Subgroups — #14415285

Order 1920.

1000 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	#11756236	C₂	1	1	yes	maximal, characteristic
2	C₂×#91291	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#65213	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748350	C₂	1	2	yes	maximal, complemented
6	#11747012	C₂	1	4	yes	maximal, complemented
7	#11746994	C₂	1	4	yes	maximal, complemented
8	#10610189	—	3	1	no	maximal, Hall
9	C₂×#6817	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	#91819	C₄	1	4	yes	—
12	#91764	C₄	1	2	yes	—
13	#91763	C₄	1	1	yes	characteristic
14	C₄×D₆₀	C₂²	1	4	yes	—
15	#91758	C₄	1	1	yes	characteristic
16	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃×#681	C₂²	1	2	yes	—
19	C₂×#6651	C₂²	1	4	yes	—
20	#91291	—	2	8	no	—
21	C₂×#6648	C₂²	1	4	yes	—
22	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
23	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
24	C₃×C₄⋊#168	C₂²	1	4	yes	—
25	#90957	C₂²	1	8	yes	—
26	#79041	—	5	1	no	maximal, Hall
27	C₂×#4047	—	3	2	no	—
28	#65811	—	3	1	no	—
29	C₂×C₄×C₅×C₈	—	3	1	no	—
30	#65399	—	3	2	no	—
31	#65213	D₃	1	1	yes	characteristic, complemented
32	#64977	—	3	4	no	—
33	C₈×D₂₀	—	3	4	no	—
34	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×#1220	C₂×C₄	1	4	yes	—
37	#6817	—	2	4	no	—
38	#6817	—	2	4	no	—
39	C₂×D₆₀	C₂×C₄	1	4	yes	—
40	C₂×D₆₀	C₂×C₄	1	2	yes	—
41	C₂×C₄×D₁₅	C₂³	1	4	yes	—
42	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
43	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄²×C₅	C₂³	1	2	yes	—
46	#6714	C₂×C₄	1	8	yes	—
47	C₄⋊Dic₁₅	C₂×C₄	1	4	yes	—
48	C₂×C₅×C₃⋊C₈	—	2	2	no	—
49	C₂×C₅×C₃⋊C₈	D₄	1	4	yes	complemented
50	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
51	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
52	C₂×C₅×C₃⋊C₈	C₂³	1	1	yes	characteristic
53	C₃×#681	—	2	4	no	—
54	C₂×C₃×#168	C₂³	1	4	yes	—
55	#6651	—	2	8	no	—
56	#6648	—	2	8	no	—
57	C₂×#907	—	5	2	no	—
58	#5477	—	5	1	no	—
59	C₃×#423	—	5	1	no	—
60	#5129	—	5	2	no	—
61	C₂×C₄×C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
62	#4838	—	5	4	no	—
63	#4834	—	5	4	no	—
64	C₂×#709	—	3	2	no	—
65	C₂²×C₅×C₈	—	3	1	no	—
66	C₂²×C₅×C₈	—	3	1	no	—
67	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
68	#4076	—	3	4	no	—
69	C₂×#681	C₂×D₃	1	2	yes	—
70	#4047	—	6	8	no	—
71	C₂×#676	—	3	4	no	—
72	C₂×C₈×D₅	—	3	4	no	—
73	#4021	—	3	2	no	—
74	#4020	—	3	1	no	—
75	C₄×D₂₀	—	3	4	no	—
76	#4015	—	3	1	no	—
77	C₄×C₅×C₈	—	3	2	no	—
78	C₄×C₅×C₈	—	3	2	no	—
79	#3952	—	3	8	no	—
80	C₄⋊#168	C₂×D₃	1	4	yes	—
81	#2931	—	15	1	no	Sylow
82	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
83	C₂²×D₁₅	C₂²×C₄	1	4	yes	—
84	C₂×C₃×C₄×C₅	—	2	4	no	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
89	C₂×C₃×C₄×C₅	#41	1	2	yes	—
