Subgroups — #14296533

Order 1920.

448 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91224	C₂	1	1	yes	maximal, characteristic
2	C₂×#91221	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	#10604078	—	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	#91224	C₄	1	4	yes	complemented
15	#91221	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	#70660	—	5	1	no	maximal, Hall
20	C₂×#3995	—	3	1	no	—
21	C₂×#3992	—	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	2	yes	—
29	C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
30	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
31	#6758	—	2	2	no	—
32	#6758	C₂×C₄	1	4	yes	—
33	D₅⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
34	D₅⋊(C₃⋊C₈)	Q₈	1	2	yes	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
37	C₃×#685	—	2	2	no	—
38	C₃×C₄⋊Dic₅	C₂×C₄	1	2	yes	—
39	C₃×C₄×Dic₅	—	2	2	no	—
40	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
41	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
42	C₃×C₈×D₅	C₂×C₄	1	2	yes	—
43	C₃×#427	—	5	1	no	—
44	C₂×#819	—	5	1	no	—
45	C₂×#815	—	5	1	no	—
46	#4997	—	5	2	no	—
47	#4532	—	5	2	no	—
48	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
49	C₂×#700	—	3	2	no	—
50	C₂×D₅⋊C₈	—	3	1	no	—
51	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
52	#4038	C₂×D₃	1	2	yes	—
53	#4002	—	6	2	no	—
54	C₄×#170	—	6	2	no	—
55	#3995	—	3	4	no	—
56	#3992	—	3	4	no	—
57	C₅×#107	Dic₃	1	1	yes	characteristic, complemented
58	#3950	Dic₃	1	1	yes	characteristic, complemented
59	#3940	Dic₃	1	2	yes	complemented
60	#1773	—	15	1	no	Sylow
61	C₂²×C₃×D₅	#41	1	1	yes	characteristic
62	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
63	C₂×ΓL(1,𝔽₁₆)	—	4	1	no	—
64	C₂×ΓL(1,𝔽₁₆)	—	4	1	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×Dic₅	—	2	2	no	—
68	C₂×C₃×Dic₅	C₂×D₄	1	1	yes	characteristic
69	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
70	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
71	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
72	C₃×C₄×D₅	#41	1	1	yes	characteristic
73	C₃⋊#170	—	2	4	no	—
74	C₃⋊#170	C₄⋊C₄	1	4	yes	—
75	C₃×C₅×C₈	C₄²	1	2	yes	—
76	C₃×#168	C₄²	1	2	yes	—
77	C₂²×C₃×C₈	—	5	1	no	—
78	C₃×#117	—	5	2	no	—
79	C₂×C₄×Dic₃	—	5	1	no	—
80	C₂×#208	—	5	2	no	—
81	C₂²×C₃⋊C₈	—	5	1	no	—
82	C₃×#107	—	5	1	no	—
83	C₃×#107	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
84	C₃×#106	—	5	2	no	—
85	#819	—	5	4	no	—
86	#815	—	5	4	no	—
87	#803	—	10	2	no	—
88	C₄×C₃⋊C₈	—	10	2	no	—
89	C₂²×AGL(1,𝔽₅)	—	6	1	no	—
90	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
91	C₂×#170	—	3	2	no	—
92	C₂×#170	—	6	2	no	—
93	C₄×AGL(1,𝔽₅)	—	3	2	no	—
94	C₄×AGL(1,𝔽₅)	—	6	1	no	—
95	#700	—	3	4	no	—
96	#700	—	6	2	no	—
97	D₅⋊C₈	—	3	2	no	—
98	D₅⋊C₈	—	3	2	no	—
99	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
100	C₅×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
101	#685	—	2	2	no	—
102	C₄⋊Dic₅	C₂×Dic₃	1	2	yes	—
103	C₄×Dic₅	—	2	2	no	—
104	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
105	C₈×D₅	C₄×D₃	1	2	yes	—
