Subgroups — #14312766

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	D₁₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	#11747668	C₂	1	1	yes	maximal, characteristic
3	#11747533	C₂	1	1	yes	maximal, characteristic
4	C₅×#4512	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64732	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745877	C₂	1	1	yes	maximal, characteristic, complemented
7	#11745876	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599880	—	3	1	no	maximal, Hall
9	C₂×C₈×D₁₅	C₂²	1	1	yes	characteristic
10	#91386	C₄	1	1	yes	characteristic
11	C₂×#6651	C₄	1	1	yes	characteristic
12	C₃×C₅×M₃₂	C₂²	1	1	yes	characteristic
13	#91096	C₂²	1	1	yes	characteristic
14	#91082	C₄	1	2	yes	complemented
15	C₁₆×D₁₅	C₄	1	2	yes	complemented
16	C₅×#815	C₂²	1	1	yes	characteristic
17	C₅×#812	C₂²	1	1	yes	characteristic
18	C₃×#3947	C₂²	1	1	yes	characteristic
19	C₃×#3944	C₂²	1	1	yes	characteristic
20	Dic₃×#168	C₄	1	1	yes	characteristic
21	Dic₅×C₃⋊C₈	C₄	1	1	yes	characteristic
22	#90936	C₄	1	2	yes	complemented
23	#90933	C₄	1	2	yes	complemented
24	#69087	—	5	1	no	maximal, Hall
25	D₅×M₃₂	—	3	1	no	—
26	D₅×M₃₂	—	3	1	no	—
27	#64997	—	3	1	no	—
28	C₅×#347	—	3	1	no	—
29	#64732	—	3	1	no	—
30	#64732	D₃	1	1	yes	characteristic, complemented
31	#64686	—	3	1	no	—
32	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
34	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
35	C₈×D₁₅	C₂×C₄	1	1	yes	characteristic
36	C₈×D₁₅	C₂×C₄	1	1	yes	characteristic
37	C₈×D₁₅	C₂×C₄	1	1	yes	characteristic
38	C₈×D₁₅	C₂×C₄	1	1	yes	characteristic
39	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
43	#6669	—	2	1	no	—
44	#6667	—	2	1	no	—
45	#6651	—	2	1	no	—
46	#6651	—	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	D₃×M₃₂	—	5	1	no	—
52	D₃×M₃₂	—	5	1	no	—
53	#4708	—	5	1	no	—
54	C₃×#347	—	5	1	no	—
55	#4512	—	5	1	no	—
56	#4512	D₅	1	1	yes	characteristic, complemented
57	#4466	—	5	1	no	—
58	C₂×#676	—	3	1	no	—
59	C₂×C₈×D₅	—	3	1	no	—
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	C₂×D₃	1	1	yes	characteristic
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	—	3	2	no	—
71	D₅×C₁₆	—	3	2	no	—
72	D₅×C₁₆	—	3	2	no	—
73	#2110	—	15	1	no	Sylow
74	C₂²×D₁₅	M₁₆	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×Dic₁₅	M₁₆	1	1	yes	characteristic
77	C₄×D₁₅	C₄²	1	1	yes	characteristic
78	C₄×D₁₅	C₄²	1	1	yes	characteristic
79	C₄×D₁₅	M₁₆	1	1	yes	characteristic
80	C₄×D₁₅	M₁₆	1	1	yes	characteristic
81	C₂×C₅×Dic₃	—	2	1	no	—
82	C₂×C₃×Dic₅	—	2	1	no	—
83	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₃⋊#168	C₄²	1	1	yes	characteristic
86	C₃⋊#168	C₄²	1	1	yes	characteristic
87	C₃×#168	—	2	1	no	—
88	C₅×C₃⋊C₈	—	2	1	no	—
89	C₂×ΓL(1,𝔽₂₅)	—	5	1	no	—
90	C₂×D₃×C₈	—	5	1	no	—
