Subgroups — #14296497

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅⋊#822	C₂	1	1	yes	maximal, characteristic
2	#11751726	C₂	1	1	yes	maximal, characteristic
3	#11751696	C₂	1	1	yes	maximal, characteristic
4	#11751695	C₂	1	1	yes	maximal, characteristic
5	C₄×D₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#11746733	C₂	1	1	yes	maximal, characteristic
7	C₃×#64862	C₂	1	1	yes	maximal, characteristic
8	#10603851	—	3	1	no	maximal, Hall
9	#91989	C₂²	1	1	yes	characteristic
10	D₅⋊#208	C₂²	1	1	yes	characteristic
11	C₃×#4130	C₂²	1	1	yes	characteristic
12	C₃×D₅⋊M₁₆	C₂²	1	1	yes	characteristic
13	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
14	C₂×D₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×D₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	#91162	C₄	1	2	yes	complemented
17	#91161	C₄	1	2	yes	complemented
18	#91160	C₄	1	2	yes	complemented
19	#91159	C₄	1	2	yes	complemented
20	#91150	—	2	1	no	—
21	#91149	—	2	1	no	—
22	ΓL(1,𝔽₁₆)⋊C₈	—	2	1	no	—
23	AGL(1,𝔽₅)×C₃⋊C₈	—	2	1	no	—
24	C₄×C₃⋊#168	C₄	1	1	yes	characteristic, complemented
25	C₄×C₅×C₃⋊C₈	C₄	1	1	yes	characteristic, complemented
26	Dic₅×C₃⋊C₈	C₄	1	1	yes	characteristic, complemented
27	Dic₅×C₃⋊C₈	C₄	1	1	yes	characteristic, complemented
28	#68741	—	5	1	no	maximal, Hall
29	D₅⋊#108	—	3	1	no	—
30	#65721	—	3	1	no	—
31	#65718	—	3	1	no	—
32	#65718	—	3	1	no	—
33	C₄×C₈×D₅	—	3	1	no	—
34	#64862	—	3	1	no	—
35	#64862	D₃	1	1	yes	characteristic
36	C₂×C₃×C₄×D₅	—	2	1	no	—
37	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
38	#6763	—	2	1	no	—
39	#6762	—	2	1	no	—
40	C₄⋊ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
41	C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
42	#6758	C₂×C₄	1	2	yes	—
43	D₅⋊(C₃⋊C₈)	—	2	1	no	—
44	C₃×C₂²⋊AGL(1,𝔽₅)	—	2	1	no	—
45	C₃×#704	—	2	1	no	—
46	C₃×C₄⋊AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
47	C₃×C₄×AGL(1,𝔽₅)	—	2	1	no	—
48	C₃×#700	C₂×C₄	1	2	yes	—
49	C₃×D₅⋊C₈	—	2	1	no	—
50	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
51	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
52	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
53	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₃×C₄×Dic₅	—	2	1	no	—
56	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
57	D₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
58	D₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
59	D₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
60	D₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
61	D₅×C₃⋊C₈	D₄	1	1	yes	characteristic
