Subgroups — #14306032

Order 1920.

640 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756236	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#91109	C₂	1	4	yes	maximal, complemented
4	#11751033	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601306	—	3	1	no	maximal, Hall
6	C₂×#6817	C₂²	1	2	yes	—
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	#91819	—	2	2	no	—
10	C₂×C₄×Dic₁₅	C₂²	1	2	yes	—
11	#91764	C₄	1	2	yes	complemented
12	#91763	C₄	1	1	yes	characteristic, complemented
13	C₄×D₆₀	—	2	2	no	—
14	#91758	C₄	1	1	yes	characteristic, complemented
15	C₃×C₅×C₄⋊C₈	—	2	2	no	—
16	#91115	—	2	2	no	—
17	#91109	—	2	8	no	—
18	#70513	—	5	1	no	maximal, Hall
19	#65811	—	3	1	no	—
20	C₅×#423	D₃	1	1	yes	characteristic, complemented
21	C₂×#3957	—	3	4	no	—
22	#65221	—	3	1	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂×#1220	—	2	2	no	—
26	C₂²×Dic₁₅	—	2	2	no	—
27	#6817	—	2	4	no	—
28	#6817	—	2	4	no	—
29	C₂×D₆₀	—	2	2	no	—
30	C₂×D₆₀	C₂×C₄	1	2	yes	—
31	C₂×C₄×D₁₅	—	2	2	no	—
32	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
33	C₃×C₅×M₁₆	—	2	4	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	D₄	1	2	yes	—
38	#6714	—	2	4	no	—
39	C₄⋊Dic₁₅	—	2	2	no	—
40	C₄×Dic₁₅	—	2	2	no	—
41	C₄×Dic₁₅	D₄	1	4	yes	complemented
42	#5477	—	5	1	no	—
43	C₃×#423	D₅	1	1	yes	characteristic, complemented
44	C₂×#825	—	5	4	no	—
45	#4932	—	5	1	no	—
46	C₂×#709	—	3	2	no	—
47	C₂×C₅×M₁₆	C₂×D₃	1	2	yes	—
48	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
49	#4076	—	6	2	no	—
50	C₂×C₄×Dic₅	—	3	2	no	—
51	#4021	—	3	2	no	—
52	#4020	—	3	1	no	—
53	C₄×D₂₀	—	6	2	no	—
54	#4015	—	3	1	no	—
55	C₅×C₄⋊C₈	—	2	2	no	—
56	#3963	—	6	2	no	—
57	#3957	—	6	8	no	—
58	#1963	—	15	1	no	Sylow
59	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂²×D₁₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	4	2	no	—
62	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	#41	1	2	yes	—
68	#1220	—	4	4	no	—
69	C₂×Dic₁₅	—	2	2	no	—
70	C₂×Dic₁₅	—	2	4	no	—
71	C₂×Dic₁₅	—	4	4	no	—
72	D₆₀	—	2	4	no	—
73	D₆₀	—	4	4	no	—
74	C₄×D₁₅	—	4	2	no	—
75	C₄×D₁₅	—	4	2	no	—
76	Dic₃₀	—	2	4	no	—
77	C₃×C₅×C₈	—	4	2	no	—
78	C₂×#235	—	5	2	no	—
79	C₂×C₃×M₁₆	C₂×D₅	1	2	yes	—
80	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
81	#930	—	10	2	no	—
82	C₂×C₄×Dic₃	—	5	2	no	—
83	#875	—	5	2	no	—
84	#874	—	5	1	no	—
85	C₄×D₁₂	—	10	2	no	—
86	#869	—	5	1	no	—
87	C₃×C₄⋊C₈	—	2	2	no	—
88	C₃⋊#95	—	10	2	no	—
89	#825	—	10	8	no	—
90	C₂²×C₄×C₅	D₁₂	1	2	yes	—
91	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
92	C₂×C₂²⋊D₅	—	6	2	no	—
93	C₂²×Dic₅	—	6	2	no	—
94	#709	—	6	4	no	—
95	#709	—	6	4	no	—
96	C₂×D₂₀	—	3	2	no	—
97	C₂×D₂₀	—	6	2	no	—
98	C₂×C₄×D₅	—	6	2	no	—
99	C₂×Dic₁₀	—	3	2	no	—
100	C₅×M₁₆	—	2	4	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
103	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
104	C₄²×C₅	#67	1	2	yes	—
105	#685	—	6	4	no	—
106	C₄⋊Dic₅	—	6	2	no	—
107	C₄×Dic₅	—	3	4	no	—
108	C₄×Dic₅	—	6	2	no	—
109	#532	—	15	1	no	—
110	#423	D₁₅	1	1	yes	characteristic, complemented