91	#872	—	5	1	no	—
92	C₃×M₃₂	—	5	1	no	—
93	C₃×M₃₂	C₂×D₅	1	1	yes	characteristic
94	C₃×#97	—	5	1	no	—
95	#815	C₂×D₅	1	1	yes	characteristic
96	C₈×Dic₃	—	5	1	no	—
97	#812	—	5	1	no	—
98	#812	C₂×D₅	1	1	yes	characteristic
99	C₄×C₃⋊C₈	—	5	1	no	—
100	#798	—	5	2	no	—
101	#798	—	5	2	no	—
102	D₃×C₁₆	—	5	2	no	—
103	D₃×C₁₆	—	5	2	no	—
104	C₂×C₄×D₅	—	3	1	no	—
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	C₄×D₃	1	1	yes	characteristic
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	—	3	1	no	—
116	C₈×D₅	—	3	1	no	—
117	C₈×D₅	—	3	1	no	—
118	C₈×D₅	—	3	1	no	—
119	C₅×C₁₆	—	3	2	no	—
120	C₅×C₁₆	C₄×D₃	1	2	yes	—
121	#672	—	3	2	no	—
122	#672	C₄×D₃	1	2	yes	—
123	C₂×M₃₂	—	15	1	no	—
124	C₂×M₃₂	—	15	1	no	—
125	#405	—	15	1	no	—
126	#347	—	3	1	no	—
127	#347	—	5	1	no	—
128	#347	—	15	1	no	—
129	#347	—	15	1	no	—
130	C₂²×C₃×C₅	C₂×M₁₆	1	1	yes	characteristic
131	C₂×D₁₅	—	4	1	no	—
132	C₂×D₁₅	#97	1	1	yes	characteristic
133	C₂×D₁₅	#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	#97	1	1	yes	characteristic
137	Dic₁₅	#97	1	1	yes	characteristic
138	C₃×Dic₅	—	4	1	no	—
139	C₅×Dic₃	—	4	1	no	—
140	C₂×C₄×D₃	—	5	1	no	—
141	C₂×C₃×C₈	—	5	1	no	—
142	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
143	C₃×C₄²	—	5	1	no	—
144	(C₂×D₃)⋊C₄	—	10	1	no	—
145	Dic₃⋊C₄	—	10	1	no	—
146	C₄×Dic₃	C₄×D₅	1	1	yes	characteristic
147	C₂×C₃⋊C₈	—	5	1	no	—
148	C₂×C₃⋊C₈	C₄×D₅	1	1	yes	characteristic
149	ΓL(1,𝔽₂₅)	—	10	1	no	—
150	ΓL(1,𝔽₂₅)	—	10	1	no	—
151	D₃×C₈	—	5	1	no	—
152	D₃×C₈	—	5	1	no	—
153	D₃×C₈	—	5	1	no	—
154	D₃×C₈	—	5	1	no	—
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	—	3	1	no	—
160	C₂×C₄×C₅	—	6	1	no	—
161	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂×Dic₅	—	2	1	no	—
163	C₂×Dic₅	—	3	1	no	—
164	C₄×D₅	—	3	1	no	—
165	C₄×D₅	—	3	1	no	—
166	C₄×D₅	—	3	1	no	—
167	C₄×D₅	—	3	1	no	—
168	C₅×C₈	—	6	1	no	—
169	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
170	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
171	#168	—	2	1	no	—
172	#168	—	3	1	no	—
173	#168	—	3	1	no	—
174	C₂×M₁₆	—	15	1	no	—
175	C₂²×C₈	—	15	1	no	—
176	#117	—	15	1	no	—
177	M₃₂	—	3	1	no	—
178	M₃₂	—	5	1	no	—
179	M₃₂	—	15	2	no	—
180	M₃₂	—	15	1	no	—
181	M₃₂	—	15	2	no	—
182	M₃₂	D₃×D₅	1	1	yes	characteristic
183	C₂×C₁₆	—	15	2	no	—
184	C₂×C₁₆	—	15	2	no	—
185	#97	—	3	1	no	—
186	#97	—	5	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	D₁₅	—	2	1	no	—
192	D₁₅	—	2	1	no	—
193	C₂²×D₃	—	5	1	no	—
194	C₂×C₃×C₄	—	10	1	no	—
195	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
196	C₂×Dic₃	—	2	1	no	—
197	C₂×Dic₃	—	5	1	no	—
198	C₄×D₃	—	5	1	no	—
199	C₄×D₃	—	5	1	no	—
200	C₄×D₃	—	5	1	no	—
201	C₄×D₃	—	5	1	no	—
202	C₃×C₈	—	10	1	no	—
203	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
204	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
