Subgroups — #14296466

Order 1920.

464 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅⋊#819	C₂	1	1	yes	maximal, characteristic
2	#11752380	C₂	1	1	yes	maximal, characteristic
3	#11752377	C₂	1	2	yes	maximal
4	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
5	#11746733	C₂	1	2	yes	maximal
6	#10603851	—	3	1	no	maximal, Hall
7	#91989	C₂²	1	2	yes	—
8	D₅⋊#208	C₂²	1	2	yes	—
9	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
12	#91225	C₄	1	2	yes	complemented
13	#91224	C₄	1	2	yes	complemented
14	#91223	C₄	1	2	yes	complemented
15	#91222	C₄	1	2	yes	complemented
16	#91221	—	2	2	no	—
17	C₈×ΓL(1,𝔽₁₆)	—	2	2	no	—
18	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic, complemented
19	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic, complemented
20	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic, complemented
21	C₃×C₄×#168	C₄	1	1	yes	characteristic, complemented
22	#69779	—	5	1	no	maximal, Hall
23	D₅⋊#108	—	3	1	no	—
24	#65721	—	3	1	no	—
25	#65718	—	3	2	no	—
26	C₄×C₈×D₅	D₃	1	1	yes	characteristic
27	#64862	—	3	2	no	—
28	C₂×C₃×C₄×D₅	—	2	1	no	—
29	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
30	#6763	—	2	2	no	—
31	#6762	—	2	2	no	—
32	C₄⋊ΓL(1,𝔽₁₆)	C₂×C₄	1	4	yes	—
33	C₄×ΓL(1,𝔽₁₆)	—	2	2	no	—
34	#6758	C₂×C₄	1	4	yes	—
35	D₅⋊(C₃⋊C₈)	—	2	2	no	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄×Dic₅	—	2	1	no	—
40	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
43	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
44	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
45	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
46	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
47	C₃×C₈×D₅	D₄	1	1	yes	characteristic
48	C₃×C₈×D₅	D₄	1	1	yes	characteristic
49	C₃×C₈×D₅	Q₈	1	1	yes	characteristic
50	C₃×C₈×D₅	Q₈	1	1	yes	characteristic
51	C₂×C₃×C₄×C₈	—	5	1	no	—
52	C₂×#819	—	5	1	no	—
53	#5106	—	5	1	no	—
54	#5101	—	5	2	no	—
55	#4523	—	5	2	no	—
56	#4130	—	3	2	no	—
57	D₅⋊M₁₆	—	3	2	no	—
58	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
59	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
60	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
61	#3996	—	3	2	no	—
62	#3995	—	3	2	no	—
63	#3994	—	3	2	no	—
64	#3993	—	3	2	no	—
65	#3992	—	6	2	no	—
66	C₈×AGL(1,𝔽₅)	—	6	2	no	—
67	C₄×C₅×C₈	Dic₃	1	1	yes	characteristic, complemented
68	C₈×Dic₅	Dic₃	1	1	yes	characteristic, complemented
69	C₈×Dic₅	Dic₃	1	1	yes	characteristic, complemented
70	C₄×#168	Dic₃	1	1	yes	characteristic, complemented
71	#1772	—	15	1	no	Sylow
72	C₂²×C₃×D₅	#41	1	1	yes	characteristic
73	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
74	C₂×C₃×C₄×C₅	—	2	1	no	—
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×Dic₅	—	2	1	no	—
77	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
78	C₃×C₄×D₅	—	2	4	no	—
79	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
80	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
