Subgroups — #14297491

Order 1920.

624 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#926	C₂	1	1	yes	maximal, characteristic
2	C₂×#91114	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₂²⋊#168	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₃⋊#168	C₂	1	2	yes	maximal, complemented
5	#11745967	C₂	1	2	yes	maximal, complemented
6	#10602376	—	3	1	no	maximal, Hall
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊#168	C₂²	1	2	yes	—
9	C₂×C₅×#217	C₄	1	2	yes	—
10	C₅×#926	—	2	4	no	—
11	C₂×C₅×C₄⋊Dic₃	C₄	1	2	yes	—
12	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃×#168	C₂²	1	2	yes	—
15	#91114	—	2	2	no	—
16	C₃×C₂²⋊#168	—	2	2	no	—
17	Dic₃⋊#168	—	2	4	no	—
18	#69909	—	5	1	no	maximal, Hall
19	C₂×C₅×#117	—	3	1	no	—
20	C₂×C₂²⋊#168	—	3	1	no	—
21	C₂×C₂²⋊#168	D₃	1	1	yes	characteristic
22	C₂×C₄⋊#168	—	3	2	no	—
23	#64744	—	3	2	no	—
24	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	Q₈	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
32	C₂²×C₅×Dic₃	C₂×C₄	1	2	yes	—
33	C₂×C₃⋊#168	—	2	2	no	—
34	C₂×C₃⋊#168	—	2	2	no	—
35	C₂×C₃⋊#168	—	2	2	no	—
36	C₅×#217	—	2	4	no	—
37	C₅×C₄⋊Dic₃	—	2	4	no	—
38	C₄×C₅×Dic₃	—	2	2	no	—
39	C₄×C₅×Dic₃	D₄	1	4	yes	—
40	C₂×C₃×#168	—	2	2	no	—
41	C₂×C₃×#168	—	2	2	no	—
42	C₂×C₃×#168	—	2	2	no	—
43	C₂×#926	D₅	1	1	yes	characteristic
44	C₂×C₃×C₂²⋊C₈	—	5	1	no	—
45	C₂×#830	—	5	1	no	—
46	C₂×Dic₃⋊C₈	—	5	2	no	—
47	#4550	—	5	2	no	—
48	C₂³×C₄×C₅	C₂×D₃	1	1	yes	characteristic
49	C₅×#117	—	6	4	no	—
50	C₂×C₅×C₄⋊C₄	—	3	2	no	—
51	C₂×C₅×C₂²⋊C₄	—	3	2	no	—
52	C₂×C₄²×C₅	—	3	1	no	—
53	C₂×C₄²×C₅	—	3	1	no	—
54	C₂²×#168	—	3	2	no	—
55	C₂²×#168	C₂×D₃	1	2	yes	—
56	C₂²⋊#168	—	2	2	no	—
57	C₂²⋊#168	—	6	2	no	—
58	C₄⋊#168	—	6	4	no	—
59	#1798	—	15	1	no	Sylow
60	C₂³×C₃×C₅	—	2	1	no	—
61	C₂³×C₃×C₅	—	2	1	no	—
62	C₂³×C₃×C₅	—	2	2	no	—
63	C₂³×C₃×C₅	—	2	2	no	—
64	C₂³×C₃×C₅	C₄⋊C₄	1	2	yes	—
65	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	—	2	1	no	—
67	C₂×C₃×C₄×C₅	—	2	1	no	—
68	C₂×C₃×C₄×C₅	—	2	4	no	—
69	C₂×C₃×C₄×C₅	—	2	4	no	—
70	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	#41	1	2	yes	—
75	C₂×C₅×Dic₃	—	2	2	no	—
76	C₂×C₅×Dic₃	—	2	2	no	—
77	C₂×C₅×Dic₃	—	2	2	no	—
78	C₂×C₅×Dic₃	—	2	2	no	—
79	C₂×C₅×Dic₃	—	2	2	no	—
80	C₂×C₅×Dic₃	C₂²⋊C₄	1	4	yes	—
81	C₃⋊#168	—	4	2	no	—
82	C₃×#168	—	4	2	no	—
83	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
84	C₂²×C₃×C₈	—	5	2	no	—
85	C₂×#217	Dic₅	1	2	yes	—
86	#926	—	2	4	no	—
87	C₂×C₄⋊Dic₃	Dic₅	1	2	yes	—
