Subgroups — #14458216

Order 1920.

952 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11753981	C₂	1	2	yes	maximal, complemented
2	#11753971	C₂	1	2	yes	maximal, complemented
3	#11750801	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4700	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64984	C₂	1	1	yes	maximal, characteristic, complemented
6	#11748088	C₂	1	2	yes	maximal, complemented
7	#11748084	C₂	1	2	yes	maximal, complemented
8	#11747114	C₂	1	4	yes	maximal, complemented
9	#10611625	—	3	1	no	maximal, Hall
10	#92017	C₂²	1	4	yes	complemented
11	#91793	C₂²	1	1	yes	characteristic
12	#91790	C₂²	1	2	yes	complemented
13	C₅×#904	C₂²	1	1	yes	characteristic
14	C₅×#901	C₂²	1	2	yes	complemented
15	C₂×C₃×D₄₀	C₂²	1	2	yes	complemented
16	C₂×C₃×D₄₀	C₂²	1	2	yes	complemented
17	C₃×#4020	C₂²	1	2	yes	complemented
18	#91312	C₂²	1	4	yes	complemented
19	#91298	C₂²	1	4	yes	complemented
20	#91277	—	2	2	no	—
21	#91273	—	2	2	no	—
22	#91258	—	2	4	no	—
23	#91256	—	2	2	no	—
24	#91254	—	2	4	no	—
25	D₃×D₄₀	—	2	2	no	—
26	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic, complemented
27	#91103	C₂²	1	1	yes	characteristic, complemented
28	#91089	C₂²	1	2	yes	complemented
29	C₅×#819	C₂²	1	1	yes	characteristic, complemented
30	C₅×#805	C₂²	1	2	yes	complemented
31	#90958	C₂²	1	4	yes	complemented
32	#81791	—	5	1	no	maximal, Hall
33	#66103	—	3	2	no	—
34	#66086	—	3	2	no	—
35	C₅×#443	—	3	1	no	—
36	#65431	—	3	2	no	—
37	#65186	—	3	2	no	—
38	#65111	—	3	4	no	—
39	#64989	—	3	1	no	—
40	#64984	D₃	1	1	yes	characteristic, complemented
41	#6817	C₂³	1	2	yes	—
42	C₅×#235	C₂³	1	2	yes	—
43	C₂×C₃×D₂₀	—	4	2	no	—
44	C₂×C₃×D₂₀	C₂³	1	4	yes	—
45	D₅×#67	—	4	4	no	—
46	#6777	—	2	4	no	—
47	D₃×D₂₀	—	2	4	no	—
48	#6766	—	2	4	no	—
49	#6765	—	2	4	no	—
50	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
52	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
53	#6710	C₂³	1	2	yes	—
54	Dic₆₀	D₄	1	1	yes	characteristic, complemented
55	D₁₂₀	D₄	1	1	yes	characteristic, complemented
56	#6706	D₄	1	2	yes	complemented
57	#6705	—	2	2	no	—
58	C₈×D₁₅	—	2	2	no	—
59	C₅×#208	C₂³	1	2	yes	—
60	C₅×Dic₁₂	D₄	1	1	yes	characteristic, complemented
61	C₅×D₂₄	D₄	1	1	yes	characteristic, complemented
62	C₅×#204	D₄	1	2	yes	complemented
63	C₅×ΓL(1,𝔽₂₅)	—	2	2	no	—
64	C₅×D₃×C₈	—	2	2	no	—
65	C₃×D₄₀	—	2	2	no	—
66	C₃×D₄₀	—	2	2	no	—
67	C₃×D₄₀	—	2	2	no	—
68	C₃×D₄₀	—	2	2	no	—
69	#6657	—	2	4	no	—
70	#6655	—	2	4	no	—
71	#6653	—	2	4	no	—
72	#6652	—	2	4	no	—
73	#5757	—	5	2	no	—
74	#5757	—	5	2	no	—
75	C₃×#493	—	5	1	no	—
76	#5160	—	5	2	no	—
77	#5160	—	5	2	no	—
78	#5077	—	5	4	no	—
79	#4700	—	5	1	no	—
80	#4700	D₅	1	1	yes	characteristic, complemented
