Subgroups — C₂×M₁₆×ΓL(1,𝔽₁₆) #14411239

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic
2	C₂×D₅⋊#208	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×D₅×M₁₆	C₂	1	1	yes	maximal, characteristic
4	M₁₆×ΓL(1,𝔽₁₆)	C₂	1	8	yes	maximal, direct product factor
5	C₂×#91221	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×ΓL(1,𝔽₁₆)	C₂	1	2	yes	maximal, complemented
7	C₂×M₁₆×AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
8	C₂³×ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
9	C₂²×C₃×C₄×D₅	C₂²	1	1	yes	characteristic
10	C₂×#6762	C₄	1	2	yes	complemented
11	C₂×C₄×ΓL(1,𝔽₁₆)	C₄	1	4	yes	—
12	C₂×C₄×ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
13	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
14	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
15	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	4	yes	—
16	D₅⋊#208	C₄	1	4	yes	complemented
17	D₅⋊#208	C₂²	1	4	yes	—
18	C₂×#6758	C₄	1	2	yes	complemented
19	C₂×D₅⋊(C₃⋊C₈)	C₂²	1	2	yes	—
20	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic, complemented
21	C₂×C₃×#681	C₄	1	1	yes	characteristic, complemented
22	C₃×D₅×M₁₆	C₄	1	4	yes	complemented
23	C₃×D₅×M₁₆	C₂²	1	4	yes	—
24	C₂×C₃×#676	C₄	1	2	yes	complemented
25	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
26	#91221	C₂²	1	8	yes	complemented
27	C₈×ΓL(1,𝔽₁₆)	C₂²	1	8	yes	complemented
28	C₂×Dic₃×M₁₆	—	5	1	no	maximal, Hall
29	C₂²×C₄×AGL(1,𝔽₅)	—	3	1	no	—
30	C₂×D₅⋊M₁₆	—	3	1	no	—
31	C₂×D₅×M₁₆	D₃	1	1	yes	characteristic
32	M₁₆×AGL(1,𝔽₅)	—	3	8	no	—
33	C₂×#3992	—	3	2	no	—
34	C₂×C₈×AGL(1,𝔽₅)	—	3	2	no	—
35	C₂³×C₃×D₅	C₂×C₄	1	1	yes	characteristic
36	C₂²×ΓL(1,𝔽₁₆)	—	2	8	no	—
37	C₂²×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
38	C₂²×ΓL(1,𝔽₁₆)	C₂×C₄	1	8	yes	—
39	C₂²×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
40	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
43	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
46	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	—
48	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
49	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
50	#6762	C₂×C₄	1	8	yes	complemented
51	C₂×C₃⋊#170	C₂×C₄	1	4	yes	complemented
52	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
53	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	8	yes	—
54	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
55	C₄×ΓL(1,𝔽₁₆)	C₂³	1	2	yes	—
56	C₄×ΓL(1,𝔽₁₆)	C₂³	1	2	yes	—
57	#6758	C₂×C₄	1	8	yes	complemented
58	D₅⋊(C₃⋊C₈)	C₂×C₄	1	4	yes	complemented
59	D₅⋊(C₃⋊C₈)	C₂³	1	4	yes	—
60	C₃×C₅×M₁₆	C₂×C₄	1	4	yes	complemented
