Subgroups — #14411549

Order 1920.

1112 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	C₄×C₅×#235	C₂	1	1	yes	maximal, characteristic
2	C₂×#91286	C₂	1	2	yes	maximal
3	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#3935	C₂	1	1	yes	maximal, characteristic, complemented
5	#11748323	C₂	1	2	yes	maximal, complemented
6	#11747005	C₂	1	2	yes	maximal, complemented
7	#11746979	C₂	1	4	yes	maximal, complemented
8	D₃×#3935	C₂	1	2	yes	maximal, complemented
9	#10614242	—	3	1	no	maximal, Hall
10	C₂×C₅×#235	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×#6710	C₂²	1	2	yes	—
13	C₄×C₅×#67	C₄	1	4	yes	—
14	C₅×#926	C₂²	1	1	yes	characteristic
15	C₄×C₅×D₁₂	C₄	1	2	yes	—
16	C₅×#872	C₂²	1	2	yes	—
17	C₄²×C₅×D₃	C₂²	1	2	yes	—
18	C₄×C₅×Dic₆	C₄	1	2	yes	—
19	C₂²×C₃×#168	C₂²	1	2	yes	—
20	C₂×#6650	C₂²	1	4	yes	—
21	#91286	C₄	1	16	yes	complemented
22	C₂×D₃×#168	C₂²	1	4	yes	—
23	#91087	C₂²	1	2	yes	—
24	C₄×C₃⋊#168	C₂²	1	2	yes	—
25	C₃×#3935	C₂²	1	1	yes	characteristic
26	C₃×#3935	C₂²	1	2	yes	—
27	C₃×#3935	C₂²	1	1	yes	characteristic
28	#90953	C₂²	1	4	yes	—
29	Dic₃×#168	C₂²	1	4	yes	—
30	#78486	—	5	1	no	maximal, Hall
31	C₄×C₅×#41	—	3	1	no	—
32	C₂×#4094	—	3	2	no	—
33	#65286	—	3	2	no	—
34	#65286	—	3	4	no	—
35	C₄×#681	—	3	2	no	—
36	C₄×#681	—	3	1	no	—
37	C₂×#3935	—	3	2	no	—
38	C₂×#3935	D₃	1	1	yes	characteristic, complemented
39	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×C₅×#67	C₂×C₄	1	2	yes	—
42	C₅×#235	C₂×C₄	1	8	yes	—
43	C₂×C₅×D₁₂	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	—
45	C₂×C₄×C₅×D₃	C₂³	1	2	yes	—
46	C₂×C₅×Dic₆	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
48	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
49	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
50	#6710	C₂×C₄	1	8	yes	complemented
51	C₂×C₃⋊#168	C₂³	1	4	yes	—
52	C₅×#217	C₂×C₄	1	2	yes	—
53	C₅×(C₂×D₃)⋊C₄	C₂×C₄	1	4	yes	—
54	C₅×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
55	C₅×Dic₃⋊C₄	C₂×C₄	1	4	yes	—
56	C₄×C₅×Dic₃	C₂×C₄	1	4	yes	—
57	C₄×C₅×Dic₃	C₂³	1	2	yes	—
58	C₂×C₃×#168	C₂×C₄	1	8	yes	complemented
59	C₂×C₃×#168	C₂³	1	2	yes	—
60	C₂×C₃×#168	C₂³	1	2	yes	—
61	#6650	C₂×C₄	1	16	yes	complemented
62	D₃×#168	C₂×C₄	1	16	yes	complemented
63	C₂×#901	—	5	2	no	—
64	C₄×#235	D₅	1	1	yes	characteristic
65	C₂×C₃×#97	—	5	1	no	—
66	#5101	—	5	2	no	—
67	C₄×#208	—	5	1	no	—
68	#4708	—	5	4	no	—
69	#4707	—	5	2	no	—
70	D₃×#97	—	5	2	no	—
71	C₂×C₅×#41	—	3	1	no	—
72	C₄×C₅×Q₈	—	3	2	no	—
73	C₄×C₅×D₄	—	3	4	no	—
74	C₄×C₅×D₄	—	3	2	no	—
75	C₅×#117	—	3	1	no	—
76	C₅×#117	—	3	2	no	—
77	C₂×C₄²×C₅	—	3	2	no	—
78	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
79	#4094	—	3	16	no	—
80	C₂×#681	—	3	2	no	—
81	C₂×#681	—	3	4	no	—
82	C₂²×#168	—	3	4	no	—
83	C₂²×#168	C₂×D₃	1	2	yes	—
84	#3935	—	3	2	no	—
85	#3935	—	3	4	no	—
86	#3935	C₂×D₃	1	2	yes	—
87	#3935	C₂×D₃	1	1	yes	characteristic
88	#3935	C₂×D₃	1	1	yes	characteristic
