Subgroups — #14296010

Order 1920.

584 conjugacy classes of subgroups.

The subgroup lattice has 396 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	1	yes	maximal, characteristic
2	C₃×C₅×#405	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91109	C₂	1	1	yes	maximal, characteristic
4	C₂×#91099	C₂	1	1	yes	maximal, characteristic, complemented
5	#11751103	C₂	1	1	yes	maximal, characteristic
6	#11746379	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746353	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600681	—	3	1	no	maximal, Hall
9	C₂×#6817	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#117	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
14	C₂×#6710	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
16	#91790	—	2	2	no	—
17	#91790	C₄	1	4	yes	complemented
18	C₂×#6705	—	2	1	no	—
19	C₂×C₈×D₁₅	—	2	1	no	—
20	C₃×C₅×#97	—	2	1	no	—
21	C₃×C₄×C₅×C₈	—	2	1	no	—
22	#91109	C₄	1	4	yes	complemented
23	#91099	—	2	2	no	—
24	#91094	C₄	1	2	yes	complemented
25	#91093	C₄	1	2	yes	complemented
26	#69888	—	5	1	no	maximal, Hall
27	C₂×#4047	—	3	1	no	—
28	C₅×#405	D₃	1	1	yes	characteristic, complemented
29	C₂×#3957	—	3	1	no	—
30	C₂×#3947	—	3	1	no	—
31	#65291	—	3	1	no	—
32	#64778	—	3	1	no	—
33	#64752	—	3	1	no	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×#1220	—	2	1	no	—
36	C₂²×Dic₁₅	—	2	1	no	—
37	#6817	—	2	1	no	—
38	#6817	—	2	1	no	—
39	#6817	C₂×C₄	1	2	yes	—
40	#6817	C₂×C₄	1	2	yes	—
41	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×D₁₅	—	2	1	no	—
43	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
44	C₃×C₅×M₁₆	—	2	1	no	—
45	C₃×C₅×M₁₆	C₂×C₄	1	2	yes	—
46	C₂×C₃×C₅×C₈	—	2	1	no	—
47	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
48	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
49	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
50	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
51	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
53	C₃×C₄²×C₅	—	2	1	no	—
54	C₄×Dic₁₅	—	2	1	no	—
55	C₄×Dic₁₅	C₂×C₄	1	2	yes	—
56	#6710	—	2	1	no	—
57	#6710	C₂×C₄	1	2	yes	—
58	C₂×C₃⋊#168	—	2	1	no	—
59	C₂×C₃⋊#168	—	2	1	no	—
60	C₂×C₃⋊#168	—	2	1	no	—
61	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
62	#6705	—	2	4	no	—
63	C₈×D₁₅	—	2	4	no	—
64	C₂×#901	—	5	1	no	—
65	C₃×#405	D₅	1	1	yes	characteristic, complemented
66	C₂×#825	—	5	1	no	—
67	C₂×#815	—	5	1	no	—
68	#5002	—	5	1	no	—
69	#4558	—	5	1	no	—
70	#4532	—	5	1	no	—
71	C₂×#709	—	3	1	no	—
72	C₂×C₅×M₁₆	C₂×D₃	1	1	yes	characteristic
73	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
74	C₅×#117	C₂×D₃	1	1	yes	characteristic
75	C₂×C₄×Dic₅	—	3	1	no	—
76	C₂×#681	—	3	1	no	—
77	C₂²×#168	—	3	1	no	—
78	#4047	—	3	4	no	—
79	#4047	—	6	2	no	—
80	C₂×#676	—	6	1	no	—
81	C₂×C₈×D₅	—	6	1	no	—
82	C₅×#97	—	2	1	no	—
83	C₄×C₅×C₈	—	2	1	no	—
84	#3957	—	3	4	no	—
85	#3947	—	6	2	no	—
86	#3942	—	3	2	no	—
87	#3941	—	3	2	no	—
88	#1765	—	15	1	no	Sylow
89	C₂³×C₃×C₅	#41	1	1	yes	characteristic
90	C₂²×D₁₅	—	2	1	no	—
91	C₂×C₃×C₄×C₅	—	2	2	no	—
92	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
98	#1220	—	2	2	no	—
99	#1220	—	2	2	no	—
100	C₂×Dic₁₅	—	2	1	no	—
101	C₂×Dic₁₅	—	2	2	no	—
102	C₂×Dic₁₅	—	4	2	no	—
