Subgroups — #14305010

Order 1920.

496 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#803	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
4	C₂×#90983	C₂	1	2	yes	maximal
5	#11745968	C₂	1	2	yes	maximal
6	#10600629	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₅	C₂²	1	2	yes	—
10	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₃×#168	C₂²	1	2	yes	—
13	#91115	C₄	1	1	yes	characteristic
14	#91115	C₄	1	1	yes	characteristic
15	C₅×#803	—	2	2	no	—
16	C₃×C₄⋊#168	—	2	2	no	—
17	#90983	—	2	4	no	—
18	#70469	—	5	1	no	maximal, Hall
19	C₂×C₅×#97	—	3	1	no	—
20	C₂×#3946	—	3	2	no	—
21	C₄×C₄⋊Dic₅	—	3	1	no	—
22	C₂×C₄⋊#168	D₃	1	1	yes	characteristic
23	#64770	—	3	2	no	—
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₂²×Dic₁₅	C₂×C₄	1	2	yes	—
28	C₂²×Dic₁₅	C₂×C₄	1	2	yes	—
29	C₃×C₄²×C₅	D₄	1	2	yes	—
30	C₃×C₄²×C₅	Q₈	1	2	yes	—
31	C₄⋊Dic₁₅	—	2	4	no	—
32	C₄×Dic₁₅	—	2	2	no	—
33	C₄×Dic₁₅	—	2	2	no	—
34	C₂×C₅×C₃⋊C₈	—	2	1	no	—
35	C₂×C₅×C₃⋊C₈	—	2	1	no	—
36	C₂×C₅×C₃⋊C₈	—	2	1	no	—
37	C₂×C₅×C₃⋊C₈	—	2	1	no	—
38	C₂×C₅×C₃⋊C₈	D₄	1	2	yes	—
39	C₂×C₅×C₃⋊C₈	Q₈	1	2	yes	—
40	C₂×C₃×#168	—	2	2	no	—
41	C₂×C₃×#168	—	2	2	no	—
42	C₂×C₃×#168	—	2	2	no	—
43	C₂×C₃×C₄⋊C₈	—	5	1	no	—
44	C₂×Dic₃⋊C₈	—	5	2	no	—
45	C₄×C₄⋊Dic₃	—	5	1	no	—
46	C₂×#803	D₅	1	1	yes	characteristic
47	#4550	—	5	2	no	—
48	C₂²×C₅×C₈	—	3	1	no	—
49	C₂²×C₅×C₈	—	3	1	no	—
50	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
51	C₂×C₄⋊Dic₅	—	3	2	no	—
52	C₂×C₄×Dic₅	—	3	2	no	—
53	C₂²×#168	C₂×D₃	1	2	yes	—
54	C₅×#97	—	6	2	no	—
55	#3963	—	3	1	no	—
56	#3963	—	3	1	no	—
57	#3946	—	6	4	no	—
58	C₄⋊#168	—	2	2	no	—
59	#1940	—	15	1	no	Sylow
60	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
65	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
71	C₂×Dic₁₅	—	2	2	no	—
72	C₂×Dic₁₅	—	2	2	no	—
73	C₂×Dic₁₅	—	2	2	no	—
74	C₂×Dic₁₅	—	2	2	no	—
75	C₂×Dic₁₅	—	2	2	no	—
76	C₂×Dic₁₅	M₁₆	1	4	yes	—
77	C₃×#168	—	4	2	no	—
78	C₅×C₃⋊C₈	—	2	2	no	—
79	C₅×C₃⋊C₈	—	2	2	no	—
80	C₂²×C₃×C₈	—	5	2	no	—
81	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
82	C₂×C₄⋊Dic₃	—	5	2	no	—
83	C₂×C₄×Dic₃	—	5	2	no	—
84	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
85	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
86	C₃×C₄⋊C₈	—	10	2	no	—
87	C₃⋊#95	—	5	1	no	—
88	C₃⋊#95	—	5	1	no	—
89	Dic₃⋊C₈	—	10	4	no	—
90	#803	—	2	2	no	—
91	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
94	C₂²×Dic₅	—	3	2	no	—
95	C₂²×Dic₅	—	3	2	no	—
96	C₂×C₅×C₈	—	3	2	no	—
97	C₂×C₅×C₈	—	3	2	no	—
98	C₂×C₅×C₈	—	6	1	no	—
99	C₂×C₅×C₈	—	6	1	no	—
100	C₂×C₅×C₈	—	6	1	no	—
101	C₂×C₅×C₈	—	6	1	no	—
102	C₄²×C₅	Dic₆	1	2	yes	—
103	C₄²×C₅	#67	1	2	yes	—
104	C₄⋊Dic₅	—	6	4	no	—
105	C₄×Dic₅	—	6	2	no	—
106	C₄×Dic₅	—	6	2	no	—
107	C₂×#168	—	2	2	no	—
108	C₂×#168	—	2	2	no	—
109	C₂×#168	—	2	2	no	—
110	C₂×C₄⋊C₈	—	5	1	no	—
111	C₂×C₄⋊C₈	—	15	2	no	—
112	C₂×#97	—	3	1	no	—
113	C₄×C₄⋊C₄	—	15	1	no	—
114	#336	—	15	2	no	—
115	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
117	C₂²×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
118	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
