Subgroups — #14411518

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	C₂×#91790	C₂	1	2	yes	maximal
2	C₄×#6817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₅×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751202	C₂	1	2	yes	maximal, complemented
5	C₄×#6710	C₂	1	1	yes	maximal, characteristic, complemented
6	#11750809	C₂	1	4	yes	maximal, complemented
7	#11750808	C₂	1	2	yes	maximal, complemented
8	D₁₅×#97	C₂	1	2	yes	maximal, complemented
9	#10610217	—	3	1	no	maximal, Hall
10	C₂×#6817	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₄×#1220	C₄	1	4	yes	—
14	#91815	C₂²	1	1	yes	characteristic
15	C₂×#6710	C₂²	1	2	yes	—
16	#91790	C₄	1	16	yes	complemented
17	C₂×#6705	C₂²	1	4	yes	—
18	C₂×C₈×D₁₅	C₂²	1	4	yes	—
19	C₄×D₆₀	C₄	1	2	yes	—
20	#91761	C₂²	1	2	yes	—
21	C₄²×D₁₅	C₂²	1	2	yes	—
22	C₄×Dic₃₀	C₄	1	2	yes	—
23	C₃×C₅×#97	C₂²	1	1	yes	characteristic
24	C₃×C₅×#97	C₂²	1	1	yes	characteristic
25	C₃×C₅×#97	C₂²	1	2	yes	—
26	#91099	C₂²	1	4	yes	—
27	C₈×Dic₁₅	C₂²	1	4	yes	—
28	#91087	C₂²	1	2	yes	—
29	C₄×C₃⋊#168	C₂²	1	2	yes	—
30	#78486	—	5	1	no	maximal, Hall
31	C₂×#4047	—	3	2	no	—
32	C₄×#709	—	3	1	no	—
33	C₂×C₅×#97	D₃	1	1	yes	characteristic, complemented
34	#65390	—	3	2	no	—
35	C₄×#681	—	3	1	no	—
36	#64997	—	3	4	no	—
37	#64996	—	3	2	no	—
38	D₅×#97	—	3	2	no	—
39	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
40	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂×#1220	C₂×C₄	1	2	yes	—
42	#6817	C₂×C₄	1	8	yes	—
43	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×D₁₅	C₂×C₄	1	4	yes	—
45	C₂×C₄×D₁₅	C₂³	1	2	yes	—
46	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×C₅×C₈	C₂×C₄	1	8	yes	complemented
48	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
49	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
50	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
51	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
53	#6719	C₂×C₄	1	2	yes	—
54	#6714	C₂×C₄	1	4	yes	—
55	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
56	#6712	C₂×C₄	1	4	yes	—
57	C₄×Dic₁₅	C₂×C₄	1	4	yes	—
58	C₄×Dic₁₅	C₂³	1	2	yes	—
59	#6710	C₂×C₄	1	8	yes	complemented
60	C₂×C₃⋊#168	C₂³	1	4	yes	—
61	#6705	C₂×C₄	1	16	yes	complemented
62	C₈×D₁₅	C₂×C₄	1	16	yes	complemented
63	C₂×#901	—	5	2	no	—
64	C₄×#235	—	5	1	no	—
65	C₂×C₃×#97	D₅	1	1	yes	characteristic, complemented
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×#709	—	3	1	no	—
72	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
73	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
74	C₄×C₂²⋊D₅	—	3	4	no	—
75	#4072	—	3	1	no	—
76	C₂×#681	—	3	2	no	—
77	#4047	—	3	16	no	—
78	C₂×#676	—	3	4	no	—
79	C₂×C₈×D₅	—	3	4	no	—
80	C₄×D₂₀	—	3	2	no	—
81	#4018	—	3	2	no	—
82	C₄²×D₅	—	3	2	no	—
83	C₄×Dic₁₀	—	3	2	no	—
84	C₅×#97	C₂×D₃	1	1	yes	characteristic
