Subgroups — #14305053

Order 1920.

496 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91222	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
3	#11752371	C₂	1	2	yes	maximal
4	C₃×D₅×#106	C₂	1	1	yes	maximal, characteristic
5	#11746745	C₂	1	2	yes	maximal
6	#10604051	—	3	1	no	maximal, Hall
7	C₂×C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
8	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
9	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
10	C₂×D₅⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
12	C₃×D₅×C₄⋊C₄	C₂²	1	2	yes	—
13	#91235	—	2	2	no	—
14	#91222	—	2	2	no	—
15	C₈×ΓL(1,𝔽₁₆)	—	2	2	no	—
16	C₃×C₅×#106	C₄	1	1	yes	characteristic, complemented
17	C₃×#3949	C₄	1	1	yes	characteristic, complemented
18	C₃×#3940	C₄	1	2	yes	complemented
19	#70560	—	5	1	no	maximal, Hall
20	C₂×#3993	—	3	1	no	—
21	C₂×C₈×AGL(1,𝔽₅)	—	3	1	no	—
22	#65712	—	3	2	no	—
23	D₅×#106	D₃	1	1	yes	characteristic
24	#64874	—	3	2	no	—
25	C₂²×ΓL(1,𝔽₁₆)	—	2	2	no	—
26	C₂²×ΓL(1,𝔽₁₆)	D₄	1	2	yes	—
27	C₂×C₃×C₄×D₅	—	2	2	no	—
28	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
29	C₂×C₃⋊#170	—	2	1	no	—
30	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
31	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
32	C₄⋊ΓL(1,𝔽₁₆)	—	4	4	no	—
33	C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
34	C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
35	D₅⋊(C₃⋊C₈)	—	2	1	no	—
36	D₅⋊(C₃⋊C₈)	Q₈	1	2	yes	—
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
39	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
40	C₃×#683	—	2	2	no	—
41	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
42	C₃×C₈×D₅	D₄	1	2	yes	—
43	C₃×C₈×D₅	Q₈	1	2	yes	—
44	C₂×C₃×#106	—	5	1	no	—
45	C₂×#817	—	5	1	no	—
46	C₂×C₈×Dic₃	—	5	1	no	—
47	#4984	—	5	2	no	—
48	#4529	—	5	2	no	—
49	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
50	C₂×C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
51	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
52	C₂×D₅⋊C₈	—	3	1	no	—
53	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
54	D₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
55	(C₂×AGL(1,𝔽₅))⋊C₄	—	6	2	no	—
56	#3993	—	6	2	no	—
57	C₈×AGL(1,𝔽₅)	—	6	2	no	—
58	C₅×#106	Dic₃	1	1	yes	characteristic, complemented
59	#3949	Dic₃	1	1	yes	characteristic, complemented
60	#3940	Dic₃	1	2	yes	complemented
61	#1941	—	15	1	no	Sylow
62	C₂²×C₃×D₅	C₂×D₄	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)	—	4	2	no	—
66	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
67	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
68	C₂×ΓL(1,𝔽₁₆)	SD₁₆	1	4	yes	—
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃×Dic₅	—	2	2	no	—
