Subgroups — #14280722

Order 1920.

428 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6710	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×#208	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#681	C₂	1	1	yes	maximal, characteristic
4	#11745833	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745832	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600642	—	3	1	no	maximal, Hall
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×#6710	C₄	1	2	yes	—
11	C₂×C₅×#208	C₄	1	2	yes	—
12	C₂×C₃×#681	C₄	1	2	yes	—
13	#91087	C₄	1	2	yes	—
14	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
15	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
16	C₅×#803	C₄	1	2	yes	—
17	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
19	C₃×#3935	C₄	1	2	yes	—
20	C₃×C₄×#168	C₂²	1	1	yes	characteristic
21	C₃×C₄×#168	C₂²	1	1	yes	characteristic
22	#69849	—	5	1	no	maximal, Hall
23	C₄×C₅×M₁₆	—	3	1	no	—
24	C₄×#681	—	3	1	no	—
25	C₄×#681	D₃	1	1	yes	characteristic
26	#64675	—	3	1	no	—
27	#64674	—	3	1	no	—
28	#64674	—	3	1	no	—
29	C₈×#168	—	3	1	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
37	#6710	—	2	4	no	—
38	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
39	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
40	C₅×#208	—	2	4	no	—
41	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
43	C₃×#681	—	2	4	no	—
44	C₂×C₃×#168	C₂×C₄	1	2	yes	—
45	C₂×C₃×#168	C₂×C₄	1	2	yes	—
46	C₃×C₄×M₁₆	—	5	1	no	—
47	C₄×#208	—	5	1	no	—
48	C₄×#208	D₅	1	1	yes	characteristic
49	#4455	—	5	1	no	—
50	#4454	—	5	1	no	—
51	#4454	—	5	1	no	—
52	C₈×C₃⋊C₈	—	5	1	no	—
53	C₂×C₅×M₁₆	—	3	2	no	—
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂×#681	—	3	2	no	—
56	C₂×#681	Dic₃	1	2	yes	—
57	C₅×#97	—	3	2	no	—
58	C₄×C₅×C₈	—	3	1	no	—
59	C₄×C₅×C₈	—	3	1	no	—
60	#3935	—	3	2	no	—
61	#3935	Dic₃	1	2	yes	—
62	C₄×#168	—	3	1	no	—
63	C₄×#168	—	3	1	no	—
64	C₄×#168	C₂×D₃	1	1	yes	characteristic
65	C₄×#168	C₂×D₃	1	1	yes	characteristic
66	#1455	—	15	1	no	Sylow
67	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
77	C₃⋊#168	—	2	2	no	—
78	C₃⋊#168	M₁₆	1	4	yes	complemented
79	C₃×#168	—	2	2	no	—
80	C₃×#168	M₁₆	1	4	yes	complemented
81	C₅×C₃⋊C₈	—	2	2	no	—
82	C₅×C₃⋊C₈	M₁₆	1	4	yes	complemented
83	C₂×C₃×M₁₆	—	5	2	no	—
84	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
85	C₂×#208	—	5	2	no	—
86	C₂×#208	Dic₅	1	2	yes	—
87	C₃×#97	—	5	2	no	—
88	C₃×C₄×C₈	—	5	1	no	—
89	C₃×C₄×C₈	—	5	1	no	—
90	#803	—	5	2	no	—
91	#803	Dic₅	1	2	yes	—
92	C₄×C₃⋊C₈	—	5	1	no	—
93	C₄×C₃⋊C₈	—	5	1	no	—
94	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
95	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
99	C₅×M₁₆	—	6	4	no	—
100	C₂×C₅×C₈	—	3	2	no	—
101	C₂×C₅×C₈	—	3	2	no	—
102	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
103	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
104	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
105	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
106	#681	—	2	4	no	—
107	#681	—	6	4	no	—
108	C₂×#168	—	3	2	no	—
109	C₂×#168	—	3	2	no	—
110	C₂×#168	C₂×Dic₃	1	2	yes	—
111	C₂×#168	C₂×Dic₃	1	2	yes	—
112	C₄×M₁₆	—	3	1	no	—
113	C₄×M₁₆	—	5	1	no	—
114	C₄×M₁₆	—	15	1	no	—
115	#322	—	15	1	no	—
116	#322	—	15	1	no	—
117	#322	—	15	1	no	—
118	C₈²	—	15	1	no	—
119	C₂²×C₃×C₅	—	2	1	no	—
120	C₂²×C₃×C₅	—	2	1	no	—
121	C₂²×C₃×C₅	—	2	1	no	—
122	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
123	C₃×C₄×C₅	—	2	1	no	—
124	C₃×C₄×C₅	—	2	1	no	—
125	C₃×C₄×C₅	—	2	1	no	—
126	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
127	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
128	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
129	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
130	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
131	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
132	C₃×M₁₆	—	10	4	no	—
133	C₂×C₃×C₈	—	5	2	no	—
134	C₂×C₃×C₈	—	5	2	no	—
135	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
136	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
137	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
138	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
139	#208	—	2	4	no	—
140	#208	—	10	4	no	—
141	C₂×C₃⋊C₈	—	5	2	no	—
142	C₂×C₃⋊C₈	—	5	2	no	—
143	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
144	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
