Subgroups — #14307232

Order 1920.

520 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	C₂×#91759	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₁₅	C₂	1	1	yes	maximal, characteristic
4	#11746372	C₂	1	2	yes	maximal
5	#11746372	C₂	1	2	yes	maximal
6	#10601254	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₅	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
12	C₂×#6712	—	2	1	no	—
13	C₂×#6712	—	2	1	no	—
14	C₂×C₄×Dic₁₅	—	2	1	no	—
15	#91759	—	2	2	no	—
16	#91759	C₄	1	4	yes	complemented
17	C₃×C₅×C₄⋊C₈	—	2	2	no	—
18	#91115	—	2	1	no	—
19	#70461	—	5	1	no	maximal, Hall
20	C₂×#4016	—	3	1	no	—
21	C₂×C₅×C₄⋊C₈	D₃	1	1	yes	characteristic
22	C₄×C₄⋊Dic₅	—	3	1	no	—
23	#64771	—	3	2	no	—
24	#64771	—	3	2	no	—
25	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂²×Dic₁₅	—	2	2	no	—
29	C₂²×Dic₁₅	—	2	1	no	—
30	C₂²×Dic₁₅	—	2	1	no	—
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	—	2	4	no	—
33	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₃×C₄²×C₅	D₄	1	2	yes	—
35	C₄⋊Dic₁₅	—	2	1	no	—
36	C₄⋊Dic₁₅	—	2	1	no	—
37	C₄⋊Dic₁₅	—	2	2	no	—
38	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₁₅	C₂×C₄	1	2	yes	—
40	C₄⋊Dic₁₅	Q₈	1	4	yes	—
41	#6712	—	2	4	no	—
42	#6712	—	2	4	no	—
43	C₄×Dic₁₅	—	4	2	no	—
44	C₂×#870	—	5	1	no	—
45	C₂×C₃×C₄⋊C₈	D₅	1	1	yes	characteristic
46	C₄×C₄⋊Dic₃	—	5	1	no	—
47	#4551	—	5	2	no	—
48	#4551	—	5	2	no	—
49	C₂²×C₅×C₈	C₂×D₃	1	2	yes	—
50	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
51	C₂×C₄⋊Dic₅	—	3	1	no	—
52	C₂×C₄⋊Dic₅	—	3	1	no	—
53	C₂×C₄⋊Dic₅	—	3	2	no	—
54	C₂×#683	—	6	1	no	—
55	C₂×#683	—	6	1	no	—
56	C₂×C₄×Dic₅	—	6	1	no	—
57	#4016	—	3	4	no	—
58	#4016	—	6	2	no	—
59	C₅×C₄⋊C₈	—	2	2	no	—
60	#3963	—	6	1	no	—
61	#1989	—	15	1	no	Sylow
62	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	—	2	2	no	—
64	C₂×C₃×C₄×C₅	—	2	1	no	—
65	C₂×C₃×C₄×C₅	—	2	1	no	—
66	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
71	C₂×C₃×C₄×C₅	#41	1	2	yes	—
72	C₂×Dic₁₅	—	2	2	no	—
73	C₂×Dic₁₅	—	2	2	no	—
74	C₂×Dic₁₅	—	2	4	no	—
75	C₂×Dic₁₅	—	4	2	no	—
76	C₂×Dic₁₅	—	4	2	no	—
77	C₂×Dic₁₅	—	4	2	no	—
78	C₂×Dic₁₅	—	4	1	no	—
79	C₂×Dic₁₅	—	4	1	no	—
80	C₃×C₅×C₈	—	4	2	no	—
81	C₂²×C₃×C₈	C₂×D₅	1	2	yes	—
82	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
83	C₂×C₄⋊Dic₃	—	5	2	no	—
84	C₂×C₄⋊Dic₃	—	5	1	no	—
85	C₂×C₄⋊Dic₃	—	5	1	no	—
86	C₂×Dic₃⋊C₄	—	10	1	no	—
87	C₂×Dic₃⋊C₄	—	10	1	no	—
88	C₂×C₄×Dic₃	—	10	1	no	—
89	#870	—	5	4	no	—
90	#870	—	10	2	no	—
91	C₃×C₄⋊C₈	—	2	2	no	—
92	C₃⋊#95	—	10	1	no	—
93	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
94	C₂²×C₄×C₅	D₁₂	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
96	C₂²×Dic₅	—	6	2	no	—
97	C₂²×Dic₅	—	6	1	no	—
98	C₂²×Dic₅	—	6	1	no	—
99	C₂×C₅×C₈	—	2	4	no	—
100	C₂×C₅×C₈	—	2	2	no	—
101	C₄²×C₅	C₄×D₃	1	2	yes	—
102	C₄²×C₅	#67	1	2	yes	—
103	C₄⋊Dic₅	—	3	2	no	—
104	C₄⋊Dic₅	—	3	2	no	—
105	C₄⋊Dic₅	—	3	4	no	—
106	C₄⋊Dic₅	—	6	2	no	—
107	C₄⋊Dic₅	—	6	1	no	—
108	C₄⋊Dic₅	—	6	1	no	—
109	#683	—	6	4	no	—
110	#683	—	6	4	no	—
111	C₄×Dic₅	—	12	2	no	—
112	C₂×#125	—	15	1	no	—
