Subgroups — #14280460

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#64974	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#11746868	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4454	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745832	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598861	—	3	1	no	maximal, Hall
9	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
10	C₂×D₅×C₃⋊C₈	C₄	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
14	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×#3952	C₄	1	2	yes	—
17	C₃×#3946	C₄	1	2	yes	—
18	C₃×C₄×#168	C₂²	1	1	yes	characteristic
19	#90982	C₄	1	2	yes	—
20	#90955	C₄	1	2	yes	—
21	Dic₅×C₃⋊C₈	C₄	1	2	yes	—
22	#70740	—	5	1	no	maximal, Hall
23	#64974	—	3	1	no	—
24	#64974	D₃	1	1	yes	characteristic
25	C₄×C₈×D₅	—	3	1	no	—
26	C₅×#322	—	3	1	no	—
27	#64675	—	3	1	no	—
28	C₈×#168	—	3	1	no	—
29	C₈×#168	—	3	1	no	—
30	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
36	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
37	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
38	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×#168	C₂×C₄	1	2	yes	—
43	D₅×C₃⋊C₈	C₈	1	8	yes	complemented
44	C₃×#431	—	5	1	no	—
45	#4925	—	5	1	no	—
46	C₂×C₄×C₃⋊C₈	—	5	1	no	—
47	#4454	—	5	1	no	—
48	#4454	D₅	1	1	yes	characteristic, complemented
49	C₈×C₃⋊C₈	—	5	1	no	—
50	C₈×C₃⋊C₈	—	5	1	no	—
51	C₂×C₈×D₅	—	3	2	no	—
52	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
53	C₄×C₅×C₈	—	3	1	no	—
54	C₄×C₅×C₈	—	3	1	no	—
55	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
56	#3952	—	3	2	no	—
57	#3952	Dic₃	1	2	yes	—
58	#3946	—	3	2	no	—
59	#3946	Dic₃	1	2	yes	—
60	C₈×Dic₅	—	3	2	no	—
61	C₄×#168	—	3	1	no	—
62	C₄×#168	—	3	1	no	—
63	C₄×#168	C₂×D₃	1	1	yes	characteristic
64	#1450	—	15	1	no	Sylow
65	C₂²×C₃×D₅	C₄²	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
70	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
71	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
72	C₃×C₄×D₅	—	2	2	no	—
73	C₃×C₄×D₅	—	2	2	no	—
74	C₃×C₄×D₅	C₂×C₈	1	4	yes	—
75	C₃×C₅×C₈	—	2	2	no	—
76	C₃⋊#168	—	2	2	no	—
77	C₃⋊#168	C₂×C₈	1	4	yes	—
78	C₃×#168	—	2	2	no	—
79	C₅×C₃⋊C₈	—	2	2	no	—
80	C₅×C₃⋊C₈	C₂×C₈	1	4	yes	—
81	C₂×C₃×C₄²	—	5	1	no	—
82	C₂²×C₃⋊C₈	—	5	2	no	—
83	C₃×C₄⋊C₈	—	5	2	no	—
84	C₃×C₂²⋊C₈	—	5	2	no	—
85	C₃×C₄×C₈	—	5	1	no	—
86	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
87	#830	—	5	2	no	—
88	C₄⋊(C₃⋊C₈)	—	5	2	no	—
89	C₄×C₃⋊C₈	—	5	1	no	—
90	C₄×C₃⋊C₈	—	5	1	no	—
91	C₄×C₃⋊C₈	—	5	2	no	—
92	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
93	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
94	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
95	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
