Subgroups — #14439872

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11756229	C₂	1	1	yes	maximal, characteristic
2	C₃×#65804	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11748451	C₂	1	4	yes	maximal
5	#11746960	C₂	1	2	yes	maximal, complemented
6	#11746950	C₂	1	4	yes	maximal, complemented
7	#11746948	C₂	1	2	yes	maximal, complemented
8	#10610173	—	3	1	no	maximal, Hall
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	#91815	C₂²	1	2	yes	—
11	#91813	—	2	2	no	—
12	#91759	C₂²	1	2	yes	—
13	#91758	C₂²	1	2	yes	—
14	C₄×Dic₃₀	—	2	2	no	—
15	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
16	C₃×#4072	C₂²	1	2	yes	—
17	C₃×#4070	—	2	2	no	—
18	C₃×#4016	C₂²	1	2	yes	—
19	C₃×#4015	C₂²	1	2	yes	—
20	C₃×C₄×Dic₁₀	—	2	2	no	—
21	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	C₄×C₅×C₃⋊C₈	C₂²	1	2	yes	—
23	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#90990	C₂²	1	8	yes	—
25	#90987	C₂²	1	8	yes	—
26	#83339	—	5	1	no	maximal, Hall
27	#65804	—	3	1	no	—
28	#65804	D₃	1	1	yes	characteristic
29	C₂×C₄×C₅×C₈	—	3	1	no	—
30	#65395	—	3	4	no	—
31	#64972	—	3	4	no	—
32	#64971	—	3	2	no	—
33	#64969	—	3	2	no	—
34	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₂×Dic₃₀	—	2	2	no	—
37	C₂×C₃×Dic₁₀	—	2	2	no	—
38	C₃×C₄²×C₅	D₄	1	2	yes	—
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	#6719	—	2	2	no	—
42	C₄⋊Dic₁₅	—	2	2	no	—
43	C₄⋊Dic₁₅	C₂³	1	4	yes	—
44	#6712	—	2	4	no	—
45	C₄×Dic₁₅	—	2	2	no	—
46	C₂×C₅×C₃⋊C₈	Q₈	1	8	yes	complemented
47	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
48	C₂×C₅×C₃⋊C₈	C₂³	1	2	yes	—
49	C₃×C₂²⋊Dic₅	—	2	2	no	—
50	C₃×C₄⋊Dic₅	—	2	2	no	—
51	C₃×C₄⋊Dic₅	C₂³	1	4	yes	—
52	C₃×#683	—	2	4	no	—
53	C₃×C₄×Dic₅	—	2	2	no	—
54	C₃×#533	—	5	1	no	—
55	#5470	—	5	1	no	—
56	#5080	—	5	2	no	—
57	#5078	—	5	2	no	—
58	#5065	—	5	4	no	—
59	#4998	—	5	4	no	—
60	C₂×C₄×C₃⋊C₈	D₅	1	1	yes	characteristic
61	C₂²×C₅×C₈	—	3	2	no	—
62	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
63	#4072	—	3	2	no	—
64	#4072	C₂×D₃	1	2	yes	—
65	#4070	—	2	2	no	—
66	#4070	—	6	2	no	—
67	#4016	—	3	2	no	—
68	#4016	C₂×D₃	1	2	yes	—
69	#4015	—	3	2	no	—
70	#4015	C₂×D₃	1	2	yes	—
71	C₄×Dic₁₀	—	2	2	no	—
72	C₄×Dic₁₀	—	6	2	no	—
73	C₄×C₅×C₈	—	3	1	no	—
74	C₄×C₅×C₈	—	3	1	no	—
75	C₄×C₅×C₈	—	3	2	no	—
76	#3950	—	3	8	no	—
77	#3949	—	3	8	no	—
78	#3160	—	15	1	no	Sylow
79	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	—	2	4	no	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂×C₃×C₄×C₅	C₂×Q₈	1	4	yes	—
85	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
86	C₂×Dic₁₅	—	2	4	no	—
87	Dic₃₀	—	4	4	no	—
88	C₂×C₃×Dic₅	—	2	4	no	—
89	C₃×Dic₁₀	—	4	4	no	—
90	C₅×C₃⋊C₈	C₂×Q₈	1	4	yes	—
91	C₅×C₃⋊C₈	C₂×Q₈	1	4	yes	—
92	C₃×C₄⋊Q₈	—	5	2	no	—
93	C₃×#125	—	5	2	no	—
94	C₃×C₂²⋊Q₈	—	10	2	no	—
95	C₃×C₄×Q₈	—	10	2	no	—
96	C₃×#117	—	5	2	no	—
97	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
98	#926	—	5	2	no	—
99	#924	—	10	2	no	—
100	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
101	#870	—	5	2	no	—
102	#869	—	5	2	no	—
