Subgroups — #135×ΓL(1,𝔽₁₆) #14416225

Order 1920.

1000 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×ΓL(1,𝔽₁₆)	C₂	1	2	yes	maximal, complemented
2	C₃×D₅×#135	C₂	1	1	yes	maximal, characteristic
3	#11752453	C₂	1	2	yes	maximal, complemented
4	Q₁₆×ΓL(1,𝔽₁₆)	C₂	1	2	yes	maximal, complemented
5	SD₁₆×ΓL(1,𝔽₁₆)	C₂	1	4	yes	maximal, complemented
6	D₈×ΓL(1,𝔽₁₆)	C₂	1	2	yes	maximal, complemented
7	#11752380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
9	AGL(1,𝔽₅)×#135	—	3	1	no	maximal, Hall
10	C₃×D₅×#41	C₂²	1	2	yes	—
11	Q₈×ΓL(1,𝔽₁₆)	C₂²	1	4	yes	—
12	D₄×ΓL(1,𝔽₁₆)	—	2	4	no	—
13	D₄×ΓL(1,𝔽₁₆)	C₂²	1	4	yes	—
14	#91989	—	2	2	no	—
15	#91989	C₂²	1	2	yes	—
16	C₂×C₄×ΓL(1,𝔽₁₆)	—	2	2	no	—
17	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
18	C₂×D₅⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
19	C₃×C₅×#135	C₄	1	1	yes	characteristic, complemented
20	C₃×#4096	C₄	1	2	yes	complemented
21	C₃×#4065	C₄	1	1	yes	characteristic, complemented
22	C₃×D₅×Q₁₆	C₂²	1	1	yes	characteristic
23	C₃×#4062	C₄	1	2	yes	complemented
24	C₃×D₅×SD₁₆	C₂²	1	2	yes	—
25	C₃×#4058	C₄	1	1	yes	characteristic, complemented
26	C₃×D₅×D₈	C₂²	1	1	yes	characteristic
27	C₃×#4050	C₄	1	1	yes	characteristic, complemented
28	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
29	#91238	C₂²	1	4	yes	—
30	#91236	C₂²	1	4	yes	—
31	#91223	C₂²	1	2	yes	—
32	#91222	C₂²	1	2	yes	—
33	C₈×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
34	C₈×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
35	Dic₃×#135	—	5	1	no	maximal, Hall
36	#41×AGL(1,𝔽₅)	—	3	2	no	—
37	D₅×#135	D₃	1	1	yes	characteristic
38	#65794	—	3	2	no	—
39	Q₁₆×AGL(1,𝔽₅)	—	3	2	no	—
40	SD₁₆×AGL(1,𝔽₅)	—	3	4	no	—
41	D₈×AGL(1,𝔽₅)	—	3	2	no	—
42	#65721	—	3	1	no	—
43	C₂×C₈×AGL(1,𝔽₅)	—	3	1	no	—
44	C₂²×ΓL(1,𝔽₁₆)	—	2	4	no	—
45	C₂²×ΓL(1,𝔽₁₆)	D₄	1	2	yes	—
46	C₃×C₅×#41	C₂×C₄	1	2	yes	complemented
47	C₃×#713	C₂×C₄	1	2	yes	complemented
48	C₃×Q₈×D₅	C₂³	1	2	yes	—
49	C₃×#711	—	2	2	no	—
50	C₃×#711	C₂×C₄	1	2	yes	complemented
51	C₃×D₄×D₅	—	2	2	no	—
52	C₃×D₄×D₅	C₂³	1	2	yes	—
53	C₃×#709	—	2	2	no	—
54	C₃×#709	C₂×C₄	1	2	yes	complemented
55	C₂×C₃×C₄×D₅	—	2	2	no	—
56	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
57	#6763	—	2	2	no	—
58	#6763	—	2	4	no	—
59	C₂×C₃⋊#170	—	2	1	no	—
60	C₄⋊ΓL(1,𝔽₁₆)	—	2	4	no	—
61	C₄⋊ΓL(1,𝔽₁₆)	C₂³	1	4	yes	—
62	C₄×ΓL(1,𝔽₁₆)	—	2	4	no	—
63	C₄×ΓL(1,𝔽₁₆)	—	2	2	no	—
64	C₄×ΓL(1,𝔽₁₆)	D₄	1	1	yes	characteristic
65	C₄×ΓL(1,𝔽₁₆)	D₄	1	1	yes	characteristic