145	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
150	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
151	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
155	C₅×C₈	—	3	4	no	—
156	C₅×C₈	—	6	2	no	—
157	#168	—	2	2	no	—
158	#168	—	3	4	no	—
159	#168	—	6	2	no	—
160	#168	#208	1	4	yes	complemented
161	C₂×M₁₆	—	3	2	no	—
162	C₂×M₁₆	—	5	2	no	—
163	C₂×M₁₆	—	15	2	no	—
164	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
165	#97	—	3	2	no	—
166	#97	—	5	2	no	—
167	#97	—	15	2	no	—
168	C₄×C₈	—	3	1	no	—
169	C₄×C₈	—	3	1	no	—
170	C₄×C₈	—	5	1	no	—
171	C₄×C₈	—	5	1	no	—
172	C₄×C₈	—	15	1	no	—
173	C₄×C₈	—	15	1	no	—
174	C₂×C₃×C₅	—	4	1	no	—
175	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
176	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
177	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
178	C₂³×C₃	C₄×Dic₅	1	1	yes	characteristic
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
183	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
184	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
187	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
188	C₃×C₈	—	5	4	no	—
189	C₃×C₈	—	10	2	no	—
190	C₃⋊C₈	—	2	2	no	—
191	C₃⋊C₈	—	5	4	no	—
192	C₃⋊C₈	—	10	2	no	—
193	C₃⋊C₈	#681	1	4	yes	complemented
194	C₂²×C₅	—	2	1	no	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	—	2	1	no	—
197	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	2	1	no	—
200	C₄×C₅	—	2	1	no	—
201	C₄×C₅	D₃×M₁₆	1	2	yes	—
202	C₄×C₅	D₃×M₁₆	1	2	yes	—
203	C₄×C₅	C₂×#208	1	2	yes	—
204	C₂²×C₄	D₅×Dic₃	1	1	yes	characteristic
205	C₂²×C₄	D₃×Dic₅	1	1	yes	characteristic
206	C₂²×C₄	#1200	1	1	yes	characteristic
207	M₁₆	—	6	4	no	—
208	M₁₆	—	10	4	no	—
209	M₁₆	—	30	4	no	—
210	C₂×C₈	—	3	2	no	—
211	C₂×C₈	—	3	2	no	—
212	C₂×C₈	—	5	2	no	—
213	C₂×C₈	—	5	2	no	—
214	C₂×C₈	—	15	2	no	—
215	C₂×C₈	—	15	2	no	—
216	C₄²	D₅×Dic₃	1	1	yes	characteristic
217	C₄²	D₃×Dic₅	1	1	yes	characteristic
218	C₄²	#1200	1	1	yes	characteristic
219	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
220	C₃×C₅	#1455	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	—	2	1	no	—
223	C₂²×C₃	—	2	1	no	—
224	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
225	C₃×C₄	—	2	1	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	1	no	—
228	C₃×C₄	D₅×M₁₆	1	2	yes	—
229	C₃×C₄	D₅×M₁₆	1	2	yes	—
230	C₃×C₄	C₂×#681	1	2	yes	—
231	C₂×C₅	—	4	1	no	—
232	C₂×C₅	C₄×#208	1	1	yes	characteristic
233	C₂×C₅	Dic₃×M₁₆	1	1	yes	characteristic
234	C₂×C₅	Dic₃×M₁₆	1	1	yes	characteristic
235	C₂³	Dic₃×Dic₅	1	1	yes	characteristic
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	Dic₃×Dic₅	1	1	yes	characteristic
240	C₂×C₄	Dic₃×Dic₅	1	1	yes	characteristic
241	C₂×C₄	Dic₃×Dic₅	1	1	yes	characteristic
242	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
243	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
244	C₂×C₄	C₂×#1200	1	1	yes	characteristic
245	C₈	—	3	4	no	—
246	C₈	—	5	4	no	—
247	C₈	—	6	2	no	—
248	C₈	—	10	2	no	—
249	C₈	—	15	4	no	—
250	C₈	—	30	2	no	—
251	C₂×C₃	—	4	1	no	—
252	C₂×C₃	C₄×#681	1	1	yes	characteristic
253	C₂×C₃	M₁₆×Dic₅	1	1	yes	characteristic
254	C₂×C₃	M₁₆×Dic₅	1	1	yes	characteristic
255	C₅	#69849	1	1	yes	characteristic, complemented, Sylow
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	C₂×Dic₃×Dic₅	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	D₅×#208	1	2	yes	—
264	C₄	D₃×#681	1	2	yes	—
265	C₄	#91292	1	2	yes	—
266	C₃	#10600642	1	1	yes	characteristic, complemented, Sylow
267	C₂	—	4	1	no	—
268	C₂	Dic₅×#208	1	1	yes	characteristic
269	C₂	Dic₃×#681	1	1	yes	characteristic
270	C₂	#11748328	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×D₃×D₅
commutator, socle	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₂×C₃×C₄²×C₅	C₂²
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1455
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄² ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
4. #11745833	C₂	semidirect
5. #11745832	C₂	semidirect
6. #11745831	C₂	semidirect
7. C₃⋊C₈×#168	C₂	semidirect
78. C₃⋊#168	M₁₆	semidirect
80. C₃×#168	M₁₆	semidirect
82. C₅×C₃⋊C₈	M₁₆	semidirect
160. #168	#208	semidirect
193. C₃⋊C₈	#681	semidirect
220. C₃×C₅	#1455	semidirect
255. C₅	#69849	semidirect
266. C₃	#10600642	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6710	2
C₄×C₅×#208	2
C₃×C₄×#681	2
#11745833	2
#11745832	2
#11745831	2
C₃⋊C₈×#168	2
#10600642	3
#69849	5