Subgroups — #14295085

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂×#91114	C₂	1	2	yes	maximal
3	C₂×#91087	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746345	C₂	1	2	yes	maximal, complemented
6	#10602375	—	3	1	no	maximal, Hall
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₃×C₅×#117	—	2	2	no	—
9	C₃×C₅×#117	—	2	2	no	—
10	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
11	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊#168	C₂²	1	2	yes	—
15	C₂²×C₃⋊#168	C₂²	1	2	yes	—
16	#91114	C₄	1	8	yes	complemented
17	#91087	—	2	2	no	—
18	C₄×C₃⋊#168	—	2	2	no	—
19	#71582	—	5	1	no	maximal, Hall
20	C₂×C₅×#117	D₃	1	1	yes	characteristic
21	C₂×C₂²⋊#168	—	3	2	no	—
22	C₂×#3935	—	3	1	no	—
23	C₂×C₄×#168	—	3	1	no	—
24	#64744	—	3	2	no	—
25	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	—	2	1	no	—
27	C₂²×C₃×C₄×C₅	—	2	1	no	—
28	C₂²×C₃×C₄×C₅	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
34	C₃×C₅×C₄⋊C₄	—	2	4	no	—
35	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
36	C₃×C₄²×C₅	—	2	1	no	—
37	C₃×C₄²×C₅	—	2	1	no	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₃×C₄²×C₅	D₄	1	2	yes	—
40	C₂×C₃⋊#168	—	2	2	no	—
41	C₂×C₃⋊#168	—	2	2	no	—
42	C₂×C₃⋊#168	—	2	2	no	—
43	C₂×C₃⋊#168	—	2	2	no	—
44	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
45	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
46	C₂×C₃×#117	D₅	1	1	yes	characteristic
47	C₂×#830	—	5	2	no	—
48	C₂×#803	—	5	1	no	—
49	C₂×C₄×C₃⋊C₈	—	5	1	no	—
50	#4524	—	5	2	no	—
51	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
52	C₅×#117	—	2	2	no	—
53	C₅×#117	—	2	2	no	—
54	C₂×C₅×C₄⋊C₄	Dic₃	1	2	yes	—
55	C₂×C₅×C₂²⋊C₄	Dic₃	1	2	yes	—
56	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
57	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
58	C₂²×#168	—	3	2	no	—
59	C₂²×#168	—	3	2	no	—
60	C₂²⋊#168	—	3	8	no	—
61	#3935	—	6	2	no	—
62	C₄×#168	—	6	2	no	—
63	#1750	—	15	1	no	Sylow
64	C₂³×C₃×C₅	—	2	1	no	—
65	C₂³×C₃×C₅	—	2	1	no	—
66	C₂³×C₃×C₅	—	2	4	no	—
67	C₂³×C₃×C₅	C₄²	1	2	yes	—
68	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	—	2	1	no	—
70	C₂×C₃×C₄×C₅	—	2	1	no	—
71	C₂×C₃×C₄×C₅	—	2	4	no	—
72	C₂×C₃×C₄×C₅	—	2	4	no	—
73	C₂×C₃×C₄×C₅	—	2	4	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	4	no	—
76	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	#41	1	2	yes	—
82	C₃⋊#168	—	2	4	no	—
83	C₃⋊#168	—	2	4	no	—
84	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
85	C₃×#117	—	2	2	no	—
86	C₃×#117	—	2	2	no	—
87	C₂×C₃×C₄⋊C₄	Dic₅	1	2	yes	—
88	C₂×C₃×C₂²⋊C₄	Dic₅	1	2	yes	—
89	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
90	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
91	C₂²×C₃⋊C₈	—	5	2	no	—
92	C₂²×C₃⋊C₈	—	5	2	no	—
93	#830	—	5	8	no	—
94	#803	—	10	2	no	—
95	C₄×C₃⋊C₈	—	10	2	no	—
96	C₂⁴×C₅	C₂×Dic₃	1	1	yes	characteristic
97	C₂²×C₄×C₅	—	2	2	no	—
98	C₂²×C₄×C₅	—	2	1	no	—
99	C₂²×C₄×C₅	—	2	1	no	—
100	C₂²×C₄×C₅	C₄×D₃	1	4	yes	—
101	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
102	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
103	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
104	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
105	C₅×C₄⋊C₄	—	2	4	no	—
106	C₅×C₂²⋊C₄	—	2	4	no	—
107	C₄²×C₅	—	2	1	no	—
108	C₄²×C₅	—	2	1	no	—