81	#4149	—	3	4	no	—
82	C₅×#135	—	3	1	no	—
83	C₅×#131	—	3	2	no	—
84	#4095	—	3	4	no	—
85	#4065	—	6	2	no	—
86	#4060	—	6	4	no	—
87	D₅×D₈	—	6	2	no	—
88	#4054	—	3	4	no	—
89	#4054	—	6	4	no	—
90	#4050	—	3	1	no	—
91	#4050	—	6	2	no	—
92	C₂×D₄₀	—	6	2	no	—
93	C₂×D₄₀	C₂×D₃	1	2	yes	complemented
94	C₂×D₄₀	C₂×D₃	1	2	yes	complemented
95	#4047	—	3	2	no	—
96	#4020	C₂×D₃	1	2	yes	complemented
97	C₅×#108	—	3	1	no	—
98	C₅×#104	—	3	2	no	—
99	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic, complemented
100	#3955	—	3	4	no	—
101	#3951	—	3	1	no	—
102	#3937	—	3	2	no	—
103	#3292	—	15	1	no	Sylow
104	C₂²×C₃×D₅	—	2	4	no	—
105	C₂×D₃×D₅	—	4	4	no	—
106	C₂×C₃×C₄×C₅	—	2	1	no	—
107	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
108	#1220	—	2	2	no	—
109	D₆₀	C₂×D₄	1	2	yes	—
110	C₄×D₁₅	—	2	2	no	—
111	Dic₃₀	C₂×D₄	1	2	yes	—
112	C₅×#67	—	2	2	no	—
113	C₅×D₁₂	C₂×D₄	1	2	yes	—
114	C₄×C₅×D₃	—	2	2	no	—
115	C₅×Dic₆	C₂×D₄	1	2	yes	—
116	C₃×D₂₀	—	2	4	no	—
117	C₃×D₂₀	—	2	4	no	—
118	C₃×D₂₀	—	8	4	no	—
119	#1202	—	4	4	no	—
120	#1201	—	4	4	no	—
121	D₅×Dic₃	—	4	4	no	—
122	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
123	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
124	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
125	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
126	C₃⋊#168	—	2	2	no	—
127	C₅×C₃⋊C₈	—	2	2	no	—
128	#1004	—	5	4	no	—
129	C₂×C₃×D₈	—	5	2	no	—
130	C₂×C₃×D₈	—	5	2	no	—
131	C₃×#127	—	5	2	no	—
132	#932	—	5	4	no	—
133	#932	—	5	4	no	—
134	#912	—	10	2	no	—
135	#912	—	10	2	no	—
136	#911	—	10	4	no	—
137	#911	—	10	4	no	—
138	D₃×D₈	—	10	2	no	—
139	D₃×D₈	—	10	2	no	—
140	#904	—	5	1	no	—
141	#904	C₂×D₅	1	1	yes	characteristic
142	#901	—	5	2	no	—
143	#901	C₂×D₅	1	2	yes	complemented
144	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic, complemented
145	#833	—	5	4	no	—
146	#819	—	5	1	no	—
147	#819	C₂×D₅	1	1	yes	characteristic, complemented
148	#805	—	5	2	no	—
149	#805	C₂×D₅	1	2	yes	complemented
150	C₅×#41	—	3	2	no	—
151	#713	—	6	4	no	—
152	D₄×D₅	—	6	4	no	—
153	D₄×D₅	—	12	4	no	—
154	#709	—	3	2	no	—
155	#709	—	6	4	no	—
156	C₂×D₂₀	—	4	2	no	—
157	C₂×D₂₀	—	6	4	no	—
158	C₂×D₂₀	C₂²×D₃	1	4	yes	—
159	C₅×Q₁₆	—	3	1	no	—
160	C₅×SD₁₆	—	3	2	no	—
161	C₅×D₈	—	3	1	no	—
162	C₅×M₁₆	—	3	2	no	—
163	C₅×M₁₆	—	6	2	no	—
164	C₂×C₅×C₈	—	6	2	no	—
165	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
166	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
167	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
168	#688	—	6	4	no	—
169	#686	—	6	4	no	—
170	#681	—	3	2	no	—
171	Dic₂₀	—	3	1	no	—
172	D₄₀	—	2	2	no	—
173	D₄₀	—	2	2	no	—
174	D₄₀	—	2	2	no	—
175	D₄₀	—	2	2	no	—
176	D₄₀	—	3	1	no	—
