Subgroups — #14284599

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#65811	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊#168)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4925	C₂	1	1	yes	maximal, characteristic, complemented
4	#11745855	C₂	1	2	yes	maximal, complemented
5	#11745852	C₂	1	2	yes	maximal, complemented
6	#10601565	—	3	1	no	maximal, Hall
7	C₂×C₃×#709	—	2	1	no	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊#168	—	2	1	no	—
10	C₃×#4076	—	2	1	no	—
11	C₃×#4076	C₄	1	2	yes	—
12	C₃×#4021	—	2	1	no	—
13	C₃×#4020	C₂²	1	1	yes	characteristic
14	C₃×C₄×D₂₀	—	2	1	no	—
15	C₃×C₄×D₂₀	C₄	1	2	yes	—
16	C₃×#4015	C₂²	1	1	yes	characteristic
17	C₄⋊(C₃⋊#168)	—	2	1	no	—
18	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	—
19	C₅×#830	—	2	1	no	—
20	C₅×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
21	C₄×C₅×C₃⋊C₈	—	2	1	no	—
22	#71804	—	5	1	no	maximal, Hall
23	#65811	D₃	1	1	yes	characteristic
24	C₅×#431	—	3	1	no	—
25	C₂×C₄⋊#168	—	3	1	no	—
26	#64706	—	3	2	no	—
27	#64705	—	3	2	no	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂×C₃×C₂²⋊D₅	—	2	2	no	—
32	C₃×#709	—	4	2	no	—
33	C₃×#709	—	4	2	no	—
34	C₂×C₃×D₂₀	—	2	1	no	—
35	C₂×C₃×D₂₀	—	2	1	no	—
36	C₂×C₃×D₂₀	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₄×D₅	—	2	2	no	—
38	C₂×C₃×Dic₁₀	—	2	1	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
43	C₂×C₃⋊#168	—	2	2	no	—
44	C₂×C₃⋊#168	—	4	2	no	—
45	C₂×C₅×C₃⋊C₈	—	2	2	no	—
46	C₃×#685	—	2	2	no	—
47	C₃×#685	—	2	2	no	—
48	C₃×C₄⋊Dic₅	—	2	1	no	—
49	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
50	C₃×#532	—	5	1	no	—
51	#4925	D₅	1	1	yes	characteristic, complemented
52	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
53	#4547	—	5	2	no	—
54	#4544	—	5	2	no	—
55	C₂×#709	—	2	1	no	—
56	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
57	#4076	—	2	1	no	—
58	#4076	Dic₃	1	2	yes	—
59	C₂²×#168	—	6	1	no	—
60	#4021	—	2	1	no	—
61	#4020	C₂×D₃	1	1	yes	characteristic
62	C₄×D₂₀	—	2	1	no	—
63	C₄×D₂₀	Dic₃	1	2	yes	—
64	#4015	C₂×D₃	1	1	yes	characteristic
65	C₅×C₄⋊C₈	—	3	2	no	—
66	C₅×C₂²⋊C₈	—	6	1	no	—
67	C₄×C₅×C₈	—	6	1	no	—
68	C₄⋊#168	—	3	2	no	—
69	C₄⋊#168	—	6	1	no	—
70	#1540	—	15	1	no	Sylow
71	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₃×D₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	4	1	no	—
75	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
79	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₃×C₂²⋊D₅	—	4	4	no	—
84	C₂×C₃×Dic₅	—	2	2	no	—
85	C₃×D₂₀	—	4	2	no	—
86	C₃×D₂₀	—	4	1	no	—
87	C₃×D₂₀	—	4	1	no	—
88	C₃×D₂₀	—	4	2	no	—
89	C₃×C₄×D₅	—	4	4	no	—
90	C₃×Dic₁₀	—	4	1	no	—
91	C₃×Dic₁₀	—	4	1	no	—
92	C₃⋊#168	—	4	2	no	—
93	C₅×C₃⋊C₈	—	4	2	no	—
94	C₂×C₃×#41	—	10	1	no	—