61	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	complemented
62	C₃×#681	C₂×C₄	1	4	yes	complemented
63	C₂×C₃×#168	C₂×C₄	1	2	yes	complemented
64	C₃×#676	C₂×C₄	1	8	yes	complemented
65	C₃×C₈×D₅	C₂×C₄	1	4	yes	complemented
66	C₃×C₈×D₅	C₂³	1	4	yes	—
67	C₂²×C₃×M₁₆	—	5	1	no	—
68	C₂²×C₄×Dic₃	—	5	1	no	—
69	C₂²×#208	—	5	1	no	—
70	Dic₃×M₁₆	—	5	8	no	—
71	C₂×#815	—	5	2	no	—
72	C₂×C₈×Dic₃	—	5	2	no	—
73	C₂³×AGL(1,𝔽₅)	—	3	2	no	—
74	C₂²×C₄×D₅	C₂×D₃	1	1	yes	characteristic
75	C₂×#704	—	3	2	no	—
76	C₂×C₄×AGL(1,𝔽₅)	—	3	2	no	—
77	C₂×C₄×AGL(1,𝔽₅)	—	3	4	no	—
78	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
79	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
80	C₂×C₄×AGL(1,𝔽₅)	—	3	4	no	—
81	D₅⋊M₁₆	—	3	4	no	—
82	D₅⋊M₁₆	—	3	4	no	—
83	C₂×#700	—	3	2	no	—
84	C₂×D₅⋊C₈	—	3	2	no	—
85	C₂×C₅×M₁₆	Dic₃	1	1	yes	characteristic, complemented
86	C₂×#681	Dic₃	1	1	yes	characteristic, complemented
87	D₅×M₁₆	Dic₃	1	4	yes	complemented
88	D₅×M₁₆	C₂×D₃	1	4	yes	—
89	C₂×#676	Dic₃	1	2	yes	complemented
90	C₂×C₈×D₅	C₂×D₃	1	2	yes	—
91	#3992	—	3	8	no	—
92	C₈×AGL(1,𝔽₅)	—	3	8	no	—
93	C₂×C₄×M₁₆	—	15	1	no	Sylow
94	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
95	C₂²×C₃×D₅	—	2	4	no	—
96	C₂²×C₃×D₅	C₄²	1	1	yes	characteristic
97	C₂²×C₃×D₅	C₄²	1	2	yes	—
98	C₂²×C₃×D₅	C₂²×C₄	1	2	yes	—
99	C₂²×C₃×D₅	C₂²×C₄	1	1	yes	characteristic
100	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
101	C₂×ΓL(1,𝔽₁₆)	—	2	16	no	—
102	C₂×ΓL(1,𝔽₁₆)	M₁₆	1	8	yes	direct product factor
103	C₂×ΓL(1,𝔽₁₆)	C₂²×C₄	1	4	yes	—
104	C₂×ΓL(1,𝔽₁₆)	C₂²×C₄	1	4	yes	—
105	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
106	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
107	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
109	C₂×C₃×Dic₅	C₄²	1	2	yes	—
110	C₂×C₃×Dic₅	C₄²	1	1	yes	characteristic
111	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
113	C₃×C₄×D₅	C₄²	1	2	yes	—
114	C₃×C₄×D₅	C₄²	1	2	yes	—
115	C₃×C₄×D₅	C₄²	1	2	yes	—
116	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
117	C₃×C₄×D₅	C₂²×C₄	1	2	yes	—
118	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
119	C₃×C₄×D₅	C₂²×C₄	1	2	yes	—
120	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
121	C₃×C₄×D₅	C₂²×C₄	1	2	yes	—
122	C₃×C₄×D₅	C₂⁴	1	1	yes	characteristic
123	C₃⋊#170	C₂²×C₄	1	8	yes	complemented
124	C₃×C₅×C₈	C₂²×C₄	1	4	yes	complemented
125	C₃×#168	C₂²×C₄	1	4	yes	complemented
126	C₂³×C₃×C₄	—	5	1	no	—
127	C₂³×Dic₃	—	5	2	no	—
128	C₂×C₃×M₁₆	—	5	1	no	—
129	C₂×C₃×M₁₆	—	5	4	no	—
130	C₂×C₃×M₁₆	—	5	4	no	—
131	C₂×C₃×M₁₆	—	5	2	no	—
132	C₂×C₃×M₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
133	C₂²×C₃×C₈	—	5	2	no	—
