Subgroups — #14437787

Order 1920.

800 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756236	C₂	1	1	yes	maximal, characteristic
2	#11756229	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751214	C₂	1	4	yes	maximal, complemented
5	#11750798	C₂	1	4	yes	maximal, complemented
6	#11750795	C₂	1	2	yes	maximal, complemented
7	#11750782	C₂	1	2	yes	maximal, complemented
8	#10610194	—	3	1	no	maximal, Hall
9	C₂×#6817	C₂²	1	2	yes	—
10	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	#91819	—	2	2	no	—
13	#91815	C₂²	1	2	yes	—
14	#91813	—	2	2	no	—
15	#91764	C₂²	1	2	yes	—
16	#91763	C₂²	1	1	yes	characteristic
17	C₄×D₆₀	—	2	2	no	—
18	#91759	C₂²	1	2	yes	—
19	#91758	C₂²	1	1	yes	characteristic
20	#91758	C₂²	1	2	yes	—
21	C₄×Dic₃₀	—	2	2	no	—
22	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
23	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
24	#91105	C₂²	1	8	yes	—
25	#91100	C₂²	1	8	yes	—
26	#78463	—	5	1	no	maximal, Hall
27	#65811	—	3	1	no	—
28	#65804	—	3	1	no	—
29	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic, complemented
30	#65402	—	3	4	no	—
31	#64986	—	3	4	no	—
32	#64983	—	3	2	no	—
33	#64970	—	3	2	no	—
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×#1220	—	2	2	no	—
37	#6817	—	2	8	no	—
38	C₂×D₆₀	—	2	2	no	—
39	C₂×D₆₀	C₂³	1	2	yes	—
40	C₂×C₄×D₁₅	—	2	2	no	—
41	C₂×Dic₃₀	—	2	2	no	—
42	C₂×Dic₃₀	C₂³	1	2	yes	—
43	C₂×C₃×C₅×C₈	—	2	4	no	—
44	C₂×C₃×C₅×C₈	C₂³	1	4	yes	—
45	C₃×C₄²×C₅	D₄	1	2	yes	—
46	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
47	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
48	#6719	—	2	2	no	—
49	#6714	—	2	4	no	—
50	C₄⋊Dic₁₅	—	2	2	no	—
51	C₄⋊Dic₁₅	—	2	2	no	—
52	C₄⋊Dic₁₅	C₂³	1	4	yes	—
53	#6712	—	2	4	no	—
54	C₄×Dic₁₅	—	2	2	no	—
55	#5477	—	5	1	no	—
56	#5470	—	5	1	no	—
57	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic, complemented
58	#5113	—	5	4	no	—
59	#4697	—	5	4	no	—
60	#4694	—	5	2	no	—
61	#4681	—	5	2	no	—
62	C₂×#709	—	3	2	no	—
63	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
64	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
65	#4076	—	6	2	no	—
66	#4072	—	3	2	no	—
67	#4070	—	6	2	no	—
68	#4021	—	3	2	no	—
69	#4020	—	3	1	no	—
70	C₄×D₂₀	—	6	2	no	—
71	#4016	—	3	2	no	—
72	#4015	—	3	2	no	—
73	#4015	—	3	1	no	—
74	C₄×Dic₁₀	—	6	2	no	—
75	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
76	C₄×C₅×C₈	C₂×D₃	1	2	yes	—
77	#3953	—	3	8	no	—
78	#3948	—	3	8	no	—
79	#3131	—	15	1	no	Sylow
80	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×D₁₅	—	2	2	no	—
82	C₂×C₃×C₄×C₅	—	2	4	no	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	#41	1	4	yes	—
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	#1220	—	4	4	no	—
89	C₂×Dic₁₅	—	2	4	no	—
90	C₂×Dic₁₅	—	2	2	no	—
91	D₆₀	—	2	4	no	—
92	D₆₀	—	4	4	no	—
93	C₄×D₁₅	—	4	4	no	—
94	Dic₃₀	—	2	4	no	—
95	Dic₃₀	—	4	4	no	—
96	C₃×C₅×C₈	—	2	4	no	—
97	C₂×#235	—	5	2	no	—
98	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
99	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
100	#930	—	10	2	no	—
101	#926	—	5	2	no	—
102	#924	—	10	2	no	—
103	#875	—	5	2	no	—
104	#874	—	5	1	no	—
105	C₄×D₁₂	—	10	2	no	—
106	#870	—	5	2	no	—
107	#869	—	5	2	no	—
108	#869	—	5	1	no	—
109	C₄×Dic₆	—	10	2	no	—
110	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
111	C₃×C₄×C₈	C₂×D₅	1	2	yes	—
112	#821	—	5	8	no	—
