Subgroups — #14296535

Order 1920.

448 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	#11752387	C₂	1	2	yes	maximal
2	#11752380	C₂	1	1	yes	maximal, characteristic
3	C₂×#91221	C₂	1	1	yes	maximal, characteristic
4	C₃×D₅×#108	C₂	1	1	yes	maximal, characteristic
5	#11746771	C₂	1	2	yes	maximal
6	#10604090	—	3	1	no	maximal, Hall
7	#91989	C₂²	1	2	yes	—
8	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
9	C₂×D₅⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₃×D₅×M₁₆	C₂²	1	2	yes	—
11	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
12	#91223	C₄	1	2	yes	complemented
13	#91222	C₄	1	2	yes	complemented
14	#91221	—	2	2	no	—
15	#91221	C₄	1	2	yes	complemented
16	#91221	C₄	1	2	yes	complemented
17	C₈×ΓL(1,𝔽₁₆)	—	2	2	no	—
18	C₃×C₅×#108	C₄	1	1	yes	characteristic, complemented
19	C₃×#3954	C₄	1	2	yes	complemented
20	C₃×#3951	C₄	1	1	yes	characteristic, complemented
21	#70703	—	5	1	no	maximal, Hall
22	#65728	—	3	2	no	—
23	#65721	—	3	1	no	—
24	C₂×#3992	—	3	1	no	—
25	D₅×#108	D₃	1	1	yes	characteristic
26	#64900	—	3	2	no	—
27	C₂²×ΓL(1,𝔽₁₆)	—	2	1	no	—
28	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
29	#6763	—	2	2	no	—
30	C₂×C₃⋊#170	Q₈	1	2	yes	—
31	C₄⋊ΓL(1,𝔽₁₆)	C₂×C₄	1	4	yes	—
32	C₄×ΓL(1,𝔽₁₆)	—	2	2	no	—
33	C₄×ΓL(1,𝔽₁₆)	—	2	1	no	—
34	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
35	D₅⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
36	D₅⋊(C₃⋊C₈)	D₄	1	2	yes	—
37	C₃×C₅×M₁₆	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
39	C₃×#681	C₂×C₄	1	2	yes	—
40	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
41	C₃×#676	—	2	2	no	—
42	C₃×C₈×D₅	—	2	2	no	—
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	C₂×C₃×#108	—	5	1	no	—
48	#5122	—	5	2	no	—
49	#5106	—	5	1	no	—
50	C₂×#815	—	5	1	no	—
51	#4555	—	5	2	no	—
52	#4130	—	3	2	no	—
53	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
54	C₂×D₅⋊C₈	—	3	1	no	—
55	D₅×M₁₆	C₂×D₃	1	2	yes	—
56	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
57	#3994	—	3	2	no	—
58	#3993	—	3	2	no	—
59	#3992	—	3	2	no	—
60	#3992	—	3	2	no	—
61	#3992	—	6	2	no	—
62	C₈×AGL(1,𝔽₅)	—	6	2	no	—
63	C₅×#108	Dic₃	1	1	yes	characteristic, complemented
64	#3954	Dic₃	1	2	yes	complemented
65	#3951	Dic₃	1	1	yes	characteristic, complemented
66	#1773	—	15	1	no	Sylow
67	C₂²×C₃×D₅	#41	1	1	yes	characteristic
68	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
69	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
70	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
71	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
73	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
74	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
75	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
76	C₃×C₄×D₅	#41	1	1	yes	characteristic
77	C₃⋊#170	C₄⋊C₄	1	4	yes	—
78	C₃×C₅×C₈	—	2	2	no	—
79	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
80	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
81	C₃×#168	—	2	2	no	—
82	C₃×#168	C₄²	1	1	yes	characteristic
83	C₃×#168	C₄²	1	1	yes	characteristic
84	C₂×C₃×M₁₆	—	5	2	no	—
85	C₂²×C₃×C₈	—	5	1	no	—
86	#926	—	5	2	no	—
87	C₂×C₄×Dic₃	—	5	1	no	—
88	C₂²×C₃⋊C₈	—	5	1	no	—
89	C₃×#108	—	5	2	no	—
90	C₃×#108	—	5	1	no	—
91	C₃×#108	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
92	#818	—	5	2	no	—
93	#817	—	5	2	no	—
94	#815	—	5	2	no	—
95	#815	—	5	2	no	—
96	#815	—	10	2	no	—
97	C₈×Dic₃	—	10	2	no	—