66	C₄×ΓL(1,𝔽₁₆)	C₂³	1	2	yes	—
67	D₅⋊(C₃⋊C₈)	—	2	1	no	—
68	D₅⋊(C₃⋊C₈)	C₂³	1	2	yes	—
69	C₃×C₅×Q₁₆	C₂×C₄	1	1	yes	characteristic, complemented
70	C₃×C₅×SD₁₆	C₂×C₄	1	2	yes	complemented
71	C₃×C₅×D₈	C₂×C₄	1	1	yes	characteristic, complemented
72	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic, complemented
73	C₃×#689	C₂×C₄	1	2	yes	complemented
74	C₃×#688	C₂×C₄	1	2	yes	complemented
75	C₃×#687	C₂×C₄	1	2	yes	complemented
76	C₃×#686	C₂×C₄	1	2	yes	complemented
77	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic, complemented
78	C₃×Dic₂₀	C₂×C₄	1	1	yes	characteristic, complemented
79	C₃×D₄₀	C₂×C₄	1	1	yes	characteristic, complemented
80	C₃×#677	C₂×C₄	1	2	yes	complemented
81	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic, complemented
82	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic, complemented
83	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
84	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
85	C₂×C₃×#135	—	5	1	no	—
86	Dic₃×#41	—	5	2	no	—
87	#5233	—	5	2	no	—
88	Dic₃×Q₁₆	—	5	2	no	—
89	Dic₃×SD₁₆	—	5	4	no	—
90	Dic₃×D₈	—	5	2	no	—
91	#5106	—	5	1	no	—
92	C₂×C₈×Dic₃	—	5	1	no	—
93	D₅×#41	C₂×D₃	1	2	yes	—
94	Q₈×AGL(1,𝔽₅)	—	3	4	no	—
95	D₄×AGL(1,𝔽₅)	—	3	4	no	—
96	D₄×AGL(1,𝔽₅)	—	6	4	no	—
97	#4130	—	3	2	no	—
98	#4130	—	6	2	no	—
99	C₂×C₄×AGL(1,𝔽₅)	—	3	1	no	—
100	C₂×C₄×AGL(1,𝔽₅)	—	6	2	no	—
101	C₂×D₅⋊C₈	—	3	1	no	—
102	C₅×#135	Dic₃	1	1	yes	characteristic, complemented
103	#4096	Dic₃	1	2	yes	complemented
104	#4065	Dic₃	1	1	yes	characteristic, complemented
105	D₅×Q₁₆	C₂×D₃	1	1	yes	characteristic
106	#4062	Dic₃	1	2	yes	complemented
107	D₅×SD₁₆	C₂×D₃	1	2	yes	—
108	#4058	Dic₃	1	1	yes	characteristic, complemented
109	D₅×D₈	C₂×D₃	1	1	yes	characteristic
110	#4050	Dic₃	1	1	yes	characteristic, complemented
111	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
112	#4009	—	3	4	no	—
113	#4007	—	3	4	no	—
114	#3994	—	3	2	no	—
115	#3993	—	3	2	no	—
116	C₈×AGL(1,𝔽₅)	—	3	2	no	—
117	C₈×AGL(1,𝔽₅)	—	3	2	no	—
118	C₄×#135	—	15	1	no	Sylow
119	C₂²×C₃×D₅	—	2	2	no	—
120	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
121	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
122	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
123	C₂×ΓL(1,𝔽₁₆)	—	4	8	no	—
124	C₂×ΓL(1,𝔽₁₆)	C₂×D₄	1	2	yes	—
125	C₂×ΓL(1,𝔽₁₆)	C₂×D₄	1	2	yes	—
126	C₃×C₅×Q₈	C₂²×C₄	1	2	yes	—
127	C₃×C₅×D₄	—	2	2	no	—
128	C₃×C₅×D₄	C₂²×C₄	1	2	yes	—
129	C₂×C₃×C₄×C₅	—	2	2	no	—
