Subgroups — #14417341

Order 1920.

1056 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	C₂²×C₃×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	#11756557	C₂	1	2	yes	maximal, complemented
3	#11751316	C₂	1	2	yes	maximal, complemented
4	#11751253	C₂	1	4	yes	maximal, complemented
5	D₈×Dic₁₅	C₂	1	4	yes	maximal, complemented
6	#11751207	C₂	1	1	yes	maximal, characteristic, complemented
7	#11751202	C₂	1	1	yes	maximal, characteristic, complemented
8	#10615554	—	3	1	no	maximal, Hall
9	C₂²×C₃×C₅×D₄	C₂²	1	2	yes	—
10	C₂×C₃×C₅×D₈	C₄	1	8	yes	complemented
11	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
12	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
13	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6719	—	2	4	no	—
15	D₄×Dic₁₅	—	2	4	no	—
16	D₄×Dic₁₅	C₂²	1	8	yes	complemented
17	#91815	C₂²	1	2	yes	—
18	#91815	C₂²	1	1	yes	characteristic
19	C₂×#6710	C₂²	1	1	yes	characteristic
20	#91116	C₂²	1	8	yes	complemented
21	#91102	C₂²	1	2	yes	complemented
22	#91101	C₂²	1	2	yes	complemented
23	#91099	C₂²	1	2	yes	complemented
24	C₈×Dic₁₅	C₂²	1	2	yes	complemented
25	#83823	—	5	1	no	maximal, Hall
26	C₂²×C₅×D₈	D₃	1	1	yes	characteristic
27	#66132	—	3	2	no	—
28	#65504	—	3	2	no	—
29	#65441	—	3	4	no	—
30	D₈×Dic₅	—	3	4	no	—
31	#65395	—	3	1	no	—
32	#65390	—	3	1	no	—
33	C₂⁴×C₃×C₅	—	2	2	no	—
34	C₂×C₃×C₅×D₄	—	2	2	no	—
35	C₂×C₃×C₅×D₄	—	2	4	no	—
36	C₂×C₃×C₅×D₄	C₂×C₄	1	8	yes	complemented
37	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
38	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
39	C₂²×Dic₁₅	—	2	8	no	—
40	C₃×C₅×D₈	C₂×C₄	1	8	yes	complemented
41	C₃×C₅×D₈	C₂×C₄	1	8	yes	complemented
42	C₂×C₃×C₅×C₈	C₂×C₄	1	4	yes	complemented
43	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
45	#6719	—	2	2	no	—
46	#6719	—	2	8	no	—
47	#6719	D₄	1	2	yes	—
48	C₄⋊Dic₁₅	D₄	1	2	yes	—
49	C₄⋊Dic₁₅	C₂³	1	4	yes	—
50	C₄×Dic₁₅	—	2	2	no	—
51	C₄×Dic₁₅	C₂³	1	2	yes	—
52	#6710	—	2	2	no	—
53	C₂×C₃⋊#168	C₂³	1	2	yes	—
54	C₂²×C₃×D₈	D₅	1	1	yes	characteristic
55	#5798	—	5	2	no	—
56	#5215	—	5	2	no	—
57	#5152	—	5	4	no	—
58	Dic₃×D₈	—	5	4	no	—
59	#5106	—	5	1	no	—
60	#5101	—	5	1	no	—
61	C₂²×C₅×D₄	C₂×D₃	1	2	yes	—
62	C₂×C₅×D₈	Dic₃	1	8	yes	complemented
63	C₂×C₅×D₈	C₂×D₃	1	2	yes	—
64	C₂×C₅×D₈	C₂×D₃	1	2	yes	—
65	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
66	C₂×C₂²⋊Dic₅	—	6	4	no	—
67	D₄×Dic₅	—	3	8	no	—
68	D₄×Dic₅	—	6	4	no	—
69	#4072	—	3	1	no	—
70	#4072	—	3	2	no	—
71	C₂×#681	—	3	1	no	—
72	#3964	—	3	8	no	—
73	#3950	—	3	2	no	—
74	#3949	—	3	2	no	—
75	#3947	—	3	2	no	—
76	C₈×Dic₅	—	3	2	no	—
77	#2948	—	15	1	no	Sylow
78	C₂³×C₃×C₅	—	2	4	no	—
79	C₂³×C₃×C₅	—	2	8	no	—
80	C₂³×C₃×C₅	—	8	4	no	—
81	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
82	C₃×C₅×D₄	—	2	8	no	—
83	C₃×C₅×D₄	—	2	4	no	—
84	C₃×C₅×D₄	C₂²×C₄	1	8	yes	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	#41	1	2	yes	—
