Subgroups — #14298816

Order 1920.

496 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#384	C₂	1	1	yes	maximal, characteristic
2	#11751036	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic
4	#11746350	C₂	1	4	yes	maximal
5	#10600367	—	3	1	no	maximal, Hall
6	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
7	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
8	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₅	—	2	1	no	—
11	C₂×C₄⋊Dic₁₅	—	2	1	no	—
12	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
13	C₂²×C₃⋊#168	C₂²	1	2	yes	—
14	C₄×C₃⋊#168	C₄	1	4	yes	complemented
15	#69574	—	5	1	no	maximal, Hall
16	C₅×#384	D₃	1	1	yes	characteristic
17	#65224	—	3	1	no	—
18	C₂×C₄×#168	—	3	1	no	—
19	#64749	—	3	4	no	—
20	C₂²×C₃×C₄×C₅	—	2	2	no	—
21	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
22	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₅	—	2	2	no	—
25	C₃×C₅×C₄⋊C₄	—	2	4	no	—
26	C₃×C₅×C₄⋊C₄	—	4	2	no	—
27	C₃×C₅×C₄⋊C₄	—	4	2	no	—
28	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₃×C₄²×C₅	Q₈	1	2	yes	—
30	C₄⋊Dic₁₅	—	2	4	no	—
31	C₄⋊Dic₁₅	—	4	2	no	—
32	C₄⋊Dic₁₅	—	4	2	no	—
33	C₂×C₃⋊#168	—	2	2	no	—
34	C₂×C₃⋊#168	—	2	2	no	—
35	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
36	C₃×#384	D₅	1	1	yes	characteristic
37	#4935	—	5	1	no	—
38	C₂×C₄×C₃⋊C₈	—	5	1	no	—
39	#4529	—	5	4	no	—
40	C₂×C₅×C₄⋊C₄	—	2	1	no	—
41	C₂×C₅×C₄⋊C₄	—	2	1	no	—
42	C₂×C₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
43	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
44	C₂×C₄⋊Dic₅	—	3	2	no	—
45	C₂×C₄⋊Dic₅	—	6	1	no	—
46	C₂×C₄⋊Dic₅	—	6	1	no	—
47	C₂²×#168	—	3	2	no	—
48	C₄×#168	—	3	4	no	—
49	#1834	—	15	1	no	Sylow
50	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃×C₄×C₅	—	4	2	no	—
52	C₂×C₃×C₄×C₅	—	4	2	no	—
53	C₂×C₃×C₄×C₅	—	4	2	no	—
54	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
56	C₂×C₃×C₄×C₅	D₈	1	2	yes	—
57	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
58	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
59	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	#41	1	2	yes	—
63	C₂×C₃×C₄×C₅	#41	1	2	yes	—
64	C₂×Dic₁₅	—	4	2	no	—
65	C₂×Dic₁₅	—	4	2	no	—
66	C₂×Dic₁₅	—	4	2	no	—
67	C₃⋊#168	—	2	4	no	—
68	C₂×C₃×C₄⋊C₄	—	2	1	no	—
69	C₂×C₃×C₄⋊C₄	—	2	1	no	—
70	C₂×C₃×C₄⋊C₄	C₂×D₅	1	2	yes	—
71	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
72	C₂×C₄⋊Dic₃	—	5	2	no	—
73	C₂×C₄⋊Dic₃	—	10	1	no	—
74	C₂×C₄⋊Dic₃	—	10	1	no	—
75	C₂²×C₃⋊C₈	—	5	2	no	—
76	C₄×C₃⋊C₈	—	5	4	no	—
77	C₂²×C₄×C₅	—	2	2	no	—
78	C₂²×C₄×C₅	#67	1	1	yes	characteristic
79	C₂²×C₄×C₅	#67	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
81	C₂²×Dic₅	—	6	2	no	—
82	C₅×C₄⋊C₄	—	2	4	no	—
83	C₅×C₄⋊C₄	—	4	2	no	—
84	C₅×C₄⋊C₄	—	4	2	no	—
85	C₄²×C₅	Dic₆	1	2	yes	—
86	C₄²×C₅	C₄×D₃	1	2	yes	—
87	C₄⋊Dic₅	—	6	4	no	—
88	C₄⋊Dic₅	—	12	2	no	—
89	C₄⋊Dic₅	—	12	2	no	—
90	C₂×#168	—	3	4	no	—
91	C₂×#168	—	6	2	no	—
92	C₂×#168	—	6	2	no	—
93	C₂×C₄×C₈	—	15	1	no	—
94	#384	—	15	1	no	—
95	#384	D₁₅	1	1	yes	characteristic
96	#340	—	15	4	no	—
97	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂²×C₃×C₅	#124	1	1	yes	characteristic
99	C₂²×C₃×C₅	#124	1	1	yes	characteristic
