	2	no	—
206	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
207	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
208	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
209	C₂×Dic₅	#217	1	1	yes	characteristic
210	C₂×Dic₅	#217	1	1	yes	characteristic
211	C₂×Dic₅	#217	1	1	yes	characteristic
212	D₂₀	—	4	2	no	—
213	D₂₀	—	4	4	no	—
214	D₂₀	—	4	1	no	—
215	D₂₀	—	4	1	no	—
216	D₂₀	—	4	1	no	—
217	D₂₀	—	4	1	no	—
218	C₄×D₅	—	2	1	no	—
219	C₄×D₅	—	2	1	no	—
220	C₄×D₅	—	4	1	no	—
221	C₄×D₅	—	4	4	no	—
222	C₄×D₅	#213	1	2	yes	—
223	C₄×D₅	#214	1	2	yes	—
224	C₅×C₈	—	12	1	no	—
225	C₅×C₈	—	12	1	no	—
226	#168	—	12	1	no	—
227	#168	—	12	1	no	—
228	C₂×#41	—	10	1	no	—
229	C₂×M₁₆	—	30	1	no	—
230	C₂²×C₈	—	30	1	no	—
231	C₄⋊Q₈	D₃×D₅	1	1	yes	characteristic
232	#127	—	5	1	no	—
233	#124	—	10	1	no	—
234	#121	—	5	2	no	—
235	#121	—	10	1	no	—
236	C₄×D₄	—	5	2	no	—
237	C₄×D₄	—	10	1	no	—
238	C₂×C₄²	—	5	1	no	—
239	C₄⋊C₈	—	3	1	no	—
240	C₄⋊C₈	—	3	1	no	—
241	C₄⋊C₈	—	15	1	no	—
242	C₄⋊C₈	—	15	1	no	—
243	C₄⋊C₈	—	30	1	no	—
244	C₄⋊C₈	—	30	1	no	—
245	C₂×C₃×C₅	#413	1	1	yes	characteristic
246	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
247	C₂×C₃×C₅	#417	1	1	yes	characteristic
248	C₃×D₅	—	2	2	no	—
249	C₃×D₅	—	8	2	no	—
250	C₂³×C₃	—	5	1	no	—
251	C₂³×C₃	—	10	2	no	—
252	C₃×Q₈	—	4	1	no	—
253	C₃×Q₈	—	4	1	no	—
254	C₃×D₄	—	20	2	no	—
255	C₃×D₄	—	20	4	no	—
256	C₃×D₄	—	20	1	no	—
257	C₃×D₄	—	20	1	no	—
258	C₃×D₄	—	20	1	no	—
259	C₃×D₄	—	20	1	no	—
260	C₂×C₃×C₄	—	2	2	no	—
261	C₂×C₃×C₄	—	5	1	no	—
262	C₂×C₃×C₄	—	5	2	no	—
263	C₂×C₃×C₄	—	5	2	no	—
264	C₂×C₃×C₄	—	5	1	no	—
265	C₂×C₃×C₄	—	5	1	no	—
266	C₂×C₃×C₄	—	10	1	no	—
267	C₂×C₃×C₄	—	10	1	no	—
268	C₂×C₃×C₄	—	20	1	no	—
269	C₂×C₃×C₄	—	20	4	no	—
270	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
271	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
272	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
273	C₃⋊C₈	—	4	1	no	—
274	C₃⋊C₈	—	4	1	no	—
275	C₃⋊C₈	—	20	1	no	—
276	C₃⋊C₈	—	20	1	no	—
277	C₂²×C₅	C₂×#217	1	1	yes	characteristic
278	C₂×D₅	—	2	1	no	—
279	C₂×D₅	—	2	1	no	—
280	C₂×D₅	—	4	2	no	—
281	C₂×D₅	—	4	2	no	—
282	C₂×D₅	—	4	2	no	—
283	C₂×D₅	#832	1	2	yes	—
284	C₄×C₅	—	2	1	no	—
285	C₄×C₅	—	4	2	no	—
286	C₄×C₅	C₂×#213	1	1	yes	characteristic
287	C₄×C₅	C₂×#214	1	1	yes	characteristic
288	C₄×C₅	#949	1	1	yes	characteristic
289	C₄×C₅	#951	1	1	yes	characteristic
290	Dic₅	—	2	1	no	—
291	Dic₅	—	2	1	no	—
292	Dic₅	#832	1	2	yes	—
293	#41	—	20	2	no	—
294	#41	—	20	2	no	—
295	C₂×Q₈	—	2	1	no	—
296	C₂×D₄	—	5	2	no	—
297	C₂×D₄	—	10	2	no	—
298	C₂×D₄	—	10	1	no	—
299	C₂×D₄	—	10	1	no	—
300	C₂²×C₄	—	5	1	no	—
301	C₂²×C₄	—	5	1	no	—
302	C₂²×C₄	—	5	1	no	—
303	C₂²×C₄	—	10	2	no	—
304	M₁₆	—	60	2	no	—
305	C₂×C₈	—	6	1	no	—
306	C₂×C₈	—	6	1	no	—
307	C₂×C₈	—	30	1	no	—
308	C₂×C₈	—	30	1	no	—
309	C₂×C₈	—	60	2	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	D₅×Dic₃	1	2	yes	—
312	C₂²⋊C₄	—	10	2	no	—
313	C₂²⋊C₄	—	10	2	no	—
314	C₄²	—	5	1	no	—
315	C₄²	—	5	1	no	—
316	C₄²	—	5	1	no	—
317	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