98	C₂²×AGL(1,𝔽₅)	—	6	1	no	—
99	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
100	C₂²⋊AGL(1,𝔽₅)	—	6	2	no	—
101	C₂×#170	—	3	2	no	—
102	C₄⋊AGL(1,𝔽₅)	—	3	4	no	—
103	C₄×AGL(1,𝔽₅)	—	3	2	no	—
104	C₄×AGL(1,𝔽₅)	—	6	2	no	—
105	C₄×AGL(1,𝔽₅)	—	6	1	no	—
106	D₅⋊C₈	—	3	2	no	—
107	D₅⋊C₈	—	3	2	no	—
108	C₅×M₁₆	C₂×Dic₃	1	2	yes	—
109	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
110	#681	C₂×Dic₃	1	2	yes	—
111	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
112	#676	—	2	2	no	—
113	C₈×D₅	—	2	2	no	—
114	C₈×D₅	C₄×D₃	1	1	yes	characteristic
115	C₈×D₅	C₄×D₃	1	1	yes	characteristic
116	C₈×D₅	C₄×D₃	1	1	yes	characteristic
117	C₈×D₅	C₄×D₃	1	1	yes	characteristic
118	C₂×#108	—	5	1	no	—
119	#427	—	15	1	no	—
120	#405	—	15	2	no	—
121	C₂×#97	—	15	1	no	—
122	#339	—	15	2	no	—
123	C₂²×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
124	C₂×C₃×D₅	—	2	2	no	—
125	C₂×C₃×D₅	#117	1	1	yes	characteristic
126	C₂×C₃×D₅	C₄×D₄	1	1	yes	characteristic
127	ΓL(1,𝔽₁₆)	—	4	2	no	—
128	ΓL(1,𝔽₁₆)	—	4	4	no	—
129	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
130	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
131	C₃×Dic₅	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₃×Dic₅	C₄×Q₈	1	1	yes	characteristic
133	C₂²×C₃×C₄	—	5	1	no	—
134	C₂²×Dic₃	—	10	1	no	—
135	C₃×M₁₆	—	5	2	no	—
136	C₃×M₁₆	—	10	2	no	—
137	C₃×M₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
138	C₂×C₃×C₈	—	5	1	no	—
139	C₂×C₃×C₈	—	5	1	no	—
140	C₂×C₃×C₈	—	5	1	no	—
141	C₂×C₃×C₈	—	5	1	no	—
142	C₂×C₃×C₈	—	5	1	no	—
143	C₂×C₃×C₈	—	10	2	no	—
144	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
145	#217	—	10	2	no	—
146	C₄⋊Dic₃	—	5	4	no	—
147	C₄×Dic₃	—	5	2	no	—
148	C₄×Dic₃	—	10	2	no	—
149	C₄×Dic₃	—	10	1	no	—
150	C₂×C₃⋊C₈	—	5	2	no	—
151	C₂×C₃⋊C₈	—	5	2	no	—
152	C₂×C₃⋊C₈	—	5	2	no	—
153	C₂²×D₅	#239	1	1	yes	characteristic
154	C₂×AGL(1,𝔽₅)	—	6	2	no	—
155	C₂×AGL(1,𝔽₅)	—	6	4	no	—
156	C₂×AGL(1,𝔽₅)	—	12	2	no	—
157	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
158	C₂×Dic₅	D₃×Q₈	1	1	yes	characteristic
159	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
160	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
161	C₄×D₅	D₃×D₄	1	1	yes	characteristic
162	C₄×D₅	#237	1	1	yes	characteristic
163	#170	—	3	4	no	—
164	C₅×C₈	—	2	2	no	—
165	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
166	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
167	#168	—	2	2	no	—
168	#168	C₄×Dic₃	1	1	yes	characteristic
169	#168	C₄×Dic₃	1	1	yes	characteristic
170	C₂×M₁₆	—	5	2	no	—
171	C₂²×C₈	—	5	1	no	—
172	C₂²×C₈	—	15	1	no	—
173	#117	—	15	2	no	—
174	C₂×C₄²	—	15	1	no	—
175	#108	—	5	2	no	—
176	#108	—	5	1	no	—
177	#108	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
178	#107	—	15	2	no	—
179	#106	—	15	2	no	—
180	#97	—	15	2	no	—
181	#97	—	15	2	no	—
182	#97	—	30	2	no	—
183	C₄×C₈	—	30	2	no	—
184	C₂×C₃×C₅	—	2	1	no	—
185	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
186	C₃×D₅	—	2	1	no	—
187	C₃×D₅	#444	1	2	yes	—
188	C₂³×C₃	—	5	1	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	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
195	C₂×Dic₃	—	10	2	no	—
196	C₂×Dic₃	—	10	4	no	—
197	C₂×Dic₃	—	20	2	no	—
198	C₃×C₈	—	2	2	no	—
199	C₃×C₈	—	5	1	no	—
200	C₃×C₈	—	5	1	no	—
201	C₃×C₈	—	10	2	no	—
202	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
204	C₃⋊C₈	—	5	4	no	—
205	C₂²×C₅	Q₈×Dic₃	1	1	yes	characteristic
206	C₂×D₅	—	2	2	no	—