85	C₅×#97	C₂×D₃	1	1	yes	characteristic
86	C₅×#97	C₂×D₃	1	2	yes	—
87	#3947	—	3	4	no	—
88	C₈×Dic₅	—	3	4	no	—
89	#3935	—	3	2	no	—
90	C₄×#168	—	3	2	no	—
91	#2875	—	15	1	no	Sylow
92	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂²×D₁₅	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	#1220	C₄²	1	8	yes	—
101	C₂×Dic₁₅	C₂²×C₄	1	4	yes	—
102	C₂×Dic₁₅	C₂²×C₄	1	2	yes	—
103	D₆₀	C₄²	1	4	yes	—
104	C₄×D₁₅	—	4	4	no	—
105	C₄×D₁₅	C₂²×C₄	1	8	yes	—
106	Dic₃₀	C₄²	1	4	yes	—
107	C₃×C₅×C₈	C₂²×C₄	1	4	yes	—
108	C₃×C₅×C₈	C₂²×C₄	1	4	yes	—
109	C₃⋊#168	C₂²×C₄	1	8	yes	—
110	C₂×#235	—	5	1	no	—
111	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
112	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
113	C₄×#67	—	5	4	no	—
114	#926	—	5	1	no	—
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	—	5	2	no	—
120	#872	—	5	2	no	—
121	C₄²×D₃	—	5	2	no	—
122	C₄×Dic₆	—	5	2	no	—
123	C₃×#97	C₂×D₅	1	1	yes	characteristic
124	C₃×#97	C₂×D₅	1	1	yes	characteristic
125	C₃×#97	C₂×D₅	1	2	yes	—
126	#815	—	5	4	no	—
127	C₈×Dic₃	—	5	4	no	—
128	#803	—	5	2	no	—
129	C₄×C₃⋊C₈	—	5	2	no	—
130	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
131	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₂²⋊D₅	—	3	2	no	—
133	#709	—	3	8	no	—
134	C₂×D₂₀	—	3	1	no	—
135	C₂×C₄×D₅	—	3	2	no	—
136	C₂×C₄×D₅	—	3	4	no	—
137	C₂×Dic₁₀	—	3	1	no	—
138	C₂×C₅×C₈	C₄×D₃	1	8	yes	complemented
139	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
140	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
141	C₄²×C₅	C₄×D₃	1	2	yes	—
142	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
143	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
144	C₂²⋊Dic₅	—	3	2	no	—
145	#685	—	3	4	no	—
146	C₄⋊Dic₅	—	3	2	no	—
147	#683	—	3	4	no	—
148	C₄×Dic₅	—	3	4	no	—
149	C₄×Dic₅	—	3	2	no	—
150	#681	—	3	8	no	—
151	C₂×#168	—	3	4	no	—
152	#676	—	3	16	no	—
153	C₈×D₅	—	3	16	no	—
154	C₂×#131	—	15	2	no	—
155	C₄×#41	—	15	1	no	—
156	#405	—	15	2	no	—
157	#405	—	15	4	no	—
158	C₄×M₁₆	—	15	1	no	—
159	C₄×M₁₆	—	15	2	no	—
160	C₂×#97	—	15	2	no	—
161	C₂×#97	D₁₅	1	1	yes	characteristic, complemented
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×D₁₅	—	2	4	no	—
166	C₂×D₁₅	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	Dic₁₅	—	2	4	no	—
172	Dic₁₅	C₂×C₄²	1	4	yes	—
173	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
174	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
175	C₂×#67	—	5	2	no	—
176	#235	—	5	8	no	—
177	C₂×D₁₂	—	5	1	no	—
178	C₂×C₄×D₃	—	5	2	no	—
179	C₂×C₄×D₃	—	5	4	no	—
180	C₂×Dic₆	—	5	1	no	—
181	C₂×C₃×C₈	C₄×D₅	1	8	yes	complemented
182	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
183	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
184	C₃×C₄²	C₄×D₅	1	2	yes	—
185	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
186	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