130	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
131	C₃×C₂²⋊D₅	—	2	2	no	—
132	C₃×C₂²⋊D₅	—	2	2	no	—
133	C₃×C₂²⋊D₅	—	2	2	no	—
134	C₂×C₃×Dic₅	—	2	2	no	—
135	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
136	C₃×D₂₀	—	2	2	no	—
137	C₃×D₂₀	C₂²×C₄	1	2	yes	—
138	C₃×C₄×D₅	—	2	2	no	—
139	C₃×C₄×D₅	—	2	2	no	—
140	C₃×C₄×D₅	—	2	2	no	—
141	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
142	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
143	C₃×C₄×D₅	#41	1	1	yes	characteristic
144	C₃×C₄×D₅	C₂⁴	1	1	yes	characteristic
145	C₃×Dic₁₀	—	2	2	no	—
146	C₃×Dic₁₀	C₂²×C₄	1	2	yes	—
147	C₃⋊#170	—	2	2	no	—
148	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
149	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
150	C₃×#168	C₂²×C₄	1	1	yes	characteristic
151	C₃×#168	C₂²×C₄	1	1	yes	characteristic
152	C₂×C₃×#41	—	5	2	no	—
153	C₃×#135	—	5	2	no	—
154	C₃×#135	—	5	1	no	—
155	C₃×#135	—	5	2	no	—
156	C₃×#135	—	5	1	no	—
157	C₃×#135	—	5	1	no	—
158	C₃×#135	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
159	C₂×C₃×Q₁₆	—	5	1	no	—
160	C₂×C₃×SD₁₆	—	5	2	no	—
161	C₂×C₃×D₈	—	5	1	no	—
162	C₂²×C₃×C₈	—	5	1	no	—
163	Q₈×Dic₃	—	5	4	no	—
164	D₄×Dic₃	—	5	4	no	—
165	D₄×Dic₃	—	10	4	no	—
166	#926	—	5	2	no	—
167	#926	—	10	2	no	—
168	C₂×C₄×Dic₃	—	5	1	no	—
169	C₂×C₄×Dic₃	—	10	2	no	—
170	C₂²×C₃⋊C₈	—	5	1	no	—
171	#835	—	5	4	no	—
172	#832	—	5	4	no	—
173	#818	—	5	2	no	—
174	#817	—	5	2	no	—
175	C₈×Dic₃	—	5	2	no	—
176	C₈×Dic₃	—	5	2	no	—
177	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
178	C₂²×AGL(1,𝔽₅)	—	6	4	no	—
179	C₅×#41	C₂×Dic₃	1	2	yes	complemented
180	#713	C₂×Dic₃	1	2	yes	complemented
181	Q₈×D₅	C₂²×D₃	1	2	yes	—
182	#711	—	2	2	no	—
183	#711	C₂×Dic₃	1	2	yes	complemented
184	D₄×D₅	—	2	2	no	—
185	D₄×D₅	C₂²×D₃	1	2	yes	—
186	#709	—	2	2	no	—
187	#709	C₂×Dic₃	1	2	yes	complemented
188	C₂×C₄×D₅	—	2	2	no	—
189	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
190	C₂²⋊AGL(1,𝔽₅)	—	6	4	no	—
191	C₂²⋊AGL(1,𝔽₅)	—	6	2	no	—
192	C₂×#170	—	6	1	no	—
193	C₄⋊AGL(1,𝔽₅)	—	3	4	no	—
194	C₄⋊AGL(1,𝔽₅)	—	6	4	no	—
195	C₄×AGL(1,𝔽₅)	—	3	1	no	—
196	C₄×AGL(1,𝔽₅)	—	3	1	no	—
197	C₄×AGL(1,𝔽₅)	—	3	2	no	—
198	C₄×AGL(1,𝔽₅)	—	6	4	no	—
199	C₄×AGL(1,𝔽₅)	—	6	2	no	—
200	D₅⋊C₈	—	3	2	no	—
201	D₅⋊C₈	—	6	1	no	—
202	C₅×Q₁₆	C₂×Dic₃	1	1	yes	characteristic, complemented
203	C₅×SD₁₆	C₂×Dic₃	1	2	yes	complemented
204	C₅×D₈	C₂×Dic₃	1	1	yes	characteristic, complemented
205	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic, complemented