81	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
82	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
83	C₃⋊#170	—	2	4	no	—
84	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
85	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
86	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
87	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
88	C₃×#168	C₄²	1	1	yes	characteristic
89	C₃×#168	C₄²	1	1	yes	characteristic
90	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
91	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
92	C₂²×C₃×C₈	—	5	1	no	—
93	C₂²×C₃×C₈	—	5	1	no	—
94	C₂×C₃×C₄²	—	5	1	no	—
95	#926	—	5	2	no	—
96	C₂×#208	—	5	2	no	—
97	C₃×C₄×C₈	—	5	1	no	—
98	C₃×C₄×C₈	—	5	1	no	—
99	C₃×C₄×C₈	—	5	1	no	—
100	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
101	#819	—	5	2	no	—
102	#819	—	5	2	no	—
103	#818	—	5	2	no	—
104	#817	—	5	2	no	—
105	#815	—	10	2	no	—
106	C₈×Dic₃	—	10	2	no	—
107	C₂×C₄×D₅	—	2	1	no	—
108	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
109	C₂²⋊AGL(1,𝔽₅)	—	6	2	no	—
110	#704	—	6	2	no	—
111	C₄⋊AGL(1,𝔽₅)	—	3	4	no	—
112	C₄×AGL(1,𝔽₅)	—	6	2	no	—
113	#700	—	3	4	no	—
114	D₅⋊C₈	—	6	2	no	—
115	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
116	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
117	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
118	C₄×Dic₅	—	2	1	no	—
119	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
120	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
121	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
122	C₈×D₅	Dic₆	1	1	yes	characteristic
123	C₈×D₅	Dic₆	1	1	yes	characteristic
124	C₈×D₅	C₄×D₃	1	1	yes	characteristic
125	C₈×D₅	C₄×D₃	1	1	yes	characteristic
126	C₈×D₅	C₄×D₃	1	1	yes	characteristic
127	C₈×D₅	C₄×D₃	1	1	yes	characteristic
128	C₈×D₅	D₁₂	1	1	yes	characteristic
129	C₈×D₅	D₁₂	1	1	yes	characteristic
130	C₂×#108	—	15	1	no	—
131	#427	—	15	1	no	—
132	#405	—	15	2	no	—
133	C₂×C₄×C₈	—	5	1	no	—
134	#339	—	15	2	no	—
135	C₂²×C₃×C₅	#117	1	1	yes	characteristic
136	C₂×C₃×D₅	—	2	2	no	—
137	C₂×C₃×D₅	C₄×D₄	1	1	yes	characteristic
138	C₂×C₃×D₅	C₄×Q₈	1	1	yes	characteristic
139	ΓL(1,𝔽₁₆)	—	4	4	no	—
140	C₃×C₄×C₅	—	2	2	no	—
141	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
143	C₃×Dic₅	—	2	2	no	—
144	C₃×Dic₅	C₄×D₄	1	1	yes	characteristic
145	C₃×Dic₅	C₄×Q₈	1	1	yes	characteristic
146	C₂²×C₃×C₄	—	5	1	no	—
147	C₂²×C₃×C₄	—	10	1	no	—
148	C₂×C₃×C₈	—	5	1	no	—
149	C₂×C₃×C₈	—	5	1	no	—
150	C₂×C₃×C₈	—	5	1	no	—
151	C₂×C₃×C₈	—	5	1	no	—
152	C₂×C₃×C₈	—	5	1	no	—
153	C₂×C₃×C₈	—	5	1	no	—
154	C₂×C₃×C₈	—	5	1	no	—
155	C₂×C₃×C₈	—	5	1	no	—
156	C₂×C₃×C₈	—	5	1	no	—
157	C₂×C₃×C₈	—	5	1	no	—
158	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
159	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
160	C₃×C₄²	—	5	1	no	—
161	C₃×C₄²	—	10	1	no	—
162	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
163	#217	—	10	2	no	—
164	C₄⋊Dic₃	—	5	4	no	—
165	C₄×Dic₃	—	10	2	no	—
166	#208	—	5	4	no	—
