Subgroups — #14478159

Order 1920.

720 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₅×#131	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751241	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751008	C₂	1	3	yes	maximal, complemented
5	D₁₅×M₃₂	C₂	1	3	yes	maximal, complemented
6	#11751002	C₂	1	3	yes	maximal, complemented
7	C₂×#91082	C₂	1	3	yes	maximal, complemented
8	#10613743	—	3	1	no	maximal, Hall
9	#41×D₁₅	C₄	1	1	yes	characteristic
10	C₃×C₅×#131	C₂²	1	1	yes	characteristic
11	#91837	C₄	1	1	yes	characteristic
12	#91796	C₄	1	3	yes	—
13	M₁₆×D₁₅	C₄	1	3	yes	—
14	#91790	C₂²	1	3	yes	—
15	C₂×C₈×D₁₅	C₂²	1	3	yes	—
16	C₃×C₅×M₃₂	C₂²	1	3	yes	complemented
17	C₂×C₃×C₅×C₁₆	C₂²	1	3	yes	complemented
18	#91096	C₂²	1	3	yes	complemented
19	C₂×C₃⋊#672	C₂²	1	3	yes	complemented
20	#91082	C₂²	1	1	yes	characteristic
21	#91082	C₂²	1	3	yes	—
22	#91082	C₂²	1	3	yes	complemented
23	#91082	C₂²	1	3	yes	complemented
24	C₁₆×D₁₅	C₂²	1	6	yes	complemented
25	#82012	—	5	1	no	maximal, Hall
26	D₅×#131	—	3	1	no	—
27	C₅×#504	D₃	1	1	yes	characteristic, complemented
28	#65429	—	3	1	no	—
29	#65196	—	3	3	no	—
30	D₅×M₃₂	—	3	3	no	—
31	#65190	—	3	3	no	—
32	C₂×#3930	—	3	3	no	—
33	C₃×C₅×#41	C₂×C₄	1	1	yes	characteristic
34	#6821	C₈	1	1	yes	characteristic
35	Q₈×D₁₅	C₈	1	1	yes	characteristic
36	#6819	C₈	1	3	yes	—
37	D₄×D₁₅	C₈	1	3	yes	—
38	#6817	C₂×C₄	1	3	yes	—
39	C₂×C₄×D₁₅	C₂×C₄	1	3	yes	—
40	C₃×C₅×M₁₆	C₂×C₄	1	3	yes	—
41	C₂×C₃×C₅×C₈	C₂³	1	3	yes	—
42	#6710	C₂×C₄	1	3	yes	—
43	C₂×C₃⋊#168	C₂×C₄	1	3	yes	—
44	#6705	C₂×C₄	1	6	yes	—
45	C₈×D₁₅	C₂×C₄	1	3	yes	—
46	C₈×D₁₅	C₂×C₄	1	3	yes	—
47	C₈×D₁₅	C₂³	1	1	yes	characteristic
48	C₈×D₁₅	C₂³	1	3	yes	—
49	C₃×C₅×C₁₆	C₂³	1	1	yes	characteristic
50	C₃×C₅×C₁₆	C₂³	1	3	yes	—
51	C₃⋊#672	C₂³	1	3	yes	—
52	C₃⋊#672	C₂³	1	1	yes	characteristic
53	D₃×#131	—	5	1	no	—
54	C₃×#504	D₅	1	1	yes	characteristic, complemented
55	#5140	—	5	1	no	—
56	#4907	—	5	3	no	—
57	D₃×M₃₂	—	5	3	no	—
58	#4901	—	5	3	no	—
59	C₂×#798	—	5	3	no	—
60	D₅×#41	—	3	1	no	—
61	C₅×#131	C₂×D₃	1	1	yes	characteristic
62	#4094	—	3	1	no	—
63	#4053	—	3	3	no	—
64	D₅×M₁₆	—	3	3	no	—
65	#4047	—	3	3	no	—
66	C₂×C₈×D₅	—	3	3	no	—
67	C₅×M₃₂	C₂×D₃	1	3	yes	complemented
68	C₂×C₅×C₁₆	C₂×D₃	1	3	yes	complemented
69	#3944	—	3	3	no	—
70	C₂×#672	—	3	3	no	—
71	#3930	—	3	1	no	—
72	#3930	—	3	3	no	—
73	#3930	—	3	3	no	—
74	#3930	—	3	3	no	—
