Subgroups — #14299161

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91087	C₂	1	1	yes	maximal, characteristic
2	C₅×#4936	C₂	1	1	yes	maximal, characteristic
3	C₃×#65224	C₂	1	1	yes	maximal, characteristic
4	C₂×#90989	C₂	1	2	yes	maximal
5	C₂×#90986	C₂	1	2	yes	maximal
6	#10600343	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊#168	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
10	C₂×C₅×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄⋊Dic₅	—	2	2	no	—
12	C₂×C₃×C₄⋊Dic₅	C₂²	1	2	yes	—
13	#91087	C₄	1	4	yes	complemented
14	C₅×C₃⋊#95	—	2	2	no	—
15	#90989	—	2	4	no	—
16	#90986	—	2	4	no	—
17	#69578	—	5	1	no	maximal, Hall
18	C₅×#382	—	3	1	no	—
19	C₂×#3940	—	3	2	no	—
20	C₂×#3939	—	3	2	no	—
21	#65224	D₃	1	1	yes	characteristic
22	C₂×#3935	—	3	1	no	—
23	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
25	C₂²×C₅×Dic₃	—	2	1	no	—
26	C₂²×C₅×Dic₃	—	2	1	no	—
27	C₂²×C₃×Dic₅	—	2	2	no	—
28	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
29	C₃×C₄²×C₅	Q₈	1	2	yes	—
30	C₂×C₃⋊#168	C₂×C₄	1	4	yes	—
31	C₂×C₃⋊#168	D₄	1	4	yes	—
32	C₂×C₃⋊#168	Q₈	1	4	yes	—
33	C₅×C₄⋊Dic₃	—	2	2	no	—
34	C₅×C₄⋊Dic₃	—	2	2	no	—
35	C₃×C₄⋊Dic₅	—	2	2	no	—
36	C₃×C₄⋊Dic₅	—	2	2	no	—
37	C₃×C₄⋊Dic₅	—	4	4	no	—
38	C₃×#384	—	5	1	no	—
39	C₂×#808	—	5	2	no	—
40	C₂×#807	—	5	2	no	—
41	#4936	D₅	1	1	yes	characteristic
42	C₂×#803	—	5	1	no	—
43	C₂×C₅×C₄⋊C₄	—	3	1	no	—
44	C₂×C₅×C₄⋊C₄	—	3	1	no	—
45	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
46	C₂×C₄⋊Dic₅	—	2	2	no	—
47	C₂×C₄⋊Dic₅	C₂×D₃	1	2	yes	—
48	C₂²×#168	—	3	2	no	—
49	C₅×#95	—	6	2	no	—
50	#3940	—	6	4	no	—
51	#3939	—	6	4	no	—
52	#3935	—	3	4	no	—
53	#1847	—	15	1	no	Sylow
54	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	2	2	no	—
57	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
60	C₂×C₃×C₄×C₅	C₂×Q₈	1	2	yes	—
61	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂×C₅×Dic₃	—	4	1	no	—
63	C₂×C₅×Dic₃	—	4	1	no	—
64	C₂×C₅×Dic₃	—	4	2	no	—
65	C₂×C₅×Dic₃	—	4	2	no	—
66	C₂×C₃×Dic₅	—	4	2	no	—
67	C₂×C₃×Dic₅	—	4	4	no	—
68	C₃⋊#168	C₄⋊C₄	1	8	yes	complemented
69	C₂×C₃×C₄⋊C₄	—	5	2	no	—
70	C₂×C₃×C₄⋊C₄	—	10	2	no	—
71	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
72	C₂×C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
73	C₂×C₄⋊Dic₃	C₂×D₅	1	1	yes	characteristic
74	C₂²×C₃⋊C₈	—	5	2	no	—
75	C₃⋊#95	—	2	2	no	—
76	#808	—	10	4	no	—
77	#807	—	10	4	no	—
78	#803	—	5	4	no	—
79	C₂²×C₄×C₅	—	6	1	no	—
80	C₂²×C₄×C₅	—	6	1	no	—
81	C₂²×C₄×C₅	#67	1	2	yes	—
82	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
83	C₂²×Dic₅	—	2	2	no	—
84	C₅×C₄⋊C₄	—	6	2	no	—
85	C₅×C₄⋊C₄	—	6	2	no	—
86	C₄²×C₅	Dic₆	1	2	yes	—
87	C₄²×C₅	C₄×D₃	1	2	yes	—
88	C₄⋊Dic₅	—	2	2	no	—