113	#816	—	5	8	no	—
114	C₂²×C₄×C₅	D₁₂	1	2	yes	—
115	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
116	C₂×C₂²⋊D₅	—	6	2	no	—
117	#709	—	6	8	no	—
118	C₂×D₂₀	—	3	2	no	—
119	C₂×D₂₀	—	6	2	no	—
120	C₂×C₄×D₅	—	6	2	no	—
121	C₂×Dic₁₀	—	3	2	no	—
122	C₂×Dic₁₀	—	6	2	no	—
123	C₂×C₅×C₈	—	2	4	no	—
124	C₂×C₅×C₈	C₂²×D₃	1	4	yes	—
125	C₄²×C₅	D₁₂	1	2	yes	—
126	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
127	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
128	C₂²⋊Dic₅	—	6	2	no	—
129	#685	—	6	4	no	—
130	C₄⋊Dic₅	—	3	4	no	—
131	C₄⋊Dic₅	—	6	2	no	—
132	C₄⋊Dic₅	—	6	2	no	—
133	#683	—	6	4	no	—
134	C₄×Dic₅	—	6	2	no	—
135	#533	—	15	1	no	—
136	#532	—	15	1	no	—
137	#488	—	15	4	no	—
138	#487	—	15	2	no	—
139	#486	—	15	2	no	—
140	#416	—	15	4	no	—
141	C₂×C₄×C₈	D₁₅	1	1	yes	characteristic, complemented
142	C₂²×C₃×C₅	—	2	2	no	—
143	C₂²×C₃×C₅	#124	1	2	yes	—
144	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₁₅	—	4	2	no	—
146	C₂×D₁₅	—	4	4	no	—
147	C₃×C₄×C₅	—	2	2	no	—
148	C₃×C₄×C₅	#124	1	2	yes	—
149	C₃×C₄×C₅	#135	1	4	yes	—
150	C₃×C₄×C₅	C₂×#41	1	2	yes	—
151	Dic₁₅	—	4	2	no	—
152	Dic₁₅	—	4	4	no	—
153	C₂²×C₃×C₄	D₂₀	1	2	yes	—
154	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
155	C₂×#67	—	10	2	no	—
156	#235	—	10	8	no	—
157	C₂×D₁₂	—	5	2	no	—
158	C₂×D₁₂	—	10	2	no	—
159	C₂×C₄×D₃	—	10	2	no	—
160	C₂×Dic₆	—	5	2	no	—
161	C₂×Dic₆	—	10	2	no	—
162	C₂×C₃×C₈	—	2	4	no	—
163	C₂×C₃×C₈	C₂²×D₅	1	4	yes	—
164	C₃×C₄²	D₂₀	1	2	yes	—
165	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
166	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
167	#217	—	10	2	no	—
168	(C₂×D₃)⋊C₄	—	10	4	no	—
169	C₄⋊Dic₃	—	5	4	no	—
170	C₄⋊Dic₃	—	10	2	no	—
171	C₄⋊Dic₃	—	10	2	no	—
172	Dic₃⋊C₄	—	10	4	no	—
173	C₄×Dic₃	—	10	2	no	—
174	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
175	C₂²×D₅	—	6	2	no	—
176	C₂×C₄×C₅	—	2	4	no	—
177	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
178	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
179	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
180	C₂×C₄×C₅	#235	1	4	yes	—
181	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
182	C₂²⋊D₅	—	12	4	no	—
183	C₂×Dic₅	—	6	4	no	—
184	C₂×Dic₅	—	6	2	no	—
185	D₂₀	—	6	4	no	—
186	D₂₀	—	12	4	no	—
187	C₄×D₅	—	12	4	no	—
188	Dic₁₀	—	6	4	no	—
189	Dic₁₀	—	12	4	no	—
190	C₅×C₈	—	2	4	no	—
191	C₂×#41	—	15	2	no	—
192	C₂²×C₈	C₂×D₁₅	1	2	yes	—
193	C₄⋊Q₈	—	15	2	no	—
194	C₄⋊Q₈	—	15	1	no	—
195	#127	—	15	1	no	—
196	#125	—	15	2	no	—
197	#124	—	15	2	no	—
198	C₂²⋊Q₈	—	30	2	no	—
199	#121	—	30	2	no	—
200	C₄×Q₈	—	30	2	no	—
201	C₄×D₄	—	30	2	no	—
202	#117	—	15	2	no	—
203	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
204	#103	—	15	8	no	—
205	#102	—	15	8	no	—
206	C₄×C₈	C₂×D₁₅	1	2	yes	—
207	C₄×C₈	C₂×D₁₅	1	2	yes	—
208	C₂×C₃×C₅	—	2	2	no	—
209	C₂×C₃×C₅	C₂×#124	1	1	yes	characteristic
210	C₂×C₃×C₅	C₂×#135	1	2	yes	—
211	D₁₅	—	8	2	no	—
212	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
213	C₂²×D₃	—	10	2	no	—
214	C₂×C₃×C₄	—	2	4	no	—
215	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
216	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
217	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
218	C₂×C₃×C₄	#709	1	4	yes	—
219	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
220	#67	—	20	4	no	—
221	C₂×Dic₃	—	10	4	no	—
