Subgroups — #14294670

Order 1920.

744 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃×#681	C₂	1	2	yes	maximal
3	#11746373	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#64772	C₂	1	1	yes	maximal, characteristic, complemented
5	#11745983	C₂	1	2	yes	maximal, complemented
6	#10602791	—	3	1	no	maximal, Hall
7	C₂²×C₄×C₅×D₃	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#6710	C₂²	1	2	yes	—
10	C₂²×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×ΓL(1,𝔽₂₅)	—	2	4	no	—
12	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	4	yes	—
13	C₂×C₃×#681	C₂²	1	2	yes	—
14	C₂×#6650	—	2	2	no	—
15	D₃×#681	—	2	4	no	—
16	D₃×#681	C₄	1	8	yes	—
17	C₂×D₃×#168	—	2	2	no	—
18	#68938	—	5	1	no	maximal, Hall
19	C₂²×C₅×M₁₆	—	3	1	no	—
20	C₂²×#681	—	3	2	no	—
21	#64772	—	3	2	no	—
22	#64772	—	3	1	no	—
23	#64772	D₃	1	1	yes	characteristic, complemented
24	C₂³×C₅×D₃	Q₈	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×C₅×Dic₃	Q₈	1	1	yes	characteristic
27	C₂×C₄×C₅×D₃	C₂×C₄	1	4	yes	—
28	C₂×C₄×C₅×D₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₅×D₃	D₄	1	2	yes	—
30	C₂×C₄×C₅×D₃	D₄	1	4	yes	—
31	C₂×C₃×C₅×C₈	—	2	2	no	—
32	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
33	#6710	—	2	2	no	—
34	#6710	C₂×C₄	1	4	yes	—
35	C₂×C₃⋊#168	—	2	2	no	—
36	C₂×C₅×C₃⋊C₈	—	2	2	no	—
37	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
38	C₅×ΓL(1,𝔽₂₅)	—	2	8	no	—
39	C₃×#681	—	2	2	no	—
40	C₃×#681	C₂×C₄	1	4	yes	—
41	C₂×C₃×#168	—	2	2	no	—
42	#6650	—	2	8	no	—
43	D₃×#168	—	2	8	no	—
44	C₂×D₃×M₁₆	—	5	2	no	—
45	C₂²×ΓL(1,𝔽₂₅)	D₅	1	1	yes	characteristic
46	C₃×#341	—	5	1	no	—
47	#4558	—	5	2	no	—
48	#4552	—	5	1	no	—
49	C₂³×C₄×C₅	—	3	1	no	—
50	C₂×C₅×M₁₆	—	3	4	no	—
51	C₂×C₅×M₁₆	—	6	4	no	—
52	C₂²×C₅×C₈	—	3	1	no	—
53	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
54	C₂×#681	—	3	2	no	—
55	C₂×#681	—	3	8	no	—
56	C₂×#681	—	6	2	no	—
57	C₂×#681	—	6	4	no	—
58	C₂×#681	C₂×D₃	1	2	yes	—
59	C₂²×#168	—	6	2	no	—
60	#1738	—	15	1	no	Sylow
61	C₂³×C₃×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₅×D₃	—	4	2	no	—
63	C₂²×C₅×D₃	C₄⋊C₄	1	2	yes	—
64	C₂²×C₅×D₃	C₄⋊C₄	1	4	yes	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂×C₅×Dic₃	C₄⋊C₄	1	4	yes	—
70	C₂×C₅×Dic₃	C₄⋊C₄	1	2	yes	—
71	C₄×C₅×D₃	C₄²	1	4	yes	—
72	C₄×C₅×D₃	C₂²⋊C₄	1	4	yes	—
73	C₄×C₅×D₃	C₂²⋊C₄	1	4	yes	—
74	C₄×C₅×D₃	C₂²⋊C₄	1	4	yes	—
75	C₃×C₅×C₈	—	2	2	no	—
76	C₃⋊#168	—	2	4	no	—
77	C₃×#168	—	2	4	no	—
78	C₅×C₃⋊C₈	—	2	2	no	—
79	C₂²×C₄×D₃	C₂×D₅	1	1	yes	characteristic
80	C₂×C₃×M₁₆	—	5	2	no	—
81	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
82	C₂×#208	—	5	2	no	—
83	C₂²×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
84	D₃×M₁₆	—	5	8	no	—
85	D₃×M₁₆	—	10	4	no	—
86	C₂×ΓL(1,𝔽₂₅)	—	2	4	no	—
87	C₂×ΓL(1,𝔽₂₅)	—	10	2	no	—
88	C₂×ΓL(1,𝔽₂₅)	Dic₅	1	4	yes	—
89	C₂×D₃×C₈	—	10	2	no	—
90	C₂⁴×C₅	—	3	1	no	—
91	C₂²×C₄×C₅	—	3	1	no	—
92	C₂²×C₄×C₅	—	3	2	no	—
93	C₂²×C₄×C₅	—	3	4	no	—
94	C₂²×C₄×C₅	—	3	4	no	—
95	C₂²×C₄×C₅	—	3	2	no	—
96	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