134	C₂×C₄×Dic₃	—	5	2	no	—
135	C₂×C₄×Dic₃	—	5	4	no	—
136	C₂×C₄×Dic₃	—	5	1	no	—
137	C₂×C₄×Dic₃	—	5	1	no	—
138	C₂×C₄×Dic₃	—	5	4	no	—
139	C₂×#208	—	5	2	no	—
140	C₂×#208	—	5	4	no	—
141	C₂×#208	—	5	4	no	—
142	C₂×#208	—	5	2	no	—
143	C₂²×C₃⋊C₈	—	5	2	no	—
144	#815	—	5	8	no	—
145	C₈×Dic₃	—	5	8	no	—
146	C₂³×D₅	C₄×D₃	1	1	yes	characteristic
147	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
148	C₂²×AGL(1,𝔽₅)	—	3	8	no	—
149	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
150	C₂²×AGL(1,𝔽₅)	—	6	8	no	—
151	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
152	C₂²×Dic₅	C₄×D₃	1	1	yes	characteristic
153	C₂×C₄×D₅	C₄×D₃	1	2	yes	—
154	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
155	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
156	C₂×C₄×D₅	C₄×D₃	1	2	yes	—
157	C₂×C₄×D₅	C₂×Dic₃	1	1	yes	characteristic
158	C₂×C₄×D₅	C₂×Dic₃	1	2	yes	—
159	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
160	C₂×C₄×D₅	C₂²×D₃	1	2	yes	—
161	#704	—	3	8	no	—
162	C₂×#170	—	3	4	no	—
163	C₄×AGL(1,𝔽₅)	—	3	2	no	—
164	C₄×AGL(1,𝔽₅)	—	3	8	no	—
165	C₄×AGL(1,𝔽₅)	—	3	2	no	—
166	C₄×AGL(1,𝔽₅)	—	3	2	no	—
167	C₄×AGL(1,𝔽₅)	—	3	2	no	—
168	#700	—	3	8	no	—
169	D₅⋊C₈	—	3	4	no	—
170	D₅⋊C₈	—	3	4	no	—
171	C₅×M₁₆	C₂×Dic₃	1	4	yes	complemented
172	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	complemented
173	#681	C₂×Dic₃	1	4	yes	complemented
174	C₂×#168	C₂×Dic₃	1	2	yes	complemented
175	#676	C₂×Dic₃	1	8	yes	complemented
176	C₈×D₅	C₂×Dic₃	1	4	yes	complemented
177	C₈×D₅	C₂²×D₃	1	4	yes	—
178	C₂²×M₁₆	—	5	1	no	—
179	C₂²×M₁₆	—	15	1	no	—
180	C₂²×C₄²	—	15	1	no	—
181	C₄×M₁₆	—	15	8	no	—
182	C₂×#97	—	15	2	no	—
183	C₂×C₄×C₈	—	15	2	no	—
184	C₂²×C₃×C₅	—	2	2	no	—
185	C₂²×C₃×C₅	C₂×C₄²	1	2	yes	—
186	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
187	C₂×C₃×D₅	—	2	2	no	—
188	C₂×C₃×D₅	—	2	4	no	—
189	C₂×C₃×D₅	—	2	4	no	—
190	C₂×C₃×D₅	C₂×C₄²	1	2	yes	—
191	C₂×C₃×D₅	C₂×C₄²	1	1	yes	characteristic
192	C₂×C₃×D₅	C₂×M₁₆	1	4	yes	—
193	C₂×C₃×D₅	C₂³×C₄	1	1	yes	characteristic
194	ΓL(1,𝔽₁₆)	—	2	4	no	—
195	ΓL(1,𝔽₁₆)	C₂×M₁₆	1	8	yes	direct product factor
196	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
197	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
198	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
199	C₃×Dic₅	C₂×C₄²	1	1	yes	characteristic
200	C₃×Dic₅	C₂×C₄²	1	2	yes	—
201	C₃×Dic₅	C₂³×C₄	1	1	yes	characteristic
202	C₂⁴×C₃	—	5	1	no	—
203	C₂²×C₃×C₄	—	5	1	no	—
204	C₂²×C₃×C₄	—	5	1	no	—
205	C₂²×C₃×C₄	—	5	2	no	—
206	C₂²×C₃×C₄	—	5	1	no	—
207	C₂²×C₃×C₄	—	5	2	no	—
208	C₂²×C₃×C₄	—	5	2	no	—
209	C₂²×C₃×C₄	—	5	1	no	—