95	C₃×C₄⋊Q₈	—	5	1	no	—
96	C₃×#127	—	5	1	no	—
97	C₃×#124	—	10	1	no	—
98	C₃×#121	—	5	2	no	—
99	C₃×#121	—	10	1	no	—
100	C₃×C₄×D₄	—	5	2	no	—
101	C₃×C₄×D₄	—	10	1	no	—
102	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
103	C₂²×C₃⋊C₈	—	10	1	no	—
104	#830	—	2	1	no	—
105	C₄⋊(C₃⋊C₈)	—	5	2	no	—
106	C₄⋊(C₃⋊C₈)	—	10	1	no	—
107	C₄⋊(C₃⋊C₈)	C₂×D₅	1	2	yes	—
108	C₄×C₃⋊C₈	—	2	1	no	—
109	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
110	C₂²×C₄×C₅	#67	1	1	yes	characteristic
111	C₂²×C₄×C₅	#67	1	1	yes	characteristic
112	C₂×C₂²⋊D₅	—	2	2	no	—
113	#709	—	4	2	no	—
114	#709	—	4	2	no	—
115	C₂×D₂₀	—	2	1	no	—
116	C₂×D₂₀	—	2	1	no	—
117	C₂×D₂₀	C₂×Dic₃	1	2	yes	—
118	C₂×C₄×D₅	—	2	2	no	—
119	C₂×Dic₁₀	—	2	1	no	—
120	C₂×C₅×C₈	—	6	2	no	—
121	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
122	C₄²×C₅	#67	1	1	yes	characteristic
123	C₄²×C₅	#67	1	1	yes	characteristic
124	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
125	#685	—	2	2	no	—
126	#685	—	2	2	no	—
127	C₄⋊Dic₅	—	2	1	no	—
128	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
129	C₂×#168	—	6	2	no	—
130	C₂×#168	—	12	2	no	—
131	#532	—	5	1	no	—
132	#431	—	3	1	no	—
133	C₂×C₄⋊C₈	—	15	1	no	—
134	#332	—	15	2	no	—
135	#331	—	15	2	no	—
136	C₂²×C₃×C₅	—	2	1	no	—
137	C₂²×C₃×C₅	—	2	1	no	—
138	C₂²×C₃×C₅	#102	1	2	yes	—
139	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
140	C₂×C₃×D₅	—	4	2	no	—
141	C₂×C₃×D₅	—	4	2	no	—
142	C₂×C₃×D₅	—	4	2	no	—
143	C₃×C₄×C₅	—	2	1	no	—
144	C₃×C₄×C₅	—	2	1	no	—
145	C₃×C₄×C₅	—	2	1	no	—
146	C₃×C₄×C₅	#102	1	2	yes	—
147	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
148	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
149	C₃×C₄×C₅	#135	1	2	yes	—
150	C₃×Dic₅	—	4	2	no	—
151	C₃×#41	—	20	2	no	—
152	C₃×#41	—	20	2	no	—
153	C₂×C₃×Q₈	—	10	1	no	—
154	C₂×C₃×D₄	—	5	2	no	—
155	C₂×C₃×D₄	—	10	2	no	—
156	C₂×C₃×D₄	—	10	1	no	—
157	C₂×C₃×D₄	—	10	1	no	—
158	C₂²×C₃×C₄	—	10	2	no	—
159	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
160	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
161	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
162	C₃×C₄⋊C₄	—	5	2	no	—
163	C₃×C₄⋊C₄	—	10	1	no	—
164	C₃×C₂²⋊C₄	—	10	2	no	—
165	C₃×C₂²⋊C₄	—	10	2	no	—
166	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
167	C₃×C₄²	D₂₀	1	1	yes	characteristic
168	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
169	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
170	C₂×C₃⋊C₈	—	2	2	no	—
171	C₂×C₃⋊C₈	—	10	2	no	—
172	C₂×C₃⋊C₈	—	20	2	no	—
173	C₂³×C₅	#217	1	1	yes	characteristic
174	C₂²×D₅	—	2	2	no	—
175	C₂×C₄×C₅	—	2	2	no	—
176	C₂×C₄×C₅	—	4	1	no	—
177	C₂×C₄×C₅	#213	1	2	yes	—
178	C₂×C₄×C₅	#214	1	2	yes	—
179	C₂×C₄×C₅	#217	1	1	yes	characteristic
