Subgroups — #14297869

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91101	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751088	C₂	1	2	yes	maximal
5	#11746350	C₂	1	2	yes	maximal, complemented
6	#10601370	—	3	1	no	maximal, Hall
7	C₂²×C₄×D₁₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×#6714	—	2	2	no	—
11	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
12	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₁₅	—	2	2	no	—
14	C₂×C₈×D₁₅	—	2	2	no	—
15	C₂×C₈×D₁₅	C₄	1	4	yes	complemented
16	C₃×C₅×#106	—	2	2	no	—
17	#91115	—	2	2	no	—
18	#91101	—	2	2	no	—
19	#70577	—	5	1	no	maximal, Hall
20	C₂²×C₈×D₅	—	3	1	no	—
21	C₂×C₅×#106	D₃	1	1	yes	characteristic, complemented
22	C₂×#3949	—	3	1	no	—
23	#65276	—	3	2	no	—
24	#64749	—	3	2	no	—
25	C₂³×D₁₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×Dic₁₅	—	2	2	no	—
29	C₂²×Dic₁₅	D₄	1	1	yes	characteristic
30	C₂×C₄×D₁₅	—	2	1	no	—
31	C₂×C₄×D₁₅	—	2	1	no	—
32	C₂×C₄×D₁₅	—	2	1	no	—
33	C₂×C₄×D₁₅	—	2	1	no	—
34	C₂×C₄×D₁₅	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₁₅	Q₈	1	2	yes	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
38	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
39	C₃×C₅×C₄⋊C₄	—	2	2	no	—
40	C₃×C₅×C₄⋊C₄	—	2	2	no	—
41	#6714	—	4	4	no	—
42	C₄⋊Dic₁₅	—	2	2	no	—
43	C₄⋊Dic₁₅	—	2	2	no	—
44	C₂×C₃⋊#168	—	2	1	no	—
45	C₂×C₃⋊#168	—	2	1	no	—
46	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
47	C₈×D₁₅	—	2	8	no	—
48	C₂²×D₃×C₈	—	5	1	no	—
49	C₂×C₃×#106	D₅	1	1	yes	characteristic, complemented
50	C₂×#817	—	5	1	no	—
51	#4987	—	5	2	no	—
52	#4529	—	5	2	no	—
53	C₂²×C₄×D₅	—	3	1	no	—
54	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
56	C₂×#685	—	6	2	no	—
57	C₂×C₄⋊Dic₅	—	3	2	no	—
58	C₂²×#168	—	3	1	no	—
59	C₂×C₈×D₅	—	3	4	no	—
60	C₂×C₈×D₅	—	6	2	no	—
61	C₂×C₈×D₅	—	6	2	no	—
62	C₅×#106	—	2	2	no	—
63	#3963	—	6	2	no	—
64	#3949	—	6	2	no	—
65	#1807	—	15	1	no	Sylow
66	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂²×D₁₅	—	2	1	no	—
68	C₂²×D₁₅	—	2	1	no	—
69	C₂²×D₁₅	—	2	2	no	—
70	C₂²×D₁₅	C₄⋊C₄	1	2	yes	—
71	C₂²×D₁₅	SD₁₆	1	4	yes	—
72	C₂×C₃×C₄×C₅	—	4	2	no	—
73	C₂×C₃×C₄×C₅	—	4	2	no	—
74	C₂×C₃×C₄×C₅	—	4	2	no	—
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
81	C₂×Dic₁₅	—	2	1	no	—
82	C₂×Dic₁₅	—	2	1	no	—
83	C₂×Dic₁₅	—	4	2	no	—
84	C₂×Dic₁₅	—	4	2	no	—
85	C₂×Dic₁₅	—	4	2	no	—
86	C₂×Dic₁₅	C₄⋊C₄	1	2	yes	—
87	C₄×D₁₅	—	2	2	no	—
88	C₄×D₁₅	—	2	2	no	—
89	C₄×D₁₅	—	2	2	no	—
90	C₄×D₁₅	—	2	2	no	—
91	C₃×C₅×C₈	C₂²⋊C₄	1	4	yes	complemented
92	C₃⋊#168	—	2	2	no	—
93	C₂²×C₄×D₃	—	5	1	no	—
94	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
95	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
96	C₂×(C₂×D₃)⋊C₄	—	10	2	no	—
97	C₂×C₄⋊Dic₃	—	5	2	no	—
98	C₂²×C₃⋊C₈	—	5	1	no	—
99	C₂×D₃×C₈	—	5	4	no	—
100	C₂×D₃×C₈	—	10	2	no	—
101	C₂×D₃×C₈	—	10	2	no	—
102	C₃×#106	—	2	2	no	—
103	C₃⋊#95	—	10	2	no	—
104	#817	—	10	2	no	—
105	C₂³×D₅	—	3	1	no	—
106	C₂²×C₄×C₅	—	2	2	no	—