97	C₅×M₁₆	—	6	8	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	3	2	no	—
100	C₂×C₅×C₈	—	6	2	no	—
101	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
102	#681	—	2	2	no	—
103	#681	—	3	4	no	—
104	#681	—	6	8	no	—
105	#681	—	6	2	no	—
106	#681	C₄×D₃	1	4	yes	—
107	C₂×#168	—	2	2	no	—
108	C₂×#168	—	6	8	no	—
109	C₂×#168	—	6	2	no	—
110	C₂²×M₁₆	—	3	1	no	—
111	C₂²×M₁₆	—	15	2	no	—
112	#341	—	5	1	no	—
113	#341	—	15	2	no	—
114	#341	—	15	1	no	—
115	C₂²×C₃×C₅	—	2	2	no	—
116	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	2	yes	—
117	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₅×D₃	—	2	4	no	—
119	C₂×C₅×D₃	—	4	4	no	—
120	C₂×C₅×D₃	#95	1	4	yes	—
121	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
122	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
124	C₅×Dic₃	#95	1	4	yes	—
125	C₂³×D₃	Dic₁₀	1	1	yes	characteristic
126	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
127	C₂²×Dic₃	Dic₁₀	1	1	yes	characteristic
128	C₂×C₄×D₃	C₄×D₅	1	4	yes	—
129	C₂×C₄×D₃	D₂₀	1	2	yes	—
130	C₂×C₄×D₃	C₂×Dic₅	1	2	yes	—
131	C₂×C₄×D₃	C₂²⋊D₅	1	4	yes	—
132	C₃×M₁₆	—	5	4	no	—
133	C₃×M₁₆	—	10	2	no	—
134	C₂×C₃×C₈	—	2	2	no	—
135	C₂×C₃×C₈	—	10	2	no	—
136	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
137	#208	—	5	4	no	—
138	#208	—	10	2	no	—
139	C₂×C₃⋊C₈	—	2	2	no	—
140	C₂×C₃⋊C₈	—	10	2	no	—
141	C₂×C₃⋊C₈	C₂×Dic₅	1	2	yes	—
142	ΓL(1,𝔽₂₅)	—	2	8	no	—
143	ΓL(1,𝔽₂₅)	—	10	8	no	—
144	D₃×C₈	—	10	8	no	—
145	C₂³×C₅	—	3	2	no	—
146	C₂³×C₅	—	3	4	no	—
147	C₂³×C₅	—	12	2	no	—
148	C₂³×C₅	D₃×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅	—	3	4	no	—
150	C₂×C₄×C₅	—	3	2	no	—
151	C₂×C₄×C₅	—	3	4	no	—
152	C₂×C₄×C₅	—	3	4	no	—
153	C₂×C₄×C₅	—	3	4	no	—
154	C₂×C₄×C₅	—	3	4	no	—
155	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
156	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
157	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅	D₃×D₄	1	2	yes	—
159	C₅×C₈	—	2	2	no	—
160	C₅×C₈	—	6	2	no	—
161	#168	—	2	4	no	—
162	#168	—	6	4	no	—
163	C₂³×C₄	—	3	1	no	—
164	C₂×M₁₆	—	3	4	no	—
165	C₂×M₁₆	—	5	2	no	—
166	C₂×M₁₆	—	6	4	no	—
167	C₂×M₁₆	—	15	8	no	—
168	C₂×M₁₆	—	15	2	no	—
169	C₂×M₁₆	—	30	2	no	—
170	C₂×M₁₆	—	30	4	no	—
171	C₂²×C₈	—	3	1	no	—
172	C₂²×C₈	—	30	2	no	—
173	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
174	C₂×C₃×C₅	—	2	2	no	—
175	C₂×C₃×C₅	C₂×#95	1	1	yes	characteristic
176	C₂×C₃×C₅	#430	1	2	yes	—
177	C₅×D₃	—	2	4	no	—
178	C₂³×C₃	C₂×Dic₁₀	1	1	yes	characteristic
179	C₂²×D₃	—	4	2	no	—
180	C₂²×D₃	#683	1	4	yes	—
181	C₂²×D₃	C₄⋊Dic₅	1	2	yes	—
182	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
183	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
186	C₂×Dic₃	#683	1	4	yes	—
187	C₂×Dic₃	C₄⋊Dic₅	1	2	yes	—
188	C₄×D₃	C₄×Dic₅	1	4	yes	—
189	C₄×D₃	#685	1	4	yes	—
190	C₄×D₃	#685	1	4	yes	—
191	C₄×D₃	C₂²⋊Dic₅	1	4	yes	—
192	C₃×C₈	—	2	2	no	—
193	C₃×C₈	—	10	4	no	—
194	C₃⋊C₈	—	2	2	no	—
195	C₃⋊C₈	—	10	4	no	—
196	C₂²×C₅	—	2	2	no	—
197	C₂²×C₅	—	3	4	no	—
198	C₂²×C₅	—	6	4	no	—