96	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
97	C₂×C₅×C₈	—	3	2	no	—
98	C₂×C₅×C₈	—	3	2	no	—
99	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
100	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
101	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
102	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
103	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
104	C₂×#168	—	3	2	no	—
105	C₂×#168	—	3	2	no	—
106	C₂×#168	C₂×Dic₃	1	2	yes	—
107	C₈×D₅	—	3	8	no	—
108	#431	—	5	1	no	—
109	#431	—	15	1	no	—
110	C₂×C₄×C₈	—	15	1	no	—
111	#322	—	3	1	no	—
112	#322	—	15	1	no	—
113	C₈²	—	15	1	no	—
114	C₈²	—	15	1	no	—
115	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₃×D₅	—	2	1	no	—
117	C₂×C₃×D₅	—	2	1	no	—
118	C₂×C₃×D₅	C₄×C₈	1	2	yes	—
119	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
120	C₃×C₄×C₅	#131	1	2	yes	—
121	C₃×C₄×C₅	#131	1	2	yes	—
122	C₃×Dic₅	—	2	1	no	—
123	C₃×Dic₅	—	2	1	no	—
124	C₃×Dic₅	C₄×C₈	1	2	yes	—
125	C₂²×C₃×C₄	—	5	1	no	—
126	C₂²×C₃×C₄	—	5	1	no	—
127	C₂²×C₃×C₄	—	5	1	no	—
128	C₂×C₃×C₈	—	5	2	no	—
129	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
130	C₃×C₄²	—	5	1	no	—
131	C₃×C₄²	—	5	1	no	—
132	C₃×C₄²	—	5	1	no	—
133	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
134	C₂×C₃⋊C₈	—	5	2	no	—
135	C₂×C₃⋊C₈	—	5	8	no	—
136	C₂×C₃⋊C₈	—	5	2	no	—
137	C₂×C₃⋊C₈	C₄×D₅	1	2	yes	—
138	C₂×C₃⋊C₈	C₄×D₅	1	2	yes	—
139	C₂²×D₅	C₄×Dic₃	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
143	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
144	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
145	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
146	C₄×D₅	—	2	2	no	—
147	C₄×D₅	—	2	2	no	—
148	C₄×D₅	D₃×C₈	1	4	yes	—
149	C₅×C₈	—	2	2	no	—
150	C₅×C₈	—	3	4	no	—
151	C₅×C₈	—	6	2	no	—
152	#168	—	2	2	no	—
153	#168	—	3	4	no	—
154	#168	—	6	2	no	—
155	C₂²×C₈	—	15	2	no	—
156	C₂×C₄²	—	5	1	no	—
157	C₄⋊C₈	—	5	2	no	—
158	C₄⋊C₈	—	15	2	no	—
159	C₂²⋊C₈	—	5	2	no	—
160	C₂²⋊C₈	—	15	2	no	—
161	C₄×C₈	—	3	1	no	—
162	C₄×C₈	—	3	1	no	—
163	C₄×C₈	—	5	1	no	—
164	C₄×C₈	—	15	2	no	—
165	C₄×C₈	—	15	1	no	—
166	C₄×C₈	—	15	1	no	—
167	C₄×C₈	D₃×D₅	1	1	yes	characteristic
168	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
169	C₂×C₃×C₅	#405	1	1	yes	characteristic
170	C₂×C₃×C₅	#405	1	1	yes	characteristic
171	C₃×D₅	—	2	2	no	—
172	C₂³×C₃	—	5	1	no	—
173	C₂×C₃×C₄	—	5	1	no	—
174	C₂×C₃×C₄	—	5	4	no	—
175	C₂×C₃×C₄	—	5	1	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	10	2	no	—
178	C₂×C₃×C₄	—	10	2	no	—
179	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
180	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
182	C₃×C₈	—	2	2	no	—
183	C₃×C₈	—	10	2	no	—
184	C₃⋊C₈	—	2	2	no	—
185	C₃⋊C₈	—	5	4	no	—
186	C₃⋊C₈	—	10	2	no	—
187	C₃⋊C₈	C₈×D₅	1	4	yes	—