187	#217	—	5	2	no	—
188	(C₂×D₃)⋊C₄	—	5	4	no	—
189	C₄⋊Dic₃	—	5	2	no	—
190	Dic₃⋊C₄	—	5	4	no	—
191	C₄×Dic₃	—	5	4	no	—
192	C₄×Dic₃	—	5	2	no	—
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	C₂×C₄×D₃	1	1	yes	characteristic
198	C₂²×D₅	—	3	2	no	—
199	C₂×C₄×C₅	—	4	2	no	—
200	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
201	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
202	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
203	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
204	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
205	C₂²⋊D₅	—	3	8	no	—
206	C₂×Dic₅	—	3	2	no	—
207	C₂×Dic₅	—	3	4	no	—
208	D₂₀	—	3	4	no	—
209	C₄×D₅	—	3	8	no	—
210	C₄×D₅	—	12	4	no	—
211	Dic₁₀	—	3	4	no	—
212	C₅×C₈	C₂×C₄×D₃	1	4	yes	—
213	C₅×C₈	C₂×C₄×D₃	1	4	yes	—
214	#168	—	3	8	no	—
215	C₂×#41	—	15	1	no	—
216	#131	—	15	16	no	—
217	C₂×M₁₆	—	15	2	no	—
218	C₂×M₁₆	—	15	4	no	—
219	C₂²×C₈	—	15	4	no	—
220	C₂²×C₈	C₂×D₁₅	1	2	yes	—
221	C₄×Q₈	—	15	2	no	—
222	C₄×D₄	—	15	4	no	—
223	C₄×D₄	—	15	2	no	—
224	#117	—	15	1	no	—
225	#117	—	15	2	no	—
226	C₂×C₄²	—	15	2	no	—
227	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
228	#97	—	15	4	no	—
229	#97	—	15	2	no	—
230	#97	C₂×D₁₅	1	1	yes	characteristic
231	#97	C₂×D₁₅	1	1	yes	characteristic
232	#97	C₂×D₁₅	1	2	yes	—
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	D₁₅	—	2	4	no	—
239	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
240	C₂²×D₃	—	5	2	no	—
241	C₂×C₃×C₄	—	4	2	no	—
242	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
243	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
244	C₂×C₃×C₄	C₂×C₄×D₅	1	4	yes	—
245	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
246	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
247	#67	—	5	8	no	—
248	C₂×Dic₃	—	5	2	no	—
249	C₂×Dic₃	—	5	4	no	—
250	D₁₂	—	5	4	no	—
251	C₄×D₃	—	5	8	no	—
252	C₄×D₃	—	20	4	no	—
253	Dic₆	—	5	4	no	—
254	C₃×C₈	C₂×C₄×D₅	1	4	yes	—
255	C₃×C₈	C₂×C₄×D₅	1	4	yes	—
256	C₃⋊C₈	—	5	8	no	—
257	C₂²×C₅	—	2	2	no	—
258	C₂²×C₅	C₄²×D₃	1	2	yes	—
259	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
260	C₂×D₅	—	3	4	no	—
261	C₂×D₅	—	6	4	no	—
262	C₄×C₅	—	2	2	no	—
263	C₄×C₅	—	2	2	no	—
264	C₄×C₅	C₄²×D₃	1	2	yes	—
265	C₄×C₅	C₂²×C₄×D₃	1	2	yes	—
266	Dic₅	—	3	4	no	—
267	Dic₅	—	6	4	no	—
268	#41	—	15	8	no	—
269	C₂×Q₈	—	15	1	no	—
270	C₂×D₄	—	15	2	no	—
271	C₂×D₄	—	15	1	no	—
272	C₂²×C₄	—	15	2	no	—
273	C₂²×C₄	—	15	4	no	—
274	C₂²×C₄	C₄×D₁₅	1	2	yes	—
275	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
276	M₁₆	—	15	8	no	—
277	M₁₆	—	15	16	no	—
278	C₂×C₈	—	15	4	no	—
279	C₂×C₈	—	15	16	no	—
280	C₂×C₈	C₄×D₁₅	1	8	yes	complemented
281	C₂×C₈	C₂²×D₁₅	1	2	yes	—
282	C₂×C₈	C₂²×D₁₅	1	2	yes	—
283	C₄⋊C₄	—	15	4	no	—
284	C₄⋊C₄	—	15	2	no	—
285	C₂²⋊C₄	—	15	2	no	—
286	C₂²⋊C₄	—	15	4	no	—
287	C₄²	—	15	4	no	—
