Subgroups — #14478162

Order 1920.

720 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	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91096	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751008	C₂	1	4	yes	maximal, complemented
5	D₁₅×M₃₂	C₂	1	4	yes	maximal, complemented
6	#11751002	C₂	1	4	yes	maximal, complemented
7	#10613738	—	3	1	no	maximal, Hall
8	C₂×#6817	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×#6710	C₄	1	1	yes	characteristic
11	#91790	C₄	1	2	yes	—
12	#91790	C₄	1	2	yes	—
13	#91790	C₂²	1	2	yes	—
14	#91790	C₂²	1	2	yes	—
15	C₂×#6705	C₄	1	2	yes	—
16	C₂×C₈×D₁₅	C₂²	1	2	yes	—
17	C₃×C₅×M₃₂	C₂²	1	2	yes	—
18	C₃×C₅×M₃₂	C₂²	1	2	yes	complemented
19	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	complemented
20	#91096	C₂²	1	2	yes	—
21	#91096	C₂²	1	2	yes	complemented
22	C₂×C₃⋊#672	C₂²	1	2	yes	complemented
23	#91082	C₂²	1	8	yes	complemented
24	C₁₆×D₁₅	C₂²	1	8	yes	complemented
25	#82007	—	5	1	no	maximal, Hall
26	C₂×#4047	—	3	1	no	—
27	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic, complemented
28	C₂×#3944	—	3	1	no	—
29	#65196	—	3	4	no	—
30	D₅×M₃₂	—	3	4	no	—
31	#65190	—	3	4	no	—
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×#1220	C₈	1	2	yes	—
34	#6817	C₈	1	2	yes	—
35	#6817	C₈	1	2	yes	—
36	#6817	C₂×C₄	1	2	yes	—
37	#6817	C₂×C₄	1	2	yes	—
38	C₂×D₆₀	C₈	1	1	yes	characteristic
39	C₂×C₄×D₁₅	C₂×C₄	1	2	yes	—
40	C₂×Dic₃₀	C₈	1	1	yes	characteristic
41	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
47	#6710	C₂×C₄	1	2	yes	—
48	#6710	C₂×C₄	1	2	yes	—
49	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
50	#6705	C₂×C₄	1	4	yes	—
51	#6705	C₂×C₄	1	4	yes	—
52	C₈×D₁₅	C₂×C₄	1	4	yes	—
53	C₈×D₁₅	C₂³	1	4	yes	—
54	C₃×C₅×C₁₆	C₂³	1	4	yes	—
55	C₃⋊#672	C₂³	1	4	yes	—
56	C₂×#901	—	5	1	no	—
57	C₂×C₃×M₃₂	D₅	1	1	yes	characteristic, complemented
58	C₂×#812	—	5	1	no	—
59	#4907	—	5	4	no	—
60	D₃×M₃₂	—	5	4	no	—
61	#4901	—	5	4	no	—
62	C₂×#709	—	3	1	no	—
63	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
64	C₂×#681	—	3	1	no	—
65	#4047	—	3	2	no	—
66	#4047	—	3	2	no	—
67	#4047	—	3	2	no	—
68	#4047	—	3	2	no	—
69	C₂×#676	—	3	2	no	—
70	C₂×C₈×D₅	—	3	2	no	—
71	C₅×M₃₂	C₂×D₃	1	2	yes	—
72	C₅×M₃₂	C₂×D₃	1	2	yes	complemented
73	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	complemented
74	#3944	—	3	2	no	—
75	#3944	—	3	2	no	—
76	C₂×#672	—	3	2	no	—
77	#3930	—	3	8	no	—
78	D₅×C₁₆	—	3	8	no	—
79	#3407	—	15	1	no	Sylow
80	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
81	C₂²×D₁₅	C₂×C₈	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
88	#1220	C₂×C₈	1	4	yes	—
89	#1220	C₂×C₈	1	4	yes	—
90	C₂×Dic₁₅	C₂×C₈	1	2	yes	—
91	D₆₀	C₂×C₈	1	2	yes	—
92	D₆₀	C₂×C₈	1	2	yes	—
93	C₄×D₁₅	C₂×C₈	1	4	yes	—
94	C₄×D₁₅	C₂²×C₄	1	4	yes	—
95	Dic₃₀	C₂×C₈	1	2	yes	—
96	Dic₃₀	C₂×C₈	1	2	yes	—
97	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
99	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
100	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
101	C₃⋊#168	C₂²×C₄	1	4	yes	—
102	C₂×#235	—	5	1	no	—
103	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
