Subgroups — #14294261

Order 1920.

656 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91221	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3992	C₂	1	1	yes	maximal, characteristic
3	#11751788	C₂	1	1	yes	maximal, characteristic
4	#11751748	C₂	1	1	yes	maximal, characteristic
5	C₂×ΓL(1,𝔽₁₆)⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×AGL(1,𝔽₅)×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₈×D₅×Dic₃	C₂	1	1	yes	maximal, characteristic
8	#10603838	—	3	1	no	maximal, Hall
9	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
10	C₂×D₅⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄×AGL(1,𝔽₅)	C₂²	1	1	yes	characteristic
12	C₂×C₃×D₅⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
14	C₄×D₅×Dic₃	C₂²	1	1	yes	characteristic
15	C₂×D₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	#91221	C₄	1	4	yes	complemented
17	C₃×#3992	C₄	1	4	yes	complemented
18	#91169	C₄	1	4	yes	complemented
19	(C₂×AGL(1,𝔽₅))⋊Dic₃	C₄	1	4	yes	—
20	ΓL(1,𝔽₁₆)⋊C₈	C₄	1	4	yes	complemented
21	AGL(1,𝔽₅)×C₃⋊C₈	C₄	1	4	yes	complemented
22	C₈×Dic₁₅	C₄	1	1	yes	characteristic, complemented
23	C₅×C₈×Dic₃	C₄	1	1	yes	characteristic, complemented
24	#90951	C₄	1	1	yes	characteristic, complemented
25	Dic₃×#168	C₄	1	1	yes	characteristic, complemented
26	#69804	—	5	1	no	maximal, Hall
27	C₂×#3992	—	3	1	no	—
28	C₂×#3992	D₃	1	1	yes	characteristic
29	C₂×C₈×AGL(1,𝔽₅)	—	3	1	no	—
30	C₂×C₈×AGL(1,𝔽₅)	—	3	1	no	—
31	#65686	—	3	1	no	—
32	#65676	—	3	1	no	—
33	C₄×C₈×D₅	—	3	1	no	—
34	C₂²×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
35	C₂²×C₃×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
36	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
37	C₂×D₅×Dic₃	C₂×C₄	1	2	yes	—
38	C₂×C₃⋊#170	C₂×C₄	1	2	yes	—
39	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
40	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
41	D₅⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
42	D₅⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
43	C₂×C₃×#170	C₂×C₄	1	2	yes	—
44	C₃×C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
45	C₃×C₄×AGL(1,𝔽₅)	C₂×C₄	1	2	yes	—
46	C₃×D₅⋊C₈	C₂×C₄	1	2	yes	—
47	C₃×D₅⋊C₈	C₂×C₄	1	2	yes	—
48	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
49	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
50	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
51	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
54	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
55	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
56	Dic₃×Dic₅	C₂×C₄	1	2	yes	—
57	D₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
58	D₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
59	C₂×C₃×#97	—	5	1	no	—
60	C₂×#815	—	5	1	no	—
61	C₂×#815	—	5	1	no	—
62	C₂×C₈×Dic₃	—	5	1	no	—
63	C₂×C₈×Dic₃	—	5	1	no	—
64	#4932	—	5	1	no	—
65	C₂×C₄×C₃⋊C₈	—	5	1	no	—
66	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
67	C₂×C₄×AGL(1,𝔽₅)	C₂×D₃	1	1	yes	characteristic
68	C₂×D₅⋊C₈	—	3	1	no	—
69	C₂×D₅⋊C₈	C₂×D₃	1	1	yes	characteristic
70	C₂×C₈×D₅	—	3	1	no	—
