Subgroups — #14285204

Order 1920.

396 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752478	C₂	1	1	yes	maximal, characteristic
2	C₅×#4835	C₂	1	1	yes	maximal, characteristic
3	#11746853	C₂	1	1	yes	maximal, characteristic
4	#11746302	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64702	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745863	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745856	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601614	—	3	1	no	maximal, Hall
9	#91758	C₂²	1	1	yes	characteristic
10	C₄²×C₅×D₃	C₂²	1	1	yes	characteristic
11	C₃×#4015	C₂²	1	1	yes	characteristic
12	#91381	—	2	1	no	—
13	#91369	—	2	1	no	—
14	#91365	C₄	1	2	yes	—
15	#91340	C₄	1	2	yes	—
16	#91338	—	2	1	no	—
17	Dic₃×Dic₁₀	—	2	1	no	—
18	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
20	C₅×(C₂×D₃)⋊C₈	—	2	1	no	—
21	C₅×#815	—	2	1	no	—
22	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₃×C₄⋊#168	C₂²	1	1	yes	characteristic
24	(C₂×D₃)⋊#168	—	2	1	no	—
25	Dic₃×#168	—	2	1	no	—
26	#68427	—	5	1	no	maximal, Hall
27	#65804	—	3	1	no	—
28	C₅×#432	—	3	1	no	—
29	#65214	—	3	1	no	—
30	#64702	—	3	1	no	—
31	#64702	D₃	1	1	yes	characteristic, complemented
32	#64701	—	3	1	no	—
33	#64701	—	3	1	no	—
34	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
37	C₂×C₄×C₅×D₃	D₄	1	1	yes	characteristic
38	C₂×C₃×Dic₁₀	—	2	1	no	—
39	C₂×C₃×C₅×C₈	—	2	1	no	—
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	C₄⋊Dic₁₅	—	2	1	no	—
42	C₄⋊Dic₁₅	—	2	1	no	—
43	C₂×C₃⋊#168	—	2	1	no	—
44	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
46	C₄×C₅×Dic₃	D₄	1	1	yes	characteristic
47	C₂×C₅×C₃⋊C₈	—	2	1	no	—
48	C₃×C₄⋊Dic₅	—	2	1	no	—
49	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
50	C₂×C₃×#168	—	2	1	no	—
51	#6669	—	2	2	no	—
52	#6668	—	2	2	no	—
53	(C₂×D₃)⋊Dic₅	—	2	2	no	—
54	Dic₃×Dic₅	—	2	2	no	—
55	#5728	—	5	1	no	—
56	#4835	D₅	1	1	yes	characteristic
57	#4833	—	5	1	no	—
58	C₃×#332	—	5	1	no	—
59	#4547	—	5	1	no	—
60	#4481	—	5	1	no	—
61	#4481	—	5	1	no	—
62	C₂×C₄²×C₅	—	3	1	no	—
63	#4072	—	6	1	no	—
64	#4070	—	3	2	no	—
65	#4070	—	6	1	no	—
66	#4016	—	6	1	no	—
67	#4015	—	3	1	no	—
68	#4015	C₂×D₃	1	1	yes	characteristic
69	C₄×Dic₁₀	—	3	2	no	—
70	C₄×Dic₁₀	—	6	1	no	—
71	C₅×C₄⋊C₈	—	3	1	no	—
72	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
73	C₅×C₂²⋊C₈	—	6	1	no	—
74	C₅×#97	—	6	1	no	—
75	C₂²⋊#168	—	6	1	no	—
76	C₄⋊#168	—	3	1	no	—
77	C₄⋊#168	C₂×D₃	1	1	yes	characteristic
78	C₄×#168	—	6	1	no	—
79	#1554	—	15	1	no	Sylow
80	C₂²×C₅×D₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×Dic₁₅	—	2	2	no	—
85	Dic₃₀	—	4	2	no	—