206	#689	C₂×Dic₃	1	2	yes	complemented
207	#688	C₂×Dic₃	1	2	yes	complemented
208	#687	C₂×Dic₃	1	2	yes	complemented
209	#686	C₂×Dic₃	1	2	yes	complemented
210	C₂×#168	C₂×Dic₃	1	1	yes	characteristic, complemented
211	Dic₂₀	C₂×Dic₃	1	1	yes	characteristic, complemented
212	D₄₀	C₂×Dic₃	1	1	yes	characteristic, complemented
213	#677	C₂×Dic₃	1	2	yes	complemented
214	C₈×D₅	C₂×Dic₃	1	1	yes	characteristic, complemented
215	C₈×D₅	C₂×Dic₃	1	1	yes	characteristic, complemented
216	C₈×D₅	C₂²×D₃	1	1	yes	characteristic
217	C₈×D₅	C₂²×D₃	1	1	yes	characteristic
218	C₂×#135	—	5	1	no	—
219	C₄×#41	—	15	2	no	—
220	C₄×Q₁₆	—	15	2	no	—
221	C₄×SD₁₆	—	15	4	no	—
222	C₄×D₈	—	15	2	no	—
223	#427	—	15	1	no	—
224	#416	—	15	2	no	—
225	C₂×C₄×C₈	—	15	1	no	—
226	C₂²×C₃×C₅	—	2	2	no	—
227	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
228	C₂×C₃×D₅	—	2	2	no	—
229	C₂×C₃×D₅	—	2	2	no	—
230	C₂×C₃×D₅	—	4	4	no	—
231	C₂×C₃×D₅	C₄×D₄	1	1	yes	characteristic
232	C₂×C₃×D₅	C₂²×D₄	1	1	yes	characteristic
233	ΓL(1,𝔽₁₆)	—	2	2	no	—
234	ΓL(1,𝔽₁₆)	—	4	4	no	—
235	ΓL(1,𝔽₁₆)	#135	1	4	yes	direct product factor
236	C₃×C₄×C₅	—	2	2	no	—
237	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
238	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
239	C₃×Dic₅	—	2	2	no	—
240	C₃×Dic₅	C₄×D₄	1	1	yes	characteristic
241	C₃×Dic₅	C₂×#41	1	1	yes	characteristic
242	C₃×#41	—	5	2	no	—
243	C₃×#41	—	5	2	no	—
244	C₃×#41	—	5	2	no	—
245	C₃×#41	—	10	2	no	—
246	C₃×#41	—	10	2	no	—
247	C₃×#41	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
248	C₂×C₃×Q₈	—	5	2	no	—
249	C₂×C₃×D₄	—	5	2	no	—
250	C₂×C₃×D₄	—	10	2	no	—
251	C₂²×C₃×C₄	—	5	1	no	—
252	C₂²×C₃×C₄	—	10	2	no	—
253	C₂²×Dic₃	—	5	2	no	—
254	C₂²×Dic₃	—	10	4	no	—
255	C₃×Q₁₆	—	5	2	no	—
256	C₃×Q₁₆	—	5	1	no	—
257	C₃×Q₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
258	C₃×SD₁₆	—	5	2	no	—
259	C₃×SD₁₆	—	5	2	no	—
260	C₃×SD₁₆	—	5	2	no	—
261	C₃×SD₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
262	C₃×D₈	—	5	2	no	—
263	C₃×D₈	—	5	1	no	—
264	C₃×D₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
265	C₂×C₃×C₈	—	5	1	no	—
266	C₂×C₃×C₈	—	5	1	no	—
267	C₂×C₃×C₈	—	5	1	no	—
268	C₂×C₃×C₈	—	5	1	no	—
269	C₂×C₃×C₈	—	5	1	no	—
270	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
271	#217	—	10	4	no	—
272	#217	—	10	2	no	—
273	C₄⋊Dic₃	—	5	4	no	—
274	C₄⋊Dic₃	—	10	4	no	—
275	C₄×Dic₃	—	5	2	no	—
276	C₄×Dic₃	—	5	1	no	—
277	C₄×Dic₃	—	5	1	no	—
278	C₄×Dic₃	—	10	4	no	—
279	C₄×Dic₃	—	10	2	no	—