88	C₂×C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
89	C₂×C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
90	C₂²×C₃⋊C₈	—	5	2	no	—
91	C₃×C₂²⋊C₈	—	10	2	no	—
92	#830	—	10	2	no	—
93	Dic₃⋊C₈	—	10	4	no	—
94	C₂⁴×C₅	C₄×D₃	1	1	yes	characteristic
95	C₂²×C₄×C₅	—	2	2	no	—
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	—	3	2	no	—
98	C₂²×C₄×C₅	—	3	2	no	—
99	C₂²×C₄×C₅	Dic₆	1	2	yes	—
100	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
101	C₂²×C₄×C₅	#67	1	2	yes	—
102	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
103	C₅×C₄⋊C₄	—	6	4	no	—
104	C₅×C₂²⋊C₄	—	6	4	no	—
105	C₄²×C₅	—	3	4	no	—
106	C₄²×C₅	—	6	2	no	—
107	C₂×#168	—	2	2	no	—
108	C₂×#168	—	2	2	no	—
109	C₂×#168	—	2	2	no	—
110	C₂×#168	—	6	2	no	—
111	C₂×#168	—	6	2	no	—
112	C₂×#168	—	6	2	no	—
113	C₂×#117	—	3	1	no	—
114	C₂×C₄⋊C₈	—	15	2	no	—
115	C₂×C₂²⋊C₈	—	5	1	no	—
116	C₂×C₂²⋊C₈	—	15	1	no	—
117	#336	—	15	2	no	—
118	C₂²×C₃×C₅	—	2	2	no	—
119	C₂²×C₃×C₅	—	2	2	no	—
120	C₂²×C₃×C₅	—	2	2	no	—
121	C₂²×C₃×C₅	—	4	2	no	—
122	C₂²×C₃×C₅	—	4	2	no	—
123	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×C₃×C₅	#117	1	1	yes	characteristic
126	C₂²×C₃×C₅	#131	1	2	yes	—
127	C₂²×C₃×C₅	#131	1	2	yes	—
128	C₃×C₄×C₅	—	2	2	no	—
129	C₃×C₄×C₅	#121	1	2	yes	—
130	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
131	C₅×Dic₃	—	2	4	no	—
132	C₅×Dic₃	—	4	2	no	—
133	C₂⁴×C₃	C₂×Dic₅	1	1	yes	characteristic
134	C₂²×C₃×C₄	—	2	2	no	—
135	C₂²×C₃×C₄	—	2	2	no	—
136	C₂²×C₃×C₄	Dic₁₀	1	2	yes	—
137	C₂²×C₃×C₄	D₂₀	1	2	yes	—
138	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
139	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
140	C₂²×Dic₃	C₂×Dic₅	1	2	yes	—
141	C₂²×Dic₃	C₂×Dic₅	1	2	yes	—
142	C₂×C₃×C₈	—	10	2	no	—
143	C₂×C₃×C₈	—	10	2	no	—
144	C₂×C₃×C₈	—	10	2	no	—
145	#217	—	2	4	no	—
146	C₄⋊Dic₃	—	2	4	no	—
147	C₄×Dic₃	—	2	2	no	—
148	C₄×Dic₃	C₂²⋊D₅	1	4	yes	—
149	C₂×C₃⋊C₈	—	10	2	no	—
150	C₂×C₃⋊C₈	—	10	2	no	—
151	C₂×C₃⋊C₈	—	10	2	no	—
152	C₂³×C₅	—	2	1	no	—
153	C₂³×C₅	—	2	1	no	—
154	C₂³×C₅	—	2	2	no	—
155	C₂³×C₅	—	2	2	no	—
156	C₂³×C₅	Dic₃⋊C₄	1	2	yes	—
157	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
158	C₂×C₄×C₅	—	2	1	no	—
159	C₂×C₄×C₅	—	2	1	no	—
160	C₂×C₄×C₅	—	2	4	no	—
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	—	3	4	no	—
163	C₂×C₄×C₅	—	6	2	no	—
164	C₂×C₄×C₅	—	6	2	no	—
165	C₂×C₄×C₅	—	6	2	no	—
166	C₂×C₄×C₅	—	6	2	no	—
167	C₂×C₄×C₅	—	6	2	no	—
168	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
