Subgroups — #97×ΓL(1,𝔽₁₆) #14294210

Order 1920.

720 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	C₂×#91221	C₂	1	2	yes	maximal
2	C₂×C₈×ΓL(1,𝔽₁₆)	C₂	1	2	yes	maximal
3	C₄²×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
4	#11752335	C₂	1	1	yes	maximal, characteristic
5	C₃×D₅×#97	C₂	1	1	yes	maximal, characteristic
6	AGL(1,𝔽₅)×#97	—	3	1	no	maximal, Hall
7	C₂×C₄×ΓL(1,𝔽₁₆)	C₄	1	4	yes	—
8	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
9	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
10	C₂×D₅⋊(C₃⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
12	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
13	#91231	C₄	1	4	yes	complemented
14	#91221	C₄	1	8	yes	complemented
15	C₈×ΓL(1,𝔽₁₆)	C₄	1	8	yes	complemented
16	C₃×C₅×#97	C₄	1	1	yes	characteristic, complemented
17	C₃×#3947	C₄	1	2	yes	complemented
18	C₃×#3935	C₄	1	1	yes	characteristic, complemented
19	Dic₃×#97	—	5	1	no	maximal, Hall
20	C₂×#3992	—	3	2	no	—
21	C₂×C₈×AGL(1,𝔽₅)	—	3	2	no	—
22	C₄²×AGL(1,𝔽₅)	—	3	1	no	—
23	#65676	—	3	1	no	—
24	D₅×#97	D₃	1	1	yes	characteristic
25	C₂²×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
26	C₂²×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
29	C₂×C₃⋊#170	C₂×C₄	1	4	yes	—
30	C₄×ΓL(1,𝔽₁₆)	—	2	8	no	—
31	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	4	yes	—
32	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	4	yes	—
33	D₅⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
34	D₅⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
38	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×#168	C₂×C₄	1	2	yes	—
40	C₃×C₈×D₅	C₂×C₄	1	4	yes	—
41	C₃×C₈×D₅	C₂×C₄	1	4	yes	—
42	C₂×C₃×#97	—	5	1	no	—
43	C₂×#815	—	5	2	no	—
44	C₂×C₈×Dic₃	—	5	2	no	—
45	C₄²×Dic₃	—	5	1	no	—
46	C₂×#803	—	5	1	no	—
47	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
48	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
49	C₂×C₄×AGL(1,𝔽₅)	—	3	4	no	—
50	C₂×D₅⋊C₈	—	3	2	no	—
51	C₂×C₈×D₅	C₂×D₃	1	2	yes	—
52	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
53	#4002	—	3	4	no	—
54	#3992	—	3	8	no	—
55	C₈×AGL(1,𝔽₅)	—	3	8	no	—
56	C₅×#97	Dic₃	1	1	yes	characteristic, complemented
57	#3947	Dic₃	1	2	yes	complemented
58	#3935	Dic₃	1	1	yes	characteristic, complemented
59	C₄×#97	—	15	1	no	Sylow
60	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
61	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
62	C₂×ΓL(1,𝔽₁₆)	C₄²	1	4	yes	—
63	C₂×ΓL(1,𝔽₁₆)	C₄²	1	4	yes	—
64	C₂×ΓL(1,𝔽₁₆)	M₁₆	1	8	yes	—
65	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×Dic₅	C₄²	1	2	yes	—
68	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
69	C₃×C₄×D₅	—	2	4	no	—
70	C₃×C₄×D₅	C₂²×C₄	1	2	yes	—
71	C₃×C₄×D₅	C₂²×C₄	1	2	yes	—
72	C₃⋊#170	C₄²	1	8	yes	complemented
73	C₃×C₅×C₈	C₄²	1	4	yes	complemented
74	C₃×#168	C₄²	1	4	yes	complemented
75	C₂²×C₃×C₈	—	5	2	no	—
76	C₂×C₃×C₄²	—	5	1	no	—
77	C₂×C₄×Dic₃	—	5	1	no	—
78	C₂×C₄×Dic₃	—	5	1	no	—
79	C₂×C₄×Dic₃	—	5	4	no	—
80	C₂²×C₃⋊C₈	—	5	2	no	—
81	C₃×#97	—	5	2	no	—
82	C₃×#97	—	5	1	no	—
83	C₃×#97	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
84	#815	—	5	8	no	—
85	C₈×Dic₃	—	5	8	no	—
86	#803	—	5	4	no	—
87	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
88	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
