Subgroups — #14283047

Order 1920.

544 conjugacy classes of subgroups.

The subgroup lattice has 336 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	#11751027	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65221	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745956	C₂	1	4	yes	maximal, complemented
5	#10602605	—	3	1	no	maximal, Hall
6	C₂×C₅×#235	—	2	1	no	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₅×#930	—	2	1	no	—
9	C₅×#930	C₂²	1	2	yes	—
10	C₅×#875	—	2	1	no	—
11	C₅×#874	C₄	1	1	yes	characteristic, complemented
12	C₄×C₅×D₁₂	—	2	1	no	—
13	C₄×C₅×D₁₂	C₄	1	2	yes	complemented
14	C₅×#869	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃×C₄×Dic₅	—	2	1	no	—
16	#91114	C₂²	1	2	yes	—
17	C₄⋊(C₃⋊#168)	—	2	1	no	—
18	#91087	—	2	1	no	—
19	C₃×#3963	—	2	1	no	—
20	C₃×#3963	C₂²	1	2	yes	—
21	#69432	—	5	1	no	maximal, Hall
22	C₅×#532	—	3	1	no	—
23	#65221	D₃	1	1	yes	characteristic, complemented
24	#65215	—	3	1	no	—
25	#64757	—	3	4	no	—
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₅×#67	—	2	2	no	—
30	C₅×#235	—	4	4	no	—
31	C₂×C₅×D₁₂	—	2	1	no	—
32	C₂×C₅×D₁₂	—	2	1	no	—
33	C₂×C₅×D₁₂	C₂×C₄	1	2	yes	—
34	C₂×C₄×C₅×D₃	—	2	2	no	—
35	C₂×C₅×Dic₆	—	2	1	no	—
36	C₂²×C₃×Dic₅	—	2	2	no	—
37	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×C₃⋊#168	—	2	2	no	—
42	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
43	C₅×(C₂×D₃)⋊C₄	—	2	2	no	—
44	C₅×C₄⋊Dic₃	—	2	1	no	—
45	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
46	C₃×C₄×Dic₅	—	2	4	no	—
47	#5477	D₅	1	1	yes	characteristic
48	C₃×#376	—	5	1	no	—
49	#4926	—	5	1	no	—
50	#4451	—	5	4	no	—
51	C₂×C₅×#41	—	6	1	no	—
52	C₅×C₄⋊Q₈	—	3	1	no	—
53	C₅×#127	—	3	1	no	—
54	C₅×#124	—	6	1	no	—
55	C₅×#121	—	3	2	no	—
56	C₅×#121	—	6	1	no	—
57	C₄×C₅×D₄	—	3	2	no	—
58	C₄×C₅×D₄	—	6	1	no	—
59	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
60	C₂×C₄×Dic₅	—	2	1	no	—
61	#3963	—	2	1	no	—
62	#3963	C₂×D₃	1	2	yes	—
63	C₂²⋊#168	—	3	2	no	—
64	C₄⋊#168	—	6	1	no	—
65	#3935	—	6	1	no	—
66	#1505	—	15	1	no	Sylow
67	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₅×D₃	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	4	1	no	—
72	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
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	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₅×#67	—	4	4	no	—
79	C₂×C₅×Dic₃	—	2	2	no	—
80	C₅×D₁₂	—	4	2	no	—
81	C₅×D₁₂	—	4	2	no	—
82	C₅×D₁₂	—	4	2	no	—
83	C₄×C₅×D₃	—	4	4	no	—
84	C₅×Dic₆	—	4	2	no	—
85	C₂×C₃×Dic₅	—	2	4	no	—
86	C₂×C₃×Dic₅	—	4	2	no	—
87	C₂×C₃×Dic₅	—	4	2	no	—
88	C₃⋊#168	—	4	2	no	—
89	C₂×#235	—	2	1	no	—
90	C₂×C₃×C₄²	—	10	1	no	—
91	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
92	#930	—	2	1	no	—
93	#930	C₂×D₅	1	2	yes	—
94	#875	—	2	1	no	—
95	#874	Dic₅	1	1	yes	characteristic, complemented
96	C₄×D₁₂	—	2	1	no	—
97	C₄×D₁₂	Dic₅	1	2	yes	complemented
98	#869	Dic₅	1	1	yes	characteristic, complemented
99	C₃×#95	—	5	2	no	—
100	C₃×#95	—	10	1	no	—
101	#830	—	5	2	no	—
102	C₄⋊(C₃⋊C₈)	—	10	1	no	—
103	#803	—	10	1	no	—
104	C₅×#41	—	12	4	no	—
105	C₂×C₅×Q₈	—	6	1	no	—
106	C₂×C₅×D₄	—	3	2	no	—
107	C₂×C₅×D₄	—	6	2	no	—
108	C₂×C₅×D₄	—	6	1	no	—
109	C₂×C₅×D₄	—	6	1	no	—
110	C₂²×C₄×C₅	—	6	2	no	—
111	C₂²×C₄×C₅	#67	1	1	yes	characteristic
112	C₂²×C₄×C₅	#67	1	1	yes	characteristic