88	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
89	C₂×Dic₁₅	—	2	4	no	—
90	C₂×Dic₁₅	—	8	8	no	—
91	C₂×Dic₁₅	C₂×D₄	1	4	yes	—
92	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
93	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₃⋊#168	—	2	2	no	—
95	C₂²×C₃×D₄	C₂×D₅	1	2	yes	—
96	C₂×C₃×D₈	Dic₅	1	8	yes	complemented
97	C₂×C₃×D₈	C₂×D₅	1	2	yes	—
98	C₂×C₃×D₈	C₂×D₅	1	2	yes	—
99	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
100	C₂×#217	—	10	4	no	—
101	D₄×Dic₃	—	5	8	no	—
102	D₄×Dic₃	—	10	4	no	—
103	#926	—	5	1	no	—
104	#926	—	5	2	no	—
105	C₂×#208	—	5	1	no	—
106	#832	—	5	8	no	—
107	#818	—	5	2	no	—
108	#817	—	5	2	no	—
109	#815	—	5	2	no	—
110	C₈×Dic₃	—	5	2	no	—
111	C₂⁴×C₅	—	2	2	no	—
112	C₂×C₅×D₄	—	2	2	no	—
113	C₂×C₅×D₄	—	2	4	no	—
114	C₂×C₅×D₄	C₂×Dic₃	1	8	yes	complemented
115	C₂×C₅×D₄	C₂²×D₃	1	4	yes	—
116	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
117	C₂²×Dic₅	—	6	8	no	—
118	C₅×D₈	C₂×Dic₃	1	8	yes	complemented
119	C₅×D₈	C₂×Dic₃	1	8	yes	complemented
120	C₂×C₅×C₈	C₂×Dic₃	1	4	yes	complemented
121	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
122	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
123	C₂²⋊Dic₅	—	3	2	no	—
124	C₂²⋊Dic₅	—	6	8	no	—
125	C₂²⋊Dic₅	—	6	2	no	—
126	C₄⋊Dic₅	—	3	2	no	—
127	C₄⋊Dic₅	—	3	4	no	—
128	C₄×Dic₅	—	3	2	no	—
129	C₄×Dic₅	—	6	2	no	—
130	#681	—	6	2	no	—
131	C₂×#168	—	3	2	no	—
132	C₂²×D₈	D₁₅	1	1	yes	characteristic
133	#518	—	15	2	no	—
134	#442	—	15	4	no	—
135	C₄×D₈	—	15	4	no	—
136	#427	—	15	1	no	—
137	#418	—	15	2	no	—
138	#405	—	15	1	no	—
139	C₂²×C₃×C₅	—	2	2	no	—
140	C₂²×C₃×C₅	—	2	8	no	—
141	C₂²×C₃×C₅	—	4	8	no	—
142	C₂²×C₃×C₅	—	8	8	no	—
143	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
144	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
145	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
146	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
147	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
148	Dic₁₅	—	2	4	no	—
149	Dic₁₅	—	4	4	no	—
150	C₂⁴×C₃	—	2	2	no	—
151	C₂×C₃×D₄	—	2	2	no	—
152	C₂×C₃×D₄	—	2	4	no	—
153	C₂×C₃×D₄	C₂×Dic₅	1	8	yes	complemented
154	C₂×C₃×D₄	C₂²×D₅	1	4	yes	—
155	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
156	C₂²×Dic₃	—	10	8	no	—
157	C₃×D₈	C₂×Dic₅	1	8	yes	complemented
158	C₃×D₈	C₂×Dic₅	1	8	yes	complemented
159	C₂×C₃×C₈	C₂×Dic₅	1	4	yes	complemented
160	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
161	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
162	#217	—	5	2	no	—
163	#217	—	10	8	no	—
164	#217	—	10	2	no	—
165	C₄⋊Dic₃	—	5	2	no	—
166	C₄⋊Dic₃	—	5	4	no	—
167	C₄×Dic₃	—	5	2	no	—
168	C₄×Dic₃	—	10	2	no	—
169	#208	—	10	2	no	—
170	C₂×C₃⋊C₈	—	5	2	no	—
171	C₂³×C₅	—	2	4	no	—
172	C₂³×C₅	—	2	8	no	—
173	C₂³×C₅	—	8	4	no	—
174	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
175	C₅×D₄	—	2	8	no	—