280	C₂×C₃⋊C₈	—	5	2	no	—
281	C₂×C₃⋊C₈	—	10	1	no	—
282	C₂×C₃⋊C₈	—	10	1	no	—
283	C₂²×D₅	—	2	2	no	—
284	C₂²×D₅	D₃×D₄	1	1	yes	characteristic
285	C₂×AGL(1,𝔽₅)	—	3	2	no	—
286	C₂×AGL(1,𝔽₅)	—	3	2	no	—
287	C₂×AGL(1,𝔽₅)	—	6	4	no	—
288	C₂×AGL(1,𝔽₅)	—	6	4	no	—
289	C₂×AGL(1,𝔽₅)	—	12	8	no	—
290	C₅×Q₈	C₂²×Dic₃	1	2	yes	—
291	C₅×D₄	—	2	2	no	—
292	C₅×D₄	C₂²×Dic₃	1	2	yes	—
293	C₂×C₄×C₅	—	2	2	no	—
294	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
295	C₂²⋊D₅	—	2	2	no	—
296	C₂²⋊D₅	—	2	2	no	—
297	C₂²⋊D₅	—	2	2	no	—
298	C₂×Dic₅	—	2	2	no	—
299	C₂×Dic₅	#237	1	1	yes	characteristic
300	D₂₀	—	2	2	no	—
301	D₂₀	C₂²×Dic₃	1	2	yes	—
302	C₄×D₅	—	2	2	no	—
303	C₄×D₅	—	2	2	no	—
304	C₄×D₅	—	2	2	no	—
305	C₄×D₅	D₃×D₄	1	1	yes	characteristic
306	C₄×D₅	#237	1	1	yes	characteristic
307	C₄×D₅	C₂²×Dic₃	1	1	yes	characteristic
308	C₄×D₅	C₂³×D₃	1	1	yes	characteristic
309	Dic₁₀	—	2	2	no	—
310	Dic₁₀	C₂²×Dic₃	1	2	yes	—
311	#170	—	6	2	no	—
312	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
313	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
314	#168	C₂²×Dic₃	1	1	yes	characteristic
315	#168	C₂²×Dic₃	1	1	yes	characteristic
316	C₂×#41	—	5	2	no	—
317	#135	—	5	2	no	—
318	#135	—	5	1	no	—
319	#135	—	5	2	no	—
320	#135	—	5	1	no	—
321	#135	—	5	1	no	—
322	#135	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor
323	C₂×Q₁₆	—	5	1	no	—
324	C₂×SD₁₆	—	5	2	no	—
325	C₂×D₈	—	5	1	no	—
326	C₂²×C₈	—	5	1	no	—
327	C₂²×C₈	—	15	1	no	—
328	C₄×Q₈	—	15	4	no	—
329	C₄×D₄	—	15	4	no	—
330	C₄×D₄	—	30	4	no	—
331	#117	—	15	2	no	—
332	#117	—	30	2	no	—
333	C₂×C₄²	—	15	1	no	—
334	C₂×C₄²	—	30	2	no	—
335	#107	—	15	2	no	—
336	#106	—	15	2	no	—
337	#103	—	15	4	no	—
338	#102	—	15	4	no	—
339	C₄×C₈	—	15	2	no	—
340	C₄×C₈	—	15	2	no	—
341	C₂×C₃×C₅	—	2	1	no	—
342	C₂×C₃×C₅	—	4	2	no	—
343	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
344	C₃×D₅	—	2	1	no	—
345	C₃×D₅	—	4	2	no	—
346	C₃×D₅	C₂×#135	1	2	yes	—
347	C₂³×C₃	—	5	1	no	—
348	C₂³×C₃	—	10	2	no	—
349	C₃×Q₈	—	5	2	no	—
350	C₃×Q₈	—	10	2	no	—
351	C₃×Q₈	C₂²×AGL(1,𝔽₅)	1	2	yes	—
352	C₃×D₄	—	2	2	no	—
353	C₃×D₄	—	5	2	no	—
354	C₃×D₄	—	10	2	no	—
355	C₃×D₄	—	10	2	no	—
356	C₃×D₄	—	10	2	no	—
357	C₃×D₄	—	10	2	no	—
358	C₃×D₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
359	C₂×C₃×C₄	—	2	2	no	—
360	C₂×C₃×C₄	—	5	1	no	—
361	C₂×C₃×C₄	—	5	1	no	—
362	C₂×C₃×C₄	—	5	1	no	—
363	C₂×C₃×C₄	—	5	1	no	—
364	C₂×C₃×C₄	—	5	1	no	—
365	C₂×C₃×C₄	—	10	2	no	—
366	C₂×C₃×C₄	—	10	2	no	—
367	C₂×C₃×C₄	—	10	2	no	—
