Subgroups — #14418025

Order 1920.

912 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756507	C₂	1	1	yes	maximal, characteristic
2	C₂×#91790	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751226	C₂	1	1	yes	maximal, characteristic, complemented
5	#11751106	C₂	1	2	yes	maximal, complemented
6	#11750818	C₂	1	4	yes	maximal, complemented
7	#11750801	C₂	1	4	yes	maximal, complemented
8	#10610310	—	3	1	no	maximal, Hall
9	C₂×#6817	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#117	C₂²	1	1	yes	characteristic
13	C₂×#6710	C₂²	1	2	yes	—
14	#91798	C₄	1	4	yes	complemented
15	#91797	C₄	1	4	yes	complemented
16	#91793	C₂²	1	4	yes	—
17	#91790	—	2	2	no	—
18	#91790	—	2	2	no	—
19	#91790	C₂²	1	4	yes	complemented
20	#91790	C₂²	1	4	yes	complemented
21	C₂×#6705	—	2	2	no	—
22	C₂×C₈×D₁₅	—	2	2	no	—
23	C₃×C₅×#97	C₂²	1	2	yes	complemented
24	C₃×C₄×C₅×C₈	C₂²	1	2	yes	complemented
25	#91103	C₂²	1	4	yes	complemented
26	#91089	C₂²	1	8	yes	complemented
27	#78579	—	5	1	no	maximal, Hall
28	#66082	—	3	1	no	—
29	C₂×#4047	—	3	2	no	—
30	C₅×#405	D₃	1	1	yes	characteristic, complemented
31	#65414	—	3	1	no	—
32	#65294	—	3	2	no	—
33	#65006	—	3	4	no	—
34	#64989	—	3	4	no	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×#1220	—	2	2	no	—
37	#6817	—	2	2	no	—
38	#6817	—	2	2	no	—
39	#6817	C₂×C₄	1	4	yes	—
40	#6817	C₂³	1	4	yes	—
41	C₂×D₆₀	C₂×C₄	1	2	yes	—
42	C₂×C₄×D₁₅	—	2	2	no	—
43	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
44	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	—
45	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
46	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
47	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
48	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₃×C₅×C₄⋊C₄	—	2	1	no	—
51	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
52	C₃×C₄²×C₅	C₂³	1	2	yes	—
53	#6710	—	2	2	no	—
54	#6710	C₂³	1	4	yes	—
55	C₂×C₃⋊#168	—	2	2	no	—
56	Dic₆₀	C₂×C₄	1	4	yes	—
57	D₁₂₀	C₂×C₄	1	4	yes	—
58	#6706	C₂×C₄	1	8	yes	—
59	#6705	—	2	4	no	—
60	#6705	—	2	4	no	—
61	C₈×D₁₅	—	2	4	no	—
62	C₈×D₁₅	—	2	4	no	—
63	#5748	—	5	1	no	—
64	C₂×#901	—	5	2	no	—
65	C₃×#405	D₅	1	1	yes	characteristic, complemented
66	#5125	—	5	1	no	—
67	#5005	—	5	2	no	—
68	#4717	—	5	4	no	—
69	#4700	—	5	4	no	—
70	C₂×#709	—	3	2	no	—
71	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
72	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
73	C₅×#117	C₂×D₃	1	1	yes	characteristic
74	C₂×#681	—	3	2	no	—
75	#4055	—	3	4	no	—
76	#4054	—	3	4	no	—
77	#4050	—	3	4	no	—
78	#4047	—	3	4	no	—
79	#4047	—	3	4	no	—
80	#4047	—	6	2	no	—
81	#4047	—	6	2	no	—
82	C₂×#676	—	6	2	no	—
83	C₂×C₈×D₅	—	6	2	no	—
84	C₅×#97	C₂×D₃	1	2	yes	complemented
85	C₄×C₅×C₈	C₂×D₃	1	2	yes	complemented
86	#3951	—	3	4	no	—
87	#3937	—	3	8	no	—
88	#2955	—	15	1	no	Sylow
89	C₂³×C₃×C₅	#41	1	1	yes	characteristic
90	C₂²×D₁₅	—	2	2	no	—
91	C₂×C₃×C₄×C₅	—	2	2	no	—
92	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
93	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
94	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