75	D₅×C₁₆	—	3	6	no	—
76	#3407	—	15	1	no	Sylow
77	C₂²×D₁₅	C₂×C₈	1	3	yes	—
78	C₃×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
79	C₃×C₅×D₄	C₂×C₈	1	3	yes	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	3	yes	—
81	#1220	C₂×C₈	1	6	yes	—
82	C₂×Dic₁₅	C₂×C₈	1	3	yes	—
83	D₆₀	C₂×C₈	1	3	yes	—
84	C₄×D₁₅	C₂×C₈	1	3	yes	—
85	C₄×D₁₅	C₂×C₈	1	3	yes	—
86	C₄×D₁₅	C₂²×C₄	1	1	yes	characteristic
87	C₄×D₁₅	C₂²×C₄	1	3	yes	—
88	Dic₃₀	C₂×C₈	1	3	yes	—
89	C₃×C₅×C₈	C₂²×C₄	1	3	yes	—
90	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
91	C₃⋊#168	C₂²×C₄	1	1	yes	characteristic
92	C₃⋊#168	C₂²×C₄	1	3	yes	—
93	D₃×#41	—	5	1	no	—
94	C₃×#131	C₂×D₅	1	1	yes	characteristic
95	#948	—	5	1	no	—
96	#907	—	5	3	no	—
97	D₃×M₁₆	—	5	3	no	—
98	#901	—	5	3	no	—
99	C₂×D₃×C₈	—	5	3	no	—
100	C₃×M₃₂	C₂×D₅	1	3	yes	complemented
101	C₂×C₃×C₁₆	C₂×D₅	1	3	yes	complemented
102	#812	—	5	3	no	—
103	C₂×C₃⋊C₁₆	—	5	3	no	—
104	#798	—	5	1	no	—
105	#798	—	5	3	no	—
106	#798	—	5	3	no	—
107	#798	—	5	3	no	—
108	D₃×C₁₆	—	5	6	no	—
109	C₅×#41	C₄×D₃	1	1	yes	characteristic
110	#713	—	3	1	no	—
111	Q₈×D₅	—	3	1	no	—
112	#711	—	3	3	no	—
113	D₄×D₅	—	3	3	no	—
114	#709	—	3	3	no	—
115	C₂×C₄×D₅	—	3	3	no	—
116	C₅×M₁₆	C₄×D₃	1	3	yes	—
117	C₂×C₅×C₈	C₂²×D₃	1	3	yes	—
118	#681	—	3	3	no	—
119	C₂×#168	—	3	3	no	—
120	#676	—	3	6	no	—
121	C₈×D₅	—	3	3	no	—
122	C₈×D₅	—	3	3	no	—
123	C₈×D₅	—	3	3	no	—
124	C₈×D₅	—	3	1	no	—
125	C₅×C₁₆	C₂²×D₃	1	3	yes	—
126	C₅×C₁₆	C₂²×D₃	1	1	yes	characteristic
127	#672	—	3	3	no	—
128	#672	—	3	1	no	—
129	C₂×#131	—	15	1	no	—
130	#504	—	15	1	no	—
131	#504	—	15	3	no	—
132	#504	—	15	3	no	—
133	#504	D₁₅	1	1	yes	characteristic, complemented
134	C₂×M₃₂	—	15	3	no	—
135	C₂×M₃₂	—	15	3	no	—
136	C₂²×C₃×C₅	C₂²×C₈	1	3	yes	—
137	C₂×D₁₅	—	4	3	no	—
138	C₂×D₁₅	C₂²×C₈	1	1	yes	characteristic
139	C₂×D₁₅	C₂²×C₈	1	3	yes	—
140	C₃×C₄×C₅	C₂²×C₈	1	3	yes	—
141	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
142	Dic₁₅	C₂²×C₈	1	3	yes	—
143	Dic₁₅	C₂²×C₈	1	1	yes	characteristic
144	C₃×#41	C₄×D₅	1	1	yes	characteristic
145	#239	—	5	1	no	—
146	D₃×Q₈	—	5	1	no	—
147	#237	—	5	3	no	—
148	D₃×D₄	—	5	3	no	—
149	#235	—	5	3	no	—
150	C₂×C₄×D₃	—	5	3	no	—
151	C₃×M₁₆	C₄×D₅	1	3	yes	—
152	C₂×C₃×C₈	C₂²×D₅	1	3	yes	—
153	#208	—	5	3	no	—
154	C₂×C₃⋊C₈	—	5	3	no	—
155	ΓL(1,𝔽₂₅)	—	5	6	no	—
156	D₃×C₈	—	5	3	no	—
157	D₃×C₈	—	5	3	no	—
158	D₃×C₈	—	5	3	no	—
159	D₃×C₈	—	5	1	no	—