104	C₂×#208	—	5	1	no	—
105	#901	—	5	2	no	—
106	#901	—	5	2	no	—
107	#901	—	5	2	no	—
108	#901	—	5	2	no	—
109	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
110	C₂×D₃×C₈	—	5	2	no	—
111	C₃×M₃₂	C₂×D₅	1	2	yes	—
112	C₃×M₃₂	C₂×D₅	1	2	yes	complemented
113	C₂×C₃×C₁₆	C₂×D₅	1	2	yes	complemented
114	#812	—	5	2	no	—
115	#812	—	5	2	no	—
116	C₂×C₃⋊C₁₆	—	5	2	no	—
117	#798	—	5	8	no	—
118	D₃×C₁₆	—	5	8	no	—
119	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
120	C₂×C₂²⋊D₅	—	3	2	no	—
121	#709	—	3	2	no	—
122	#709	—	3	2	no	—
123	#709	—	3	2	no	—
124	#709	—	3	2	no	—
125	C₂×D₂₀	—	3	1	no	—
126	C₂×C₄×D₅	—	3	2	no	—
127	C₂×Dic₁₀	—	3	1	no	—
128	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
129	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
130	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
131	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
132	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
133	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
134	#681	—	3	2	no	—
135	#681	—	3	2	no	—
136	C₂×#168	—	3	2	no	—
137	#676	—	3	4	no	—
138	#676	—	3	4	no	—
139	C₈×D₅	—	3	4	no	—
140	C₈×D₅	—	3	4	no	—
141	C₅×C₁₆	C₂²×D₃	1	4	yes	—
142	#672	—	3	4	no	—
143	C₂×#131	—	15	1	no	—
144	#504	—	15	4	no	—
145	#504	—	15	4	no	—
146	C₂×M₃₂	—	15	1	no	—
147	C₂×M₃₂	—	15	4	no	—
148	C₂×M₃₂	D₁₅	1	1	yes	characteristic, complemented
149	C₂²×C₃×C₅	—	4	1	no	—
150	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
151	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
152	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
153	C₂×D₁₅	—	4	2	no	—
154	C₂×D₁₅	C₂²×C₈	1	4	yes	—
155	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
156	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
157	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
158	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
159	Dic₁₅	C₂²×C₈	1	4	yes	—
160	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
161	C₂×#67	—	5	2	no	—
162	#235	—	5	2	no	—
163	#235	—	5	2	no	—
164	#235	—	5	2	no	—
165	#235	—	5	2	no	—
166	C₂×D₁₂	—	5	1	no	—
167	C₂×C₄×D₃	—	5	2	no	—
168	C₂×Dic₆	—	5	1	no	—
169	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
170	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
171	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
172	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
173	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
174	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
175	#208	—	5	2	no	—
176	#208	—	5	2	no	—
177	C₂×C₃⋊C₈	—	5	2	no	—
178	ΓL(1,𝔽₂₅)	—	5	4	no	—
179	ΓL(1,𝔽₂₅)	—	5	4	no	—
180	D₃×C₈	—	5	4	no	—
181	D₃×C₈	—	5	4	no	—
182	C₃×C₁₆	C₂²×D₅	1	4	yes	—
183	C₃⋊C₁₆	—	5	4	no	—
184	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
185	C₂²×D₅	—	3	2	no	—
186	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
187	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
188	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
189	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
191	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