109	C₄²×C₅	#67	1	2	yes	—
110	C₄²×C₅	#67	1	2	yes	—
111	C₂×#168	—	3	4	no	—
112	C₂×#168	—	3	4	no	—
113	C₂×#168	—	6	2	no	—
114	C₂×#168	—	6	2	no	—
115	C₂×#168	—	6	2	no	—
116	C₂×#168	—	6	2	no	—
117	C₂×#117	D₁₅	1	1	yes	characteristic
118	C₂×C₂²⋊C₈	—	15	2	no	—
119	C₂×#97	—	15	1	no	—
120	C₂×C₄×C₈	—	15	1	no	—
121	#336	—	15	2	no	—
122	C₂²×C₃×C₅	—	2	2	no	—
123	C₂²×C₃×C₅	—	2	4	no	—
124	C₂²×C₃×C₅	—	4	2	no	—
125	C₂²×C₃×C₅	—	4	2	no	—
126	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
127	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
128	C₂²×C₃×C₅	#117	1	1	yes	characteristic
129	C₂²×C₃×C₅	#131	1	4	yes	—
130	C₃×C₄×C₅	—	2	2	no	—
131	C₃×C₄×C₅	—	2	4	no	—
132	C₃×C₄×C₅	—	4	2	no	—
133	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
134	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
135	C₂⁴×C₃	C₂×Dic₅	1	1	yes	characteristic
136	C₂²×C₃×C₄	—	2	2	no	—
137	C₂²×C₃×C₄	—	2	1	no	—
138	C₂²×C₃×C₄	—	2	1	no	—
139	C₂²×C₃×C₄	C₄×D₅	1	4	yes	—
140	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
141	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
142	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
143	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	2	4	no	—
145	C₃×C₂²⋊C₄	—	2	4	no	—
146	C₃×C₄²	—	2	1	no	—
147	C₃×C₄²	—	2	1	no	—
148	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
149	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
150	C₂×C₃⋊C₈	—	5	4	no	—
151	C₂×C₃⋊C₈	—	5	4	no	—
152	C₂×C₃⋊C₈	—	10	2	no	—
153	C₂×C₃⋊C₈	—	10	2	no	—
154	C₂×C₃⋊C₈	—	10	2	no	—
155	C₂×C₃⋊C₈	—	10	2	no	—
156	C₂³×C₅	—	2	1	no	—
157	C₂³×C₅	—	2	4	no	—
158	C₂³×C₅	—	2	1	no	—
159	C₂³×C₅	C₄×Dic₃	1	2	yes	—
160	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	—	2	4	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	2	4	no	—
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
169	C₂×C₄×C₅	#217	1	4	yes	—
170	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
171	C₂×C₄×C₅	#235	1	2	yes	—
172	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
174	#168	—	6	4	no	—
175	#168	—	6	4	no	—
176	C₂³×C₄	C₂×D₁₅	1	1	yes	characteristic
177	C₂²×C₈	—	15	2	no	—
178	C₂²×C₈	—	15	2	no	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	C₂×C₄⋊C₄	Dic₁₅	1	2	yes	—
182	C₂×C₂²⋊C₄	Dic₁₅	1	2	yes	—
183	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
184	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
185	C₂²⋊C₈	—	15	8	no	—
186	#97	—	30	2	no	—
187	C₄×C₈	—	30	2	no	—
188	C₂×C₃×C₅	—	4	2	no	—
189	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
190	C₂×C₃×C₅	#405	1	2	yes	—
191	C₂×C₃×C₅	#408	1	2	yes	—
192	C₂×C₃×C₅	#433	1	2	yes	—
193	C₂³×C₃	—	2	1	no	—
194	C₂³×C₃	—	2	1	no	—
195	C₂³×C₃	—	2	4	no	—
196	C₂³×C₃	C₄×Dic₅	1	2	yes	—
197	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	4	no	—
205	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
206	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
207	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
208	C₂×C₃×C₄	#709	1	2	yes	—
209	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
211	C₃⋊C₈	—	10	4	no	—
212	C₃⋊C₈	—	10	4	no	—
213	C₂²×C₅	—	2	2	no	—
214	C₂²×C₅	—	2	4	no	—
215	C₂²×C₅	—	4	2	no	—
216	C₂²×C₅	—	4	2	no	—
217	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
218	C₂²×C₅	#926	1	1	yes	characteristic