113	C₂×C₄⋊C₈	D₁₅	1	1	yes	characteristic
114	C₄×C₄⋊C₄	—	15	1	no	—
115	#340	—	15	2	no	—
116	#340	—	15	2	no	—
117	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂²×C₃×C₅	C₂²⋊Q₈	1	2	yes	—
119	C₂²×C₃×C₅	#135	1	2	yes	—
120	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂²×C₃×C₅	#137	1	1	yes	characteristic
122	C₃×C₄×C₅	—	2	2	no	—
123	C₃×C₄×C₅	—	2	2	no	—
124	C₃×C₄×C₅	C₄×Q₈	1	2	yes	—
125	C₃×C₄×C₅	#124	1	1	yes	characteristic
126	C₃×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
127	Dic₁₅	—	4	2	no	—
128	Dic₁₅	—	4	2	no	—
129	Dic₁₅	—	4	4	no	—
130	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
131	C₂²×C₃×C₄	D₂₀	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
133	C₂²×Dic₃	—	10	2	no	—
134	C₂²×Dic₃	—	10	1	no	—
135	C₂²×Dic₃	—	10	1	no	—
136	C₂×C₃×C₈	—	2	4	no	—
137	C₂×C₃×C₈	—	2	2	no	—
138	C₃×C₄²	C₄×D₅	1	2	yes	—
139	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
140	C₄⋊Dic₃	—	5	2	no	—
141	C₄⋊Dic₃	—	5	2	no	—
142	C₄⋊Dic₃	—	5	4	no	—
143	C₄⋊Dic₃	—	10	2	no	—
144	C₄⋊Dic₃	—	10	1	no	—
145	C₄⋊Dic₃	—	10	1	no	—
146	Dic₃⋊C₄	—	10	4	no	—
147	Dic₃⋊C₄	—	10	4	no	—
148	C₄×Dic₃	—	20	2	no	—
149	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
154	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
155	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
156	C₂×C₄×C₅	#237	1	2	yes	—
157	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
158	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
159	C₂×Dic₅	—	6	4	no	—
160	C₂×Dic₅	—	6	2	no	—
161	C₂×Dic₅	—	6	2	no	—
162	C₂×Dic₅	—	12	2	no	—
163	C₂×Dic₅	—	12	2	no	—
164	C₂×Dic₅	—	12	2	no	—
165	C₂×Dic₅	—	12	1	no	—
166	C₂×Dic₅	—	12	1	no	—
167	C₅×C₈	—	4	2	no	—
168	C₂²×C₈	C₂×D₁₅	1	2	yes	—
169	#125	—	15	4	no	—
170	#125	—	30	2	no	—
171	C₂×C₄⋊C₄	—	15	1	no	—
172	C₂×C₄⋊C₄	—	15	1	no	—
173	C₂×C₄⋊C₄	—	15	2	no	—
174	C₂×C₄⋊C₄	—	30	1	no	—
175	C₂×C₄⋊C₄	—	30	1	no	—
176	C₂×C₄²	—	30	1	no	—
177	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
178	C₄⋊C₈	—	2	2	no	—
179	#95	—	30	1	no	—
180	C₂×C₃×C₅	#391	1	1	yes	characteristic
181	C₂×C₃×C₅	#416	1	1	yes	characteristic
182	C₂×C₃×C₅	#417	1	1	yes	characteristic
183	C₂×C₃×C₅	#478	1	2	yes	—
184	C₂×C₃×C₅	#479	1	2	yes	—
185	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	#685	1	2	yes	—
190	C₂×C₃×C₄	#685	1	2	yes	—
191	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
192	C₂×C₃×C₄	#711	1	2	yes	—
193	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
194	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
195	C₂×Dic₃	—	10	4	no	—
196	C₂×Dic₃	—	10	2	no	—
197	C₂×Dic₃	—	10	2	no	—
198	C₂×Dic₃	—	20	2	no	—
199	C₂×Dic₃	—	20	2	no	—
200	C₂×Dic₃	—	20	2	no	—
201	C₂×Dic₃	—	20	1	no	—
202	C₂×Dic₃	—	20	1	no	—
203	C₃×C₈	—	4	2	no	—
204	C₂²×C₅	#897	1	2	yes	—
205	C₂²×C₅	#904	1	2	yes	—
206	C₂²×C₅	#908	1	1	yes	characteristic
207	C₂²×C₅	#909	1	1	yes	characteristic
208	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
209	C₄×C₅	—	2	2	no	—
210	C₄×C₅	—	2	2	no	—
211	C₄×C₅	#887	1	2	yes	—
212	C₄×C₅	#936	1	1	yes	characteristic