177	D₄₀	—	6	4	no	—
178	#677	—	3	2	no	—
179	#677	—	6	4	no	—
180	#676	—	6	2	no	—
181	C₈×D₅	—	6	2	no	—
182	#576	—	15	2	no	—
183	#576	—	15	2	no	—
184	#493	—	5	1	no	—
185	#472	—	15	2	no	—
186	#472	—	15	2	no	—
187	#453	—	15	4	no	—
188	#443	—	3	1	no	—
189	#443	—	15	1	no	—
190	C₂²×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
191	C₂×D₁₅	—	2	2	no	—
192	C₂×C₅×D₃	—	2	2	no	—
193	C₂×C₃×D₅	—	4	4	no	—
194	C₂×C₃×D₅	—	8	4	no	—
195	D₃×D₅	—	16	4	no	—
196	C₃×C₄×C₅	—	2	2	no	—
197	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
198	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
199	Dic₁₅	—	2	2	no	—
200	C₅×Dic₃	—	2	2	no	—
201	C₂×C₃×D₄	—	5	4	no	—
202	C₂×C₃×D₄	—	20	2	no	—
203	C₂×#67	—	20	4	no	—
204	#237	—	10	4	no	—
205	#237	—	10	4	no	—
206	D₃×D₄	—	10	4	no	—
207	D₃×D₄	—	10	4	no	—
208	#235	—	5	2	no	—
209	#235	C₂²×D₅	1	2	yes	—
210	C₃×D₈	—	10	2	no	—
211	C₃×D₈	—	10	2	no	—
212	C₃×D₈	—	10	2	no	—
213	C₃×D₈	—	10	2	no	—
214	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
215	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
216	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
217	#214	—	10	4	no	—
218	#214	—	10	4	no	—
219	#213	—	10	4	no	—
220	#213	—	10	4	no	—
221	#208	—	5	2	no	—
222	#208	C₂²×D₅	1	2	yes	—
223	Dic₁₂	—	5	1	no	—
224	Dic₁₂	D₂₀	1	1	yes	characteristic, complemented
225	D₂₄	—	5	1	no	—
226	D₂₄	D₂₀	1	1	yes	characteristic, complemented
227	#204	—	5	2	no	—
228	#204	D₂₀	1	2	yes	complemented
229	ΓL(1,𝔽₂₅)	—	2	2	no	—
230	ΓL(1,𝔽₂₅)	—	10	2	no	—
231	D₃×C₈	—	2	2	no	—
232	D₃×C₈	—	10	2	no	—
233	C₂²×D₅	—	2	4	no	—
234	C₂²×D₅	—	12	4	no	—
235	C₅×Q₈	—	3	2	no	—
236	C₅×D₄	—	3	2	no	—
237	C₅×D₄	—	6	2	no	—
238	C₂×C₄×C₅	—	2	1	no	—
239	C₂×C₄×C₅	—	6	2	no	—
240	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
241	C₂²⋊D₅	—	6	2	no	—
242	C₂²⋊D₅	—	12	4	no	—
243	D₂₀	—	2	4	no	—
244	D₂₀	—	2	4	no	—
245	D₂₀	—	3	2	no	—
246	D₂₀	—	8	4	no	—
247	D₂₀	—	12	4	no	—
248	C₄×D₅	—	6	2	no	—
249	C₄×D₅	—	12	4	no	—
250	Dic₁₀	—	3	2	no	—
251	C₅×C₈	—	6	2	no	—
252	C₅×C₈	D₃×D₄	1	1	yes	characteristic
253	C₅×C₈	D₃×D₄	1	1	yes	characteristic
254	C₅×C₈	D₃×D₄	1	1	yes	characteristic
255	C₅×C₈	D₃×D₄	1	1	yes	characteristic
256	#168	—	6	2	no	—
257	H(5,𝔽₂)	—	15	4	no	—
258	AGL(1,ℤ/8)	—	15	4	no	—
259	AGL(1,ℤ/8)	—	15	4	no	—
260	AGL(1,ℤ/8)	—	30	4	no	—
261	AGL(1,ℤ/8)	—	30	4	no	—
262	#135	—	3	1	no	—
263	#135	—	15	1	no	—
264	#135	—	30	2	no	—
265	#135	—	30	2	no	—
266	C₂×D₈	—	5	2	no	—
267	C₂×D₈	—	5	2	no	—
268	C₂×D₈	—	30	2	no	—
269	C₂×D₈	—	30	2	no	—
270	#131	—	3	2	no	—
271	#131	—	15	2	no	—
272	#127	—	5	2	no	—
273	#108	—	3	1	no	—
274	#108	—	15	1	no	—
275	#104	—	3	2	no	—
276	#104	—	15	2	no	—
277	#100	—	15	4	no	—