107	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
108	C₂²×Dic₅	—	3	1	no	—
109	C₂²×Dic₅	—	6	2	no	—
110	C₂×C₄×D₅	—	3	2	no	—
111	C₂×C₄×D₅	—	3	2	no	—
112	C₂×C₄×D₅	—	6	1	no	—
113	C₂×C₄×D₅	—	6	1	no	—
114	C₂×C₄×D₅	—	6	1	no	—
115	C₂×C₄×D₅	—	6	1	no	—
116	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
117	C₂×C₅×C₈	D₁₂	1	2	yes	—
118	C₂×C₅×C₈	#67	1	2	yes	—
119	C₅×C₄⋊C₄	—	2	2	no	—
120	C₅×C₄⋊C₄	—	2	2	no	—
121	#685	—	12	4	no	—
122	C₄⋊Dic₅	—	6	2	no	—
123	C₄⋊Dic₅	—	6	2	no	—
124	C₂×#168	—	3	2	no	—
125	C₂×#168	—	6	1	no	—
126	C₂×#168	—	6	1	no	—
127	C₈×D₅	—	6	8	no	—
128	C₂³×C₈	—	15	1	no	—
129	C₂×#106	—	15	1	no	—
130	C₂×#106	D₁₅	1	1	yes	characteristic, complemented
131	#380	—	15	2	no	—
132	#340	—	15	2	no	—
133	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₃×C₅	#121	1	1	yes	characteristic
135	C₂²×C₃×C₅	#121	1	1	yes	characteristic
136	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
137	C₂²×C₃×C₅	#135	1	2	yes	—
138	C₂×D₁₅	—	2	2	no	—
139	C₂×D₁₅	—	2	2	no	—
140	C₂×D₁₅	—	2	4	no	—
141	C₂×D₁₅	—	2	4	no	—
142	C₂×D₁₅	#106	1	4	yes	complemented
143	C₃×C₄×C₅	—	8	2	no	—
144	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₃×C₄×C₅	#117	1	1	yes	characteristic
146	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
147	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
148	Dic₁₅	—	2	2	no	—
149	Dic₁₅	—	8	2	no	—
150	C₂³×D₃	—	5	1	no	—
151	C₂²×C₃×C₄	—	2	2	no	—
152	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
153	C₂²×Dic₃	—	5	1	no	—
154	C₂²×Dic₃	—	10	2	no	—
155	C₂×C₄×D₃	—	5	2	no	—
156	C₂×C₄×D₃	—	5	2	no	—
157	C₂×C₄×D₃	—	10	1	no	—
158	C₂×C₄×D₃	—	10	1	no	—
159	C₂×C₄×D₃	—	10	1	no	—
160	C₂×C₄×D₃	—	10	1	no	—
161	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
162	C₂×C₃×C₈	D₂₀	1	2	yes	—
163	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
164	C₃×C₄⋊C₄	—	2	2	no	—
165	C₃×C₄⋊C₄	—	2	2	no	—
166	(C₂×D₃)⋊C₄	—	20	4	no	—
167	C₄⋊Dic₃	—	10	2	no	—
168	C₄⋊Dic₃	—	10	2	no	—
169	C₂×C₃⋊C₈	—	5	2	no	—
170	C₂×C₃⋊C₈	—	10	1	no	—
171	C₂×C₃⋊C₈	—	10	1	no	—
172	D₃×C₈	—	10	8	no	—
173	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
174	C₂²×D₅	—	3	2	no	—
175	C₂²×D₅	—	3	4	no	—
176	C₂²×D₅	—	6	1	no	—
177	C₂²×D₅	—	6	2	no	—
178	C₂²×D₅	—	6	1	no	—
179	C₂×C₄×C₅	—	4	2	no	—
180	C₂×C₄×C₅	—	4	2	no	—
181	C₂×C₄×C₅	—	4	2	no	—
182	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
184	C₂×C₄×C₅	#237	1	1	yes	characteristic
185	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
186	C₂×C₄×C₅	#239	1	1	yes	characteristic
187	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
188	C₂×Dic₅	—	3	2	no	—
189	C₂×Dic₅	—	6	1	no	—
190	C₂×Dic₅	—	6	1	no	—
191	C₂×Dic₅	—	12	2	no	—
192	C₂×Dic₅	—	12	2	no	—
193	C₂×Dic₅	—	12	2	no	—
194	C₄×D₅	—	6	2	no	—
195	C₄×D₅	—	6	2	no	—
196	C₄×D₅	—	6	2	no	—
197	C₄×D₅	—	6	2	no	—
198	C₅×C₈	(C₂×D₃)⋊C₄	1	4	yes	complemented
199	#168	—	6	2	no	—
200	C₂³×C₄	—	15	1	no	—
201	C₂²×C₈	—	15	1	no	—