219	C₂²×C₅	#948	1	4	yes	—
220	C₂²×C₅	C₂×#217	1	1	yes	characteristic
221	C₄×C₅	—	2	2	no	—
222	C₄×C₅	—	2	4	no	—
223	C₄×C₅	—	4	2	no	—
224	C₄×C₅	C₄×#67	1	2	yes	—
225	C₄×C₅	C₄×#67	1	2	yes	—
226	C₂⁴	C₂×Dic₁₅	1	1	yes	characteristic
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	C₄×D₁₅	1	4	yes	—
231	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
232	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
233	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
234	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
235	C₂×C₈	—	15	4	no	—
236	C₂×C₈	—	15	4	no	—
237	C₂×C₈	—	30	2	no	—
238	C₂×C₈	—	30	2	no	—
239	C₂×C₈	—	30	2	no	—
240	C₂×C₈	—	30	2	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	#1220	1	2	yes	—
246	C₄²	#1220	1	2	yes	—
247	C₃×C₅	#1750	1	1	yes	characteristic, complemented, Hall
248	C₂²×C₃	—	2	4	no	—
249	C₂²×C₃	—	2	2	no	—
250	C₂²×C₃	—	4	2	no	—
251	C₂²×C₃	—	4	2	no	—
252	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
253	C₂²×C₃	#4072	1	1	yes	characteristic
254	C₂²×C₃	#4094	1	4	yes	—
255	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
256	C₃×C₄	—	2	2	no	—
257	C₃×C₄	—	2	4	no	—
258	C₃×C₄	—	4	2	no	—
259	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
260	C₃×C₄	C₄×C₂²⋊D₅	1	2	yes	—
261	C₂×C₅	—	4	2	no	—
262	C₂×C₅	#4997	1	2	yes	—
263	C₂×C₅	#5000	1	2	yes	—
264	C₂×C₅	C₄×#217	1	1	yes	characteristic
265	C₂×C₅	#5234	1	2	yes	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	4	no	—
268	C₂³	—	2	1	no	—
269	C₂³	C₄×Dic₁₅	1	2	yes	—
270	C₂³	C₂²×Dic₁₅	1	1	yes	characteristic
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	C₄×Dic₁₅	1	2	yes	—
279	C₂×C₄	#6719	1	4	yes	—
280	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
281	C₂×C₄	#6817	1	2	yes	—
282	C₂×C₄	C₂×#1220	1	1	yes	characteristic
283	C₂×C₄	C₂×#1220	1	1	yes	characteristic
284	C₈	—	30	4	no	—
285	C₈	—	30	4	no	—
286	C₂×C₃	—	4	2	no	—
287	C₂×C₃	#65286	1	2	yes	—
288	C₂×C₃	#65289	1	2	yes	—
289	C₂×C₃	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
290	C₂×C₃	#65523	1	2	yes	—
291	C₅	#71582	1	1	yes	characteristic, complemented, Sylow
292	C₂²	—	2	2	no	—
293	C₂²	—	2	4	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	2	no	—
296	C₂²	C₂×C₄×Dic₁₅	1	1	yes	characteristic
297	C₂²	#91815	1	1	yes	characteristic
298	C₂²	#91837	1	4	yes	—
299	C₂²	C₂×#6719	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	4	no	—
302	C₄	—	4	2	no	—
303	C₄	C₄×#1220	1	2	yes	—
304	C₄	C₄×#1220	1	2	yes	—
305	C₃	#10602375	1	1	yes	characteristic, complemented, Sylow
306	C₂	—	4	2	no	—
307	C₂	#11751098	1	2	yes	—
308	C₂	#11751101	1	2	yes	—
309	C₂	C₄×#6719	1	1	yes	characteristic
310	C₂	#11751335	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅
commutator	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₂×C₃×C₅×#117	C₂
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×#117	D₁₅
hypocenter	C₃×C₅	#1750
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
3. C₂×#91087	C₂	semidirect
4. C₂×C₄×C₃⋊#168	C₂	semidirect
5. #11746345	C₂	semidirect
16. #91114	C₄	semidirect
247. C₃×C₅	#1750	semidirect
291. C₅	#71582	semidirect
305. C₃	#10602375	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×#117	2
C₂×#91114	2
C₂×#91087	2
C₂×C₄×C₃⋊#168	2
#11746345	2
#10602375	3
#71582	5