188	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
189	C₂×D₅	—	2	1	no	—
190	C₂×D₅	—	2	1	no	—
191	C₂×D₅	C₈×Dic₃	1	2	yes	—
192	C₄×C₅	C₂×D₃×C₈	1	2	yes	—
193	C₄×C₅	#907	1	2	yes	—
194	C₄×C₅	#948	1	2	yes	—
195	Dic₅	—	2	1	no	—
196	Dic₅	—	2	1	no	—
197	Dic₅	C₈×Dic₃	1	2	yes	—
198	C₂²×C₄	—	5	1	no	—
199	C₂²×C₄	—	5	1	no	—
200	C₂²×C₄	—	5	1	no	—
201	C₂×C₈	—	3	2	no	—
202	C₂×C₈	—	3	2	no	—
203	C₂×C₈	—	5	2	no	—
204	C₂×C₈	—	15	8	no	—
205	C₂×C₈	—	15	2	no	—
206	C₂×C₈	—	15	2	no	—
207	C₂×C₈	D₅×Dic₃	1	2	yes	—
208	C₄²	—	5	1	no	—
209	C₄²	—	5	1	no	—
210	C₄²	—	5	1	no	—
211	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
212	C₃×C₅	#1450	1	1	yes	characteristic, complemented, Hall
213	C₂²×C₃	—	5	2	no	—
214	C₂²×C₃	—	10	1	no	—
215	C₂²×C₃	—	10	1	no	—
216	C₂²×C₃	C₄²×D₅	1	1	yes	characteristic
217	C₃×C₄	—	5	2	no	—
218	C₃×C₄	—	10	1	no	—
219	C₃×C₄	—	10	1	no	—
220	C₃×C₄	C₂×C₈×D₅	1	2	yes	—
221	C₃×C₄	#4053	1	2	yes	—
222	C₃×C₄	#4053	1	2	yes	—
223	C₂×C₅	#4997	1	1	yes	characteristic
224	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
225	C₂×C₅	#5122	1	1	yes	characteristic
226	D₅	—	2	2	no	—
227	C₂³	—	5	1	no	—
228	C₂×C₄	—	5	1	no	—
229	C₂×C₄	—	5	4	no	—
230	C₂×C₄	—	5	1	no	—
231	C₂×C₄	—	5	1	no	—
232	C₂×C₄	—	10	2	no	—
233	C₂×C₄	—	10	2	no	—
234	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
235	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
236	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	3	4	no	—
239	C₈	—	6	2	no	—
240	C₈	—	10	2	no	—
241	C₈	—	15	4	no	—
242	C₈	—	30	2	no	—
243	C₂×C₃	—	10	2	no	—
244	C₂×C₃	C₄×C₈×D₅	1	1	yes	characteristic
245	C₂×C₃	#64997	1	1	yes	characteristic
246	C₂×C₃	#64997	1	1	yes	characteristic
247	C₅	#70740	1	1	yes	characteristic, complemented, Sylow
248	C₂²	—	5	2	no	—
249	C₂²	—	10	1	no	—
250	C₂²	—	10	1	no	—
251	C₂²	C₄×D₅×Dic₃	1	1	yes	characteristic
252	C₄	—	5	2	no	—
253	C₄	—	10	1	no	—
254	C₄	—	10	1	no	—
255	C₄	D₃×C₈×D₅	1	2	yes	—
256	C₄	#91268	1	2	yes	—
257	C₄	#91284	1	2	yes	—
258	C₃	#10598861	1	1	yes	characteristic, complemented, Sylow
259	C₂	—	10	2	no	—
260	C₂	#11746975	1	1	yes	characteristic
261	C₂	C₈×D₅×Dic₃	1	1	yes	characteristic
262	C₂	#11748051	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₃×D₅	C₄²
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#1450
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₈×C₃⋊#168	C₂	semidirect
5. C₅×#4454	C₂	semidirect
6. #11745832	C₂	semidirect
7. C₃⋊C₈×#168	C₂	semidirect
43. D₅×C₃⋊C₈	C₈	semidirect
48. #4454	D₅	semidirect
212. C₃×C₅	#1450	semidirect
247. C₅	#70740	semidirect
258. C₃	#10598861	semidirect

Maximal subgroups

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

subgroup	index
C₃×#64974	2
C₄×D₅×C₃⋊C₈	2
#11746868	2
C₈×C₃⋊#168	2
C₅×#4454	2
#11745832	2
C₃⋊C₈×#168	2
#10598861	3
#70740	5