Subgroups — #14296468

Order 1920.

464 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	C₂×#91225	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
3	#11752360	C₂	1	2	yes	maximal
4	C₃×#65169	C₂	1	1	yes	maximal, characteristic
5	#11746746	C₂	1	2	yes	maximal
6	#10604077	—	3	1	no	maximal, Hall
7	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
8	C₂×#6758	C₂²	1	2	yes	—
9	C₂×D₅⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₃×#4038	C₂²	1	2	yes	—
12	#91231	—	2	2	no	—
13	C₄×C₃⋊#170	—	2	2	no	—
14	#91225	C₄	1	4	yes	complemented
15	C₈×ΓL(1,𝔽₁₆)	C₄	1	4	yes	complemented
16	C₃×C₅×#107	C₄	1	1	yes	characteristic, complemented
17	C₃×#3950	C₄	1	1	yes	characteristic, complemented
18	C₃×#3940	C₄	1	2	yes	complemented
19	#70659	—	5	1	no	maximal, Hall
20	C₂×#3996	—	3	1	no	—
21	C₂×C₈×AGL(1,𝔽₅)	—	3	1	no	—
22	#65701	—	3	2	no	—
23	#65169	D₃	1	1	yes	characteristic
24	#64875	—	3	2	no	—
25	C₂²×ΓL(1,𝔽₁₆)	—	2	1	no	—
26	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
27	C₂×C₃⋊#170	—	2	2	no	—
28	C₂×C₃⋊#170	D₄	1	1	yes	characteristic
29	C₂×C₃⋊#170	D₄	1	1	yes	characteristic
30	C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
31	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
32	#6758	—	2	2	no	—
33	#6758	C₂×C₄	1	4	yes	—
34	D₅⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
35	D₅⋊(C₃⋊C₈)	Q₈	1	2	yes	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
38	C₃×#685	—	2	2	no	—
39	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
40	C₃×C₄×Dic₅	—	2	2	no	—
41	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
42	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
43	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
44	C₃×#427	—	5	1	no	—
45	C₂×#819	—	5	1	no	—
46	C₂×C₈×Dic₃	—	5	1	no	—
47	#4997	—	5	2	no	—
48	#4532	—	5	2	no	—
49	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
50	C₂×#700	—	3	2	no	—
51	C₂×D₅⋊C₈	—	3	1	no	—
52	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
53	#4038	C₂×D₃	1	2	yes	—
54	#4002	—	6	2	no	—
55	C₄×#170	—	6	2	no	—
56	#3996	—	3	4	no	—
57	C₈×AGL(1,𝔽₅)	—	3	4	no	—
58	C₅×#107	Dic₃	1	1	yes	characteristic, complemented
59	#3950	Dic₃	1	1	yes	characteristic, complemented
60	#3940	Dic₃	1	2	yes	complemented
61	#1772	—	15	1	no	Sylow
62	C₂²×C₃×D₅	#41	1	1	yes	characteristic
63	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
64	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
65	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×Dic₅	—	2	2	no	—
69	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
70	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
72	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
73	C₃×C₄×D₅	#41	1	1	yes	characteristic
74	C₃⋊#170	—	2	4	no	—
75	C₃⋊#170	C₄⋊C₄	1	2	yes	—
76	C₃⋊#170	C₄⋊C₄	1	2	yes	—
77	C₃×C₅×C₈	C₄²	1	2	yes	—
78	C₃×#168	C₄²	1	2	yes	—
79	C₂²×C₃×C₈	—	5	1	no	—
80	C₃×#117	—	5	2	no	—
81	C₂×C₄×Dic₃	—	5	1	no	—
82	C₂×#208	—	5	2	no	—
83	C₂²×C₃⋊C₈	—	5	1	no	—
84	C₃×#107	—	5	1	no	—
85	C₃×#107	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
86	C₃×#106	—	5	2	no	—
87	#819	—	5	4	no	—
88	C₈×Dic₃	—	5	4	no	—
89	#803	—	10	2	no	—
90	C₄×C₃⋊C₈	—	10	2	no	—
91	C₂²×AGL(1,𝔽₅)	—	6	1	no	—
92	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
93	C₂×#170	—	3	1	no	—
94	C₂×#170	—	3	1	no	—
95	C₂×#170	—	6	2	no	—
96	C₄×AGL(1,𝔽₅)	—	3	2	no	—
97	C₄×AGL(1,𝔽₅)	—	6	1	no	—
98	#700	—	3	4	no	—
