Subgroups — #14307131

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#533	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic
3	C₄×#6710	C₂	1	1	yes	maximal, characteristic
4	#11746353	C₂	1	4	yes	maximal
5	#10602634	—	3	1	no	maximal, Hall
6	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
7	C₃×C₅×C₄⋊Q₈	C₄	1	1	yes	characteristic, complemented
8	C₃×C₅×#125	C₄	1	2	yes	complemented
9	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
10	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
11	C₃×C₄×C₅×Q₈	—	2	1	no	—
12	C₃×C₄×C₅×Q₈	—	2	1	no	—
13	C₃×C₅×#117	C₂²	1	2	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₅	—	2	1	no	—
16	C₂×C₄×Dic₁₅	C₂²	1	2	yes	—
17	C₂×#6710	C₂²	1	2	yes	—
18	#91115	—	2	1	no	—
19	#91087	—	2	1	no	—
20	C₄×C₃⋊#168	—	2	1	no	—
21	#71841	—	5	1	no	maximal, Hall
22	C₅×#533	D₃	1	1	yes	characteristic
23	C₄×C₄⋊Dic₅	—	3	1	no	—
24	C₄×#681	—	3	1	no	—
25	#64752	—	3	4	no	—
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₂×C₃×C₅×Q₈	—	2	1	no	—
28	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×Dic₁₅	—	2	2	no	—
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	1	no	—
35	C₃×C₅×C₄⋊C₄	—	2	1	no	—
36	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
39	C₃×C₄²×C₅	—	2	2	no	—
40	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
43	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₄⋊Dic₁₅	—	2	4	no	—
45	C₄×Dic₁₅	—	2	4	no	—
46	#6710	—	2	2	no	—
47	#6710	Q₈	1	4	yes	complemented
48	C₂×C₃⋊#168	—	2	2	no	—
49	C₃×#533	D₅	1	1	yes	characteristic
50	C₄×C₄⋊Dic₃	—	5	1	no	—
51	C₄×#208	—	5	1	no	—
52	#4532	—	5	4	no	—
53	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic, complemented
54	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic, complemented
55	C₅×#125	Dic₃	1	2	yes	complemented
56	C₅×C₂²⋊Q₈	—	2	1	no	—
57	C₅×C₂²⋊Q₈	—	2	1	no	—
58	C₄×C₅×Q₈	—	2	1	no	—
59	C₄×C₅×Q₈	—	2	1	no	—
60	C₅×#117	C₂×D₃	1	2	yes	—
61	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
62	C₂×C₄⋊Dic₅	—	6	1	no	—
63	C₂×C₄×Dic₅	—	3	2	no	—
64	C₂×#681	—	3	2	no	—
65	#3963	—	6	1	no	—
66	#3935	—	6	1	no	—
67	C₄×#168	—	6	1	no	—
68	#1986	—	15	1	no	Sylow
69	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
70	C₃×C₅×Q₈	—	4	2	no	—
71	C₃×C₅×Q₈	—	4	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	4	1	no	—
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
83	C₂×C₃×C₄×C₅	#41	1	2	yes	—
84	C₂×Dic₁₅	—	2	4	no	—
85	C₂×Dic₁₅	—	4	2	no	—
86	C₂×Dic₁₅	—	4	2	no	—
87	C₃⋊#168	—	2	4	no	—
88	C₃×C₄⋊Q₈	Dic₅	1	1	yes	characteristic, complemented
89	C₃×C₄⋊Q₈	Dic₅	1	1	yes	characteristic, complemented
90	C₃×#125	Dic₅	1	2	yes	complemented
91	C₃×C₂²⋊Q₈	—	2	1	no	—
92	C₃×C₂²⋊Q₈	—	2	1	no	—
93	C₃×C₄×Q₈	—	2	1	no	—
94	C₃×C₄×Q₈	—	2	1	no	—
95	C₃×#117	C₂×D₅	1	2	yes	—
96	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
97	C₂×C₄⋊Dic₃	—	10	1	no	—
98	C₂×C₄×Dic₃	—	5	2	no	—
99	C₂×#208	—	5	2	no	—
100	C₃⋊#95	—	10	1	no	—
101	#803	—	10	1	no	—
102	C₄×C₃⋊C₈	—	10	1	no	—
103	C₂×C₅×Q₈	—	2	1	no	—
104	C₂×C₅×Q₈	—	2	1	no	—
105	C₂²×C₄×C₅	#67	1	1	yes	characteristic
106	C₂²×C₄×C₅	#67	1	1	yes	characteristic
107	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
108	C₂²×Dic₅	—	6	2	no	—
109	C₅×C₄⋊C₄	—	2	2	no	—
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	—	2	1	no	—
112	C₅×C₄⋊C₄	—	2	1	no	—
113	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
114	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
115	C₅×C₂²⋊C₄	—	2	2	no	—
116	C₄²×C₅	—	2	2	no	—
117	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
118	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
