Subgroups — M₃₂×ΓL(1,𝔽₁₆) #14312588

Order 1920.

392 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
2	D₅⋊#812	C₂	1	1	yes	maximal, characteristic
3	C₃×D₅×M₃₂	C₂	1	1	yes	maximal, characteristic
4	#11746715	C₂	1	2	yes	maximal, complemented
5	C₁₆×ΓL(1,𝔽₁₆)	C₂	1	2	yes	maximal, complemented
6	AGL(1,𝔽₅)×M₃₂	—	3	1	no	maximal, Hall
7	C₂×C₄×ΓL(1,𝔽₁₆)	C₄	1	1	yes	characteristic
8	C₂×D₅⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
10	#91227	C₄	1	2	yes	complemented
11	C₈×ΓL(1,𝔽₁₆)	C₄	1	1	yes	characteristic
12	C₈×ΓL(1,𝔽₁₆)	C₄	1	1	yes	characteristic
13	C₈×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
14	C₈×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
15	#91219	C₄	1	2	yes	complemented
16	D₅⋊(C₃⋊C₁₆)	C₂²	1	2	yes	—
17	C₃×C₅×M₃₂	C₄	1	1	yes	characteristic, complemented
18	C₃×#3944	C₄	1	1	yes	characteristic, complemented
19	C₃×#3930	C₄	1	2	yes	complemented
20	C₃×D₅×C₁₆	C₂²	1	2	yes	—
21	Dic₃×M₃₂	—	5	1	no	maximal, Hall
22	C₂×C₈×AGL(1,𝔽₅)	—	3	1	no	—
23	D₅⋊M₃₂	—	3	1	no	—
24	D₅×M₃₂	D₃	1	1	yes	characteristic
25	#64844	—	3	2	no	—
26	C₁₆×AGL(1,𝔽₅)	—	3	2	no	—
27	C₂²×ΓL(1,𝔽₁₆)	C₈	1	2	yes	—
28	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃⋊#170	C₈	1	2	yes	—
30	C₄×ΓL(1,𝔽₁₆)	C₈	1	2	yes	—
31	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	1	yes	characteristic
32	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	1	yes	characteristic
33	D₅⋊(C₃⋊C₈)	C₈	1	2	yes	—
34	D₅⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
35	D₅⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
38	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
40	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
41	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
42	C₃⋊#674	C₂×C₄	1	4	yes	complemented
43	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
44	C₃×#672	C₂×C₄	1	2	yes	complemented
45	C₂×C₃×M₃₂	—	5	1	no	—
46	C₂×C₈×Dic₃	—	5	1	no	—
47	C₂×#812	—	5	1	no	—
48	#4502	—	5	2	no	—
49	Dic₃×C₁₆	—	5	2	no	—
50	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
51	C₂×D₅⋊C₈	—	3	1	no	—
52	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
53	#3998	—	3	2	no	—
54	C₈×AGL(1,𝔽₅)	—	3	1	no	—
55	C₈×AGL(1,𝔽₅)	—	3	1	no	—
56	C₈×AGL(1,𝔽₅)	—	3	1	no	—
57	C₈×AGL(1,𝔽₅)	—	3	1	no	—
58	#3990	—	3	2	no	—
59	D₅⋊C₁₆	—	3	2	no	—
60	C₅×M₃₂	Dic₃	1	1	yes	characteristic, complemented
61	#3944	Dic₃	1	1	yes	characteristic, complemented
62	#3930	Dic₃	1	2	yes	complemented
63	D₅×C₁₆	C₂×D₃	1	2	yes	—
64	C₄×M₃₂	—	15	1	no	Sylow
65	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
66	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
67	C₂×ΓL(1,𝔽₁₆)	C₂×C₈	1	2	yes	—
