Subgroups — #14284725

Order 1920.

492 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756236	C₂	1	1	yes	maximal, characteristic
2	C₅×#4925	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65214	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745863	C₂	1	2	yes	maximal, complemented
5	#11745854	C₂	1	2	yes	maximal, complemented
6	#10601566	—	3	1	no	maximal, Hall
7	C₂×#6817	C₄	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	#91819	—	2	1	no	—
10	#91819	—	2	1	no	—
11	#91764	C₄	1	2	yes	—
12	#91763	C₂²	1	1	yes	characteristic
13	C₄×D₆₀	—	2	1	no	—
14	C₄×D₆₀	—	2	1	no	—
15	#91758	C₂²	1	1	yes	characteristic
16	C₅×#830	—	2	1	no	—
17	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
18	C₄×C₅×C₃⋊C₈	—	2	1	no	—
19	C₃×C₂²⋊#168	—	2	1	no	—
20	C₃×C₄⋊#168	C₂²	1	2	yes	—
21	C₃×C₄×#168	—	2	1	no	—
22	#70773	—	5	1	no	maximal, Hall
23	#65811	—	3	1	no	—
24	C₅×#431	—	3	1	no	—
25	#65214	D₃	1	1	yes	characteristic, complemented
26	#64705	—	3	2	no	—
27	#64701	—	3	2	no	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂×#1220	—	2	2	no	—
32	#6817	—	4	4	no	—
33	C₂×D₆₀	—	2	1	no	—
34	C₂×D₆₀	—	2	1	no	—
35	C₂×D₆₀	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₅	—	2	2	no	—
37	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
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	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
42	#6714	—	2	2	no	—
43	#6714	—	2	2	no	—
44	C₄⋊Dic₁₅	—	2	1	no	—
45	C₄⋊Dic₁₅	—	2	1	no	—
46	C₂×C₅×C₃⋊C₈	—	2	2	no	—
47	C₂×C₃×#168	—	2	2	no	—
48	#5477	—	5	1	no	—
49	C₃×#431	—	5	1	no	—
50	#4925	D₅	1	1	yes	characteristic, complemented
51	#4485	—	5	2	no	—
52	#4481	—	5	2	no	—
53	C₂×#709	—	3	2	no	—
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	#4076	—	6	1	no	—
56	#4076	—	6	1	no	—
57	#4021	—	3	2	no	—
58	#4020	—	3	1	no	—
59	C₄×D₂₀	—	6	1	no	—
60	C₄×D₂₀	—	6	1	no	—
61	#4015	—	3	1	no	—
62	C₅×C₄⋊C₈	—	3	2	no	—
63	C₅×C₂²⋊C₈	—	6	1	no	—
64	C₄×C₅×C₈	—	6	1	no	—
65	C₂²⋊#168	—	2	1	no	—
66	C₄⋊#168	C₂×D₃	1	2	yes	—
67	C₄×#168	—	2	1	no	—
68	#1543	—	15	1	no	Sylow
69	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×D₁₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	#1220	—	4	4	no	—
81	C₂×Dic₁₅	—	2	2	no	—
82	D₆₀	—	4	2	no	—
83	D₆₀	—	4	2	no	—
84	D₆₀	—	4	2	no	—
85	C₄×D₁₅	—	4	4	no	—
86	Dic₃₀	—	4	2	no	—
87	C₃×#168	—	4	2	no	—
88	C₅×C₃⋊C₈	—	4	2	no	—
89	C₂×#235	—	5	2	no	—
90	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
91	#930	—	10	1	no	—
92	#930	—	10	1	no	—
93	#875	—	5	2	no	—
94	#874	—	5	1	no	—
95	C₄×D₁₂	—	10	1	no	—
96	C₄×D₁₂	—	10	1	no	—
97	#869	—	5	1	no	—
98	C₃×C₄⋊C₈	—	5	2	no	—
99	C₃×C₂²⋊C₈	—	10	1	no	—
100	C₃×C₄×C₈	—	10	1	no	—
101	#830	—	2	1	no	—
102	C₄⋊(C₃⋊C₈)	C₂×D₅	1	2	yes	—
103	C₄×C₃⋊C₈	—	2	1	no	—
104	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
105	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
106	C₂²×C₄×C₅	#67	1	1	yes	characteristic
107	C₂×C₂²⋊D₅	—	6	2	no	—
108	#709	—	12	4	no	—
109	C₂×D₂₀	—	3	2	no	—
110	C₂×D₂₀	—	6	1	no	—
111	C₂×D₂₀	—	6	1	no	—
112	C₂×C₄×D₅	—	6	2	no	—
113	C₂×Dic₁₀	—	3	2	no	—