100	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
101	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
102	C₂²×C₃×C₅	C₂×SD₁₆	1	1	yes	characteristic
103	C₂²×C₃×C₅	C₂×Q₁₆	1	1	yes	characteristic
104	C₃×C₄×C₅	—	8	2	no	—
105	C₃×C₄×C₅	#106	1	4	yes	—
106	C₃×C₄×C₅	#107	1	4	yes	—
107	C₃×C₄×C₅	#117	1	2	yes	—
108	C₃×C₄×C₅	#125	1	2	yes	—
109	Dic₁₅	—	8	2	no	—
110	C₂²×C₃×C₄	—	2	2	no	—
111	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
112	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
113	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
114	C₂²×Dic₃	—	10	2	no	—
115	C₃×C₄⋊C₄	—	2	4	no	—
116	C₃×C₄⋊C₄	—	4	2	no	—
117	C₃×C₄⋊C₄	—	4	2	no	—
118	C₃×C₄²	Dic₁₀	1	2	yes	—
119	C₃×C₄²	C₄×D₅	1	2	yes	—
120	C₄⋊Dic₃	—	10	4	no	—
121	C₄⋊Dic₃	—	20	2	no	—
122	C₄⋊Dic₃	—	20	2	no	—
123	C₂×C₃⋊C₈	—	5	4	no	—
124	C₂×C₃⋊C₈	—	10	2	no	—
125	C₂×C₃⋊C₈	—	10	2	no	—
126	C₂³×C₅	C₂×#67	1	1	yes	characteristic
127	C₂×C₄×C₅	—	4	2	no	—
128	C₂×C₄×C₅	—	4	2	no	—
129	C₂×C₄×C₅	—	4	2	no	—
130	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
131	C₂×C₄×C₅	Dic₃⋊C₄	1	2	yes	—
132	C₂×C₄×C₅	#213	1	2	yes	—
133	C₂×C₄×C₅	#214	1	2	yes	—
134	C₂×C₄×C₅	#215	1	2	yes	—
135	C₂×C₄×C₅	#216	1	2	yes	—
136	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
138	C₂×C₄×C₅	#237	1	2	yes	—
139	C₂×C₄×C₅	#239	1	2	yes	—
140	C₂×Dic₅	—	12	2	no	—
141	C₂×Dic₅	—	12	2	no	—
142	C₂×Dic₅	—	12	2	no	—
143	#168	—	6	4	no	—
144	C₂²×C₈	—	15	2	no	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	15	2	no	—
148	C₂×C₄⋊C₄	—	30	1	no	—
149	C₂×C₄⋊C₄	—	30	1	no	—
150	C₂×C₄⋊C₄	C₂×D₁₅	1	2	yes	—
151	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
152	C₄×C₈	—	15	4	no	—
153	C₂×C₃×C₅	#382	1	1	yes	characteristic
154	C₂×C₃×C₅	C₂×#106	1	1	yes	characteristic
155	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
156	C₂×C₃×C₅	#486	1	1	yes	characteristic
157	C₂×C₃×C₅	#486	1	1	yes	characteristic
158	C₂×C₃×C₅	#487	1	1	yes	characteristic
159	C₂×C₃×C₅	#487	1	1	yes	characteristic
160	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
161	C₂×C₃×C₄	—	4	2	no	—
162	C₂×C₃×C₄	—	4	2	no	—
163	C₂×C₃×C₄	—	4	2	no	—
164	C₂×C₃×C₄	#683	1	2	yes	—
165	C₂×C₃×C₄	#683	1	2	yes	—
166	C₂×C₃×C₄	#686	1	2	yes	—
167	C₂×C₃×C₄	#687	1	2	yes	—
168	C₂×C₃×C₄	#688	1	2	yes	—
169	C₂×C₃×C₄	#689	1	2	yes	—
170	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
172	C₂×C₃×C₄	#711	1	2	yes	—
173	C₂×C₃×C₄	#713	1	2	yes	—
174	C₂×Dic₃	—	20	2	no	—
175	C₂×Dic₃	—	20	2	no	—
176	C₂×Dic₃	—	20	2	no	—
177	C₃⋊C₈	—	10	4	no	—
178	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
179	C₂²×C₅	C₂×#213	1	1	yes	characteristic
180	C₂²×C₅	C₂×#214	1	1	yes	characteristic
181	C₂²×C₅	#936	1	1	yes	characteristic
182	C₂²×C₅	C₂×#215	1	1	yes	characteristic
183	C₂²×C₅	C₂×#216	1	1	yes	characteristic
184	C₂²×C₅	#947	1	1	yes	characteristic
185	C₄×C₅	—	8	2	no	—
186	C₄×C₅	#807	1	4	yes	—
187	C₄×C₅	#808	1	4	yes	—
188	C₄×C₅	#890	1	2	yes	—
189	C₄×C₅	#892	1	2	yes	—
190	Dic₅	—	24	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	30	2	no	—
193	C₂²×C₄	#1220	1	1	yes	characteristic
194	C₂²×C₄	#1220	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
196	C₂×C₈	—	15	4	no	—
197	C₂×C₈	—	30	2	no	—