368	C₂×C₃×C₄	—	10	2	no	—
369	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
370	C₂×Dic₃	—	5	2	no	—
371	C₂×Dic₃	—	5	2	no	—
372	C₂×Dic₃	—	10	4	no	—
373	C₂×Dic₃	—	10	4	no	—
374	C₂×Dic₃	—	20	8	no	—
375	C₃×C₈	—	5	1	no	—
376	C₃×C₈	—	5	1	no	—
377	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
378	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
379	C₃⋊C₈	—	10	2	no	—
380	C₂²×C₅	—	2	2	no	—
381	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
382	C₂×D₅	—	2	2	no	—
383	C₂×D₅	—	2	2	no	—
384	C₂×D₅	—	4	4	no	—
385	C₂×D₅	D₄×Dic₃	1	1	yes	characteristic
386	C₂×D₅	C₂×D₃×D₄	1	1	yes	characteristic
387	AGL(1,𝔽₅)	—	3	4	no	—
388	AGL(1,𝔽₅)	—	6	2	no	—
389	AGL(1,𝔽₅)	—	12	4	no	—
390	C₄×C₅	—	2	2	no	—
391	C₄×C₅	D₄×Dic₃	1	1	yes	characteristic
392	C₄×C₅	C₂³×Dic₃	1	1	yes	characteristic
393	Dic₅	—	2	2	no	—
394	Dic₅	D₄×Dic₃	1	1	yes	characteristic
395	Dic₅	C₂×#237	1	1	yes	characteristic
396	#41	—	5	2	no	—
397	#41	—	5	2	no	—
398	#41	—	5	2	no	—
399	#41	—	10	2	no	—
400	#41	—	10	2	no	—
401	#41	C₂×ΓL(1,𝔽₁₆)	1	2	yes	complemented
402	C₂×Q₈	—	5	2	no	—
403	C₂×D₄	—	5	2	no	—
404	C₂×D₄	—	10	2	no	—
405	C₂²×C₄	—	5	1	no	—
406	C₂²×C₄	—	10	2	no	—
407	C₂²×C₄	—	15	2	no	—
408	C₂²×C₄	—	30	4	no	—
409	Q₁₆	—	5	2	no	—
410	Q₁₆	—	5	1	no	—
411	Q₁₆	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
412	SD₁₆	—	5	2	no	—
413	SD₁₆	—	5	2	no	—
414	SD₁₆	—	5	2	no	—
415	SD₁₆	C₂×ΓL(1,𝔽₁₆)	1	2	yes	complemented
416	D₈	—	5	2	no	—
417	D₈	—	5	1	no	—
418	D₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
419	C₂×C₈	—	5	1	no	—
420	C₂×C₈	—	5	1	no	—
421	C₂×C₈	—	5	1	no	—
422	C₂×C₈	—	5	1	no	—
423	C₂×C₈	—	5	1	no	—
424	C₂×C₈	—	15	2	no	—
425	C₂×C₈	—	30	1	no	—
426	C₂×C₈	—	30	1	no	—
427	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
428	C₄⋊C₄	—	15	4	no	—
429	C₄⋊C₄	—	30	4	no	—
430	C₂²⋊C₄	—	30	4	no	—
431	C₂²⋊C₄	—	30	2	no	—
432	C₄²	—	15	2	no	—
433	C₄²	—	15	1	no	—
434	C₄²	—	15	1	no	—
435	C₄²	—	30	2	no	—
436	C₄²	—	30	4	no	—
437	C₃×C₅	C₄×#135	1	1	yes	characteristic, complemented, Hall
438	C₂²×C₃	—	2	2	no	—
439	C₂²×C₃	—	5	1	no	—
440	C₂²×C₃	—	5	1	no	—
441	C₂²×C₃	—	10	2	no	—
442	C₂²×C₃	—	10	2	no	—
443	C₂²×C₃	—	20	4	no	—
444	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
445	C₃×C₄	—	2	2	no	—
446	C₃×C₄	—	5	1	no	—
447	C₃×C₄	—	5	1	no	—
448	C₃×C₄	—	10	2	no	—
449	C₃×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
450	C₃×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
451	Dic₃	—	5	4	no	—
452	Dic₃	—	10	2	no	—