72	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
73	C₃×C₄×D₅	—	4	2	no	—
74	C₃×C₄×D₅	—	4	2	no	—
75	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
77	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
78	C₃×C₄×D₅	#41	1	1	yes	characteristic
79	C₃⋊#170	—	2	2	no	—
80	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	complemented
81	C₃×#168	C₄⋊C₄	1	2	yes	complemented
82	C₂²×C₃×C₈	—	5	1	no	—
83	C₂×C₃×C₄⋊C₄	—	5	2	no	—
84	C₂×C₄⋊Dic₃	—	5	2	no	—
85	C₂×C₄⋊Dic₃	—	10	2	no	—
86	C₂×C₄×Dic₃	—	5	1	no	—
87	C₂²×C₃⋊C₈	—	5	1	no	—
88	C₃×#106	—	5	1	no	—
89	C₃×#106	—	5	2	no	—
90	C₃×#106	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
91	C₃⋊#95	—	10	2	no	—
92	#817	—	10	2	no	—
93	C₈×Dic₃	—	10	2	no	—
94	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
95	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
96	C₂×C₄×D₅	—	2	2	no	—
97	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
98	C₂×#170	—	6	1	no	—
99	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
100	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
101	C₄⋊AGL(1,𝔽₅)	—	12	4	no	—
102	C₄×AGL(1,𝔽₅)	—	6	1	no	—
103	C₄×AGL(1,𝔽₅)	—	6	1	no	—
104	D₅⋊C₈	—	3	2	no	—
105	D₅⋊C₈	—	6	1	no	—
106	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
107	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
108	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
109	#683	—	2	2	no	—
110	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
111	C₈×D₅	Dic₆	1	2	yes	—
112	C₈×D₅	D₁₂	1	2	yes	—
113	C₂×#106	—	5	1	no	—
114	C₂×#106	—	15	1	no	—
115	C₂×C₄×C₈	—	15	1	no	—
116	#389	—	15	2	no	—
117	#340	—	15	2	no	—
118	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
119	C₂×C₃×D₅	#121	1	1	yes	characteristic
120	C₂×C₃×D₅	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₃×D₅	C₂×SD₁₆	1	2	yes	—
122	C₂×C₃×D₅	#135	1	2	yes	—
123	ΓL(1,𝔽₁₆)	—	2	4	no	—
124	ΓL(1,𝔽₁₆)	—	8	2	no	—
125	C₃×C₄×C₅	—	4	2	no	—
126	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
128	C₃×Dic₅	—	4	2	no	—
129	C₃×Dic₅	C₂²⋊Q₈	1	1	yes	characteristic
130	C₃×Dic₅	#125	1	1	yes	characteristic
131	C₂²×C₃×C₄	—	5	1	no	—
132	C₂²×C₃×C₄	—	10	2	no	—
133	C₂²×Dic₃	—	5	2	no	—
134	C₂²×Dic₃	—	10	2	no	—
135	C₂×C₃×C₈	—	5	1	no	—
136	C₂×C₃×C₈	—	5	2	no	—
137	C₂×C₃×C₈	—	5	2	no	—
138	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
139	C₃×C₄⋊C₄	—	5	2	no	—
140	C₃×C₄⋊C₄	—	10	2	no	—
141	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
142	C₄⋊Dic₃	—	10	2	no	—
143	C₄⋊Dic₃	—	10	2	no	—
144	C₄⋊Dic₃	—	20	4	no	—
145	C₄×Dic₃	—	10	1	no	—
146	C₄×Dic₃	—	10	1	no	—
147	C₂×C₃⋊C₈	—	5	2	no	—
148	C₂×C₃⋊C₈	—	10	1	no	—
149	C₂×C₃⋊C₈	—	10	1	no	—
150	C₂²×D₅	D₃×D₄	1	1	yes	characteristic
151	C₂×AGL(1,𝔽₅)	—	3	4	no	—
152	C₂×AGL(1,𝔽₅)	—	6	2	no	—