95	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	#1220	—	2	4	no	—
97	#1220	—	2	4	no	—
98	C₂×Dic₁₅	—	2	2	no	—
99	D₆₀	—	2	2	no	—
100	D₆₀	C₂²×C₄	1	4	yes	—
101	C₄×D₁₅	—	2	4	no	—
102	C₄×D₁₅	—	2	4	no	—
103	Dic₃₀	—	2	2	no	—
104	Dic₃₀	C₂²×C₄	1	4	yes	—
105	C₃×C₅×C₈	C₂²×C₄	1	4	yes	—
106	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
107	C₃×C₅×C₈	C₂×D₄	1	2	yes	—
108	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
109	C₃⋊#168	—	2	4	no	—
110	C₂×#235	—	5	2	no	—
111	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
112	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
113	C₃×#117	C₂×D₅	1	1	yes	characteristic
114	C₂×#208	—	5	2	no	—
115	#909	—	5	4	no	—
116	#908	—	5	4	no	—
117	#904	—	5	4	no	—
118	#901	—	5	4	no	—
119	#901	—	5	4	no	—
120	#901	—	10	2	no	—
121	#901	—	10	2	no	—
122	C₂×ΓL(1,𝔽₂₅)	—	10	2	no	—
123	C₂×D₃×C₈	—	10	2	no	—
124	C₃×#97	C₂×D₅	1	2	yes	complemented
125	C₃×C₄×C₈	C₂×D₅	1	2	yes	complemented
126	#819	—	5	4	no	—
127	#805	—	5	8	no	—
128	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
129	C₂×C₂²⋊D₅	—	6	2	no	—
130	#709	—	3	4	no	—
131	#709	—	3	4	no	—
132	#709	—	6	2	no	—
133	#709	—	6	2	no	—
134	C₂×D₂₀	—	3	2	no	—
135	C₂×C₄×D₅	—	6	2	no	—
136	C₂×Dic₁₀	—	3	2	no	—
137	C₅×M₁₆	C₄×D₃	1	4	yes	—
138	C₂×C₅×C₈	D₁₂	1	2	yes	—
139	C₂×C₅×C₈	D₁₂	1	2	yes	—
140	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
141	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
142	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
143	C₅×C₄⋊C₄	—	2	1	no	—
144	C₅×C₂²⋊C₄	—	2	1	no	—
145	C₄²×C₅	C₂²×D₃	1	2	yes	—
146	#681	—	3	4	no	—
147	#681	—	6	2	no	—
148	C₂×#168	—	6	2	no	—
149	Dic₂₀	—	3	4	no	—
150	D₄₀	—	3	4	no	—
151	#677	—	3	8	no	—
152	#676	—	6	4	no	—
153	#676	—	6	4	no	—
154	C₈×D₅	—	6	4	no	—
155	C₈×D₅	—	6	4	no	—
156	#575	—	15	1	no	—
157	C₂×#131	—	15	2	no	—
158	#444	—	15	4	no	—
159	#443	—	15	4	no	—
160	#430	—	15	1	no	—
161	#421	—	15	2	no	—
162	#405	D₁₅	1	1	yes	characteristic, complemented
163	C₂²×C₃×C₅	—	4	1	no	—
164	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
165	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
166	C₂×D₁₅	—	2	4	no	—
167	C₂×D₁₅	—	4	4	no	—
168	C₃×C₄×C₅	—	4	2	no	—
169	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
170	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
171	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
172	Dic₁₅	—	2	4	no	—
173	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
174	C₂×#67	—	10	2	no	—
175	#235	—	5	4	no	—
176	#235	—	5	4	no	—
177	#235	—	10	2	no	—
178	#235	—	10	2	no	—
179	C₂×D₁₂	—	5	2	no	—
180	C₂×C₄×D₃	—	10	2	no	—
181	C₂×Dic₆	—	5	2	no	—
182	C₃×M₁₆	C₄×D₅	1	4	yes	—
183	C₂×C₃×C₈	D₂₀	1	2	yes	—
184	C₂×C₃×C₈	D₂₀	1	2	yes	—
185	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
186	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
187	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
188	C₃×C₄⋊C₄	—	2	1	no	—
189	C₃×C₂²⋊C₄	—	2	1	no	—
190	C₃×C₄²	C₂²×D₅	1	2	yes	—
191	#208	—	5	4	no	—
192	#208	—	10	2	no	—
193	C₂×C₃⋊C₈	—	10	2	no	—
194	Dic₁₂	—	5	4	no	—
195	D₂₄	—	5	4	no	—