222	C₂×Dic₃	—	10	2	no	—
223	D₁₂	—	10	4	no	—
224	D₁₂	—	20	4	no	—
225	C₄×D₃	—	20	4	no	—
226	Dic₆	—	10	4	no	—
227	Dic₆	—	20	4	no	—
228	C₃×C₈	—	2	4	no	—
229	C₂²×C₅	—	2	2	no	—
230	C₂²×C₅	#875	1	2	yes	—
231	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
232	C₂×D₅	—	12	2	no	—
233	C₂×D₅	—	12	4	no	—
234	C₄×C₅	—	2	2	no	—
235	C₄×C₅	#875	1	2	yes	—
236	C₄×C₅	#904	1	4	yes	—
237	C₄×C₅	C₂×#235	1	2	yes	—
238	Dic₅	—	12	2	no	—
239	Dic₅	—	12	4	no	—
240	#41	—	30	8	no	—
241	C₂×Q₈	—	15	2	no	—
242	C₂×Q₈	—	30	2	no	—
243	C₂×D₄	—	15	2	no	—
244	C₂×D₄	—	30	2	no	—
245	C₂×D₄	—	30	2	no	—
246	C₂²×C₄	—	30	2	no	—
247	C₂²×C₄	D₆₀	1	2	yes	—
248	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	C₂²×D₁₅	1	4	yes	—
251	C₄⋊C₄	—	15	4	no	—
252	C₄⋊C₄	—	30	2	no	—
253	C₄⋊C₄	—	30	4	no	—
254	C₄⋊C₄	—	30	2	no	—
255	C₂²⋊C₄	—	30	2	no	—
256	C₂²⋊C₄	—	30	4	no	—
257	C₄²	—	30	2	no	—
258	C₄²	D₆₀	1	2	yes	—
259	C₄²	C₂²×D₁₅	1	1	yes	characteristic
260	C₄²	C₂²×D₁₅	1	1	yes	characteristic
261	C₃×C₅	#3131	1	1	yes	characteristic, complemented, Hall
262	C₂²×C₃	—	2	2	no	—
263	C₂²×C₃	#4021	1	2	yes	—
264	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
265	C₂×D₃	—	20	2	no	—
266	C₂×D₃	—	20	4	no	—
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	#4021	1	2	yes	—
269	C₃×C₄	#4050	1	4	yes	—
270	C₃×C₄	C₂×#709	1	2	yes	—
271	Dic₃	—	20	2	no	—
272	Dic₃	—	20	4	no	—
273	C₂×C₅	—	2	2	no	—
274	C₂×C₅	C₂×#875	1	1	yes	characteristic
275	C₂×C₅	C₂×#904	1	2	yes	—
276	D₅	—	24	2	no	—
277	C₂³	—	30	2	no	—
278	C₂³	C₂×D₆₀	1	1	yes	characteristic
279	Q₈	—	30	4	no	—
280	Q₈	—	60	4	no	—
281	D₄	—	30	4	no	—
282	D₄	—	60	4	no	—
283	D₄	—	60	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	30	4	no	—
286	C₂×C₄	—	30	2	no	—
287	C₂×C₄	—	60	4	no	—
288	C₂×C₄	C₂×D₆₀	1	2	yes	—
289	C₂×C₄	C₂×D₆₀	1	2	yes	—
290	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
291	C₂×C₄	#6817	1	4	yes	—
292	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
293	C₈	—	2	4	no	—
294	C₂×C₃	—	2	2	no	—
295	C₂×C₃	C₂×#4021	1	1	yes	characteristic
296	C₂×C₃	C₂×#4050	1	2	yes	—
297	D₃	—	40	2	no	—
298	C₅	#78463	1	1	yes	characteristic, complemented, Sylow
299	C₂²	—	2	2	no	—
300	C₂²	—	60	2	no	—
301	C₂²	—	60	4	no	—
302	C₂²	#91764	1	2	yes	—
303	C₂²	C₂²×D₆₀	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	60	2	no	—
306	C₄	—	60	4	no	—
307	C₄	#91764	1	2	yes	—
308	C₄	#91793	1	4	yes	—
309	C₄	C₂×#6817	1	2	yes	—
310	C₃	#10610194	1	1	yes	characteristic, complemented, Sylow
311	C₂	—	2	2	no	—
312	C₂	—	120	2	no	—
313	C₂	C₂×#91764	1	1	yes	characteristic
314	C₂	C₂×#91793	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₀
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	#11756236	C₂
hypercenter	C₂×C₄×C₈	D₁₅
hypocenter	C₃×C₅	#3131
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #11756229	C₂	semidirect
3. C₂×C₃×C₄×C₅×C₈	C₂	semidirect
4. #11751214	C₂	semidirect
5. #11750798	C₂	semidirect
6. #11750795	C₂	semidirect
7. #11750782	C₂	semidirect
29. C₂×C₄×C₅×C₈	D₃	semidirect
57. C₂×C₃×C₄×C₈	D₅	semidirect
141. C₂×C₄×C₈	D₁₅	semidirect
261. C₃×C₅	#3131	semidirect
298. C₅	#78463	semidirect
310. C₃	#10610194	semidirect

Maximal subgroups

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

subgroup	index
#11756236	2
#11756229	2
C₂×C₃×C₄×C₅×C₈	2
#11751214	2
#11750798	2
#11750795	2
#11750782	2
#10610194	3
#78463	5