Subgroups — #14439864

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752498	C₂	1	2	yes	maximal
2	#11750784	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4683	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
5	#11748019	C₂	1	2	yes	maximal, complemented
6	#11748016	C₂	1	2	yes	maximal, complemented
7	#11747672	C₂	1	2	yes	maximal
8	#11747671	C₂	1	2	yes	maximal
9	#11746956	C₂	1	2	yes	maximal, complemented
10	#10613647	—	3	1	no	maximal, Hall
11	#91759	C₂²	1	2	yes	—
12	C₅×#870	C₂²	1	2	yes	—
13	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
15	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
16	#91468	C₂²	1	4	yes	—
17	#91358	C₂²	1	4	yes	—
18	#91349	—	2	4	no	—
19	#91323	—	2	4	no	—
20	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	#91102	C₂²	1	2	yes	—
22	#91101	C₂²	1	2	yes	—
23	C₅×#818	C₂²	1	2	yes	—
24	C₅×#817	C₂²	1	2	yes	—
25	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
26	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
27	C₃×C₄×#168	C₂²	1	1	yes	characteristic
28	#90988	C₂²	1	4	yes	—
29	#90985	C₂²	1	4	yes	—
30	#78442	—	5	1	no	maximal, Hall
31	#66022	—	3	2	no	—
32	C₅×#499	—	3	1	no	—
33	#65353	—	3	2	no	—
34	#65169	—	3	2	no	—
35	#65163	—	3	2	no	—
36	#65149	—	3	2	no	—
37	#65144	—	3	2	no	—
38	C₄×C₈×D₅	D₃	1	1	yes	characteristic
39	#64972	—	3	1	no	—
40	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
41	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
42	C₂×#1204	—	2	4	no	—
43	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
45	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
46	C₄⋊Dic₁₅	C₂³	1	4	yes	—
47	#6712	—	2	4	no	—
48	C₅×C₄⋊Dic₃	C₂³	1	4	yes	—
49	C₅×Dic₃⋊C₄	—	2	4	no	—
50	C₃×C₄×Dic₅	D₄	1	2	yes	—
51	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
52	C₂×C₃×#168	C₂³	1	1	yes	characteristic
53	C₂×C₃×#168	C₂³	1	1	yes	characteristic
54	C₃×C₈×D₅	Q₈	1	4	yes	complemented
55	C₃×C₈×D₅	Q₈	1	4	yes	complemented
56	#6668	—	2	4	no	—
57	#6665	—	2	4	no	—
58	Dic₃×Dic₅	—	2	4	no	—
59	#5470	—	5	2	no	—
60	C₂×C₃×C₄×C₈	—	5	1	no	—
61	#5106	—	5	2	no	—
62	#5106	—	5	2	no	—
63	#4683	—	5	2	no	—
64	#4683	—	5	1	no	—
65	#4683	D₅	1	1	yes	characteristic, complemented
66	#4682	—	5	2	no	—
67	#4680	—	5	2	no	—
68	C₅×#125	—	3	2	no	—
69	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
70	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
71	#4042	—	6	4	no	—
72	#4038	—	3	4	no	—
73	#4034	—	3	4	no	—
74	#4033	—	6	4	no	—
75	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
76	#4016	—	3	2	no	—
77	C₅×#107	—	3	2	no	—
78	C₅×#106	—	3	2	no	—
79	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
80	#3950	—	3	2	no	—
81	#3949	—	3	2	no	—
82	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
83	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
84	#3940	—	3	4	no	—
85	#3939	—	3	4	no	—
86	C₄×#168	C₂×D₃	1	1	yes	characteristic
87	#3160	—	15	1	no	Sylow
88	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
90	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₂×Dic₁₅	—	2	4	no	—
92	C₂×C₅×Dic₃	—	2	4	no	—
93	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
95	C₃×C₄×D₅	—	2	4	no	—
96	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
97	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
98	#1204	—	4	8	no	—
99	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
100	C₃×C₅×C₈	C₂×Q₈	1	2	yes	—
101	C₃×#168	C₂×Q₈	1	2	yes	—