90	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	#1220	—	2	8	no	—
92	C₂×Dic₁₅	C₂²×C₄	1	4	yes	—
93	D₆₀	—	2	4	no	—
94	D₆₀	—	2	8	no	—
95	C₄×D₁₅	—	2	4	no	—
96	C₄×D₁₅	—	2	4	no	—
97	Dic₃₀	—	2	4	no	—
98	C₃×#168	—	2	4	no	—
99	C₅×C₃⋊C₈	—	2	2	no	—
100	C₅×C₃⋊C₈	C₂×D₄	1	2	yes	—
101	C₅×C₃⋊C₈	C₂×D₄	1	2	yes	—
102	C₂×#235	—	5	2	no	—
103	C₂×C₃×M₁₆	—	5	2	no	—
104	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
105	#930	—	5	4	no	—
106	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
107	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
108	#907	—	10	8	no	—
109	C₂×ΓL(1,𝔽₂₅)	—	5	4	no	—
110	C₂×D₃×C₈	—	5	4	no	—
111	#875	—	5	2	no	—
112	#874	—	5	1	no	—
113	C₄×D₁₂	—	5	4	no	—
114	#869	—	5	1	no	—
115	C₃×C₄⋊C₈	—	5	4	no	—
116	(C₂×D₃)⋊C₈	—	5	8	no	—
117	C₄×C₃⋊C₈	C₂×D₅	1	2	yes	—
118	C₄×C₃⋊C₈	C₂×D₅	1	2	yes	—
119	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
120	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
121	C₂×C₂²⋊D₅	—	3	4	no	—
122	#709	—	6	4	no	—
123	#709	—	6	4	no	—
124	C₂×D₂₀	—	3	2	no	—
125	C₂×D₂₀	—	3	4	no	—
126	C₂×C₄×D₅	—	3	4	no	—
127	C₂×Dic₁₀	—	3	2	no	—
128	C₂×C₅×C₈	—	3	4	no	—
129	C₂×C₅×C₈	—	3	1	no	—
130	C₂×C₅×C₈	—	3	1	no	—
131	C₂×C₅×C₈	—	3	2	no	—
132	C₂×C₅×C₈	—	6	2	no	—
133	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
134	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
135	C₄²×C₅	C₂²×D₃	1	2	yes	—
136	#685	—	3	8	no	—
137	C₄⋊Dic₅	—	3	4	no	—
138	#681	—	2	4	no	—
139	C₂×#168	C₂²×D₃	1	4	yes	—
140	#676	—	6	8	no	—
141	C₈×D₅	—	6	8	no	—
142	C₂×#131	—	15	2	no	—
143	#532	—	15	1	no	—
144	#436	—	15	4	no	—
145	C₈×D₄	—	15	4	no	—
146	#423	—	5	1	no	—
147	#408	—	15	2	no	—
148	C₂×C₄×C₈	—	3	1	no	—
149	C₂²×C₃×C₅	—	2	2	no	—
150	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
151	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
152	C₂×D₁₅	—	2	4	no	—
153	C₂×D₁₅	—	2	8	no	—
154	C₃×C₄×C₅	—	2	2	no	—
155	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
156	C₃×C₄×C₅	#131	1	4	yes	complemented
157	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
158	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
159	Dic₁₅	—	2	4	no	—
160	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
161	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
162	C₂×#67	—	5	4	no	—
163	#235	—	10	4	no	—
164	#235	—	10	4	no	—
165	C₂×D₁₂	—	5	4	no	—
166	C₂×D₁₂	—	5	2	no	—
167	C₂×C₄×D₃	—	5	4	no	—
168	C₂×Dic₆	—	5	2	no	—
169	C₃×M₁₆	—	10	4	no	—
170	C₂×C₃×C₈	—	5	4	no	—
171	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
172	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
173	C₃×C₄²	C₂²×D₅	1	2	yes	—
174	(C₂×D₃)⋊C₄	—	5	8	no	—
175	C₄⋊Dic₃	—	5	4	no	—
176	C₂×C₃⋊C₈	—	2	2	no	—
177	C₂×C₃⋊C₈	D₂₀	1	4	yes	complemented
178	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
179	C₂×C₃⋊C₈	C₂²×D₅	1	1	yes	characteristic