202	C₂²×C₈	—	15	4	no	—
203	C₂²×C₈	—	30	2	no	—
204	C₂²×C₈	—	30	2	no	—
205	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
206	C₂×C₄⋊C₄	—	15	2	no	—
207	C₂×C₄⋊C₄	C₂×D₁₅	1	2	yes	—
208	C₂×C₂²⋊C₄	—	30	2	no	—
209	#106	—	2	2	no	—
210	#106	—	30	2	no	—
211	#95	—	30	2	no	—
212	C₂×C₃×C₅	#380	1	1	yes	characteristic
213	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
214	C₂×C₃×C₅	#427	1	1	yes	characteristic
215	C₂×C₃×C₅	#465	1	2	yes	—
216	C₂×C₃×C₅	#465	1	2	yes	—
217	D₁₅	—	2	4	no	—
218	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
219	C₂²×D₃	—	5	2	no	—
220	C₂²×D₃	—	5	4	no	—
221	C₂²×D₃	—	10	1	no	—
222	C₂²×D₃	—	10	2	no	—
223	C₂²×D₃	—	10	1	no	—
224	C₂×C₃×C₄	—	4	2	no	—
225	C₂×C₃×C₄	—	4	2	no	—
226	C₂×C₃×C₄	—	4	2	no	—
227	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
229	C₂×C₃×C₄	#711	1	1	yes	characteristic
230	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
231	C₂×C₃×C₄	#713	1	1	yes	characteristic
232	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
233	C₂×Dic₃	—	5	2	no	—
234	C₂×Dic₃	—	10	1	no	—
235	C₂×Dic₃	—	10	1	no	—
236	C₂×Dic₃	—	20	2	no	—
237	C₂×Dic₃	—	20	2	no	—
238	C₂×Dic₃	—	20	2	no	—
239	C₄×D₃	—	10	2	no	—
240	C₄×D₃	—	10	2	no	—
241	C₄×D₃	—	10	2	no	—
242	C₄×D₃	—	10	2	no	—
243	C₃×C₈	#685	1	4	yes	complemented
244	C₃⋊C₈	—	10	2	no	—
245	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
246	C₂²×C₅	#895	1	1	yes	characteristic
247	C₂²×C₅	D₃×SD₁₆	1	2	yes	—
248	C₂²×C₅	#916	1	2	yes	—
249	C₂²×C₅	#938	1	1	yes	characteristic
250	C₂×D₅	—	3	4	no	—
251	C₂×D₅	—	6	4	no	—
252	C₂×D₅	—	6	4	no	—
253	C₂×D₅	—	6	2	no	—
254	C₂×D₅	—	6	2	no	—
255	C₄×C₅	—	8	2	no	—
256	C₄×C₅	#892	1	1	yes	characteristic
257	C₄×C₅	#897	1	1	yes	characteristic
258	C₄×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
259	C₄×C₅	#946	1	1	yes	characteristic
260	Dic₅	—	6	2	no	—
261	Dic₅	—	24	2	no	—
262	C₂⁴	—	15	1	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	15	1	no	—
265	C₂²×C₄	—	15	2	no	—
266	C₂²×C₄	—	15	2	no	—
267	C₂²×C₄	—	30	1	no	—
268	C₂²×C₄	—	30	1	no	—
269	C₂²×C₄	—	30	1	no	—
270	C₂²×C₄	—	30	1	no	—
271	C₂²×C₄	—	30	2	no	—
272	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
273	C₂×C₈	—	15	2	no	—
274	C₂×C₈	—	30	1	no	—
275	C₂×C₈	—	30	1	no	—
276	C₂×C₈	—	30	8	no	—
277	C₂×C₈	C₄×D₁₅	1	2	yes	—
278	C₂×C₈	D₆₀	1	2	yes	—
279	C₂×C₈	#1220	1	2	yes	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	30	2	no	—
283	C₄⋊C₄	—	30	2	no	—
284	C₂²⋊C₄	—	60	4	no	—
285	C₃×C₅	#1807	1	1	yes	characteristic, complemented, Hall
286	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
287	C₂²×C₃	#4041	1	1	yes	characteristic
288	C₂²×C₃	D₅×SD₁₆	1	2	yes	—
289	C₂²×C₃	#4062	1	2	yes	—
290	C₂²×C₃	#4084	1	1	yes	characteristic
291	C₂×D₃	—	5	4	no	—
292	C₂×D₃	—	10	2	no	—
293	C₂×D₃	—	10	4	no	—
294	C₂×D₃	—	10	4	no	—
295	C₂×D₃	—	10	2	no	—
296	C₃×C₄	—	8	2	no	—
297	C₃×C₄	#4038	1	1	yes	characteristic
298	C₃×C₄	#4043	1	1	yes	characteristic