196	#204	—	5	8	no	—
197	ΓL(1,𝔽₂₅)	—	10	4	no	—
198	ΓL(1,𝔽₂₅)	—	10	4	no	—
199	D₃×C₈	—	10	4	no	—
200	D₃×C₈	—	10	4	no	—
201	C₂³×C₅	#235	1	1	yes	characteristic
202	C₂²×D₅	—	6	2	no	—
203	C₂×C₄×C₅	—	2	2	no	—
204	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
205	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
206	C₂×C₄×C₅	#235	1	1	yes	characteristic
207	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
208	C₂²⋊D₅	—	6	4	no	—
209	C₂²⋊D₅	—	6	4	no	—
210	C₂×Dic₅	—	6	2	no	—
211	D₂₀	—	3	4	no	—
212	D₂₀	—	6	2	no	—
213	C₄×D₅	—	6	4	no	—
214	C₄×D₅	—	6	4	no	—
215	Dic₁₀	—	3	4	no	—
216	Dic₁₀	—	6	2	no	—
217	C₅×C₈	C₂×C₄×D₃	1	4	yes	—
218	C₅×C₈	C₂×D₁₂	1	2	yes	—
219	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
220	C₅×C₈	C₂×D₁₂	1	1	yes	characteristic
221	#168	—	6	4	no	—
222	C₂×#41	—	15	2	no	—
223	#137	—	15	4	no	—
224	AGL(1,ℤ/8)	—	15	4	no	—
225	#135	—	15	4	no	—
226	#131	—	15	4	no	—
227	#131	—	15	4	no	—
228	#131	—	30	2	no	—
229	#131	—	30	2	no	—
230	C₂×M₁₆	—	15	2	no	—
231	C₂×M₁₆	—	30	2	no	—
232	C₂×M₁₆	C₂×D₁₅	1	1	yes	characteristic
233	C₂²×C₈	—	30	2	no	—
234	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
235	#117	C₂×D₁₅	1	1	yes	characteristic
236	#108	—	15	4	no	—
237	#104	—	15	8	no	—
238	#97	C₂×D₁₅	1	2	yes	complemented
239	C₄×C₈	C₂×D₁₅	1	2	yes	complemented
240	C₂×C₃×C₅	—	2	1	no	—
241	C₂×C₃×C₅	—	2	2	no	—
242	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
243	D₁₅	—	4	4	no	—
244	C₂³×C₃	#709	1	1	yes	characteristic
245	C₂²×D₃	—	10	2	no	—
246	C₂×C₃×C₄	—	2	2	no	—
247	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
248	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
249	C₂×C₃×C₄	#709	1	1	yes	characteristic
250	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
251	#67	—	10	4	no	—
252	#67	—	10	4	no	—
253	C₂×Dic₃	—	10	2	no	—
254	D₁₂	—	5	4	no	—
255	D₁₂	—	10	2	no	—
256	C₄×D₃	—	10	4	no	—
257	C₄×D₃	—	10	4	no	—
258	Dic₆	—	5	4	no	—
259	Dic₆	—	10	2	no	—
260	C₃×C₈	C₂×C₄×D₅	1	4	yes	—
261	C₃×C₈	C₂×D₂₀	1	2	yes	—
262	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
263	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
264	C₃⋊C₈	—	10	4	no	—
265	C₂²×C₅	—	4	1	no	—
266	C₂²×C₅	C₄×D₁₂	1	2	yes	—
267	C₂²×C₅	C₂×#235	1	1	yes	characteristic
268	C₂×D₅	—	6	4	no	—
269	C₂×D₅	—	12	4	no	—
270	C₄×C₅	—	4	2	no	—
271	C₄×C₅	C₄×D₁₂	1	2	yes	—
272	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
273	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
274	Dic₅	—	6	4	no	—
275	#41	—	15	4	no	—
276	#41	—	15	4	no	—
277	#41	—	30	2	no	—
278	#41	—	30	2	no	—
279	C₂×Q₈	—	15	2	no	—
280	C₂×D₄	—	15	2	no	—
281	C₂×D₄	—	30	2	no	—
282	C₂²×C₄	—	30	2	no	—
283	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
284	Q₁₆	—	15	4	no	—
285	SD₁₆	—	15	8	no	—
286	D₈	—	15	4	no	—
287	M₁₆	—	15	4	no	—
288	M₁₆	—	30	2	no	—
289	M₁₆	—	30	4	no	—
290	M₁₆	—	30	4	no	—
291	M₁₆	C₄×D₁₅	1	4	yes	—
292	C₂×C₈	—	30	2	no	—
293	C₂×C₈	—	30	4	no	—
294	C₂×C₈	—	30	4	no	—
295	C₂×C₈	D₆₀	1	2	yes	—
296	C₂×C₈	D₆₀	1	2	yes	—
297	C₂×C₈	C₂²×D₁₅	1	2	yes	—
298	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