210	C₂²×C₃×C₄	—	5	1	no	—
211	C₂²×C₃×C₄	—	5	2	no	—
212	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₂²×Dic₃	—	5	2	no	—
214	C₂²×Dic₃	—	5	8	no	—
215	C₂²×Dic₃	—	5	2	no	—
216	C₂²×Dic₃	—	10	8	no	—
217	C₃×M₁₆	—	5	4	no	—
218	C₃×M₁₆	—	5	8	no	—
219	C₃×M₁₆	C₂×AGL(1,𝔽₅)	1	4	yes	complemented
220	C₂×C₃×C₈	—	5	2	no	—
221	C₂×C₃×C₈	—	5	4	no	—
222	C₂×C₃×C₈	—	5	4	no	—
223	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
224	C₄×Dic₃	—	5	2	no	—
225	C₄×Dic₃	—	5	2	no	—
226	C₄×Dic₃	—	5	2	no	—
227	C₄×Dic₃	—	5	8	no	—
228	C₄×Dic₃	—	5	2	no	—
229	#208	—	5	8	no	—
230	#208	—	5	8	no	—
231	C₂×C₃⋊C₈	—	5	4	no	—
232	C₂×C₃⋊C₈	—	5	4	no	—
233	C₂×C₃⋊C₈	—	5	4	no	—
234	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
235	C₂²×D₅	—	2	4	no	—
236	C₂²×D₅	C₄×Dic₃	1	1	yes	characteristic
237	C₂²×D₅	C₄×Dic₃	1	2	yes	—
238	C₂²×D₅	C₂×C₄×D₃	1	2	yes	—
239	C₂²×D₅	C₂×C₄×D₃	1	1	yes	characteristic
240	C₂×AGL(1,𝔽₅)	—	3	8	no	—
241	C₂×AGL(1,𝔽₅)	—	3	4	no	—
242	C₂×AGL(1,𝔽₅)	—	3	4	no	—
243	C₂×AGL(1,𝔽₅)	—	6	16	no	—
244	C₂×AGL(1,𝔽₅)	—	6	4	no	—
245	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
246	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
247	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
248	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
249	C₂×Dic₅	C₄×Dic₃	1	2	yes	—
250	C₂×Dic₅	C₄×Dic₃	1	1	yes	characteristic
251	C₂×Dic₅	C₂×C₄×D₃	1	1	yes	characteristic
252	C₂×Dic₅	C₂×C₄×D₃	1	2	yes	—
253	C₄×D₅	C₄×Dic₃	1	2	yes	—
254	C₄×D₅	C₄×Dic₃	1	2	yes	—
255	C₄×D₅	C₄×Dic₃	1	2	yes	—
256	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
257	C₄×D₅	C₂×C₄×D₃	1	2	yes	—
258	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
259	C₄×D₅	C₂×C₄×D₃	1	2	yes	—
260	C₄×D₅	C₂²×Dic₃	1	1	yes	characteristic
261	C₄×D₅	C₂²×Dic₃	1	2	yes	—
262	C₄×D₅	C₂³×D₃	1	1	yes	characteristic
263	#170	—	3	8	no	—
264	C₅×C₈	C₂²×Dic₃	1	4	yes	complemented
265	#168	C₂²×Dic₃	1	4	yes	complemented
266	C₂³×C₄	—	5	1	no	—
267	C₂³×C₄	—	15	2	no	—
268	C₂×M₁₆	—	5	1	no	—
269	C₂×M₁₆	—	5	4	no	—
270	C₂×M₁₆	—	5	4	no	—
271	C₂×M₁₆	—	5	2	no	—
272	C₂×M₁₆	—	15	2	no	—
273	C₂×M₁₆	—	15	4	no	—
274	C₂×M₁₆	—	15	4	no	—
275	C₂×M₁₆	—	15	2	no	—
276	C₂×M₁₆	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor
277	C₂²×C₈	—	5	2	no	—
278	C₂²×C₈	—	15	2	no	—
279	C₂×C₄²	—	15	1	no	—
280	C₂×C₄²	—	15	1	no	—
281	C₂×C₄²	—	15	4	no	—
282	C₂×C₄²	—	15	2	no	—
283	C₂×C₄²	—	15	4	no	—
284	#97	—	15	8	no	—
285	C₄×C₈	—	15	8	no	—
286	C₂×C₃×C₅	—	2	2	no	—
287	C₂×C₃×C₅	C₄×M₁₆	1	2	yes	complemented
288	C₂×C₃×C₅	C₂²×C₄²	1	1	yes	characteristic
289	C₃×D₅	—	2	2	no	—
290	C₃×D₅	C₄×M₁₆	1	2	yes	complemented
291	C₃×D₅	C₂²×M₁₆	1	2	yes	—