299	C₃×C₄	C₂×#685	1	1	yes	characteristic
300	C₃×C₄	#4092	1	1	yes	characteristic
301	Dic₃	—	10	2	no	—
302	Dic₃	—	40	2	no	—
303	C₂×C₅	D₃×#106	1	1	yes	characteristic
304	C₂×C₅	#4860	1	1	yes	characteristic
305	C₂×C₅	#4864	1	2	yes	—
306	C₂×C₅	#4987	1	1	yes	characteristic
307	C₂×C₅	#5171	1	2	yes	—
308	D₅	—	6	4	no	—
309	C₂³	—	15	2	no	—
310	C₂³	—	15	4	no	—
311	C₂³	—	30	1	no	—
312	C₂³	—	30	2	no	—
313	C₂³	—	30	1	no	—
314	C₂³	D₄×D₁₅	1	1	yes	characteristic
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	15	2	no	—
319	C₂×C₄	—	30	1	no	—
320	C₂×C₄	—	30	1	no	—
321	C₂×C₄	—	30	2	no	—
322	C₂×C₄	—	30	2	no	—
323	C₂×C₄	—	30	2	no	—
324	C₂×C₄	—	30	2	no	—
325	C₂×C₄	—	60	2	no	—
326	C₂×C₄	—	60	2	no	—
327	C₂×C₄	—	60	2	no	—
328	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
329	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
330	C₂×C₄	#6819	1	1	yes	characteristic
331	C₂×C₄	Q₈×D₁₅	1	1	yes	characteristic
332	C₂×C₄	#6821	1	1	yes	characteristic
333	C₂×C₄	C₂×#1220	1	1	yes	characteristic
334	C₈	—	30	2	no	—
335	C₈	#6714	1	4	yes	complemented
336	C₂×C₃	D₅×#106	1	1	yes	characteristic
337	C₂×C₃	#65149	1	1	yes	characteristic
338	C₂×C₃	#65153	1	2	yes	—
339	C₂×C₃	#65276	1	1	yes	characteristic
340	C₂×C₃	#65460	1	2	yes	—
341	D₃	—	10	4	no	—
342	C₅	#70577	1	1	yes	characteristic, complemented, Sylow
343	C₂²	—	15	4	no	—
344	C₂²	—	30	2	no	—
345	C₂²	—	30	4	no	—
346	C₂²	—	30	4	no	—
347	C₂²	—	30	2	no	—
348	C₂²	C₄⋊C₄×D₁₅	1	1	yes	characteristic
349	C₂²	#91784	1	1	yes	characteristic
350	C₂²	SD₁₆×D₁₅	1	2	yes	—
351	C₂²	#91805	1	2	yes	—
352	C₂²	#91827	1	1	yes	characteristic
353	C₄	—	8	2	no	—
354	C₄	—	30	2	no	—
355	C₄	—	120	2	no	—
356	C₄	#91781	1	1	yes	characteristic
357	C₄	#91786	1	1	yes	characteristic
358	C₄	C₂×#6714	1	1	yes	characteristic
359	C₄	#91835	1	1	yes	characteristic
360	C₃	#10601370	1	1	yes	characteristic, complemented, Sylow
361	C₂	—	30	4	no	—
362	C₂	D₁₅×#106	1	1	yes	characteristic
363	C₂	#11750961	1	1	yes	characteristic
364	C₂	#11750965	1	2	yes	—
365	C₂	#11751088	1	1	yes	characteristic
366	C₂	#11751272	1	2	yes	—

	subgroup	G/H
center	C₂³	D₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂×C₃×C₅×#106	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂³×D₁₅	D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×#106	D₁₅
hypocenter	C₃×C₅	#1807
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×#106	C₂	semidirect
3. C₂×#91101	C₂	semidirect
5. #11746350	C₂	semidirect
15. C₂×C₈×D₁₅	C₄	semidirect
21. C₂×C₅×#106	D₃	semidirect
49. C₂×C₃×#106	D₅	semidirect
91. C₃×C₅×C₈	C₂²⋊C₄	semidirect
130. C₂×#106	D₁₅	semidirect
142. C₂×D₁₅	#106	semidirect
198. C₅×C₈	(C₂×D₃)⋊C₄	semidirect
243. C₃×C₈	#685	semidirect
285. C₃×C₅	#1807	semidirect
335. C₈	#6714	semidirect
342. C₅	#70577	semidirect
360. C₃	#10601370	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×D₁₅	2
C₂×C₃×C₅×#106	2
C₂×#91101	2
#11751088	2
#11746350	2
#10601370	3
#70577	5