102	C₃×#168	C₂×Q₈	1	2	yes	—
103	C₂²×C₃×C₈	—	5	1	no	—
104	C₂²×C₃×C₈	—	5	1	no	—
105	C₂×C₃×C₄²	—	5	1	no	—
106	#926	—	5	4	no	—
107	#924	—	10	4	no	—
108	#870	—	5	2	no	—
109	#870	C₂×D₅	1	2	yes	—
110	#869	—	5	4	no	—
111	C₄×Dic₆	—	10	4	no	—
112	C₃×C₄×C₈	—	5	1	no	—
113	C₃×C₄×C₈	—	5	1	no	—
114	C₃×C₄×C₈	—	5	1	no	—
115	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
116	#818	—	5	2	no	—
117	#818	—	5	4	no	—
118	#818	C₂×D₅	1	2	yes	—
119	#817	—	5	2	no	—
120	#817	—	5	4	no	—
121	#817	C₂×D₅	1	2	yes	—
122	C₂×C₄×D₅	D₁₂	1	2	yes	—
123	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
124	C₂×Dic₁₀	—	6	4	no	—
125	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
126	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
127	C₅×C₄⋊C₄	—	3	4	no	—
128	C₅×C₄⋊C₄	—	6	4	no	—
129	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
130	#685	—	6	4	no	—
131	C₄⋊Dic₅	—	3	4	no	—
132	#683	—	6	4	no	—
133	#683	—	6	4	no	—
134	C₄×Dic₅	—	6	4	no	—
135	C₄×Dic₅	D₁₂	1	2	yes	—
136	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
137	C₂×#168	C₂²×D₃	1	1	yes	characteristic
138	C₂×#168	C₂²×D₃	1	1	yes	characteristic
139	C₈×D₅	Dic₆	1	4	yes	complemented
140	C₈×D₅	Dic₆	1	4	yes	complemented
141	#533	—	15	2	no	—
142	#500	—	15	2	no	—
143	#499	—	3	1	no	—
144	#499	—	15	2	no	—
145	#499	—	15	1	no	—
146	#498	—	15	2	no	—
147	#427	—	15	2	no	—
148	#427	—	15	2	no	—
149	C₂×C₄×C₈	—	5	1	no	—
150	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₃×D₅	—	2	2	no	—
152	C₂×C₃×D₅	C₄⋊Q₈	1	2	yes	—
153	C₃×C₄×C₅	—	2	2	no	—
154	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
155	C₃×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
156	Dic₁₅	—	4	4	no	—
157	C₃×Dic₅	C₄⋊Q₈	1	2	yes	—
158	C₃×Dic₅	#135	1	4	yes	—
159	C₅×Dic₃	—	4	4	no	—
160	C₂²×C₃×C₄	—	5	1	no	—
161	C₂²×C₃×C₄	—	5	2	no	—
162	C₂×Dic₆	—	10	4	no	—
163	C₂×C₃×C₈	—	5	1	no	—
164	C₂×C₃×C₈	—	5	1	no	—
165	C₂×C₃×C₈	—	5	4	no	—
166	C₂×C₃×C₈	—	5	4	no	—
167	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
168	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
169	C₃×C₄²	—	5	2	no	—
170	C₃×C₄²	—	5	1	no	—
171	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
172	#217	—	10	4	no	—
173	C₄⋊Dic₃	—	5	4	no	—
174	C₄⋊Dic₃	—	10	4	no	—
175	C₄⋊Dic₃	C₂²×D₅	1	4	yes	—
176	Dic₃⋊C₄	—	2	4	no	—
177	Dic₃⋊C₄	—	10	4	no	—
178	C₄×Dic₃	—	10	4	no	—
179	C₂²×D₅	C₂×D₁₂	1	1	yes	characteristic
180	C₂×C₄×C₅	—	6	4	no	—
181	C₂×C₄×C₅	C₂×D₁₂	1	2	yes	—
182	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
183	C₂×Dic₅	—	6	4	no	—
184	C₂×Dic₅	C₂×D₁₂	1	1	yes	characteristic
185	C₂×Dic₅	C₂×D₁₂	1	2	yes	—
186	C₄×D₅	—	2	4	no	—
187	C₄×D₅	C₂×Dic₆	1	2	yes	—
188	C₄×D₅	C₂×Dic₆	1	2	yes	—
189	Dic₁₀	—	12	8	no	—
190	C₅×C₈	C₂×Dic₆	1	2	yes	—
191	C₅×C₈	C₂×Dic₆	1	2	yes	—
192	#168	C₂×Dic₆	1	2	yes	—
193	#168	C₂×Dic₆	1	2	yes	—
194	C₂²×C₈	—	5	1	no	—
195	C₂²×C₈	—	5	1	no	—
196	C₄⋊Q₈	—	15	4	no	—
197	#125	—	3	2	no	—
198	#125	—	15	2	no	—
199	C₂²⋊Q₈	—	30	4	no	—
200	C₄×Q₈	—	30	4	no	—
201	#117	—	15	4	no	—
202	C₂×C₄²	—	5	1	no	—
203	#107	—	3	2	no	—
204	#107	—	15	4	no	—
205	#107	—	15	2	no	—
206	#106	—	3	2	no	—
207	#106	—	15	4	no	—
208	#106	—	15	2	no	—
209	C₄×C₈	—	5	1	no	—
210	C₄×C₈	—	5	1	no	—
211	C₄×C₈	—	5	1	no	—
212	C₄×C₈	D₃×D₅	1	1	yes	characteristic
213	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
214	C₂×C₃×C₅	C₂×#135	1	2	yes	—
215	C₃×D₅	—	2	2	no	—
216	C₂³×C₃	—	5	1	no	—
217	C₂×C₃×C₄	—	5	1	no	—
218	C₂×C₃×C₄	—	5	2	no	—