169	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
170	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
171	C₂×C₄×C₅	#237	1	2	yes	—
172	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
173	#168	—	4	2	no	—
174	#168	—	12	2	no	—
175	C₂³×C₄	D₃×D₅	1	1	yes	characteristic
176	C₂²×C₈	—	5	2	no	—
177	C₂²×C₈	—	15	2	no	—
178	#117	—	6	4	no	—
179	C₂×C₄⋊C₄	—	3	2	no	—
180	C₂×C₂²⋊C₄	—	3	2	no	—
181	C₂×C₄²	—	3	1	no	—
182	C₂×C₄²	—	3	1	no	—
183	C₄⋊C₈	—	30	4	no	—
184	C₂²⋊C₈	—	10	2	no	—
185	C₂²⋊C₈	—	30	2	no	—
186	C₂×C₃×C₅	—	4	2	no	—
187	C₂×C₃×C₅	#380	1	1	yes	characteristic
188	C₂×C₃×C₅	#408	1	2	yes	—
189	C₂×C₃×C₅	#424	1	1	yes	characteristic
190	C₂×C₃×C₅	#424	1	1	yes	characteristic
191	C₂×C₃×C₅	#433	1	2	yes	—
192	C₂³×C₃	—	2	1	no	—
193	C₂³×C₃	—	2	1	no	—
194	C₂³×C₃	—	2	2	no	—
195	C₂³×C₃	—	2	2	no	—
196	C₂³×C₃	C₄⋊Dic₅	1	2	yes	—
197	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	C₄⋊Dic₅	1	2	yes	—
203	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
205	C₂×C₃×C₄	#709	1	2	yes	—
206	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
207	C₂×Dic₃	—	2	2	no	—
208	C₂×Dic₃	—	2	2	no	—
209	C₂×Dic₃	—	2	2	no	—
210	C₂×Dic₃	—	2	2	no	—
211	C₂×Dic₃	—	2	2	no	—
212	C₂×Dic₃	C₂²⋊Dic₅	1	4	yes	—
213	C₃×C₈	—	20	2	no	—
214	C₃⋊C₈	—	20	2	no	—
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	—	2	2	no	—
218	C₂²×C₅	—	4	2	no	—
219	C₂²×C₅	—	4	2	no	—
220	C₂²×C₅	#877	1	1	yes	characteristic
221	C₂²×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
222	C₂²×C₅	#901	1	2	yes	—
223	C₂²×C₅	#907	1	2	yes	—
224	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
225	C₄×C₅	—	2	2	no	—
226	C₄×C₅	—	6	4	no	—
227	C₄×C₅	—	12	2	no	—
228	C₄×C₅	#878	1	2	yes	—
229	C₄×C₅	#939	1	2	yes	—
230	C₂⁴	D₃×Dic₅	1	1	yes	characteristic
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	3	2	no	—
234	C₂²×C₄	—	3	2	no	—
235	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
236	C₂²×C₄	#1202	1	2	yes	—
237	C₂²×C₄	#1204	1	2	yes	—
238	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
239	C₂×C₈	—	10	2	no	—
240	C₂×C₈	—	10	2	no	—
241	C₂×C₈	—	10	2	no	—
242	C₂×C₈	—	30	2	no	—
243	C₂×C₈	—	30	2	no	—
244	C₂×C₈	—	30	2	no	—
245	C₄⋊C₄	—	6	4	no	—
246	C₂²⋊C₄	—	6	4	no	—
247	C₄²	—	3	4	no	—
248	C₄²	—	6	2	no	—
249	C₃×C₅	#1798	1	1	yes	characteristic, complemented, Hall
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	2	2	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	4	2	no	—
254	C₂²×C₃	—	4	2	no	—