199	C₂²×C₅	—	12	4	no	—
200	C₂²×C₅	D₃×C₄⋊C₄	1	2	yes	—
201	C₂²×C₅	D₃×C₄⋊C₄	1	1	yes	characteristic
202	C₄×C₅	—	3	4	no	—
203	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
204	C₄×C₅	D₃×C₂²⋊C₄	1	2	yes	—
205	C₄×C₅	D₃×C₂²⋊C₄	1	1	yes	characteristic
206	C₂⁴	—	3	1	no	—
207	C₂²×C₄	—	3	1	no	—
208	C₂²×C₄	—	3	4	no	—
209	C₂²×C₄	—	3	2	no	—
210	C₂²×C₄	—	3	2	no	—
211	C₂²×C₄	—	3	4	no	—
212	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
213	M₁₆	—	5	4	no	—
214	M₁₆	—	6	8	no	—
215	M₁₆	—	10	2	no	—
216	M₁₆	—	15	4	no	—
217	M₁₆	—	30	8	no	—
218	M₁₆	—	30	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	3	2	no	—
221	C₂×C₈	—	6	2	no	—
222	C₂×C₈	—	10	2	no	—
223	C₂×C₈	—	30	8	no	—
224	C₂×C₈	—	30	2	no	—
225	C₂×C₈	D₃×Dic₅	1	2	yes	—
226	C₃×C₅	#1738	1	1	yes	characteristic, complemented, Hall
227	C₂²×C₃	—	2	2	no	—
228	C₂²×C₃	C₂×#683	1	2	yes	—
229	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
230	C₂×D₃	—	2	4	no	—
231	C₂×D₃	—	4	4	no	—
232	C₂×D₃	#3963	1	4	yes	—
233	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
234	C₃×C₄	C₂×#685	1	2	yes	—
235	C₃×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
236	Dic₃	#3963	1	4	yes	—
237	C₂×C₅	—	2	2	no	—
238	C₂×C₅	—	6	4	no	—
239	C₂×C₅	D₃×#95	1	1	yes	characteristic
240	C₂×C₅	#4893	1	2	yes	—
241	C₂³	—	3	2	no	—
242	C₂³	—	3	4	no	—
243	C₂³	—	12	2	no	—
244	C₂³	D₃×Dic₁₀	1	1	yes	characteristic
245	C₂×C₄	—	3	4	no	—
246	C₂×C₄	—	3	2	no	—
247	C₂×C₄	—	3	4	no	—
248	C₂×C₄	—	3	4	no	—
249	C₂×C₄	—	3	4	no	—
250	C₂×C₄	—	3	4	no	—
251	C₂×C₄	C₄×D₃×D₅	1	2	yes	—
252	C₂×C₄	D₃×D₂₀	1	1	yes	characteristic
253	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
254	C₂×C₄	D₃×C₂²⋊D₅	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	6	2	no	—
257	C₈	—	10	4	no	—
258	C₈	—	30	4	no	—
259	C₂×C₃	—	2	2	no	—
260	C₂×C₃	#65414	1	2	yes	—
261	C₂×C₃	C₂×#3963	1	1	yes	characteristic
262	D₃	—	2	4	no	—
263	C₅	#68938	1	1	yes	characteristic, complemented, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	3	4	no	—
266	C₂²	—	6	4	no	—
267	C₂²	—	12	4	no	—
268	C₂²	D₃×#683	1	2	yes	—
269	C₂²	D₃×C₄⋊Dic₅	1	1	yes	characteristic
270	C₄	—	3	4	no	—
271	C₄	C₄×D₃×Dic₅	1	1	yes	characteristic
272	C₄	D₃×#685	1	2	yes	—
273	C₄	D₃×C₂²⋊Dic₅	1	1	yes	characteristic
274	C₃	#10602791	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	2	2	no	—
276	C₂	—	6	4	no	—
277	C₂	#11748059	1	2	yes	—
278	C₂	D₃×#3963	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₃×D₂₀
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂²×C₄	C₂×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₅×D₃	Q₈
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	#1738
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #11746373	C₂	semidirect
4. C₃×#64772	C₂	semidirect
5. #11745983	C₂	semidirect
23. #64772	D₃	semidirect
226. C₃×C₅	#1738	semidirect
263. C₅	#68938	semidirect
274. C₃	#10602791	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×ΓL(1,𝔽₂₅)	2
C₂×D₃×#681	2
#11746373	2
C₃×#64772	2
#11745983	2
#10602791	3
#68938	5