68	C₂×ΓL(1,𝔽₁₆)	C₂×C₈	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
70	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
71	C₃×C₄×D₅	C₄²	1	1	yes	characteristic
72	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
73	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
74	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
75	C₃⋊#170	C₂×C₈	1	2	yes	—
76	C₃⋊#170	C₂×C₈	1	2	yes	—
77	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
78	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₃×#168	C₄²	1	1	yes	characteristic
80	C₃×#168	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₃×C₈	—	5	1	no	—
82	C₂×C₄×Dic₃	—	5	1	no	—
83	C₂²×C₃⋊C₈	—	5	1	no	—
84	C₃×M₃₂	—	5	1	no	—
85	C₃×M₃₂	—	5	2	no	—
86	C₃×M₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
87	C₂×C₃×C₁₆	—	5	2	no	—
88	C₈×Dic₃	—	5	1	no	—
89	C₈×Dic₃	—	5	1	no	—
90	C₈×Dic₃	—	5	1	no	—
91	C₈×Dic₃	—	5	1	no	—
92	#812	—	5	2	no	—
93	#812	—	5	2	no	—
94	C₂×C₃⋊C₁₆	—	5	2	no	—
95	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
96	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
97	C₂×#170	—	3	2	no	—
98	C₄×AGL(1,𝔽₅)	—	3	2	no	—
99	C₄×AGL(1,𝔽₅)	—	3	1	no	—
100	C₄×AGL(1,𝔽₅)	—	3	1	no	—
101	D₅⋊C₈	—	3	2	no	—
102	D₅⋊C₈	—	3	1	no	—
103	D₅⋊C₈	—	3	1	no	—
104	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
105	C₂×#168	C₄×D₃	1	1	yes	characteristic
106	C₈×D₅	C₄×D₃	1	1	yes	characteristic
107	C₈×D₅	C₄×D₃	1	1	yes	characteristic
108	C₈×D₅	C₂×Dic₃	1	1	yes	characteristic
109	C₈×D₅	C₂²×D₃	1	1	yes	characteristic
110	#674	—	3	4	no	—
111	C₅×C₁₆	C₂×Dic₃	1	2	yes	complemented
112	#672	C₂×Dic₃	1	2	yes	complemented
113	C₂×M₃₂	—	5	1	no	—
114	C₂×M₃₂	—	15	1	no	—
115	C₂×C₄×C₈	—	15	1	no	—
116	#346	—	15	2	no	—
117	C₄×C₁₆	—	15	2	no	—
118	C₂²×C₃×C₅	C₄×C₈	1	1	yes	characteristic
119	C₂×C₃×D₅	—	2	2	no	—
120	C₂×C₃×D₅	C₄×C₈	1	1	yes	characteristic
121	C₂×C₃×D₅	C₂²×C₈	1	1	yes	characteristic
122	ΓL(1,𝔽₁₆)	—	2	2	no	—
123	ΓL(1,𝔽₁₆)	M₃₂	1	4	yes	direct product factor
124	C₃×C₄×C₅	C₄×C₈	1	1	yes	characteristic
125	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
126	C₃×Dic₅	C₄×C₈	1	1	yes	characteristic
127	C₃×Dic₅	C₂²×C₈	1	1	yes	characteristic
128	C₂²×C₃×C₄	—	5	1	no	—
129	C₂²×Dic₃	—	5	2	no	—
130	C₂×C₃×C₈	—	5	1	no	—
131	C₂×C₃×C₈	—	5	1	no	—
132	C₂×C₃×C₈	—	5	1	no	—
133	C₂×C₃×C₈	—	5	1	no	—
134	C₂×C₃×C₈	—	5	1	no	—
135	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
136	C₄×Dic₃	—	5	1	no	—
137	C₄×Dic₃	—	5	1	no	—
138	C₄×Dic₃	—	5	2	no	—
139	C₂×C₃⋊C₈	—	5	1	no	—
140	C₂×C₃⋊C₈	—	5	1	no	—
141	C₂×C₃⋊C₈	—	5	2	no	—
142	C₂×C₃⋊C₈	—	5	2	no	—
143	C₃×C₁₆	—	5	2	no	—
144	C₃×C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
145	C₃⋊C₁₆	—	5	4	no	—
