Subgroups — #14312679

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752377	C₂	1	1	yes	maximal, characteristic
2	C₃×D₅⋊M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×D₅×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#11746451	C₂	1	2	yes	maximal, complemented
5	ΓL(1,𝔽₁₆)⋊C₁₆	C₂	1	2	yes	maximal, complemented
6	#10604098	—	3	1	no	maximal, Hall
7	#91989	C₄	1	1	yes	characteristic
8	D₅⋊#208	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
10	#91221	C₄	1	2	yes	—
11	C₈×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
12	C₃×#3998	C₄	1	2	yes	complemented
13	C₃×#3990	C₄	1	2	yes	complemented
14	C₃×D₅⋊C₁₆	C₂²	1	2	yes	—
15	C₂×C₃⋊#672	C₄	1	1	yes	characteristic, complemented
16	C₂×C₅×C₃⋊C₁₆	C₄	1	1	yes	characteristic, complemented
17	D₅×C₃⋊C₁₆	C₄	1	1	yes	characteristic, complemented
18	D₅×C₃⋊C₁₆	C₄	1	1	yes	characteristic, complemented
19	D₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
20	D₅×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	#71419	—	5	1	no	maximal, Hall
22	#65718	—	3	1	no	—
23	D₅⋊M₃₂	D₃	1	1	yes	characteristic
24	C₂×D₅×C₁₆	—	3	1	no	—
25	#64844	—	3	2	no	—
26	C₁₆×AGL(1,𝔽₅)	—	3	2	no	—
27	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
28	#6763	C₈	1	2	yes	—
29	#6762	C₈	1	2	yes	—
30	C₄⋊ΓL(1,𝔽₁₆)	C₈	1	2	yes	—
31	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
32	#6758	C₈	1	2	yes	—
33	D₅⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
36	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
40	C₃×#674	C₂×C₄	1	4	yes	—
41	C₃⋊#672	C₂×C₄	1	1	yes	characteristic
42	C₃⋊#672	C₂×C₄	1	1	yes	characteristic
43	C₅×C₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₅×C₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×M₃₂	—	5	1	no	—
46	#5101	—	5	1	no	—
47	C₂²×C₃⋊C₁₆	—	5	1	no	—
48	#4502	—	5	2	no	—
49	Dic₃×C₁₆	—	5	2	no	—
50	#4130	—	3	1	no	—
51	D₅⋊M₁₆	—	3	1	no	—
52	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
53	#3998	Dic₃	1	2	yes	complemented
54	#3992	—	3	2	no	—
55	C₈×AGL(1,𝔽₅)	—	3	2	no	—
56	#3990	Dic₃	1	2	yes	complemented
57	D₅⋊C₁₆	C₂×D₃	1	2	yes	—
58	C₂×C₅×C₁₆	—	3	1	no	—
59	C₂×#672	—	3	1	no	—
60	D₅×C₁₆	—	3	1	no	—
61	D₅×C₁₆	—	3	1	no	—
62	D₅×C₁₆	—	3	1	no	—
63	D₅×C₁₆	—	3	1	no	—
64	#2108	—	15	1	no	Sylow
65	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
66	C₂×ΓL(1,𝔽₁₆)	C₂×C₈	1	4	yes	—
67	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
68	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
69	C₃×C₄×D₅	C₄²	1	1	yes	characteristic
70	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
71	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
72	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
73	C₃⋊#170	C₂×C₈	1	4	yes	—
74	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
75	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₃×#168	C₄²	1	1	yes	characteristic
77	C₃×#168	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₃×C₈	—	5	1	no	—
79	#926	—	5	1	no	—
80	C₂×#208	—	5	1	no	—
81	C₃×M₃₂	—	5	2	no	—
82	C₃×M₃₂	—	5	2	no	—
83	C₂×C₃×C₁₆	—	5	2	no	—
84	#815	—	5	2	no	—
85	C₈×Dic₃	—	5	2	no	—
86	C₂×C₃⋊C₁₆	—	5	1	no	—
87	C₂×C₃⋊C₁₆	—	5	1	no	—
88	C₂×C₃⋊C₁₆	—	5	1	no	—
89	C₂×C₃⋊C₁₆	—	5	1	no	—