119	C₄²×C₅	#67	1	1	yes	characteristic
120	C₄²×C₅	#67	1	1	yes	characteristic
121	C₄⋊Dic₅	—	6	4	no	—
122	C₄×Dic₅	—	6	4	no	—
123	#681	—	3	4	no	—
124	#681	—	6	2	no	—
125	C₂×#168	—	6	2	no	—
126	#533	D₁₅	1	1	yes	characteristic
127	C₄×M₁₆	—	15	1	no	—
128	C₄×C₄⋊C₄	—	15	1	no	—
129	#339	—	15	4	no	—
130	C₂²×C₃×C₅	—	2	1	no	—
131	C₂²×C₃×C₅	—	2	1	no	—
132	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₃×C₅	#124	1	1	yes	characteristic
134	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
135	C₃×C₄×C₅	—	2	1	no	—
136	C₃×C₄×C₅	—	2	1	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	4	2	no	—
139	C₃×C₄×C₅	—	4	2	no	—
140	C₃×C₄×C₅	#104	1	2	yes	—
141	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
142	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
143	Dic₁₅	—	4	4	no	—
144	C₂×C₃×Q₈	—	2	1	no	—
145	C₂×C₃×Q₈	—	2	1	no	—
146	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
147	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
148	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
149	C₂²×Dic₃	—	10	2	no	—
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	—	2	2	no	—
152	C₃×C₄⋊C₄	—	2	1	no	—
153	C₃×C₄⋊C₄	—	2	1	no	—
154	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
155	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
156	C₃×C₂²⋊C₄	—	2	2	no	—
157	C₃×C₄²	—	2	2	no	—
158	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
159	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
160	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
161	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
162	C₄⋊Dic₃	—	10	4	no	—
163	C₄×Dic₃	—	10	4	no	—
164	#208	—	5	4	no	—
165	#208	—	10	2	no	—
166	C₂×C₃⋊C₈	—	10	2	no	—
167	C₂³×C₅	C₂×#67	1	1	yes	characteristic
168	C₅×Q₈	—	4	2	no	—
169	C₅×Q₈	—	4	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	4	1	no	—
174	C₂×C₄×C₅	#217	1	1	yes	characteristic
175	C₂×C₄×C₅	#217	1	1	yes	characteristic
176	C₂×C₄×C₅	#217	1	1	yes	characteristic
177	C₂×C₄×C₅	#217	1	1	yes	characteristic
178	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
179	C₂×C₄×C₅	#239	1	2	yes	—
180	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
182	C₂×Dic₅	—	6	4	no	—
183	C₂×Dic₅	—	12	2	no	—
184	C₂×Dic₅	—	12	2	no	—
185	#168	—	6	4	no	—
186	C₂×M₁₆	—	15	2	no	—
187	C₄⋊Q₈	Dic₁₅	1	1	yes	characteristic, complemented
188	C₄⋊Q₈	Dic₁₅	1	1	yes	characteristic, complemented
189	#125	Dic₁₅	1	2	yes	complemented
190	C₂²⋊Q₈	—	2	1	no	—
191	C₂²⋊Q₈	—	2	1	no	—
192	C₄×Q₈	—	2	1	no	—
193	C₄×Q₈	—	2	1	no	—
194	#117	C₂×D₁₅	1	2	yes	—
195	C₂×C₄⋊C₄	—	30	1	no	—
196	C₂×C₄²	—	15	2	no	—
197	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
198	#97	—	30	1	no	—
199	C₄×C₈	—	30	1	no	—
200	#95	—	30	1	no	—
201	C₂×C₃×C₅	—	2	1	no	—
202	C₂×C₃×C₅	—	2	1	no	—
203	C₂×C₃×C₅	#391	1	1	yes	characteristic
204	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
205	C₂×C₃×C₅	#421	1	1	yes	characteristic
206	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
207	C₃×Q₈	—	4	2	no	—
208	C₃×Q₈	—	4	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	4	1	no	—
213	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
217	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
218	C₂×C₃×C₄	#713	1	2	yes	—
219	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
221	C₂×Dic₃	—	10	4	no	—
222	C₂×Dic₃	—	20	2	no	—
223	C₂×Dic₃	—	20	2	no	—
224	C₃⋊C₈	—	10	4	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	#944	1	1	yes	characteristic
228	C₂²×C₅	#947	1	1	yes	characteristic
229	C₂²×C₅	C₂×#217	1	1	yes	characteristic