113	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
114	C₂²×Dic₅	—	2	2	no	—
115	C₅×C₄⋊C₄	—	3	2	no	—
116	C₅×C₄⋊C₄	—	6	1	no	—
117	C₅×C₂²⋊C₄	—	6	2	no	—
118	C₅×C₂²⋊C₄	—	6	2	no	—
119	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
120	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
121	C₄²×C₅	D₁₂	1	1	yes	characteristic
122	C₄²×C₅	D₁₂	1	1	yes	characteristic
123	C₄×Dic₅	—	2	4	no	—
124	C₂×#168	—	6	2	no	—
125	#532	—	3	1	no	—
126	#432	—	15	1	no	—
127	#376	—	5	1	no	—
128	#327	—	15	4	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
133	C₂×C₅×D₃	—	4	2	no	—
134	C₂×C₅×D₃	—	4	2	no	—
135	C₂×C₅×D₃	—	4	2	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	2	1	no	—
140	C₃×C₄×C₅	—	2	1	no	—
141	C₃×C₄×C₅	#104	1	2	yes	—
142	C₃×Dic₅	—	4	4	no	—
143	C₅×Dic₃	—	4	2	no	—
144	C₂²×C₃×C₄	—	10	2	no	—
145	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
146	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
147	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
148	C₂×#67	—	2	2	no	—
149	#235	—	4	4	no	—
150	C₂×D₁₂	—	2	1	no	—
151	C₂×D₁₂	—	2	1	no	—
152	C₂×D₁₂	C₂×Dic₅	1	2	yes	—
153	C₂×C₄×D₃	—	2	2	no	—
154	C₂×Dic₆	—	2	1	no	—
155	C₃×C₄²	—	10	4	no	—
156	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
157	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
158	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
159	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
160	(C₂×D₃)⋊C₄	—	2	2	no	—
161	(C₂×D₃)⋊C₄	—	2	2	no	—
162	C₄⋊Dic₃	—	2	1	no	—
163	C₄⋊Dic₃	C₂×Dic₅	1	2	yes	—
164	C₂×C₃⋊C₈	—	10	2	no	—
165	C₂³×C₅	—	6	2	no	—
166	C₂³×C₅	C₂×#67	1	1	yes	characteristic
167	C₅×Q₈	—	12	2	no	—
168	C₅×D₄	—	12	2	no	—
169	C₅×D₄	—	12	2	no	—
170	C₅×D₄	—	12	4	no	—
171	C₅×D₄	—	12	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	4	1	no	—
175	C₂×C₄×C₅	—	6	2	no	—
176	C₂×C₄×C₅	—	12	4	no	—
177	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
183	C₂×Dic₅	—	2	4	no	—
184	C₂×Dic₅	—	4	2	no	—
185	C₂×Dic₅	—	4	2	no	—
186	#168	—	12	2	no	—
187	C₂×#41	—	6	1	no	—
188	C₄⋊Q₈	—	3	1	no	—
189	#127	—	3	1	no	—
190	#124	—	6	1	no	—
191	#121	—	3	2	no	—
192	#121	—	6	1	no	—
193	C₄×D₄	—	3	2	no	—
194	C₄×D₄	—	6	1	no	—
195	C₂×C₄²	—	10	1	no	—
196	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
197	C₄⋊C₈	—	30	1	no	—
198	C₂²⋊C₈	—	15	2	no	—
199	#97	—	30	1	no	—
200	#95	—	5	2	no	—
201	#95	—	10	1	no	—
202	C₂×C₃×C₅	—	2	2	no	—
203	C₂×C₃×C₅	#410	1	1	yes	characteristic
204	C₂×C₃×C₅	#418	1	1	yes	characteristic
205	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
206	C₅×D₃	—	8	2	no	—
207	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
208	C₂²×D₃	—	2	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	—	10	4	no	—
213	C₂×C₃×C₄	—	20	2	no	—
214	C₂×C₃×C₄	—	20	2	no	—
215	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
218	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
221	#67	—	4	4	no	—
222	C₂×Dic₃	—	2	2	no	—
223	D₁₂	—	4	2	no	—
224	D₁₂	—	4	2	no	—
225	D₁₂	—	4	2	no	—
226	C₄×D₃	—	4	4	no	—
227	Dic₆	—	4	2	no	—
228	C₃⋊C₈	—	20	2	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	—	12	2	no	—
232	C₂²×C₅	—	12	2	no	—
233	C₂²×C₅	—	12	2	no	—
234	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
235	C₂²×C₅	#949	1	2	yes	—
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	12	2	no	—