292	C₂³×C₃	—	5	1	no	—
293	C₂³×C₃	—	5	2	no	—
294	C₂³×C₃	—	5	1	no	—
295	C₂³×C₃	—	5	2	no	—
296	C₂³×C₃	—	10	4	no	—
297	C₂³×C₃	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
298	C₂×C₃×C₄	—	5	2	no	—
299	C₂×C₃×C₄	—	5	1	no	—
300	C₂×C₃×C₄	—	5	1	no	—
301	C₂×C₃×C₄	—	5	2	no	—
302	C₂×C₃×C₄	—	5	2	no	—
303	C₂×C₃×C₄	—	5	2	no	—
304	C₂×C₃×C₄	—	5	2	no	—
305	C₂×C₃×C₄	—	5	1	no	—
306	C₂×C₃×C₄	—	5	2	no	—
307	C₂×C₃×C₄	—	5	1	no	—
308	C₂×C₃×C₄	—	5	2	no	—
309	C₂×C₃×C₄	—	5	1	no	—
310	C₂×C₃×C₄	—	5	2	no	—
311	C₂×C₃×C₄	—	5	1	no	—
312	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	2	yes	—
313	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
314	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
315	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
316	C₂×Dic₃	—	5	8	no	—
317	C₂×Dic₃	—	5	4	no	—
318	C₂×Dic₃	—	5	4	no	—
319	C₂×Dic₃	—	10	16	no	—
320	C₂×Dic₃	—	10	4	no	—
321	C₃×C₈	—	5	4	no	—
322	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	4	yes	complemented
323	C₃⋊C₈	—	5	8	no	—
324	C₂²×C₅	—	2	2	no	—
325	C₂²×C₅	C₂×C₄×Dic₃	1	2	yes	—
326	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
327	C₂×D₅	—	2	2	no	—
328	C₂×D₅	—	2	4	no	—
329	C₂×D₅	—	2	4	no	—
330	C₂×D₅	D₃×M₁₆	1	4	yes	—
331	C₂×D₅	C₂×C₄×Dic₃	1	2	yes	—
332	C₂×D₅	C₂×C₄×Dic₃	1	1	yes	characteristic
333	C₂×D₅	C₂²×C₄×D₃	1	1	yes	characteristic
334	AGL(1,𝔽₅)	—	3	8	no	—
335	AGL(1,𝔽₅)	—	6	4	no	—
336	C₄×C₅	C₂×C₄×Dic₃	1	2	yes	—
337	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
338	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
339	Dic₅	C₂×C₄×Dic₃	1	1	yes	characteristic
340	Dic₅	C₂×C₄×Dic₃	1	2	yes	—
341	Dic₅	C₂²×C₄×D₃	1	1	yes	characteristic
342	C₂⁴	—	5	1	no	—
343	C₂²×C₄	—	5	1	no	—
344	C₂²×C₄	—	5	1	no	—
345	C₂²×C₄	—	5	2	no	—
346	C₂²×C₄	—	5	1	no	—
347	C₂²×C₄	—	5	2	no	—
348	C₂²×C₄	—	5	2	no	—
349	C₂²×C₄	—	5	2	no	—
350	C₂²×C₄	—	5	1	no	—
351	C₂²×C₄	—	5	1	no	—
352	C₂²×C₄	—	15	2	no	—
353	C₂²×C₄	—	15	8	no	—
354	C₂²×C₄	—	15	2	no	—
355	C₂²×C₄	—	30	8	no	—
356	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
357	M₁₆	—	5	4	no	—
358	M₁₆	—	5	8	no	—
359	M₁₆	—	15	8	no	—
360	M₁₆	—	15	8	no	—
361	M₁₆	C₂×ΓL(1,𝔽₁₆)	1	4	yes	direct product factor
362	C₂×C₈	—	5	2	no	—
363	C₂×C₈	—	5	4	no	—
364	C₂×C₈	—	5	4	no	—
365	C₂×C₈	—	15	4	no	—
366	C₂×C₈	—	15	4	no	—
367	C₂×C₈	—	15	4	no	—
368	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	complemented
369	C₄²	—	15	2	no	—
370	C₄²	—	15	2	no	—
371	C₄²	—	15	2	no	—
372	C₄²	—	15	8	no	—
373	C₄²	—	15	2	no	—
374	C₃×C₅	C₂×C₄×M₁₆	1	1	yes	characteristic, complemented, Hall
375	C₂²×C₃	—	2	2	no	—
376	C₂²×C₃	—	5	4	no	—
377	C₂²×C₃	—	5	1	no	—