119	C₂²×C₃×C₅	#131	1	2	yes	—
120	C₃×C₄×C₅	—	2	2	no	—
121	C₃×C₄×C₅	—	2	2	no	—
122	C₃×C₄×C₅	#121	1	1	yes	characteristic
123	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
124	C₃×C₄×C₅	#125	1	1	yes	characteristic
125	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
126	Dic₁₅	—	2	4	no	—
127	Dic₁₅	—	4	2	no	—
128	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
129	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
131	C₂²×Dic₃	—	5	2	no	—
132	C₂²×Dic₃	—	5	2	no	—
133	C₂×C₃×C₈	—	10	2	no	—
134	C₂×C₃×C₈	—	10	2	no	—
135	C₂×C₃×C₈	—	10	2	no	—
136	C₃×C₄²	Dic₁₀	1	2	yes	—
137	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
138	C₄⋊Dic₃	—	10	4	no	—
139	C₄×Dic₃	—	10	2	no	—
140	C₄×Dic₃	—	10	2	no	—
141	C₂×C₃⋊C₈	—	2	1	no	—
142	C₂×C₃⋊C₈	—	2	1	no	—
143	C₂×C₃⋊C₈	—	2	1	no	—
144	C₂×C₃⋊C₈	—	2	1	no	—
145	C₂×C₃⋊C₈	Dic₁₀	1	2	yes	—
146	C₂×C₃⋊C₈	D₂₀	1	2	yes	—
147	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
151	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
152	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
153	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅	#237	1	1	yes	characteristic
155	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
156	C₂×C₄×C₅	#239	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
158	C₂×Dic₅	—	3	4	no	—
159	C₂×Dic₅	—	6	2	no	—
160	C₂×Dic₅	—	6	2	no	—
161	C₂×Dic₅	—	6	2	no	—
162	C₂×Dic₅	—	6	2	no	—
163	C₂×Dic₅	—	6	2	no	—
164	C₅×C₈	—	6	2	no	—
165	C₅×C₈	—	6	2	no	—
166	#168	—	4	2	no	—
167	C₂²×C₈	—	3	1	no	—
168	C₂²×C₈	—	3	1	no	—
169	C₂²×C₈	—	5	2	no	—
170	C₂×C₄⋊C₄	—	15	2	no	—
171	C₂×C₄²	—	15	2	no	—
172	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
173	C₄⋊C₈	—	10	2	no	—
174	C₄⋊C₈	—	30	4	no	—
175	#97	—	6	2	no	—
176	#95	—	15	1	no	—
177	#95	—	15	1	no	—
178	C₂×C₃×C₅	#389	1	1	yes	characteristic
179	C₂×C₃×C₅	#423	1	1	yes	characteristic
180	C₂×C₃×C₅	#424	1	1	yes	characteristic
181	C₂×C₃×C₅	#435	1	2	yes	—
182	C₂×C₃×C₅	#446	1	2	yes	—
183	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	#683	1	2	yes	—
187	C₂×C₃×C₄	#683	1	2	yes	—
188	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
189	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
191	C₂×C₃×C₄	#709	1	1	yes	characteristic
192	C₂×C₃×C₄	#709	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
194	C₂×Dic₃	—	5	4	no	—
195	C₂×Dic₃	—	10	2	no	—
196	C₂×Dic₃	—	10	2	no	—
197	C₂×Dic₃	—	10	2	no	—
198	C₂×Dic₃	—	10	2	no	—
199	C₂×Dic₃	—	10	2	no	—
200	C₃×C₈	—	20	2	no	—
201	C₃⋊C₈	—	2	2	no	—
202	C₃⋊C₈	—	2	2	no	—
203	C₂²×C₅	#887	1	1	yes	characteristic
204	C₂²×C₅	#893	1	1	yes	characteristic
205	C₂²×C₅	#901	1	2	yes	—
206	C₂²×C₅	D₃×M₁₆	1	2	yes	—
207	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
208	C₄×C₅	—	2	2	no	—
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	#888	1	1	yes	characteristic
211	C₄×C₅	#890	1	1	yes	characteristic
212	C₄×C₅	#938	1	1	yes	characteristic
213	C₄×C₅	#946	1	1	yes	characteristic
214	Dic₅	—	6	4	no	—
215	Dic₅	—	12	2	no	—
216	C₂²×C₄	—	15	2	no	—
217	C₂²×C₄	—	15	2	no	—