89	C₄×#168	—	3	2	no	—
90	C₄×#168	—	3	4	no	—
91	#2875	—	15	1	no	Sylow
92	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₅×D₃	C₂²×C₄	1	2	yes	—
94	C₂×C₃×C₄×C₅	—	4	2	no	—
95	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
96	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
97	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
99	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
100	C₅×#67	C₄²	1	8	yes	—
101	C₂×C₅×Dic₃	C₂²×C₄	1	4	yes	—
102	C₂×C₅×Dic₃	C₂²×C₄	1	2	yes	—
103	C₅×D₁₂	C₄²	1	4	yes	—
104	C₄×C₅×D₃	—	4	4	no	—
105	C₄×C₅×D₃	C₂²×C₄	1	8	yes	—
106	C₅×Dic₆	C₄²	1	4	yes	—
107	C₃⋊#168	C₂²×C₄	1	8	yes	—
108	C₃×#168	C₂²×C₄	1	4	yes	—
109	C₃×#168	C₂²×C₄	1	4	yes	—
110	C₂×#235	C₂×D₅	1	1	yes	characteristic
111	C₂²×C₃×C₈	—	5	2	no	—
112	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
113	C₄×#67	Dic₅	1	4	yes	—
114	#926	C₂×D₅	1	1	yes	characteristic
115	C₂×#208	—	5	2	no	—
116	#901	—	5	16	no	—
117	C₂×ΓL(1,𝔽₂₅)	—	5	4	no	—
118	C₂×D₃×C₈	—	5	4	no	—
119	C₄×D₁₂	Dic₅	1	2	yes	—
120	#872	C₂×D₅	1	2	yes	—
121	C₄²×D₃	C₂×D₅	1	2	yes	—
122	C₄×Dic₆	Dic₅	1	2	yes	—
123	C₃×#97	—	5	1	no	—
124	C₃×#97	—	5	1	no	—
125	C₃×#97	—	5	2	no	—
126	#815	—	5	4	no	—
127	C₈×Dic₃	—	5	4	no	—
128	#803	—	5	2	no	—
129	C₄×C₃⋊C₈	—	5	2	no	—
130	C₅×#41	—	3	8	no	—
131	C₂×C₅×Q₈	—	3	1	no	—
132	C₂×C₅×D₄	—	3	2	no	—
133	C₂×C₅×D₄	—	3	1	no	—
134	C₂²×C₄×C₅	—	3	2	no	—
135	C₂²×C₄×C₅	—	3	4	no	—
136	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
137	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
138	C₅×C₄⋊C₄	—	3	4	no	—
139	C₅×C₄⋊C₄	—	3	2	no	—
140	C₅×C₂²⋊C₄	—	3	2	no	—
141	C₅×C₂²⋊C₄	—	3	4	no	—
142	C₄²×C₅	—	3	4	no	—
143	C₄²×C₅	—	3	2	no	—
144	C₄²×C₅	C₄×D₃	1	2	yes	—
145	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
146	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
147	#681	—	3	8	no	—
148	#681	—	3	16	no	—
149	C₂×#168	—	3	4	no	—
150	C₂×#168	—	3	16	no	—
151	C₂×#168	C₄×D₃	1	8	yes	complemented
152	C₂×#168	C₂²×D₃	1	2	yes	—
153	C₂×#168	C₂²×D₃	1	2	yes	—
154	C₂×#131	—	15	2	no	—
155	C₄×#41	—	3	1	no	—
156	#405	—	15	4	no	—
157	#405	—	15	2	no	—
158	C₄×M₁₆	—	15	2	no	—
159	C₄×M₁₆	—	15	1	no	—
160	C₂×#97	—	5	1	no	—
161	C₂×#97	—	15	2	no	—
162	C₂²×C₃×C₅	—	2	2	no	—
163	C₂²×C₃×C₅	C₂×C₄²	1	2	yes	—
164	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
165	C₂×C₅×D₃	—	2	4	no	—
166	C₂×C₅×D₃	C₂×C₄²	1	4	yes	—
167	C₃×C₄×C₅	—	2	2	no	—
168	C₃×C₄×C₅	—	2	2	no	—
169	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
170	C₃×C₄×C₅	C₂³×C₄	1	2	yes	—
171	C₅×Dic₃	—	2	4	no	—
172	C₅×Dic₃	C₂×C₄²	1	4	yes	—
173	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
174	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
175	C₂×#67	C₂×Dic₅	1	2	yes	—
176	#235	C₄×D₅	1	8	yes	—
177	C₂×D₁₂	C₂×Dic₅	1	1	yes	characteristic
178	C₂×C₄×D₃	C₂×Dic₅	1	4	yes	—
179	C₂×C₄×D₃	C₂²×D₅	1	2	yes	—
180	C₂×Dic₆	C₂×Dic₅	1	1	yes	characteristic