288	C₄²	—	15	2	no	—
289	C₄²	C₄×D₁₅	1	2	yes	—
290	C₄²	C₂²×D₁₅	1	1	yes	characteristic
291	C₄²	C₂²×D₁₅	1	1	yes	characteristic
292	C₃×C₅	#2875	1	1	yes	characteristic, complemented, Hall
293	C₂²×C₃	—	2	2	no	—
294	C₂²×C₃	C₄²×D₅	1	2	yes	—
295	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
296	C₂×D₃	—	5	4	no	—
297	C₂×D₃	—	10	4	no	—
298	C₃×C₄	—	2	2	no	—
299	C₃×C₄	—	2	2	no	—
300	C₃×C₄	C₄²×D₅	1	2	yes	—
301	C₃×C₄	C₂²×C₄×D₅	1	2	yes	—
302	Dic₃	—	5	4	no	—
303	Dic₃	—	10	4	no	—
304	C₂×C₅	—	2	2	no	—
305	C₂×C₅	C₂×C₄²×D₃	1	1	yes	characteristic
306	C₂×C₅	#5745	1	2	yes	—
307	D₅	—	6	4	no	—
308	C₂³	—	15	2	no	—
309	C₂³	C₂×C₄×D₁₅	1	1	yes	characteristic
310	Q₈	—	15	4	no	—
311	D₄	—	15	8	no	—
312	D₄	—	15	4	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	15	2	no	—
315	C₂×C₄	—	15	8	no	—
316	C₂×C₄	—	15	4	no	—
317	C₂×C₄	—	60	4	no	—
318	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
319	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
320	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
321	C₂×C₄	C₂×C₄×D₁₅	1	4	yes	—
322	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
323	C₈	—	15	8	no	—
324	C₈	C₂×C₄×D₁₅	1	4	yes	—
325	C₈	C₂×C₄×D₁₅	1	4	yes	—
326	C₂×C₃	—	2	2	no	—
327	C₂×C₃	C₂×C₄²×D₅	1	1	yes	characteristic
328	C₂×C₃	#66079	1	2	yes	—
329	D₃	—	10	4	no	—
330	C₅	#78486	1	1	yes	characteristic, complemented, Sylow
331	C₂²	—	2	2	no	—
332	C₂²	—	15	4	no	—
333	C₂²	—	30	4	no	—
334	C₂²	C₄²×D₁₅	1	2	yes	—
335	C₂²	C₂²×C₄×D₁₅	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	15	4	no	—
339	C₄	—	30	4	no	—
340	C₄	C₄²×D₁₅	1	2	yes	—
341	C₄	C₂²×C₄×D₁₅	1	2	yes	—
342	C₃	#10610217	1	1	yes	characteristic, complemented, Sylow
343	C₂	—	2	2	no	—
344	C₂	—	30	4	no	—
345	C₂	C₂×C₄²×D₁₅	1	1	yes	characteristic
346	C₂	#11756504	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₅
commutator	C₂×C₃×C₅	C₂²×C₄²
Fitting	C₂×C₃×C₅×#97	C₂
socle	C₂²×C₃×C₅	C₂³×C₄
prime radical	C₂×#6817	C₂²
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂×#97	D₁₅
hypocenter	C₃×C₅	#2875
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×C₅×#97	C₂	semidirect
4. #11751202	C₂	semidirect
5. C₄×#6710	C₂	semidirect
6. #11750809	C₂	semidirect
7. #11750808	C₂	semidirect
8. D₁₅×#97	C₂	semidirect
16. #91790	C₄	semidirect
33. C₂×C₅×#97	D₃	semidirect
47. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
59. #6710	C₂×C₄	semidirect
61. #6705	C₂×C₄	semidirect
62. C₈×D₁₅	C₂×C₄	semidirect
65. C₂×C₃×#97	D₅	semidirect
138. C₂×C₅×C₈	C₄×D₃	semidirect
161. C₂×#97	D₁₅	semidirect
181. C₂×C₃×C₈	C₄×D₅	semidirect
280. C₂×C₈	C₄×D₁₅	semidirect
292. C₃×C₅	#2875	semidirect
330. C₅	#78486	semidirect
342. C₃	#10610217	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91790	2
C₄×#6817	2
C₂×C₃×C₅×#97	2
#11751202	2
C₄×#6710	2
#11750809	2
#11750808	2
D₁₅×#97	2
#10610217	3
#78486	5