Subgroups — #14280730

Order 1920.

428 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6762	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#704	C₂	1	1	yes	maximal, characteristic
3	Dic₅×#208	C₂	1	1	yes	maximal, characteristic
4	#11746487	C₂	1	1	yes	maximal, characteristic, complemented
5	#170⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746485	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
8	#10604737	—	3	1	no	maximal, Hall
9	C₂×#6762	C₄	1	2	yes	—
10	C₂×C₃×#704	C₄	1	2	yes	—
11	C₂×#6710	C₄	1	1	yes	characteristic
12	C₂×C₅×#208	C₄	1	1	yes	characteristic
13	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
14	#91231	C₄	1	2	yes	—
15	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
16	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
17	C₃×#4002	C₄	1	2	yes	—
18	C₃×C₄×#170	C₂²	1	1	yes	characteristic
19	C₃×C₄×#170	C₂²	1	1	yes	characteristic
20	#90952	C₄	1	1	yes	characteristic
21	#90952	C₄	1	1	yes	characteristic
22	Dic₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
23	Dic₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#69849	—	5	1	no	maximal, Hall
25	C₄×#704	—	3	1	no	—
26	C₄×#704	D₃	1	1	yes	characteristic
27	M₁₆×Dic₅	—	3	1	no	—
28	#170⋊C₈	—	3	1	no	—
29	#170⋊C₈	—	3	1	no	—
30	#64879	—	3	1	no	—
31	C₈×#170	—	3	1	no	—
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
35	#6762	—	2	4	no	—
36	C₂×C₃⋊#170	C₂×C₄	1	2	yes	—
37	C₂×C₃⋊#170	C₂×C₄	1	2	yes	—
38	C₃×#704	—	2	4	no	—
39	C₂×C₃×#170	C₂×C₄	1	2	yes	—
40	C₂×C₃×#170	C₂×C₄	1	2	yes	—
41	#6710	—	2	2	no	—
42	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
44	C₅×#208	—	2	2	no	—
45	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
46	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
48	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
49	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
50	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
51	C₃×C₄×M₁₆	—	5	1	no	—
52	C₄×#208	—	5	1	no	—
53	C₄×#208	—	5	1	no	—
54	#4455	—	5	1	no	—
55	#4454	—	5	1	no	—
56	#4454	—	5	1	no	—
57	C₈×C₃⋊C₈	—	5	1	no	—
58	C₂×#704	—	3	2	no	—
59	C₂×#704	Dic₃	1	2	yes	—
60	C₂×C₅×M₁₆	—	3	1	no	—
61	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
62	C₂×#681	—	3	1	no	—
63	#4002	—	3	2	no	—
64	#4002	Dic₃	1	2	yes	—
65	C₄×#170	—	3	1	no	—
66	C₄×#170	—	3	1	no	—
67	C₄×#170	C₂×D₃	1	1	yes	characteristic
68	C₄×#170	C₂×D₃	1	1	yes	characteristic
69	#3947	—	3	1	no	—
70	#3947	—	3	1	no	—
71	C₈×Dic₅	—	3	1	no	—
72	C₈×Dic₅	—	3	1	no	—
73	#1455	—	15	1	no	Sylow
74	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	—	2	1	no	—
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×Dic₅	—	2	2	no	—
80	C₂×C₃×Dic₅	—	2	2	no	—
81	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
82	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
83	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
85	C₃⋊#170	—	2	2	no	—
86	C₃⋊#170	M₁₆	1	4	yes	complemented
87	C₃×#170	—	2	2	no	—
88	C₃×#170	M₁₆	1	4	yes	complemented
89	C₃⋊#168	—	2	1	no	—
90	C₃⋊#168	M₁₆	1	2	yes	complemented
91	C₅×C₃⋊C₈	—	2	1	no	—
92	C₅×C₃⋊C₈	M₁₆	1	2	yes	complemented
93	C₂×C₃×M₁₆	—	5	2	no	—
94	C₂×C₃×C₄²	—	5	1	no	—
95	C₂×#208	—	5	2	no	—
96	C₂×#208	—	5	1	no	—
97	C₂×#208	AGL(1,𝔽₅)	1	1	yes	characteristic
98	C₃×#97	—	5	2	no	—
99	C₃×C₄×C₈	—	5	1	no	—
100	C₃×C₄×C₈	—	5	1	no	—
101	#803	—	5	2	no	—
