Subgroups — #14312765

Order 1920.

320 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	C₃×D₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#11747646	C₂	1	1	yes	maximal, characteristic
3	D₅×#812	C₂	1	1	yes	maximal, characteristic
4	#11746333	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4512	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745878	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745877	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599880	—	3	1	no	maximal, Hall
9	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
10	#91357	C₄	1	1	yes	characteristic
11	C₂×#6649	C₄	1	1	yes	characteristic
12	C₃×C₅×M₃₂	C₂²	1	1	yes	characteristic
13	#91099	C₂²	1	1	yes	characteristic
14	#91096	C₂²	1	1	yes	characteristic
15	C₅×#815	C₂²	1	1	yes	characteristic
16	C₅×#812	C₂²	1	1	yes	characteristic
17	C₃×#3944	C₂²	1	1	yes	characteristic
18	C₃×#3930	C₄	1	2	yes	complemented
19	C₃×D₅×C₁₆	C₄	1	2	yes	complemented
20	#90951	C₄	1	1	yes	characteristic
21	Dic₃×#168	C₄	1	1	yes	characteristic
22	#90934	C₄	1	2	yes	complemented
23	D₅×C₃⋊C₁₆	C₄	1	2	yes	complemented
24	#71425	—	5	1	no	maximal, Hall
25	D₅×M₃₂	—	3	1	no	—
26	D₅×M₃₂	D₃	1	1	yes	characteristic
27	#64997	—	3	1	no	—
28	C₅×#347	—	3	1	no	—
29	#64732	—	3	1	no	—
30	#64732	—	3	1	no	—
31	#64686	—	3	1	no	—
32	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
36	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
39	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
42	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
43	#6668	—	2	1	no	—
44	#6665	—	2	1	no	—
45	#6649	—	2	1	no	—
46	#6649	—	2	1	no	—
47	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	—
48	C₃⋊#672	C₂×C₄	1	2	yes	—
49	C₃×#672	C₂×C₄	1	2	yes	—
50	C₅×C₃⋊C₁₆	C₂×C₄	1	2	yes	—
51	C₂×C₃×M₃₂	—	5	1	no	—
52	#5101	—	5	1	no	—
53	C₂×#812	—	5	1	no	—
54	#4512	—	5	1	no	—
55	#4512	—	5	1	no	—
56	#4512	—	5	1	no	—
57	#4512	D₅	1	1	yes	characteristic, complemented
58	C₂×#676	—	3	1	no	—
59	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
60	#4018	—	3	1	no	—
61	C₅×M₃₂	—	3	1	no	—
62	C₅×M₃₂	C₂×D₃	1	1	yes	characteristic
63	C₅×#97	—	3	1	no	—
64	#3947	—	3	1	no	—
65	C₈×Dic₅	—	3	1	no	—
66	#3944	—	3	1	no	—
67	#3944	C₂×D₃	1	1	yes	characteristic
68	C₄×#168	—	3	1	no	—
69	#3930	—	3	2	no	—
70	#3930	Dic₃	1	2	yes	complemented
71	D₅×C₁₆	—	3	2	no	—
72	D₅×C₁₆	Dic₃	1	2	yes	complemented
73	#2110	—	15	1	no	Sylow
74	C₂²×C₃×D₅	M₁₆	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₁₅	—	2	1	no	—
77	C₂×C₅×Dic₃	—	2	1	no	—
78	C₂×C₃×Dic₅	M₁₆	1	1	yes	characteristic
79	C₃×C₄×D₅	C₄²	1	1	yes	characteristic
80	C₃×C₄×D₅	C₄²	1	1	yes	characteristic
81	C₃×C₄×D₅	M₁₆	1	1	yes	characteristic
82	C₃×C₄×D₅	M₁₆	1	1	yes	characteristic
83	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₃⋊#168	—	2	1	no	—
86	C₃×#168	C₄²	1	1	yes	characteristic
87	C₃×#168	C₄²	1	1	yes	characteristic
88	C₅×C₃⋊C₈	—	2	1	no	—
89	C₂²×C₃×C₈	—	5	1	no	—
90	#926	—	5	1	no	—
91	C₂×#208	—	5	1	no	—
92	C₃×M₃₂	—	5	1	no	—
93	C₃×M₃₂	—	5	2	no	—
94	C₃×M₃₂	C₂×D₅	1	1	yes	characteristic
95	C₂×C₃×C₁₆	—	5	2	no	—
96	#815	—	5	1	no	—
97	#815	C₂×D₅	1	1	yes	characteristic
98	C₈×Dic₃	—	5	1	no	—
99	C₈×Dic₃	—	5	1	no	—
100	#812	—	5	1	no	—
101	#812	—	5	2	no	—
102	#812	C₂×D₅	1	1	yes	characteristic
103	C₂×C₃⋊C₁₆	—	5	2	no	—
