Subgroups — #14285898

Order 1920.

300 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11750936	C₂	1	1	yes	maximal, characteristic
2	#11750934	C₂	1	1	yes	maximal, characteristic
3	#11750786	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746301	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746301	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746279	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599366	—	3	1	no	maximal, Hall
9	C₄²×D₁₅	C₂²	1	1	yes	characteristic
10	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#91104	—	2	1	no	—
14	#91104	—	2	1	no	—
15	#91104	C₄	1	1	yes	characteristic
16	#91104	C₄	1	1	yes	characteristic
17	#91099	—	2	1	no	—
18	#91098	C₄	1	1	yes	characteristic
19	#91098	C₄	1	1	yes	characteristic
20	C₈×Dic₁₅	—	2	1	no	—
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
23	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
24	#68573	—	5	1	no	maximal, Hall
25	#65124	—	3	1	no	—
26	#65122	—	3	1	no	—
27	#64974	—	3	1	no	—
28	C₅×#335	D₃	1	1	yes	characteristic, complemented
29	#64700	—	3	1	no	—
30	#64700	—	3	1	no	—
31	#64678	—	3	1	no	—
32	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×D₁₅	D₄	1	1	yes	characteristic
34	C₂×C₄×D₁₅	Q₈	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
41	C₄×Dic₁₅	D₄	1	1	yes	characteristic
42	C₄×Dic₁₅	Q₈	1	1	yes	characteristic
43	C₂×C₃⋊#168	—	2	1	no	—
44	C₂×C₃⋊#168	—	2	1	no	—
45	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
47	#4835	—	5	1	no	—
48	#4833	—	5	1	no	—
49	#4685	—	5	1	no	—
50	C₃×#335	D₅	1	1	yes	characteristic, complemented
51	#4480	—	5	1	no	—
52	#4480	—	5	1	no	—
53	#4458	—	5	1	no	—
54	C₄²×D₅	—	3	1	no	—
55	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
56	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
57	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
58	#3952	—	3	1	no	—
59	#3952	—	3	1	no	—
60	#3952	—	6	1	no	—
61	#3952	—	6	1	no	—
62	#3947	—	6	1	no	—
63	#3946	—	3	1	no	—
64	#3946	—	3	1	no	—
65	C₈×Dic₅	—	6	1	no	—
66	C₄⋊#168	—	3	1	no	—
67	C₄⋊#168	—	3	1	no	—
68	C₄×#168	—	3	1	no	—
69	#1572	—	15	1	no	Sylow
70	C₂²×D₁₅	C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
75	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
76	C₂×Dic₁₅	C₄⋊C₄	1	1	yes	characteristic
77	C₄×D₁₅	—	2	1	no	—
78	C₄×D₁₅	—	2	1	no	—
79	C₄×D₁₅	—	2	1	no	—
80	C₄×D₁₅	—	2	1	no	—
81	C₄×D₁₅	—	4	1	no	—
82	C₃×C₅×C₈	—	2	1	no	—
83	C₃×C₅×C₈	—	2	1	no	—
84	C₃×C₅×C₈	—	4	1	no	—
85	C₃×C₅×C₈	—	4	1	no	—
86	C₃⋊#168	—	2	1	no	—
87	C₃⋊#168	—	2	1	no	—
88	C₃⋊#168	—	4	1	no	—
89	C₃⋊#168	—	4	1	no	—
90	C₄²×D₃	—	5	1	no	—
91	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
92	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
93	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
94	(C₂×D₃)⋊C₈	—	5	1	no	—
95	(C₂×D₃)⋊C₈	—	5	1	no	—
96	(C₂×D₃)⋊C₈	—	10	1	no	—
97	(C₂×D₃)⋊C₈	—	10	1	no	—
98	#815	—	10	1	no	—
99	Dic₃⋊C₈	—	5	1	no	—
100	Dic₃⋊C₈	—	5	1	no	—
101	C₈×Dic₃	—	10	1	no	—
102	C₄⋊(C₃⋊C₈)	—	5	1	no	—
103	C₄⋊(C₃⋊C₈)	—	5	1	no	—
104	C₄×C₃⋊C₈	—	5	1	no	—
105	C₂×C₄×D₅	—	3	1	no	—
106	C₂×C₄×D₅	—	3	1	no	—
107	C₂×C₄×D₅	—	3	1	no	—
108	C₂×C₅×C₈	—	2	1	no	—