89	C₂×C₄×D₅	C₄×D₃	1	2	yes	—
90	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
91	C₂×#170	—	3	4	no	—
92	C₄×AGL(1,𝔽₅)	—	3	4	no	—
93	C₄×AGL(1,𝔽₅)	—	3	4	no	—
94	C₄×AGL(1,𝔽₅)	—	6	8	no	—
95	D₅⋊C₈	—	3	4	no	—
96	D₅⋊C₈	—	3	4	no	—
97	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
98	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
99	C₄×Dic₅	C₄×D₃	1	2	yes	—
100	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
101	C₂×#168	C₂×Dic₃	1	2	yes	—
102	C₈×D₅	C₄×D₃	1	4	yes	—
103	C₈×D₅	C₄×D₃	1	4	yes	—
104	C₂×#97	—	5	1	no	—
105	C₂×#97	—	15	1	no	—
106	C₂×#97	—	15	2	no	—
107	C₂×C₄×C₈	—	15	2	no	—
108	C₄³	—	15	1	no	—
109	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₃×D₅	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₃×D₅	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₃×D₅	C₂×M₁₆	1	4	yes	—
113	ΓL(1,𝔽₁₆)	—	2	4	no	—
114	ΓL(1,𝔽₁₆)	#97	1	8	yes	direct product factor
115	C₃×C₄×C₅	—	2	2	no	—
116	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
117	C₃×Dic₅	—	2	2	no	—
118	C₃×Dic₅	C₂×C₄²	1	2	yes	—
119	C₂²×C₃×C₄	—	5	1	no	—
120	C₂²×C₃×C₄	—	5	2	no	—
121	C₂²×Dic₃	—	5	2	no	—
122	C₂²×Dic₃	—	5	2	no	—
123	C₂×C₃×C₈	—	5	2	no	—
124	C₂×C₃×C₈	—	5	4	no	—
125	C₂×C₃×C₈	—	5	4	no	—
126	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
127	C₃×C₄²	—	5	2	no	—
128	C₃×C₄²	—	5	1	no	—
129	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
130	C₄×Dic₃	—	5	4	no	—
131	C₄×Dic₃	—	5	4	no	—
132	C₄×Dic₃	—	10	8	no	—
133	C₂×C₃⋊C₈	—	5	4	no	—
134	C₂×C₃⋊C₈	—	5	4	no	—
135	C₂×C₃⋊C₈	—	5	4	no	—
136	C₂²×D₅	C₂×C₄×D₃	1	1	yes	characteristic
137	C₂×AGL(1,𝔽₅)	—	3	4	no	—
138	C₂×AGL(1,𝔽₅)	—	3	8	no	—
139	C₂×AGL(1,𝔽₅)	—	3	4	no	—
140	C₂×AGL(1,𝔽₅)	—	6	4	no	—
141	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
142	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
143	C₂×Dic₅	C₄×Dic₃	1	2	yes	—
144	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
145	C₄×D₅	—	2	4	no	—
146	C₄×D₅	C₂×C₄×D₃	1	2	yes	—
147	C₄×D₅	C₂×C₄×D₃	1	2	yes	—
148	#170	—	3	8	no	—
149	C₅×C₈	C₄×Dic₃	1	4	yes	complemented
150	#168	C₄×Dic₃	1	4	yes	complemented
151	C₂²×C₈	—	5	2	no	—
152	C₂²×C₈	—	15	2	no	—
153	C₂×C₄²	—	5	1	no	—
154	C₂×C₄²	—	15	1	no	—
155	C₂×C₄²	—	15	1	no	—
156	C₂×C₄²	—	15	4	no	—
157	#97	—	5	2	no	—
158	#97	—	5	1	no	—
159	#97	—	15	4	no	—
160	#97	—	15	8	no	—
161	#97	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor
162	C₄×C₈	—	15	8	no	—
163	C₂×C₃×C₅	C₄³	1	1	yes	characteristic
164	C₂×C₃×C₅	C₄×M₁₆	1	2	yes	—
165	C₃×D₅	C₂×#97	1	2	yes	—
166	C₃×D₅	C₄×M₁₆	1	2	yes	—
167	C₂³×C₃	—	5	1	no	—
168	C₂×C₃×C₄	—	5	1	no	—
169	C₂×C₃×C₄	—	5	2	no	—
170	C₂×C₃×C₄	—	5	2	no	—
171	C₂×C₃×C₄	—	5	2	no	—
172	C₂×C₃×C₄	—	10	4	no	—
173	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
174	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
175	C₂×Dic₃	—	5	4	no	—
176	C₂×Dic₃	—	5	8	no	—
177	C₂×Dic₃	—	5	4	no	—
178	C₂×Dic₃	—	10	4	no	—
179	C₃×C₈	—	5	4	no	—
180	C₃×C₈	C₄×AGL(1,𝔽₅)	1	4	yes	complemented
181	C₃⋊C₈	—	5	8	no	—
182	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
183	C₂×D₅	C₄²×D₃	1	1	yes	characteristic
184	C₂×D₅	C₄²×D₃	1	1	yes	characteristic
185	C₂×D₅	D₃×M₁₆	1	4	yes	—
186	AGL(1,𝔽₅)	—	3	8	no	—
187	AGL(1,𝔽₅)	—	6	4	no	—
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	C₂×C₄×Dic₃	1	2	yes	—