89	C₄⋊Dic₅	—	2	2	no	—
90	C₄⋊Dic₅	—	4	4	no	—
91	C₂×#168	—	3	4	no	—
92	C₂×#168	—	3	4	no	—
93	C₂×#168	—	3	4	no	—
94	C₂×#107	—	15	2	no	—
95	C₂×#106	—	15	2	no	—
96	C₂×#97	—	15	1	no	—
97	#384	—	5	1	no	—
98	#382	—	3	1	no	—
99	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂²×C₃×C₅	#127	1	1	yes	characteristic
101	C₂²×C₃×C₅	C₄⋊Q₈	1	1	yes	characteristic
102	C₂²×C₃×C₅	AGL(1,ℤ/8)	1	2	yes	—
103	C₂²×C₃×C₅	#137	1	2	yes	—
104	C₃×C₄×C₅	—	2	4	no	—
105	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
106	C₃×C₄×C₅	C₄⋊Q₈	1	2	yes	—
107	C₃×Dic₅	—	8	2	no	—
108	C₅×Dic₃	—	8	1	no	—
109	C₅×Dic₃	—	8	1	no	—
110	C₂²×C₃×C₄	—	10	2	no	—
111	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
112	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
113	C₂²×Dic₃	—	2	1	no	—
114	C₂²×Dic₃	—	2	1	no	—
115	C₃×C₄⋊C₄	—	10	2	no	—
116	C₃×C₄⋊C₄	—	10	2	no	—
117	C₃×C₄⋊C₄	—	20	4	no	—
118	C₃×C₄²	Dic₁₀	1	2	yes	—
119	C₃×C₄²	C₄×D₅	1	2	yes	—
120	C₄⋊Dic₃	—	2	2	no	—
121	C₄⋊Dic₃	—	2	2	no	—
122	C₂×C₃⋊C₈	—	5	4	no	—
123	C₂×C₃⋊C₈	—	5	4	no	—
124	C₂×C₃⋊C₈	—	5	4	no	—
125	C₂³×C₅	C₂×#67	1	1	yes	characteristic
126	C₂×C₄×C₅	—	2	2	no	—
127	C₂×C₄×C₅	—	2	2	no	—
128	C₂×C₄×C₅	—	12	2	no	—
129	C₂×C₄×C₅	—	12	1	no	—
130	C₂×C₄×C₅	—	12	1	no	—
131	C₂×C₄×C₅	—	12	2	no	—
132	C₂×C₄×C₅	Dic₃⋊C₄	1	4	yes	—
133	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
135	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
136	C₂×C₄×C₅	D₃×Q₈	1	2	yes	—
137	C₂×Dic₅	—	4	4	no	—
138	C₂×Dic₅	—	4	2	no	—
139	#168	—	3	8	no	—
140	C₂²×C₈	—	15	2	no	—
141	C₂×C₄⋊C₄	—	3	1	no	—
142	C₂×C₄⋊C₄	—	3	1	no	—
143	C₂×C₄⋊C₄	—	5	2	no	—
144	C₂×C₄⋊C₄	—	10	2	no	—
145	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
146	#107	—	30	4	no	—
147	#106	—	30	4	no	—
148	#97	—	15	4	no	—
149	#95	—	6	2	no	—
150	C₂×C₃×C₅	#384	1	1	yes	characteristic
151	C₂×C₃×C₅	#428	1	2	yes	—
152	C₂×C₃×C₅	#496	1	1	yes	characteristic
153	C₂×C₃×C₅	#497	1	1	yes	characteristic
154	C₂×C₃×C₅	#501	1	2	yes	—
155	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	2	2	no	—
158	C₂×C₃×C₄	—	20	2	no	—
159	C₂×C₃×C₄	—	20	4	no	—
160	C₂×C₃×C₄	#683	1	4	yes	—
161	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
162	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
163	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
164	C₂×C₃×C₄	Q₈×D₅	1	2	yes	—
165	C₂×Dic₃	—	4	2	no	—
166	C₂×Dic₃	—	4	1	no	—
167	C₂×Dic₃	—	4	1	no	—
168	C₂×Dic₃	—	4	2	no	—
169	C₃⋊C₈	—	5	8	no	—
170	C₂²×C₅	C₂×Dic₃⋊C₄	1	1	yes	characteristic
171	C₂²×C₅	#932	1	2	yes	—
172	C₂²×C₅	#940	1	1	yes	characteristic
173	C₂²×C₅	#942	1	2	yes	—
174	C₂²×C₅	#944	1	1	yes	characteristic
175	C₄×C₅	—	2	4	no	—
176	C₄×C₅	—	24	1	no	—
177	C₄×C₅	—	24	1	no	—