146	C₂²×D₅	D₃×C₈	1	1	yes	characteristic
147	C₂×AGL(1,𝔽₅)	—	3	2	no	—
148	C₂×AGL(1,𝔽₅)	—	3	2	no	—
149	C₂×AGL(1,𝔽₅)	—	6	4	no	—
150	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
151	C₂×Dic₅	D₃×C₈	1	1	yes	characteristic
152	C₄×D₅	D₃×C₈	1	1	yes	characteristic
153	C₄×D₅	D₃×C₈	1	1	yes	characteristic
154	C₄×D₅	C₄×Dic₃	1	1	yes	characteristic
155	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
156	#170	—	3	2	no	—
157	#170	—	3	2	no	—
158	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
159	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
160	#168	C₄×Dic₃	1	1	yes	characteristic
161	#168	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂²×C₈	—	5	1	no	—
163	C₂²×C₈	—	15	1	no	—
164	C₂×C₄²	—	15	1	no	—
165	M₃₂	—	5	1	no	—
166	M₃₂	—	5	2	no	—
167	M₃₂	—	15	2	no	—
168	M₃₂	—	15	2	no	—
169	M₃₂	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor
170	C₂×C₁₆	—	5	2	no	—
171	C₂×C₁₆	—	15	2	no	—
172	C₄×C₈	—	15	1	no	—
173	C₄×C₈	—	15	1	no	—
174	C₄×C₈	—	15	1	no	—
175	C₄×C₈	—	15	1	no	—
176	C₂×C₃×C₅	—	2	1	no	—
177	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
178	C₃×D₅	—	2	1	no	—
179	C₃×D₅	C₂×M₃₂	1	2	yes	—
180	C₂³×C₃	—	5	1	no	—
181	C₂×C₃×C₄	—	5	1	no	—
182	C₂×C₃×C₄	—	5	1	no	—
183	C₂×C₃×C₄	—	5	1	no	—
184	C₂×C₃×C₄	—	5	1	no	—
185	C₂×C₃×C₄	—	5	1	no	—
186	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
187	C₂×Dic₃	—	5	2	no	—
188	C₂×Dic₃	—	5	2	no	—
189	C₂×Dic₃	—	10	4	no	—
190	C₃×C₈	—	5	1	no	—
191	C₃×C₈	—	5	1	no	—
192	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
193	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
194	C₃⋊C₈	—	5	2	no	—
195	C₃⋊C₈	—	5	2	no	—
196	C₂²×C₅	C₈×Dic₃	1	1	yes	characteristic
197	C₂×D₅	—	2	2	no	—
198	C₂×D₅	C₈×Dic₃	1	1	yes	characteristic
199	C₂×D₅	C₂×D₃×C₈	1	1	yes	characteristic
200	AGL(1,𝔽₅)	—	3	4	no	—
201	AGL(1,𝔽₅)	—	6	2	no	—
202	C₄×C₅	C₈×Dic₃	1	1	yes	characteristic
203	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
204	Dic₅	C₈×Dic₃	1	1	yes	characteristic
205	Dic₅	C₂×D₃×C₈	1	1	yes	characteristic
206	C₂²×C₄	—	5	1	no	—
207	C₂²×C₄	—	15	2	no	—
208	C₂×C₈	—	5	1	no	—
209	C₂×C₈	—	5	1	no	—
210	C₂×C₈	—	5	1	no	—
211	C₂×C₈	—	5	1	no	—
212	C₂×C₈	—	5	1	no	—
213	C₂×C₈	—	15	1	no	—
214	C₂×C₈	—	15	1	no	—
215	C₂×C₈	—	15	2	no	—
216	C₂×C₈	—	15	2	no	—
217	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
218	C₄²	—	15	1	no	—
219	C₄²	—	15	1	no	—
220	C₄²	—	15	2	no	—
221	C₁₆	—	5	2	no	—
222	C₁₆	—	15	4	no	—
223	C₁₆	C₂×ΓL(1,𝔽₁₆)	1	2	yes	complemented
224	C₃×C₅	C₄×M₃₂	1	1	yes	characteristic, complemented, Hall
225	C₂²×C₃	—	5	1	no	—
226	C₂²×C₃	—	5	1	no	—
227	C₂²×C₃	—	10	2	no	—
228	C₂²×C₃	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