192	C₂²⋊D₅	—	3	4	no	—
193	C₂²⋊D₅	—	3	4	no	—
194	C₂×Dic₅	—	3	2	no	—
195	D₂₀	—	3	2	no	—
196	D₂₀	—	3	2	no	—
197	C₄×D₅	—	3	4	no	—
198	C₄×D₅	—	3	4	no	—
199	Dic₁₀	—	3	2	no	—
200	Dic₁₀	—	3	2	no	—
201	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
202	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
203	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
204	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
205	#168	—	3	4	no	—
206	C₂×#41	—	15	1	no	—
207	#131	—	15	2	no	—
208	#131	—	15	2	no	—
209	#131	—	15	2	no	—
210	#131	—	15	2	no	—
211	C₂×M₁₆	—	15	1	no	—
212	C₂×M₁₆	—	15	2	no	—
213	C₂²×C₈	—	15	2	no	—
214	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
215	M₃₂	—	15	2	no	—
216	M₃₂	—	15	2	no	—
217	M₃₂	—	15	8	no	—
218	M₃₂	C₂×D₁₅	1	2	yes	—
219	M₃₂	C₂×D₁₅	1	2	yes	complemented
220	C₂×C₁₆	—	15	2	no	—
221	C₂×C₁₆	—	15	8	no	—
222	C₂×C₁₆	C₂×D₁₅	1	2	yes	complemented
223	C₂×C₃×C₅	—	2	1	no	—
224	C₂×C₃×C₅	—	2	1	no	—
225	C₂×C₃×C₅	—	2	1	no	—
226	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
227	D₁₅	—	2	4	no	—
228	C₂³×C₃	C₈×D₅	1	1	yes	characteristic
229	C₂²×D₃	—	5	2	no	—
230	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
231	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
232	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
236	#67	—	5	4	no	—
237	#67	—	5	4	no	—
238	C₂×Dic₃	—	5	2	no	—
239	D₁₂	—	5	2	no	—
240	D₁₂	—	5	2	no	—
241	C₄×D₃	—	5	4	no	—
242	C₄×D₃	—	5	4	no	—
243	Dic₆	—	5	2	no	—
244	Dic₆	—	5	2	no	—
245	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
246	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
247	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
248	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
249	C₃⋊C₈	—	5	4	no	—
250	C₂²×C₅	—	4	1	no	—
251	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
252	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
253	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
254	C₂×D₅	—	3	4	no	—
255	C₂×D₅	—	12	2	no	—
256	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
257	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
258	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
259	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
260	Dic₅	—	3	4	no	—
261	#41	—	15	2	no	—
262	#41	—	15	2	no	—
263	#41	—	15	2	no	—
264	#41	—	15	2	no	—
265	C₂×Q₈	—	15	1	no	—
266	C₂×D₄	—	15	2	no	—
267	C₂×D₄	—	15	1	no	—
268	C₂²×C₄	—	15	2	no	—
269	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
270	M₁₆	—	15	2	no	—
271	M₁₆	—	15	2	no	—
272	M₁₆	—	15	4	no	—
273	M₁₆	—	15	4	no	—
274	C₂×C₈	—	15	2	no	—
275	C₂×C₈	—	15	4	no	—
276	C₂×C₈	—	15	4	no	—
277	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
278	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
279	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
280	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
281	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
282	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
283	C₁₆	—	15	4	no	—
284	C₁₆	C₂²×D₁₅	1	4	yes	—