219	C₂×C₃×C₄	—	5	2	no	—
220	C₂×C₃×C₄	—	5	2	no	—
221	C₂×C₃×C₄	—	10	4	no	—
222	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
223	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
224	C₂×Dic₃	—	2	4	no	—
225	C₂×Dic₃	—	10	4	no	—
226	Dic₆	—	20	8	no	—
227	C₃×C₈	—	5	2	no	—
228	C₃×C₈	—	5	2	no	—
229	C₃×C₈	Q₈×D₅	1	2	yes	—
230	C₃×C₈	Q₈×D₅	1	2	yes	—
231	C₂²×C₅	C₂²×D₁₂	1	1	yes	characteristic
232	C₂×D₅	—	2	2	no	—
233	C₂×D₅	#869	1	2	yes	—
234	C₄×C₅	—	2	2	no	—
235	C₄×C₅	—	12	4	no	—
236	C₄×C₅	C₂²×Dic₆	1	1	yes	characteristic
237	C₄×C₅	C₂²×Dic₆	1	1	yes	characteristic
238	Dic₅	—	12	4	no	—
239	Dic₅	#869	1	2	yes	—
240	Dic₅	#904	1	4	yes	—
241	C₂×Q₈	—	30	4	no	—
242	C₂²×C₄	—	5	1	no	—
243	C₂²×C₄	—	5	2	no	—
244	C₂×C₈	—	5	1	no	—
245	C₂×C₈	—	5	1	no	—
246	C₂×C₈	—	5	4	no	—
247	C₂×C₈	—	5	4	no	—
248	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
249	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
250	C₄⋊C₄	—	3	4	no	—
251	C₄⋊C₄	—	6	4	no	—
252	C₄⋊C₄	—	15	4	no	—
253	C₄⋊C₄	—	30	4	no	—
254	C₄⋊C₄	—	30	4	no	—
255	C₂²⋊C₄	—	30	4	no	—
256	C₄²	—	5	2	no	—
257	C₄²	—	5	1	no	—
258	C₄²	—	30	4	no	—
259	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
260	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
261	C₂²×C₃	—	5	2	no	—
262	C₂²×C₃	—	10	2	no	—
263	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
264	C₃×C₄	—	2	2	no	—
265	C₃×C₄	—	5	2	no	—
266	C₃×C₄	—	5	4	no	—
267	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
268	C₃×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
269	Dic₃	—	4	4	no	—
270	Dic₃	—	20	4	no	—
271	C₂×C₅	C₂×#869	1	1	yes	characteristic
272	C₂×C₅	C₂×#904	1	2	yes	—
273	D₅	—	2	2	no	—
274	C₂³	—	5	1	no	—
275	Q₈	—	60	8	no	—
276	C₂×C₄	—	5	1	no	—
277	C₂×C₄	—	5	2	no	—
278	C₂×C₄	—	5	2	no	—
279	C₂×C₄	—	5	2	no	—
280	C₂×C₄	—	6	4	no	—
281	C₂×C₄	—	10	4	no	—
282	C₂×C₄	—	30	4	no	—
283	C₂×C₄	D₅×D₁₂	1	2	yes	—
284	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
285	C₈	—	5	2	no	—
286	C₈	—	5	2	no	—
287	C₈	D₅×Dic₆	1	2	yes	—
288	C₈	D₅×Dic₆	1	2	yes	—
289	C₂×C₃	—	10	2	no	—
290	C₂×C₃	D₅×C₄⋊Q₈	1	1	yes	characteristic
291	C₂×C₃	D₅×#135	1	2	yes	—
292	C₅	#78442	1	1	yes	characteristic, complemented, Sylow
293	C₂²	—	5	2	no	—
294	C₂²	—	10	2	no	—
295	C₂²	C₂×D₅×D₁₂	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	5	2	no	—
298	C₄	—	5	4	no	—
299	C₄	—	12	4	no	—
300	C₄	—	60	4	no	—
301	C₄	C₂×D₅×Dic₆	1	1	yes	characteristic
302	C₄	C₂×D₅×Dic₆	1	1	yes	characteristic
303	C₃	#10613647	1	1	yes	characteristic, complemented, Sylow
304	C₂	—	10	2	no	—
305	C₂	D₅×#869	1	1	yes	characteristic
306	C₂	D₅×#904	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₅×D₁₂
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂²×C₃×D₅	C₂×D₄
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#3160
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #11750784	C₂	semidirect
3. C₅×#4683	C₂	semidirect
5. #11748019	C₂	semidirect
6. #11748016	C₂	semidirect
9. #11746956	C₂	semidirect
54. C₃×C₈×D₅	Q₈	semidirect
55. C₃×C₈×D₅	Q₈	semidirect
65. #4683	D₅	semidirect
139. C₈×D₅	Dic₆	semidirect
140. C₈×D₅	Dic₆	semidirect
260. C₃×C₅	#3160	semidirect
292. C₅	#78442	semidirect
303. C₃	#10613647	semidirect

Maximal subgroups

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

subgroup	index
#11752498	2
#11750784	2
C₅×#4683	2
C₃×C₄×C₈×D₅	2
#11748019	2
#11748016	2
#11747672	2
#11747671	2
#11746956	2
#10613647	3
#78442	5