Subgroups — #14312872

Order 1920.

456 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#4047	C₂	1	1	yes	maximal, characteristic
2	C₂×#90934	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#11746370	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4549	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745879	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745879	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599900	—	3	1	no	maximal, Hall
9	C₂×C₃×#709	C₄	1	1	yes	characteristic
10	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×#681	C₄	1	1	yes	characteristic
12	C₃×#4047	—	2	2	no	—
13	C₃×#4047	—	2	2	no	—
14	C₂×C₃×#676	C₄	1	1	yes	characteristic
15	C₂×C₃×#676	C₄	1	1	yes	characteristic
16	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
18	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
20	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
22	#90934	—	2	2	no	—
23	D₅×C₃⋊C₁₆	—	2	2	no	—
24	#72004	—	5	1	no	maximal, Hall
25	C₂×#4047	D₃	1	1	yes	characteristic
26	C₂×#3930	—	3	1	no	—
27	C₂×D₅×C₁₆	—	3	1	no	—
28	C₅×C₂²⋊C₁₆	—	3	1	no	—
29	C₂²⋊#672	—	3	1	no	—
30	#64727	—	3	1	no	—
31	#64727	—	3	1	no	—
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₂²⋊D₅	C₈	1	1	yes	characteristic
34	C₂×C₃×C₂²⋊D₅	C₈	1	1	yes	characteristic
35	C₃×#709	—	2	2	no	—
36	C₃×#709	—	2	2	no	—
37	C₂×C₃×D₂₀	C₈	1	1	yes	characteristic
38	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×Dic₁₀	C₈	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	—	2	1	no	—
42	C₂×C₃×C₅×C₈	—	2	1	no	—
43	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₃×#681	—	2	2	no	—
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
48	C₃×#676	—	2	2	no	—
49	C₃×#676	—	2	2	no	—
50	C₃×C₈×D₅	—	2	1	no	—
51	C₃×C₈×D₅	—	2	1	no	—
52	C₃×C₈×D₅	D₄	1	2	yes	—
53	C₃×C₈×D₅	D₄	1	2	yes	—
54	C₃⋊#672	—	2	1	no	—
55	C₃⋊#672	—	2	1	no	—
56	C₅×C₃⋊C₁₆	—	2	1	no	—
57	C₅×C₃⋊C₁₆	—	2	1	no	—
58	C₂×C₃×#131	—	5	1	no	—
59	C₂×#812	—	5	1	no	—
60	C₂²×C₃⋊C₁₆	—	5	1	no	—
61	#4549	—	5	1	no	—
62	#4549	—	5	1	no	—
63	#4549	—	5	1	no	—
64	#4549	D₅	1	1	yes	characteristic, complemented
65	C₂×#709	Dic₃	1	1	yes	characteristic
66	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
67	C₂×#681	Dic₃	1	1	yes	characteristic
68	#4047	—	2	2	no	—
69	#4047	—	2	2	no	—
70	C₂×#676	Dic₃	1	1	yes	characteristic
71	C₂×#676	Dic₃	1	1	yes	characteristic
72	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
73	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
74	C₂×C₅×C₁₆	—	3	1	no	—
75	C₂×C₅×C₁₆	—	3	1	no	—
76	C₂×#672	—	3	1	no	—
77	C₂×#672	—	3	1	no	—
78	#3930	—	6	2	no	—
79	D₅×C₁₆	—	6	2	no	—
80	#2113	—	15	1	no	Sylow
81	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	—	2	1	no	—
85	C₂×C₃×C₄×C₅	—	2	1	no	—
86	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	C₃×C₂²⋊D₅	—	2	2	no	—
89	C₃×C₂²⋊D₅	—	2	2	no	—
90	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
