Subgroups — #14305188

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91223	C₂	1	1	yes	maximal, characteristic
2	C₂×#91221	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	#10604052	—	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	#91223	—	2	2	no	—
15	#91221	—	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	#70562	—	5	1	no	maximal, Hall
20	C₂×#3994	—	3	1	no	—
21	C₂×#3992	—	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	1	yes	characteristic
27	C₂²×ΓL(1,𝔽₁₆)	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄×D₅	—	2	2	no	—
29	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
30	C₂×C₃⋊#170	—	2	1	no	—
31	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
32	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
33	C₄⋊ΓL(1,𝔽₁₆)	—	4	4	no	—
34	C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
35	C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
36	D₅⋊(C₃⋊C₈)	—	2	1	no	—
37	D₅⋊(C₃⋊C₈)	Q₈	1	2	yes	—
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
40	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
41	C₃×#683	—	2	2	no	—
42	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
43	C₃×C₈×D₅	D₄	1	2	yes	—
44	C₃×C₈×D₅	Q₈	1	2	yes	—
45	C₂×C₃×#106	—	5	1	no	—
46	C₂×#818	—	5	1	no	—
47	C₂×#815	—	5	1	no	—
48	#4984	—	5	2	no	—
49	#4529	—	5	2	no	—
50	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
51	C₂×C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
52	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
53	C₂×D₅⋊C₈	—	3	1	no	—
54	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
55	D₅×C₄⋊C₄	C₂×D₃	1	2	yes	—
56	(C₂×AGL(1,𝔽₅))⋊C₄	—	6	2	no	—
57	#3994	—	6	2	no	—
58	#3992	—	6	2	no	—
59	C₅×#106	Dic₃	1	1	yes	characteristic, complemented
60	#3949	Dic₃	1	1	yes	characteristic, complemented
61	#3940	Dic₃	1	2	yes	complemented
62	#1944	—	15	1	no	Sylow
63	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
64	C₂×ΓL(1,𝔽₁₆)	—	2	1	no	—
65	C₂×ΓL(1,𝔽₁₆)	—	2	1	no	—
66	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
67	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
68	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
69	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
70	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
71	C₂×C₃×C₄×C₅	—	2	2	no	—
72	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₃×Dic₅	—	2	2	no	—
74	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
75	C₃×C₄×D₅	—	4	2	no	—
76	C₃×C₄×D₅	—	4	2	no	—
77	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
78	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
79	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
80	C₃×C₄×D₅	#41	1	1	yes	characteristic
81	C₃⋊#170	—	2	2	no	—
82	C₃×C₅×C₈	C₄⋊C₄	1	2	yes	complemented
83	C₃×#168	C₄⋊C₄	1	2	yes	complemented
84	C₂²×C₃×C₈	—	5	1	no	—
85	C₂×C₃×C₄⋊C₄	—	5	2	no	—
86	C₂×C₄⋊Dic₃	—	5	2	no	—
87	C₂×C₄⋊Dic₃	—	10	2	no	—
88	C₂×C₄×Dic₃	—	5	1	no	—
89	C₂²×C₃⋊C₈	—	5	1	no	—
90	C₃×#106	—	5	1	no	—
91	C₃×#106	—	5	2	no	—
92	C₃×#106	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
93	C₃⋊#95	—	10	2	no	—
94	#818	—	10	2	no	—
95	#815	—	10	2	no	—
96	C₂²×AGL(1,𝔽₅)	—	3	1	no	—
97	C₂²×AGL(1,𝔽₅)	—	3	1	no	—
98	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
99	C₂×C₄×D₅	—	2	2	no	—
100	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
101	C₂×#170	—	6	1	no	—
102	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
103	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