160	C₃×C₁₆	C₂²×D₅	1	3	yes	—
161	C₃×C₁₆	C₂²×D₅	1	1	yes	characteristic
162	C₃⋊C₁₆	—	5	3	no	—
163	C₃⋊C₁₆	—	5	1	no	—
164	C₂²×D₅	—	3	3	no	—
165	C₅×Q₈	D₃×C₈	1	1	yes	characteristic
166	C₅×D₄	D₃×C₈	1	3	yes	—
167	C₂×C₄×C₅	C₂×C₄×D₃	1	3	yes	—
168	C₂²⋊D₅	—	3	6	no	—
169	C₂×Dic₅	—	3	3	no	—
170	D₂₀	—	3	3	no	—
171	C₄×D₅	—	3	3	no	—
172	C₄×D₅	—	3	3	no	—
173	C₄×D₅	—	3	1	no	—
174	C₄×D₅	—	3	3	no	—
175	Dic₁₀	—	3	3	no	—
176	C₅×C₈	C₂×C₄×D₃	1	3	yes	—
177	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
178	#168	—	3	1	no	—
179	#168	—	3	3	no	—
180	C₂×#41	—	15	1	no	—
181	#131	—	15	1	no	—
182	#131	—	15	3	no	—
183	#131	—	15	3	no	—
184	#131	C₂×D₁₅	1	1	yes	characteristic
185	C₂×M₁₆	—	15	3	no	—
186	C₂²×C₈	—	15	3	no	—
187	M₃₂	—	15	3	no	—
188	M₃₂	—	15	3	no	—
189	M₃₂	—	15	1	no	—
190	M₃₂	—	15	3	no	—
191	M₃₂	—	15	3	no	—
192	M₃₂	C₂×D₁₅	1	3	yes	complemented
193	C₂×C₁₆	—	15	3	no	—
194	C₂×C₁₆	—	15	6	no	—
195	C₂×C₁₆	C₂×D₁₅	1	3	yes	complemented
196	C₂×C₃×C₅	—	2	3	no	—
197	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
198	D₁₅	—	2	1	no	—
199	D₁₅	—	2	3	no	—
200	C₂²×D₃	—	5	3	no	—
201	C₃×Q₈	C₈×D₅	1	1	yes	characteristic
202	C₃×D₄	C₈×D₅	1	3	yes	—
203	C₂×C₃×C₄	C₂×C₄×D₅	1	3	yes	—
204	#67	—	5	6	no	—
205	C₂×Dic₃	—	5	3	no	—
206	D₁₂	—	5	3	no	—
207	C₄×D₃	—	5	1	no	—
208	C₄×D₃	—	5	3	no	—
209	C₄×D₃	—	5	3	no	—
210	C₄×D₃	—	5	3	no	—
211	Dic₆	—	5	3	no	—
212	C₃×C₈	C₂×C₄×D₅	1	3	yes	—
213	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
214	C₃⋊C₈	—	5	1	no	—
215	C₃⋊C₈	—	5	3	no	—
216	C₂²×C₅	C₂×D₃×C₈	1	3	yes	—
217	C₂×D₅	—	3	1	no	—
218	C₂×D₅	—	3	3	no	—
219	C₂×D₅	—	12	3	no	—
220	C₄×C₅	C₂×D₃×C₈	1	3	yes	—
221	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
222	Dic₅	—	3	3	no	—
223	Dic₅	—	3	1	no	—
224	#41	—	15	1	no	—
225	#41	—	15	3	no	—
226	#41	—	15	3	no	—
227	#41	C₄×D₁₅	1	1	yes	characteristic
228	C₂×Q₈	—	15	1	no	—
229	C₂×D₄	—	15	3	no	—
230	C₂²×C₄	—	15	3	no	—
231	M₁₆	—	15	3	no	—
232	M₁₆	—	15	6	no	—
233	M₁₆	C₄×D₁₅	1	3	yes	—
234	C₂×C₈	—	15	3	no	—
235	C₂×C₈	—	15	3	no	—
236	C₂×C₈	—	15	3	no	—
237	C₂×C₈	—	15	3	no	—
238	C₂×C₈	—	15	1	no	—
239	C₂×C₈	C₂²×D₁₅	1	3	yes	—
240	C₁₆	—	15	1	no	—
241	C₁₆	—	15	3	no	—
242	C₁₆	C₂²×D₁₅	1	3	yes	—
243	C₁₆	C₂²×D₁₅	1	1	yes	characteristic
244	C₃×C₅	#3407	1	1	yes	characteristic, complemented, Hall
245	C₂²×C₃	C₂×C₈×D₅	1	3	yes	—
246	C₂×D₃	—	5	1	no	—
247	C₂×D₃	—	5	3	no	—
248	C₂×D₃	—	20	3	no	—
