Subgroups — #14282923

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 320 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	#11751026	C₂	1	1	yes	maximal, characteristic
4	#11746359	C₂	1	4	yes	maximal
5	#10602628	—	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	—	2	1	no	—
9	C₃×C₅×C₂²⋊Q₈	—	2	1	no	—
10	C₃×C₅×C₂²⋊Q₈	C₂²	1	2	yes	—
11	C₃×C₄×C₅×Q₈	—	2	1	no	—
12	C₃×C₄×C₅×Q₈	C₄	1	2	yes	complemented
13	C₃×C₅×#117	—	2	1	no	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₁₅	—	2	1	no	—
16	C₂×C₄×Dic₁₅	—	2	1	no	—
17	#91115	C₂²	1	2	yes	—
18	#91114	C₂²	1	2	yes	—
19	C₄⋊(C₃⋊#168)	—	2	1	no	—
20	C₄×C₃⋊#168	—	2	1	no	—
21	#71835	—	5	1	no	maximal, Hall
22	C₅×#533	D₃	1	1	yes	characteristic
23	C₄×C₄⋊Dic₅	—	3	1	no	—
24	#65214	—	3	1	no	—
25	#64758	—	3	4	no	—
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
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	1	no	—
33	C₃×C₅×C₄⋊C₄	—	2	1	no	—
34	C₃×C₅×C₄⋊C₄	—	2	1	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₅×C₄⋊C₄	—	2	2	no	—
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	—	4	2	no	—
46	C₂×C₃⋊#168	—	2	2	no	—
47	C₃×#533	D₅	1	1	yes	characteristic
48	C₄×C₄⋊Dic₃	—	5	1	no	—
49	#4925	—	5	1	no	—
50	#4538	—	5	4	no	—
51	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic, complemented
52	C₅×C₄⋊Q₈	Dic₃	1	1	yes	characteristic, complemented
53	C₅×#125	—	2	1	no	—
54	C₅×C₂²⋊Q₈	—	2	1	no	—
55	C₅×C₂²⋊Q₈	C₂×D₃	1	2	yes	—
56	C₄×C₅×Q₈	—	2	1	no	—
57	C₄×C₅×Q₈	Dic₃	1	2	yes	complemented
58	C₅×#117	—	2	1	no	—
59	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
60	C₂×C₄⋊Dic₅	—	6	1	no	—
61	C₂×C₄×Dic₅	—	6	1	no	—
62	#3963	—	3	2	no	—
63	C₂²⋊#168	—	3	2	no	—
64	C₄⋊#168	—	6	1	no	—
65	C₄×#168	—	6	1	no	—
66	#1503	—	15	1	no	Sylow
67	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
68	C₃×C₅×Q₈	—	4	2	no	—
69	C₃×C₅×Q₈	—	4	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	4	1	no	—
74	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
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	SD₁₆	1	2	yes	—
79	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×Dic₁₅	—	2	4	no	—
83	C₂×Dic₁₅	—	4	2	no	—
84	C₂×Dic₁₅	—	4	2	no	—
85	C₃⋊#168	—	4	2	no	—
86	C₃×C₄⋊Q₈	Dic₅	1	1	yes	characteristic, complemented
87	C₃×C₄⋊Q₈	Dic₅	1	1	yes	characteristic, complemented
88	C₃×#125	—	2	1	no	—
89	C₃×C₂²⋊Q₈	—	2	1	no	—
90	C₃×C₂²⋊Q₈	C₂×D₅	1	2	yes	—
91	C₃×C₄×Q₈	—	2	1	no	—
92	C₃×C₄×Q₈	Dic₅	1	2	yes	complemented
93	C₃×#117	—	2	1	no	—
94	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
95	C₂×C₄⋊Dic₃	—	10	1	no	—
96	C₂×C₄×Dic₃	—	10	1	no	—
97	C₃⋊#95	—	5	2	no	—
98	#830	—	5	2	no	—
99	C₄⋊(C₃⋊C₈)	—	10	1	no	—
100	C₄×C₃⋊C₈	—	10	1	no	—
101	C₂×C₅×Q₈	—	2	1	no	—
102	C₂×C₅×Q₈	C₂×Dic₃	1	2	yes	—
103	C₂²×C₄×C₅	#67	1	1	yes	characteristic
104	C₂²×C₄×C₅	#67	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
106	C₂²×Dic₅	—	6	2	no	—
107	C₅×C₄⋊C₄	—	2	1	no	—
108	C₅×C₄⋊C₄	—	2	1	no	—
109	C₅×C₄⋊C₄	—	2	1	no	—
110	C₅×C₄⋊C₄	—	2	2	no	—
111	C₅×C₄⋊C₄	—	2	2	no	—
112	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
113	C₅×C₂²⋊C₄	—	2	2	no	—
114	C₄²×C₅	—	2	2	no	—
115	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
116	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