229	C₃×C₄	—	5	1	no	—
230	C₃×C₄	—	5	1	no	—
231	C₃×C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
232	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
233	Dic₃	—	5	4	no	—
234	Dic₃	—	10	2	no	—
235	C₂×C₅	—	2	1	no	—
236	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
237	D₅	—	2	1	no	—
238	D₅	D₃×M₃₂	1	2	yes	—
239	C₂³	—	5	1	no	—
240	C₂×C₄	—	5	1	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	—	15	2	no	—
246	C₂×C₄	—	15	2	no	—
247	C₂×C₄	—	30	4	no	—
248	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
249	C₈	—	5	1	no	—
250	C₈	—	5	1	no	—
251	C₈	—	15	2	no	—
252	C₈	—	15	2	no	—
253	C₈	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
254	C₈	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	5	2	no	—
257	C₂×C₃	—	10	1	no	—
258	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
259	C₅	Dic₃×M₃₂	1	1	yes	characteristic, complemented, Sylow
260	C₂²	—	5	1	no	—
261	C₂²	—	5	1	no	—
262	C₂²	—	10	2	no	—
263	C₂²	C₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
264	C₄	—	5	1	no	—
265	C₄	—	5	1	no	—
266	C₄	—	15	4	no	—
267	C₄	—	30	2	no	—
268	C₄	C₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
269	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
270	C₃	AGL(1,𝔽₅)×M₃₂	1	1	yes	characteristic, complemented, Sylow
271	C₂	—	2	1	no	—
272	C₂	—	5	2	no	—
273	C₂	—	10	1	no	—
274	C₂	C₂×C₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×ΓL(1,𝔽₁₆)
commutator, socle	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₅×M₃₂	C₄
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₃×C₈×D₅	C₂³
hypercenter	M₃₂	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₄×M₃₂
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
4. #11746715	C₂	semidirect
5. C₁₆×ΓL(1,𝔽₁₆)	C₂	semidirect
10. #91227	C₄	semidirect
15. #91219	C₄	semidirect
17. C₃×C₅×M₃₂	C₄	semidirect
18. C₃×#3944	C₄	semidirect
19. C₃×#3930	C₄	semidirect
42. C₃⋊#674	C₂×C₄	semidirect
43. C₃×C₅×C₁₆	C₂×C₄	semidirect
44. C₃×#672	C₂×C₄	semidirect
60. C₅×M₃₂	Dic₃	semidirect
61. #3944	Dic₃	semidirect
62. #3930	Dic₃	semidirect
86. C₃×M₃₂	AGL(1,𝔽₅)	semidirect
111. C₅×C₁₆	C₂×Dic₃	semidirect
112. #672	C₂×Dic₃	semidirect
123. ΓL(1,𝔽₁₆)	M₃₂	direct (N × H)
144. C₃×C₁₆	C₂×AGL(1,𝔽₅)	semidirect
169. M₃₂	ΓL(1,𝔽₁₆)	direct (N × H)
223. C₁₆	C₂×ΓL(1,𝔽₁₆)	semidirect
224. C₃×C₅	C₄×M₃₂	semidirect
259. C₅	Dic₃×M₃₂	semidirect
270. C₃	AGL(1,𝔽₅)×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×ΓL(1,𝔽₁₆)	2
D₅⋊#812	2
C₃×D₅×M₃₂	2
#11746715	2
C₁₆×ΓL(1,𝔽₁₆)	2
AGL(1,𝔽₅)×M₃₂	3
Dic₃×M₃₂	5

Subgroups — M32×ΓL(1,𝔽16) #14312588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₃₂×ΓL(1,𝔽₁₆) #14312588