71	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
72	C₄²×D₅	—	3	1	no	—
73	(C₂×AGL(1,𝔽₅))⋊C₄	—	3	4	no	—
74	#4002	—	3	4	no	—
75	#3992	—	3	4	no	—
76	#3992	Dic₃	1	4	yes	complemented
77	C₈×AGL(1,𝔽₅)	—	3	4	no	—
78	C₈×AGL(1,𝔽₅)	—	3	4	no	—
79	C₄×C₅×C₈	—	3	1	no	—
80	C₈×Dic₅	—	3	1	no	—
81	C₈×Dic₅	—	3	1	no	—
82	C₄×#168	—	3	1	no	—
83	#1726	—	15	1	no	Sylow
84	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
85	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
86	C₂×ΓL(1,𝔽₁₆)	C₄²	1	4	yes	—
87	C₂×C₃×AGL(1,𝔽₅)	—	2	4	no	—
88	C₂×C₃×AGL(1,𝔽₅)	C₄²	1	4	yes	—
89	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂×Dic₁₅	C₄²	1	2	yes	—
91	C₂×C₅×Dic₃	C₄²	1	2	yes	—
92	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
93	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
94	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
95	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
96	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
97	D₅×Dic₃	—	2	4	no	—
98	C₃⋊#170	C₄²	1	4	yes	—
99	C₃×#170	C₄²	1	4	yes	—
100	C₃×C₅×C₈	C₄²	1	2	yes	—
101	C₃⋊#168	C₄²	1	2	yes	—
102	C₃×#168	C₄²	1	2	yes	—
103	C₅×C₃⋊C₈	C₄²	1	2	yes	—
104	C₂²×C₃×C₈	—	5	1	no	—
105	C₂²×C₃×C₈	—	5	1	no	—
106	C₂×C₃×C₄²	—	5	1	no	—
107	C₂×C₄×Dic₃	—	5	1	no	—
108	C₂×C₄×Dic₃	—	5	1	no	—
109	C₂²×C₃⋊C₈	—	5	1	no	—
110	C₂²×C₃⋊C₈	—	5	1	no	—
111	C₃×#97	—	5	4	no	—
112	C₃⋊#95	—	5	4	no	—
113	#815	—	5	4	no	—
114	#815	—	5	4	no	—
115	C₈×Dic₃	—	5	4	no	—
116	C₈×Dic₃	—	5	1	no	—
117	C₈×Dic₃	—	5	1	no	—
118	C₈×Dic₃	—	5	1	no	—
119	C₈×Dic₃	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
120	C₄×C₃⋊C₈	—	5	4	no	—
121	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
122	C₂²×AGL(1,𝔽₅)	C₄×D₃	1	2	yes	—
123	C₂×C₄×D₅	—	3	2	no	—
124	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
125	C₂×#170	—	3	2	no	—
126	C₂×#170	C₄×D₃	1	2	yes	—
127	C₄×AGL(1,𝔽₅)	—	3	2	no	—
128	C₄×AGL(1,𝔽₅)	—	3	2	no	—
129	C₄×AGL(1,𝔽₅)	C₄×D₃	1	2	yes	—
130	C₄×AGL(1,𝔽₅)	C₂×Dic₃	1	2	yes	—
131	D₅⋊C₈	—	3	2	no	—
132	D₅⋊C₈	—	3	2	no	—
133	D₅⋊C₈	C₄×D₃	1	2	yes	—
134	D₅⋊C₈	C₂×Dic₃	1	2	yes	—
135	C₂×C₅×C₈	—	3	1	no	—
136	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
137	C₄²×C₅	—	3	1	no	—
138	C₄×Dic₅	—	3	1	no	—
139	C₄×Dic₅	—	3	2	no	—
140	C₂×#168	—	3	1	no	—
141	C₂×#168	C₄×D₃	1	1	yes	characteristic
142	C₈×D₅	—	3	2	no	—
143	C₈×D₅	—	3	2	no	—
144	C₈×D₅	C₄×D₃	1	2	yes	—
145	C₈×D₅	C₂×Dic₃	1	2	yes	—
146	C₂×#97	—	5	1	no	—
147	C₂×#97	—	15	1	no	—
148	C₂×#97	—	15	1	no	—
149	C₂×C₄×C₈	—	15	1	no	—
150	C₂×C₄×C₈	—	15	1	no	—
151	C₂×C₄×C₈	—	15	1	no	—
152	#376	—	15	1	no	—
153	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
154	C₂×C₃×D₅	C₂×C₄²	1	1	yes	characteristic
155	C₂×C₃×D₅	C₂×C₄²	1	1	yes	characteristic
156	C₂×C₃×D₅	#131	1	4	yes	—
157	ΓL(1,𝔽₁₆)	—	2	4	no	—
158	C₃×AGL(1,𝔽₅)	—	2	4	no	—
159	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
160	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
161	Dic₁₅	—	2	2	no	—
162	C₃×Dic₅	C₂×C₄²	1	1	yes	characteristic
163	C₃×Dic₅	C₂×C₄²	1	1	yes	characteristic
164	C₅×Dic₃	—	2	2	no	—
165	C₂²×C₃×C₄	—	5	1	no	—
166	C₂²×C₃×C₄	—	5	2	no	—
167	C₂²×Dic₃	—	5	2	no	—