153	C₂×AGL(1,𝔽₅)	—	6	2	no	—
154	C₂×AGL(1,𝔽₅)	—	12	2	no	—
155	C₂×AGL(1,𝔽₅)	—	12	2	no	—
156	C₂×AGL(1,𝔽₅)	—	12	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
159	C₂×Dic₅	—	2	2	no	—
160	C₂×Dic₅	#237	1	1	yes	characteristic
161	C₄×D₅	—	4	2	no	—
162	C₄×D₅	—	4	2	no	—
163	C₄×D₅	C₂×Dic₆	1	1	yes	characteristic
164	C₄×D₅	C₂×D₁₂	1	1	yes	characteristic
165	C₄×D₅	D₃×Q₈	1	1	yes	characteristic
166	C₄×D₅	#239	1	1	yes	characteristic
167	#170	—	6	2	no	—
168	C₅×C₈	C₄⋊Dic₃	1	2	yes	complemented
169	#168	C₄⋊Dic₃	1	2	yes	complemented
170	C₂²×C₈	—	5	1	no	—
171	C₂²×C₈	—	15	1	no	—
172	C₂×C₄⋊C₄	—	5	2	no	—
173	C₂×C₄⋊C₄	—	15	2	no	—
174	C₂×C₄⋊C₄	—	30	2	no	—
175	C₂×C₄²	—	15	1	no	—
176	#106	—	5	1	no	—
177	#106	—	5	2	no	—
178	#106	—	30	2	no	—
179	#106	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
180	C₄×C₈	—	30	2	no	—
181	#95	—	30	2	no	—
182	C₂×C₃×C₅	#389	1	1	yes	characteristic
183	C₂×C₃×C₅	C₄×SD₁₆	1	2	yes	—
184	C₃×D₅	#465	1	2	yes	—
185	C₃×D₅	#498	1	1	yes	characteristic
186	C₃×D₅	#499	1	1	yes	characteristic
187	C₂³×C₃	—	5	1	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	5	1	no	—
190	C₂×C₃×C₄	—	5	1	no	—
191	C₂×C₃×C₄	—	5	1	no	—
192	C₂×C₃×C₄	—	5	1	no	—
193	C₂×C₃×C₄	—	5	1	no	—
194	C₂×C₃×C₄	—	10	2	no	—
195	C₂×C₃×C₄	—	20	2	no	—
196	C₂×C₃×C₄	—	20	2	no	—
197	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
198	C₂×Dic₃	—	5	4	no	—
199	C₂×Dic₃	—	10	2	no	—
200	C₂×Dic₃	—	10	2	no	—
201	C₂×Dic₃	—	20	2	no	—
202	C₂×Dic₃	—	20	2	no	—
203	C₂×Dic₃	—	20	2	no	—
204	C₃×C₈	—	5	2	no	—
205	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	complemented
206	C₃⋊C₈	—	10	2	no	—
207	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
208	C₂×D₅	#888	1	1	yes	characteristic
209	C₂×D₅	#895	1	1	yes	characteristic
210	C₂×D₅	D₃×SD₁₆	1	2	yes	—
211	C₂×D₅	#916	1	2	yes	—
212	AGL(1,𝔽₅)	—	6	4	no	—
213	AGL(1,𝔽₅)	—	24	2	no	—
214	C₄×C₅	—	4	2	no	—
215	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
216	C₄×C₅	Q₈×Dic₃	1	1	yes	characteristic
217	Dic₅	—	4	2	no	—
218	Dic₅	#890	1	1	yes	characteristic
219	Dic₅	#897	1	1	yes	characteristic
220	C₂²×C₄	—	5	1	no	—
221	C₂²×C₄	—	10	2	no	—
222	C₂²×C₄	—	15	2	no	—
223	C₂²×C₄	—	30	2	no	—
224	C₂×C₈	—	5	1	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	5	2	no	—
227	C₂×C₈	—	15	2	no	—
228	C₂×C₈	—	30	1	no	—
229	C₂×C₈	—	30	1	no	—
230	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
231	C₄⋊C₄	—	5	2	no	—
232	C₄⋊C₄	—	10	2	no	—
233	C₄⋊C₄	—	30	2	no	—
234	C₄⋊C₄	—	30	2	no	—
235	C₄⋊C₄	—	60	4	no	—
236	C₄⋊C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
237	C₄²	—	30	1	no	—
238	C₄²	—	30	1	no	—
239	C₃×C₅	#1941	1	1	yes	characteristic, complemented, Hall
240	C₂²×C₃	—	5	1	no	—
241	C₂²×C₃	—	5	1	no	—
242	C₂²×C₃	—	5	2	no	—