213	C₄×C₅	#944	1	1	yes	characteristic
214	Dic₅	—	12	2	no	—
215	Dic₅	—	12	4	no	—
216	Dic₅	—	12	2	no	—
217	C₂²×C₄	—	30	2	no	—
218	C₂²×C₄	—	30	1	no	—
219	C₂²×C₄	—	30	1	no	—
220	C₂²×C₄	D₆₀	1	1	yes	characteristic
221	C₂²×C₄	D₆₀	1	1	yes	characteristic
222	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₄⋊C₄	—	15	2	no	—
226	C₄⋊C₄	—	15	2	no	—
227	C₄⋊C₄	—	15	4	no	—
228	C₄⋊C₄	—	30	4	no	—
229	C₄⋊C₄	—	30	4	no	—
230	C₄⋊C₄	—	30	2	no	—
231	C₄⋊C₄	—	30	1	no	—
232	C₄⋊C₄	—	30	1	no	—
233	C₄²	—	60	2	no	—
234	C₄²	C₄×D₁₅	1	2	yes	—
235	C₄²	#1220	1	2	yes	—
236	C₃×C₅	#1989	1	1	yes	characteristic, complemented, Hall
237	C₂²×C₃	#4043	1	2	yes	—
238	C₂²×C₃	#4050	1	2	yes	—
239	C₂²×C₃	#4054	1	1	yes	characteristic
240	C₂²×C₃	#4055	1	1	yes	characteristic
241	C₂²×C₃	C₂×#685	1	1	yes	characteristic
242	C₃×C₄	—	2	2	no	—
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	#4033	1	2	yes	—
245	C₃×C₄	#4082	1	1	yes	characteristic
246	C₃×C₄	#4090	1	1	yes	characteristic
247	Dic₃	—	20	4	no	—
248	Dic₃	—	20	2	no	—
249	Dic₃	—	20	2	no	—
250	C₂×C₅	#4829	1	2	yes	—
251	C₂×C₅	#4847	1	2	yes	—
252	C₂×C₅	#4973	1	1	yes	characteristic
253	C₂×C₅	#5113	1	1	yes	characteristic
254	C₂×C₅	#5133	1	1	yes	characteristic
255	C₂³	C₂×D₆₀	1	1	yes	characteristic
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	30	4	no	—
260	C₂×C₄	—	30	2	no	—
261	C₂×C₄	—	30	2	no	—
262	C₂×C₄	—	60	2	no	—
263	C₂×C₄	—	60	2	no	—
264	C₂×C₄	—	60	2	no	—
265	C₂×C₄	—	60	1	no	—
266	C₂×C₄	—	60	1	no	—
267	C₂×C₄	#6714	1	2	yes	—
268	C₂×C₄	#6714	1	2	yes	—
269	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
270	C₂×C₄	#6819	1	2	yes	—
271	C₂×C₄	Q₈×D₁₅	1	2	yes	—
272	C₂×C₄	C₂×#1220	1	1	yes	characteristic
273	C₈	—	4	2	no	—
274	C₂×C₃	#65118	1	2	yes	—
275	C₂×C₃	#65136	1	2	yes	—
276	C₂×C₃	#65263	1	1	yes	characteristic
277	C₂×C₃	#65402	1	1	yes	characteristic
278	C₂×C₃	#65422	1	1	yes	characteristic
279	C₅	#70461	1	1	yes	characteristic, complemented, Sylow
280	C₂²	#91786	1	2	yes	—
281	C₂²	#91793	1	2	yes	—
282	C₂²	#91797	1	1	yes	characteristic
283	C₂²	#91798	1	1	yes	characteristic
284	C₂²	C₂×#6714	1	1	yes	characteristic
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	60	4	no	—
288	C₄	—	60	2	no	—
289	C₄	—	60	2	no	—
290	C₄	#91776	1	2	yes	—
291	C₄	#91825	1	1	yes	characteristic
292	C₄	#91833	1	1	yes	characteristic
293	C₃	#10601254	1	1	yes	characteristic, complemented, Sylow
294	C₂	#11750930	1	2	yes	—
295	C₂	#11750948	1	2	yes	—
296	C₂	#11751074	1	1	yes	characteristic
297	C₂	#11751214	1	1	yes	characteristic
298	C₂	#11751234	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₀
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂×C₃×C₅×C₄⋊C₈	C₂
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₅
hypocenter	C₃×C₅	#1989
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
16. #91759	C₄	semidirect
236. C₃×C₅	#1989	semidirect
279. C₅	#70461	semidirect
293. C₃	#10601254	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91759	2
C₂×C₃×C₅×C₄⋊C₈	2
C₄×C₄⋊Dic₁₅	2
#11746372	2
#10601254	3
#70461	5