103	C₄×Dic₆	—	10	2	no	—
104	#808	—	5	8	no	—
105	#807	—	5	8	no	—
106	C₄×C₃⋊C₈	C₂×D₅	1	2	yes	—
107	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
108	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
109	C₂²×C₄×C₅	#67	1	2	yes	—
110	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
111	C₂×Dic₁₀	—	2	2	no	—
112	C₂×Dic₁₀	—	6	2	no	—
113	C₂×C₅×C₈	—	3	8	no	—
114	C₂×C₅×C₈	—	3	2	no	—
115	C₂×C₅×C₈	—	3	2	no	—
116	C₄²×C₅	#67	1	2	yes	—
117	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
118	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
119	C₂²⋊Dic₅	—	2	2	no	—
120	C₂²⋊Dic₅	—	6	2	no	—
121	C₄⋊Dic₅	—	2	2	no	—
122	C₄⋊Dic₅	—	3	4	no	—
123	C₄⋊Dic₅	—	6	2	no	—
124	C₄⋊Dic₅	C₂²×D₃	1	4	yes	—
125	#683	—	2	4	no	—
126	#683	—	6	4	no	—
127	C₄×Dic₅	—	2	2	no	—
128	C₄×Dic₅	—	6	2	no	—
129	#533	—	5	1	no	—
130	#533	—	15	1	no	—
131	#500	—	15	2	no	—
132	#499	—	15	4	no	—
133	#498	—	15	2	no	—
134	#427	—	15	4	no	—
135	C₂×C₄×C₈	—	3	1	no	—
136	C₂²×C₃×C₅	—	2	2	no	—
137	C₂²×C₃×C₅	C₄⋊Q₈	1	2	yes	—
138	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₃×C₄×C₅	—	2	2	no	—
140	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
141	C₃×C₄×C₅	#135	1	4	yes	—
142	C₃×C₄×C₅	C₂²×Q₈	1	2	yes	—
143	Dic₁₅	—	4	4	no	—
144	C₃×Dic₅	—	4	4	no	—
145	C₂×C₃×Q₈	—	10	2	no	—
146	C₂²×C₃×C₄	D₂₀	1	2	yes	—
147	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
148	C₂×Dic₆	—	10	2	no	—
149	C₃×C₄⋊C₄	—	5	4	no	—
150	C₃×C₄⋊C₄	—	10	4	no	—
151	C₃×C₄⋊C₄	—	10	2	no	—
152	C₃×C₂²⋊C₄	—	10	2	no	—
153	C₃×C₄²	—	10	2	no	—
154	C₃×C₄²	D₂₀	1	2	yes	—
155	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
156	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
157	#217	—	10	2	no	—
158	C₄⋊Dic₃	—	5	4	no	—
159	C₄⋊Dic₃	—	10	2	no	—
160	Dic₃⋊C₄	—	10	4	no	—
161	C₄×Dic₃	—	10	2	no	—
162	C₂×C₃⋊C₈	Dic₁₀	1	8	yes	complemented
163	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
164	C₂×C₃⋊C₈	C₂²×D₅	1	2	yes	—
165	C₂³×C₅	C₂×#67	1	1	yes	characteristic
166	C₂×C₄×C₅	—	2	4	no	—
167	C₂×C₄×C₅	D₃×Q₈	1	4	yes	—
168	C₂×C₄×C₅	C₂×#67	1	2	yes	—
169	C₂×C₄×C₅	C₂×#67	1	2	yes	—
170	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
172	C₂×Dic₅	—	2	4	no	—
173	C₂×Dic₅	—	6	4	no	—
174	Dic₁₀	—	4	4	no	—
175	Dic₁₀	—	12	4	no	—
176	C₅×C₈	—	3	4	no	—
177	C₅×C₈	—	3	4	no	—
178	C₂²×C₈	—	3	2	no	—
179	C₄⋊Q₈	—	5	2	no	—
180	C₄⋊Q₈	—	15	2	no	—
181	#125	—	5	2	no	—
182	#125	—	15	2	no	—
183	C₂²⋊Q₈	—	10	2	no	—
184	C₂²⋊Q₈	—	30	2	no	—
185	C₄×Q₈	—	10	2	no	—
186	C₄×Q₈	—	30	2	no	—
187	#117	—	5	2	no	—
188	#117	—	15	2	no	—
189	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
190	#107	—	15	8	no	—
191	#106	—	15	8	no	—
192	C₄×C₈	—	3	1	no	—
193	C₄×C₈	—	3	2	no	—
194	C₄×C₈	—	3	1	no	—
195	C₂×C₃×C₅	—	2	2	no	—
196	C₂×C₃×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
197	C₂×C₃×C₅	C₂×#135	1	2	yes	—
198	C₂³×C₃	C₂×D₂₀	1	1	yes	characteristic
199	C₃×Q₈	—	20	4	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	10	4	no	—
202	C₂×C₃×C₄	C₂×Dic₁₀	1	4	yes	—
203	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