278	C₄×C₈	D₃×D₅	1	1	yes	characteristic, complemented
279	C₂×C₃×C₅	—	2	1	no	—
280	C₂×C₃×C₅	D₄²	1	1	yes	characteristic
281	D₁₅	—	4	2	no	—
282	C₃×D₅	—	8	4	no	—
283	C₅×D₃	—	4	2	no	—
284	C₂³×C₃	—	10	4	no	—
285	C₂²×D₃	—	20	4	no	—
286	C₃×D₄	—	10	4	no	—
287	C₃×D₄	—	10	4	no	—
288	C₃×D₄	—	40	4	no	—
289	C₂×C₃×C₄	—	2	1	no	—
290	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
291	#67	—	2	2	no	—
292	#67	—	10	2	no	—
293	#67	—	20	4	no	—
294	#67	—	20	4	no	—
295	C₂×Dic₃	—	20	4	no	—
296	D₁₂	—	5	2	no	—
297	D₁₂	C₂×D₂₀	1	2	yes	—
298	C₄×D₃	—	2	2	no	—
299	C₄×D₃	—	10	2	no	—
300	Dic₆	—	5	2	no	—
301	Dic₆	C₂×D₂₀	1	2	yes	—
302	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
303	C₃×C₈	C₂×D₂₀	1	1	yes	characteristic
304	C₃×C₈	D₄×D₅	1	1	yes	characteristic
305	C₃×C₈	D₄×D₅	1	1	yes	characteristic
306	C₃⋊C₈	—	2	2	no	—
307	C₃⋊C₈	—	10	2	no	—
308	C₂²×C₅	—	6	2	no	—
309	C₂²×C₅	#1004	1	1	yes	characteristic
310	C₂×D₅	—	4	4	no	—
311	C₂×D₅	—	6	2	no	—
312	C₂×D₅	—	8	4	no	—
313	C₂×D₅	—	48	4	no	—
314	C₄×C₅	—	2	2	no	—
315	C₄×C₅	—	6	2	no	—
316	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
317	C₄×C₅	C₂×D₃×D₄	1	1	yes	characteristic
318	Dic₅	—	6	2	no	—
319	#41	—	3	2	no	—
320	#41	—	15	2	no	—
321	#41	—	30	4	no	—
322	#41	—	30	4	no	—
323	C₂×D₄	—	5	4	no	—
324	C₂×D₄	—	20	2	no	—
325	C₂×D₄	—	30	4	no	—
326	C₂×D₄	—	30	4	no	—
327	C₂×D₄	—	60	4	no	—
328	Q₁₆	—	3	1	no	—
329	Q₁₆	—	15	1	no	—
330	SD₁₆	—	3	2	no	—
331	SD₁₆	—	15	2	no	—
332	SD₁₆	—	30	4	no	—
333	SD₁₆	—	30	4	no	—
334	D₈	—	3	1	no	—
335	D₈	—	10	2	no	—
336	D₈	—	10	2	no	—
337	D₈	—	10	2	no	—
338	D₈	—	10	2	no	—
339	D₈	—	15	1	no	—
340	D₈	—	30	4	no	—
341	D₈	—	30	4	no	—
342	M₁₆	—	3	2	no	—
343	M₁₆	—	6	2	no	—
344	M₁₆	—	15	2	no	—
345	M₁₆	—	30	2	no	—
346	C₂×C₈	—	6	2	no	—
347	C₂×C₈	—	30	2	no	—
348	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
349	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
350	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
351	C₃×C₅	#3292	1	1	yes	characteristic, complemented, Hall
352	C₂²×C₃	—	20	4	no	—
353	C₂²×C₃	—	40	4	no	—
354	C₂²×C₃	#4149	1	1	yes	characteristic
355	C₂×D₃	—	2	2	no	—
356	C₂×D₃	—	10	2	no	—
357	C₂×D₃	—	80	4	no	—
358	C₃×C₄	—	2	2	no	—
359	C₃×C₄	C₂²×D₂₀	1	1	yes	characteristic
360	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
361	Dic₃	—	2	2	no	—
362	Dic₃	—	10	2	no	—
363	C₂×C₅	—	2	1	no	—
364	C₂×C₅	—	12	2	no	—
365	C₂×C₅	#5717	1	1	yes	characteristic
366	D₅	—	8	4	no	—
367	D₅	—	12	2	no	—
368	C₂³	—	10	4	no	—
369	C₂³	—	60	4	no	—
370	Q₈	—	3	2	no	—
371	Q₈	—	15	2	no	—
372	D₄	—	3	2	no	—
373	D₄	—	6	2	no	—
374	D₄	—	10	4	no	—
375	D₄	—	10	4	no	—
376	D₄	—	15	2	no	—
377	D₄	—	30	2	no	—
378	D₄	—	40	4	no	—
379	D₄	—	60	4	no	—