99	#700	—	6	2	no	—
100	D₅⋊C₈	—	3	2	no	—
101	D₅⋊C₈	—	3	2	no	—
102	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
103	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
104	#685	—	2	2	no	—
105	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
106	C₄×Dic₅	—	2	2	no	—
107	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
108	C₈×D₅	C₄×D₃	1	2	yes	—
109	C₈×D₅	C₄×D₃	1	2	yes	—
110	C₂×#108	—	15	1	no	—
111	#427	—	5	1	no	—
112	#405	—	15	2	no	—
113	C₂×C₄×C₈	—	15	1	no	—
114	#339	—	15	2	no	—
115	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
116	C₂×C₃×D₅	—	2	1	no	—
117	C₂×C₃×D₅	—	2	1	no	—
118	C₂×C₃×D₅	#117	1	1	yes	characteristic
119	C₂×C₃×D₅	C₄×Q₈	1	1	yes	characteristic
120	ΓL(1,𝔽₁₆)	—	2	4	no	—
121	C₃×C₄×C₅	—	4	2	no	—
122	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
123	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
124	C₃×Dic₅	—	4	2	no	—
125	C₃×Dic₅	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₃×Dic₅	C₄×D₄	1	1	yes	characteristic
127	C₂²×C₃×C₄	—	5	1	no	—
128	C₂²×Dic₃	—	10	1	no	—
129	C₂×C₃×C₈	—	5	1	no	—
130	C₂×C₃×C₈	—	5	2	no	—
131	C₂×C₃×C₈	—	5	2	no	—
132	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
133	C₃×C₄⋊C₄	—	5	2	no	—
134	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
135	C₃×C₂²⋊C₄	—	10	2	no	—
136	C₃×C₄²	—	10	2	no	—
137	C₄×Dic₃	—	5	2	no	—
138	C₄×Dic₃	—	10	1	no	—
139	#208	—	5	4	no	—
140	#208	—	10	2	no	—
141	C₂×C₃⋊C₈	—	5	1	no	—
142	C₂×C₃⋊C₈	—	5	1	no	—
143	C₂×C₃⋊C₈	—	5	2	no	—
144	C₂×C₃⋊C₈	—	5	2	no	—
145	C₂×C₃⋊C₈	—	10	2	no	—
146	C₂²×D₅	#237	1	1	yes	characteristic
147	C₂×AGL(1,𝔽₅)	—	6	2	no	—
148	C₂×AGL(1,𝔽₅)	—	6	2	no	—
149	C₂×AGL(1,𝔽₅)	—	6	2	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
152	C₂×Dic₅	—	2	2	no	—
153	C₂×Dic₅	D₃×D₄	1	1	yes	characteristic
154	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
155	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
156	C₄×D₅	D₃×Q₈	1	1	yes	characteristic
157	C₄×D₅	#239	1	1	yes	characteristic
158	#170	—	3	2	no	—
159	#170	—	3	2	no	—
160	#170	—	6	4	no	—
161	C₅×C₈	C₄×Dic₃	1	2	yes	—
162	#168	C₄×Dic₃	1	2	yes	—
163	C₂×M₁₆	—	15	2	no	—
164	C₂²×C₈	—	5	1	no	—
165	C₂²×C₈	—	15	1	no	—
166	#117	—	5	2	no	—
167	C₂×C₄²	—	15	1	no	—
168	#108	—	15	4	no	—
169	#107	—	5	1	no	—
170	#107	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
171	#106	—	5	2	no	—
172	#97	—	30	2	no	—
173	C₄×C₈	—	15	4	no	—
174	C₄×C₈	—	30	2	no	—
175	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₃×C₅	#443	1	1	yes	characteristic
177	C₂×C₃×C₅	#443	1	1	yes	characteristic
178	C₃×D₅	—	2	1	no	—
179	C₃×D₅	—	2	1	no	—
180	C₂³×C₃	—	5	1	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	5	1	no	—
183	C₂×C₃×C₄	—	5	1	no	—
184	C₂×C₃×C₄	—	5	1	no	—
185	C₂×C₃×C₄	—	5	1	no	—
186	C₂×C₃×C₄	—	5	1	no	—
187	C₂×C₃×C₄	—	10	2	no	—
188	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
189	C₂×Dic₃	—	10	2	no	—
190	C₂×Dic₃	—	10	2	no	—
191	C₂×Dic₃	—	10	2	no	—
192	C₃×C₈	—	5	2	no	—
193	C₃×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
194	C₃⋊C₈	—	5	2	no	—
195	C₃⋊C₈	—	5	2	no	—
196	C₃⋊C₈	—	10	4	no	—
197	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
198	C₂×D₅	—	2	1	no	—
199	C₂×D₅	—	2	1	no	—
200	C₂×D₅	#887	1	1	yes	characteristic
201	C₂×D₅	#892	1	1	yes	characteristic
202	AGL(1,𝔽₅)	—	6	4	no	—
203	C₄×C₅	—	4	2	no	—
204	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
205	C₄×C₅	Q₈×Dic₃	1	1	yes	characteristic