103	D₆₀	—	2	1	no	—
104	D₆₀	C₄²	1	2	yes	—
105	C₄×D₁₅	—	2	2	no	—
106	C₄×D₁₅	—	2	2	no	—
107	Dic₃₀	—	2	1	no	—
108	Dic₃₀	C₄²	1	2	yes	—
109	C₃×C₅×C₈	—	2	2	no	—
110	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
111	C₃×C₅×C₈	C₂²⋊C₄	1	2	yes	—
112	C₃⋊#168	—	2	2	no	—
113	C₃⋊#168	—	2	2	no	—
114	C₂×#235	—	5	1	no	—
115	C₂×C₃×M₁₆	C₂×D₅	1	1	yes	characteristic
116	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
117	C₃×#117	C₂×D₅	1	1	yes	characteristic
118	C₂×C₄×Dic₃	—	5	1	no	—
119	C₂×#208	—	5	1	no	—
120	C₂²×C₃⋊C₈	—	5	1	no	—
121	#901	—	5	4	no	—
122	#901	—	10	2	no	—
123	C₂×ΓL(1,𝔽₂₅)	—	10	1	no	—
124	C₂×D₃×C₈	—	10	1	no	—
125	C₃×#97	—	2	1	no	—
126	C₃×C₄×C₈	—	2	1	no	—
127	#825	—	5	4	no	—
128	#815	—	10	2	no	—
129	#810	—	5	2	no	—
130	#809	—	5	2	no	—
131	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₂²⋊D₅	—	6	1	no	—
133	C₂²×Dic₅	—	6	1	no	—
134	#709	—	3	2	no	—
135	#709	—	3	2	no	—
136	#709	—	6	1	no	—
137	#709	—	6	1	no	—
138	C₂×D₂₀	—	3	1	no	—
139	C₂×C₄×D₅	—	6	1	no	—
140	C₂×Dic₁₀	—	3	1	no	—
141	C₅×M₁₆	—	2	1	no	—
142	C₅×M₁₆	C₄×D₃	1	2	yes	—
143	C₂×C₅×C₈	—	2	1	no	—
144	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
145	C₂×C₅×C₈	D₁₂	1	2	yes	—
146	C₂×C₅×C₈	#67	1	1	yes	characteristic
147	C₂×C₅×C₈	#67	1	1	yes	characteristic
148	C₅×C₄⋊C₄	C₄×D₃	1	2	yes	—
149	C₅×C₂²⋊C₄	—	2	1	no	—
150	C₄²×C₅	—	2	1	no	—
151	C₄×Dic₅	—	3	2	no	—
152	C₄×Dic₅	—	6	1	no	—
153	#681	—	3	2	no	—
154	#681	—	6	1	no	—
155	C₂×#168	—	3	2	no	—
156	C₂×#168	—	6	1	no	—
157	C₂×#168	—	6	1	no	—
158	C₂×#168	—	6	1	no	—
159	#676	—	6	4	no	—
160	C₈×D₅	—	6	4	no	—
161	C₂×#131	—	15	1	no	—
162	C₂×#104	—	15	1	no	—
163	#416	—	15	1	no	—
164	#405	D₁₅	1	1	yes	characteristic, complemented
165	C₂×#97	—	15	1	no	—
166	#341	—	15	1	no	—
167	#339	—	15	1	no	—
168	C₂²×C₃×C₅	—	2	2	no	—
169	C₂²×C₃×C₅	#117	1	1	yes	characteristic
170	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
171	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
172	C₂×D₁₅	—	2	2	no	—
173	C₂×D₁₅	—	4	2	no	—
174	C₃×C₄×C₅	—	4	2	no	—
175	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
176	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
178	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
179	Dic₁₅	—	2	2	no	—
180	Dic₁₅	—	4	2	no	—
181	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
182	C₂×#67	—	10	1	no	—
183	C₂²×Dic₃	—	10	1	no	—
184	#235	—	5	2	no	—
185	#235	—	5	2	no	—
186	#235	—	10	1	no	—
187	#235	—	10	1	no	—
188	C₂×D₁₂	—	5	1	no	—
189	C₂×C₄×D₃	—	10	1	no	—
190	C₂×Dic₆	—	5	1	no	—
191	C₃×M₁₆	—	2	1	no	—
192	C₃×M₁₆	C₄×D₅	1	2	yes	—
193	C₂×C₃×C₈	—	2	1	no	—
194	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
195	C₂×C₃×C₈	D₂₀	1	2	yes	—
196	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
197	C₂×C₃×C₈	C₂²⋊D₅	1	1	yes	characteristic
198	C₃×C₄⋊C₄	C₄×D₅	1	2	yes	—
199	C₃×C₂²⋊C₄	—	2	1	no	—
200	C₃×C₄²	—	2	1	no	—
201	C₄×Dic₃	—	5	2	no	—
202	C₄×Dic₃	—	10	1	no	—
203	#208	—	5	2	no	—
204	#208	—	10	1	no	—
205	C₂×C₃⋊C₈	—	5	2	no	—
206	C₂×C₃⋊C₈	—	10	1	no	—
207	C₂×C₃⋊C₈	—	10	1	no	—
208	C₂×C₃⋊C₈	—	10	1	no	—
209	ΓL(1,𝔽₂₅)	—	10	4	no	—
210	D₃×C₈	—	10	4	no	—
211	C₂³×C₅	#235	1	1	yes	characteristic
212	C₂²×D₅	—	6	1	no	—
213	C₂×C₄×C₅	—	2	2	no	—
