Subgroups — #14280429

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6740	C₂	1	1	yes	maximal, characteristic
2	C₄×#6738	C₂	1	1	yes	maximal, characteristic
3	#11747583	C₂	1	1	yes	maximal, characteristic
4	C₈×C₃⋊#170	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#170⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#170	C₂	1	1	yes	maximal, characteristic, complemented
8	#10604009	—	3	1	no	maximal, Hall
9	C₂×#6740	C₄	1	2	yes	—
10	C₂×#6738	C₄	1	2	yes	—
11	C₄×#1200	C₂²	1	1	yes	characteristic
12	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
13	C₄×C₃⋊#170	C₂²	1	1	yes	characteristic
14	C₃×C₄×#170	C₂²	1	1	yes	characteristic
15	C₃×C₄×#170	C₂²	1	1	yes	characteristic
16	#91169	C₄	1	2	yes	—
17	Dic₃×#170	C₄	1	2	yes	—
18	#91104	C₄	1	1	yes	characteristic
19	C₅×Dic₃⋊C₈	C₄	1	1	yes	characteristic
20	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
21	Dic₃⋊#168	C₄	1	1	yes	characteristic
22	#90957	C₄	1	1	yes	characteristic
23	Dic₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#68067	—	5	1	no	maximal, Hall
25	C₄×#700	—	3	1	no	—
26	C₄×D₅⋊C₈	—	3	1	no	—
27	#65122	—	3	1	no	—
28	#170⋊C₈	—	3	1	no	—
29	#170⋊C₈	D₃	1	1	yes	characteristic, complemented
30	C₈×#170	—	3	1	no	—
31	C₈×#170	—	3	1	no	—
32	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
33	C₂×#1200	C₂×C₄	1	1	yes	characteristic
34	C₂×#1200	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃⋊#170	C₂×C₄	1	2	yes	—
36	C₂×C₃⋊#170	C₂×C₄	1	2	yes	—
37	C₂×C₃×#170	C₂×C₄	1	2	yes	—
38	C₂×C₃×#170	C₂×C₄	1	2	yes	—
39	#6740	C₈	1	8	yes	complemented
40	#6738	—	2	4	no	—
41	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
45	C₃×C₄×Dic₅	C₂³	1	1	yes	characteristic
46	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
47	Dic₃×Dic₅	C₂×C₄	1	1	yes	characteristic
48	Dic₃×Dic₅	C₂×C₄	1	1	yes	characteristic
49	C₄×ΓL(1,𝔽₂₅)	—	5	1	no	—
50	#4685	—	5	1	no	—
51	C₄×D₃×C₈	—	5	1	no	—
52	C₃×#322	—	5	1	no	—
53	#4454	—	5	1	no	—
54	C₈×C₃⋊C₈	—	5	1	no	—
55	C₈×C₃⋊C₈	—	5	1	no	—
56	C₂×#700	—	3	2	no	—
57	C₂×D₅⋊C₈	—	3	2	no	—
58	C₄²×D₅	—	3	1	no	—
59	#4002	—	3	2	no	—
60	C₄×#170	—	3	2	no	—
61	C₄×#170	—	3	1	no	—
62	C₄×#170	—	3	1	no	—
63	C₄×#170	C₂×D₃	1	1	yes	characteristic
64	C₄×#170	C₂×D₃	1	1	yes	characteristic
65	C₅×C₄⋊C₈	—	3	1	no	—
66	#3952	—	3	1	no	—
67	#3952	—	3	1	no	—
68	C₈×Dic₅	—	3	1	no	—
69	C₈×Dic₅	C₂×D₃	1	1	yes	characteristic
70	C₄⋊#168	—	3	1	no	—
71	C₈×M₁₆	—	15	1	no	Sylow
72	C₂²×D₁₅	C₄²	1	1	yes	characteristic