380	D₄	—	60	4	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	6	2	no	—
383	C₂×C₄	—	30	2	no	—
384	C₂×C₄	—	60	4	no	—
385	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
386	C₈	—	6	2	no	—
387	C₈	—	30	2	no	—
388	C₈	D₃×D₂₀	1	1	yes	characteristic
389	C₈	D₃×D₂₀	1	1	yes	characteristic
390	C₈	#6777	1	1	yes	characteristic
391	C₈	#6777	1	1	yes	characteristic
392	C₂×C₃	—	2	1	no	—
393	C₂×C₃	—	40	4	no	—
394	C₂×C₃	D₄×D₂₀	1	1	yes	characteristic
395	D₃	—	4	2	no	—
396	D₃	—	20	2	no	—
397	C₅	#81791	1	1	yes	characteristic, complemented, Sylow
398	C₂²	—	6	2	no	—
399	C₂²	—	20	4	no	—
400	C₂²	—	30	2	no	—
401	C₂²	—	40	4	no	—
402	C₂²	—	240	4	no	—
403	C₂²	#92017	1	1	yes	characteristic
404	C₄	—	2	2	no	—
405	C₄	—	6	2	no	—
406	C₄	—	30	2	no	—
407	C₄	C₂×D₃×D₂₀	1	1	yes	characteristic
408	C₄	C₂×#6777	1	1	yes	characteristic
409	C₃	#10611625	1	1	yes	characteristic, complemented, Sylow
410	C₂	—	2	1	no	—
411	C₂	—	12	2	no	—
412	C₂	—	40	4	no	—
413	C₂	—	60	2	no	—
414	C₂	#11752745	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#11752745
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂×C₃×C₅	D₄²
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#3292
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #11753981	C₂	semidirect
2. #11753971	C₂	semidirect
3. #11750801	C₂	semidirect
4. C₅×#4700	C₂	semidirect
5. C₃×#64984	C₂	semidirect
6. #11748088	C₂	semidirect
7. #11748084	C₂	semidirect
8. #11747114	C₂	semidirect
10. #92017	C₂²	semidirect
12. #91790	C₂²	semidirect
14. C₅×#901	C₂²	semidirect
15. C₂×C₃×D₄₀	C₂²	semidirect
16. C₂×C₃×D₄₀	C₂²	semidirect
17. C₃×#4020	C₂²	semidirect
18. #91312	C₂²	semidirect
19. #91298	C₂²	semidirect
26. C₃×C₄×C₅×C₈	C₂²	semidirect
27. #91103	C₂²	semidirect
28. #91089	C₂²	semidirect
29. C₅×#819	C₂²	semidirect
30. C₅×#805	C₂²	semidirect
31. #90958	C₂²	semidirect
40. #64984	D₃	semidirect
54. Dic₆₀	D₄	semidirect
55. D₁₂₀	D₄	semidirect
56. #6706	D₄	semidirect
60. C₅×Dic₁₂	D₄	semidirect
61. C₅×D₂₄	D₄	semidirect
62. C₅×#204	D₄	semidirect
80. #4700	D₅	semidirect
93. C₂×D₄₀	C₂×D₃	semidirect
94. C₂×D₄₀	C₂×D₃	semidirect
96. #4020	C₂×D₃	semidirect
99. C₄×C₅×C₈	C₂×D₃	semidirect
143. #901	C₂×D₅	semidirect
144. C₃×C₄×C₈	C₂×D₅	semidirect
147. #819	C₂×D₅	semidirect
149. #805	C₂×D₅	semidirect
224. Dic₁₂	D₂₀	semidirect
226. D₂₄	D₂₀	semidirect
228. #204	D₂₀	semidirect
278. C₄×C₈	D₃×D₅	semidirect
351. C₃×C₅	#3292	semidirect
397. C₅	#81791	semidirect
409. C₃	#10611625	semidirect

Maximal subgroups

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

subgroup	index
#11753981	2
#11753971	2
#11750801	2
C₅×#4700	2
C₃×#64984	2
#11748088	2
#11748084	2
#11747114	2
#10611625	3
#81791	5