102	#803	—	5	1	no	—
103	#803	—	5	1	no	—
104	C₄×C₃⋊C₈	—	5	1	no	—
105	C₄×C₃⋊C₈	—	5	1	no	—
106	C₄×C₃⋊C₈	—	5	1	no	—
107	C₄×C₃⋊C₈	—	5	1	no	—
108	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
109	C₂²×Dic₅	C₄×D₃	1	1	yes	characteristic
110	C₂²×Dic₅	C₂×Dic₃	1	1	yes	characteristic
111	#704	—	2	4	no	—
112	#704	—	6	4	no	—
113	C₂×#170	—	3	2	no	—
114	C₂×#170	—	3	2	no	—
115	C₂×#170	C₂×Dic₃	1	2	yes	—
116	C₂×#170	C₂×Dic₃	1	2	yes	—
117	C₅×M₁₆	—	6	2	no	—
118	C₂×C₅×C₈	—	3	1	no	—
119	C₂×C₅×C₈	—	3	1	no	—
120	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
121	C₄×Dic₅	C₄×D₃	1	1	yes	characteristic
122	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
123	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
124	#681	—	6	2	no	—
125	C₂×#168	—	3	1	no	—
126	C₂×#168	—	3	1	no	—
127	C₄×M₁₆	—	5	1	no	—
128	C₄×M₁₆	—	15	1	no	—
129	C₄×M₁₆	—	15	1	no	—
130	#322	—	15	1	no	—
131	#322	—	15	1	no	—
132	#322	—	15	1	no	—
133	C₈²	—	15	1	no	—
134	C₂²×C₃×C₅	—	2	1	no	—
135	C₂²×C₃×C₅	—	2	1	no	—
136	C₂²×C₃×C₅	—	2	1	no	—
137	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
140	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
141	C₃×Dic₅	—	2	1	no	—
142	C₃×Dic₅	—	2	1	no	—
143	C₃×Dic₅	C₂×M₁₆	1	2	yes	—
144	C₃×Dic₅	C₂×M₁₆	1	2	yes	—
145	C₂²×C₃×C₄	—	5	1	no	—
146	C₂²×C₃×C₄	—	5	1	no	—
147	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
148	C₃×M₁₆	—	10	4	no	—
149	C₂×C₃×C₈	—	5	2	no	—
150	C₂×C₃×C₈	—	5	2	no	—
151	C₃×C₄²	—	5	1	no	—
152	C₃×C₄²	—	5	1	no	—
153	C₃×C₄²	—	5	1	no	—
154	C₃×C₄²	—	5	1	no	—
155	#208	—	2	2	no	—
156	#208	—	10	2	no	—
157	#208	—	10	4	no	—
158	C₂×C₃⋊C₈	—	5	2	no	—
159	C₂×C₃⋊C₈	—	5	1	no	—
160	C₂×C₃⋊C₈	—	5	2	no	—
161	C₂×C₃⋊C₈	—	5	1	no	—
162	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
163	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
164	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
165	C₂×C₄×C₅	—	2	1	no	—
166	C₂×C₄×C₅	—	2	1	no	—
167	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×Dic₅	—	2	2	no	—
170	C₂×Dic₅	—	2	2	no	—
171	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
172	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
173	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×Dic₅	C₂²×Dic₃	1	1	yes	characteristic
175	#170	—	2	2	no	—
176	#170	—	3	4	no	—
177	#170	—	6	2	no	—
178	#170	#208	1	4	yes	complemented
179	C₅×C₈	—	3	2	no	—
180	C₅×C₈	—	6	1	no	—
181	#168	—	3	2	no	—
182	#168	—	6	1	no	—
183	C₂×M₁₆	—	3	1	no	—
184	C₂×M₁₆	—	5	2	no	—
185	C₂×M₁₆	—	15	2	no	—
186	C₂×M₁₆	—	15	1	no	—
187	C₂×C₄²	—	5	1	no	—
188	#97	—	5	2	no	—
189	#97	—	15	1	no	—
190	#97	—	15	1	no	—
191	#97	—	15	2	no	—
192	C₄×C₈	—	5	1	no	—
193	C₄×C₈	—	5	1	no	—
194	C₄×C₈	—	15	1	no	—
195	C₄×C₈	—	15	1	no	—
196	C₄×C₈	—	15	1	no	—
197	C₄×C₈	—	15	1	no	—
198	C₂×C₃×C₅	—	4	1	no	—
199	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
200	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
201	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
202	C₂³×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	2	1	no	—
205	C₂×C₃×C₄	—	5	1	no	—
206	C₂×C₃×C₄	—	5	1	no	—
207	C₂×C₃×C₄	—	5	1	no	—
208	C₂×C₃×C₄	—	5	1	no	—
209	C₂×C₃×C₄	—	10	2	no	—
210	C₂×C₃×C₄	—	10	2	no	—
211	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₃×C₈	—	5	4	no	—
214	C₃×C₈	—	10	2	no	—
215	C₃⋊C₈	—	2	1	no	—
216	C₃⋊C₈	—	5	4	no	—