299	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
300	C₄⋊C₄	—	2	1	no	—
301	C₂²⋊C₄	—	2	1	no	—
302	C₄²	C₂²×D₁₅	1	2	yes	—
303	C₃×C₅	#2955	1	1	yes	characteristic, complemented, Hall
304	C₂²×C₃	—	4	1	no	—
305	C₂²×C₃	C₄×D₂₀	1	2	yes	—
306	C₂²×C₃	C₂×#709	1	1	yes	characteristic
307	C₂×D₃	—	10	4	no	—
308	C₂×D₃	—	20	4	no	—
309	C₃×C₄	—	4	2	no	—
310	C₃×C₄	C₄×D₂₀	1	2	yes	—
311	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
312	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
313	Dic₃	—	10	4	no	—
314	C₂×C₅	—	2	1	no	—
315	C₂×C₅	—	2	2	no	—
316	C₂×C₅	C₂×C₄×D₁₂	1	1	yes	characteristic
317	D₅	—	12	4	no	—
318	C₂³	—	30	2	no	—
319	C₂³	#6817	1	1	yes	characteristic
320	Q₈	—	15	4	no	—
321	Q₈	—	30	2	no	—
322	D₄	—	15	4	no	—
323	D₄	—	30	4	no	—
324	D₄	—	30	4	no	—
325	D₄	—	30	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	30	2	no	—
328	C₂×C₄	—	30	4	no	—
329	C₂×C₄	—	30	4	no	—
330	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
331	C₂×C₄	C₂×D₆₀	1	2	yes	—
332	C₂×C₄	#6817	1	1	yes	characteristic
333	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
334	C₈	—	30	4	no	—
335	C₈	C₂×C₄×D₁₅	1	4	yes	—
336	C₈	C₂×D₆₀	1	2	yes	—
337	C₈	C₂×D₆₀	1	1	yes	characteristic
338	C₈	C₂×D₆₀	1	1	yes	characteristic
339	C₂×C₃	—	2	1	no	—
340	C₂×C₃	—	2	2	no	—
341	C₂×C₃	C₂×C₄×D₂₀	1	1	yes	characteristic
342	D₃	—	20	4	no	—
343	C₅	#78579	1	1	yes	characteristic, complemented, Sylow
344	C₂²	—	4	1	no	—
345	C₂²	—	30	4	no	—
346	C₂²	—	60	4	no	—
347	C₂²	C₄×D₆₀	1	2	yes	—
348	C₂²	C₂×#6817	1	1	yes	characteristic
349	C₄	—	4	2	no	—
350	C₄	—	30	4	no	—
351	C₄	C₄×D₆₀	1	2	yes	—
352	C₄	C₂²×C₄×D₁₅	1	1	yes	characteristic
353	C₄	C₂²×D₆₀	1	1	yes	characteristic
354	C₃	#10610310	1	1	yes	characteristic, complemented, Sylow
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	60	4	no	—
358	C₂	C₂×C₄×D₆₀	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₆₀
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₃×C₅×#405	C₂
socle	C₂×C₃×C₅	C₂×C₄×D₄
prime radical	#11756507	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	#405	D₁₅
hypocenter	C₃×C₅	#2955
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. C₂×#91790	C₂	semidirect
3. C₃×C₅×#405	C₂	semidirect
4. #11751226	C₂	semidirect
5. #11751106	C₂	semidirect
6. #11750818	C₂	semidirect
7. #11750801	C₂	semidirect
14. #91798	C₄	semidirect
15. #91797	C₄	semidirect
19. #91790	C₂²	semidirect
20. #91790	C₂²	semidirect
23. C₃×C₅×#97	C₂²	semidirect
24. C₃×C₄×C₅×C₈	C₂²	semidirect
25. #91103	C₂²	semidirect
26. #91089	C₂²	semidirect
30. C₅×#405	D₃	semidirect
65. C₃×#405	D₅	semidirect
84. C₅×#97	C₂×D₃	semidirect
85. C₄×C₅×C₈	C₂×D₃	semidirect
124. C₃×#97	C₂×D₅	semidirect
125. C₃×C₄×C₈	C₂×D₅	semidirect
162. #405	D₁₅	semidirect
238. #97	C₂×D₁₅	semidirect
239. C₄×C₈	C₂×D₁₅	semidirect
303. C₃×C₅	#2955	semidirect
343. C₅	#78579	semidirect
354. C₃	#10610310	semidirect

Maximal subgroups

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

subgroup	index
#11756507	2
C₂×#91790	2
C₃×C₅×#405	2
#11751226	2
#11751106	2
#11750818	2
#11750801	2
#10610310	3
#78579	5