111	C₂×#104	—	15	4	no	—
112	#376	—	15	1	no	—
113	C₂²×C₃×C₅	—	2	2	no	—
114	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂²×C₃×C₅	#121	1	2	yes	—
116	C₂×D₁₅	—	4	2	no	—
117	C₂×D₁₅	—	4	4	no	—
118	C₃×C₄×C₅	—	2	2	no	—
119	C₃×C₄×C₅	—	2	2	no	—
120	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
121	C₃×C₄×C₅	#124	1	1	yes	characteristic
122	C₃×C₄×C₅	#127	1	1	yes	characteristic
123	Dic₁₅	—	4	4	no	—
124	Dic₁₅	—	4	2	no	—
125	C₂²×C₃×C₄	D₂₀	1	2	yes	—
126	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
127	C₂×#67	—	10	2	no	—
128	C₂²×Dic₃	—	10	2	no	—
129	#235	—	10	4	no	—
130	#235	—	10	4	no	—
131	C₂×D₁₂	—	5	2	no	—
132	C₂×D₁₂	—	10	2	no	—
133	C₂×C₄×D₃	—	10	2	no	—
134	C₂×Dic₆	—	5	2	no	—
135	C₃×M₁₆	—	2	4	no	—
136	C₂×C₃×C₈	—	2	2	no	—
137	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
138	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
139	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
140	(C₂×D₃)⋊C₄	—	10	4	no	—
141	C₄⋊Dic₃	—	10	2	no	—
142	C₄×Dic₃	—	5	4	no	—
143	C₄×Dic₃	—	10	2	no	—
144	C₂³×C₅	C₂×D₁₂	1	1	yes	characteristic
145	C₂²×D₅	—	6	2	no	—
146	C₂×C₄×C₅	—	4	2	no	—
147	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
148	C₂×C₄×C₅	(C₂×D₃)⋊C₄	1	2	yes	—
149	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
150	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
151	C₂×C₄×C₅	#239	1	2	yes	—
152	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
153	C₂²⋊D₅	—	12	4	no	—
154	C₂×Dic₅	—	6	4	no	—
155	C₂×Dic₅	—	6	2	no	—
156	C₂×Dic₅	—	12	4	no	—
157	D₂₀	—	6	4	no	—
158	D₂₀	—	12	4	no	—
159	C₄×D₅	—	12	2	no	—
160	C₄×D₅	—	12	2	no	—
161	Dic₁₀	—	6	4	no	—
162	C₅×C₈	—	4	2	no	—
163	C₂×#41	—	15	2	no	—
164	C₂×M₁₆	C₂×D₁₅	1	2	yes	—
165	C₄⋊Q₈	—	15	1	no	—
166	#127	—	15	1	no	—
167	#124	—	15	2	no	—
168	#121	—	30	2	no	—
169	C₄×D₄	—	30	2	no	—
170	C₂×C₄²	—	15	2	no	—
171	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
172	C₄⋊C₈	—	2	2	no	—
173	#104	—	30	8	no	—
174	#95	—	30	2	no	—
175	C₂×C₃×C₅	—	2	2	no	—
176	C₂×C₃×C₅	#390	1	1	yes	characteristic
177	C₂×C₃×C₅	#421	1	2	yes	—
178	D₁₅	—	8	2	no	—
179	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
180	C₂²×D₃	—	10	2	no	—
181	C₂×C₃×C₄	—	4	2	no	—
182	C₂×C₃×C₄	#685	1	2	yes	—
183	C₂×C₃×C₄	#685	1	2	yes	—
184	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
185	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
186	C₂×C₃×C₄	#713	1	2	yes	—
187	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
188	#67	—	20	4	no	—
189	C₂×Dic₃	—	10	4	no	—
190	C₂×Dic₃	—	10	2	no	—
191	C₂×Dic₃	—	20	4	no	—
192	D₁₂	—	10	4	no	—
193	D₁₂	—	20	4	no	—
194	C₄×D₃	—	20	2	no	—
195	C₄×D₃	—	20	2	no	—
196	Dic₆	—	10	4	no	—
197	C₃×C₈	—	4	2	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	#895	1	2	yes	—
200	C₂²×C₅	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
201	C₂×D₅	—	12	4	no	—
202	C₂×D₅	—	12	2	no	—
203	C₄×C₅	—	2	2	no	—
204	C₄×C₅	—	2	2	no	—
205	C₄×C₅	#893	1	2	yes	—
206	C₄×C₅	#940	1	1	yes	characteristic
207	C₄×C₅	#947	1	1	yes	characteristic
208	Dic₅	—	12	4	no	—