104	C₄⋊AGL(1,𝔽₅)	—	12	4	no	—
105	C₄×AGL(1,𝔽₅)	—	6	1	no	—
106	C₄×AGL(1,𝔽₅)	—	6	1	no	—
107	D₅⋊C₈	—	3	2	no	—
108	D₅⋊C₈	—	6	1	no	—
109	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
110	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
111	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
112	#683	—	2	2	no	—
113	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
114	C₈×D₅	Dic₆	1	2	yes	—
115	C₈×D₅	D₁₂	1	2	yes	—
116	C₂×#107	—	15	1	no	—
117	C₂×#106	—	5	1	no	—
118	C₂×#97	—	15	1	no	—
119	#389	—	15	2	no	—
120	#340	—	15	2	no	—
121	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
122	C₂×C₃×D₅	#121	1	1	yes	characteristic
123	C₂×C₃×D₅	C₄⋊Q₈	1	1	yes	characteristic
124	C₂×C₃×D₅	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₃×D₅	#137	1	2	yes	—
126	ΓL(1,𝔽₁₆)	—	4	1	no	—
127	ΓL(1,𝔽₁₆)	—	4	1	no	—
128	ΓL(1,𝔽₁₆)	—	8	2	no	—
129	C₃×C₄×C₅	—	4	2	no	—
130	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
132	C₃×Dic₅	—	4	2	no	—
133	C₃×Dic₅	C₂²⋊Q₈	1	1	yes	characteristic
134	C₃×Dic₅	#125	1	1	yes	characteristic
135	C₂²×C₃×C₄	—	5	1	no	—
136	C₂²×C₃×C₄	—	10	2	no	—
137	C₂²×Dic₃	—	5	1	no	—
138	C₂²×Dic₃	—	5	1	no	—
139	C₂²×Dic₃	—	10	2	no	—
140	C₂×C₃×C₈	—	5	1	no	—
141	C₂×C₃×C₈	—	5	2	no	—
142	C₂×C₃×C₈	—	5	2	no	—
143	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	5	2	no	—
145	C₃×C₄⋊C₄	—	10	2	no	—
146	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
147	C₄⋊Dic₃	—	10	2	no	—
148	C₄⋊Dic₃	—	10	2	no	—
149	C₄⋊Dic₃	—	20	4	no	—
150	C₄×Dic₃	—	10	1	no	—
151	C₄×Dic₃	—	10	1	no	—
152	C₂×C₃⋊C₈	—	5	2	no	—
153	C₂×C₃⋊C₈	—	10	1	no	—
154	C₂×C₃⋊C₈	—	10	1	no	—
155	C₂²×D₅	D₃×D₄	1	1	yes	characteristic
156	C₂×AGL(1,𝔽₅)	—	6	1	no	—
157	C₂×AGL(1,𝔽₅)	—	6	1	no	—
158	C₂×AGL(1,𝔽₅)	—	6	2	no	—
159	C₂×AGL(1,𝔽₅)	—	6	2	no	—
160	C₂×AGL(1,𝔽₅)	—	12	2	no	—
161	C₂×AGL(1,𝔽₅)	—	12	2	no	—
162	C₂×AGL(1,𝔽₅)	—	12	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
165	C₂×Dic₅	—	2	2	no	—
166	C₂×Dic₅	#237	1	1	yes	characteristic
167	C₄×D₅	—	4	2	no	—
168	C₄×D₅	—	4	2	no	—
169	C₄×D₅	C₂×Dic₆	1	1	yes	characteristic
170	C₄×D₅	C₂×D₁₂	1	1	yes	characteristic
171	C₄×D₅	D₃×Q₈	1	1	yes	characteristic
172	C₄×D₅	#239	1	1	yes	characteristic
173	#170	—	6	2	no	—
174	C₅×C₈	C₄⋊Dic₃	1	2	yes	complemented
175	#168	C₄⋊Dic₃	1	2	yes	complemented
176	C₂²×C₈	—	5	1	no	—
177	C₂²×C₈	—	15	1	no	—
178	C₂×C₄⋊C₄	—	5	2	no	—
179	C₂×C₄⋊C₄	—	15	2	no	—
180	C₂×C₄⋊C₄	—	30	2	no	—
181	C₂×C₄²	—	15	1	no	—
182	#107	—	30	2	no	—
183	#106	—	5	1	no	—
184	#106	—	5	2	no	—
185	#106	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
186	#97	—	30	2	no	—
187	#95	—	30	2	no	—
188	C₂×C₃×C₅	#389	1	1	yes	characteristic
189	C₂×C₃×C₅	#440	1	2	yes	—
190	C₃×D₅	#469	1	1	yes	characteristic
191	C₃×D₅	#470	1	1	yes	characteristic
192	C₃×D₅	#501	1	2	yes	—
193	C₂³×C₃	—	5	1	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	5	1	no	—
196	C₂×C₃×C₄	—	5	1	no	—
197	C₂×C₃×C₄	—	5	1	no	—
198	C₂×C₃×C₄	—	5	1	no	—
199	C₂×C₃×C₄	—	5	1	no	—
200	C₂×C₃×C₄	—	10	2	no	—
201	C₂×C₃×C₄	—	20	2	no	—
202	C₂×C₃×C₄	—	20	2	no	—
203	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
204	C₂×Dic₃	—	10	1	no	—
205	C₂×Dic₃	—	10	1	no	—
206	C₂×Dic₃	—	10	2	no	—
207	C₂×Dic₃	—	10	2	no	—
208	C₂×Dic₃	—	20	2	no	—
209	C₂×Dic₃	—	20	2	no	—
210	C₂×Dic₃	—	20	2	no	—
211	C₃×C₈	—	5	2	no	—
212	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	2	yes	complemented
213	C₃⋊C₈	—	10	2	no	—
214	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
215	C₂×D₅	#888	1	1	yes	characteristic