249	C₃×C₄	C₂×C₈×D₅	1	3	yes	—
250	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
251	Dic₃	—	5	3	no	—
252	Dic₃	—	5	1	no	—
253	C₂×C₅	—	2	3	no	—
254	C₂×C₅	C₂²×D₃×C₈	1	1	yes	characteristic
255	D₅	—	6	1	no	—
256	D₅	—	6	3	no	—
257	C₂³	—	15	3	no	—
258	Q₈	—	15	3	no	—
259	Q₈	C₈×D₁₅	1	1	yes	characteristic
260	D₄	—	15	6	no	—
261	D₄	—	15	3	no	—
262	D₄	C₈×D₁₅	1	3	yes	—
263	C₂×C₄	—	15	3	no	—
264	C₂×C₄	—	15	3	no	—
265	C₂×C₄	—	15	3	no	—
266	C₂×C₄	—	15	1	no	—
267	C₂×C₄	—	15	3	no	—
268	C₂×C₄	C₂×C₄×D₁₅	1	3	yes	—
269	C₈	—	15	1	no	—
270	C₈	—	15	3	no	—
271	C₈	C₂×C₄×D₁₅	1	3	yes	—
272	C₈	C₂³×D₁₅	1	1	yes	characteristic
273	C₂×C₃	—	2	3	no	—
274	C₂×C₃	C₂²×C₈×D₅	1	1	yes	characteristic
275	D₃	—	10	1	no	—
276	D₃	—	10	3	no	—
277	C₅	#82012	1	1	yes	characteristic, complemented, Sylow
278	C₂²	—	15	1	no	—
279	C₂²	—	15	3	no	—
280	C₂²	—	60	3	no	—
281	C₂²	C₂×C₈×D₁₅	1	3	yes	—
282	C₄	—	15	1	no	—
283	C₄	—	15	3	no	—
284	C₄	C₂×C₈×D₁₅	1	3	yes	—
285	C₄	C₂²×C₄×D₁₅	1	1	yes	characteristic
286	C₃	#10613743	1	1	yes	characteristic, complemented, Sylow
287	C₂	—	2	3	no	—
288	C₂	—	30	1	no	—
289	C₂	—	30	3	no	—
290	C₂	C₂²×C₈×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂³×D₁₅
commutator, socle	C₂×C₃×C₅	C₂³×C₈
Fitting	C₃×C₅×#504	C₂
prime radical	#41×D₁₅	C₄
cosocle	C₃×C₅×C₈	C₂⁴
hypercenter	#504	D₁₅
hypocenter	C₃×C₅	#3407
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₈ ◁ #504

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#504	C₂	semidirect
3. #11751241	C₂	semidirect
4. #11751008	C₂	semidirect
5. D₁₅×M₃₂	C₂	semidirect
6. #11751002	C₂	semidirect
7. C₂×#91082	C₂	semidirect
16. C₃×C₅×M₃₂	C₂²	semidirect
17. C₂×C₃×C₅×C₁₆	C₂²	semidirect
18. #91096	C₂²	semidirect
19. C₂×C₃⋊#672	C₂²	semidirect
22. #91082	C₂²	semidirect
23. #91082	C₂²	semidirect
24. C₁₆×D₁₅	C₂²	semidirect
27. C₅×#504	D₃	semidirect
54. C₃×#504	D₅	semidirect
67. C₅×M₃₂	C₂×D₃	semidirect
68. C₂×C₅×C₁₆	C₂×D₃	semidirect
100. C₃×M₃₂	C₂×D₅	semidirect
101. C₂×C₃×C₁₆	C₂×D₅	semidirect
133. #504	D₁₅	semidirect
192. M₃₂	C₂×D₁₅	semidirect
195. C₂×C₁₆	C₂×D₁₅	semidirect
244. C₃×C₅	#3407	semidirect
277. C₅	#82012	semidirect
286. C₃	#10613743	semidirect

Maximal subgroups

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

subgroup	index
D₁₅×#131	2
C₃×C₅×#504	2
#11751241	2
#11751008	2
D₁₅×M₃₂	2
#11751002	2
C₂×#91082	2
#10613743	3
#82012	5