209	Dic₅	—	12	2	no	—
210	#41	—	30	4	no	—
211	#41	—	30	4	no	—
212	C₂×Q₈	—	15	2	no	—
213	C₂×D₄	—	15	2	no	—
214	C₂×D₄	—	30	2	no	—
215	C₂×D₄	—	30	2	no	—
216	C₂²×C₄	—	30	2	no	—
217	C₂²×C₄	—	30	2	no	—
218	C₂²×C₄	D₆₀	1	2	yes	—
219	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
220	M₁₆	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₄⋊C₄	—	30	2	no	—
223	C₂²⋊C₄	—	30	4	no	—
224	C₄²	—	15	4	no	—
225	C₄²	—	30	2	no	—
226	C₄²	C₄×D₁₅	1	1	yes	characteristic
227	C₄²	C₄×D₁₅	1	1	yes	characteristic
228	C₄²	#1220	1	2	yes	—
229	C₃×C₅	#1963	1	1	yes	characteristic, complemented, Hall
230	C₂²×C₃	—	2	2	no	—
231	C₂²×C₃	#4041	1	2	yes	—
232	C₂²×C₃	C₂×#685	1	1	yes	characteristic
233	C₂×D₃	—	20	4	no	—
234	C₂×D₃	—	20	2	no	—
235	C₃×C₄	—	2	2	no	—
236	C₃×C₄	—	2	2	no	—
237	C₃×C₄	#4039	1	2	yes	—
238	C₃×C₄	#4086	1	1	yes	characteristic
239	C₃×C₄	#4093	1	1	yes	characteristic
240	Dic₃	—	20	4	no	—
241	Dic₃	—	20	2	no	—
242	C₂×C₅	—	2	2	no	—
243	C₂×C₅	#4988	1	1	yes	characteristic
244	C₂×C₅	#5139	1	2	yes	—
245	D₅	—	24	2	no	—
246	C₂³	—	30	2	no	—
247	C₂³	C₂×D₆₀	1	1	yes	characteristic
248	Q₈	—	30	4	no	—
249	D₄	—	30	4	no	—
250	D₄	—	60	4	no	—
251	D₄	—	60	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	30	4	no	—
254	C₂×C₄	—	30	2	no	—
255	C₂×C₄	—	60	4	no	—
256	C₂×C₄	—	60	2	no	—
257	C₂×C₄	—	60	2	no	—
258	C₂×C₄	#6714	1	2	yes	—
259	C₂×C₄	#6714	1	2	yes	—
260	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
261	C₂×C₄	D₄×D₁₅	1	2	yes	—
262	C₂×C₄	#6821	1	2	yes	—
263	C₂×C₄	C₂×#1220	1	1	yes	characteristic
264	C₈	—	4	2	no	—
265	C₂×C₃	—	2	2	no	—
266	C₂×C₃	#65277	1	1	yes	characteristic
267	C₂×C₃	#65428	1	2	yes	—
268	D₃	—	40	2	no	—
269	C₅	#70513	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	—	60	4	no	—
272	C₂²	—	60	2	no	—
273	C₂²	#91784	1	2	yes	—
274	C₂²	C₂×#6714	1	1	yes	characteristic
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	60	4	no	—
278	C₄	—	60	2	no	—
279	C₄	#91782	1	2	yes	—
280	C₄	#91829	1	1	yes	characteristic
281	C₄	#91836	1	1	yes	characteristic
282	C₃	#10601306	1	1	yes	characteristic, complemented, Sylow
283	C₂	—	2	2	no	—
284	C₂	—	120	2	no	—
285	C₂	#11751089	1	1	yes	characteristic
286	C₂	#11751240	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#1220
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₅×#423	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	#11756236	C₂
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	#423	D₁₅
hypocenter	C₃×C₅	#1963
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#423	C₂	semidirect
3. C₂×#91109	C₂	semidirect
4. #11751033	C₂	semidirect
11. #91764	C₄	semidirect
12. #91763	C₄	semidirect
14. #91758	C₄	semidirect
20. C₅×#423	D₃	semidirect
41. C₄×Dic₁₅	D₄	semidirect
43. C₃×#423	D₅	semidirect
110. #423	D₁₅	semidirect
229. C₃×C₅	#1963	semidirect
269. C₅	#70513	semidirect
282. C₃	#10601306	semidirect

Maximal subgroups

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

subgroup	index
#11756236	2
C₃×C₅×#423	2
C₂×#91109	2
#11751033	2
#10601306	3
#70513	5