62	D₅×C₃⋊C₈	D₄	1	1	yes	characteristic
63	D₅×C₃⋊C₈	Q₈	1	1	yes	characteristic
64	D₅×C₃⋊C₈	Q₈	1	1	yes	characteristic
65	C₂×#822	—	5	1	no	—
66	#5122	—	5	1	no	—
67	#4998	—	5	1	no	—
68	#4997	—	5	1	no	—
69	C₂×C₄×C₃⋊C₈	—	5	1	no	—
70	C₃×#339	—	5	1	no	—
71	#4523	—	5	1	no	—
72	#4130	—	3	1	no	—
73	#4130	C₂×D₃	1	1	yes	characteristic
74	D₅⋊M₁₆	—	3	1	no	—
75	D₅⋊M₁₆	C₂×D₃	1	1	yes	characteristic
76	C₂×C₈×D₅	—	3	1	no	—
77	C₂×C₈×D₅	—	3	1	no	—
78	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
79	#3996	—	3	2	no	—
80	#3995	—	3	2	no	—
81	#3994	—	3	2	no	—
82	#3993	—	3	2	no	—
83	#3992	—	6	1	no	—
84	#3992	—	6	1	no	—
85	C₈×AGL(1,𝔽₅)	—	6	1	no	—
86	C₈×AGL(1,𝔽₅)	—	6	1	no	—
87	C₄×C₅×C₈	—	3	1	no	—
88	C₈×Dic₅	—	3	1	no	—
89	C₈×Dic₅	—	3	1	no	—
90	C₄×#168	—	3	1	no	—
91	#1772	—	15	1	no	Sylow
92	C₂²×C₃×D₅	#41	1	1	yes	characteristic
93	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
94	C₂×C₃×AGL(1,𝔽₅)	—	2	2	no	—
95	C₂×C₃×C₄×C₅	—	2	1	no	—
96	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₃×Dic₅	—	2	1	no	—
98	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
99	C₃×C₄×D₅	—	2	4	no	—
100	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
101	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
102	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
103	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
104	C₃⋊#170	—	2	2	no	—
105	C₃×#170	—	2	2	no	—
106	C₃⋊#168	C₄²	1	1	yes	characteristic
107	C₃⋊#168	C₄²	1	1	yes	characteristic
108	C₃⋊#168	C₄⋊C₄	1	1	yes	characteristic
109	C₃⋊#168	C₄⋊C₄	1	1	yes	characteristic
110	C₅×C₃⋊C₈	C₄²	1	1	yes	characteristic
111	C₅×C₃⋊C₈	C₄²	1	1	yes	characteristic
112	C₅×C₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
113	C₅×C₃⋊C₈	C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₃×M₁₆	—	5	1	no	—
115	C₃×#117	—	5	1	no	—
116	C₂×C₃×C₄²	—	5	1	no	—
117	#926	—	5	1	no	—
118	C₂×#208	—	5	1	no	—
119	C₂²×C₃⋊C₈	—	5	1	no	—
120	C₂²×C₃⋊C₈	—	5	1	no	—
121	#822	—	5	2	no	—
122	#822	—	5	2	no	—
123	#815	—	10	1	no	—
124	C₈×Dic₃	—	10	1	no	—
125	#808	—	5	2	no	—
126	#807	—	5	2	no	—
127	#803	—	10	1	no	—
128	C₄×C₃⋊C₈	—	5	1	no	—
129	C₄×C₃⋊C₈	—	5	1	no	—
130	C₄×C₃⋊C₈	—	5	1	no	—
131	C₄×C₃⋊C₈	—	10	1	no	—
132	C₄×C₃⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
133	C₂×C₄×D₅	—	2	1	no	—
134	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
135	C₂²⋊AGL(1,𝔽₅)	—	2	1	no	—
136	C₂²⋊AGL(1,𝔽₅)	—	6	1	no	—
137	#704	—	2	1	no	—
138	#704	—	6	1	no	—
139	C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
140	C₄⋊AGL(1,𝔽₅)	C₄×D₃	1	2	yes	—