285	C₃×C₅	#3407	1	1	yes	characteristic, complemented, Hall
286	C₂²×C₃	—	4	1	no	—
287	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
288	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
289	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
290	C₂×D₃	—	5	4	no	—
291	C₂×D₃	—	20	2	no	—
292	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
293	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
294	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
295	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
296	Dic₃	—	5	4	no	—
297	C₂×C₅	—	2	1	no	—
298	C₂×C₅	—	2	1	no	—
299	C₂×C₅	—	2	1	no	—
300	C₂×C₅	C₂²×D₃×C₈	1	1	yes	characteristic
301	D₅	—	6	4	no	—
302	C₂³	—	15	2	no	—
303	C₂³	C₈×D₁₅	1	1	yes	characteristic
304	Q₈	—	15	2	no	—
305	Q₈	—	15	2	no	—
306	D₄	—	15	4	no	—
307	D₄	—	15	4	no	—
308	D₄	—	15	2	no	—
309	D₄	—	15	2	no	—
310	C₂×C₄	—	15	2	no	—
311	C₂×C₄	—	15	4	no	—
312	C₂×C₄	—	15	4	no	—
313	C₂×C₄	C₈×D₁₅	1	1	yes	characteristic
314	C₂×C₄	C₈×D₁₅	1	1	yes	characteristic
315	C₂×C₄	C₈×D₁₅	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
319	C₈	—	15	4	no	—
320	C₈	C₂×C₄×D₁₅	1	1	yes	characteristic
321	C₈	C₂×C₄×D₁₅	1	1	yes	characteristic
322	C₈	C₂×C₄×D₁₅	1	1	yes	characteristic
323	C₈	C₂³×D₁₅	1	1	yes	characteristic
324	C₂×C₃	—	2	1	no	—
325	C₂×C₃	—	2	1	no	—
326	C₂×C₃	—	2	1	no	—
327	C₂×C₃	C₂²×C₈×D₅	1	1	yes	characteristic
328	D₃	—	10	4	no	—
329	C₅	#82007	1	1	yes	characteristic, complemented, Sylow
330	C₂²	—	4	1	no	—
331	C₂²	—	15	4	no	—
332	C₂²	—	60	2	no	—
333	C₂²	C₂×C₈×D₁₅	1	1	yes	characteristic
334	C₂²	C₂×C₈×D₁₅	1	1	yes	characteristic
335	C₂²	C₂×C₈×D₁₅	1	1	yes	characteristic
336	C₄	—	15	4	no	—
337	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
338	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
339	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
340	C₄	C₂²×C₄×D₁₅	1	1	yes	characteristic
341	C₃	#10613738	1	1	yes	characteristic, complemented, Sylow
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	2	1	no	—
345	C₂	—	30	4	no	—
346	C₂	C₂²×C₈×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×D₁₅
commutator, socle	C₂×C₃×C₅	C₂³×C₈
Fitting	C₂×C₃×C₅×M₃₂	C₂
prime radical	C₂×#6817	C₄
cosocle	C₃×C₅×C₈	C₂⁴
hypercenter	C₂×M₃₂	D₁₅
hypocenter	C₃×C₅	#3407
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₂×#91096	C₂	semidirect
4. #11751008	C₂	semidirect
5. D₁₅×M₃₂	C₂	semidirect
6. #11751002	C₂	semidirect
18. C₃×C₅×M₃₂	C₂²	semidirect
19. C₂×C₃×C₅×C₁₆	C₂²	semidirect
21. #91096	C₂²	semidirect
22. C₂×C₃⋊#672	C₂²	semidirect
23. #91082	C₂²	semidirect
24. C₁₆×D₁₅	C₂²	semidirect
27. C₂×C₅×M₃₂	D₃	semidirect
57. C₂×C₃×M₃₂	D₅	semidirect
72. C₅×M₃₂	C₂×D₃	semidirect
73. C₂×C₅×C₁₆	C₂×D₃	semidirect
112. C₃×M₃₂	C₂×D₅	semidirect
113. C₂×C₃×C₁₆	C₂×D₅	semidirect
148. C₂×M₃₂	D₁₅	semidirect
219. M₃₂	C₂×D₁₅	semidirect
222. C₂×C₁₆	C₂×D₁₅	semidirect
285. C₃×C₅	#3407	semidirect
329. C₅	#82007	semidirect
341. C₃	#10613738	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
C₂×C₃×C₅×M₃₂	2
C₂×#91096	2
#11751008	2
D₁₅×M₃₂	2
#11751002	2
#10613738	3
#82007	5