117	C₄²×C₅	#67	1	1	yes	characteristic
118	C₄²×C₅	#67	1	1	yes	characteristic
119	C₄⋊Dic₅	—	6	4	no	—
120	C₄×Dic₅	—	12	2	no	—
121	C₂×#168	—	6	2	no	—
122	#533	D₁₅	1	1	yes	characteristic
123	#431	—	15	1	no	—
124	C₄×C₄⋊C₄	—	15	1	no	—
125	#328	—	15	4	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
130	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
131	C₃×C₄×C₅	—	2	1	no	—
132	C₃×C₄×C₅	—	2	1	no	—
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	—	4	2	no	—
135	C₃×C₄×C₅	—	4	2	no	—
136	C₃×C₄×C₅	#103	1	2	yes	—
137	C₃×C₄×C₅	#103	1	2	yes	—
138	C₃×C₄×C₅	#104	1	2	yes	—
139	Dic₁₅	—	4	4	no	—
140	C₂×C₃×Q₈	—	2	1	no	—
141	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
142	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
143	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
145	C₂²×Dic₃	—	10	2	no	—
146	C₃×C₄⋊C₄	—	2	1	no	—
147	C₃×C₄⋊C₄	—	2	1	no	—
148	C₃×C₄⋊C₄	—	2	1	no	—
149	C₃×C₄⋊C₄	—	2	2	no	—
150	C₃×C₄⋊C₄	—	2	2	no	—
151	C₃×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
152	C₃×C₂²⋊C₄	—	2	2	no	—
153	C₃×C₄²	—	2	2	no	—
154	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
155	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
156	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
157	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
158	C₄⋊Dic₃	—	10	4	no	—
159	C₄×Dic₃	—	20	2	no	—
160	C₂×C₃⋊C₈	—	10	2	no	—
161	C₂³×C₅	C₂×#67	1	1	yes	characteristic
162	C₅×Q₈	—	4	2	no	—
163	C₅×Q₈	—	4	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	4	1	no	—
168	C₂×C₄×C₅	#215	1	2	yes	—
169	C₂×C₄×C₅	#216	1	2	yes	—
170	C₂×C₄×C₅	#217	1	1	yes	characteristic
171	C₂×C₄×C₅	#217	1	1	yes	characteristic
172	C₂×C₄×C₅	#217	1	1	yes	characteristic
173	C₂×C₄×C₅	#217	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
176	C₂×Dic₅	—	6	4	no	—
177	C₂×Dic₅	—	12	2	no	—
178	C₂×Dic₅	—	12	2	no	—
179	#168	—	12	2	no	—
180	C₄⋊Q₈	Dic₁₅	1	1	yes	characteristic, complemented
181	C₄⋊Q₈	Dic₁₅	1	1	yes	characteristic, complemented
182	#125	—	2	1	no	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₂²⋊Q₈	C₂×D₁₅	1	2	yes	—
185	C₄×Q₈	—	2	1	no	—
186	C₄×Q₈	Dic₁₅	1	2	yes	complemented
187	#117	—	2	1	no	—
188	C₂×C₄⋊C₄	—	30	1	no	—
189	C₂×C₄²	—	30	1	no	—
190	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
191	C₄⋊C₈	—	30	1	no	—
192	C₂²⋊C₈	—	15	2	no	—
193	C₄×C₈	—	30	1	no	—
194	#95	—	15	2	no	—
195	C₂×C₃×C₅	—	2	1	no	—
196	C₂×C₃×C₅	—	2	1	no	—
197	C₂×C₃×C₅	#410	1	1	yes	characteristic
198	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
199	C₂×C₃×C₅	C₂×#104	1	1	yes	characteristic
200	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
201	C₃×Q₈	—	4	2	no	—
202	C₃×Q₈	—	4	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	#688	1	2	yes	—
208	C₂×C₃×C₄	#689	1	2	yes	—
209	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
215	C₂×Dic₃	—	10	4	no	—
216	C₂×Dic₃	—	20	2	no	—
217	C₂×Dic₃	—	20	2	no	—
218	C₃⋊C₈	—	20	2	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	—	2	1	no	—
221	C₂²×C₅	C₂×#215	1	1	yes	characteristic
222	C₂²×C₅	C₂×#216	1	1	yes	characteristic
223	C₂²×C₅	C₂×#217	1	1	yes	characteristic
224	C₄×C₅	—	2	1	no	—
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	4	2	no	—
228	C₄×C₅	—	4	2	no	—
229	C₄×C₅	#835	1	2	yes	—