453	Dic₃	—	20	4	no	—
454	C₂×C₅	—	2	1	no	—
455	C₂×C₅	—	4	2	no	—
456	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
457	D₅	—	2	1	no	—
458	D₅	—	4	2	no	—
459	D₅	D₃×#135	1	2	yes	—
460	C₂³	—	5	1	no	—
461	C₂³	—	10	2	no	—
462	Q₈	—	5	2	no	—
463	Q₈	—	10	2	no	—
464	Q₈	C₂²×ΓL(1,𝔽₁₆)	1	2	yes	—
465	D₄	—	2	2	no	—
466	D₄	—	5	2	no	—
467	D₄	—	10	2	no	—
468	D₄	—	10	2	no	—
469	D₄	—	10	2	no	—
470	D₄	—	10	2	no	—
471	D₄	C₂²×ΓL(1,𝔽₁₆)	1	2	yes	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	5	1	no	—
474	C₂×C₄	—	5	1	no	—
475	C₂×C₄	—	5	1	no	—
476	C₂×C₄	—	5	1	no	—
477	C₂×C₄	—	5	1	no	—
478	C₂×C₄	—	10	2	no	—
479	C₂×C₄	—	10	2	no	—
480	C₂×C₄	—	10	2	no	—
481	C₂×C₄	—	10	2	no	—
482	C₂×C₄	—	15	2	no	—
483	C₂×C₄	—	15	2	no	—
484	C₂×C₄	—	30	4	no	—
485	C₂×C₄	—	30	4	no	—
486	C₂×C₄	—	60	8	no	—
487	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
488	C₈	—	5	1	no	—
489	C₈	—	5	1	no	—
490	C₈	—	30	2	no	—
491	C₈	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
492	C₈	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
493	C₂×C₃	—	2	1	no	—
494	C₂×C₃	—	4	2	no	—
495	C₂×C₃	—	5	2	no	—
496	C₂×C₃	—	10	1	no	—
497	C₂×C₃	—	20	2	no	—
498	C₂×C₃	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
499	C₅	Dic₃×#135	1	1	yes	characteristic, complemented, Sylow
500	C₂²	—	2	2	no	—
501	C₂²	—	5	1	no	—
502	C₂²	—	5	1	no	—
503	C₂²	—	10	2	no	—
504	C₂²	—	10	2	no	—
505	C₂²	—	20	4	no	—
506	C₂²	D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
507	C₄	—	2	2	no	—
508	C₄	—	5	1	no	—
509	C₄	—	5	1	no	—
510	C₄	—	10	2	no	—
511	C₄	—	15	4	no	—
512	C₄	—	30	2	no	—
513	C₄	—	60	4	no	—
514	C₄	D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
515	C₄	C₂³×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
516	C₃	AGL(1,𝔽₅)×#135	1	1	yes	characteristic, complemented, Sylow
517	C₂	—	2	1	no	—
518	C₂	—	4	2	no	—
519	C₂	—	5	2	no	—
520	C₂	—	10	1	no	—
521	C₂	—	20	2	no	—
522	C₂	C₂×D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×ΓL(1,𝔽₁₆)
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₄	C₂³×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×#135	C₄
socle	C₂×C₃×C₅	C₂×C₄×D₄
prime radical	C₃×D₅×#135	C₂
cosocle	C₃×C₄×D₅	C₂⁴
hypercenter	#135	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₄×#135
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #41×ΓL(1,𝔽₁₆)	C₂	semidirect
3. #11752453	C₂	semidirect
4. Q₁₆×ΓL(1,𝔽₁₆)	C₂	semidirect
5. SD₁₆×ΓL(1,𝔽₁₆)	C₂	semidirect
6. D₈×ΓL(1,𝔽₁₆)	C₂	semidirect