114	C₂×C₅×C₈	—	6	2	no	—
115	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
116	C₄²×C₅	D₁₂	1	1	yes	characteristic
117	C₄²×C₅	#67	1	1	yes	characteristic
118	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
119	#685	—	6	2	no	—
120	#685	—	6	2	no	—
121	C₄⋊Dic₅	—	6	1	no	—
122	C₄⋊Dic₅	—	6	1	no	—
123	C₂×#168	—	2	2	no	—
124	#532	—	15	1	no	—
125	#431	—	3	1	no	—
126	#431	—	5	1	no	—
127	#333	—	15	2	no	—
128	#331	—	15	2	no	—
129	C₂²×C₃×C₅	—	2	1	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×D₁₅	—	4	2	no	—
134	C₂×D₁₅	—	4	2	no	—
135	C₂×D₁₅	—	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	#100	1	2	yes	—
140	C₃×C₄×C₅	#135	1	2	yes	—
141	C₃×C₄×C₅	#135	1	2	yes	—
142	Dic₁₅	—	4	2	no	—
143	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
144	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
145	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
146	C₂×#67	—	10	2	no	—
147	#235	—	20	4	no	—
148	C₂×D₁₂	—	5	2	no	—
149	C₂×D₁₂	—	10	1	no	—
150	C₂×D₁₂	—	10	1	no	—
151	C₂×C₄×D₃	—	10	2	no	—
152	C₂×Dic₆	—	5	2	no	—
153	C₂×C₃×C₈	—	10	2	no	—
154	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
155	C₃×C₄²	D₂₀	1	1	yes	characteristic
156	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
157	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
158	(C₂×D₃)⋊C₄	—	10	2	no	—
159	(C₂×D₃)⋊C₄	—	10	2	no	—
160	C₄⋊Dic₃	—	10	1	no	—
161	C₄⋊Dic₃	—	10	1	no	—
162	C₂×C₃⋊C₈	—	2	2	no	—
163	C₂³×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
164	C₂²×D₅	—	6	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
169	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
174	C₂²⋊D₅	—	12	4	no	—
175	C₂×Dic₅	—	6	2	no	—
176	D₂₀	—	12	2	no	—
177	D₂₀	—	12	2	no	—
178	D₂₀	—	12	2	no	—
179	C₄×D₅	—	12	4	no	—
180	Dic₁₀	—	12	2	no	—
181	C₅×C₈	—	12	2	no	—
182	#168	—	4	2	no	—
183	C₂×#41	—	15	2	no	—
184	C₄⋊Q₈	—	15	1	no	—
185	#127	—	15	1	no	—
186	#124	—	15	2	no	—
187	#121	—	30	1	no	—
188	#121	—	30	1	no	—
189	C₄×D₄	—	30	1	no	—
190	C₄×D₄	—	30	1	no	—
191	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
192	C₄⋊C₈	—	3	2	no	—
193	C₄⋊C₈	—	5	2	no	—
194	C₂²⋊C₈	—	6	1	no	—
195	C₂²⋊C₈	—	10	1	no	—
196	C₄×C₈	—	6	1	no	—
197	C₄×C₈	—	10	1	no	—
198	C₂×C₃×C₅	—	4	1	no	—
199	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
200	C₂×C₃×C₅	#416	1	1	yes	characteristic
201	C₂×C₃×C₅	#416	1	1	yes	characteristic
202	D₁₅	—	8	2	no	—
203	C₂³×C₃	#685	1	1	yes	characteristic
204	C₂²×D₃	—	10	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	2	no	—
208	C₂×C₃×C₄	#685	1	1	yes	characteristic
209	C₂×C₃×C₄	#685	1	1	yes	characteristic
210	C₂×C₃×C₄	#685	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
214	#67	—	20	4	no	—
215	C₂×Dic₃	—	10	2	no	—
216	D₁₂	—	20	2	no	—
217	D₁₂	—	20	2	no	—
218	D₁₂	—	20	2	no	—
219	C₄×D₃	—	20	4	no	—
220	Dic₆	—	20	2	no	—
221	C₃×C₈	—	20	2	no	—
222	C₃⋊C₈	—	4	2	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	—	2	1	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
227	C₂×D₅	—	12	2	no	—
228	C₂×D₅	—	12	2	no	—
229	C₂×D₅	—	12	2	no	—