181	C₂×C₃×C₈	—	5	8	no	—
182	C₂×C₃×C₈	—	5	2	no	—
183	C₂×C₃×C₈	—	5	2	no	—
184	C₃×C₄²	C₂×Dic₅	1	2	yes	—
185	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
186	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
187	#217	C₂×Dic₅	1	2	yes	—
188	(C₂×D₃)⋊C₄	C₂×Dic₅	1	4	yes	—
189	C₄⋊Dic₃	C₂×Dic₅	1	2	yes	—
190	Dic₃⋊C₄	C₂×Dic₅	1	4	yes	—
191	C₄×Dic₃	C₂×Dic₅	1	4	yes	—
192	C₄×Dic₃	C₂²×D₅	1	2	yes	—
193	#208	—	5	8	no	—
194	C₂×C₃⋊C₈	—	5	4	no	—
195	ΓL(1,𝔽₂₅)	—	5	16	no	—
196	D₃×C₈	—	5	16	no	—
197	C₂³×C₅	—	3	2	no	—
198	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
199	C₅×Q₈	—	3	4	no	—
200	C₅×D₄	—	3	8	no	—
201	C₅×D₄	—	3	4	no	—
202	C₂×C₄×C₅	—	3	2	no	—
203	C₂×C₄×C₅	—	3	8	no	—
204	C₂×C₄×C₅	—	3	4	no	—
205	C₂×C₄×C₅	—	4	2	no	—
206	C₂×C₄×C₅	—	12	4	no	—
207	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
208	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
209	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
210	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
211	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
212	#168	—	3	8	no	—
213	#168	C₂×C₄×D₃	1	4	yes	—
214	#168	C₂×C₄×D₃	1	4	yes	—
215	C₂×#41	—	3	1	no	—
216	#131	—	15	16	no	—
217	C₂×M₁₆	—	15	4	no	—
218	C₂×M₁₆	—	15	2	no	—
219	C₂²×C₈	—	5	2	no	—
220	C₂²×C₈	—	15	4	no	—
221	C₄×Q₈	—	3	2	no	—
222	C₄×D₄	—	3	4	no	—
223	C₄×D₄	—	3	2	no	—
224	#117	—	3	1	no	—
225	#117	—	3	2	no	—
226	C₂×C₄²	—	3	2	no	—
227	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
228	#97	—	5	1	no	—
229	#97	—	5	1	no	—
230	#97	—	5	2	no	—
231	#97	—	15	4	no	—
232	#97	—	15	2	no	—
233	C₄×C₈	—	15	4	no	—
234	C₄×C₈	—	15	2	no	—
235	C₂×C₃×C₅	—	2	2	no	—
236	C₂×C₃×C₅	C₂²×C₄²	1	1	yes	characteristic
237	C₂×C₃×C₅	#568	1	2	yes	—
238	C₅×D₃	—	2	4	no	—
239	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
240	C₂²×D₃	C₂²×Dic₅	1	2	yes	—
241	C₂×C₃×C₄	—	4	2	no	—
242	C₂×C₃×C₄	C₂×C₄×D₅	1	4	yes	—
243	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
244	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
245	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
246	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
247	#67	C₄×Dic₅	1	8	yes	—
248	C₂×Dic₃	C₂²×Dic₅	1	4	yes	—
249	C₂×Dic₃	C₂²×Dic₅	1	2	yes	—
250	D₁₂	C₄×Dic₅	1	4	yes	—
251	C₄×D₃	—	4	4	no	—
252	C₄×D₃	C₂×C₄×D₅	1	8	yes	—
253	Dic₆	C₄×Dic₅	1	4	yes	—
254	C₃×C₈	—	5	4	no	—
255	C₃×C₈	—	5	4	no	—
256	C₃⋊C₈	—	5	8	no	—
257	C₂²×C₅	—	2	2	no	—
258	C₂²×C₅	—	3	4	no	—
259	C₂²×C₅	—	6	4	no	—
260	C₂²×C₅	C₄²×D₃	1	2	yes	—
261	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
262	C₄×C₅	—	2	2	no	—
263	C₄×C₅	—	2	2	no	—
264	C₄×C₅	—	3	4	no	—
265	C₄×C₅	—	6	4	no	—
266	C₄×C₅	C₄²×D₃	1	2	yes	—
267	C₄×C₅	C₂²×C₄×D₃	1	2	yes	—
268	#41	—	3	8	no	—
269	C₂×Q₈	—	3	1	no	—
270	C₂×D₄	—	3	2	no	—
271	C₂×D₄	—	3	1	no	—
272	C₂²×C₄	—	3	2	no	—
273	C₂²×C₄	—	3	4	no	—
274	C₂²×C₄	D₃×Dic₅	1	2	yes	—
275	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