90	C₂×C₃⋊C₁₆	—	5	1	no	—
91	C₂×C₃⋊C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
92	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
93	C₂²⋊AGL(1,𝔽₅)	—	3	2	no	—
94	#704	—	3	2	no	—
95	C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
96	C₄×AGL(1,𝔽₅)	—	3	2	no	—
97	#700	—	3	2	no	—
98	D₅⋊C₈	—	3	2	no	—
99	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
100	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
101	C₈×D₅	C₄×D₃	1	1	yes	characteristic
102	C₈×D₅	C₄×D₃	1	1	yes	characteristic
103	C₈×D₅	C₂×Dic₃	1	1	yes	characteristic
104	C₈×D₅	C₂²×D₃	1	1	yes	characteristic
105	#674	C₂×Dic₃	1	4	yes	—
106	C₅×C₁₆	—	3	1	no	—
107	C₅×C₁₆	—	3	1	no	—
108	#672	—	3	1	no	—
109	#672	—	3	1	no	—
110	C₂×M₃₂	—	5	1	no	—
111	C₂²×C₁₆	—	15	1	no	—
112	#405	—	15	1	no	—
113	#346	—	15	2	no	—
114	C₄×C₁₆	—	15	2	no	—
115	C₂²×C₃×C₅	C₄×C₈	1	1	yes	characteristic
116	C₂×C₃×D₅	—	2	2	no	—
117	C₂×C₃×D₅	C₄×C₈	1	1	yes	characteristic
118	C₂×C₃×D₅	C₂²×C₈	1	1	yes	characteristic
119	ΓL(1,𝔽₁₆)	—	2	4	no	—
120	C₃×C₄×C₅	C₄×C₈	1	1	yes	characteristic
121	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
122	C₃×Dic₅	C₄×C₈	1	1	yes	characteristic
123	C₃×Dic₅	C₂²×C₈	1	1	yes	characteristic
124	C₂²×C₃×C₄	—	5	1	no	—
125	C₂×C₃×C₈	—	5	1	no	—
126	C₂×C₃×C₈	—	5	1	no	—
127	C₂×C₃×C₈	—	5	1	no	—
128	C₂×C₃×C₈	—	5	1	no	—
129	C₂×C₃×C₈	—	5	1	no	—
130	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
131	#217	—	5	2	no	—
132	C₄⋊Dic₃	—	5	2	no	—
133	C₄×Dic₃	—	5	2	no	—
134	#208	—	5	2	no	—
135	#208	—	5	2	no	—
136	C₂×C₃⋊C₈	—	5	2	no	—
137	C₃×C₁₆	—	5	4	no	—
138	C₃⋊C₁₆	—	5	1	no	—
139	C₃⋊C₁₆	—	5	1	no	—
140	C₃⋊C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
141	C₃⋊C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₂²×D₅	D₃×C₈	1	1	yes	characteristic
143	C₂×AGL(1,𝔽₅)	—	3	4	no	—
144	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
145	C₂×Dic₅	D₃×C₈	1	1	yes	characteristic
146	C₄×D₅	D₃×C₈	1	1	yes	characteristic
147	C₄×D₅	D₃×C₈	1	1	yes	characteristic
148	C₄×D₅	C₄×Dic₃	1	1	yes	characteristic
149	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
150	#170	—	3	4	no	—
151	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
152	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
153	#168	C₄×Dic₃	1	1	yes	characteristic
154	#168	C₂²×Dic₃	1	1	yes	characteristic
155	C₂×M₁₆	—	15	1	no	—
156	C₂²×C₈	—	5	1	no	—
157	#117	—	15	1	no	—
158	M₃₂	—	5	2	no	—
159	M₃₂	—	5	2	no	—
160	C₂×C₁₆	—	3	1	no	—
161	C₂×C₁₆	—	5	2	no	—
162	C₂×C₁₆	—	15	1	no	—
163	C₂×C₁₆	—	15	1	no	—
164	C₂×C₁₆	—	15	1	no	—
165	C₂×C₁₆	—	15	1	no	—
166	C₂×C₁₆	—	15	1	no	—
167	#97	—	15	2	no	—
168	C₄×C₈	—	15	2	no	—
169	C₂×C₃×C₅	—	2	1	no	—
170	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
171	C₃×D₅	—	2	1	no	—
172	C₃×D₅	#504	1	2	yes	—
173	C₂³×C₃	—	5	1	no	—
174	C₂×C₃×C₄	—	5	1	no	—
175	C₂×C₃×C₄	—	5	1	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	5	1	no	—
178	C₂×C₃×C₄	—	5	1	no	—
179	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
180	C₂×Dic₃	—	5	4	no	—
181	C₃×C₈	—	5	1	no	—
182	C₃×C₈	—	5	1	no	—
183	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
184	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
185	C₃⋊C₈	—	5	4	no	—