178	C₄×C₅	#888	1	2	yes	—
179	C₄×C₅	D₃×C₄⋊C₄	1	2	yes	—
180	Dic₅	—	8	2	no	—
181	C₂²×C₄	—	6	1	no	—
182	C₂²×C₄	—	6	1	no	—
183	C₂²×C₄	—	10	2	no	—
184	C₂²×C₄	#1201	1	2	yes	—
185	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
186	C₂×C₈	—	15	4	no	—
187	C₂×C₈	—	15	4	no	—
188	C₂×C₈	—	15	4	no	—
189	C₄⋊C₄	—	6	2	no	—
190	C₄⋊C₄	—	6	2	no	—
191	C₄⋊C₄	—	10	2	no	—
192	C₄⋊C₄	—	10	2	no	—
193	C₄⋊C₄	—	20	4	no	—
194	C₄²	#1200	1	2	yes	—
195	C₄²	#1204	1	2	yes	—
196	C₃×C₅	#1847	1	1	yes	characteristic, complemented, Hall
197	C₂²×C₃	C₂×#683	1	1	yes	characteristic
198	C₂²×C₃	#4086	1	1	yes	characteristic
199	C₂²×C₃	#4090	1	1	yes	characteristic
200	C₂²×C₃	#4095	1	2	yes	—
201	C₂²×C₃	#4097	1	2	yes	—
202	C₃×C₄	—	2	4	no	—
203	C₃×C₄	—	40	2	no	—
204	C₃×C₄	#4034	1	2	yes	—
205	C₃×C₄	D₅×C₄⋊C₄	1	2	yes	—
206	Dic₃	—	8	1	no	—
207	Dic₃	—	8	1	no	—
208	C₂×C₅	#4964	1	2	yes	—
209	C₂×C₅	#4975	1	1	yes	characteristic
210	C₂×C₅	#5074	1	1	yes	characteristic
211	C₂×C₅	#5081	1	2	yes	—
212	C₂×C₅	#5086	1	1	yes	characteristic
213	C₂³	C₂×#1201	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	12	1	no	—
217	C₂×C₄	—	12	1	no	—
218	C₂×C₄	—	12	2	no	—
219	C₂×C₄	—	12	2	no	—
220	C₂×C₄	—	20	2	no	—
221	C₂×C₄	—	20	4	no	—
222	C₂×C₄	#6669	1	4	yes	—
223	C₂×C₄	D₁₅⋊Q₈	1	2	yes	—
224	C₂×C₄	#6777	1	2	yes	—
225	C₂×C₄	C₂×#1200	1	1	yes	characteristic
226	C₂×C₄	C₂×#1204	1	1	yes	characteristic
227	C₈	—	15	8	no	—
228	C₂×C₃	#65265	1	1	yes	characteristic
229	C₂×C₃	#65284	1	2	yes	—
230	C₂×C₃	#65347	1	1	yes	characteristic
231	C₂×C₃	#65349	1	1	yes	characteristic
232	C₂×C₃	#65354	1	2	yes	—
233	C₅	#69578	1	1	yes	characteristic, complemented, Sylow
234	C₂²	#91298	1	2	yes	—
235	C₂²	#91309	1	2	yes	—
236	C₂²	#91385	1	1	yes	characteristic
237	C₂²	#91463	1	1	yes	characteristic
238	C₂²	C₂×#6669	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	24	1	no	—
241	C₄	—	24	1	no	—
242	C₄	—	40	2	no	—
243	C₄	#91437	1	2	yes	—
244	C₄	#91470	1	2	yes	—
245	C₃	#10600343	1	1	yes	characteristic, complemented, Sylow
246	C₂	#11746969	1	2	yes	—
247	C₂	#11747041	1	1	yes	characteristic
248	C₂	#11747082	1	1	yes	characteristic
249	C₂	#11748470	1	2	yes	—
250	C₂	#11748866	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1201
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle, prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₃×D₅
hypocenter	C₃×C₅	#1847
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
13. #91087	C₄	semidirect
68. C₃⋊#168	C₄⋊C₄	semidirect
196. C₃×C₅	#1847	semidirect
233. C₅	#69578	semidirect
245. C₃	#10600343	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91087	2
C₅×#4936	2
C₃×#65224	2
C₂×#90989	2
C₂×#90986	2
#10600343	3
#69578	5