205	C₃⋊C₈	—	2	1	no	—
206	C₃⋊C₈	—	5	1	no	—
207	C₃⋊C₈	—	5	1	no	—
208	C₂²×C₅	D₃×M₁₆	1	1	yes	characteristic
209	C₂×D₅	—	3	1	no	—
210	C₂×D₅	—	3	1	no	—
211	C₂×D₅	—	12	1	no	—
212	C₄×C₅	—	12	1	no	—
213	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
214	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
215	Dic₅	—	3	1	no	—
216	Dic₅	—	3	1	no	—
217	Dic₅	—	4	1	no	—
218	C₂²×C₄	—	15	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	—	15	1	no	—
224	C₂×C₈	—	15	1	no	—
225	C₂×C₈	—	15	1	no	—
226	C₂×C₈	—	15	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	—	5	1	no	—
233	C₁₆	—	3	2	no	—
234	C₁₆	—	5	2	no	—
235	C₁₆	—	15	2	no	—
236	C₁₆	#1200	1	2	yes	—
237	C₃×C₅	#2110	1	1	yes	characteristic, complemented, Hall
238	C₂²×C₃	D₅×M₁₆	1	1	yes	characteristic
239	C₂×D₃	—	5	1	no	—
240	C₂×D₃	—	5	1	no	—
241	C₂×D₃	—	20	1	no	—
242	C₃×C₄	—	20	1	no	—
243	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
244	C₃×C₄	D₅×M₁₆	1	1	yes	characteristic
245	Dic₃	—	4	1	no	—
246	Dic₃	—	5	1	no	—
247	Dic₃	—	5	1	no	—
248	C₂×C₅	—	2	1	no	—
249	C₂×C₅	D₃×#97	1	1	yes	characteristic
250	D₅	—	6	1	no	—
251	D₅	—	6	1	no	—
252	C₂³	—	15	1	no	—
253	C₂×C₄	—	6	1	no	—
254	C₂×C₄	—	10	1	no	—
255	C₂×C₄	—	15	1	no	—
256	C₂×C₄	—	15	1	no	—
257	C₂×C₄	—	15	1	no	—
258	C₂×C₄	—	15	1	no	—
259	C₂×C₄	—	15	1	no	—
260	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
261	C₈	—	6	1	no	—
262	C₈	—	10	1	no	—
263	C₈	—	15	1	no	—
264	C₈	—	15	1	no	—
265	C₈	C₄×D₃×D₅	1	1	yes	characteristic
266	C₈	C₂×#1200	1	1	yes	characteristic
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	D₅×#97	1	1	yes	characteristic
269	D₃	—	10	1	no	—
270	D₃	—	10	1	no	—
271	C₅	#69087	1	1	yes	characteristic, complemented, Sylow
272	C₂²	—	15	1	no	—
273	C₂²	—	15	1	no	—
274	C₂²	—	60	1	no	—
275	C₂²	#91246	1	1	yes	characteristic
276	C₄	—	12	1	no	—
277	C₄	—	15	1	no	—
278	C₄	—	15	1	no	—
279	C₄	—	20	1	no	—
280	C₄	#91246	1	1	yes	characteristic
281	C₄	C₄×#1200	1	1	yes	characteristic
282	C₃	#10599880	1	1	yes	characteristic, complemented, Sylow
283	C₂	—	2	1	no	—
284	C₂	—	30	1	no	—
285	C₂	—	30	1	no	—
286	C₂	#11747603	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#1200
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×D₁₅	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. C₅×#4512	C₂	semidirect
5. C₃×#64732	C₂	semidirect
6. #11745877	C₂	semidirect
7. #11745876	C₂	semidirect
14. #91082	C₄	semidirect
15. C₁₆×D₁₅	C₄	semidirect
22. #90936	C₄	semidirect
23. #90933	C₄	semidirect
30. #64732	D₃	semidirect
56. #4512	D₅	semidirect
237. C₃×C₅	#2110	semidirect
271. C₅	#69087	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
D₁₅×M₃₂	2
#11747668	2
#11747533	2
C₅×#4512	2
C₃×#64732	2
#11745877	2
#11745876	2
#10599880	3
#69087	5