Subgroups — #14305156

Order 1920.

480 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	C₂×#91101	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#815	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3950	C₂	1	1	yes	maximal, characteristic
4	#11748857	C₂	1	2	yes	maximal
5	#11745980	C₂	1	2	yes	maximal
6	#10600612	—	3	1	no	maximal, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
9	C₂×C₄×C₅×Dic₃	C₂²	1	1	yes	characteristic
10	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
12	C₂×#6668	—	2	2	no	—
13	#91101	—	2	2	no	—
14	C₅×#815	—	2	2	no	—
15	C₃×#3950	C₄	1	4	yes	complemented
16	#90994	—	2	2	no	—
17	#70613	—	5	1	no	maximal, Hall
18	C₂×C₅×#97	—	3	1	no	—
19	C₂×#3950	D₃	1	1	yes	characteristic
20	C₂×#3949	—	3	1	no	—
21	#65217	—	3	2	no	—
22	#64771	—	3	2	no	—
23	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×Dic₁₅	—	2	2	no	—
25	C₂²×C₅×Dic₃	D₄	1	2	yes	—
26	C₂²×C₃×Dic₅	—	2	2	no	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	D₄	1	2	yes	—
29	C₂×C₃×C₅×C₈	Q₈	1	2	yes	—
30	C₄⋊Dic₁₅	—	2	2	no	—
31	C₄⋊Dic₁₅	—	2	2	no	—
32	C₄×C₅×Dic₃	—	2	1	no	—
33	C₄×C₅×Dic₃	—	2	1	no	—
34	C₂×C₅×C₃⋊C₈	—	2	1	no	—
35	C₂×C₅×C₃⋊C₈	—	2	1	no	—
36	C₂×C₅×C₃⋊C₈	Q₈	1	2	yes	—
37	C₃×C₄⋊Dic₅	—	2	2	no	—
38	C₃×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
39	#6668	—	4	4	no	—
40	C₂×C₃×#107	—	5	1	no	—
41	C₂×#817	—	5	1	no	—
42	C₂×#815	D₅	1	1	yes	characteristic
43	#4984	—	5	2	no	—
44	#4529	—	5	2	no	—
45	C₂²×C₅×C₈	—	3	1	no	—
46	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
47	C₂×C₄²×C₅	—	3	1	no	—
48	C₂×C₄⋊Dic₅	—	3	2	no	—
49	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
50	C₂×#683	—	6	2	no	—
51	C₅×#97	—	6	2	no	—
52	#3963	—	6	2	no	—
53	#3950	Dic₃	1	4	yes	complemented
54	#3949	—	6	2	no	—
55	#1943	—	15	1	no	Sylow
56	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
62	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
63	C₂×Dic₁₅	—	4	2	no	—
64	C₂×Dic₁₅	—	4	2	no	—
65	C₂×Dic₁₅	—	4	2	no	—
66	C₂×C₅×Dic₃	—	2	2	no	—
67	C₂×C₅×Dic₃	—	2	2	no	—
68	C₂×C₅×Dic₃	—	2	2	no	—
69	C₂×C₃×Dic₅	—	2	4	no	—
70	C₂×C₃×Dic₅	—	4	2	no	—
71	C₂×C₃×Dic₅	—	4	2	no	—
72	C₃×C₅×C₈	C₄⋊C₄	1	4	yes	complemented
73	C₅×C₃⋊C₈	—	2	2	no	—
74	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
75	C₂×C₃×C₄⋊C₄	—	5	2	no	—
76	C₂×C₄⋊Dic₃	—	5	2	no	—
77	C₂×C₄⋊Dic₃	—	10	2	no	—
78	C₂×C₄×Dic₃	C₂×D₅	1	1	yes	characteristic
79	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
80	C₃×#107	—	5	4	no	—
81	C₃⋊#95	—	10	2	no	—
82	#817	—	10	2	no	—
83	#815	—	2	2	no	—
84	C₂²×C₄×C₅	—	3	2	no	—
85	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
86	C₂²×Dic₅	—	2	2	no	—
87	C₂²×Dic₅	—	6	2	no	—
88	C₂×C₅×C₈	—	3	2	no	—
89	C₂×C₅×C₈	—	6	1	no	—
90	C₂×C₅×C₈	—	6	1	no	—
91	C₂×C₅×C₈	Dic₆	1	2	yes	—
92	C₂×C₅×C₈	D₁₂	1	2	yes	—
93	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
94	C₄²×C₅	—	6	1	no	—
95	C₄²×C₅	—	6	1	no	—
96	C₄⋊Dic₅	—	2	2	no	—
97	C₄⋊Dic₅	—	6	2	no	—
98	C₄⋊Dic₅	—	6	2	no	—
99	C₄⋊Dic₅	C₂×Dic₃	1	4	yes	—
100	#683	—	12	4	no	—
101	C₂×#107	—	5	1	no	—
102	C₂×#106	—	15	1	no	—