378	C₂²×C₃	—	5	1	no	—
379	C₂²×C₃	—	5	2	no	—
380	C₂²×C₃	—	10	4	no	—
381	C₂²×C₃	—	10	4	no	—
382	C₂²×C₃	—	10	2	no	—
383	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
384	C₂²×C₃	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
385	C₃×C₄	—	5	1	no	—
386	C₃×C₄	—	5	2	no	—
387	C₃×C₄	—	5	1	no	—
388	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	2	yes	—
389	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
390	C₃×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
391	Dic₃	—	5	8	no	—
392	Dic₃	—	10	4	no	—
393	C₂×C₅	—	2	2	no	—
394	C₂×C₅	Dic₃×M₁₆	1	2	yes	complemented
395	C₂×C₅	C₂²×C₄×Dic₃	1	1	yes	characteristic
396	D₅	—	2	2	no	—
397	D₅	Dic₃×M₁₆	1	2	yes	complemented
398	D₅	C₂×D₃×M₁₆	1	2	yes	—
399	C₂³	—	5	1	no	—
400	C₂³	—	5	2	no	—
401	C₂³	—	5	2	no	—
402	C₂³	—	5	1	no	—
403	C₂³	—	10	4	no	—
404	C₂³	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
405	C₂×C₄	—	5	1	no	—
406	C₂×C₄	—	5	2	no	—
407	C₂×C₄	—	5	2	no	—
408	C₂×C₄	—	5	1	no	—
409	C₂×C₄	—	5	2	no	—
410	C₂×C₄	—	5	1	no	—
411	C₂×C₄	—	5	1	no	—
412	C₂×C₄	—	5	2	no	—
413	C₂×C₄	—	5	2	no	—
414	C₂×C₄	—	5	2	no	—
415	C₂×C₄	—	5	2	no	—
416	C₂×C₄	—	5	1	no	—
417	C₂×C₄	—	5	1	no	—
418	C₂×C₄	—	5	2	no	—
419	C₂×C₄	—	15	4	no	—
420	C₂×C₄	—	15	8	no	—
421	C₂×C₄	—	15	4	no	—
422	C₂×C₄	—	30	16	no	—
423	C₂×C₄	—	30	4	no	—
424	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
425	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
426	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	2	yes	—
427	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
428	C₈	—	5	4	no	—
429	C₈	—	15	8	no	—
430	C₈	C₂²×ΓL(1,𝔽₁₆)	1	4	yes	complemented
431	C₂×C₃	—	2	2	no	—
432	C₂×C₃	—	5	2	no	—
433	C₂×C₃	—	5	2	no	—
434	C₂×C₃	—	10	2	no	—
435	C₂×C₃	M₁₆×AGL(1,𝔽₅)	1	2	yes	complemented
436	C₂×C₃	C₂²×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
437	C₅	C₂×Dic₃×M₁₆	1	1	yes	characteristic, complemented, Sylow
438	C₂²	—	2	2	no	—
439	C₂²	—	5	1	no	—
440	C₂²	—	5	1	no	—
441	C₂²	—	5	4	no	—
442	C₂²	—	5	2	no	—
443	C₂²	—	10	4	no	—
444	C₂²	—	10	4	no	—
445	C₂²	—	10	2	no	—
446	C₂²	C₂×C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
447	C₂²	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
448	C₄	—	5	1	no	—
449	C₄	—	5	1	no	—
450	C₄	—	5	2	no	—
451	C₄	—	15	8	no	—
452	C₄	—	30	4	no	—
453	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	2	yes	—
454	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
455	C₄	C₂³×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
456	C₃	C₂×M₁₆×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
457	C₂	—	2	2	no	—
458	C₂	—	5	2	no	—
459	C₂	—	5	2	no	—