180	C₂×C₄×C₅	#217	1	1	yes	characteristic
181	C₂×C₄×C₅	#217	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
183	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
184	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
185	C₂²⋊D₅	—	4	4	no	—
186	C₂×Dic₅	—	2	2	no	—
187	D₂₀	—	4	2	no	—
188	D₂₀	—	4	1	no	—
189	D₂₀	—	4	1	no	—
190	D₂₀	—	4	2	no	—
191	C₄×D₅	—	4	4	no	—
192	Dic₁₀	—	4	1	no	—
193	Dic₁₀	—	4	1	no	—
194	C₅×C₈	—	12	2	no	—
195	#168	—	12	2	no	—
196	C₂×#41	—	10	1	no	—
197	C₂²×C₈	—	30	1	no	—
198	C₄⋊Q₈	—	5	1	no	—
199	#127	—	5	1	no	—
200	#124	—	10	1	no	—
201	#121	—	5	2	no	—
202	#121	—	10	1	no	—
203	C₄×D₄	—	5	2	no	—
204	C₄×D₄	—	10	1	no	—
205	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
206	C₄⋊C₈	—	3	2	no	—
207	C₄⋊C₈	—	15	2	no	—
208	C₄⋊C₈	—	30	1	no	—
209	C₂²⋊C₈	—	6	1	no	—
210	C₄×C₈	—	6	1	no	—
211	C₂×C₃×C₅	—	2	2	no	—
212	C₂×C₃×C₅	#413	1	1	yes	characteristic
213	C₂×C₃×C₅	C₂×#102	1	1	yes	characteristic
214	C₂×C₃×C₅	#416	1	1	yes	characteristic
215	C₃×D₅	—	8	2	no	—
216	C₂³×C₃	—	10	2	no	—
217	C₂³×C₃	#685	1	1	yes	characteristic
218	C₃×Q₈	—	20	1	no	—
219	C₃×Q₈	—	20	1	no	—
220	C₃×D₄	—	20	2	no	—
221	C₃×D₄	—	20	4	no	—
222	C₃×D₄	—	20	2	no	—
223	C₃×D₄	—	20	1	no	—
224	C₃×D₄	—	20	1	no	—
225	C₂×C₃×C₄	—	2	2	no	—
226	C₂×C₃×C₄	—	4	1	no	—
227	C₂×C₃×C₄	—	10	2	no	—
228	C₂×C₃×C₄	—	20	4	no	—
229	C₂×C₃×C₄	#677	1	2	yes	—
230	C₂×C₃×C₄	D₄₀	1	2	yes	—
231	C₂×C₃×C₄	#685	1	1	yes	characteristic
232	C₂×C₃×C₄	#685	1	1	yes	characteristic
233	C₂×C₃×C₄	#685	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
237	C₃⋊C₈	—	4	2	no	—
238	C₃⋊C₈	—	20	2	no	—
239	C₂²×C₅	—	2	1	no	—
240	C₂²×C₅	—	2	1	no	—
241	C₂²×C₅	#832	1	2	yes	—
242	C₂²×C₅	C₂×#217	1	1	yes	characteristic
243	C₂×D₅	—	4	2	no	—
244	C₂×D₅	—	4	2	no	—
245	C₂×D₅	—	4	2	no	—
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	—	2	1	no	—
248	C₄×C₅	—	2	1	no	—
249	C₄×C₅	#832	1	2	yes	—
250	C₄×C₅	C₂×#213	1	1	yes	characteristic
251	C₄×C₅	C₂×#214	1	1	yes	characteristic
252	C₄×C₅	#950	1	2	yes	—
253	Dic₅	—	4	2	no	—
254	#41	—	20	2	no	—
255	#41	—	20	2	no	—
256	C₂×Q₈	—	10	1	no	—
257	C₂×D₄	—	5	2	no	—
258	C₂×D₄	—	10	2	no	—
259	C₂×D₄	—	10	1	no	—
260	C₂×D₄	—	10	1	no	—
261	C₂²×C₄	—	10	2	no	—
262	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
263	C₂²×C₄	#1201	1	1	yes	characteristic
264	C₂²×C₄	#1202	1	1	yes	characteristic
265	C₂×C₈	—	6	2	no	—
266	C₂×C₈	—	30	2	no	—
267	C₂×C₈	—	60	2	no	—
268	C₄⋊C₄	—	5	2	no	—
269	C₄⋊C₄	—	10	1	no	—
270	C₂²⋊C₄	—	10	2	no	—
271	C₂²⋊C₄	—	10	2	no	—
272	C₄²	D₅×Dic₃	1	1	yes	characteristic
273	C₄²	#1201	1	1	yes	characteristic
274	C₄²	#1202	1	1	yes	characteristic
275	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