104	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
105	C₂×C₅×C₈	—	3	1	no	—
106	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
107	C₄²×C₅	—	3	1	no	—
108	#685	—	6	1	no	—
109	#683	—	6	1	no	—
110	C₄×Dic₅	—	3	1	no	—
111	C₂×#168	—	3	1	no	—
112	C₂×#168	C₄×D₃	1	1	yes	characteristic
113	#676	—	6	1	no	—
114	#676	—	6	1	no	—
115	C₈×D₅	C₄×D₃	1	1	yes	characteristic
116	C₈×D₅	C₄×D₃	1	1	yes	characteristic
117	C₈×D₅	C₂×Dic₃	1	1	yes	characteristic
118	C₈×D₅	C₂×Dic₃	1	1	yes	characteristic
119	C₅×C₁₆	—	3	2	no	—
120	C₅×C₁₆	C₂×Dic₃	1	2	yes	—
121	#672	—	3	2	no	—
122	#672	C₂×Dic₃	1	2	yes	—
123	C₂×M₃₂	—	5	1	no	—
124	C₂×M₃₂	—	15	1	no	—
125	#405	—	15	1	no	—
126	#347	—	3	1	no	—
127	#347	—	15	1	no	—
128	#347	—	15	1	no	—
129	#347	—	15	1	no	—
130	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₃×D₅	—	4	1	no	—
132	C₂×C₃×D₅	#97	1	1	yes	characteristic
133	C₂×C₃×D₅	#97	1	1	yes	characteristic
134	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
135	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
136	Dic₁₅	—	4	1	no	—
137	C₃×Dic₅	#97	1	1	yes	characteristic
138	C₃×Dic₅	#97	1	1	yes	characteristic
139	C₅×Dic₃	—	4	1	no	—
140	C₂²×C₃×C₄	—	5	1	no	—
141	C₂×C₃×C₈	—	5	1	no	—
142	C₂×C₃×C₈	—	5	1	no	—
143	C₂×C₃×C₈	—	5	1	no	—
144	C₂×C₃×C₈	—	5	1	no	—
145	C₂×C₃×C₈	—	5	1	no	—
146	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
147	#217	—	10	1	no	—
148	C₄⋊Dic₃	—	10	1	no	—
149	C₄×Dic₃	—	5	1	no	—
150	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
151	#208	—	10	1	no	—
152	#208	—	10	1	no	—
153	C₂×C₃⋊C₈	—	5	1	no	—
154	C₂×C₃⋊C₈	C₄×D₅	1	1	yes	characteristic
155	C₃×C₁₆	—	5	2	no	—
156	C₃×C₁₆	C₄×D₅	1	2	yes	—
157	C₃⋊C₁₆	—	5	2	no	—
158	C₃⋊C₁₆	C₄×D₅	1	2	yes	—
159	C₂²×D₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
160	C₂×C₄×C₅	—	6	1	no	—
161	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂×Dic₅	—	6	1	no	—
163	C₂×Dic₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
164	C₄×D₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
165	C₄×D₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
166	C₄×D₅	C₄×Dic₃	1	1	yes	characteristic
167	C₄×D₅	C₄×Dic₃	1	1	yes	characteristic
168	C₅×C₈	—	6	1	no	—
169	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
170	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
171	#168	—	6	1	no	—
172	#168	C₄×Dic₃	1	1	yes	characteristic
173	#168	C₄×Dic₃	1	1	yes	characteristic
174	C₂×M₁₆	—	15	1	no	—
175	C₂²×C₈	—	5	1	no	—
176	#117	—	15	1	no	—
177	M₃₂	—	3	1	no	—
178	M₃₂	—	5	1	no	—
179	M₃₂	—	5	2	no	—
180	M₃₂	—	15	2	no	—
181	M₃₂	—	15	1	no	—
182	M₃₂	D₃×D₅	1	1	yes	characteristic
183	C₂×C₁₆	—	5	2	no	—
184	C₂×C₁₆	—	15	2	no	—
185	#97	—	3	1	no	—
186	#97	—	15	1	no	—
187	C₄×C₈	—	15	1	no	—
188	C₄×C₈	—	15	1	no	—
189	C₂×C₃×C₅	—	2	1	no	—
190	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
191	C₃×D₅	—	2	1	no	—
192	C₃×D₅	—	2	1	no	—
193	C₂³×C₃	—	5	1	no	—
194	C₂×C₃×C₄	—	5	1	no	—
195	C₂×C₃×C₄	—	5	1	no	—
196	C₂×C₃×C₄	—	5	1	no	—
197	C₂×C₃×C₄	—	5	1	no	—
198	C₂×C₃×C₄	—	5	1	no	—
199	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
200	C₂×Dic₃	—	2	1	no	—
201	C₂×Dic₃	—	10	1	no	—
202	C₃×C₈	—	5	1	no	—
203	C₃×C₈	—	5	1	no	—
204	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
205	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