214	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
215	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
216	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
217	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
218	C₂×C₄×C₅	#235	1	1	yes	characteristic
219	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
220	C₂²⋊D₅	—	6	2	no	—
221	C₂²⋊D₅	—	6	2	no	—
222	C₂×Dic₅	—	6	1	no	—
223	C₂×Dic₅	—	6	2	no	—
224	C₂×Dic₅	—	12	2	no	—
225	D₂₀	—	3	2	no	—
226	D₂₀	—	6	1	no	—
227	C₄×D₅	—	6	2	no	—
228	C₄×D₅	—	6	2	no	—
229	Dic₁₀	—	3	2	no	—
230	Dic₁₀	—	6	1	no	—
231	C₅×C₈	—	2	2	no	—
232	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
233	C₅×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
234	#168	—	6	2	no	—
235	#168	—	6	2	no	—
236	C₂×#41	—	15	1	no	—
237	#131	—	15	4	no	—
238	#131	—	30	2	no	—
239	C₂×M₁₆	—	15	1	no	—
240	C₂×M₁₆	—	30	1	no	—
241	C₂×M₁₆	C₂×D₁₅	1	1	yes	characteristic
242	C₂²×C₈	—	15	1	no	—
243	C₂²×C₈	—	30	1	no	—
244	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
245	#117	C₂×D₁₅	1	1	yes	characteristic
246	C₂×C₄²	—	15	1	no	—
247	#104	—	15	4	no	—
248	#103	—	15	2	no	—
249	#102	—	15	2	no	—
250	#97	—	2	1	no	—
251	#97	—	30	2	no	—
252	C₄×C₈	—	2	1	no	—
253	C₂×C₃×C₅	—	2	1	no	—
254	C₂×C₃×C₅	—	2	1	no	—
255	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₃×C₅	#444	1	2	yes	—
257	D₁₅	—	4	2	no	—
258	C₂³×C₃	#709	1	1	yes	characteristic
259	C₂²×D₃	—	10	1	no	—
260	C₂×C₃×C₄	—	2	2	no	—
261	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
262	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
263	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
264	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
265	C₂×C₃×C₄	#709	1	1	yes	characteristic
266	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
267	#67	—	10	2	no	—
268	#67	—	10	2	no	—
269	C₂×Dic₃	—	10	1	no	—
270	C₂×Dic₃	—	10	2	no	—
271	C₂×Dic₃	—	20	2	no	—
272	D₁₂	—	5	2	no	—
273	D₁₂	—	10	1	no	—
274	C₄×D₃	—	10	2	no	—
275	C₄×D₃	—	10	2	no	—
276	Dic₆	—	5	2	no	—
277	Dic₆	—	10	1	no	—
278	C₃×C₈	—	2	2	no	—
279	C₃×C₈	#685	1	2	yes	—
280	C₃×C₈	#685	1	2	yes	—
281	C₃⋊C₈	—	10	2	no	—
282	C₃⋊C₈	—	10	2	no	—
283	C₂²×C₅	—	2	2	no	—
284	C₂²×C₅	#872	1	1	yes	characteristic
285	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
286	C₂²×C₅	C₄×#67	1	1	yes	characteristic
287	C₂×D₅	—	6	2	no	—
288	C₂×D₅	—	12	2	no	—
289	C₄×C₅	—	4	2	no	—
290	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
291	C₄×C₅	C₄×D₁₂	1	1	yes	characteristic
292	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
293	C₄×C₅	C₄×#67	1	1	yes	characteristic
294	Dic₅	—	6	2	no	—
295	Dic₅	—	12	2	no	—
296	#41	—	15	2	no	—
297	#41	—	15	2	no	—
298	#41	—	30	1	no	—
299	#41	—	30	1	no	—
300	C₂×Q₈	—	15	1	no	—
301	C₂×D₄	—	15	1	no	—
302	C₂×D₄	—	30	1	no	—
303	C₂²×C₄	—	30	1	no	—
304	C₂²×C₄	—	30	1	no	—
305	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
306	M₁₆	—	2	1	no	—
307	M₁₆	—	15	2	no	—
308	M₁₆	—	30	1	no	—
309	M₁₆	—	30	4	no	—
310	M₁₆	C₄×D₁₅	1	2	yes	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	15	2	no	—
313	C₂×C₈	—	30	1	no	—
314	C₂×C₈	—	30	1	no	—
315	C₂×C₈	—	30	1	no	—
316	C₂×C₈	—	30	4	no	—
317	C₂×C₈	C₄×D₁₅	1	2	yes	—
318	C₂×C₈	D₆₀	1	2	yes	—
319	C₂×C₈	#1220	1	1	yes	characteristic
