Subgroups — #14297875

Order 1920.

672 conjugacy classes of subgroups.

The subgroup lattice has 324 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	#11751088	C₂	1	2	yes	maximal
3	C₂×#91092	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#340	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746372	C₂	1	1	yes	maximal, characteristic, complemented
6	#10599661	—	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	4	no	—
14	C₂×C₈×D₁₅	C₄	1	4	yes	complemented
15	#91115	—	2	2	no	—
16	#91092	—	2	4	no	—
17	#68868	—	5	1	no	maximal, Hall
18	C₂²×C₈×D₅	—	3	1	no	—
19	#65276	—	3	2	no	—
20	C₂×#3940	—	3	2	no	—
21	C₅×#340	D₃	1	1	yes	characteristic, complemented
22	#64771	—	3	1	no	—
23	C₂³×D₁₅	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×Dic₁₅	—	2	2	no	—
27	C₂²×Dic₁₅	D₄	1	1	yes	characteristic
28	C₂×C₄×D₁₅	—	2	2	no	—
29	C₂×C₄×D₁₅	—	2	2	no	—
30	C₂×C₄×D₁₅	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₁₅	Q₈	1	2	yes	—
32	C₂×C₃×C₅×C₈	—	2	2	no	—
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	#6714	—	4	4	no	—
37	C₄⋊Dic₁₅	—	2	2	no	—
38	C₄⋊Dic₁₅	—	2	2	no	—
39	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
40	C₂×C₃⋊#168	D₄	1	4	yes	—
41	C₈×D₁₅	—	2	8	no	—
42	C₂²×D₃×C₈	—	5	1	no	—
43	#4987	—	5	2	no	—
44	C₂×#808	—	5	2	no	—
45	C₃×#340	D₅	1	1	yes	characteristic, complemented
46	#4551	—	5	1	no	—
47	C₂²×C₄×D₅	—	3	1	no	—
48	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
49	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
50	C₂×#685	—	6	2	no	—
51	C₂×C₄⋊Dic₅	—	3	2	no	—
52	C₂²×#168	—	3	1	no	—
53	C₂×C₈×D₅	—	3	4	no	—
54	C₂×C₈×D₅	—	6	4	no	—
55	#3963	—	6	2	no	—
56	#3940	—	6	4	no	—
57	#1807	—	15	1	no	Sylow
58	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂²×D₁₅	—	2	2	no	—
60	C₂²×D₁₅	—	2	1	no	—
61	C₂²×D₁₅	—	2	1	no	—
62	C₂²×D₁₅	C₄⋊C₄	1	2	yes	—
63	C₂²×D₁₅	SD₁₆	1	4	yes	—
64	C₂×C₃×C₄×C₅	—	4	2	no	—
65	C₂×C₃×C₄×C₅	—	4	2	no	—
66	C₂×C₃×C₄×C₅	—	4	2	no	—
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	#41	1	2	yes	—
71	C₂×Dic₁₅	—	2	2	no	—
72	C₂×Dic₁₅	—	4	2	no	—
73	C₂×Dic₁₅	—	4	2	no	—
74	C₂×Dic₁₅	—	4	2	no	—
75	C₂×Dic₁₅	C₄⋊C₄	1	2	yes	—
76	C₄×D₁₅	—	2	2	no	—
77	C₄×D₁₅	—	2	2	no	—
78	C₄×D₁₅	—	2	4	no	—
79	C₃×C₅×C₈	—	2	2	no	—
80	C₃⋊#168	C₂²⋊C₄	1	4	yes	complemented
81	C₂²×C₄×D₃	—	5	1	no	—
82	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
83	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
84	C₂×(C₂×D₃)⋊C₄	—	10	2	no	—
85	C₂×C₄⋊Dic₃	—	5	2	no	—
86	C₂²×C₃⋊C₈	—	5	1	no	—
87	C₂×D₃×C₈	—	5	4	no	—