198	C₂×C₈	—	30	2	no	—
199	C₄⋊C₄	—	2	4	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	30	4	no	—
203	C₄⋊C₄	—	60	2	no	—
204	C₄⋊C₄	—	60	2	no	—
205	C₄²	Dic₃₀	1	2	yes	—
206	C₄²	C₄×D₁₅	1	2	yes	—
207	C₃×C₅	#1834	1	1	yes	characteristic, complemented, Hall
208	C₂²×C₃	C₂×#683	1	1	yes	characteristic
209	C₂²×C₃	C₂×#686	1	1	yes	characteristic
210	C₂²×C₃	C₂×#687	1	1	yes	characteristic
211	C₂²×C₃	#4082	1	1	yes	characteristic
212	C₂²×C₃	C₂×#688	1	1	yes	characteristic
213	C₂²×C₃	C₂×#689	1	1	yes	characteristic
214	C₂²×C₃	#4093	1	1	yes	characteristic
215	C₃×C₄	—	8	2	no	—
216	C₃×C₄	#3939	1	4	yes	—
217	C₃×C₄	#3940	1	4	yes	—
218	C₃×C₄	#4036	1	2	yes	—
219	C₃×C₄	#4038	1	2	yes	—
220	Dic₃	—	40	2	no	—
221	C₂×C₅	C₂×#807	1	1	yes	characteristic
222	C₂×C₅	C₂×#808	1	1	yes	characteristic
223	C₂×C₅	#4977	1	1	yes	characteristic
224	C₂×C₅	#5070	1	1	yes	characteristic
225	C₂×C₅	#5079	1	1	yes	characteristic
226	C₂×C₅	#5088	1	1	yes	characteristic
227	C₂×C₅	#5093	1	1	yes	characteristic
228	C₂³	C₂×#1220	1	1	yes	characteristic
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	60	2	no	—
233	C₂×C₄	—	60	2	no	—
234	C₂×C₄	—	60	2	no	—
235	C₂×C₄	#6712	1	2	yes	—
236	C₂×C₄	#6712	1	2	yes	—
237	C₂×C₄	#6715	1	2	yes	—
238	C₂×C₄	#6716	1	2	yes	—
239	C₂×C₄	#6717	1	2	yes	—
240	C₂×C₄	#6718	1	2	yes	—
241	C₂×C₄	C₂×Dic₃₀	1	1	yes	characteristic
242	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
243	C₂×C₄	#6819	1	2	yes	—
244	C₂×C₄	#6821	1	2	yes	—
245	C₈	—	30	4	no	—
246	C₂×C₃	C₂×#3939	1	1	yes	characteristic
247	C₂×C₃	C₂×#3940	1	1	yes	characteristic
248	C₂×C₃	#65266	1	1	yes	characteristic
249	C₂×C₃	#65359	1	1	yes	characteristic
250	C₂×C₃	#65370	1	1	yes	characteristic
251	C₂×C₃	#65377	1	1	yes	characteristic
252	C₂×C₃	#65382	1	1	yes	characteristic
253	C₅	#69574	1	1	yes	characteristic, complemented, Sylow
254	C₂²	C₂×#6712	1	1	yes	characteristic
255	C₂²	C₂×#6715	1	1	yes	characteristic
256	C₂²	C₂×#6716	1	1	yes	characteristic
257	C₂²	#91825	1	1	yes	characteristic
258	C₂²	C₂×#6717	1	1	yes	characteristic
259	C₂²	C₂×#6718	1	1	yes	characteristic
260	C₂²	#91836	1	1	yes	characteristic
261	C₄	—	8	2	no	—
262	C₄	—	120	2	no	—
263	C₄	#91091	1	4	yes	—
264	C₄	#91092	1	4	yes	—
265	C₄	#91779	1	2	yes	—
266	C₄	#91781	1	2	yes	—
267	C₃	#10600367	1	1	yes	characteristic, complemented, Sylow
268	C₂	C₂×#91091	1	1	yes	characteristic
269	C₂	C₂×#91092	1	1	yes	characteristic
270	C₂	#11751078	1	1	yes	characteristic
271	C₂	#11751171	1	1	yes	characteristic
272	C₂	#11751180	1	1	yes	characteristic
273	C₂	#11751189	1	1	yes	characteristic
274	C₂	#11751194	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1220
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#384	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#384	D₁₅
hypocenter	C₃×C₅	#1834
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

Complements

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

N	H ≅ G/N	type
14. C₄×C₃⋊#168	C₄	semidirect
207. C₃×C₅	#1834	semidirect
253. C₅	#69574	semidirect
267. C₃	#10600367	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₅×#384	2
#11751036	2
C₂×C₄×C₃⋊#168	2
#11746350	2
#10600367	3
#69574	5