460	C₂	—	10	2	no	—
461	C₂	M₁₆×ΓL(1,𝔽₁₆)	1	2	yes	direct product factor
462	C₂	C₂²×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₅	C₂²×C₄²
Frattini	C₄	C₂³×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₅×M₁₆	C₄
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂³×C₃×D₅	C₂×C₄
cosocle	C₃×C₄×D₅	C₂⁴
hypercenter	C₂×M₁₆	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₂×C₄×M₁₆
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. M₁₆×ΓL(1,𝔽₁₆)	C₂	direct (N × H)
5. C₂×#91221	C₂	semidirect
6. C₂×C₈×ΓL(1,𝔽₁₆)	C₂	semidirect
10. C₂×#6762	C₄	semidirect
16. D₅⋊#208	C₄	semidirect
18. C₂×#6758	C₄	semidirect
20. C₂×C₃×C₅×M₁₆	C₄	semidirect
21. C₂×C₃×#681	C₄	semidirect
22. C₃×D₅×M₁₆	C₄	semidirect
24. C₂×C₃×#676	C₄	semidirect
26. #91221	C₂²	semidirect
27. C₈×ΓL(1,𝔽₁₆)	C₂²	semidirect
50. #6762	C₂×C₄	semidirect
51. C₂×C₃⋊#170	C₂×C₄	semidirect
57. #6758	C₂×C₄	semidirect
58. D₅⋊(C₃⋊C₈)	C₂×C₄	semidirect
60. C₃×C₅×M₁₆	C₂×C₄	semidirect
61. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
62. C₃×#681	C₂×C₄	semidirect
63. C₂×C₃×#168	C₂×C₄	semidirect
64. C₃×#676	C₂×C₄	semidirect
65. C₃×C₈×D₅	C₂×C₄	semidirect
85. C₂×C₅×M₁₆	Dic₃	semidirect
86. C₂×#681	Dic₃	semidirect
87. D₅×M₁₆	Dic₃	semidirect
89. C₂×#676	Dic₃	semidirect
102. C₂×ΓL(1,𝔽₁₆)	M₁₆	direct (N × H)
123. C₃⋊#170	C₂²×C₄	semidirect
124. C₃×C₅×C₈	C₂²×C₄	semidirect
125. C₃×#168	C₂²×C₄	semidirect
132. C₂×C₃×M₁₆	AGL(1,𝔽₅)	semidirect
171. C₅×M₁₆	C₂×Dic₃	semidirect
172. C₂×C₅×C₈	C₂×Dic₃	semidirect
173. #681	C₂×Dic₃	semidirect
174. C₂×#168	C₂×Dic₃	semidirect
175. #676	C₂×Dic₃	semidirect
176. C₈×D₅	C₂×Dic₃	semidirect
195. ΓL(1,𝔽₁₆)	C₂×M₁₆	direct (N × H)
219. C₃×M₁₆	C₂×AGL(1,𝔽₅)	semidirect
223. C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	semidirect
264. C₅×C₈	C₂²×Dic₃	semidirect
265. #168	C₂²×Dic₃	semidirect
276. C₂×M₁₆	ΓL(1,𝔽₁₆)	direct (N × H)
287. C₂×C₃×C₅	C₄×M₁₆	semidirect
290. C₃×D₅	C₄×M₁₆	semidirect
322. C₃×C₈	C₂²×AGL(1,𝔽₅)	semidirect
361. M₁₆	C₂×ΓL(1,𝔽₁₆)	direct (N × H)
368. C₂×C₈	C₂×ΓL(1,𝔽₁₆)	semidirect
374. C₃×C₅	C₂×C₄×M₁₆	semidirect
394. C₂×C₅	Dic₃×M₁₆	semidirect
397. D₅	Dic₃×M₁₆	semidirect
430. C₈	C₂²×ΓL(1,𝔽₁₆)	semidirect
435. C₂×C₃	M₁₆×AGL(1,𝔽₅)	semidirect
437. C₅	C₂×Dic₃×M₁₆	semidirect
456. C₃	C₂×M₁₆×AGL(1,𝔽₅)	semidirect
461. C₂	M₁₆×ΓL(1,𝔽₁₆)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×ΓL(1,𝔽₁₆)	2
C₂×D₅⋊#208	2
C₂×C₃×D₅×M₁₆	2
M₁₆×ΓL(1,𝔽₁₆)	2
C₂×#91221	2
C₂×C₈×ΓL(1,𝔽₁₆)	2
C₂×M₁₆×AGL(1,𝔽₅)	3
C₂×Dic₃×M₁₆	5

Subgroups — C2×M16×ΓL(1,𝔽16) #14411239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₁₆×ΓL(1,𝔽₁₆) #14411239