180	C₂×C₃⋊C₈	C₂²×D₅	1	1	yes	characteristic
181	ΓL(1,𝔽₂₅)	—	10	8	no	—
182	D₃×C₈	—	10	8	no	—
183	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
184	C₂²×D₅	—	3	4	no	—
185	C₂×C₄×C₅	—	2	4	no	—
186	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
187	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
188	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
189	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
190	C₂×C₄×C₅	#239	1	2	yes	—
191	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
192	C₂²⋊D₅	—	6	8	no	—
193	C₂×Dic₅	—	3	4	no	—
194	D₂₀	—	6	4	no	—
195	D₂₀	—	6	8	no	—
196	C₄×D₅	—	6	4	no	—
197	C₄×D₅	—	6	4	no	—
198	Dic₁₀	—	6	4	no	—
199	C₅×C₈	—	3	2	no	—
200	C₅×C₈	—	3	2	no	—
201	C₅×C₈	—	6	2	no	—
202	#168	—	2	4	no	—
203	C₂×#41	—	15	2	no	—
204	#131	—	30	8	no	—
205	C₂×M₁₆	—	5	2	no	—
206	C₂×M₁₆	—	15	4	no	—
207	C₂²×C₈	—	3	1	no	—
208	C₂²×C₈	—	3	1	no	—
209	C₂²×C₈	—	15	4	no	—
210	C₄⋊Q₈	—	15	1	no	—
211	#127	—	15	1	no	—
212	#124	—	15	2	no	—
213	#121	—	15	4	no	—
214	C₄×D₄	—	15	4	no	—
215	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
216	C₄⋊C₈	—	5	4	no	—
217	C₂²⋊C₈	—	15	8	no	—
218	C₄×C₈	—	3	2	no	—
219	C₄×C₈	—	3	2	no	—
220	C₂×C₃×C₅	—	2	2	no	—
221	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
222	C₂×C₃×C₅	C₂×#131	1	2	yes	—
223	D₁₅	—	4	4	no	—
224	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
225	C₂²×D₃	—	5	4	no	—
226	C₂×C₃×C₄	—	2	4	no	—
227	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
228	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
230	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
231	C₂×C₃×C₄	#709	1	2	yes	—
232	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
233	#67	—	10	8	no	—
234	C₂×Dic₃	—	5	4	no	—
235	D₁₂	—	10	4	no	—
236	D₁₂	—	10	8	no	—
237	C₄×D₃	—	10	4	no	—
238	C₄×D₃	—	10	4	no	—
239	Dic₆	—	10	4	no	—
240	C₃×C₈	—	10	4	no	—
241	C₃⋊C₈	—	2	2	no	—
242	C₃⋊C₈	C₂×D₂₀	1	2	yes	—
243	C₃⋊C₈	C₂×D₂₀	1	2	yes	—
244	C₂²×C₅	—	2	2	no	—
245	C₂²×C₅	#893	1	2	yes	—
246	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
247	C₂×D₅	—	6	8	no	—
248	C₂×D₅	—	6	4	no	—
249	C₄×C₅	—	2	2	no	—
250	C₄×C₅	#893	1	2	yes	—
251	C₄×C₅	#907	1	4	yes	complemented
252	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
253	C₄×C₅	C₂×#239	1	1	yes	characteristic
254	Dic₅	—	6	4	no	—
255	#41	—	30	4	no	—
256	#41	—	30	4	no	—
257	C₂×Q₈	—	15	2	no	—
258	C₂×D₄	—	15	4	no	—
259	C₂×D₄	—	15	4	no	—
260	C₂×D₄	—	15	2	no	—
261	C₂²×C₄	—	15	4	no	—
262	C₂²×C₄	#1200	1	2	yes	—
263	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
264	M₁₆	—	10	4	no	—
265	M₁₆	—	30	8	no	—
266	C₂×C₈	—	3	4	no	—
267	C₂×C₈	—	3	1	no	—
268	C₂×C₈	—	3	1	no	—
269	C₂×C₈	—	3	2	no	—
270	C₂×C₈	—	5	4	no	—
271	C₂×C₈	—	6	2	no	—
272	C₂×C₈	—	30	8	no	—
273	C₄⋊C₄	—	15	4	no	—
274	C₂²⋊C₄	—	15	8	no	—