86	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₅×Dic₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₄×C₅×D₃	—	2	1	no	—
90	C₄×C₅×D₃	—	2	1	no	—
91	C₄×C₅×D₃	—	2	1	no	—
92	C₄×C₅×D₃	—	2	1	no	—
93	C₄×C₅×D₃	—	4	1	no	—
94	C₂×C₃×Dic₅	—	2	2	no	—
95	C₃×Dic₁₀	—	4	1	no	—
96	C₃×Dic₁₀	—	4	1	no	—
97	C₃×C₅×C₈	—	4	1	no	—
98	C₃⋊#168	—	4	1	no	—
99	C₃×#168	—	4	1	no	—
100	C₅×C₃⋊C₈	—	4	1	no	—
101	C₃×C₄⋊Q₈	—	5	1	no	—
102	#946	—	10	1	no	—
103	Q₈×Dic₃	—	10	1	no	—
104	#897	—	5	2	no	—
105	#892	—	10	1	no	—
106	#890	—	10	1	no	—
107	#887	—	5	2	no	—
108	C₄²×D₃	C₂×D₅	1	1	yes	characteristic
109	#869	—	5	1	no	—
110	C₃×C₄⋊C₈	—	5	1	no	—
111	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
112	(C₂×D₃)⋊C₈	—	2	1	no	—
113	(C₂×D₃)⋊C₈	—	10	1	no	—
114	#815	—	2	1	no	—
115	C₈×Dic₃	—	10	1	no	—
116	C₄⋊(C₃⋊C₈)	—	5	1	no	—
117	C₄⋊(C₃⋊C₈)	C₂×D₅	1	1	yes	characteristic
118	C₂²×C₄×C₅	—	3	1	no	—
119	C₂²×C₄×C₅	—	3	1	no	—
120	C₂²×C₄×C₅	—	3	1	no	—
121	C₂×Dic₁₀	—	2	1	no	—
122	C₂×Dic₁₀	—	3	2	no	—
123	C₂×C₅×C₈	—	2	1	no	—
124	C₂×C₅×C₈	—	6	1	no	—
125	C₄²×C₅	—	3	1	no	—
126	C₄²×C₅	—	3	1	no	—
127	C₄²×C₅	—	3	1	no	—
128	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
129	C₂²⋊Dic₅	—	6	2	no	—
130	C₄⋊Dic₅	—	2	1	no	—
131	C₄⋊Dic₅	—	6	1	no	—
132	C₄⋊Dic₅	—	6	1	no	—
133	C₄⋊Dic₅	C₄×D₃	1	2	yes	—
134	#683	—	6	2	no	—
135	#683	—	6	2	no	—
136	C₄×Dic₅	—	6	2	no	—
137	C₂×#168	—	2	1	no	—
138	C₂×#168	—	6	1	no	—
139	#533	—	15	1	no	—
140	#432	—	3	1	no	—
141	#431	—	15	1	no	—
142	#333	—	15	1	no	—
143	#333	—	15	1	no	—
144	#332	—	5	1	no	—
145	#332	—	15	1	no	—
146	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₅×D₃	—	2	1	no	—
148	C₂×C₅×D₃	—	2	1	no	—
149	C₂×C₅×D₃	—	2	1	no	—
150	C₃×C₄×C₅	—	2	1	no	—
151	C₃×C₄×C₅	#135	1	1	yes	characteristic
152	C₃×C₄×C₅	#135	1	1	yes	characteristic
153	C₃×C₄×C₅	#137	1	1	yes	characteristic
154	C₃×C₄×C₅	#137	1	1	yes	characteristic
155	Dic₁₅	—	4	2	no	—
156	C₃×Dic₅	—	4	2	no	—
157	C₅×Dic₃	—	2	1	no	—
158	C₅×Dic₃	—	2	1	no	—
159	C₅×Dic₃	—	2	1	no	—
160	C₂×C₃×Q₈	—	10	1	no	—
161	C₂×C₄×D₃	C₄×D₅	1	1	yes	characteristic
162	C₂×C₄×D₃	D₂₀	1	1	yes	characteristic
163	C₂×C₄×D₃	C₂²⋊D₅	1	1	yes	characteristic
164	C₂×Dic₆	—	5	2	no	—
165	C₂×C₃×C₈	—	2	1	no	—
166	C₂×C₃×C₈	—	10	1	no	—
167	C₃×C₄⋊C₄	—	5	2	no	—
168	C₃×C₄⋊C₄	—	10	1	no	—
169	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
170	(C₂×D₃)⋊C₄	—	10	2	no	—
171	C₄⋊Dic₃	—	10	1	no	—
172	C₄⋊Dic₃	—	10	1	no	—
173	C₄⋊Dic₃	—	10	2	no	—
174	Dic₃⋊C₄	—	10	2	no	—
175	C₄×Dic₃	—	10	2	no	—
176	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
177	C₄×Dic₃	D₂₀	1	1	yes	characteristic
178	C₄×Dic₃	C₂²⋊D₅	1	1	yes	characteristic
179	C₂×C₃⋊C₈	—	2	1	no	—
180	C₂×C₃⋊C₈	—	10	1	no	—
181	C₂³×C₅	—	3	1	no	—
182	C₂×C₄×C₅	—	3	1	no	—
