Subgroups — #14282389

Order 1920.

552 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	C₅×#5477	C₂	1	1	yes	maximal, characteristic
2	#11751025	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745837	C₂	1	4	yes	maximal, complemented
5	#10602595	—	3	1	no	maximal, Hall
6	C₂×C₅×#235	—	2	1	no	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×#6710	—	2	1	no	—
9	C₅×#930	—	2	1	no	—
10	C₅×#930	C₄	1	2	yes	—
11	C₅×#875	—	2	1	no	—
12	C₅×#874	C₄	1	1	yes	characteristic
13	C₄×C₅×D₁₂	—	2	1	no	—
14	C₄×C₅×D₁₂	C₂²	1	2	yes	—
15	C₅×#869	C₄	1	1	yes	characteristic
16	C₂²×C₃×#168	—	2	1	no	—
17	C₄⋊(C₃⋊#168)	—	2	1	no	—
18	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	—
19	C₃×C₄×#168	—	2	1	no	—
20	C₃×C₄×#168	C₂²	1	2	yes	—
21	#69735	—	5	1	no	maximal, Hall
22	C₅×#532	—	3	1	no	—
23	#65213	—	3	1	no	—
24	C₂×C₄×#168	D₃	1	1	yes	characteristic, complemented
25	D₄⋊#168	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂×C₅×#67	—	2	2	no	—
30	C₅×#235	—	4	2	no	—
31	C₅×#235	—	4	2	no	—
32	C₂×C₅×D₁₂	—	2	1	no	—
33	C₂×C₅×D₁₂	—	2	1	no	—
34	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₅×D₃	—	2	2	no	—
36	C₂×C₅×Dic₆	—	2	1	no	—
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	#6710	—	4	2	no	—
42	C₂×C₃⋊#168	—	2	2	no	—
43	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
44	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
45	C₅×C₄⋊Dic₃	—	2	1	no	—
46	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
47	C₂×C₃×#168	—	2	4	no	—
48	C₂×C₃×#168	—	2	2	no	—
49	#5477	D₅	1	1	yes	characteristic
50	C₂×C₃×C₄×C₈	—	5	1	no	—
51	#4924	—	5	1	no	—
52	#4459	—	5	4	no	—
53	C₂×C₅×#41	—	6	1	no	—
54	C₅×C₄⋊Q₈	—	3	1	no	—
55	C₅×#127	—	3	1	no	—
56	C₅×#124	—	6	1	no	—
57	C₅×#121	—	3	2	no	—
58	C₅×#121	—	6	1	no	—
59	C₄×C₅×D₄	—	3	2	no	—
60	C₄×C₅×D₄	—	6	1	no	—
61	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
62	C₂×#681	—	6	1	no	—
63	C₂²×#168	—	2	1	no	—
64	C₄⋊#168	—	3	2	no	—
65	C₄⋊#168	—	6	1	no	—
66	C₄×#168	—	2	1	no	—
67	C₄×#168	C₂×D₃	1	2	yes	—
68	#1492	—	15	1	no	Sylow
69	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₅×D₃	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
77	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₅×#67	—	4	4	no	—
82	C₂×C₅×Dic₃	—	2	2	no	—
83	C₅×D₁₂	—	2	4	no	—
84	C₅×D₁₂	—	4	2	no	—
85	C₅×D₁₂	—	4	2	no	—
86	C₄×C₅×D₃	—	4	2	no	—
87	C₄×C₅×D₃	—	4	2	no	—
88	C₅×Dic₆	—	4	2	no	—
89	C₃⋊#168	—	4	2	no	—
90	C₃×#168	—	2	4	no	—
91	C₂×#235	—	2	1	no	—
92	C₂²×C₃×C₈	—	10	1	no	—
93	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
94	#930	—	2	1	no	—
95	#930	Dic₅	1	2	yes	—
96	C₂×#208	—	10	1	no	—
97	#875	—	2	1	no	—
98	#874	Dic₅	1	1	yes	characteristic
99	C₄×D₁₂	—	2	1	no	—
100	C₄×D₁₂	C₂×D₅	1	2	yes	—
101	#869	Dic₅	1	1	yes	characteristic
102	C₃×C₄×C₈	—	5	2	no	—
103	C₃×C₄×C₈	—	10	1	no	—
104	C₄⋊(C₃⋊C₈)	—	5	2	no	—
105	C₄⋊(C₃⋊C₈)	—	10	1	no	—
106	C₅×#41	—	12	2	no	—
107	C₅×#41	—	12	2	no	—
108	C₂×C₅×Q₈	—	6	1	no	—
109	C₂×C₅×D₄	—	3	2	no	—
110	C₂×C₅×D₄	—	6	2	no	—
111	C₂×C₅×D₄	—	6	1	no	—
112	C₂×C₅×D₄	—	6	1	no	—
113	C₂²×C₄×C₅	—	6	2	no	—
114	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
115	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
116	C₂²×C₄×C₅	#67	1	1	yes	characteristic
117	C₅×C₄⋊C₄	—	3	2	no	—
118	C₅×C₄⋊C₄	—	6	1	no	—
119	C₅×C₂²⋊C₄	—	6	2	no	—
120	C₅×C₂²⋊C₄	—	6	2	no	—
121	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
122	C₄²×C₅	D₁₂	1	1	yes	characteristic
123	C₄²×C₅	#67	1	1	yes	characteristic
124	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