318	C₃×C₅	#1547	1	1	yes	characteristic, complemented, Hall
319	C₂²×C₃	—	5	2	no	—
320	C₂²×C₃	—	10	1	no	—
321	C₂²×C₃	—	10	1	no	—
322	C₂²×C₃	—	20	2	no	—
323	C₂²×C₃	—	20	2	no	—
324	C₂²×C₃	—	20	2	no	—
325	C₂²×C₃	D₅×C₂²⋊C₄	1	1	yes	characteristic
326	C₃×C₄	—	2	1	no	—
327	C₃×C₄	—	4	2	no	—
328	C₃×C₄	—	5	2	no	—
329	C₃×C₄	—	10	1	no	—
330	C₃×C₄	—	10	1	no	—
331	C₃×C₄	D₅×D₈	1	1	yes	characteristic
332	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
333	C₃×C₄	#4060	1	1	yes	characteristic
334	C₃×C₄	#4064	1	1	yes	characteristic
335	C₂×C₅	C₂×#832	1	1	yes	characteristic
336	C₂×C₅	#5232	1	1	yes	characteristic
337	C₂×C₅	#5237	1	1	yes	characteristic
338	D₅	—	2	2	no	—
339	D₅	—	8	2	no	—
340	C₂³	—	5	1	no	—
341	C₂³	—	10	2	no	—
342	Q₈	—	4	1	no	—
343	Q₈	—	4	1	no	—
344	D₄	—	20	2	no	—
345	D₄	—	20	4	no	—
346	D₄	—	20	1	no	—
347	D₄	—	20	1	no	—
348	D₄	—	20	1	no	—
349	D₄	—	20	1	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	5	1	no	—
352	C₂×C₄	—	5	2	no	—
353	C₂×C₄	—	5	2	no	—
354	C₂×C₄	—	5	1	no	—
355	C₂×C₄	—	5	1	no	—
356	C₂×C₄	—	10	1	no	—
357	C₂×C₄	—	10	1	no	—
358	C₂×C₄	—	20	1	no	—
359	C₂×C₄	—	20	4	no	—
360	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
361	C₂×C₄	D₅×#67	1	1	yes	characteristic
362	C₂×C₄	D₅×#67	1	1	yes	characteristic
363	C₈	—	12	1	no	—
364	C₈	—	12	1	no	—
365	C₈	—	60	1	no	—
366	C₈	—	60	1	no	—
367	C₂×C₃	—	10	2	no	—
368	C₂×C₃	—	40	2	no	—
369	C₂×C₃	#65040	1	1	yes	characteristic
370	C₂×C₃	D₅×#102	1	1	yes	characteristic
371	C₂×C₃	#65092	1	1	yes	characteristic
372	C₅	#71806	1	1	yes	characteristic, complemented, Sylow
373	C₂²	—	5	2	no	—
374	C₂²	—	10	1	no	—
375	C₂²	—	10	1	no	—
376	C₂²	—	20	2	no	—
377	C₂²	—	20	2	no	—
378	C₂²	—	20	2	no	—
379	C₂²	D₅×#217	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	4	2	no	—
382	C₄	—	5	2	no	—
383	C₄	—	10	1	no	—
384	C₄	—	10	1	no	—
385	C₄	D₅×#213	1	1	yes	characteristic
386	C₄	D₅×#214	1	1	yes	characteristic
387	C₄	#91502	1	1	yes	characteristic
388	C₄	#91508	1	1	yes	characteristic
389	C₃	#10599214	1	1	yes	characteristic, complemented, Sylow
390	C₂	—	10	2	no	—
391	C₂	—	40	2	no	—
392	C₂	D₅×#832	1	1	yes	characteristic
393	C₂	#11749198	1	1	yes	characteristic
394	C₂	#11749369	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₅×#217
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊Q₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#66059	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊Q₈	D₃×D₅
hypocenter	C₃×C₅	#1547
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #11747007	C₂	semidirect
3. D₅×C₄⋊(C₃⋊C₈)	C₂	semidirect
4. #11746368	C₂	semidirect
5. C₅×#4547	C₂	semidirect
6. #11745852	C₂	semidirect
7. #11745852	C₂	semidirect
63. #4547	D₅	semidirect
318. C₃×C₅	#1547	semidirect
372. C₅	#71806	semidirect
389. C₃	#10599214	semidirect

Maximal subgroups

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

subgroup	index
C₃×#66059	2
#11747007	2
D₅×C₄⋊(C₃⋊C₈)	2
#11746368	2
C₅×#4547	2
#11745852	2
#10599214	3
#71806	5