88	C₂×D₃×C₈	—	10	4	no	—
89	C₃⋊#95	—	10	2	no	—
90	#808	—	10	4	no	—
91	C₂³×D₅	—	3	1	no	—
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
94	C₂²×Dic₅	—	3	1	no	—
95	C₂²×Dic₅	—	6	2	no	—
96	C₂×C₄×D₅	—	3	2	no	—
97	C₂×C₄×D₅	—	3	2	no	—
98	C₂×C₄×D₅	—	6	2	no	—
99	C₂×C₄×D₅	—	6	2	no	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	#685	—	12	4	no	—
105	C₄⋊Dic₅	—	6	2	no	—
106	C₄⋊Dic₅	—	6	2	no	—
107	C₂×#168	—	3	2	no	—
108	C₂×#168	—	3	4	no	—
109	C₈×D₅	—	6	8	no	—
110	C₂³×C₈	—	15	1	no	—
111	C₂×#106	—	15	2	no	—
112	#380	—	15	2	no	—
113	#340	—	15	1	no	—
114	#340	D₁₅	1	1	yes	characteristic, complemented
115	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₃×C₅	#121	1	2	yes	—
117	C₂²×C₃×C₅	C₂×SD₁₆	1	2	yes	—
118	C₂²×C₃×C₅	#135	1	1	yes	characteristic
119	C₂²×C₃×C₅	#135	1	1	yes	characteristic
120	C₂×D₁₅	—	2	2	no	—
121	C₂×D₁₅	—	2	2	no	—
122	C₂×D₁₅	—	2	4	no	—
123	C₂×D₁₅	—	2	4	no	—
124	C₂×D₁₅	#106	1	4	yes	complemented
125	C₃×C₄×C₅	—	8	2	no	—
126	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₃×C₄×C₅	#117	1	1	yes	characteristic
128	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
129	Dic₁₅	—	2	2	no	—
130	Dic₁₅	—	8	2	no	—
131	C₂³×D₃	—	5	1	no	—
132	C₂²×C₃×C₄	—	2	2	no	—
133	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
134	C₂²×Dic₃	—	5	1	no	—
135	C₂²×Dic₃	—	10	2	no	—
136	C₂×C₄×D₃	—	5	2	no	—
137	C₂×C₄×D₃	—	5	2	no	—
138	C₂×C₄×D₃	—	10	2	no	—
139	C₂×C₄×D₃	—	10	2	no	—
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
142	C₃×C₄⋊C₄	—	2	2	no	—
143	C₃×C₄⋊C₄	—	2	2	no	—
144	(C₂×D₃)⋊C₄	—	20	4	no	—
145	C₄⋊Dic₃	—	10	2	no	—
146	C₄⋊Dic₃	—	10	2	no	—
147	C₂×C₃⋊C₈	—	5	2	no	—
148	C₂×C₃⋊C₈	—	5	4	no	—
149	D₃×C₈	—	10	8	no	—
150	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
151	C₂²×D₅	—	3	2	no	—
152	C₂²×D₅	—	3	4	no	—
153	C₂²×D₅	—	6	2	no	—
154	C₂²×D₅	—	6	1	no	—
155	C₂²×D₅	—	6	1	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	—	4	2	no	—
159	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
160	C₂×C₄×C₅	#235	1	2	yes	—
161	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
162	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
163	C₂×Dic₅	—	3	2	no	—
164	C₂×Dic₅	—	6	2	no	—
165	C₂×Dic₅	—	12	2	no	—
166	C₂×Dic₅	—	12	2	no	—
167	C₂×Dic₅	—	12	2	no	—
168	C₄×D₅	—	6	2	no	—
169	C₄×D₅	—	6	2	no	—
170	C₄×D₅	—	6	4	no	—
171	C₅×C₈	—	2	2	no	—
172	#168	—	3	4	no	—
173	C₂³×C₄	—	15	1	no	—
174	C₂²×C₈	—	15	1	no	—
175	C₂²×C₈	—	15	4	no	—
176	C₂²×C₈	—	30	4	no	—
177	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