230	C₄×C₅	#835	1	2	yes	—
231	C₄×C₅	#837	1	2	yes	—
232	Dic₅	—	12	4	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	C₂×Dic₁₅	1	2	yes	—
235	C₂²×C₄	—	30	2	no	—
236	C₂²×C₄	#1220	1	1	yes	characteristic
237	C₂²×C₄	#1220	1	1	yes	characteristic
238	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
239	C₂×C₈	—	30	2	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	30	4	no	—
246	C₄⋊C₄	C₂×Dic₁₅	1	2	yes	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	60	2	no	—
250	C₄²	C₂×Dic₁₅	1	1	yes	characteristic
251	C₄²	C₂×Dic₁₅	1	1	yes	characteristic
252	C₄²	#1220	1	1	yes	characteristic
253	C₄²	#1220	1	1	yes	characteristic
254	C₃×C₅	#1503	1	1	yes	characteristic, complemented, Hall
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	C₂×#688	1	1	yes	characteristic
258	C₂²×C₃	C₂×#689	1	1	yes	characteristic
259	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
260	C₃×C₄	—	2	1	no	—
261	C₃×C₄	—	2	1	no	—
262	C₃×C₄	—	2	1	no	—
263	C₃×C₄	—	4	2	no	—
264	C₃×C₄	—	4	2	no	—
265	C₃×C₄	#3967	1	2	yes	—
266	C₃×C₄	#3967	1	2	yes	—
267	C₃×C₄	#3969	1	2	yes	—
268	Dic₃	—	20	4	no	—
269	C₂×C₅	—	2	1	no	—
270	C₂×C₅	—	2	1	no	—
271	C₂×C₅	C₂×#835	1	1	yes	characteristic
272	C₂×C₅	C₂×#837	1	1	yes	characteristic
273	C₂×C₅	#5240	1	1	yes	characteristic
274	C₂³	C₂×#1220	1	1	yes	characteristic
275	Q₈	—	4	2	no	—
276	Q₈	—	4	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	30	4	no	—
282	C₂×C₄	—	60	2	no	—
283	C₂×C₄	—	60	2	no	—
284	C₂×C₄	#6717	1	2	yes	—
285	C₂×C₄	#6718	1	2	yes	—
286	C₂×C₄	#6719	1	1	yes	characteristic
287	C₂×C₄	#6719	1	1	yes	characteristic
288	C₂×C₄	#6719	1	1	yes	characteristic
289	C₂×C₄	#6719	1	1	yes	characteristic
290	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
291	C₂×C₄	C₂×#1220	1	1	yes	characteristic
292	C₈	—	60	2	no	—
293	C₂×C₃	—	2	1	no	—
294	C₂×C₃	—	2	1	no	—
295	C₂×C₃	C₂×#3967	1	1	yes	characteristic
296	C₂×C₃	C₂×#3969	1	1	yes	characteristic
297	C₂×C₃	#65529	1	1	yes	characteristic
298	C₅	#71835	1	1	yes	characteristic, complemented, Sylow
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	C₂×#6717	1	1	yes	characteristic
302	C₂²	C₂×#6718	1	1	yes	characteristic
303	C₂²	C₂×#6719	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	60	4	no	—
310	C₄	#91119	1	2	yes	—
311	C₄	#91119	1	2	yes	—
312	C₄	#91121	1	2	yes	—
313	C₃	#10602628	1	1	yes	characteristic, complemented, Sylow
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	C₂×#91119	1	1	yes	characteristic
317	C₂	C₂×#91121	1	1	yes	characteristic
318	C₂	#11751341	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6719
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	#1503
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×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
12. C₃×C₄×C₅×Q₈	C₄	semidirect
51. C₅×C₄⋊Q₈	Dic₃	semidirect
52. C₅×C₄⋊Q₈	Dic₃	semidirect
57. C₄×C₅×Q₈	Dic₃	semidirect
86. C₃×C₄⋊Q₈	Dic₅	semidirect
87. C₃×C₄⋊Q₈	Dic₅	semidirect
92. C₃×C₄×Q₈	Dic₅	semidirect
180. C₄⋊Q₈	Dic₁₅	semidirect
181. C₄⋊Q₈	Dic₁₅	semidirect
186. C₄×Q₈	Dic₁₅	semidirect
254. C₃×C₅	#1503	semidirect
298. C₅	#71835	semidirect
313. C₃	#10602628	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
#11751026	2
#11746359	2
#10602628	3
#71835	5