168	C₂²×Dic₃	—	5	2	no	—
169	C₂×C₃×C₈	—	5	2	no	—
170	C₂×C₃×C₈	—	5	2	no	—
171	C₂×C₃×C₈	—	5	2	no	—
172	C₂×C₃×C₈	—	5	1	no	—
173	C₂×C₃×C₈	—	5	2	no	—
174	C₂×C₃×C₈	—	5	2	no	—
175	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
176	C₃×C₄²	—	5	2	no	—
177	C₃×C₄²	—	5	2	no	—
178	C₄×Dic₃	—	5	2	no	—
179	C₄×Dic₃	—	5	2	no	—
180	C₄×Dic₃	—	5	1	no	—
181	C₄×Dic₃	—	5	2	no	—
182	C₄×Dic₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
183	C₂×C₃⋊C₈	—	5	2	no	—
184	C₂×C₃⋊C₈	—	5	2	no	—
185	C₂×C₃⋊C₈	—	5	2	no	—
186	C₂×C₃⋊C₈	—	5	1	no	—
187	C₂×C₃⋊C₈	—	5	2	no	—
188	C₂×C₃⋊C₈	—	5	2	no	—
189	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
190	C₂²×D₅	C₂×C₄×D₃	1	1	yes	characteristic
191	C₂×AGL(1,𝔽₅)	—	2	4	no	—
192	C₂×AGL(1,𝔽₅)	—	3	4	no	—
193	C₂×AGL(1,𝔽₅)	—	6	4	no	—
194	C₂×AGL(1,𝔽₅)	C₄×Dic₃	1	4	yes	—
195	C₂×C₄×C₅	—	3	2	no	—
196	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
197	C₂×Dic₅	—	3	2	no	—
198	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
199	C₄×D₅	—	6	4	no	—
200	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
201	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
202	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
203	C₄×D₅	C₂²×Dic₃	1	1	yes	characteristic
204	#170	—	3	4	no	—
205	#170	C₄×Dic₃	1	4	yes	—
206	C₅×C₈	—	3	2	no	—
207	C₅×C₈	C₄×Dic₃	1	2	yes	—
208	#168	—	3	2	no	—
209	#168	C₄×Dic₃	1	2	yes	—
210	C₂²×C₈	—	5	1	no	—
211	C₂²×C₈	—	5	1	no	—
212	C₂²×C₈	—	15	1	no	—
213	C₂²×C₈	—	15	1	no	—
214	C₂×C₄²	—	5	1	no	—
215	C₂×C₄²	—	15	1	no	—
216	C₂×C₄²	—	15	1	no	—
217	#97	—	5	4	no	—
218	#97	—	15	4	no	—
219	#97	—	15	4	no	—
220	C₄×C₈	—	3	1	no	—
221	C₄×C₈	—	15	1	no	—
222	C₄×C₈	—	15	4	no	—
223	C₄×C₈	—	15	1	no	—
224	C₄×C₈	—	15	4	no	—
225	C₄×C₈	—	15	1	no	—
226	#95	—	15	4	no	—
227	C₂×C₃×C₅	C₄³	1	1	yes	characteristic
228	C₂×C₃×C₅	#405	1	2	yes	—
229	C₃×D₅	#405	1	2	yes	—
230	C₃×D₅	#405	1	2	yes	—
231	C₂³×C₃	—	5	1	no	—
232	C₂×C₃×C₄	—	5	1	no	—
233	C₂×C₃×C₄	—	5	1	no	—
234	C₂×C₃×C₄	—	5	1	no	—
235	C₂×C₃×C₄	—	5	1	no	—
236	C₂×C₃×C₄	—	5	1	no	—
237	C₂×C₃×C₄	—	5	4	no	—
238	C₂×C₃×C₄	—	10	4	no	—
239	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
240	C₂×Dic₃	—	5	4	no	—
241	C₂×Dic₃	—	5	2	no	—
242	C₂×Dic₃	—	10	4	no	—
243	C₂×Dic₃	—	10	4	no	—
244	C₂×Dic₃	C₄×AGL(1,𝔽₅)	1	2	yes	—
245	C₃×C₈	—	5	4	no	—
246	C₃×C₈	—	5	2	no	—
247	C₃×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
248	C₃⋊C₈	—	5	4	no	—
249	C₃⋊C₈	—	5	2	no	—
250	C₃⋊C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
251	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
252	C₂×D₅	C₄²×D₃	1	1	yes	characteristic
253	C₂×D₅	#907	1	4	yes	—
254	C₂×D₅	C₂×C₄×Dic₃	1	1	yes	characteristic
255	AGL(1,𝔽₅)	—	2	4	no	—
256	AGL(1,𝔽₅)	—	6	4	no	—
257	C₄×C₅	—	6	2	no	—
258	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
259	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
260	Dic₅	—	6	2	no	—
261	Dic₅	C₄²×D₃	1	1	yes	characteristic
262	Dic₅	C₂×C₄×Dic₃	1	1	yes	characteristic
263	C₂²×C₄	—	5	1	no	—
264	C₂²×C₄	—	5	2	no	—
265	C₂²×C₄	—	15	2	no	—
266	C₂²×C₄	—	15	2	no	—
267	C₂×C₈	—	3	1	no	—
268	C₂×C₈	—	5	2	no	—