207	C₂×D₅	#892	1	1	yes	characteristic
208	C₂×D₅	#893	1	1	yes	characteristic
209	AGL(1,𝔽₅)	—	12	2	no	—
210	AGL(1,𝔽₅)	—	12	4	no	—
211	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
212	C₄×C₅	D₄×Dic₃	1	1	yes	characteristic
213	Dic₅	#887	1	1	yes	characteristic
214	Dic₅	D₃×C₄⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	—	5	1	no	—
216	C₂²×C₄	—	30	1	no	—
217	M₁₆	—	5	2	no	—
218	M₁₆	—	10	2	no	—
219	M₁₆	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
220	C₂×C₈	—	5	1	no	—
221	C₂×C₈	—	5	1	no	—
222	C₂×C₈	—	5	1	no	—
223	C₂×C₈	—	5	1	no	—
224	C₂×C₈	—	5	1	no	—
225	C₂×C₈	—	10	2	no	—
226	C₂×C₈	—	15	2	no	—
227	C₂×C₈	—	15	2	no	—
228	C₂×C₈	—	15	2	no	—
229	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
230	C₄⋊C₄	—	15	4	no	—
231	C₂²⋊C₄	—	30	2	no	—
232	C₄²	—	15	2	no	—
233	C₄²	—	30	2	no	—
234	C₄²	—	30	1	no	—
235	C₃×C₅	#1773	1	1	yes	characteristic, complemented, Hall
236	C₂²×C₃	—	5	1	no	—
237	C₂²×C₃	—	5	1	no	—
238	C₂²×C₃	—	10	2	no	—
239	C₂²×C₃	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
240	C₃×C₄	—	5	1	no	—
241	C₃×C₄	—	5	1	no	—
242	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
243	C₃×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
244	Dic₃	—	20	4	no	—
245	Dic₃	—	20	2	no	—
246	C₂×C₅	—	2	1	no	—
247	C₂×C₅	Dic₃×C₄⋊C₄	1	1	yes	characteristic
248	D₅	—	2	1	no	—
249	D₅	#4894	1	2	yes	—
250	C₂³	—	5	1	no	—
251	C₂×C₄	—	5	1	no	—
252	C₂×C₄	—	5	1	no	—
253	C₂×C₄	—	5	1	no	—
254	C₂×C₄	—	5	1	no	—
255	C₂×C₄	—	5	1	no	—
256	C₂×C₄	—	30	2	no	—
257	C₂×C₄	—	30	4	no	—
258	C₂×C₄	—	60	2	no	—
259	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	5	1	no	—
262	C₈	—	5	1	no	—
263	C₈	—	10	2	no	—
264	C₈	—	15	4	no	—
265	C₈	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
266	C₈	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
267	C₂×C₃	—	2	1	no	—
268	C₂×C₃	—	5	2	no	—
269	C₂×C₃	—	10	1	no	—
270	C₂×C₃	C₄⋊C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
271	C₅	#70703	1	1	yes	characteristic, complemented, Sylow
272	C₂²	—	5	1	no	—
273	C₂²	—	5	1	no	—
274	C₂²	—	10	2	no	—
275	C₂²	Q₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
276	C₄	—	5	1	no	—
277	C₄	—	5	1	no	—
278	C₄	—	60	4	no	—
279	C₄	—	60	2	no	—
280	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
281	C₄	D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
282	C₃	#10604090	1	1	yes	characteristic, complemented, Sylow
283	C₂	—	2	1	no	—
284	C₂	—	5	2	no	—
285	C₂	—	10	1	no	—
286	C₂	C₄⋊C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×ΓL(1,𝔽₁₆)
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×#108	C₄
socle	C₂×C₃×C₅	C₄×C₄⋊C₄
prime radical	C₂²×C₃×D₅	#41
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	#108	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1773
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
12. #91223	C₄	semidirect
13. #91222	C₄	semidirect
15. #91221	C₄	semidirect
16. #91221	C₄	semidirect
18. C₃×C₅×#108	C₄	semidirect
19. C₃×#3954	C₄	semidirect
20. C₃×#3951	C₄	semidirect
63. C₅×#108	Dic₃	semidirect
64. #3954	Dic₃	semidirect
65. #3951	Dic₃	semidirect
91. C₃×#108	AGL(1,𝔽₅)	semidirect
177. #108	ΓL(1,𝔽₁₆)	semidirect
235. C₃×C₅	#1773	semidirect
271. C₅	#70703	semidirect
282. C₃	#10604090	semidirect

Maximal subgroups

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

subgroup	index
#11752387	2
#11752380	2
C₂×#91221	2
C₃×D₅×#108	2
#11746771	2
#10604090	3
#70703	5