109	C₂×C₅×C₈	—	2	1	no	—
110	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
111	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
112	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
113	C₄×Dic₅	—	3	1	no	—
114	C₄×Dic₅	—	3	1	no	—
115	C₄×Dic₅	—	3	1	no	—
116	C₂×#168	—	3	1	no	—
117	C₂×#168	—	3	1	no	—
118	C₂×#168	—	6	1	no	—
119	C₂×#168	—	6	1	no	—
120	#432	—	15	1	no	—
121	#431	—	15	1	no	—
122	#431	—	15	1	no	—
123	#335	—	15	1	no	—
124	#335	D₁₅	1	1	yes	characteristic, complemented
125	#334	—	15	1	no	—
126	#334	—	15	1	no	—
127	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×D₁₅	—	2	1	no	—
129	C₂×D₁₅	—	2	1	no	—
130	C₂×D₁₅	—	2	1	no	—
131	C₃×C₄×C₅	—	2	1	no	—
132	C₃×C₄×C₅	#131	1	1	yes	characteristic
133	C₃×C₄×C₅	#131	1	1	yes	characteristic
134	C₃×C₄×C₅	#135	1	1	yes	characteristic
135	C₃×C₄×C₅	#135	1	1	yes	characteristic
136	Dic₁₅	—	2	1	no	—
137	Dic₁₅	—	2	1	no	—
138	Dic₁₅	—	2	1	no	—
139	C₂×C₄×D₃	—	5	1	no	—
140	C₂×C₄×D₃	—	5	1	no	—
141	C₂×C₄×D₃	—	5	1	no	—
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
145	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
146	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
147	C₄×Dic₃	—	5	1	no	—
148	C₄×Dic₃	—	5	1	no	—
149	C₄×Dic₃	—	5	1	no	—
150	C₂×C₃⋊C₈	—	5	1	no	—
151	C₂×C₃⋊C₈	—	5	1	no	—
152	C₂×C₃⋊C₈	—	10	1	no	—
153	C₂×C₃⋊C₈	—	10	1	no	—
154	C₂²×D₅	—	3	1	no	—
155	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
157	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
158	C₂×Dic₅	—	3	1	no	—
159	C₂×Dic₅	—	3	1	no	—
160	C₂×Dic₅	—	3	1	no	—
161	C₄×D₅	—	6	1	no	—
162	C₄×D₅	—	6	1	no	—
163	C₄×D₅	—	6	1	no	—
164	C₄×D₅	—	6	1	no	—
165	C₄×D₅	—	12	1	no	—
166	C₅×C₈	—	2	1	no	—
167	C₅×C₈	—	2	1	no	—
168	C₅×C₈	—	4	1	no	—
169	C₅×C₈	—	4	1	no	—
170	#168	—	6	1	no	—
171	#168	—	6	1	no	—
172	#168	—	12	1	no	—
173	#168	—	12	1	no	—
174	C₂×C₄²	—	15	1	no	—
175	C₄⋊C₈	—	15	1	no	—
176	C₄⋊C₈	—	15	1	no	—
177	C₄⋊C₈	—	15	1	no	—
178	C₄⋊C₈	—	15	1	no	—
179	C₄⋊C₈	C₂×D₁₅	1	1	yes	characteristic
180	C₄⋊C₈	C₂×D₁₅	1	1	yes	characteristic
181	C₂²⋊C₈	—	15	1	no	—
182	C₂²⋊C₈	—	15	1	no	—
183	C₂²⋊C₈	—	30	1	no	—
184	C₂²⋊C₈	—	30	1	no	—
185	#97	—	30	1	no	—
186	C₄×C₈	—	15	1	no	—
187	C₄×C₈	—	30	1	no	—
188	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
189	C₂×C₃×C₅	#424	1	1	yes	characteristic
190	C₂×C₃×C₅	#427	1	1	yes	characteristic
191	C₂×C₃×C₅	#430	1	1	yes	characteristic
192	D₁₅	—	4	1	no	—
193	C₂²×D₃	—	5	1	no	—
194	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
195	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
196	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
197	C₂×Dic₃	—	5	1	no	—
198	C₂×Dic₃	—	5	1	no	—
199	C₂×Dic₃	—	5	1	no	—
200	C₄×D₃	—	10	1	no	—
201	C₄×D₃	—	10	1	no	—
202	C₄×D₃	—	10	1	no	—
203	C₄×D₃	—	10	1	no	—
204	C₄×D₃	—	20	1	no	—
205	C₃×C₈	—	2	1	no	—
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	—	4	1	no	—
208	C₃×C₈	—	4	1	no	—
209	C₃⋊C₈	—	10	1	no	—
210	C₃⋊C₈	—	10	1	no	—
211	C₃⋊C₈	—	20	1	no	—
212	C₃⋊C₈	—	20	1	no	—
213	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
214	C₂×D₅	—	6	1	no	—
215	C₂×D₅	—	6	1	no	—
216	C₂×D₅	—	6	1	no	—