183	C₂×C₄×C₅	—	3	1	no	—
184	C₂×C₄×C₅	—	3	1	no	—
185	C₂×C₄×C₅	—	6	1	no	—
186	C₂×C₄×C₅	—	6	1	no	—
187	C₂×C₄×C₅	—	6	1	no	—
188	C₂×C₄×C₅	—	6	1	no	—
189	C₂×C₄×C₅	—	12	1	no	—
190	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
191	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
192	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
193	C₂×Dic₅	—	2	2	no	—
194	C₂×Dic₅	—	6	2	no	—
195	Dic₁₀	—	4	1	no	—
196	Dic₁₀	—	4	1	no	—
197	Dic₁₀	—	12	2	no	—
198	C₅×C₈	—	4	1	no	—
199	C₅×C₈	—	12	1	no	—
200	#168	—	4	1	no	—
201	#168	—	12	1	no	—
202	C₄⋊Q₈	—	5	1	no	—
203	C₄⋊Q₈	—	15	1	no	—
204	#125	—	30	1	no	—
205	C₂²⋊Q₈	—	15	2	no	—
206	C₂²⋊Q₈	—	30	1	no	—
207	C₄×Q₈	—	15	2	no	—
208	C₄×Q₈	—	30	1	no	—
209	#117	—	30	1	no	—
210	C₂×C₄²	—	3	1	no	—
211	C₄⋊C₈	—	3	1	no	—
212	C₄⋊C₈	—	5	1	no	—
213	C₄⋊C₈	—	15	1	no	—
214	C₄⋊C₈	D₃×D₅	1	1	yes	characteristic
215	C₂²⋊C₈	—	6	1	no	—
216	C₂²⋊C₈	—	30	1	no	—
217	#97	—	6	1	no	—
218	C₄×C₈	—	30	1	no	—
219	C₂×C₃×C₅	#413	1	1	yes	characteristic
220	C₂×C₃×C₅	#416	1	1	yes	characteristic
221	C₂×C₃×C₅	#419	1	1	yes	characteristic
222	C₅×D₃	—	4	1	no	—
223	C₂²×D₃	#685	1	1	yes	characteristic
224	C₃×Q₈	—	20	1	no	—
225	C₃×Q₈	—	20	1	no	—
226	C₂×C₃×C₄	—	10	2	no	—
227	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
229	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
230	C₂×Dic₃	—	10	2	no	—
231	C₂×Dic₃	#685	1	1	yes	characteristic
232	C₂×Dic₃	#685	1	1	yes	characteristic
233	C₂×Dic₃	#685	1	1	yes	characteristic
234	C₄×D₃	—	2	1	no	—
235	C₄×D₃	—	2	1	no	—
236	C₄×D₃	—	2	1	no	—
237	C₄×D₃	—	2	1	no	—
238	C₄×D₃	—	4	1	no	—
239	Dic₆	—	20	2	no	—
240	C₃×C₈	—	4	1	no	—
241	C₃×C₈	—	20	1	no	—
242	C₃⋊C₈	—	4	1	no	—
243	C₃⋊C₈	—	20	1	no	—
244	C₂²×C₅	—	6	1	no	—
245	C₂²×C₅	—	6	1	no	—
246	C₂²×C₅	—	6	1	no	—
247	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
248	C₄×C₅	—	2	1	no	—
249	C₄×C₅	—	6	1	no	—
250	C₄×C₅	—	6	1	no	—
251	C₄×C₅	—	6	1	no	—
252	C₄×C₅	#912	1	1	yes	characteristic
253	C₄×C₅	#915	1	1	yes	characteristic
254	C₄×C₅	#916	1	1	yes	characteristic
255	C₄×C₅	#918	1	1	yes	characteristic
256	Dic₅	—	4	2	no	—
257	Dic₅	—	12	2	no	—
258	C₂×Q₈	—	10	1	no	—
259	C₂×Q₈	—	15	2	no	—
260	C₂²×C₄	—	3	1	no	—
261	C₂²×C₄	—	3	1	no	—
262	C₂²×C₄	—	3	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	6	1	no	—
265	C₂×C₈	—	10	1	no	—
266	C₂×C₈	—	30	1	no	—
267	C₄⋊C₄	—	5	2	no	—
268	C₄⋊C₄	—	10	1	no	—
269	C₄⋊C₄	—	30	2	no	—
270	C₄⋊C₄	—	30	2	no	—
271	C₄⋊C₄	—	30	1	no	—
272	C₄⋊C₄	—	30	1	no	—
273	C₂²⋊C₄	—	30	2	no	—
274	C₄²	—	3	1	no	—
275	C₄²	—	3	1	no	—
276	C₄²	—	3	1	no	—
277	C₄²	—	30	2	no	—
278	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
279	C₃×C₅	#1554	1	1	yes	characteristic, complemented, Hall