167	#208	—	10	2	no	—
168	C₂×C₃⋊C₈	—	10	2	no	—
169	C₂²×D₅	#235	1	1	yes	characteristic
170	C₂×AGL(1,𝔽₅)	—	6	4	no	—
171	C₂×C₄×C₅	—	2	1	no	—
172	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
173	C₂×Dic₅	—	2	1	no	—
174	C₂×Dic₅	#235	1	1	yes	characteristic
175	C₄×D₅	—	2	4	no	—
176	C₄×D₅	C₂×Dic₆	1	1	yes	characteristic
177	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
178	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
179	C₄×D₅	C₂×D₁₂	1	1	yes	characteristic
180	#170	—	6	4	no	—
181	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
182	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
183	C₅×C₈	C₄⋊Dic₃	1	1	yes	characteristic
184	C₅×C₈	C₄⋊Dic₃	1	1	yes	characteristic
185	#168	C₄×Dic₃	1	1	yes	characteristic
186	#168	C₄×Dic₃	1	1	yes	characteristic
187	#168	C₄⋊Dic₃	1	1	yes	characteristic
188	#168	C₄⋊Dic₃	1	1	yes	characteristic
189	C₂×M₁₆	—	15	2	no	—
190	C₂²×C₈	—	5	1	no	—
191	C₂²×C₈	—	5	1	no	—
192	#117	—	15	2	no	—
193	C₂×C₄²	—	5	1	no	—
194	#108	—	15	2	no	—
195	#108	—	15	2	no	—
196	#107	—	15	2	no	—
197	#106	—	15	2	no	—
198	#97	—	30	2	no	—
199	C₄×C₈	—	5	1	no	—
200	C₄×C₈	—	5	1	no	—
201	C₄×C₈	—	5	1	no	—
202	C₄×C₈	—	30	2	no	—
203	C₄×C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
204	C₂×C₃×C₅	—	2	1	no	—
205	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
206	C₃×D₅	—	2	1	no	—
207	C₃×D₅	#443	1	2	yes	—
208	C₂³×C₃	—	5	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	5	1	no	—
211	C₂×C₃×C₄	—	5	1	no	—
212	C₂×C₃×C₄	—	5	1	no	—
213	C₂×C₃×C₄	—	5	1	no	—
214	C₂×C₃×C₄	—	5	1	no	—
215	C₂×C₃×C₄	—	10	4	no	—
216	C₂×C₃×C₄	—	10	1	no	—
217	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
218	C₂×Dic₃	—	10	4	no	—
219	C₃×C₈	—	5	1	no	—
220	C₃×C₈	—	5	1	no	—
221	C₃×C₈	—	5	1	no	—
222	C₃×C₈	—	5	1	no	—
223	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
224	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
225	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
226	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
227	C₃⋊C₈	—	10	4	no	—
228	C₂²×C₅	#926	1	1	yes	characteristic
229	C₂×D₅	—	2	2	no	—
230	C₂×D₅	C₄×Dic₆	1	1	yes	characteristic
231	C₂×D₅	C₄×D₁₂	1	1	yes	characteristic
232	AGL(1,𝔽₅)	—	12	4	no	—
233	C₄×C₅	—	2	2	no	—
234	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
235	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
236	Dic₅	—	2	2	no	—
237	Dic₅	C₄×Dic₆	1	1	yes	characteristic
238	Dic₅	C₄×D₁₂	1	1	yes	characteristic
239	C₂²×C₄	—	5	1	no	—
240	C₂²×C₄	—	10	1	no	—
241	M₁₆	—	15	4	no	—
242	M₁₆	—	30	2	no	—
243	C₂×C₈	—	5	1	no	—
244	C₂×C₈	—	5	1	no	—
245	C₂×C₈	—	5	1	no	—
246	C₂×C₈	—	5	1	no	—
247	C₂×C₈	—	5	1	no	—
248	C₂×C₈	—	5	1	no	—
249	C₂×C₈	—	5	1	no	—
250	C₂×C₈	—	5	1	no	—
251	C₂×C₈	—	5	1	no	—
252	C₂×C₈	—	5	1	no	—
253	C₂×C₈	—	30	2	no	—
254	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
255	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
256	C₄⋊C₄	—	15	4	no	—
257	C₂²⋊C₄	—	30	2	no	—