276	C₃×C₅	#1540	1	1	yes	characteristic, complemented, Hall
277	C₂²×C₃	—	2	1	no	—
278	C₂²×C₃	—	2	1	no	—
279	C₂²×C₃	—	20	2	no	—
280	C₂²×C₃	—	20	2	no	—
281	C₂²×C₃	—	20	2	no	—
282	C₂²×C₃	#3953	1	2	yes	—
283	C₂²×C₃	C₂×#685	1	1	yes	characteristic
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	2	1	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	20	2	no	—
288	C₃×C₄	#3953	1	2	yes	—
289	C₃×C₄	C₂×#677	1	1	yes	characteristic
290	C₃×C₄	C₂×D₄₀	1	1	yes	characteristic
291	C₃×C₄	#4096	1	2	yes	—
292	C₂×C₅	—	2	2	no	—
293	C₂×C₅	C₂×#832	1	1	yes	characteristic
294	C₂×C₅	#5233	1	1	yes	characteristic
295	C₂×C₅	#5237	1	1	yes	characteristic
296	D₅	—	8	2	no	—
297	C₂³	—	10	2	no	—
298	C₂³	#6665	1	1	yes	characteristic
299	Q₈	—	20	1	no	—
300	Q₈	—	20	1	no	—
301	D₄	—	20	2	no	—
302	D₄	—	20	4	no	—
303	D₄	—	20	2	no	—
304	D₄	—	20	1	no	—
305	D₄	—	20	1	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	10	2	no	—
309	C₂×C₄	—	20	4	no	—
310	C₂×C₄	#6653	1	2	yes	—
311	C₂×C₄	#6657	1	2	yes	—
312	C₂×C₄	#6665	1	1	yes	characteristic
313	C₂×C₄	#6665	1	1	yes	characteristic
314	C₂×C₄	#6665	1	1	yes	characteristic
315	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
316	C₂×C₄	C₂×#1201	1	1	yes	characteristic
317	C₂×C₄	C₂×#1202	1	1	yes	characteristic
318	C₈	—	12	2	no	—
319	C₈	—	60	2	no	—
320	C₂×C₃	—	2	2	no	—
321	C₂×C₃	—	40	2	no	—
322	C₂×C₃	#65291	1	1	yes	characteristic
323	C₂×C₃	C₂×#3953	1	1	yes	characteristic
324	C₂×C₃	#65421	1	1	yes	characteristic
325	C₅	#71804	1	1	yes	characteristic, complemented, Sylow
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	20	2	no	—
329	C₂²	—	20	2	no	—
330	C₂²	—	20	2	no	—
331	C₂²	#90965	1	2	yes	—
332	C₂²	C₂×#6665	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	20	2	no	—
337	C₄	#90965	1	2	yes	—
338	C₄	#91297	1	2	yes	—
339	C₄	C₂×#6653	1	1	yes	characteristic
340	C₄	C₂×#6657	1	1	yes	characteristic
341	C₃	#10601565	1	1	yes	characteristic, complemented, Sylow
342	C₂	—	2	2	no	—
343	C₂	—	40	2	no	—
344	C₂	#11748353	1	1	yes	characteristic
345	C₂	C₂×#90965	1	1	yes	characteristic
346	C₂	#11748364	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#6665
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₃×#65811	C₂
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1540
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
2. C₂×C₄⋊(C₃⋊#168)	C₂	semidirect
3. C₅×#4925	C₂	semidirect
4. #11745855	C₂	semidirect
5. #11745852	C₂	semidirect
51. #4925	D₅	semidirect
276. C₃×C₅	#1540	semidirect
325. C₅	#71804	semidirect
341. C₃	#10601565	semidirect

Maximal subgroups

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

subgroup	index
C₃×#65811	2
C₂×C₄⋊(C₃⋊#168)	2
C₅×#4925	2
#11745855	2
#11745852	2
#10601565	3
#71804	5