7. #11752380	C₂	semidirect
8. C₂×C₈×ΓL(1,𝔽₁₆)	C₂	semidirect
19. C₃×C₅×#135	C₄	semidirect
20. C₃×#4096	C₄	semidirect
21. C₃×#4065	C₄	semidirect
23. C₃×#4062	C₄	semidirect
25. C₃×#4058	C₄	semidirect
27. C₃×#4050	C₄	semidirect
46. C₃×C₅×#41	C₂×C₄	semidirect
47. C₃×#713	C₂×C₄	semidirect
50. C₃×#711	C₂×C₄	semidirect
54. C₃×#709	C₂×C₄	semidirect
69. C₃×C₅×Q₁₆	C₂×C₄	semidirect
70. C₃×C₅×SD₁₆	C₂×C₄	semidirect
71. C₃×C₅×D₈	C₂×C₄	semidirect
72. C₂×C₃×C₅×C₈	C₂×C₄	semidirect
73. C₃×#689	C₂×C₄	semidirect
74. C₃×#688	C₂×C₄	semidirect
75. C₃×#687	C₂×C₄	semidirect
76. C₃×#686	C₂×C₄	semidirect
77. C₂×C₃×#168	C₂×C₄	semidirect
78. C₃×Dic₂₀	C₂×C₄	semidirect
79. C₃×D₄₀	C₂×C₄	semidirect
80. C₃×#677	C₂×C₄	semidirect
81. C₃×C₈×D₅	C₂×C₄	semidirect
82. C₃×C₈×D₅	C₂×C₄	semidirect
102. C₅×#135	Dic₃	semidirect
103. #4096	Dic₃	semidirect
104. #4065	Dic₃	semidirect
106. #4062	Dic₃	semidirect
108. #4058	Dic₃	semidirect
110. #4050	Dic₃	semidirect
158. C₃×#135	AGL(1,𝔽₅)	semidirect
179. C₅×#41	C₂×Dic₃	semidirect
180. #713	C₂×Dic₃	semidirect
183. #711	C₂×Dic₃	semidirect
187. #709	C₂×Dic₃	semidirect
202. C₅×Q₁₆	C₂×Dic₃	semidirect
203. C₅×SD₁₆	C₂×Dic₃	semidirect
204. C₅×D₈	C₂×Dic₃	semidirect
205. C₂×C₅×C₈	C₂×Dic₃	semidirect
206. #689	C₂×Dic₃	semidirect
207. #688	C₂×Dic₃	semidirect
208. #687	C₂×Dic₃	semidirect
209. #686	C₂×Dic₃	semidirect
210. C₂×#168	C₂×Dic₃	semidirect
211. Dic₂₀	C₂×Dic₃	semidirect
212. D₄₀	C₂×Dic₃	semidirect
213. #677	C₂×Dic₃	semidirect
214. C₈×D₅	C₂×Dic₃	semidirect
215. C₈×D₅	C₂×Dic₃	semidirect
235. ΓL(1,𝔽₁₆)	#135	direct (N × H)
247. C₃×#41	C₂×AGL(1,𝔽₅)	semidirect
257. C₃×Q₁₆	C₂×AGL(1,𝔽₅)	semidirect
261. C₃×SD₁₆	C₂×AGL(1,𝔽₅)	semidirect
264. C₃×D₈	C₂×AGL(1,𝔽₅)	semidirect
270. C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	semidirect
322. #135	ΓL(1,𝔽₁₆)	direct (N × H)
401. #41	C₂×ΓL(1,𝔽₁₆)	semidirect
411. Q₁₆	C₂×ΓL(1,𝔽₁₆)	semidirect
415. SD₁₆	C₂×ΓL(1,𝔽₁₆)	semidirect
418. D₈	C₂×ΓL(1,𝔽₁₆)	semidirect
427. C₂×C₈	C₂×ΓL(1,𝔽₁₆)	semidirect
437. C₃×C₅	C₄×#135	semidirect
499. C₅	Dic₃×#135	semidirect
516. C₃	AGL(1,𝔽₅)×#135	semidirect

Maximal subgroups

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

subgroup	index
#41×ΓL(1,𝔽₁₆)	2
C₃×D₅×#135	2
#11752453	2
Q₁₆×ΓL(1,𝔽₁₆)	2
SD₁₆×ΓL(1,𝔽₁₆)	2
D₈×ΓL(1,𝔽₁₆)	2
#11752380	2
C₂×C₈×ΓL(1,𝔽₁₆)	2
AGL(1,𝔽₅)×#135	3
Dic₃×#135	5

Subgroups — #135×ΓL(1,𝔽16) #14416225

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #135×ΓL(1,𝔽₁₆) #14416225