141	C₄×AGL(1,𝔽₅)	—	2	1	no	—
142	C₄×AGL(1,𝔽₅)	—	6	1	no	—
143	#700	—	3	2	no	—
144	#700	C₄×D₃	1	2	yes	—
145	D₅⋊C₈	—	2	1	no	—
146	D₅⋊C₈	—	6	1	no	—
147	C₂×C₅×C₈	—	3	1	no	—
148	C₂×C₅×C₈	—	3	1	no	—
149	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
150	C₄×Dic₅	—	2	1	no	—
151	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
152	C₂×#168	—	3	1	no	—
153	C₂×#168	—	3	1	no	—
154	C₈×D₅	—	3	1	no	—
155	C₈×D₅	—	3	1	no	—
156	C₈×D₅	—	3	1	no	—
157	C₈×D₅	—	3	1	no	—
158	C₈×D₅	—	3	1	no	—
159	C₈×D₅	—	3	1	no	—
160	C₈×D₅	—	3	1	no	—
161	C₈×D₅	—	3	1	no	—
162	C₂×#108	—	15	1	no	—
163	#427	—	15	1	no	—
164	#405	—	15	1	no	—
165	#405	—	15	1	no	—
166	C₂×C₄×C₈	—	15	1	no	—
167	#339	—	5	1	no	—
168	#339	—	15	1	no	—
169	C₂²×C₃×C₅	#117	1	1	yes	characteristic
170	C₂×C₃×D₅	—	2	2	no	—
171	C₂×C₃×D₅	C₄×D₄	1	1	yes	characteristic
172	C₂×C₃×D₅	C₄×Q₈	1	1	yes	characteristic
173	ΓL(1,𝔽₁₆)	—	4	2	no	—
174	C₃×AGL(1,𝔽₅)	—	4	2	no	—
175	C₃×C₄×C₅	—	2	2	no	—
176	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
177	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₃×Dic₅	—	2	2	no	—
179	C₃×Dic₅	C₄×D₄	1	1	yes	characteristic
180	C₃×Dic₅	C₄×Q₈	1	1	yes	characteristic
181	C₂²×C₃×C₄	—	5	1	no	—
182	C₂²×C₃×C₄	—	10	1	no	—
183	C₃×M₁₆	—	5	2	no	—
184	C₃×M₁₆	—	10	1	no	—
185	C₂×C₃×C₈	—	10	1	no	—
186	C₃×C₄⋊C₄	—	5	2	no	—
187	C₃×C₂²⋊C₄	—	10	1	no	—
188	C₃×C₄²	—	5	1	no	—
189	C₃×C₄²	—	10	1	no	—
190	C₃×C₄²	—	10	1	no	—
191	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
192	#217	—	10	1	no	—
193	C₄⋊Dic₃	—	5	2	no	—
194	C₄×Dic₃	—	10	1	no	—
195	#208	—	5	2	no	—
196	#208	—	10	1	no	—
197	C₂×C₃⋊C₈	—	5	1	no	—
198	C₂×C₃⋊C₈	—	5	1	no	—
199	C₂×C₃⋊C₈	—	5	1	no	—
200	C₂×C₃⋊C₈	—	5	1	no	—
201	C₂×C₃⋊C₈	—	5	1	no	—
202	C₂×C₃⋊C₈	—	5	1	no	—
203	C₂×C₃⋊C₈	—	5	1	no	—
204	C₂×C₃⋊C₈	—	5	1	no	—
205	C₂×C₃⋊C₈	—	5	1	no	—
206	C₂×C₃⋊C₈	—	5	1	no	—
207	C₂×C₃⋊C₈	—	10	1	no	—
208	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
209	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
210	C₂²×D₅	#237	1	1	yes	characteristic
211	C₂×AGL(1,𝔽₅)	—	2	2	no	—
212	C₂×AGL(1,𝔽₅)	—	6	2	no	—
213	C₂×C₄×C₅	—	2	1	no	—
214	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
215	C₂×Dic₅	—	2	1	no	—
216	C₂×Dic₅	#237	1	1	yes	characteristic
217	C₄×D₅	—	2	4	no	—
218	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
219	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
220	C₄×D₅	D₃×D₄	1	1	yes	characteristic
221	C₄×D₅	D₃×Q₈	1	1	yes	characteristic
222	#170	—	2	2	no	—
223	#170	—	6	2	no	—
224	C₅×C₈	—	3	1	no	—
225	C₅×C₈	—	3	1	no	—
226	C₅×C₈	—	3	1	no	—
227	C₅×C₈	—	3	1	no	—