206	C₃⋊C₈	—	2	1	no	—
207	C₃⋊C₈	—	10	1	no	—
208	C₂²×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
209	C₂×D₅	—	4	1	no	—
210	C₂×D₅	#815	1	1	yes	characteristic
211	C₂×D₅	#815	1	1	yes	characteristic
212	C₄×C₅	—	12	1	no	—
213	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
214	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
215	Dic₅	—	12	1	no	—
216	Dic₅	#815	1	1	yes	characteristic
217	Dic₅	#815	1	1	yes	characteristic
218	C₂²×C₄	—	5	1	no	—
219	M₁₆	—	30	1	no	—
220	M₁₆	—	30	1	no	—
221	C₂×C₈	—	3	1	no	—
222	C₂×C₈	—	5	1	no	—
223	C₂×C₈	—	5	1	no	—
224	C₂×C₈	—	5	1	no	—
225	C₂×C₈	—	5	1	no	—
226	C₂×C₈	—	5	1	no	—
227	C₂×C₈	—	15	1	no	—
228	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
229	C₄⋊C₄	—	30	1	no	—
230	C₂²⋊C₄	—	30	1	no	—
231	C₄²	—	3	1	no	—
232	C₄²	—	15	1	no	—
233	C₁₆	—	3	2	no	—
234	C₁₆	—	5	2	no	—
235	C₁₆	—	15	2	no	—
236	C₁₆	D₅×Dic₃	1	2	yes	—
237	C₃×C₅	#2110	1	1	yes	characteristic, complemented, Hall
238	C₂²×C₃	—	5	1	no	—
239	C₂²×C₃	—	5	1	no	—
240	C₂²×C₃	—	20	1	no	—
241	C₂²×C₃	D₅×M₁₆	1	1	yes	characteristic
242	C₃×C₄	—	5	1	no	—
243	C₃×C₄	—	5	1	no	—
244	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
245	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
246	Dic₃	—	4	1	no	—
247	Dic₃	—	20	1	no	—
248	C₂×C₅	—	2	1	no	—
249	C₂×C₅	C₂×#815	1	1	yes	characteristic
250	D₅	—	2	1	no	—
251	D₅	—	2	1	no	—
252	C₂³	—	5	1	no	—
253	C₂×C₄	—	5	1	no	—
254	C₂×C₄	—	5	1	no	—
255	C₂×C₄	—	5	1	no	—
256	C₂×C₄	—	5	1	no	—
257	C₂×C₄	—	5	1	no	—
258	C₂×C₄	—	6	1	no	—
259	C₂×C₄	—	30	1	no	—
260	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
261	C₈	—	5	1	no	—
262	C₈	—	5	1	no	—
263	C₈	—	6	1	no	—
264	C₈	—	30	1	no	—
265	C₈	C₄×D₃×D₅	1	1	yes	characteristic
266	C₈	C₂×D₅×Dic₃	1	1	yes	characteristic
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	10	1	no	—
269	C₂×C₃	—	10	1	no	—
270	C₂×C₃	D₅×#97	1	1	yes	characteristic
271	C₅	#71425	1	1	yes	characteristic, complemented, Sylow
272	C₂²	—	5	1	no	—
273	C₂²	—	5	1	no	—
274	C₂²	—	20	1	no	—
275	C₂²	D₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
276	C₄	—	5	1	no	—
277	C₄	—	5	1	no	—
278	C₄	—	12	1	no	—
279	C₄	—	60	1	no	—
280	C₄	D₅×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
281	C₄	C₄×D₅×Dic₃	1	1	yes	characteristic
282	C₃	#10599880	1	1	yes	characteristic, complemented, Sylow
283	C₂	—	2	1	no	—
284	C₂	—	10	1	no	—
285	C₂	—	10	1	no	—
286	C₂	D₅×#815	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×D₅×Dic₃
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂×C₈	C₂×D₃×D₅
Fitting	C₃×C₅×M₃₂	C₂²
socle	C₂×C₃×C₅	C₂×#97
prime radical	C₂²×C₃×D₅	M₁₆
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	M₃₂	D₃×D₅
hypocenter	C₃×C₅	#2110
LCS	G ▷ C₃×C₄×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
4. #11746333	C₂	semidirect
5. C₅×#4512	C₂	semidirect
6. #11745878	C₂	semidirect
7. #11745877	C₂	semidirect
18. C₃×#3930	C₄	semidirect
19. C₃×D₅×C₁₆	C₄	semidirect
22. #90934	C₄	semidirect
23. D₅×C₃⋊C₁₆	C₄	semidirect
57. #4512	D₅	semidirect
70. #3930	Dic₃	semidirect
72. D₅×C₁₆	Dic₃	semidirect
237. C₃×C₅	#2110	semidirect
271. C₅	#71425	semidirect
282. C₃	#10599880	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₅×M₃₂	2
#11747646	2
D₅×#812	2
#11746333	2
C₅×#4512	2
#11745878	2
#11745877	2
#10599880	3
#71425	5