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×Dic₁₅	C₄²	1	1	yes	characteristic
75	C₄×D₁₅	—	2	1	no	—
76	C₄×D₁₅	—	2	1	no	—
77	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
78	C₂×C₅×Dic₃	C₄²	1	1	yes	characteristic
79	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
81	#1200	—	2	2	no	—
82	#1200	C₂×C₈	1	4	yes	—
83	C₃⋊#170	C₂×C₈	1	4	yes	complemented
84	C₃⋊#170	M₁₆	1	4	yes	complemented
85	C₃×#170	—	2	2	no	—
86	C₃×#170	C₂×C₈	1	4	yes	complemented
87	C₃×C₅×C₈	—	2	1	no	—
88	C₃⋊#168	—	2	1	no	—
89	C₃×#168	—	2	1	no	—
90	C₅×C₃⋊C₈	—	2	1	no	—
91	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
92	C₂×D₃×C₈	—	5	2	no	—
93	C₄²×D₃	—	5	1	no	—
94	C₃×C₄×C₈	—	5	1	no	—
95	C₃×C₄×C₈	—	5	1	no	—
96	C₃×C₄×C₈	—	5	1	no	—
97	(C₂×D₃)⋊C₈	—	5	1	no	—
98	(C₂×D₃)⋊C₈	—	5	1	no	—
99	#815	—	5	2	no	—
100	Dic₃⋊C₈	—	5	1	no	—
101	Dic₃⋊C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
102	C₈×Dic₃	—	5	2	no	—
103	C₄×C₃⋊C₈	—	5	1	no	—
104	C₄×C₃⋊C₈	—	5	1	no	—
105	C₄×C₃⋊C₈	—	5	1	no	—
106	C₂×C₄×D₅	—	3	1	no	—
107	C₂×C₄×D₅	—	3	1	no	—
108	C₂×C₄×D₅	—	3	1	no	—
109	C₂×#170	—	3	2	no	—
110	C₂×#170	—	3	2	no	—
111	C₂×#170	C₄×D₃	1	2	yes	—
112	C₂×#170	C₄×D₃	1	2	yes	—
113	#700	—	3	8	no	—
114	D₅⋊C₈	—	6	4	no	—
115	C₂×C₅×C₈	—	3	1	no	—
116	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
117	C₄²×C₅	—	3	1	no	—
118	C₄×Dic₅	—	3	1	no	—
119	C₄×Dic₅	—	3	1	no	—
120	C₄×Dic₅	C₂²×D₃	1	1	yes	characteristic
121	C₂×#168	—	3	1	no	—
122	C₂×#168	C₄×D₃	1	1	yes	characteristic
123	#431	—	15	1	no	—
124	C₄×M₁₆	—	15	1	no	—
125	C₂×C₄×C₈	—	15	1	no	—
126	#322	—	5	1	no	—
127	#322	—	15	1	no	—
128	C₈²	—	15	1	no	—
129	C₈²	—	15	1	no	—
130	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
131	C₂×D₁₅	—	2	1	no	—
132	C₂×D₁₅	—	2	1	no	—
133	C₂×D₁₅	C₄×C₈	1	2	yes	—
134	C₃×C₄×C₅	#131	1	1	yes	characteristic
135	C₃×C₄×C₅	#131	1	1	yes	characteristic
136	Dic₁₅	—	2	1	no	—
137	C₃×Dic₅	C₂²×C₈	1	2	yes	—
138	C₃×Dic₅	C₂×M₁₆	1	2	yes	—
139	C₅×Dic₃	—	2	1	no	—
140	C₅×Dic₃	C₄×C₈	1	2	yes	—
141	C₂×C₄×D₃	—	5	1	no	—
142	C₂×C₄×D₃	—	5	1	no	—
143	C₂×C₄×D₃	—	5	1	no	—
144	C₂×C₃×C₈	—	5	2	no	—
145	C₂×C₃×C₈	—	5	2	no	—
146	C₂×C₃×C₈	—	5	1	no	—
147	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
148	C₃×C₄²	—	5	1	no	—
149	C₄×Dic₃	—	5	1	no	—
150	C₄×Dic₃	—	5	1	no	—
151	C₄×Dic₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
152	C₂×C₃⋊C₈	—	5	2	no	—
153	C₂×C₃⋊C₈	—	5	2	no	—
154	C₂×C₃⋊C₈	—	5	1	no	—
155	C₂×C₃⋊C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