204	C₂×C₃×C₄	C₂×D₂₀	1	2	yes	—
205	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
207	C₂×Dic₃	—	10	4	no	—
208	Dic₆	—	20	4	no	—
209	C₃⋊C₈	C₂×Dic₁₀	1	4	yes	—
210	C₃⋊C₈	C₂×Dic₁₀	1	4	yes	—
211	C₂²×C₅	—	2	2	no	—
212	C₂²×C₅	#944	1	2	yes	—
213	C₂²×C₅	C₂²×#67	1	1	yes	characteristic
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	#944	1	2	yes	—
216	C₄×C₅	#950	1	4	yes	—
217	C₄×C₅	C₂×D₃×Q₈	1	2	yes	—
218	Dic₅	—	4	4	no	—
219	Dic₅	—	12	4	no	—
220	C₂×Q₈	—	10	2	no	—
221	C₂×Q₈	—	30	2	no	—
222	C₂²×C₄	#1202	1	2	yes	—
223	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
224	C₂×C₈	—	3	8	no	—
225	C₂×C₈	—	3	2	no	—
226	C₂×C₈	—	3	2	no	—
227	C₄⋊C₄	—	5	4	no	—
228	C₄⋊C₄	—	10	4	no	—
229	C₄⋊C₄	—	10	2	no	—
230	C₄⋊C₄	—	15	4	no	—
231	C₄⋊C₄	—	30	4	no	—
232	C₄⋊C₄	—	30	2	no	—
233	C₂²⋊C₄	—	10	2	no	—
234	C₂²⋊C₄	—	30	2	no	—
235	C₄²	—	10	2	no	—
236	C₄²	—	30	2	no	—
237	C₄²	#1202	1	2	yes	—
238	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
239	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
240	C₃×C₅	#3160	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	#4015	1	2	yes	—
243	C₂²×C₃	C₂²×D₂₀	1	1	yes	characteristic
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	20	4	no	—
246	C₃×C₄	#4015	1	2	yes	—
247	C₃×C₄	#4050	1	4	yes	—
248	C₃×C₄	C₂²×Dic₁₀	1	2	yes	—
249	Dic₃	—	20	4	no	—
250	C₂×C₅	—	2	2	no	—
251	C₂×C₅	C₂×#944	1	1	yes	characteristic
252	C₂×C₅	C₂×#950	1	2	yes	—
253	C₂³	C₂×#1202	1	1	yes	characteristic
254	Q₈	—	20	4	no	—
255	Q₈	—	60	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	10	4	no	—
258	C₂×C₄	—	30	4	no	—
259	C₂×C₄	D₃×Dic₁₀	1	4	yes	—
260	C₂×C₄	C₂×#1202	1	2	yes	—
261	C₂×C₄	C₂×#1202	1	2	yes	—
262	C₂×C₄	C₂×#1202	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
264	C₈	—	3	4	no	—
265	C₈	—	3	4	no	—
266	C₂×C₃	—	2	2	no	—
267	C₂×C₃	C₂×#4015	1	1	yes	characteristic
268	C₂×C₃	C₂×#4050	1	2	yes	—
269	C₅	#83339	1	1	yes	characteristic, complemented, Sylow
270	C₂²	—	2	2	no	—
271	C₂²	#91389	1	2	yes	—
272	C₂²	C₂²×#1202	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	20	4	no	—
275	C₄	—	60	4	no	—
276	C₄	#91311	1	4	yes	—
277	C₄	#91389	1	2	yes	—
278	C₄	C₂×D₃×Dic₁₀	1	2	yes	—
279	C₃	#10610173	1	1	yes	characteristic, complemented, Sylow
280	C₂	—	2	2	no	—
281	C₂	C₂×#91311	1	2	yes	—
282	C₂	C₂×#91389	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#1202
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂²×C₃×C₅	C₂²×D₄
prime radical	C₂³×C₃×C₅	C₂×D₄
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#3160
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
5. #11746960	C₂	semidirect
6. #11746950	C₂	semidirect
7. #11746948	C₂	semidirect
46. C₂×C₅×C₃⋊C₈	Q₈	semidirect
162. C₂×C₃⋊C₈	Dic₁₀	semidirect
240. C₃×C₅	#3160	semidirect
269. C₅	#83339	semidirect
279. C₃	#10610173	semidirect

Maximal subgroups

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

subgroup	index
#11756229	2
C₃×#65804	2
C₂×C₄×C₅×C₃⋊C₈	2
#11748451	2
#11746960	2
#11746950	2
#11746948	2
#10610173	3
#83339	5