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	#901	1	1	yes	characteristic
219	C₄×C₅	#907	1	1	yes	characteristic
220	C₄×C₅	#912	1	1	yes	characteristic
221	C₄×C₅	#919	1	1	yes	characteristic
222	Dic₅	—	6	1	no	—
223	Dic₅	—	6	1	no	—
224	Dic₅	—	6	1	no	—
225	C₂²×C₄	—	15	1	no	—
226	C₂²×C₄	—	15	1	no	—
227	C₂²×C₄	—	15	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	15	1	no	—
231	C₂×C₈	—	15	1	no	—
232	C₂×C₈	—	30	1	no	—
233	C₂×C₈	—	30	1	no	—
234	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
235	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
236	C₄²	—	15	1	no	—
237	C₄²	—	15	1	no	—
238	C₄²	—	15	1	no	—
239	C₄²	C₂²×D₁₅	1	1	yes	characteristic
240	C₃×C₅	#1572	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
242	C₂×D₃	—	10	1	no	—
243	C₂×D₃	—	10	1	no	—
244	C₂×D₃	—	10	1	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	#4047	1	1	yes	characteristic
247	C₃×C₄	#4053	1	1	yes	characteristic
248	C₃×C₄	#4058	1	1	yes	characteristic
249	C₃×C₄	#4065	1	1	yes	characteristic
250	Dic₃	—	10	1	no	—
251	Dic₃	—	10	1	no	—
252	Dic₃	—	10	1	no	—
253	C₂×C₅	#4839	1	1	yes	characteristic
254	C₂×C₅	#4880	1	1	yes	characteristic
255	C₂×C₅	#4893	1	1	yes	characteristic
256	D₅	—	12	1	no	—
257	C₂³	—	15	1	no	—
258	C₂×C₄	—	15	1	no	—
259	C₂×C₄	—	15	1	no	—
260	C₂×C₄	—	15	1	no	—
261	C₂×C₄	—	30	1	no	—
262	C₂×C₄	—	30	1	no	—
263	C₂×C₄	—	30	1	no	—
264	C₂×C₄	—	30	1	no	—
265	C₂×C₄	—	60	1	no	—
266	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
267	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
268	C₂×C₄	Q₈×D₁₅	1	1	yes	characteristic
269	C₈	—	2	1	no	—
270	C₈	—	2	1	no	—
271	C₈	—	4	1	no	—
272	C₈	—	4	1	no	—
273	C₈	—	30	1	no	—
274	C₈	—	30	1	no	—
275	C₈	—	60	1	no	—
276	C₈	—	60	1	no	—
277	C₂×C₃	#65128	1	1	yes	characteristic
278	C₂×C₃	#65169	1	1	yes	characteristic
279	C₂×C₃	#65182	1	1	yes	characteristic
280	D₃	—	20	1	no	—
281	C₅	#68573	1	1	yes	characteristic, complemented, Sylow
282	C₂²	—	30	1	no	—
283	C₂²	—	30	1	no	—
284	C₂²	—	30	1	no	—
285	C₂²	C₄⋊C₄×D₁₅	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	30	1	no	—
288	C₄	—	30	1	no	—
289	C₄	—	30	1	no	—
290	C₄	#91790	1	1	yes	characteristic
291	C₄	#91796	1	1	yes	characteristic
292	C₄	#91801	1	1	yes	characteristic
293	C₄	#91808	1	1	yes	characteristic
294	C₃	#10599366	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	60	1	no	—
296	C₂	#11750940	1	1	yes	characteristic
297	C₂	#11750981	1	1	yes	characteristic
298	C₂	#11750994	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×#335	C₂
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×D₁₅	C₄⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	#335	D₁₅
hypocenter	C₃×C₅	#1572
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#335	C₂	semidirect
5. #11746301	C₂	semidirect
6. #11746301	C₂	semidirect
7. #11746279	C₂	semidirect
28. C₅×#335	D₃	semidirect
50. C₃×#335	D₅	semidirect
124. #335	D₁₅	semidirect
240. C₃×C₅	#1572	semidirect
281. C₅	#68573	semidirect
294. C₃	#10599366	semidirect

Maximal subgroups

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

subgroup	index
#11750936	2
#11750934	2
#11750786	2
C₃×C₅×#335	2
#11746301	2
#11746279	2
#10599366	3
#68573	5