156	ΓL(1,𝔽₂₅)	—	5	8	no	—
157	D₃×C₈	—	10	4	no	—
158	C₂²×D₅	—	3	1	no	—
159	C₂×C₄×C₅	—	3	1	no	—
160	C₂×C₄×C₅	—	3	1	no	—
161	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂×Dic₅	—	3	1	no	—
163	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
164	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
165	C₄×D₅	—	3	4	no	—
166	C₄×D₅	—	6	1	no	—
167	C₄×D₅	—	6	1	no	—
168	C₄×D₅	—	6	2	no	—
169	#170	—	2	2	no	—
170	#170	—	3	4	no	—
171	#170	—	3	4	no	—
172	#170	D₃×C₈	1	4	yes	complemented
173	C₅×C₈	—	2	1	no	—
174	C₅×C₈	—	6	1	no	—
175	#168	—	2	1	no	—
176	#168	—	6	1	no	—
177	C₂×M₁₆	—	15	2	no	—
178	C₂²×C₈	—	15	2	no	—
179	C₂×C₄²	—	15	1	no	—
180	C₄⋊C₈	—	3	1	no	—
181	C₄⋊C₈	—	15	1	no	—
182	C₂²⋊C₈	—	15	1	no	—
183	C₂²⋊C₈	—	15	1	no	—
184	#97	—	15	2	no	—
185	C₄×C₈	—	5	1	no	—
186	C₄×C₈	—	5	1	no	—
187	C₄×C₈	—	5	1	no	—
188	C₄×C₈	—	15	1	no	—
189	C₄×C₈	—	15	1	no	—
190	C₄×C₈	—	15	1	no	—
191	C₄×C₈	—	15	2	no	—
192	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
193	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
194	C₂×C₃×C₅	#405	1	1	yes	characteristic
195	D₁₅	—	2	2	no	—
196	C₂²×D₃	—	5	1	no	—
197	C₂×C₃×C₄	—	5	1	no	—
198	C₂×C₃×C₄	—	5	1	no	—
199	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₂×Dic₃	—	5	1	no	—
201	C₂×Dic₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
202	C₂×Dic₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₄×D₃	—	5	4	no	—
204	C₄×D₃	—	10	2	no	—
205	C₄×D₃	—	10	1	no	—
206	C₄×D₃	—	10	1	no	—
207	C₃×C₈	—	2	1	no	—
208	C₃×C₈	—	5	4	no	—
209	C₃×C₈	—	10	2	no	—
210	C₃×C₈	—	10	1	no	—
211	C₃⋊C₈	—	2	1	no	—
212	C₃⋊C₈	—	5	4	no	—
213	C₃⋊C₈	—	5	4	no	—
214	C₃⋊C₈	—	10	1	no	—
215	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
216	C₂×D₅	—	3	2	no	—
217	C₂×D₅	—	6	1	no	—
218	C₂×D₅	—	6	1	no	—
219	C₄×C₅	—	3	2	no	—
220	C₄×C₅	—	6	1	no	—
221	C₄×C₅	#907	1	1	yes	characteristic
222	C₄×C₅	#907	1	1	yes	characteristic
223	Dic₅	—	6	1	no	—
224	Dic₅	C₂×D₃×C₈	1	2	yes	—
225	Dic₅	D₃×M₁₆	1	2	yes	—
226	C₂²×C₄	—	15	1	no	—
227	C₂²×C₄	—	15	1	no	—
228	C₂²×C₄	—	15	1	no	—
229	M₁₆	—	15	8	no	—
230	C₂×C₈	—	3	1	no	—
231	C₂×C₈	—	5	2	no	—
232	C₂×C₈	—	5	2	no	—
233	C₂×C₈	—	5	1	no	—
234	C₂×C₈	—	15	2	no	—
235	C₂×C₈	—	15	2	no	—
236	C₂×C₈	—	15	1	no	—
237	C₂×C₈	—	30	4	no	—
238	C₂×C₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
239	C₄²	—	3	1	no	—
240	C₄²	—	5	1	no	—
241	C₄²	—	15	1	no	—
242	C₄²	—	15	1	no	—
243	C₃×C₅	C₈×M₁₆	1	1	yes	characteristic, complemented, Hall