91	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
92	C₃×D₂₀	—	2	2	no	—
93	C₃×C₄×D₅	—	2	1	no	—
94	C₃×C₄×D₅	—	2	1	no	—
95	C₃×C₄×D₅	C₂²⋊C₄	1	2	yes	—
96	C₃×C₄×D₅	M₁₆	1	2	yes	—
97	C₃×Dic₁₀	—	2	2	no	—
98	C₃×C₅×C₈	—	2	1	no	—
99	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
100	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
101	C₃×#168	—	2	1	no	—
102	C₃×#168	C₂²⋊C₄	1	2	yes	—
103	C₂×C₃×#41	—	5	1	no	—
104	C₃×#131	—	10	2	no	—
105	C₃×#131	—	10	2	no	—
106	C₂×C₃×M₁₆	—	5	1	no	—
107	C₂×C₃×M₁₆	—	5	1	no	—
108	C₂×C₃×M₁₆	—	5	1	no	—
109	C₂²×C₃×C₈	—	5	1	no	—
110	C₂²×C₃×C₈	—	5	1	no	—
111	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
112	#812	—	10	2	no	—
113	C₂×C₃⋊C₁₆	—	5	1	no	—
114	C₂×C₃⋊C₁₆	—	5	1	no	—
115	C₂×C₃⋊C₁₆	—	10	2	no	—
116	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic
117	C₂×C₃⋊C₁₆	C₂×D₅	1	1	yes	characteristic
118	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
119	C₂×C₂²⋊D₅	C₃⋊C₈	1	1	yes	characteristic
120	C₂×C₂²⋊D₅	C₃⋊C₈	1	1	yes	characteristic
121	#709	—	2	2	no	—
122	#709	—	2	2	no	—
123	C₂×D₂₀	C₃⋊C₈	1	1	yes	characteristic
124	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
125	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
126	C₂×Dic₁₀	C₃⋊C₈	1	1	yes	characteristic
127	C₂×C₅×C₈	—	2	1	no	—
128	C₂×C₅×C₈	—	2	1	no	—
129	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
130	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
131	#681	—	2	2	no	—
132	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
133	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
134	#676	—	2	2	no	—
135	#676	—	2	2	no	—
136	C₈×D₅	—	2	1	no	—
137	C₈×D₅	—	2	1	no	—
138	C₈×D₅	#67	1	2	yes	—
139	C₈×D₅	#67	1	2	yes	—
140	C₅×C₁₆	—	6	1	no	—
141	C₅×C₁₆	—	6	1	no	—
142	#672	—	6	1	no	—
143	#672	—	6	1	no	—
144	C₂×#131	—	5	1	no	—
145	C₂×M₃₂	—	15	1	no	—
146	C₂²×C₁₆	—	15	1	no	—
147	C₂²⋊C₁₆	—	3	1	no	—
148	C₂²⋊C₁₆	—	15	1	no	—
149	C₂²⋊C₁₆	—	15	1	no	—
150	C₂²⋊C₁₆	—	15	1	no	—
151	C₂²×C₃×C₅	—	2	1	no	—
152	C₂²×C₃×C₅	—	2	1	no	—
153	C₂²×C₃×C₅	—	2	1	no	—
154	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
155	C₂×C₃×D₅	—	2	1	no	—
156	C₂×C₃×D₅	—	2	1	no	—
157	C₂×C₃×D₅	—	2	1	no	—
158	C₂×C₃×D₅	—	2	1	no	—
159	C₂×C₃×D₅	—	2	1	no	—
160	C₂×C₃×D₅	C₂²⋊C₈	1	2	yes	—
161	C₃×C₄×C₅	—	2	1	no	—
162	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
164	C₃×Dic₅	—	2	1	no	—
165	C₃×Dic₅	C₂²⋊C₈	1	2	yes	—
166	C₃×#41	—	10	2	no	—
167	C₃×#41	—	10	2	no	—
168	C₂×C₃×Q₈	—	5	1	no	—
169	C₂×C₃×D₄	—	5	1	no	—
170	C₂×C₃×D₄	—	5	1	no	—
171	C₂×C₃×D₄	—	5	1	no	—
172	C₂²×C₃×C₄	—	5	1	no	—
173	C₂²×C₃×C₄	—	5	1	no	—
174	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
175	C₃×M₁₆	—	10	2	no	—
176	C₃×M₁₆	—	10	2	no	—
177	C₃×M₁₆	—	10	2	no	—
178	C₂×C₃×C₈	—	2	1	no	—
179	C₂×C₃×C₈	—	2	1	no	—
180	C₂×C₃×C₈	—	5	1	no	—
181	C₂×C₃×C₈	—	5	2	no	—
182	C₂×C₃×C₈	—	5	2	no	—
183	C₂×C₃×C₈	—	5	1	no	—
184	C₂×C₃×C₈	—	10	1	no	—
185	C₂×C₃×C₈	—	10	1	no	—
186	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