320	C₂×C₈	#1220	1	1	yes	characteristic
321	C₄⋊C₄	C₄×D₁₅	1	2	yes	—
322	C₂²⋊C₄	—	2	1	no	—
323	C₄²	—	2	1	no	—
324	C₄²	—	15	2	no	—
325	C₄²	—	30	1	no	—
326	C₃×C₅	#1765	1	1	yes	characteristic, complemented, Hall
327	C₂²×C₃	—	2	2	no	—
328	C₂²×C₃	#4018	1	1	yes	characteristic
329	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
330	C₂²×C₃	C₄×C₂²⋊D₅	1	1	yes	characteristic
331	C₂×D₃	—	10	2	no	—
332	C₂×D₃	—	20	2	no	—
333	C₃×C₄	—	4	2	no	—
334	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
335	C₃×C₄	C₄×D₂₀	1	1	yes	characteristic
336	C₃×C₄	C₂×#685	1	1	yes	characteristic
337	C₃×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
338	Dic₃	—	10	2	no	—
339	Dic₃	—	20	2	no	—
340	C₂×C₅	—	2	1	no	—
341	C₂×C₅	—	2	1	no	—
342	C₂×C₅	C₄×(C₂×D₃)⋊C₄	1	1	yes	characteristic
343	C₂×C₅	#5145	1	2	yes	—
344	D₅	—	12	2	no	—
345	C₂³	—	30	1	no	—
346	C₂³	#6817	1	1	yes	characteristic
347	Q₈	—	15	2	no	—
348	Q₈	—	30	1	no	—
349	D₄	—	15	2	no	—
350	D₄	—	30	2	no	—
351	D₄	—	30	2	no	—
352	D₄	—	30	1	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	30	1	no	—
355	C₂×C₄	—	30	2	no	—
356	C₂×C₄	—	30	2	no	—
357	C₂×C₄	—	30	2	no	—
358	C₂×C₄	—	60	2	no	—
359	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
360	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
361	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
362	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
363	C₂×C₄	#6817	1	1	yes	characteristic
364	C₂×C₄	C₂×#1220	1	1	yes	characteristic
365	C₈	—	2	2	no	—
366	C₈	—	30	2	no	—
367	C₈	—	30	2	no	—
368	C₈	#6714	1	2	yes	—
369	C₈	#6714	1	2	yes	—
370	C₂×C₃	—	2	1	no	—
371	C₂×C₃	—	2	1	no	—
372	C₂×C₃	C₄×#685	1	1	yes	characteristic
373	C₂×C₃	#65434	1	2	yes	—
374	D₃	—	20	2	no	—
375	C₅	#69888	1	1	yes	characteristic, complemented, Sylow
376	C₂²	—	2	2	no	—
377	C₂²	—	30	2	no	—
378	C₂²	—	60	2	no	—
379	C₂²	#91761	1	1	yes	characteristic
380	C₂²	C₄×D₆₀	1	1	yes	characteristic
381	C₂²	C₄×#1220	1	1	yes	characteristic
382	C₄	—	4	2	no	—
383	C₄	—	30	2	no	—
384	C₄	—	60	2	no	—
385	C₄	C₄²×D₁₅	1	1	yes	characteristic
386	C₄	C₄×D₆₀	1	1	yes	characteristic
387	C₄	C₂×#6714	1	1	yes	characteristic
388	C₄	C₄×#1220	1	1	yes	characteristic
389	C₃	#10600681	1	1	yes	characteristic, complemented, Sylow
390	C₂	—	2	1	no	—
391	C₂	—	2	1	no	—
392	C₂	—	60	2	no	—
393	C₂	C₄×#6714	1	1	yes	characteristic
394	C₂	#11751246	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#1220
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#405	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×#6817	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#405	D₁₅
hypocenter	C₃×C₅	#1765
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#405	C₂	semidirect
4. C₂×#91099	C₂	semidirect
6. #11746379	C₂	semidirect
7. #11746353	C₂	semidirect
17. #91790	C₄	semidirect
22. #91109	C₄	semidirect
24. #91094	C₄	semidirect
25. #91093	C₄	semidirect
28. C₅×#405	D₃	semidirect
65. C₃×#405	D₅	semidirect
164. #405	D₁₅	semidirect
326. C₃×C₅	#1765	semidirect
375. C₅	#69888	semidirect
389. C₃	#10600681	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
C₃×C₅×#405	2
C₂×#91109	2
C₂×#91099	2
#11751103	2
#11746379	2
#11746353	2
#10600681	3
#69888	5