275	C₄²	#1200	1	1	yes	characteristic
276	C₄²	#1200	1	1	yes	characteristic
277	C₄²	C₂×D₃×D₅	1	2	yes	—
278	C₃×C₅	#2931	1	1	yes	characteristic, complemented, Hall
279	C₂²×C₃	—	2	2	no	—
280	C₂²×C₃	C₄×D₂₀	1	2	yes	—
281	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
282	C₂×D₃	—	10	4	no	—
283	C₂×D₃	—	10	8	no	—
284	C₃×C₄	—	2	2	no	—
285	C₃×C₄	C₄×D₂₀	1	2	yes	—
286	C₃×C₄	#4047	1	4	yes	complemented
287	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
288	C₃×C₄	C₂×#709	1	1	yes	characteristic
289	Dic₃	—	10	4	no	—
290	C₂×C₅	—	2	2	no	—
291	C₂×C₅	C₂×#893	1	1	yes	characteristic
292	C₂×C₅	C₂×#907	1	2	yes	—
293	D₅	—	12	4	no	—
294	C₂³	—	15	4	no	—
295	C₂³	C₂×#1200	1	1	yes	characteristic
296	Q₈	—	30	4	no	—
297	D₄	—	30	8	no	—
298	D₄	—	30	4	no	—
299	D₄	—	30	8	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	15	4	no	—
302	C₂×C₄	—	30	4	no	—
303	C₂×C₄	—	30	4	no	—
304	C₂×C₄	#6769	1	2	yes	—
305	C₂×C₄	D₃×D₂₀	1	2	yes	—
306	C₂×C₄	C₂×#1200	1	2	yes	—
307	C₂×C₄	C₂×#1200	1	1	yes	characteristic
308	C₂×C₄	C₂×#1200	1	2	yes	—
309	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
310	C₈	—	3	2	no	—
311	C₈	—	3	2	no	—
312	C₈	—	6	2	no	—
313	C₈	—	10	4	no	—
314	C₂×C₃	—	2	2	no	—
315	C₂×C₃	C₂×C₄×D₂₀	1	1	yes	characteristic
316	C₂×C₃	C₂×#4047	1	2	yes	—
317	D₃	—	20	4	no	—
318	C₅	#79041	1	1	yes	characteristic, complemented, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	30	4	no	—
321	C₂²	—	30	8	no	—
322	C₂²	#91428	1	2	yes	—
323	C₂²	C₂²×#1200	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	30	4	no	—
326	C₄	#91291	1	4	yes	complemented
327	C₄	#91428	1	2	yes	—
328	C₄	C₂×#6769	1	1	yes	characteristic
329	C₄	C₂×D₃×D₂₀	1	1	yes	characteristic
330	C₃	#10610189	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	2	2	no	—
332	C₂	—	60	4	no	—
333	C₂	C₂×#91291	1	2	yes	—
334	C₂	C₂×#91428	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂³×C₄
Fitting	C₂×C₃×C₄²×C₅	C₂²
prime radical	#11756236	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#2931
LCS	G ▷ 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. C₂×#91291	C₂	semidirect
3. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
4. C₃×#65213	C₂	semidirect
5. #11748350	C₂	semidirect
6. #11747012	C₂	semidirect
7. #11746994	C₂	semidirect
31. #65213	D₃	semidirect
49. C₂×C₅×C₃⋊C₈	D₄	semidirect
61. C₂×C₄×C₃⋊C₈	D₅	semidirect
156. C₃×C₄×C₅	#131	semidirect
177. C₂×C₃⋊C₈	D₂₀	semidirect
251. C₄×C₅	#907	semidirect
278. C₃×C₅	#2931	semidirect
286. C₃×C₄	#4047	semidirect
318. C₅	#79041	semidirect
326. C₄	#91291	semidirect
330. C₃	#10610189	semidirect

Maximal subgroups

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

subgroup	index
#11756236	2
C₂×#91291	2
C₂×C₄×C₅×C₃⋊C₈	2
C₃×#65213	2
#11748350	2
#11747012	2
#11746994	2
#10610189	3
#79041	5