103	C₂×#97	—	3	1	no	—
104	#389	—	15	2	no	—
105	#340	—	15	2	no	—
106	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
107	C₂²×C₃×C₅	#121	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
109	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
110	C₂²×C₃×C₅	#137	1	2	yes	—
111	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
113	C₃×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
114	C₃×C₄×C₅	#125	1	1	yes	characteristic
115	Dic₁₅	—	8	2	no	—
116	C₃×Dic₅	—	4	4	no	—
117	C₅×Dic₃	—	4	2	no	—
118	C₂²×C₃×C₄	—	10	2	no	—
119	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
120	C₂²×Dic₃	—	10	2	no	—
121	C₂²×Dic₃	D₂₀	1	2	yes	—
122	C₂×C₃×C₈	Dic₁₀	1	2	yes	—
123	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
124	C₂×C₃×C₈	C₂²⋊D₅	1	2	yes	—
125	C₃×C₄⋊C₄	—	5	4	no	—
126	C₃×C₄⋊C₄	—	10	2	no	—
127	C₄⋊Dic₃	—	10	2	no	—
128	C₄⋊Dic₃	—	10	2	no	—
129	C₄⋊Dic₃	—	20	4	no	—
130	C₄×Dic₃	—	2	1	no	—
131	C₄×Dic₃	—	2	1	no	—
132	C₂×C₃⋊C₈	—	2	1	no	—
133	C₂×C₃⋊C₈	—	2	1	no	—
134	C₂×C₃⋊C₈	Dic₁₀	1	2	yes	—
135	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅	—	6	2	no	—
137	C₂×C₄×C₅	—	6	2	no	—
138	C₂×C₄×C₅	—	6	2	no	—
139	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
140	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
141	C₂×C₄×C₅	#237	1	1	yes	characteristic
142	C₂×C₄×C₅	D₃×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₅	#239	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
145	C₂×Dic₅	—	2	4	no	—
146	C₂×Dic₅	—	4	2	no	—
147	C₂×Dic₅	—	4	2	no	—
148	C₂×Dic₅	—	12	2	no	—
149	C₂×Dic₅	—	12	2	no	—
150	C₂×Dic₅	—	12	2	no	—
151	C₅×C₈	—	6	2	no	—
152	C₅×C₈	C₄⋊Dic₃	1	4	yes	complemented
153	C₂²×C₈	—	3	1	no	—
154	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
155	C₂×C₄⋊C₄	—	5	2	no	—
156	C₂×C₄⋊C₄	—	15	2	no	—
157	C₂×C₄⋊C₄	—	30	2	no	—
158	C₂×C₄²	—	3	1	no	—
159	#107	—	5	4	no	—
160	#106	—	30	2	no	—
161	#97	—	6	2	no	—
162	#95	—	30	2	no	—
163	C₂×C₃×C₅	#389	1	1	yes	characteristic
164	C₂×C₃×C₅	#441	1	1	yes	characteristic
165	C₂×C₃×C₅	#442	1	1	yes	characteristic
166	C₂×C₃×C₅	#468	1	2	yes	—
167	C₂×C₃×C₅	#501	1	2	yes	—
168	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
169	C₂×C₃×C₄	—	10	4	no	—
170	C₂×C₃×C₄	—	20	2	no	—
171	C₂×C₃×C₄	—	20	2	no	—
172	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
175	C₂×C₃×C₄	#709	1	1	yes	characteristic
176	C₂×C₃×C₄	#709	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
178	C₂×Dic₃	—	2	2	no	—
179	C₂×Dic₃	—	2	2	no	—
180	C₂×Dic₃	—	2	2	no	—
181	C₂×Dic₃	—	20	2	no	—
182	C₂×Dic₃	—	20	2	no	—
183	C₂×Dic₃	—	20	2	no	—
184	C₃×C₈	#683	1	4	yes	complemented
185	C₃⋊C₈	—	2	2	no	—
186	C₂²×C₅	#888	1	1	yes	characteristic
187	C₂²×C₅	#895	1	1	yes	characteristic
188	C₂²×C₅	#911	1	2	yes	—
189	C₂²×C₅	#918	1	2	yes	—
190	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
191	C₄×C₅	—	12	2	no	—
192	C₄×C₅	#890	1	1	yes	characteristic
193	C₄×C₅	#897	1	1	yes	characteristic
194	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
195	C₄×C₅	Q₈×Dic₃	1	1	yes	characteristic
196	Dic₅	—	4	4	no	—