258	C₄²	—	5	1	no	—
259	C₄²	—	10	1	no	—
260	C₄²	—	30	2	no	—
261	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
262	C₃×C₅	#1772	1	1	yes	characteristic, complemented, Hall
263	C₂²×C₃	—	5	1	no	—
264	C₂²×C₃	—	5	1	no	—
265	C₂²×C₃	—	10	2	no	—
266	C₂²×C₃	#4130	1	1	yes	characteristic
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	5	1	no	—
269	C₃×C₄	—	5	1	no	—
270	C₃×C₄	—	10	2	no	—
271	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
272	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
273	Dic₃	—	20	4	no	—
274	C₂×C₅	—	2	1	no	—
275	C₂×C₅	C₄×C₄⋊Dic₃	1	1	yes	characteristic
276	D₅	—	2	1	no	—
277	D₅	#4700	1	2	yes	—
278	C₂³	—	5	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	5	1	no	—
281	C₂×C₄	—	5	1	no	—
282	C₂×C₄	—	5	1	no	—
283	C₂×C₄	—	5	1	no	—
284	C₂×C₄	—	5	1	no	—
285	C₂×C₄	—	10	4	no	—
286	C₂×C₄	—	10	1	no	—
287	C₂×C₄	—	30	4	no	—
288	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
289	C₈	—	5	1	no	—
290	C₈	—	5	1	no	—
291	C₈	—	5	1	no	—
292	C₈	—	5	1	no	—
293	C₈	—	30	4	no	—
294	C₈	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
295	C₈	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
296	C₈	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
297	C₈	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
298	C₂×C₃	—	2	1	no	—
299	C₂×C₃	—	5	2	no	—
300	C₂×C₃	—	10	1	no	—
301	C₂×C₃	C₄×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
302	C₅	#69779	1	1	yes	characteristic, complemented, Sylow
303	C₂²	—	5	1	no	—
304	C₂²	—	5	1	no	—
305	C₂²	—	10	2	no	—
306	C₂²	#91989	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	5	1	no	—
309	C₄	—	5	1	no	—
310	C₄	—	10	2	no	—
311	C₄	—	60	4	no	—
312	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
313	C₄	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
314	C₃	#10603851	1	1	yes	characteristic, complemented, Sylow
315	C₂	—	2	1	no	—
316	C₂	—	5	2	no	—
317	C₂	—	10	1	no	—
318	C₂	C₄×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄⋊ΓL(1,𝔽₁₆)
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₄×C₅×C₈	C₄
socle	C₂×C₃×C₅	C₄×C₄⋊C₄
prime radical	C₂²×C₃×D₅	#41
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	C₄×C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1772
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
12. #91225	C₄	semidirect
13. #91224	C₄	semidirect
14. #91223	C₄	semidirect
15. #91222	C₄	semidirect
18. C₃×C₄×C₅×C₈	C₄	semidirect
19. C₃×C₈×Dic₅	C₄	semidirect
20. C₃×C₈×Dic₅	C₄	semidirect
21. C₃×C₄×#168	C₄	semidirect
67. C₄×C₅×C₈	Dic₃	semidirect
68. C₈×Dic₅	Dic₃	semidirect
69. C₈×Dic₅	Dic₃	semidirect
70. C₄×#168	Dic₃	semidirect
100. C₃×C₄×C₈	AGL(1,𝔽₅)	semidirect
203. C₄×C₈	ΓL(1,𝔽₁₆)	semidirect
262. C₃×C₅	#1772	semidirect
302. C₅	#69779	semidirect
314. C₃	#10603851	semidirect

Maximal subgroups

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

subgroup	index
D₅⋊#819	2
#11752380	2
#11752377	2
C₃×C₄×C₈×D₅	2
#11746733	2
#10603851	3
#69779	5