106	C₈×D₅	C₄×D₃	1	2	yes	—
107	C₂×#108	—	15	1	no	—
108	#427	—	5	1	no	—
109	#405	—	15	2	no	—
110	C₂×#97	—	15	1	no	—
111	#339	—	15	2	no	—
112	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
113	C₂×C₃×D₅	—	2	1	no	—
114	C₂×C₃×D₅	—	2	1	no	—
115	C₂×C₃×D₅	#117	1	1	yes	characteristic
116	C₂×C₃×D₅	C₄×Q₈	1	1	yes	characteristic
117	ΓL(1,𝔽₁₆)	—	4	2	no	—
118	C₃×C₄×C₅	—	4	2	no	—
119	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
120	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
121	C₃×Dic₅	—	4	2	no	—
122	C₃×Dic₅	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₃×Dic₅	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₃×C₄	—	5	1	no	—
125	C₂²×Dic₃	—	10	1	no	—
126	C₂×C₃×C₈	—	5	1	no	—
127	C₂×C₃×C₈	—	5	2	no	—
128	C₂×C₃×C₈	—	5	2	no	—
129	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
130	C₃×C₄⋊C₄	—	5	2	no	—
131	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
132	C₃×C₂²⋊C₄	—	10	2	no	—
133	C₃×C₄²	—	10	2	no	—
134	C₄×Dic₃	—	5	2	no	—
135	C₄×Dic₃	—	10	1	no	—
136	#208	—	5	4	no	—
137	#208	—	10	2	no	—
138	C₂×C₃⋊C₈	—	5	2	no	—
139	C₂×C₃⋊C₈	—	5	2	no	—
140	C₂×C₃⋊C₈	—	5	2	no	—
141	C₂×C₃⋊C₈	—	10	2	no	—
142	C₂²×D₅	#237	1	1	yes	characteristic
143	C₂×AGL(1,𝔽₅)	—	6	2	no	—
144	C₂×AGL(1,𝔽₅)	—	12	1	no	—
145	C₂×AGL(1,𝔽₅)	—	12	1	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
148	C₂×Dic₅	—	2	2	no	—
149	C₂×Dic₅	D₃×D₄	1	1	yes	characteristic
150	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
151	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
152	C₄×D₅	D₃×Q₈	1	1	yes	characteristic
153	C₄×D₅	#239	1	1	yes	characteristic
154	#170	—	3	4	no	—
155	#170	—	6	4	no	—
156	C₅×C₈	C₄×Dic₃	1	2	yes	—
157	#168	C₄×Dic₃	1	2	yes	—
158	C₂×M₁₆	—	15	2	no	—
159	C₂²×C₈	—	5	1	no	—
160	C₂²×C₈	—	15	1	no	—
161	#117	—	5	2	no	—
162	C₂×C₄²	—	15	1	no	—
163	#108	—	15	4	no	—
164	#107	—	5	1	no	—
165	#107	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
166	#106	—	5	2	no	—
167	#97	—	15	4	no	—
168	#97	—	30	2	no	—
169	C₄×C₈	—	30	2	no	—
170	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₃×C₅	#444	1	1	yes	characteristic
172	C₂×C₃×C₅	#444	1	1	yes	characteristic
173	C₃×D₅	—	2	1	no	—
174	C₃×D₅	—	2	1	no	—
175	C₂³×C₃	—	5	1	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	5	1	no	—
178	C₂×C₃×C₄	—	5	1	no	—
179	C₂×C₃×C₄	—	5	1	no	—
180	C₂×C₃×C₄	—	5	1	no	—
181	C₂×C₃×C₄	—	5	1	no	—
182	C₂×C₃×C₄	—	10	2	no	—
183	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
184	C₂×Dic₃	—	10	2	no	—
185	C₂×Dic₃	—	20	1	no	—
186	C₂×Dic₃	—	20	1	no	—
187	C₃×C₈	—	5	2	no	—
188	C₃×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
189	C₃⋊C₈	—	5	4	no	—
190	C₃⋊C₈	—	10	4	no	—
191	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
192	C₂×D₅	—	2	1	no	—
193	C₂×D₅	—	2	1	no	—
194	C₂×D₅	#887	1	1	yes	characteristic
195	C₂×D₅	#892	1	1	yes	characteristic
196	AGL(1,𝔽₅)	—	12	2	no	—
197	C₄×C₅	—	4	2	no	—
198	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
199	C₄×C₅	Q₈×Dic₃	1	1	yes	characteristic
200	Dic₅	—	4	2	no	—
201	Dic₅	D₃×C₄⋊C₄	1	1	yes	characteristic