244	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
245	C₂×D₃	—	5	2	no	—
246	C₂×D₃	—	10	1	no	—
247	C₂×D₃	—	10	1	no	—
248	C₃×C₄	—	5	2	no	—
249	C₃×C₄	—	5	2	no	—
250	C₃×C₄	#4131	1	1	yes	characteristic
251	C₃×C₄	#4133	1	1	yes	characteristic
252	Dic₃	—	2	1	no	—
253	Dic₃	—	10	1	no	—
254	Dic₃	C₈×AGL(1,𝔽₅)	1	2	yes	—
255	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
256	C₂×C₅	#4707	1	1	yes	characteristic
257	C₂×C₅	#4708	1	1	yes	characteristic
258	D₅	—	6	2	no	—
259	C₂³	—	15	1	no	—
260	C₂×C₄	—	3	1	no	—
261	C₂×C₄	—	3	1	no	—
262	C₂×C₄	—	5	1	no	—
263	C₂×C₄	—	5	1	no	—
264	C₂×C₄	—	15	4	no	—
265	C₂×C₄	—	15	1	no	—
266	C₂×C₄	—	30	2	no	—
267	C₂×C₄	—	30	1	no	—
268	C₂×C₄	—	30	1	no	—
269	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
270	C₈	—	2	1	no	—
271	C₈	—	5	4	no	—
272	C₈	—	6	1	no	—
273	C₈	—	10	2	no	—
274	C₈	—	10	1	no	—
275	C₈	—	15	4	no	—
276	C₈	—	15	4	no	—
277	C₈	—	30	1	no	—
278	C₂×C₃	#65701	1	1	yes	characteristic
279	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
280	C₂×C₃	M₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic
281	D₃	—	10	2	no	—
282	C₅	#68067	1	1	yes	characteristic, complemented, Sylow
283	C₂²	—	15	2	no	—
284	C₂²	—	30	1	no	—
285	C₂²	—	30	1	no	—
286	C₂²	C₄×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
287	C₄	—	3	2	no	—
288	C₄	—	5	2	no	—
289	C₄	—	5	2	no	—
290	C₄	—	6	1	no	—
291	C₄	—	30	1	no	—
292	C₄	#91916	1	1	yes	characteristic
293	C₄	#91929	1	1	yes	characteristic
294	C₃	#10604009	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	30	2	no	—
296	C₂	D₃×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
297	C₂	#11751650	1	1	yes	characteristic
298	C₂	#11751874	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₂×C₃×C₅×C₈	C₂×C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×D₁₅	C₄²
cosocle	C₃×C₄×Dic₅	C₂³
hypercenter	C₂×C₈	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	C₈×M₁₆
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. C₈×C₃⋊#170	C₂	semidirect
5. C₃×#170⋊C₈	C₂	semidirect
6. #11746487	C₂	semidirect
7. C₃⋊C₈×#170	C₂	semidirect
29. #170⋊C₈	D₃	semidirect
39. #6740	C₈	semidirect
83. C₃⋊#170	C₂×C₈	semidirect
84. C₃⋊#170	M₁₆	semidirect
86. C₃×#170	C₂×C₈	semidirect
172. #170	D₃×C₈	semidirect
243. C₃×C₅	C₈×M₁₆	semidirect
282. C₅	#68067	semidirect
294. C₃	#10604009	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6740	2
C₄×#6738	2
#11747583	2
C₈×C₃⋊#170	2
C₃×#170⋊C₈	2
#11746487	2
C₃⋊C₈×#170	2
#10604009	3
#68067	5