190	Dic₅	—	2	2	no	—
191	Dic₅	C₄²×D₃	1	2	yes	—
192	C₂²×C₄	—	5	1	no	—
193	C₂²×C₄	—	5	2	no	—
194	C₂²×C₄	—	15	2	no	—
195	C₂²×C₄	—	15	2	no	—
196	C₂×C₈	—	5	2	no	—
197	C₂×C₈	—	5	4	no	—
198	C₂×C₈	—	5	4	no	—
199	C₂×C₈	—	15	4	no	—
200	C₂×C₈	—	15	4	no	—
201	C₂×C₈	—	15	4	no	—
202	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
203	C₄²	—	5	2	no	—
204	C₄²	—	5	1	no	—
205	C₄²	—	15	4	no	—
206	C₄²	—	15	4	no	—
207	C₄²	—	30	8	no	—
208	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
209	C₃×C₅	C₄×#97	1	1	yes	characteristic, complemented, Hall
210	C₂²×C₃	—	5	1	no	—
211	C₂²×C₃	—	5	1	no	—
212	C₂²×C₃	—	5	4	no	—
213	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	5	2	no	—
216	C₃×C₄	—	10	2	no	—
217	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
218	Dic₃	—	5	8	no	—
219	Dic₃	—	10	4	no	—
220	C₂×C₅	C₄²×Dic₃	1	1	yes	characteristic
221	C₂×C₅	Dic₃×M₁₆	1	2	yes	—
222	D₅	D₃×#97	1	2	yes	—
223	D₅	#4707	1	2	yes	—
224	C₂³	—	5	1	no	—
225	C₂×C₄	—	5	1	no	—
226	C₂×C₄	—	5	2	no	—
227	C₂×C₄	—	5	2	no	—
228	C₂×C₄	—	5	2	no	—
229	C₂×C₄	—	10	4	no	—
230	C₂×C₄	—	15	4	no	—
231	C₂×C₄	—	15	8	no	—
232	C₂×C₄	—	15	4	no	—
233	C₂×C₄	—	30	4	no	—
234	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
235	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
236	C₈	—	5	4	no	—
237	C₈	—	15	8	no	—
238	C₈	C₄×ΓL(1,𝔽₁₆)	1	4	yes	complemented
239	C₂×C₃	—	5	2	no	—
240	C₂×C₃	—	5	2	no	—
241	C₂×C₃	C₄²×AGL(1,𝔽₅)	1	1	yes	characteristic
242	C₂×C₃	M₁₆×AGL(1,𝔽₅)	1	2	yes	—
243	C₅	Dic₃×#97	1	1	yes	characteristic, complemented, Sylow
244	C₂²	—	5	1	no	—
245	C₂²	—	5	1	no	—
246	C₂²	—	5	4	no	—
247	C₂²	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	5	2	no	—
250	C₄	—	10	2	no	—
251	C₄	—	15	8	no	—
252	C₄	—	30	4	no	—
253	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
254	C₃	AGL(1,𝔽₅)×#97	1	1	yes	characteristic, complemented, Sylow
255	C₂	—	5	2	no	—
256	C₂	—	5	2	no	—
257	C₂	C₄²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
258	C₂	M₁₆×ΓL(1,𝔽₁₆)	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₅	C₄³
Fitting	C₃×C₅×#97	C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×C₃×D₅	C₂²×C₄
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	#97	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₄×#97
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
13. #91231	C₄	semidirect
14. #91221	C₄	semidirect
15. C₈×ΓL(1,𝔽₁₆)	C₄	semidirect
16. C₃×C₅×#97	C₄	semidirect
17. C₃×#3947	C₄	semidirect
18. C₃×#3935	C₄	semidirect
56. C₅×#97	Dic₃	semidirect
57. #3947	Dic₃	semidirect
58. #3935	Dic₃	semidirect
72. C₃⋊#170	C₄²	semidirect
73. C₃×C₅×C₈	C₄²	semidirect
74. C₃×#168	C₄²	semidirect
83. C₃×#97	AGL(1,𝔽₅)	semidirect
114. ΓL(1,𝔽₁₆)	#97	direct (N × H)
149. C₅×C₈	C₄×Dic₃	semidirect
150. #168	C₄×Dic₃	semidirect
161. #97	ΓL(1,𝔽₁₆)	direct (N × H)
180. C₃×C₈	C₄×AGL(1,𝔽₅)	semidirect
209. C₃×C₅	C₄×#97	semidirect
238. C₈	C₄×ΓL(1,𝔽₁₆)	semidirect
243. C₅	Dic₃×#97	semidirect
254. C₃	AGL(1,𝔽₅)×#97	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91221	2
C₂×C₈×ΓL(1,𝔽₁₆)	2
C₄²×ΓL(1,𝔽₁₆)	2
#11752335	2
C₃×D₅×#97	2
AGL(1,𝔽₅)×#97	3
Dic₃×#97	5

Subgroups — #97×ΓL(1,𝔽16) #14294210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #97×ΓL(1,𝔽₁₆) #14294210