202	Dic₅	#893	1	1	yes	characteristic
203	C₂²×C₄	—	5	1	no	—
204	C₂²×C₄	—	30	1	no	—
205	M₁₆	—	15	4	no	—
206	M₁₆	—	30	2	no	—
207	C₂×C₈	—	5	1	no	—
208	C₂×C₈	—	5	2	no	—
209	C₂×C₈	—	5	2	no	—
210	C₂×C₈	—	15	2	no	—
211	C₂×C₈	—	15	2	no	—
212	C₂×C₈	—	15	2	no	—
213	C₂×C₈	—	30	2	no	—
214	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
215	C₄⋊C₄	—	5	2	no	—
216	C₄⋊C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
217	C₂²⋊C₄	—	10	2	no	—
218	C₄²	—	10	2	no	—
219	C₄²	—	15	2	no	—
220	C₄²	—	30	1	no	—
221	C₃×C₅	#1773	1	1	yes	characteristic, complemented, Hall
222	C₂²×C₃	—	5	1	no	—
223	C₂²×C₃	—	5	1	no	—
224	C₂²×C₃	—	10	1	no	—
225	C₂²×C₃	—	10	1	no	—
226	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
227	C₃×C₄	—	4	2	no	—
228	C₃×C₄	—	5	1	no	—
229	C₃×C₄	—	5	1	no	—
230	C₃×C₄	—	20	2	no	—
231	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
232	C₃×C₄	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
233	Dic₃	—	20	2	no	—
234	C₂×C₅	Dic₃×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₅	#5197	1	1	yes	characteristic
236	C₂×C₅	#5197	1	1	yes	characteristic
237	D₅	—	2	1	no	—
238	D₅	—	2	1	no	—
239	C₂³	—	5	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	5	1	no	—
242	C₂×C₄	—	5	1	no	—
243	C₂×C₄	—	5	1	no	—
244	C₂×C₄	—	5	1	no	—
245	C₂×C₄	—	5	1	no	—
246	C₂×C₄	—	10	2	no	—
247	C₂×C₄	—	30	2	no	—
248	C₂×C₄	—	60	1	no	—
249	C₂×C₄	—	60	1	no	—
250	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
251	C₈	—	5	2	no	—
252	C₈	—	15	4	no	—
253	C₈	—	30	4	no	—
254	C₈	C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
255	C₂×C₃	—	10	1	no	—
256	C₂×C₃	—	10	1	no	—
257	C₂×C₃	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
258	C₂×C₃	#65733	1	1	yes	characteristic
259	C₂×C₃	#65741	1	1	yes	characteristic
260	C₅	#70660	1	1	yes	characteristic, complemented, Sylow
261	C₂²	—	5	1	no	—
262	C₂²	—	5	1	no	—
263	C₂²	—	10	1	no	—
264	C₂²	—	10	1	no	—
265	C₂²	D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
266	C₄	—	4	2	no	—
267	C₄	—	5	1	no	—
268	C₄	—	5	1	no	—
269	C₄	—	20	2	no	—
270	C₄	—	60	2	no	—
271	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
272	C₄	Q₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
273	C₃	#10604078	1	1	yes	characteristic, complemented, Sylow
274	C₂	—	10	1	no	—
275	C₂	—	10	1	no	—
276	C₂	C₄⋊C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
277	C₂	#11752392	1	1	yes	characteristic
278	C₂	#11752400	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	#1773
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. #91224	C₄	semidirect
15. #91221	C₄	semidirect
16. C₃×C₅×#107	C₄	semidirect
17. C₃×#3950	C₄	semidirect
18. C₃×#3940	C₄	semidirect
57. C₅×#107	Dic₃	semidirect
58. #3950	Dic₃	semidirect
59. #3940	Dic₃	semidirect
83. C₃×#107	AGL(1,𝔽₅)	semidirect
165. #107	ΓL(1,𝔽₁₆)	semidirect
221. C₃×C₅	#1773	semidirect
260. C₅	#70660	semidirect
273. C₃	#10604078	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91224	2
C₂×#91221	2
#11752360	2
C₃×#65169	2
#11746746	2
#10604078	3
#70660	5