176	C₅×D₄	—	2	4	no	—
177	C₅×D₄	C₂²×Dic₃	1	8	yes	—
178	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
179	C₂×C₄×C₅	#237	1	2	yes	—
180	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
181	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
182	C₂×Dic₅	—	3	4	no	—
183	C₂×Dic₅	—	6	4	no	—
184	C₂×Dic₅	—	24	8	no	—
185	C₅×C₈	C₂²×Dic₃	1	2	yes	—
186	C₅×C₈	C₂²×Dic₃	1	2	yes	—
187	#168	—	6	2	no	—
188	C₂²×D₄	C₂×D₁₅	1	2	yes	—
189	C₂×D₈	Dic₁₅	1	8	yes	complemented
190	C₂×D₈	C₂×D₁₅	1	2	yes	—
191	C₂×D₈	C₂×D₁₅	1	2	yes	—
192	C₂×M₁₆	—	15	1	no	—
193	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
194	C₄×D₄	—	15	8	no	—
195	C₄×D₄	—	30	4	no	—
196	#117	—	15	1	no	—
197	#117	—	15	2	no	—
198	C₂×C₂²⋊C₄	—	30	4	no	—
199	#107	—	15	2	no	—
200	#106	—	15	2	no	—
201	#102	—	15	8	no	—
202	#97	—	15	2	no	—
203	C₄×C₈	—	15	2	no	—
204	C₂×C₃×C₅	—	2	2	no	—
205	C₂×C₃×C₅	—	4	8	no	—
206	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
207	C₂×C₃×C₅	#576	1	2	yes	—
208	C₂³×C₃	—	2	4	no	—
209	C₂³×C₃	—	2	8	no	—
210	C₂³×C₃	—	8	4	no	—
211	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
212	C₃×D₄	—	2	4	no	—
213	C₃×D₄	—	2	8	no	—
214	C₃×D₄	C₂²×Dic₅	1	8	yes	—
215	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
216	C₂×C₃×C₄	#711	1	2	yes	—
217	C₂×C₃×C₄	C₂²×Dic₅	1	2	yes	—
218	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
219	C₂×Dic₃	—	5	4	no	—
220	C₂×Dic₃	—	10	4	no	—
221	C₂×Dic₃	—	40	8	no	—
222	C₃×C₈	C₂²×Dic₅	1	2	yes	—
223	C₃×C₈	C₂²×Dic₅	1	2	yes	—
224	C₃⋊C₈	—	10	2	no	—
225	C₂²×C₅	—	2	2	no	—
226	C₂²×C₅	—	2	8	no	—
227	C₂²×C₅	—	4	8	no	—
228	C₂²×C₅	—	8	8	no	—
229	C₂²×C₅	D₄×Dic₃	1	2	yes	—
230	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
231	C₄×C₅	D₄×Dic₃	1	2	yes	—
232	C₄×C₅	C₂×#237	1	1	yes	characteristic
233	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
234	Dic₅	—	6	4	no	—
235	Dic₅	—	12	4	no	—
236	C₂⁴	—	2	2	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	C₂×Dic₁₅	1	8	yes	complemented
240	C₂×D₄	C₂²×D₁₅	1	4	yes	—
241	C₂²×C₄	—	30	8	no	—
242	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
243	D₈	C₂×Dic₁₅	1	8	yes	complemented
244	D₈	C₂×Dic₁₅	1	8	yes	complemented
245	M₁₆	—	30	2	no	—
246	C₂×C₈	—	15	2	no	—
247	C₂×C₈	C₂×Dic₁₅	1	4	yes	complemented
248	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
250	C₄⋊C₄	—	15	2	no	—
251	C₄⋊C₄	—	15	4	no	—
252	C₂²⋊C₄	—	15	2	no	—
253	C₂²⋊C₄	—	30	8	no	—
254	C₂²⋊C₄	—	30	2	no	—
255	C₄²	—	15	2	no	—
256	C₄²	—	30	2	no	—
257	C₃×C₅	#2948	1	1	yes	characteristic, complemented, Hall
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	8	no	—
260	C₂²×C₃	—	4	8	no	—
261	C₂²×C₃	—	8	8	no	—
262	C₂²×C₃	D₄×Dic₅	1	2	yes	—
263	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
264	C₃×C₄	D₄×Dic₅	1	2	yes	—
265	C₃×C₄	C₂×#711	1	1	yes	characteristic
266	C₃×C₄	C₂³×Dic₅	1	1	yes	characteristic