125	#681	—	12	2	no	—
126	C₂×#168	—	2	4	no	—
127	C₂×#168	—	2	2	no	—
128	C₂×#168	—	6	2	no	—
129	#532	—	3	1	no	—
130	#423	—	15	1	no	—
131	C₂×C₄×C₈	—	5	1	no	—
132	D₄⋊C₈	—	15	4	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	—	2	1	no	—
135	C₂²×C₃×C₅	#102	1	2	yes	—
136	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₅×D₃	—	4	2	no	—
138	C₂×C₅×D₃	—	4	2	no	—
139	C₂×C₅×D₃	—	4	2	no	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	—	2	1	no	—
142	C₃×C₄×C₅	#102	1	2	yes	—
143	C₃×C₄×C₅	#104	1	2	yes	—
144	C₃×C₄×C₅	#131	1	2	yes	—
145	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
146	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
147	C₅×Dic₃	—	4	2	no	—
148	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
149	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
150	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
151	C₂×#67	—	2	2	no	—
152	#235	—	4	2	no	—
153	#235	—	4	2	no	—
154	C₂×D₁₂	—	2	1	no	—
155	C₂×D₁₂	—	2	1	no	—
156	C₂×D₁₂	C₂×Dic₅	1	2	yes	—
157	C₂×C₄×D₃	—	2	2	no	—
158	C₂×Dic₆	—	2	1	no	—
159	C₂×C₃×C₈	—	10	4	no	—
160	C₂×C₃×C₈	—	10	2	no	—
161	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
162	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
163	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
164	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
165	(C₂×D₃)⋊C₄	—	2	2	no	—
166	(C₂×D₃)⋊C₄	—	2	2	no	—
167	C₄⋊Dic₃	—	2	1	no	—
168	C₄⋊Dic₃	C₂×Dic₅	1	2	yes	—
169	#208	—	20	2	no	—
170	C₂×C₃⋊C₈	—	10	2	no	—
171	C₂³×C₅	—	6	2	no	—
172	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
173	C₅×Q₈	—	12	2	no	—
174	C₅×D₄	—	6	4	no	—
175	C₅×D₄	—	12	4	no	—
176	C₅×D₄	—	12	2	no	—
177	C₅×D₄	—	12	2	no	—
178	C₂×C₄×C₅	—	2	2	no	—
179	C₂×C₄×C₅	—	2	2	no	—
180	C₂×C₄×C₅	—	6	2	no	—
181	C₂×C₄×C₅	—	12	2	no	—
182	C₂×C₄×C₅	—	12	2	no	—
183	C₂×C₄×C₅	#204	1	2	yes	—
184	C₂×C₄×C₅	D₂₄	1	2	yes	—
185	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
186	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
187	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
191	#168	—	2	4	no	—
192	#168	—	12	2	no	—
193	C₂×#41	—	6	1	no	—
194	C₂×M₁₆	—	30	1	no	—
195	C₂²×C₈	—	10	1	no	—
196	C₄⋊Q₈	—	3	1	no	—
197	#127	—	3	1	no	—
198	#124	—	6	1	no	—
199	#121	—	3	2	no	—
200	#121	—	6	1	no	—
201	C₄×D₄	—	3	2	no	—
202	C₄×D₄	—	6	1	no	—
203	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
204	C₄⋊C₈	—	15	2	no	—
205	C₄⋊C₈	—	30	1	no	—
206	C₄×C₈	—	5	2	no	—
207	C₄×C₈	—	10	1	no	—
208	C₂×C₃×C₅	—	2	2	no	—
209	C₂×C₃×C₅	#408	1	1	yes	characteristic
210	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
211	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
212	C₅×D₃	—	8	2	no	—
213	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
214	C₂²×D₃	—	2	2	no	—
215	C₂×C₃×C₄	—	2	2	no	—
216	C₂×C₃×C₄	—	2	2	no	—
217	C₂×C₃×C₄	#686	1	2	yes	—
218	C₂×C₃×C₄	#687	1	2	yes	—
219	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
225	#67	—	4	4	no	—
226	C₂×Dic₃	—	2	2	no	—
227	D₁₂	—	2	4	no	—
228	D₁₂	—	4	2	no	—
229	D₁₂	—	4	2	no	—
230	C₄×D₃	—	4	2	no	—
231	C₄×D₃	—	4	2	no	—
232	Dic₆	—	4	2	no	—
233	C₃×C₈	—	10	4	no	—
234	C₃⋊C₈	—	20	2	no	—
235	C₂²×C₅	—	2	1	no	—
236	C₂²×C₅	—	2	1	no	—
237	C₂²×C₅	—	12	2	no	—
238	C₂²×C₅	—	12	2	no	—
239	C₂²×C₅	—	12	2	no	—
240	C₂²×C₅	#821	1	2	yes	—
241	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
242	C₄×C₅	—	2	1	no	—
243	C₄×C₅	—	2	1	no	—
244	C₄×C₅	—	12	2	no	—
245	C₄×C₅	#805	1	2	yes	—
246	C₄×C₅	#821	1	2	yes	—
247	C₄×C₅	#901	1	2	yes	—