228	#168	—	3	1	no	—
229	#168	—	3	1	no	—
230	#168	—	3	1	no	—
231	#168	—	3	1	no	—
232	C₂×M₁₆	—	5	1	no	—
233	C₂×M₁₆	—	15	1	no	—
234	C₂²×C₈	—	15	1	no	—
235	C₂²×C₈	—	15	1	no	—
236	#117	—	5	1	no	—
237	#117	—	15	1	no	—
238	C₂×C₄²	—	5	1	no	—
239	#108	—	15	2	no	—
240	#108	—	15	2	no	—
241	#107	—	15	2	no	—
242	#106	—	15	2	no	—
243	#97	—	30	1	no	—
244	#97	—	30	1	no	—
245	C₄×C₈	—	3	1	no	—
246	C₄×C₈	—	15	1	no	—
247	C₄×C₈	—	15	1	no	—
248	C₄×C₈	—	15	1	no	—
249	C₄×C₈	—	30	1	no	—
250	C₄×C₈	—	30	1	no	—
251	C₂×C₃×C₅	—	2	1	no	—
252	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
253	C₃×D₅	—	2	1	no	—
254	C₃×D₅	#443	1	2	yes	—
255	C₂³×C₃	—	5	1	no	—
256	C₂×C₃×C₄	—	2	1	no	—
257	C₂×C₃×C₄	—	5	1	no	—
258	C₂×C₃×C₄	—	5	1	no	—
259	C₂×C₃×C₄	—	5	1	no	—
260	C₂×C₃×C₄	—	5	1	no	—
261	C₂×C₃×C₄	—	5	1	no	—
262	C₂×C₃×C₄	—	10	4	no	—
263	C₂×C₃×C₄	—	10	1	no	—
264	C₂×C₃×C₄	—	10	2	no	—
265	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
266	C₂×Dic₃	—	10	2	no	—
267	C₃×C₈	—	10	2	no	—
268	C₃⋊C₈	—	5	1	no	—
269	C₃⋊C₈	—	5	1	no	—
270	C₃⋊C₈	—	5	1	no	—
271	C₃⋊C₈	—	5	1	no	—
272	C₃⋊C₈	—	10	2	no	—
273	C₃⋊C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
274	C₃⋊C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
275	C₃⋊C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
276	C₃⋊C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
277	C₂²×C₅	#877	1	1	yes	characteristic
278	C₂×D₅	—	2	2	no	—
279	C₂×D₅	#67⋊C₄	1	1	yes	characteristic
280	C₂×D₅	#887	1	1	yes	characteristic
281	AGL(1,𝔽₅)	—	4	2	no	—
282	AGL(1,𝔽₅)	—	12	2	no	—
283	C₄×C₅	—	2	2	no	—
284	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
285	C₄×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
286	Dic₅	—	2	2	no	—
287	Dic₅	#67⋊C₄	1	1	yes	characteristic
288	Dic₅	#887	1	1	yes	characteristic
289	C₂²×C₄	—	5	1	no	—
290	C₂²×C₄	—	10	1	no	—
291	M₁₆	—	5	2	no	—
292	M₁₆	—	10	1	no	—
293	M₁₆	—	15	2	no	—
294	M₁₆	—	30	1	no	—
295	C₂×C₈	—	3	1	no	—
296	C₂×C₈	—	3	1	no	—
297	C₂×C₈	—	10	1	no	—
298	C₂×C₈	—	15	1	no	—
299	C₂×C₈	—	15	1	no	—
300	C₂×C₈	—	15	1	no	—
301	C₂×C₈	—	15	1	no	—
302	C₂×C₈	—	15	1	no	—
303	C₂×C₈	—	15	1	no	—
304	C₂×C₈	—	15	1	no	—
305	C₂×C₈	—	15	1	no	—
306	C₂×C₈	—	15	1	no	—
307	C₂×C₈	—	15	1	no	—
308	C₂×C₈	—	30	1	no	—
309	C₄⋊C₄	—	5	2	no	—
310	C₄⋊C₄	—	15	2	no	—
311	C₂²⋊C₄	—	10	1	no	—
312	C₂²⋊C₄	—	30	1	no	—
313	C₄²	—	5	1	no	—
314	C₄²	—	10	1	no	—
315	C₄²	—	10	1	no	—
316	C₄²	—	30	1	no	—
317	C₄²	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
318	C₃×C₅	#1772	1	1	yes	characteristic, complemented, Hall
319	C₂²×C₃	—	5	1	no	—