187	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
188	C₃⋊C₁₆	—	2	1	no	—
189	C₃⋊C₁₆	—	2	1	no	—
190	C₃⋊C₁₆	—	10	1	no	—
191	C₃⋊C₁₆	—	10	1	no	—
192	C₂³×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
193	C₂²×D₅	C₂×C₃⋊C₈	1	1	yes	characteristic
194	C₂²×D₅	C₂×C₃⋊C₈	1	1	yes	characteristic
195	C₂×C₄×C₅	—	2	1	no	—
196	C₂×C₄×C₅	—	2	1	no	—
197	C₂×C₄×C₅	C₂×C₃⋊C₈	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
199	C₂²⋊D₅	—	2	2	no	—
200	C₂²⋊D₅	—	2	2	no	—
201	C₂×Dic₅	C₂×C₃⋊C₈	1	1	yes	characteristic
202	C₂×Dic₅	C₂×C₃⋊C₈	1	1	yes	characteristic
203	D₂₀	—	2	2	no	—
204	C₄×D₅	—	2	1	no	—
205	C₄×D₅	—	2	1	no	—
206	C₄×D₅	#208	1	2	yes	—
207	C₄×D₅	#217	1	2	yes	—
208	Dic₁₀	—	2	2	no	—
209	C₅×C₈	—	2	1	no	—
210	C₅×C₈	C₂×#67	1	1	yes	characteristic
211	C₅×C₈	C₂×#67	1	1	yes	characteristic
212	#168	—	2	1	no	—
213	#168	#217	1	2	yes	—
214	C₂×#41	—	5	1	no	—
215	#131	—	10	2	no	—
216	#131	—	10	2	no	—
217	C₂×M₁₆	—	5	1	no	—
218	C₂×M₁₆	—	5	1	no	—
219	C₂×M₁₆	—	5	1	no	—
220	C₂²×C₈	—	5	1	no	—
221	C₂²×C₈	—	5	1	no	—
222	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
223	M₃₂	—	30	2	no	—
224	C₂×C₁₆	—	3	1	no	—
225	C₂×C₁₆	—	3	1	no	—
226	C₂×C₁₆	—	15	1	no	—
227	C₂×C₁₆	—	15	1	no	—
228	C₂×C₁₆	—	30	2	no	—
229	C₂×C₃×C₅	—	4	1	no	—
230	C₂×C₃×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
231	C₂×C₃×C₅	#504	1	1	yes	characteristic
232	C₂×C₃×C₅	#504	1	1	yes	characteristic
233	C₃×D₅	—	2	2	no	—
234	C₃×D₅	—	4	1	no	—
235	C₂³×C₃	—	5	1	no	—
236	C₂³×C₃	—	5	1	no	—
237	C₂³×C₃	C₈×D₅	1	1	yes	characteristic
238	C₃×Q₈	—	10	2	no	—
239	C₃×D₄	—	10	2	no	—
240	C₃×D₄	—	10	2	no	—
241	C₃×D₄	—	10	2	no	—
242	C₂×C₃×C₄	—	2	1	no	—
243	C₂×C₃×C₄	—	2	1	no	—
244	C₂×C₃×C₄	—	5	1	no	—
245	C₂×C₃×C₄	—	5	2	no	—
246	C₂×C₃×C₄	—	5	2	no	—
247	C₂×C₃×C₄	—	5	1	no	—
248	C₂×C₃×C₄	—	10	1	no	—
249	C₂×C₃×C₄	—	10	1	no	—
250	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
251	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
252	C₃×C₈	—	2	1	no	—
253	C₃×C₈	—	5	2	no	—
254	C₃×C₈	—	10	1	no	—
255	C₃×C₈	D₄×D₅	1	1	yes	characteristic
256	C₃×C₈	D₄×D₅	1	1	yes	characteristic
257	C₂²×C₅	—	2	1	no	—
258	C₂²×C₅	—	2	1	no	—
259	C₂²×C₅	—	2	1	no	—
260	C₂²×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
261	C₂×D₅	—	2	1	no	—
262	C₂×D₅	—	2	1	no	—
263	C₂×D₅	—	2	1	no	—
264	C₂×D₅	—	2	1	no	—
265	C₂×D₅	—	2	1	no	—
266	C₂×D₅	#830	1	2	yes	—
267	C₄×C₅	—	2	1	no	—
268	C₄×C₅	C₂×#208	1	1	yes	characteristic
269	C₄×C₅	C₂×#217	1	1	yes	characteristic
270	Dic₅	—	2	1	no	—
271	Dic₅	#830	1	2	yes	—
272	#41	—	10	2	no	—
273	#41	—	10	2	no	—
274	C₂×Q₈	—	5	1	no	—
275	C₂×D₄	—	5	1	no	—
276	C₂×D₄	—	5	1	no	—
277	C₂×D₄	—	5	1	no	—
278	C₂²×C₄	—	5	1	no	—
279	C₂²×C₄	—	5	1	no	—
280	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
281	M₁₆	—	10	2	no	—
282	M₁₆	—	10	2	no	—
283	M₁₆	—	10	2	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	5	1	no	—
287	C₂×C₈	—	5	2	no	—
288	C₂×C₈	—	5	2	no	—
289	C₂×C₈	—	5	1	no	—