280	C₂²×C₃	C₂×#685	1	1	yes	characteristic
281	C₂×D₃	—	2	1	no	—
282	C₂×D₃	—	2	1	no	—
283	C₂×D₃	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	20	2	no	—
286	C₃×C₄	#4055	1	1	yes	characteristic
287	C₃×C₄	#4055	1	1	yes	characteristic
288	C₃×C₄	#4096	1	1	yes	characteristic
289	C₃×C₄	#4096	1	1	yes	characteristic
290	Dic₃	—	2	1	no	—
291	Dic₃	—	2	1	no	—
292	Dic₃	—	2	1	no	—
293	Dic₃	—	20	2	no	—
294	C₂×C₅	—	12	1	no	—
295	C₂×C₅	#4751	1	1	yes	characteristic
296	C₂×C₅	#4772	1	1	yes	characteristic
297	C₂×C₅	#4802	1	1	yes	characteristic
298	C₂³	—	3	1	no	—
299	Q₈	—	20	1	no	—
300	Q₈	—	20	1	no	—
301	Q₈	—	60	2	no	—
302	C₂×C₄	—	3	1	no	—
303	C₂×C₄	—	3	1	no	—
304	C₂×C₄	—	3	1	no	—
305	C₂×C₄	—	6	1	no	—
306	C₂×C₄	—	6	1	no	—
307	C₂×C₄	—	6	1	no	—
308	C₂×C₄	—	6	1	no	—
309	C₂×C₄	—	10	2	no	—
310	C₂×C₄	—	12	1	no	—
311	C₂×C₄	—	30	2	no	—
312	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
313	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
314	C₂×C₄	D₃×C₂²⋊D₅	1	1	yes	characteristic
315	C₈	—	4	1	no	—
316	C₈	—	12	1	no	—
317	C₈	—	20	1	no	—
318	C₈	—	60	1	no	—
319	C₂×C₃	#65291	1	1	yes	characteristic
320	C₂×C₃	#65409	1	1	yes	characteristic
321	C₂×C₃	#65421	1	1	yes	characteristic
322	D₃	—	4	1	no	—
323	C₅	#68427	1	1	yes	characteristic, complemented, Sylow
324	C₂²	—	6	1	no	—
325	C₂²	—	6	1	no	—
326	C₂²	—	6	1	no	—
327	C₂²	D₃×#685	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	6	1	no	—
330	C₄	—	6	1	no	—
331	C₄	—	6	1	no	—
332	C₄	—	20	2	no	—
333	C₄	—	60	2	no	—
334	C₄	#91263	1	1	yes	characteristic
335	C₄	#91274	1	1	yes	characteristic
336	C₄	#91493	1	1	yes	characteristic
337	C₄	#91517	1	1	yes	characteristic
338	C₃	#10601614	1	1	yes	characteristic, complemented, Sylow
339	C₂	—	12	1	no	—
340	C₂	#11747502	1	1	yes	characteristic
341	C₂	#11748123	1	1	yes	characteristic
342	C₂	#11748235	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₃×#685
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₃×C₅×C₄⋊C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂²×C₅×D₃	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄⋊C₈	D₃×D₅
hypocenter	C₃×C₅	#1554
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×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
4. #11746302	C₂	semidirect
5. C₃×#64702	C₂	semidirect
6. #11745863	C₂	semidirect
7. #11745856	C₂	semidirect
31. #64702	D₃	semidirect
279. C₃×C₅	#1554	semidirect
323. C₅	#68427	semidirect
338. C₃	#10601614	semidirect

Maximal subgroups

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

subgroup	index
#11752478	2
C₅×#4835	2
#11746853	2
#11746302	2
C₃×#64702	2
#11745863	2
#11745856	2
#10601614	3
#68427	5