243	C₂²×C₃	—	5	2	no	—
244	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
245	C₃×C₄	—	4	2	no	—
246	C₃×C₄	—	5	1	no	—
247	C₃×C₄	—	5	1	no	—
248	C₃×C₄	—	20	2	no	—
249	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
250	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
251	Dic₃	—	10	4	no	—
252	Dic₃	—	40	2	no	—
253	C₂×C₅	#4984	1	1	yes	characteristic
254	C₂×C₅	Dic₃×SD₁₆	1	2	yes	—
255	D₅	#4855	1	1	yes	characteristic
256	D₅	#4858	1	1	yes	characteristic
257	D₅	#4864	1	2	yes	—
258	C₂³	—	5	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	5	1	no	—
261	C₂×C₄	—	5	1	no	—
262	C₂×C₄	—	5	1	no	—
263	C₂×C₄	—	5	1	no	—
264	C₂×C₄	—	5	1	no	—
265	C₂×C₄	—	10	2	no	—
266	C₂×C₄	—	15	4	no	—
267	C₂×C₄	—	20	2	no	—
268	C₂×C₄	—	20	2	no	—
269	C₂×C₄	—	30	2	no	—
270	C₂×C₄	—	30	2	no	—
271	C₂×C₄	—	60	2	no	—
272	C₂×C₄	—	60	2	no	—
273	C₂×C₄	—	60	2	no	—
274	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
275	C₈	—	5	2	no	—
276	C₈	—	30	2	no	—
277	C₈	C₄⋊ΓL(1,𝔽₁₆)	1	2	yes	complemented
278	C₂×C₃	—	5	2	no	—
279	C₂×C₃	—	5	1	no	—
280	C₂×C₃	—	5	1	no	—
281	C₂×C₃	#65712	1	1	yes	characteristic
282	C₂×C₃	SD₁₆×AGL(1,𝔽₅)	1	2	yes	—
283	C₅	#70560	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	5	1	no	—
285	C₂²	—	5	1	no	—
286	C₂²	—	5	2	no	—
287	C₂²	—	5	2	no	—
288	C₂²	D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
289	C₄	—	4	2	no	—
290	C₄	—	5	1	no	—
291	C₄	—	5	1	no	—
292	C₄	—	20	2	no	—
293	C₄	—	30	4	no	—
294	C₄	—	120	2	no	—
295	C₄	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
296	C₄	Q₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
297	C₃	#10604051	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	5	2	no	—
299	C₂	—	5	1	no	—
300	C₂	—	5	1	no	—
301	C₂	#11752371	1	1	yes	characteristic
302	C₂	SD₁₆×ΓL(1,𝔽₁₆)	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×#106	C₄
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂²×C₃×D₅	C₂×D₄
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	#106	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1941
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
16. C₃×C₅×#106	C₄	semidirect
17. C₃×#3949	C₄	semidirect
18. C₃×#3940	C₄	semidirect
58. C₅×#106	Dic₃	semidirect
59. #3949	Dic₃	semidirect
60. #3940	Dic₃	semidirect
80. C₃×C₅×C₈	C₄⋊C₄	semidirect
81. C₃×#168	C₄⋊C₄	semidirect
90. C₃×#106	AGL(1,𝔽₅)	semidirect
168. C₅×C₈	C₄⋊Dic₃	semidirect
169. #168	C₄⋊Dic₃	semidirect
179. #106	ΓL(1,𝔽₁₆)	semidirect
205. C₃×C₈	C₄⋊AGL(1,𝔽₅)	semidirect
239. C₃×C₅	#1941	semidirect
277. C₈	C₄⋊ΓL(1,𝔽₁₆)	semidirect
283. C₅	#70560	semidirect
297. C₃	#10604051	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91222	2
C₂×C₈×ΓL(1,𝔽₁₆)	2
#11752371	2
C₃×D₅×#106	2
#11746745	2
#10604051	3
#70560	5