276	M₁₆	—	15	16	no	—
277	M₁₆	—	15	8	no	—
278	C₂×C₈	—	5	8	no	—
279	C₂×C₈	—	5	2	no	—
280	C₂×C₈	—	5	2	no	—
281	C₂×C₈	—	15	16	no	—
282	C₂×C₈	—	15	4	no	—
283	C₄⋊C₄	—	3	4	no	—
284	C₄⋊C₄	—	3	2	no	—
285	C₂²⋊C₄	—	3	2	no	—
286	C₂²⋊C₄	—	3	4	no	—
287	C₄²	—	3	4	no	—
288	C₄²	—	3	2	no	—
289	C₄²	D₃×Dic₅	1	2	yes	—
290	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
291	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
292	C₃×C₅	#2875	1	1	yes	characteristic, complemented, Hall
293	C₂²×C₃	—	2	2	no	—
294	C₂²×C₃	C₂×C₄×Dic₅	1	2	yes	—
295	C₂²×C₃	C₂³×Dic₅	1	1	yes	characteristic
296	C₂×D₃	—	2	4	no	—
297	C₂×D₃	C₂×C₄×Dic₅	1	4	yes	—
298	C₃×C₄	—	2	2	no	—
299	C₃×C₄	—	2	2	no	—
300	C₃×C₄	C₂×C₄×Dic₅	1	2	yes	—
301	C₃×C₄	C₂²×C₄×D₅	1	2	yes	—
302	Dic₃	—	2	4	no	—
303	Dic₃	C₂×C₄×Dic₅	1	4	yes	—
304	C₂×C₅	—	2	2	no	—
305	C₂×C₅	—	6	4	no	—
306	C₂×C₅	C₂×C₄²×D₃	1	1	yes	characteristic
307	C₂×C₅	#5745	1	2	yes	—
308	C₂³	—	3	2	no	—
309	C₂³	C₂×D₃×Dic₅	1	1	yes	characteristic
310	Q₈	—	3	4	no	—
311	D₄	—	3	8	no	—
312	D₄	—	3	4	no	—
313	C₂×C₄	—	3	2	no	—
314	C₂×C₄	—	3	8	no	—
315	C₂×C₄	—	3	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	12	4	no	—
318	C₂×C₄	C₄×D₃×D₅	1	4	yes	—
319	C₂×C₄	C₂×D₃×Dic₅	1	2	yes	—
320	C₂×C₄	C₂×D₃×Dic₅	1	2	yes	—
321	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
322	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
323	C₈	—	5	4	no	—
324	C₈	—	5	4	no	—
325	C₈	—	15	8	no	—
326	C₂×C₃	—	2	2	no	—
327	C₂×C₃	C₂²×C₄×Dic₅	1	1	yes	characteristic
328	C₂×C₃	#66153	1	2	yes	—
329	D₃	—	2	4	no	—
330	C₅	#78486	1	1	yes	characteristic, complemented, Sylow
331	C₂²	—	2	2	no	—
332	C₂²	—	3	4	no	—
333	C₂²	—	6	4	no	—
334	C₂²	C₄×D₃×Dic₅	1	2	yes	—
335	C₂²	C₂²×D₃×Dic₅	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	3	4	no	—
339	C₄	—	6	4	no	—
340	C₄	C₄×D₃×Dic₅	1	2	yes	—
341	C₄	C₂×C₄×D₃×D₅	1	2	yes	—
342	C₃	#10614242	1	1	yes	characteristic, complemented, Sylow
343	C₂	—	2	2	no	—
344	C₂	—	6	4	no	—
345	C₂	#11754125	1	2	yes	—
346	C₂	C₂×C₄×D₃×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃×D₅
commutator	C₂×C₃×C₅	C₂²×C₄²
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂×C₅×#235	C₂²
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#2875
LCS	G ▷ C₂×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
3. C₄×#6710	C₂	semidirect
4. C₂×C₃×#3935	C₂	semidirect
5. #11748323	C₂	semidirect
6. #11747005	C₂	semidirect
7. #11746979	C₂	semidirect
8. D₃×#3935	C₂	semidirect
21. #91286	C₄	semidirect
38. C₂×#3935	D₃	semidirect
50. #6710	C₂×C₄	semidirect
58. C₂×C₃×#168	C₂×C₄	semidirect
61. #6650	C₂×C₄	semidirect
62. D₃×#168	C₂×C₄	semidirect
151. C₂×#168	C₄×D₃	semidirect
292. C₃×C₅	#2875	semidirect
330. C₅	#78486	semidirect
342. C₃	#10614242	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₅×#235	2
C₂×#91286	2
C₄×#6710	2
C₂×C₃×#3935	2
#11748323	2
#11747005	2
#11746979	2
D₃×#3935	2
#10614242	3
#78486	5