248	C₄×C₅	C₂×#204	1	1	yes	characteristic
249	C₄×C₅	C₂×D₂₄	1	1	yes	characteristic
250	#41	—	12	2	no	—
251	#41	—	12	2	no	—
252	C₂×Q₈	—	6	1	no	—
253	C₂×D₄	—	3	2	no	—
254	C₂×D₄	—	6	2	no	—
255	C₂×D₄	—	6	1	no	—
256	C₂×D₄	—	6	1	no	—
257	C₂²×C₄	—	6	2	no	—
258	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
259	C₂²×C₄	#1201	1	1	yes	characteristic
260	C₂²×C₄	#1203	1	1	yes	characteristic
261	M₁₆	—	60	2	no	—
262	C₂×C₈	—	10	4	no	—
263	C₂×C₈	—	10	2	no	—
264	C₂×C₈	—	30	2	no	—
265	C₄⋊C₄	—	3	2	no	—
266	C₄⋊C₄	—	6	1	no	—
267	C₂²⋊C₄	—	6	2	no	—
268	C₂²⋊C₄	—	6	2	no	—
269	C₄²	D₃×Dic₅	1	1	yes	characteristic
270	C₄²	#1201	1	1	yes	characteristic
271	C₄²	#1203	1	1	yes	characteristic
272	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
273	C₃×C₅	#1492	1	1	yes	characteristic, complemented, Hall
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	#3964	1	2	yes	—
277	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
278	C₂×D₃	—	4	2	no	—
279	C₂×D₃	—	4	2	no	—
280	C₂×D₃	—	4	2	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	#3964	1	2	yes	—
284	C₃×C₄	#3969	1	2	yes	—
285	C₃×C₄	C₂×#686	1	1	yes	characteristic
286	C₃×C₄	C₂×#687	1	1	yes	characteristic
287	C₃×C₄	#4094	1	2	yes	—
288	Dic₃	—	4	2	no	—
289	C₂×C₅	—	2	2	no	—
290	C₂×C₅	—	24	2	no	—
291	C₂×C₅	C₂×#805	1	1	yes	characteristic
292	C₂×C₅	#5110	1	1	yes	characteristic
293	C₂×C₅	C₂×#821	1	1	yes	characteristic
294	C₂³	—	6	2	no	—
295	C₂³	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
296	Q₈	—	12	2	no	—
297	D₄	—	6	4	no	—
298	D₄	—	12	4	no	—
299	D₄	—	12	2	no	—
300	D₄	—	12	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	6	2	no	—
304	C₂×C₄	—	12	2	no	—
305	C₂×C₄	—	12	2	no	—
306	C₂×C₄	#6654	1	2	yes	—
307	C₂×C₄	#6659	1	2	yes	—
308	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
309	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
310	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
311	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
312	C₂×C₄	C₂×#1201	1	1	yes	characteristic
313	C₂×C₄	C₂×#1203	1	1	yes	characteristic
314	C₈	—	10	4	no	—
315	C₈	—	60	2	no	—
316	C₂×C₃	—	2	2	no	—
317	C₂×C₃	C₂×#3964	1	1	yes	characteristic
318	C₂×C₃	#65523	1	1	yes	characteristic
319	C₂×C₃	C₂×#3969	1	1	yes	characteristic
320	D₃	—	8	2	no	—
321	C₅	#69735	1	1	yes	characteristic, complemented, Sylow
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	12	2	no	—
325	C₂²	—	12	2	no	—
326	C₂²	—	12	2	no	—
327	C₂²	#90967	1	2	yes	—
328	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	12	2	no	—
332	C₄	#90967	1	2	yes	—
333	C₄	#90979	1	2	yes	—
334	C₄	#91286	1	2	yes	—
335	C₄	C₂×#6654	1	1	yes	characteristic
336	C₄	C₂×#6659	1	1	yes	characteristic
337	C₃	#10602595	1	1	yes	characteristic, complemented, Sylow
338	C₂	—	2	2	no	—
339	C₂	—	24	2	no	—
340	C₂	#11748345	1	1	yes	characteristic
341	C₂	C₂×#90967	1	1	yes	characteristic
342	C₂	C₂×#90979	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1203
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#5477	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1492
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #11751025	C₂	semidirect
3. C₂×C₃×C₄×#168	C₂	semidirect
4. #11745837	C₂	semidirect
24. C₂×C₄×#168	D₃	semidirect
273. C₃×C₅	#1492	semidirect
321. C₅	#69735	semidirect
337. C₃	#10602595	semidirect

Maximal subgroups

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

subgroup	index
C₅×#5477	2
#11751025	2
C₂×C₃×C₄×#168	2
#11745837	2
#10602595	3
#69735	5