186	C₂²×C₅	C₈×Dic₃	1	1	yes	characteristic
187	C₂×D₅	—	2	2	no	—
188	C₂×D₅	C₈×Dic₃	1	1	yes	characteristic
189	C₂×D₅	C₂×D₃×C₈	1	1	yes	characteristic
190	AGL(1,𝔽₅)	—	6	4	no	—
191	C₄×C₅	C₈×Dic₃	1	1	yes	characteristic
192	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
193	Dic₅	C₈×Dic₃	1	1	yes	characteristic
194	Dic₅	C₂×D₃×C₈	1	1	yes	characteristic
195	C₂²×C₄	—	5	1	no	—
196	M₁₆	—	15	2	no	—
197	M₁₆	—	15	2	no	—
198	C₂×C₈	—	5	1	no	—
199	C₂×C₈	—	5	1	no	—
200	C₂×C₈	—	5	1	no	—
201	C₂×C₈	—	5	1	no	—
202	C₂×C₈	—	5	1	no	—
203	C₂×C₈	—	15	2	no	—
204	C₂×C₈	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₄⋊C₄	—	15	2	no	—
206	C₂²⋊C₄	—	15	2	no	—
207	C₄²	—	15	2	no	—
208	C₁₆	—	3	1	no	—
209	C₁₆	—	3	1	no	—
210	C₁₆	—	5	4	no	—
211	C₁₆	—	15	1	no	—
212	C₁₆	—	15	1	no	—
213	C₃×C₅	#2108	1	1	yes	characteristic, complemented, Hall
214	C₂²×C₃	—	5	1	no	—
215	C₂²×C₃	—	5	1	no	—
216	C₂²×C₃	—	10	2	no	—
217	C₂²×C₃	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
218	C₃×C₄	—	5	1	no	—
219	C₃×C₄	—	5	1	no	—
220	C₃×C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
221	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
222	Dic₃	—	10	4	no	—
223	C₂×C₅	—	2	1	no	—
224	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
225	D₅	—	2	1	no	—
226	D₅	#4907	1	2	yes	—
227	C₂³	—	5	1	no	—
228	C₂×C₄	—	5	1	no	—
229	C₂×C₄	—	5	1	no	—
230	C₂×C₄	—	5	1	no	—
231	C₂×C₄	—	5	1	no	—
232	C₂×C₄	—	5	1	no	—
233	C₂×C₄	—	15	4	no	—
234	C₂×C₄	Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
235	C₈	—	5	1	no	—
236	C₈	—	5	1	no	—
237	C₈	—	15	4	no	—
238	C₈	Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
239	C₈	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	5	2	no	—
242	C₂×C₃	—	10	1	no	—
243	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
244	C₅	#71419	1	1	yes	characteristic, complemented, Sylow
245	C₂²	—	5	1	no	—
246	C₂²	—	5	1	no	—
247	C₂²	—	10	2	no	—
248	C₂²	ΓL(1,𝔽₁₆)⋊C₈	1	1	yes	characteristic
249	C₄	—	5	1	no	—
250	C₄	—	5	1	no	—
251	C₄	—	30	4	no	—
252	C₄	ΓL(1,𝔽₁₆)⋊C₈	1	1	yes	characteristic
253	C₄	C₂×Dic₃×AGL(1,𝔽₅)	1	1	yes	characteristic
254	C₃	#10604098	1	1	yes	characteristic, complemented, Sylow
255	C₂	—	2	1	no	—
256	C₂	—	5	2	no	—
257	C₂	—	10	1	no	—
258	C₂	C₂×ΓL(1,𝔽₁₆)⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂×D₃×AGL(1,𝔽₅)
commutator, socle	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₂×C₃×C₅×C₈	C₂×C₄
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₃×C₈×D₅	C₂³
hypercenter	C₂×C₈	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#2108
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
4. #11746451	C₂	semidirect
5. ΓL(1,𝔽₁₆)⋊C₁₆	C₂	semidirect
12. C₃×#3998	C₄	semidirect
13. C₃×#3990	C₄	semidirect
15. C₂×C₃⋊#672	C₄	semidirect
16. C₂×C₅×C₃⋊C₁₆	C₄	semidirect
17. D₅×C₃⋊C₁₆	C₄	semidirect
18. D₅×C₃⋊C₁₆	C₄	semidirect
53. #3998	Dic₃	semidirect
56. #3990	Dic₃	semidirect
91. C₂×C₃⋊C₁₆	AGL(1,𝔽₅)	semidirect
213. C₃×C₅	#2108	semidirect
244. C₅	#71419	semidirect
254. C₃	#10604098	semidirect

Maximal subgroups

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

subgroup	index
#11752377	2
C₃×D₅⋊M₃₂	2
C₂×D₅×C₃⋊C₁₆	2
#11746451	2
ΓL(1,𝔽₁₆)⋊C₁₆	2
#10604098	3
#71419	5