218	C₂²×C₄	#1200	1	1	yes	characteristic
219	C₂²×C₄	#1200	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
221	C₂×C₈	—	3	2	no	—
222	C₂×C₈	—	3	2	no	—
223	C₂×C₈	—	6	1	no	—
224	C₂×C₈	—	6	1	no	—
225	C₂×C₈	—	6	1	no	—
226	C₂×C₈	—	6	1	no	—
227	C₂×C₈	—	10	2	no	—
228	C₂×C₈	—	10	2	no	—
229	C₂×C₈	—	10	2	no	—
230	C₄⋊C₄	—	30	4	no	—
231	C₄²	—	30	2	no	—
232	C₄²	—	30	2	no	—
233	C₄²	#1201	1	2	yes	—
234	C₄²	#1204	1	2	yes	—
235	C₃×C₅	#1940	1	1	yes	characteristic, complemented, Hall
236	C₂²×C₃	C₄×Dic₁₀	1	1	yes	characteristic
237	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
238	C₂²×C₃	D₅×M₁₆	1	2	yes	—
239	C₂²×C₃	#4053	1	2	yes	—
240	C₂²×C₃	C₂×#683	1	1	yes	characteristic
241	C₃×C₄	—	2	2	no	—
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	#4015	1	1	yes	characteristic
244	C₃×C₄	#4016	1	1	yes	characteristic
245	C₃×C₄	#4070	1	1	yes	characteristic
246	C₃×C₄	#4076	1	1	yes	characteristic
247	Dic₃	—	10	4	no	—
248	Dic₃	—	20	2	no	—
249	C₂×C₅	#4828	1	2	yes	—
250	C₂×C₅	#4836	1	2	yes	—
251	C₂×C₅	#4972	1	1	yes	characteristic
252	C₂×C₅	#5102	1	1	yes	characteristic
253	C₂×C₅	#5118	1	1	yes	characteristic
254	C₂³	C₂×#1200	1	1	yes	characteristic
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	15	4	no	—
258	C₂×C₄	—	30	2	no	—
259	C₂×C₄	—	30	2	no	—
260	C₂×C₄	—	30	2	no	—
261	C₂×C₄	—	30	2	no	—
262	C₂×C₄	—	30	2	no	—
263	C₂×C₄	#6669	1	2	yes	—
264	C₂×C₄	#6669	1	2	yes	—
265	C₂×C₄	#6765	1	1	yes	characteristic
266	C₂×C₄	D₃×Dic₁₀	1	1	yes	characteristic
267	C₂×C₄	#6769	1	1	yes	characteristic
268	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
269	C₂×C₄	C₂×#1201	1	1	yes	characteristic
270	C₂×C₄	C₂×#1204	1	1	yes	characteristic
271	C₈	—	6	2	no	—
272	C₈	—	6	2	no	—
273	C₈	—	20	2	no	—
274	C₂×C₃	#64990	1	2	yes	—
275	C₂×C₃	#64995	1	2	yes	—
276	C₂×C₃	#65217	1	1	yes	characteristic
277	C₂×C₃	#65407	1	1	yes	characteristic
278	C₂×C₃	#65408	1	1	yes	characteristic
279	C₅	#70469	1	1	yes	characteristic, complemented, Sylow
280	C₂²	#91290	1	2	yes	—
281	C₂²	#91292	1	2	yes	—
282	C₂²	#91340	1	1	yes	characteristic
283	C₂²	#91428	1	1	yes	characteristic
284	C₂²	C₂×#6669	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	30	4	no	—
288	C₄	—	60	2	no	—
289	C₄	#91369	1	1	yes	characteristic
290	C₄	#91381	1	1	yes	characteristic
291	C₄	#91456	1	1	yes	characteristic
292	C₄	#91469	1	1	yes	characteristic
293	C₃	#10600629	1	1	yes	characteristic, complemented, Sylow
294	C₂	#11746856	1	2	yes	—
295	C₂	#11746995	1	2	yes	—
296	C₂	#11748435	1	1	yes	characteristic
297	C₂	#11748437	1	1	yes	characteristic
298	C₂	#11748858	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂×D₃×D₅
commutator, socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
Fitting	C₂×C₃×C₄²×C₅	C₂²
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1940
LCS	G ▷ C₂³×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
235. C₃×C₅	#1940	semidirect
279. C₅	#70469	semidirect
293. C₃	#10600629	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Dic₁₅	2
C₂×C₅×#803	2
C₂×C₃×C₄⋊#168	2
C₂×#90983	2
#11745968	2
#10600629	3
#70469	5