230	C₄×C₅	—	2	1	no	—
231	C₄×C₅	—	2	1	no	—
232	C₄×C₅	—	2	1	no	—
233	C₄×C₅	—	4	2	no	—
234	C₄×C₅	—	4	2	no	—
235	C₄×C₅	#837	1	2	yes	—
236	C₄×C₅	Q₈×Dic₃	1	2	yes	—
237	C₄×C₅	#946	1	2	yes	—
238	Dic₅	—	12	4	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	2	1	no	—
241	C₂²×C₄	—	30	2	no	—
242	C₂²×C₄	#1220	1	1	yes	characteristic
243	C₂²×C₄	#1220	1	1	yes	characteristic
244	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
245	M₁₆	—	15	4	no	—
246	M₁₆	—	30	2	no	—
247	C₂×C₈	—	30	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	30	4	no	—
253	C₄⋊C₄	C₂×Dic₁₅	1	2	yes	—
254	C₄⋊C₄	C₂×Dic₁₅	1	2	yes	—
255	C₂²⋊C₄	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	30	4	no	—
258	C₄²	C₂×Dic₁₅	1	1	yes	characteristic
259	C₄²	C₂×Dic₁₅	1	1	yes	characteristic
260	C₄²	#1220	1	1	yes	characteristic
261	C₄²	#1220	1	1	yes	characteristic
262	C₃×C₅	#1986	1	1	yes	characteristic, complemented, Hall
263	C₂²×C₃	—	2	1	no	—
264	C₂²×C₃	—	2	1	no	—
265	C₂²×C₃	#4090	1	1	yes	characteristic
266	C₂²×C₃	#4093	1	1	yes	characteristic
267	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	4	2	no	—
272	C₃×C₄	—	4	2	no	—
273	C₃×C₄	#3969	1	2	yes	—
274	C₃×C₄	Q₈×Dic₅	1	2	yes	—
275	C₃×C₄	#4092	1	2	yes	—
276	Dic₃	—	20	4	no	—
277	C₂×C₅	—	2	1	no	—
278	C₂×C₅	—	2	1	no	—
279	C₂×C₅	#5229	1	1	yes	characteristic
280	C₂×C₅	C₂×#837	1	1	yes	characteristic
281	C₂×C₅	#5236	1	1	yes	characteristic
282	C₂³	C₂×#1220	1	1	yes	characteristic
283	Q₈	—	4	2	no	—
284	Q₈	—	4	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	30	4	no	—
290	C₂×C₄	—	60	2	no	—
291	C₂×C₄	—	60	2	no	—
292	C₂×C₄	#6719	1	1	yes	characteristic
293	C₂×C₄	#6719	1	1	yes	characteristic
294	C₂×C₄	#6719	1	1	yes	characteristic
295	C₂×C₄	#6719	1	1	yes	characteristic
296	C₂×C₄	Q₈×D₁₅	1	2	yes	—
297	C₂×C₄	#6821	1	2	yes	—
298	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
299	C₂×C₄	C₂×#1220	1	1	yes	characteristic
300	C₈	—	30	4	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	#65519	1	1	yes	characteristic
304	C₂×C₃	C₂×#3969	1	1	yes	characteristic
305	C₂×C₃	#65525	1	1	yes	characteristic
306	C₅	#71841	1	1	yes	characteristic, complemented, Sylow
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	#91833	1	1	yes	characteristic
310	C₂²	#91836	1	1	yes	characteristic
311	C₂²	C₂×#6719	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	60	4	no	—
318	C₄	#91121	1	2	yes	—
319	C₄	Q₈×Dic₁₅	1	2	yes	—
320	C₄	#91835	1	2	yes	—
321	C₃	#10602634	1	1	yes	characteristic, complemented, Sylow
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	#11751330	1	1	yes	characteristic
325	C₂	C₂×#91121	1	1	yes	characteristic
326	C₂	#11751337	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1220
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#533	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#533	D₁₅
hypocenter	C₃×C₅	#1986
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

Complements

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

N	H ≅ G/N	type
6. C₃×C₅×C₄⋊Q₈	C₄	semidirect
7. C₃×C₅×C₄⋊Q₈	C₄	semidirect
8. C₃×C₅×#125	C₄	semidirect
47. #6710	Q₈	semidirect
53. C₅×C₄⋊Q₈	Dic₃	semidirect
54. C₅×C₄⋊Q₈	Dic₃	semidirect
55. C₅×#125	Dic₃	semidirect
88. C₃×C₄⋊Q₈	Dic₅	semidirect
89. C₃×C₄⋊Q₈	Dic₅	semidirect
90. C₃×#125	Dic₅	semidirect
187. C₄⋊Q₈	Dic₁₅	semidirect
188. C₄⋊Q₈	Dic₁₅	semidirect
189. #125	Dic₁₅	semidirect
262. C₃×C₅	#1986	semidirect
306. C₅	#71841	semidirect
321. C₃	#10602634	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#533	2
C₄×C₄⋊Dic₁₅	2
C₄×#6710	2
#11746353	2
#10602634	3
#71841	5