178	C₂×C₄⋊C₄	—	15	2	no	—
179	C₂×C₄⋊C₄	C₂×D₁₅	1	2	yes	—
180	C₂×C₂²⋊C₄	—	30	2	no	—
181	#106	—	30	4	no	—
182	#95	—	30	2	no	—
183	C₂×C₃×C₅	#380	1	1	yes	characteristic
184	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
185	C₂×C₃×C₅	#427	1	1	yes	characteristic
186	C₂×C₃×C₅	#465	1	2	yes	—
187	C₂×C₃×C₅	#465	1	2	yes	—
188	D₁₅	—	2	4	no	—
189	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
190	C₂²×D₃	—	5	2	no	—
191	C₂²×D₃	—	5	4	no	—
192	C₂²×D₃	—	10	2	no	—
193	C₂²×D₃	—	10	1	no	—
194	C₂²×D₃	—	10	1	no	—
195	C₂×C₃×C₄	—	4	2	no	—
196	C₂×C₃×C₄	—	4	2	no	—
197	C₂×C₃×C₄	—	4	2	no	—
198	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
199	C₂×C₃×C₄	#709	1	2	yes	—
200	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
201	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
202	C₂×Dic₃	—	5	2	no	—
203	C₂×Dic₃	—	10	2	no	—
204	C₂×Dic₃	—	20	2	no	—
205	C₂×Dic₃	—	20	2	no	—
206	C₂×Dic₃	—	20	2	no	—
207	C₄×D₃	—	10	2	no	—
208	C₄×D₃	—	10	2	no	—
209	C₄×D₃	—	10	4	no	—
210	C₃×C₈	—	2	2	no	—
211	C₃⋊C₈	—	5	4	no	—
212	C₂²×C₅	#884	1	2	yes	—
213	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
214	C₂²×C₅	#912	1	1	yes	characteristic
215	C₂²×C₅	D₃×SD₁₆	1	2	yes	—
216	C₂²×C₅	#919	1	1	yes	characteristic
217	C₂×D₅	—	3	4	no	—
218	C₂×D₅	—	6	2	no	—
219	C₂×D₅	—	6	2	no	—
220	C₂×D₅	—	6	4	no	—
221	C₂×D₅	—	6	4	no	—
222	C₄×C₅	—	8	2	no	—
223	C₄×C₅	#872	1	1	yes	characteristic
224	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
225	C₄×C₅	#896	1	2	yes	—
226	Dic₅	—	6	2	no	—
227	Dic₅	—	24	2	no	—
228	C₂⁴	—	15	1	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	15	1	no	—
231	C₂²×C₄	—	15	2	no	—
232	C₂²×C₄	—	15	2	no	—
233	C₂²×C₄	—	30	2	no	—
234	C₂²×C₄	—	30	2	no	—
235	C₂²×C₄	—	30	2	no	—
236	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	15	2	no	—
239	C₂×C₈	—	15	4	no	—
240	C₂×C₈	—	30	8	no	—
241	C₂×C₈	C₄×D₁₅	1	2	yes	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	30	2	no	—
245	C₄⋊C₄	—	30	2	no	—
246	C₂²⋊C₄	—	60	4	no	—
247	C₃×C₅	#1807	1	1	yes	characteristic, complemented, Hall
248	C₂²×C₃	#4030	1	2	yes	—
249	C₂²×C₃	D₅×C₄⋊C₄	1	1	yes	characteristic
250	C₂²×C₃	#4058	1	1	yes	characteristic
251	C₂²×C₃	D₅×SD₁₆	1	2	yes	—
252	C₂²×C₃	#4065	1	1	yes	characteristic
253	C₂×D₃	—	5	4	no	—
254	C₂×D₃	—	10	4	no	—
255	C₂×D₃	—	10	4	no	—
256	C₂×D₃	—	10	2	no	—
257	C₂×D₃	—	10	2	no	—
258	C₃×C₄	—	8	2	no	—
259	C₃×C₄	#4018	1	1	yes	characteristic
260	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
261	C₃×C₄	#4042	1	2	yes	—
262	Dic₃	—	10	2	no	—