217	C₃⋊C₈	—	5	2	no	—
218	C₃⋊C₈	—	10	2	no	—
219	C₃⋊C₈	—	10	1	no	—
220	C₃⋊C₈	#704	1	2	yes	complemented
221	C₂²×C₅	—	2	1	no	—
222	C₂²×C₅	—	2	1	no	—
223	C₂²×C₅	—	2	1	no	—
224	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
225	C₄×C₅	—	2	1	no	—
226	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
227	C₄×C₅	D₃×M₁₆	1	1	yes	characteristic
228	Dic₅	—	2	1	no	—
229	Dic₅	—	2	1	no	—
230	Dic₅	D₃×M₁₆	1	2	yes	—
231	Dic₅	C₂×#208	1	2	yes	—
232	C₂²×C₄	—	5	1	no	—
233	C₂²×C₄	—	5	1	no	—
234	C₂²×C₄	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
235	M₁₆	—	6	2	no	—
236	M₁₆	—	10	4	no	—
237	M₁₆	—	30	2	no	—
238	M₁₆	—	30	4	no	—
239	C₂×C₈	—	3	1	no	—
240	C₂×C₈	—	3	1	no	—
241	C₂×C₈	—	5	2	no	—
242	C₂×C₈	—	5	2	no	—
243	C₂×C₈	—	15	2	no	—
244	C₂×C₈	—	15	1	no	—
245	C₂×C₈	—	15	2	no	—
246	C₂×C₈	—	15	1	no	—
247	C₄²	—	5	1	no	—
248	C₄²	—	5	1	no	—
249	C₄²	—	5	1	no	—
250	C₄²	—	5	1	no	—
251	C₃×C₅	#1455	1	1	yes	characteristic, complemented, Hall
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
256	C₃×C₄	—	2	1	no	—
257	C₃×C₄	—	5	2	no	—
258	C₃×C₄	—	5	2	no	—
259	C₃×C₄	—	10	1	no	—
260	C₃×C₄	—	10	1	no	—
261	C₃×C₄	D₅⋊M₁₆	1	1	yes	characteristic
262	C₃×C₄	C₂×#704	1	1	yes	characteristic
263	C₂×C₅	—	4	1	no	—
264	C₂×C₅	C₄×#208	1	1	yes	characteristic
265	C₂×C₅	Dic₃×M₁₆	1	1	yes	characteristic
266	C₂×C₅	Dic₃×M₁₆	1	1	yes	characteristic
267	C₂³	Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	5	1	no	—
271	C₂×C₄	—	5	1	no	—
272	C₂×C₄	—	5	1	no	—
273	C₂×C₄	—	5	1	no	—
274	C₂×C₄	—	10	2	no	—
275	C₂×C₄	—	10	2	no	—
276	C₂×C₄	Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
277	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
278	C₈	—	3	2	no	—
279	C₈	—	5	4	no	—
280	C₈	—	6	1	no	—
281	C₈	—	10	2	no	—
282	C₈	—	15	4	no	—
283	C₈	—	15	2	no	—
284	C₈	—	30	2	no	—
285	C₈	—	30	1	no	—
286	C₂×C₃	—	4	1	no	—
287	C₂×C₃	M₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
288	C₂×C₃	M₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
289	C₂×C₃	C₄×#704	1	1	yes	characteristic
290	C₅	#69849	1	1	yes	characteristic, complemented, Sylow
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	C₂×Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	5	2	no	—
297	C₄	—	5	2	no	—
298	C₄	—	10	1	no	—
299	C₄	—	10	1	no	—
300	C₄	D₃×#704	1	1	yes	characteristic
301	C₄	#91931	1	1	yes	characteristic
302	C₃	#10604737	1	1	yes	characteristic, complemented, Sylow
303	C₂	—	4	1	no	—
304	C₂	AGL(1,𝔽₅)×#208	1	1	yes	characteristic
305	C₂	#11751698	1	1	yes	characteristic
306	C₂	Dic₃×#704	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator, socle	C₂²×C₃×C₅	C₂×C₄²
Fitting	C₂²×C₃×C₄×C₅	C₂×C₄
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	C₂²×C₄	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#1455
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
4. #11746487	C₂	semidirect
5. #170⋊(C₃⋊C₈)	C₂	semidirect
6. #11746485	C₂	semidirect
7. C₃⋊C₈×#170	C₂	semidirect
86. C₃⋊#170	M₁₆	semidirect
88. C₃×#170	M₁₆	semidirect
90. C₃⋊#168	M₁₆	semidirect
92. C₅×C₃⋊C₈	M₁₆	semidirect
178. #170	#208	semidirect
220. C₃⋊C₈	#704	semidirect
251. C₃×C₅	#1455	semidirect
290. C₅	#69849	semidirect
302. C₃	#10604737	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6762	2
C₃×C₄×#704	2
Dic₅×#208	2
#11746487	2
#170⋊(C₃⋊C₈)	2
#11746485	2
C₃⋊C₈×#170	2
#10604737	3
#69849	5