320	C₂²×C₃	—	5	1	no	—
321	C₂²×C₃	—	10	2	no	—
322	C₂²×C₃	#4130	1	1	yes	characteristic
323	C₃×C₄	—	2	2	no	—
324	C₃×C₄	—	5	1	no	—
325	C₃×C₄	—	5	1	no	—
326	C₃×C₄	—	10	2	no	—
327	C₃×C₄	—	20	2	no	—
328	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
329	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
330	Dic₃	—	20	2	no	—
331	C₂×C₅	—	2	1	no	—
332	C₂×C₅	#4649	1	1	yes	characteristic
333	D₅	—	2	1	no	—
334	D₅	#4824	1	2	yes	—
335	C₂³	—	5	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	5	1	no	—
338	C₂×C₄	—	5	1	no	—
339	C₂×C₄	—	5	1	no	—
340	C₂×C₄	—	5	1	no	—
341	C₂×C₄	—	5	1	no	—
342	C₂×C₄	—	10	4	no	—
343	C₂×C₄	—	10	1	no	—
344	C₂×C₄	—	10	2	no	—
345	C₂×C₄	—	30	2	no	—
346	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
347	C₈	—	3	1	no	—
348	C₈	—	3	1	no	—
349	C₈	—	3	1	no	—
350	C₈	—	3	1	no	—
351	C₈	—	10	2	no	—
352	C₈	—	15	1	no	—
353	C₈	—	15	1	no	—
354	C₈	—	15	1	no	—
355	C₈	—	15	1	no	—
356	C₈	—	30	2	no	—
357	C₂×C₃	—	2	1	no	—
358	C₂×C₃	—	5	2	no	—
359	C₂×C₃	—	10	1	no	—
360	C₂×C₃	C₄×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
361	C₅	#68741	1	1	yes	characteristic, complemented, Sylow
362	C₂²	—	5	1	no	—
363	C₂²	—	5	1	no	—
364	C₂²	—	10	2	no	—
365	C₂²	#91923	1	1	yes	characteristic
366	C₄	—	2	2	no	—
367	C₄	—	5	1	no	—
368	C₄	—	5	1	no	—
369	C₄	—	10	2	no	—
370	C₄	—	20	2	no	—
371	C₄	—	60	2	no	—
372	C₄	C₄×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
373	C₄	D₃×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
374	C₃	#10603851	1	1	yes	characteristic, complemented, Sylow
375	C₂	—	2	1	no	—
376	C₂	—	5	2	no	—
377	C₂	—	10	1	no	—
378	C₂	#11751834	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₃×C₄⋊AGL(1,𝔽₅)
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
Fitting	C₃×C₄²×C₅	C₂×C₄
socle	C₂×C₃×C₅	C₄×C₄⋊C₄
prime radical	C₂²×C₃×D₅	#41
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	C₄²	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#1772
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
16. #91162	C₄	semidirect
17. #91161	C₄	semidirect
18. #91160	C₄	semidirect
19. #91159	C₄	semidirect
24. C₄×C₃⋊#168	C₄	semidirect
25. C₄×C₅×C₃⋊C₈	C₄	semidirect
26. Dic₅×C₃⋊C₈	C₄	semidirect
27. Dic₅×C₃⋊C₈	C₄	semidirect
132. C₄×C₃⋊C₈	AGL(1,𝔽₅)	semidirect
318. C₃×C₅	#1772	semidirect
361. C₅	#68741	semidirect
374. C₃	#10603851	semidirect

Maximal subgroups

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

subgroup	index
D₅⋊#822	2
#11751726	2
#11751696	2
#11751695	2
C₄×D₅×C₃⋊C₈	2
#11746733	2
C₃×#64862	2
#10603851	3
#68741	5