242	C₄×C₅	#805	1	2	yes	—
243	Dic₅	—	4	4	no	—
244	#41	—	12	4	no	—
245	C₂×Q₈	—	6	1	no	—
246	C₂×D₄	—	3	2	no	—
247	C₂×D₄	—	6	2	no	—
248	C₂×D₄	—	6	1	no	—
249	C₂×D₄	—	6	1	no	—
250	C₂²×C₄	—	6	2	no	—
251	C₂²×C₄	—	10	2	no	—
252	C₂²×C₄	#1201	1	1	yes	characteristic
253	C₂²×C₄	#1201	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
255	C₂×C₈	—	30	2	no	—
256	C₄⋊C₄	—	3	2	no	—
257	C₄⋊C₄	—	6	1	no	—
258	C₂²⋊C₄	—	6	2	no	—
259	C₂²⋊C₄	—	6	2	no	—
260	C₄²	—	10	4	no	—
261	C₄²	D₃×Dic₅	1	1	yes	characteristic
262	C₄²	D₃×Dic₅	1	1	yes	characteristic
263	C₄²	#1203	1	1	yes	characteristic
264	C₄²	#1203	1	1	yes	characteristic
265	C₃×C₅	#1505	1	1	yes	characteristic, complemented, Hall
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	#4078	1	2	yes	—
269	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
270	C₂×D₃	—	4	2	no	—
271	C₂×D₃	—	4	2	no	—
272	C₂×D₃	—	4	2	no	—
273	C₃×C₄	—	2	1	no	—
274	C₃×C₄	—	2	1	no	—
275	C₃×C₄	—	2	1	no	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	20	4	no	—
279	C₃×C₄	#3969	1	2	yes	—
280	Dic₃	—	4	2	no	—
281	C₂×C₅	—	2	2	no	—
282	C₂×C₅	—	24	2	no	—
283	C₂×C₅	C₂×#805	1	1	yes	characteristic
284	C₂×C₅	#5001	1	1	yes	characteristic
285	C₂×C₅	#5006	1	1	yes	characteristic
286	C₂³	—	6	2	no	—
287	C₂³	C₂×#1201	1	1	yes	characteristic
288	Q₈	—	12	2	no	—
289	D₄	—	12	2	no	—
290	D₄	—	12	2	no	—
291	D₄	—	12	4	no	—
292	D₄	—	12	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	6	2	no	—
297	C₂×C₄	—	10	4	no	—
298	C₂×C₄	—	12	4	no	—
299	C₂×C₄	—	20	2	no	—
300	C₂×C₄	—	20	2	no	—
301	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
302	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
303	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
304	C₂×C₄	(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
305	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
306	C₂×C₄	C₂×#1203	1	1	yes	characteristic
307	C₈	—	60	2	no	—
308	C₂×C₃	—	2	2	no	—
309	C₂×C₃	#65504	1	1	yes	characteristic
310	C₂×C₃	C₂×#3969	1	1	yes	characteristic
311	C₂×C₃	#65529	1	1	yes	characteristic
312	D₃	—	8	2	no	—
313	C₅	#69432	1	1	yes	characteristic, complemented, Sylow
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	12	2	no	—
317	C₂²	—	12	2	no	—
318	C₂²	—	12	2	no	—
319	C₂²	#91299	1	2	yes	—
320	C₂²	C₂×(C₂×D₃)⋊Dic₅	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	12	2	no	—
327	C₄	—	20	4	no	—
328	C₄	#90979	1	2	yes	—
329	C₃	#10602605	1	1	yes	characteristic, complemented, Sylow
330	C₂	—	2	2	no	—
331	C₂	—	24	2	no	—
332	C₂	#11748386	1	1	yes	characteristic
333	C₂	C₂×#90979	1	1	yes	characteristic
334	C₂	#11748625	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₃)⋊Dic₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₅×#5477	C₂
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1505
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. #11751027	C₂	semidirect
3. C₃×#65221	C₂	semidirect
4. #11745956	C₂	semidirect
11. C₅×#874	C₄	semidirect
13. C₄×C₅×D₁₂	C₄	semidirect
14. C₅×#869	C₄	semidirect
23. #65221	D₃	semidirect
95. #874	Dic₅	semidirect
97. C₄×D₁₂	Dic₅	semidirect
98. #869	Dic₅	semidirect
265. C₃×C₅	#1505	semidirect
313. C₅	#69432	semidirect
329. C₃	#10602605	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
#11751027	2
C₃×#65221	2
#11745956	2
#10602605	3
#69432	5