269	C₂×C₈	—	5	2	no	—
270	C₂×C₈	—	5	2	no	—
271	C₂×C₈	—	5	1	no	—
272	C₂×C₈	—	5	2	no	—
273	C₂×C₈	—	5	2	no	—
274	C₂×C₈	—	15	2	no	—
275	C₂×C₈	—	15	2	no	—
276	C₂×C₈	—	15	2	no	—
277	C₂×C₈	—	15	2	no	—
278	C₂×C₈	—	15	2	no	—
279	C₂×C₈	—	15	1	no	—
280	C₂×C₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
281	C₄²	—	3	1	no	—
282	C₄²	—	5	2	no	—
283	C₄²	—	5	2	no	—
284	C₄²	—	15	2	no	—
285	C₄²	—	15	2	no	—
286	C₄²	—	15	2	no	—
287	C₄²	—	15	1	no	—
288	C₃×C₅	#1726	1	1	yes	characteristic, complemented, Hall
289	C₂²×C₃	—	5	1	no	—
290	C₂²×C₃	—	5	1	no	—
291	C₂²×C₃	—	5	4	no	—
292	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
293	C₃×C₄	—	5	1	no	—
294	C₃×C₄	—	5	1	no	—
295	C₃×C₄	—	10	4	no	—
296	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
297	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
298	Dic₃	—	2	2	no	—
299	Dic₃	—	10	4	no	—
300	Dic₃	—	10	2	no	—
301	C₂×C₅	#4708	1	2	yes	—
302	C₂×C₅	C₄²×Dic₃	1	1	yes	characteristic
303	D₅	#4708	1	2	yes	—
304	D₅	#5122	1	2	yes	—
305	C₂³	—	5	1	no	—
306	C₂×C₄	—	3	2	no	—
307	C₂×C₄	—	5	1	no	—
308	C₂×C₄	—	5	1	no	—
309	C₂×C₄	—	5	1	no	—
310	C₂×C₄	—	5	1	no	—
311	C₂×C₄	—	5	1	no	—
312	C₂×C₄	—	5	4	no	—
313	C₂×C₄	—	10	4	no	—
314	C₂×C₄	—	15	4	no	—
315	C₂×C₄	—	15	2	no	—
316	C₂×C₄	—	30	4	no	—
317	C₂×C₄	—	30	4	no	—
318	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
319	C₈	—	3	2	no	—
320	C₈	—	5	4	no	—
321	C₈	—	5	2	no	—
322	C₈	—	15	4	no	—
323	C₈	—	15	2	no	—
324	C₈	Dic₃×AGL(1,𝔽₅)	1	2	yes	—
325	C₂×C₃	—	5	2	no	—
326	C₂×C₃	—	5	2	no	—
327	C₂×C₃	C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic
328	C₂×C₃	#65718	1	2	yes	—
329	C₅	#69804	1	1	yes	characteristic, complemented, Sylow
330	C₂²	—	5	1	no	—
331	C₂²	—	5	1	no	—
332	C₂²	—	5	4	no	—
333	C₂²	C₄×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
334	C₄	—	5	1	no	—
335	C₄	—	5	1	no	—
336	C₄	—	6	2	no	—
337	C₄	—	10	4	no	—
338	C₄	—	30	4	no	—
339	C₄	—	30	2	no	—
340	C₄	C₄×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
341	C₄	C₂×Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
342	C₃	#10603838	1	1	yes	characteristic, complemented, Sylow
343	C₂	—	5	2	no	—
344	C₂	—	5	2	no	—
345	C₂	#11751656	1	2	yes	—
346	C₂	C₄×Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator	C₂×C₃×C₅	C₄³
Fitting	C₂×C₃×C₅×C₈	C₂×C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₃×D₅	C₂²×C₄
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	C₂×C₈	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#1726
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
16. #91221	C₄	semidirect
17. C₃×#3992	C₄	semidirect
18. #91169	C₄	semidirect
20. ΓL(1,𝔽₁₆)⋊C₈	C₄	semidirect
21. AGL(1,𝔽₅)×C₃⋊C₈	C₄	semidirect
22. C₈×Dic₁₅	C₄	semidirect
23. C₅×C₈×Dic₃	C₄	semidirect
24. #90951	C₄	semidirect
25. Dic₃×#168	C₄	semidirect
76. #3992	Dic₃	semidirect
119. C₈×Dic₃	AGL(1,𝔽₅)	semidirect
288. C₃×C₅	#1726	semidirect
329. C₅	#69804	semidirect
342. C₃	#10603838	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91221	2
C₂×C₃×#3992	2
#11751788	2
#11751748	2
C₂×ΓL(1,𝔽₁₆)⋊C₈	2
C₂×AGL(1,𝔽₅)×C₃⋊C₈	2
C₈×D₅×Dic₃	2
#10603838	3
#69804	5