255	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
256	C₂²×C₃	#4072	1	1	yes	characteristic
257	C₂²×C₃	#4094	1	2	yes	—
258	C₂²×C₃	#4094	1	2	yes	—
259	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
260	C₃×C₄	—	2	2	no	—
261	C₃×C₄	#4070	1	2	yes	—
262	C₃×C₄	#4076	1	2	yes	—
263	Dic₃	—	2	4	no	—
264	Dic₃	—	4	2	no	—
265	C₂×C₅	—	4	2	no	—
266	C₂×C₅	#4720	1	2	yes	—
267	C₂×C₅	#4727	1	2	yes	—
268	C₂×C₅	#4942	1	1	yes	characteristic
269	C₂×C₅	#5102	1	1	yes	characteristic
270	C₂×C₅	#5119	1	1	yes	characteristic
271	C₂³	—	2	1	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	1	no	—
275	C₂³	Dic₃⋊Dic₅	1	2	yes	—
276	C₂³	C₂×D₃×Dic₅	1	1	yes	characteristic
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	3	4	no	—
282	C₂×C₄	—	6	2	no	—
283	C₂×C₄	—	6	2	no	—
284	C₂×C₄	—	6	2	no	—
285	C₂×C₄	—	6	2	no	—
286	C₂×C₄	—	6	2	no	—
287	C₂×C₄	Dic₃⋊Dic₅	1	2	yes	—
288	C₂×C₄	#6779	1	2	yes	—
289	C₂×C₄	C₂×#1202	1	1	yes	characteristic
290	C₂×C₄	C₂×#1204	1	1	yes	characteristic
291	C₂×C₄	D₃×C₂²⋊D₅	1	2	yes	—
292	C₈	—	20	2	no	—
293	C₈	—	60	2	no	—
294	C₂×C₃	—	4	2	no	—
295	C₂×C₃	#65288	1	1	yes	characteristic
296	C₂×C₃	#65288	1	1	yes	characteristic
297	C₂×C₃	#65289	1	2	yes	—
298	C₂×C₃	#65501	1	1	yes	characteristic
299	C₂×C₃	#65523	1	2	yes	—
300	C₅	#69909	1	1	yes	characteristic, complemented, Sylow
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	#91286	1	2	yes	—
307	C₂²	#91288	1	2	yes	—
308	C₂²	#91529	1	1	yes	characteristic
309	C₂²	C₂×Dic₃⋊Dic₅	1	1	yes	characteristic
310	C₂²	D₃×C₂²⋊Dic₅	1	1	yes	characteristic
311	C₄	—	2	2	no	—
312	C₄	—	6	4	no	—
313	C₄	—	12	2	no	—
314	C₄	#91569	1	2	yes	—
315	C₄	#91575	1	2	yes	—
316	C₃	#10602376	1	1	yes	characteristic, complemented, Sylow
317	C₂	—	4	2	no	—
318	C₂	#11748330	1	2	yes	—
319	C₂	#11748439	1	1	yes	characteristic
320	C₂	#11748441	1	1	yes	characteristic
321	C₂	#11748490	1	2	yes	—
322	C₂	#11749513	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃×D₅
commutator, socle	C₂³×C₃×C₅	C₂²×C₄
Fitting	C₂³×C₃×C₄×C₅	C₂²
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂³×C₄	D₃×D₅
hypocenter	C₃×C₅	#1798
LCS	G ▷ C₂³×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
4. C₂×Dic₃⋊#168	C₂	semidirect
5. #11745967	C₂	semidirect
249. C₃×C₅	#1798	semidirect
300. C₅	#69909	semidirect
316. C₃	#10602376	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#926	2
C₂×#91114	2
C₂×C₃×C₂²⋊#168	2
C₂×Dic₃⋊#168	2
#11745967	2
#10602376	3
#69909	5