290	C₂×C₈	—	10	1	no	—
291	C₂×C₈	—	10	1	no	—
292	C₂×C₈	D₅×Dic₃	1	1	yes	characteristic
293	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
294	C₁₆	—	6	1	no	—
295	C₁₆	—	6	1	no	—
296	C₁₆	—	30	1	no	—
297	C₁₆	—	30	1	no	—
298	C₃×C₅	#2113	1	1	yes	characteristic, complemented, Hall
299	C₂²×C₃	—	2	1	no	—
300	C₂²×C₃	—	2	1	no	—
301	C₂²×C₃	—	2	1	no	—
302	C₂²×C₃	—	5	2	no	—
303	C₂²×C₃	—	10	1	no	—
304	C₂²×C₃	—	10	1	no	—
305	C₂²×C₃	—	10	1	no	—
306	C₂²×C₃	—	10	1	no	—
307	C₂²×C₃	—	10	1	no	—
308	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
309	C₃×C₄	—	2	1	no	—
310	C₃×C₄	—	5	2	no	—
311	C₃×C₄	—	10	1	no	—
312	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
313	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
314	C₂×C₅	—	4	1	no	—
315	C₂×C₅	#5140	1	1	yes	characteristic
316	C₂×C₅	#5140	1	1	yes	characteristic
317	C₂×C₅	C₂×#830	1	1	yes	characteristic
318	D₅	—	2	2	no	—
319	D₅	—	4	1	no	—
320	C₂³	—	5	1	no	—
321	C₂³	—	5	1	no	—
322	C₂³	D₅×C₃⋊C₈	1	1	yes	characteristic
323	Q₈	—	10	2	no	—
324	D₄	—	10	2	no	—
325	D₄	—	10	2	no	—
326	D₄	—	10	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	5	1	no	—
330	C₂×C₄	—	5	2	no	—
331	C₂×C₄	—	5	2	no	—
332	C₂×C₄	—	5	1	no	—
333	C₂×C₄	—	10	1	no	—
334	C₂×C₄	—	10	1	no	—
335	C₂×C₄	D₅×C₃⋊C₈	1	1	yes	characteristic
336	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
337	C₈	—	2	1	no	—
338	C₈	—	5	2	no	—
339	C₈	—	10	1	no	—
340	C₈	D₅×#67	1	1	yes	characteristic
341	C₈	D₅×#67	1	1	yes	characteristic
342	C₂×C₃	—	4	1	no	—
343	C₂×C₃	—	10	2	no	—
344	C₂×C₃	—	20	1	no	—
345	C₂×C₃	D₅×C₂²⋊C₈	1	1	yes	characteristic
346	C₂×C₃	#65190	1	1	yes	characteristic
347	C₂×C₃	#65196	1	1	yes	characteristic
348	C₅	#72004	1	1	yes	characteristic, complemented, Sylow
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	5	2	no	—
353	C₂²	—	10	1	no	—
354	C₂²	—	10	1	no	—
355	C₂²	—	10	1	no	—
356	C₂²	—	10	1	no	—
357	C₂²	—	10	1	no	—
358	C₂²	C₂×D₅×C₃⋊C₈	1	1	yes	characteristic
359	C₄	—	2	1	no	—
360	C₄	—	5	2	no	—
361	C₄	—	10	1	no	—
362	C₄	D₅×#208	1	1	yes	characteristic
363	C₄	D₅×#217	1	1	yes	characteristic
364	C₃	#10599900	1	1	yes	characteristic, complemented, Sylow
365	C₂	—	4	1	no	—
366	C₂	—	10	2	no	—
367	C₂	—	20	1	no	—
368	C₂	#11747524	1	1	yes	characteristic
369	C₂	#11747525	1	1	yes	characteristic
370	C₂	D₅×#830	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂²×C₃×C₅×C₈	C₂²
prime radical	C₂×C₃×#709	C₄
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#2113
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#90934	C₂	semidirect
3. C₂×D₅×C₃⋊C₁₆	C₂	semidirect
4. #11746370	C₂	semidirect
5. C₅×#4549	C₂	semidirect
6. #11745879	C₂	semidirect
7. #11745879	C₂	semidirect
64. #4549	D₅	semidirect
298. C₃×C₅	#2113	semidirect
348. C₅	#72004	semidirect
364. C₃	#10599900	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#4047	2
C₂×#90934	2
C₂×D₅×C₃⋊C₁₆	2
#11746370	2
C₅×#4549	2
#11745879	2
#10599900	3
#72004	5