263	Dic₃	—	40	2	no	—
264	C₂×C₅	#4667	1	1	yes	characteristic
265	C₂×C₅	#4778	1	2	yes	—
266	C₂×C₅	#4807	1	2	yes	—
267	C₂×C₅	D₃×#106	1	1	yes	characteristic
268	C₂×C₅	#4880	1	1	yes	characteristic
269	D₅	—	6	4	no	—
270	C₂³	—	15	2	no	—
271	C₂³	—	15	4	no	—
272	C₂³	—	30	2	no	—
273	C₂³	—	30	1	no	—
274	C₂³	—	30	1	no	—
275	C₂³	D₄×D₁₅	1	1	yes	characteristic
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	15	2	no	—
280	C₂×C₄	—	30	2	no	—
281	C₂×C₄	—	30	2	no	—
282	C₂×C₄	—	30	2	no	—
283	C₂×C₄	—	30	4	no	—
284	C₂×C₄	—	60	2	no	—
285	C₂×C₄	—	60	2	no	—
286	C₂×C₄	—	60	2	no	—
287	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
288	C₂×C₄	#6817	1	2	yes	—
289	C₂×C₄	D₄×D₁₅	1	2	yes	—
290	C₂×C₄	Q₈×D₁₅	1	1	yes	characteristic
291	C₈	—	2	2	no	—
292	C₈	—	15	4	no	—
293	C₂×C₃	#64956	1	1	yes	characteristic
294	C₂×C₃	#65067	1	2	yes	—
295	C₂×C₃	#65096	1	2	yes	—
296	C₂×C₃	D₅×#106	1	1	yes	characteristic
297	C₂×C₃	#65169	1	1	yes	characteristic
298	D₃	—	10	4	no	—
299	C₅	#68868	1	1	yes	characteristic, complemented, Sylow
300	C₂²	—	15	4	no	—
301	C₂²	—	30	2	no	—
302	C₂²	—	30	2	no	—
303	C₂²	—	30	4	no	—
304	C₂²	—	30	4	no	—
305	C₂²	#91773	1	2	yes	—
306	C₂²	C₄⋊C₄×D₁₅	1	1	yes	characteristic
307	C₂²	#91801	1	1	yes	characteristic
308	C₂²	SD₁₆×D₁₅	1	2	yes	—
309	C₂²	#91808	1	1	yes	characteristic
310	C₄	—	8	2	no	—
311	C₄	—	30	2	no	—
312	C₄	—	120	2	no	—
313	C₄	#91761	1	1	yes	characteristic
314	C₄	C₂²⋊C₄×D₁₅	1	1	yes	characteristic
315	C₄	#91785	1	2	yes	—
316	C₃	#10599661	1	1	yes	characteristic, complemented, Sylow
317	C₂	—	30	4	no	—
318	C₂	#11750768	1	1	yes	characteristic
319	C₂	#11750879	1	2	yes	—
320	C₂	#11750908	1	2	yes	—
321	C₂	D₁₅×#106	1	1	yes	characteristic
322	C₂	#11750981	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#340	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂³×D₁₅	D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#340	D₁₅
hypocenter	C₃×C₅	#1807
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

Complements

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

N	H ≅ G/N	type
3. C₂×#91092	C₂	semidirect
4. C₃×C₅×#340	C₂	semidirect
5. #11746372	C₂	semidirect
14. C₂×C₈×D₁₅	C₄	semidirect
21. C₅×#340	D₃	semidirect
45. C₃×#340	D₅	semidirect
80. C₃⋊#168	C₂²⋊C₄	semidirect
114. #340	D₁₅	semidirect
124. C₂×D₁₅	#106	semidirect
247. C₃×C₅	#1807	semidirect
299. C₅	#68868	semidirect
316. C₃	#10599661	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
#11751088	2
C₂×#91092	2
C₃×C₅×#340	2
#11746372	2
#10599661	3
#68868	5