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	#823	1	2	yes	—
234	C₄×C₅	#904	1	2	yes	—
235	C₄×C₅	#950	1	2	yes	—
236	Dic₅	—	12	2	no	—
237	#41	—	60	4	no	—
238	C₂×Q₈	—	15	2	no	—
239	C₂×D₄	—	15	2	no	—
240	C₂×D₄	—	30	2	no	—
241	C₂×D₄	—	30	1	no	—
242	C₂×D₄	—	30	1	no	—
243	C₂²×C₄	—	30	2	no	—
244	C₂²×C₄	#1200	1	1	yes	characteristic
245	C₂²×C₄	#1202	1	1	yes	characteristic
246	C₂²×C₄	#1203	1	1	yes	characteristic
247	C₂×C₈	—	6	2	no	—
248	C₂×C₈	—	10	2	no	—
249	C₄⋊C₄	—	30	1	no	—
250	C₄⋊C₄	—	30	1	no	—
251	C₂²⋊C₄	—	30	2	no	—
252	C₂²⋊C₄	—	30	2	no	—
253	C₄²	#1200	1	1	yes	characteristic
254	C₄²	#1202	1	1	yes	characteristic
255	C₄²	#1203	1	1	yes	characteristic
256	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
257	C₃×C₅	#1543	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	C₂×#685	1	1	yes	characteristic
262	C₂×D₃	—	20	2	no	—
263	C₂×D₃	—	20	2	no	—
264	C₂×D₃	—	20	2	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	#3955	1	2	yes	—
269	C₃×C₄	#4050	1	2	yes	—
270	C₃×C₄	#4096	1	2	yes	—
271	Dic₃	—	20	2	no	—
272	C₂×C₅	—	4	1	no	—
273	C₂×C₅	#5002	1	1	yes	characteristic
274	C₂×C₅	#5113	1	1	yes	characteristic
275	C₂×C₅	C₂×#823	1	1	yes	characteristic
276	D₅	—	24	2	no	—
277	C₂³	—	30	2	no	—
278	C₂³	#6667	1	1	yes	characteristic
279	Q₈	—	60	2	no	—
280	D₄	—	60	2	no	—
281	D₄	—	60	4	no	—
282	D₄	—	60	2	no	—
283	D₄	—	60	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	30	2	no	—
288	C₂×C₄	—	60	4	no	—
289	C₂×C₄	#6667	1	1	yes	characteristic
290	C₂×C₄	#6667	1	1	yes	characteristic
291	C₂×C₄	#6667	1	1	yes	characteristic
292	C₂×C₄	C₂×#1200	1	1	yes	characteristic
293	C₂×C₄	C₂×#1202	1	1	yes	characteristic
294	C₂×C₄	C₂×#1203	1	1	yes	characteristic
295	C₈	—	12	2	no	—
296	C₈	—	20	2	no	—
297	C₂×C₃	—	4	1	no	—
298	C₂×C₃	#65291	1	1	yes	characteristic
299	C₂×C₃	#65402	1	1	yes	characteristic
300	C₂×C₃	C₂×#3955	1	1	yes	characteristic
301	D₃	—	40	2	no	—
302	C₅	#70773	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	60	2	no	—
307	C₂²	—	60	2	no	—
308	C₂²	—	60	2	no	—
309	C₂²	C₂×#6667	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	60	2	no	—
314	C₄	#90960	1	2	yes	—
315	C₄	#91303	1	2	yes	—
316	C₄	#91311	1	2	yes	—
317	C₃	#10601566	1	1	yes	characteristic, complemented, Sylow
318	C₂	—	4	1	no	—
319	C₂	—	120	2	no	—
320	C₂	C₂×#90960	1	1	yes	characteristic
321	C₂	#11748393	1	1	yes	characteristic
322	C₂	#11748411	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6667
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	#11756236	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1543
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×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. C₅×#4925	C₂	semidirect
3. C₃×#65214	C₂	semidirect
4. #11745863	C₂	semidirect
5. #11745854	C₂	semidirect
25. #65214	D₃	semidirect
50. #4925	D₅	semidirect
257. C₃×C₅	#1543	semidirect
302. C₅	#70773	semidirect
317. C₃	#10601566	semidirect

Maximal subgroups

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

subgroup	index
#11756236	2
C₅×#4925	2
C₃×#65214	2
#11745863	2
#11745854	2
#10601566	3
#70773	5