216	C₂×D₅	#895	1	1	yes	characteristic
217	C₂×D₅	#914	1	2	yes	—
218	C₂×D₅	#915	1	2	yes	—
219	AGL(1,𝔽₅)	—	12	1	no	—
220	AGL(1,𝔽₅)	—	12	1	no	—
221	AGL(1,𝔽₅)	—	24	2	no	—
222	C₄×C₅	—	4	2	no	—
223	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
224	C₄×C₅	Q₈×Dic₃	1	1	yes	characteristic
225	Dic₅	—	4	2	no	—
226	Dic₅	#890	1	1	yes	characteristic
227	Dic₅	#897	1	1	yes	characteristic
228	C₂²×C₄	—	5	1	no	—
229	C₂²×C₄	—	10	2	no	—
230	C₂²×C₄	—	15	1	no	—
231	C₂²×C₄	—	15	1	no	—
232	C₂²×C₄	—	30	2	no	—
233	C₂×C₈	—	5	1	no	—
234	C₂×C₈	—	5	2	no	—
235	C₂×C₈	—	5	2	no	—
236	C₂×C₈	—	15	2	no	—
237	C₂×C₈	—	30	1	no	—
238	C₂×C₈	—	30	1	no	—
239	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
240	C₄⋊C₄	—	5	2	no	—
241	C₄⋊C₄	—	10	2	no	—
242	C₄⋊C₄	—	30	2	no	—
243	C₄⋊C₄	—	30	2	no	—
244	C₄⋊C₄	—	60	4	no	—
245	C₄⋊C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
246	C₄²	—	30	1	no	—
247	C₄²	—	30	1	no	—
248	C₃×C₅	#1944	1	1	yes	characteristic, complemented, Hall
249	C₂²×C₃	—	5	1	no	—
250	C₂²×C₃	—	5	1	no	—
251	C₂²×C₃	—	5	2	no	—
252	C₂²×C₃	—	5	2	no	—
253	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
254	C₃×C₄	—	4	2	no	—
255	C₃×C₄	—	5	1	no	—
256	C₃×C₄	—	5	1	no	—
257	C₃×C₄	—	20	2	no	—
258	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
259	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
260	Dic₃	—	20	1	no	—
261	Dic₃	—	20	1	no	—
262	Dic₃	—	40	2	no	—
263	C₂×C₅	#4984	1	1	yes	characteristic
264	C₂×C₅	#5164	1	2	yes	—
265	D₅	#4856	1	2	yes	—
266	D₅	#4865	1	1	yes	characteristic
267	D₅	#4867	1	1	yes	characteristic
268	C₂³	—	5	1	no	—
269	C₂×C₄	—	2	2	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	—	5	1	no	—
275	C₂×C₄	—	10	2	no	—
276	C₂×C₄	—	20	2	no	—
277	C₂×C₄	—	20	2	no	—
278	C₂×C₄	—	30	1	no	—
279	C₂×C₄	—	30	1	no	—
280	C₂×C₄	—	30	2	no	—
281	C₂×C₄	—	30	2	no	—
282	C₂×C₄	—	60	2	no	—
283	C₂×C₄	—	60	2	no	—
284	C₂×C₄	—	60	2	no	—
285	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
286	C₈	—	5	2	no	—
287	C₈	—	30	2	no	—
288	C₈	C₄⋊ΓL(1,𝔽₁₆)	1	2	yes	complemented
289	C₂×C₃	—	5	1	no	—
290	C₂×C₃	—	5	1	no	—
291	C₂×C₃	—	5	2	no	—
292	C₂×C₃	#65712	1	1	yes	characteristic
293	C₂×C₃	#65735	1	2	yes	—
294	C₅	#70562	1	1	yes	characteristic, complemented, Sylow
295	C₂²	—	5	1	no	—
296	C₂²	—	5	1	no	—
297	C₂²	—	5	2	no	—
298	C₂²	—	5	2	no	—
299	C₂²	D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
300	C₄	—	4	2	no	—
301	C₄	—	5	1	no	—
302	C₄	—	5	1	no	—
303	C₄	—	20	2	no	—
304	C₄	—	60	1	no	—
305	C₄	—	60	1	no	—
306	C₄	—	120	2	no	—
307	C₄	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
308	C₄	Q₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
309	C₃	#10604052	1	1	yes	characteristic, complemented, Sylow
310	C₂	—	5	1	no	—
311	C₂	—	5	1	no	—
312	C₂	—	5	2	no	—
313	C₂	#11752371	1	1	yes	characteristic
314	C₂	#11752394	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	#1944
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
59. C₅×#106	Dic₃	semidirect
60. #3949	Dic₃	semidirect
61. #3940	Dic₃	semidirect
82. C₃×C₅×C₈	C₄⋊C₄	semidirect
83. C₃×#168	C₄⋊C₄	semidirect
92. C₃×#106	AGL(1,𝔽₅)	semidirect
174. C₅×C₈	C₄⋊Dic₃	semidirect
175. #168	C₄⋊Dic₃	semidirect
185. #106	ΓL(1,𝔽₁₆)	semidirect
212. C₃×C₈	C₄⋊AGL(1,𝔽₅)	semidirect
248. C₃×C₅	#1944	semidirect
288. C₈	C₄⋊ΓL(1,𝔽₁₆)	semidirect
294. C₅	#70562	semidirect
309. C₃	#10604052	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91223	2
C₂×#91221	2
#11752371	2
C₃×D₅×#106	2
#11746745	2
#10604052	3
#70562	5