197	Dic₅	—	24	2	no	—
198	C₂²×C₄	—	3	2	no	—
199	C₂²×C₄	—	10	2	no	—
200	C₂²×C₄	—	30	2	no	—
201	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
202	C₂×C₈	—	3	2	no	—
203	C₂×C₈	—	6	1	no	—
204	C₂×C₈	—	6	1	no	—
205	C₂×C₈	D₅×Dic₃	1	2	yes	—
206	C₂×C₈	#1203	1	2	yes	—
207	C₂×C₈	#1204	1	2	yes	—
208	C₄⋊C₄	—	5	4	no	—
209	C₄⋊C₄	—	10	2	no	—
210	C₄⋊C₄	—	30	2	no	—
211	C₄⋊C₄	—	30	2	no	—
212	C₄⋊C₄	—	60	4	no	—
213	C₄²	—	6	1	no	—
214	C₄²	—	6	1	no	—
215	C₃×C₅	#1943	1	1	yes	characteristic, complemented, Hall
216	C₂²×C₃	#4015	1	1	yes	characteristic
217	C₂²×C₃	C₄×D₂₀	1	1	yes	characteristic
218	C₂²×C₃	#4054	1	2	yes	—
219	C₂²×C₃	#4055	1	2	yes	—
220	C₂²×C₃	#4076	1	1	yes	characteristic
221	C₃×C₄	—	20	4	no	—
222	C₃×C₄	C₄×Dic₁₀	1	1	yes	characteristic
223	C₃×C₄	#4016	1	1	yes	characteristic
224	C₃×C₄	C₂×#683	1	1	yes	characteristic
225	C₃×C₄	#4070	1	1	yes	characteristic
226	Dic₃	—	4	2	no	—
227	Dic₃	—	40	2	no	—
228	C₂×C₅	#4876	1	2	yes	—
229	C₂×C₅	#4886	1	2	yes	—
230	C₂×C₅	#4984	1	1	yes	characteristic
231	C₂×C₅	#5152	1	1	yes	characteristic
232	C₂×C₅	#5184	1	1	yes	characteristic
233	C₂³	D₃×D₂₀	1	1	yes	characteristic
234	C₂×C₄	—	6	2	no	—
235	C₂×C₄	—	6	2	no	—
236	C₂×C₄	—	6	2	no	—
237	C₂×C₄	—	10	4	no	—
238	C₂×C₄	—	20	2	no	—
239	C₂×C₄	—	20	2	no	—
240	C₂×C₄	—	60	2	no	—
241	C₂×C₄	—	60	2	no	—
242	C₂×C₄	—	60	2	no	—
243	C₂×C₄	#6765	1	1	yes	characteristic
244	C₂×C₄	D₃×Dic₁₀	1	1	yes	characteristic
245	C₂×C₄	#6769	1	1	yes	characteristic
246	C₂×C₄	C₂×D₅×Dic₃	1	1	yes	characteristic
247	C₂×C₄	C₂×#1203	1	1	yes	characteristic
248	C₂×C₄	C₂×#1204	1	1	yes	characteristic
249	C₈	—	6	2	no	—
250	C₈	#6668	1	4	yes	complemented
251	C₂×C₃	#64991	1	2	yes	—
252	C₂×C₃	#65000	1	1	yes	characteristic
253	C₂×C₃	#65005	1	1	yes	characteristic
254	C₂×C₃	#65217	1	1	yes	characteristic
255	C₂×C₃	#65423	1	2	yes	—
256	C₅	#70613	1	1	yes	characteristic, complemented, Sylow
257	C₂²	#91254	1	2	yes	—
258	C₂²	#91263	1	2	yes	—
259	C₂²	Dic₃×D₂₀	1	1	yes	characteristic
260	C₂²	#91460	1	1	yes	characteristic
261	C₂²	#91469	1	1	yes	characteristic
262	C₄	—	12	2	no	—
263	C₄	—	20	4	no	—
264	C₄	—	120	2	no	—
265	C₄	Dic₃×Dic₁₀	1	1	yes	characteristic
266	C₄	#91365	1	1	yes	characteristic
267	C₄	#91381	1	1	yes	characteristic
268	C₄	C₂×#6668	1	1	yes	characteristic
269	C₃	#10600612	1	1	yes	characteristic, complemented, Sylow
270	C₂	#11747799	1	1	yes	characteristic
271	C₂	#11747870	1	1	yes	characteristic
272	C₂	#11747923	1	2	yes	—
273	C₂	#11748032	1	2	yes	—
274	C₂	#11748857	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₃×D₂₀
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#1943
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
15. C₃×#3950	C₄	semidirect
53. #3950	Dic₃	semidirect
72. C₃×C₅×C₈	C₄⋊C₄	semidirect
152. C₅×C₈	C₄⋊Dic₃	semidirect
184. C₃×C₈	#683	semidirect
215. C₃×C₅	#1943	semidirect
250. C₈	#6668	semidirect
256. C₅	#70613	semidirect
269. C₃	#10600612	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91101	2
C₂×C₅×#815	2
C₂×C₃×#3950	2
#11748857	2
#11745980	2
#10600612	3
#70613	5