206	Dic₅	—	4	2	no	—
207	Dic₅	D₃×C₄⋊C₄	1	1	yes	characteristic
208	Dic₅	#893	1	1	yes	characteristic
209	C₂²×C₄	—	5	1	no	—
210	C₂²×C₄	—	30	1	no	—
211	M₁₆	—	15	4	no	—
212	M₁₆	—	30	2	no	—
213	C₂×C₈	—	5	1	no	—
214	C₂×C₈	—	5	2	no	—
215	C₂×C₈	—	5	2	no	—
216	C₂×C₈	—	15	2	no	—
217	C₂×C₈	—	15	1	no	—
218	C₂×C₈	—	15	1	no	—
219	C₂×C₈	—	15	2	no	—
220	C₂×C₈	—	30	2	no	—
221	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
222	C₄⋊C₄	—	5	2	no	—
223	C₄⋊C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
224	C₂²⋊C₄	—	10	2	no	—
225	C₄²	—	10	2	no	—
226	C₄²	—	15	2	no	—
227	C₄²	—	30	1	no	—
228	C₃×C₅	#1772	1	1	yes	characteristic, complemented, Hall
229	C₂²×C₃	—	5	1	no	—
230	C₂²×C₃	—	5	1	no	—
231	C₂²×C₃	—	10	1	no	—
232	C₂²×C₃	—	10	1	no	—
233	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
234	C₃×C₄	—	4	2	no	—
235	C₃×C₄	—	5	1	no	—
236	C₃×C₄	—	5	1	no	—
237	C₃×C₄	—	20	2	no	—
238	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
239	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
240	Dic₃	—	10	4	no	—
241	C₂×C₅	Dic₃×C₄⋊C₄	1	1	yes	characteristic
242	C₂×C₅	#5196	1	1	yes	characteristic
243	C₂×C₅	#5196	1	1	yes	characteristic
244	D₅	—	2	1	no	—
245	D₅	—	2	1	no	—
246	C₂³	—	5	1	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	5	1	no	—
249	C₂×C₄	—	5	1	no	—
250	C₂×C₄	—	5	1	no	—
251	C₂×C₄	—	5	1	no	—
252	C₂×C₄	—	5	1	no	—
253	C₂×C₄	—	10	2	no	—
254	C₂×C₄	—	30	2	no	—
255	C₂×C₄	—	30	2	no	—
256	C₂×C₄	—	30	2	no	—
257	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
258	C₈	—	5	2	no	—
259	C₈	—	15	2	no	—
260	C₈	—	15	2	no	—
261	C₈	—	30	4	no	—
262	C₈	C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
263	C₂×C₃	—	10	1	no	—
264	C₂×C₃	—	10	1	no	—
265	C₂×C₃	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
266	C₂×C₃	#65732	1	1	yes	characteristic
267	C₂×C₃	#65740	1	1	yes	characteristic
268	C₅	#70659	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	5	1	no	—
270	C₂²	—	5	1	no	—
271	C₂²	—	10	1	no	—
272	C₂²	—	10	1	no	—
273	C₂²	D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
274	C₄	—	4	2	no	—
275	C₄	—	5	1	no	—
276	C₄	—	5	1	no	—
277	C₄	—	20	2	no	—
278	C₄	—	30	4	no	—
279	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
280	C₄	Q₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
281	C₃	#10604077	1	1	yes	characteristic, complemented, Sylow
282	C₂	—	10	1	no	—
283	C₂	—	10	1	no	—
284	C₂	C₄⋊C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
285	C₂	#11752391	1	1	yes	characteristic
286	C₂	#11752399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×ΓL(1,𝔽₁₆)
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×#107	C₄
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂²×C₃×D₅	#41
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	#107	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1772
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
14. #91225	C₄	semidirect
15. C₈×ΓL(1,𝔽₁₆)	C₄	semidirect
16. C₃×C₅×#107	C₄	semidirect
17. C₃×#3950	C₄	semidirect
18. C₃×#3940	C₄	semidirect
58. C₅×#107	Dic₃	semidirect
59. #3950	Dic₃	semidirect
60. #3940	Dic₃	semidirect
85. C₃×#107	AGL(1,𝔽₅)	semidirect
170. #107	ΓL(1,𝔽₁₆)	semidirect
228. C₃×C₅	#1772	semidirect
268. C₅	#70659	semidirect
281. C₃	#10604077	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91225	2
C₂×C₈×ΓL(1,𝔽₁₆)	2
#11752360	2
C₃×#65169	2
#11746746	2
#10604077	3
#70659	5