267	Dic₃	—	10	4	no	—
268	Dic₃	—	20	4	no	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	—	4	8	no	—
271	C₂×C₅	#5757	1	2	yes	—
272	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
273	C₂³	—	2	4	no	—
274	C₂³	—	2	8	no	—
275	C₂³	—	8	4	no	—
276	C₂³	D₄×D₁₅	1	1	yes	characteristic
277	D₄	—	2	8	no	—
278	D₄	—	2	4	no	—
279	D₄	C₂²×Dic₁₅	1	8	yes	—
280	C₂×C₄	—	15	4	no	—
281	C₂×C₄	—	30	4	no	—
282	C₂×C₄	—	120	8	no	—
283	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
284	C₂×C₄	#6819	1	2	yes	—
285	C₂×C₄	C₂²×Dic₁₅	1	2	yes	—
286	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
287	C₈	—	30	2	no	—
288	C₈	C₂²×Dic₁₅	1	2	yes	—
289	C₈	C₂²×Dic₁₅	1	2	yes	—
290	C₂×C₃	—	2	2	no	—
291	C₂×C₃	—	4	8	no	—
292	C₂×C₃	#66091	1	2	yes	—
293	C₂×C₃	C₂×D₄×Dic₅	1	1	yes	characteristic
294	C₅	#83823	1	1	yes	characteristic, complemented, Sylow
295	C₂²	—	2	2	no	—
296	C₂²	—	2	8	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	8	8	no	—
299	C₂²	D₄×Dic₁₅	1	2	yes	—
300	C₂²	C₂×D₄×D₁₅	1	1	yes	characteristic
301	C₄	—	30	4	no	—
302	C₄	—	60	4	no	—
303	C₄	D₄×Dic₁₅	1	2	yes	—
304	C₄	C₂×#6819	1	1	yes	characteristic
305	C₄	C₂³×Dic₁₅	1	1	yes	characteristic
306	C₃	#10615554	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	2	2	no	—
308	C₂	—	4	8	no	—
309	C₂	#11756516	1	2	yes	—
310	C₂	C₂×D₄×Dic₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅
Fitting, prime radical	C₂²×C₃×C₅×D₈	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×D₈	D₁₅
hypocenter	C₃×C₅	#2948
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₈

Complements

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

N	H ≅ G/N	type
2. #11756557	C₂	semidirect
3. #11751316	C₂	semidirect
4. #11751253	C₂	semidirect
5. D₈×Dic₁₅	C₂	semidirect
6. #11751207	C₂	semidirect
7. #11751202	C₂	semidirect
10. C₂×C₃×C₅×D₈	C₄	semidirect
16. D₄×Dic₁₅	C₂²	semidirect
20. #91116	C₂²	semidirect
21. #91102	C₂²	semidirect
22. #91101	C₂²	semidirect
23. #91099	C₂²	semidirect
24. C₈×Dic₁₅	C₂²	semidirect
36. C₂×C₃×C₅×D₄	C₂×C₄	semidirect
40. C₃×C₅×D₈	C₂×C₄	semidirect
41. C₃×C₅×D₈	C₂×C₄	semidirect
42. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
62. C₂×C₅×D₈	Dic₃	semidirect
96. C₂×C₃×D₈	Dic₅	semidirect
114. C₂×C₅×D₄	C₂×Dic₃	semidirect
118. C₅×D₈	C₂×Dic₃	semidirect
119. C₅×D₈	C₂×Dic₃	semidirect
120. C₂×C₅×C₈	C₂×Dic₃	semidirect
153. C₂×C₃×D₄	C₂×Dic₅	semidirect
157. C₃×D₈	C₂×Dic₅	semidirect
158. C₃×D₈	C₂×Dic₅	semidirect
159. C₂×C₃×C₈	C₂×Dic₅	semidirect
189. C₂×D₈	Dic₁₅	semidirect
239. C₂×D₄	C₂×Dic₁₅	semidirect
243. D₈	C₂×Dic₁₅	semidirect
244. D₈	C₂×Dic₁₅	semidirect
247. C₂×C₈	C₂×Dic₁₅	semidirect
257. C₃×C₅	#2948	semidirect
294. C₅	#83823	semidirect
306. C₃	#10615554	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×C₅×D₈	2
#11756557	2
#11751316	2
#11751253	2
D₈×Dic₁₅	2
#11751207	2
#11751202	2
#10615554	3
#83823	5