Subgroups — D₄×C₄⋊ΓL(1,𝔽₁₆) #14327017

Order 1920.

1552 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×ΓL(1,𝔽₁₆)	C₂	1	2	yes	maximal
2	C₂²×C₄⋊ΓL(1,𝔽₁₆)	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×D₄×D₅	C₂	1	1	yes	maximal, characteristic
4	#11752424	C₂	1	2	yes	maximal, complemented
5	#11752371	C₂	1	2	yes	maximal, complemented
6	#11752358	C₂	1	4	yes	maximal, complemented
7	#11752347	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊ΓL(1,𝔽₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₄⋊AGL(1,𝔽₅)	—	3	1	no	maximal, Hall
10	C₂³×ΓL(1,𝔽₁₆)	C₂²	1	4	yes	—
11	C₂×C₃×D₄×D₅	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	—
13	C₂×#6763	C₂²	1	4	yes	—
14	D₄×ΓL(1,𝔽₁₆)	—	2	8	no	—
15	C₂×C₄⋊ΓL(1,𝔽₁₆)	—	2	8	no	—
16	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
18	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
19	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	4	yes	—
20	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
21	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
22	C₂×C₄×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
23	C₃×C₄×C₅×D₄	C₄	1	1	yes	characteristic, complemented
24	C₃×D₄×Dic₅	C₄	1	1	yes	characteristic, complemented
25	C₃×C₄×C₂²⋊D₅	C₄	1	2	yes	complemented
26	C₃×#4039	C₄	1	1	yes	characteristic, complemented
27	C₃×D₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
28	C₃×#4027	C₄	1	2	yes	complemented
29	C₃×D₅×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₃×C₄×D₂₀	C₄	1	1	yes	characteristic, complemented
31	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
32	#91235	C₂²	1	2	yes	—
33	#91235	C₂²	1	4	yes	—
34	D₄×C₄⋊Dic₃	—	5	1	no	maximal, Hall
35	C₂×D₄×AGL(1,𝔽₅)	—	3	2	no	—
36	C₂²×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
37	C₄×D₄×D₅	D₃	1	1	yes	characteristic
38	#65765	—	3	2	no	—
39	#65712	—	3	2	no	—
40	#65699	—	3	4	no	—
41	#65688	—	3	1	no	—
42	C₄×C₄⋊AGL(1,𝔽₅)	—	3	1	no	—
43	C₂³×C₃×D₅	C₂³	1	2	yes	—
44	C₂²×ΓL(1,𝔽₁₆)	—	2	8	no	—
45	C₂²×ΓL(1,𝔽₁₆)	—	2	8	no	—
46	C₂²×ΓL(1,𝔽₁₆)	—	4	4	no	—
47	C₂²×ΓL(1,𝔽₁₆)	C₂³	1	2	yes	—
48	C₂²×ΓL(1,𝔽₁₆)	C₂³	1	4	yes	—
49	C₂²×ΓL(1,𝔽₁₆)	C₂³	1	2	yes	—
50	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	complemented
52	C₂×C₃×C₂²⋊D₅	C₂×C₄	1	2	yes	—
53	C₂²×C₃×Dic₅	C₂×C₄	1	2	yes	complemented
54	C₃×D₄×D₅	D₄	1	4	yes	—
55	C₃×D₄×D₅	Q₈	1	4	yes	—
56	C₂×C₃×D₂₀	C₂×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄×D₅	—	2	4	no	—
58	C₂×C₃×C₄×D₅	—	2	4	no	—
59	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	complemented
60	C₂×C₃×C₄×D₅	C₂×C₄	1	2	yes	complemented
61	C₂×C₃×C₄×D₅	C₂³	1	2	yes	—
62	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
65	#6763	—	2	8	no	—
66	C₄⋊ΓL(1,𝔽₁₆)	—	2	8	no	—
67	C₄⋊ΓL(1,𝔽₁₆)	—	2	4	no	—
68	C₄⋊ΓL(1,𝔽₁₆)	—	2	4	no	—
69	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
70	C₄⋊ΓL(1,𝔽₁₆)	—	4	2	no	—
71	C₄⋊ΓL(1,𝔽₁₆)	D₄	1	4	yes	direct product factor
72	C₄×ΓL(1,𝔽₁₆)	—	2	4	no	—
73	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic, complemented
74	C₃×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	complemented
75	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic, complemented
76	C₃×C₂²⋊Dic₅	C₂×C₄	1	2	yes	complemented
77	C₃×#685	C₂×C₄	1	2	yes	complemented
78	C₃×#685	C₂×C₄	1	2	yes	complemented
79	C₃×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic, complemented
80	C₃×#683	C₂×C₄	1	2	yes	complemented
81	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic, complemented
82	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic, complemented
83	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic, complemented
84	C₂×C₃×C₄×D₄	—	5	1	no	—
85	C₂×D₄×Dic₃	—	5	2	no	—
86	C₂²×C₄⋊Dic₃	—	5	2	no	—
87	#5212	—	5	2	no	—
88	#4984	—	5	2	no	—
89	#4950	—	5	4	no	—
90	#4935	—	5	1	no	—
91	C₄×C₄⋊Dic₃	—	5	1	no	—
92	C₂³×AGL(1,𝔽₅)	—	3	4	no	—
93	C₂×D₄×D₅	C₂×D₃	1	1	yes	characteristic
94	C₂²×C₄×D₅	C₂×D₃	1	2	yes	—
95	C₂×C₂²⋊AGL(1,𝔽₅)	—	3	4	no	—
96	D₄×AGL(1,𝔽₅)	—	6	8	no	—
97	C₂×C₄⋊AGL(1,𝔽₅)	—	3	1	no	—
98	C₂×C₄⋊AGL(1,𝔽₅)	—	3	1	no	—
99	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
100	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
101	C₂×C₄⋊AGL(1,𝔽₅)	—	3	4	no	—
102	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
103	C₂×C₄⋊AGL(1,𝔽₅)	—	6	8	no	—
104	C₂×C₄×AGL(1,𝔽₅)	—	3	2	no	—
105	C₄×C₅×D₄	Dic₃	1	1	yes	characteristic, complemented
106	D₄×Dic₅	Dic₃	1	1	yes	characteristic, complemented
107	C₄×C₂²⋊D₅	Dic₃	1	2	yes	complemented
108	#4039	Dic₃	1	1	yes	characteristic, complemented
109	D₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
110	#4027	Dic₃	1	2	yes	complemented
111	D₅×C₂²⋊C₄	C₂×D₃	1	2	yes	—
112	C₄×D₂₀	Dic₃	1	1	yes	characteristic, complemented
113	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
114	(C₂×AGL(1,𝔽₅))⋊C₄	—	3	2	no	—
115	(C₂×AGL(1,𝔽₅))⋊C₄	—	3	4	no	—
116	D₄×C₄⋊C₄	—	15	1	no	Sylow
117	C₂³×C₃×C₅	C₂²×C₄	1	2	yes	—
118	C₂²×C₃×D₅	—	2	2	no	—
119	C₂²×C₃×D₅	—	2	2	no	—
120	C₂²×C₃×D₅	—	2	2	no	—
121	C₂²×C₃×D₅	—	2	2	no	—
122	C₂²×C₃×D₅	C₂²×C₄	1	2	yes	—
123	C₂²×C₃×D₅	C₂×D₄	1	4	yes	—
124	C₂²×C₃×D₅	C₂×Q₈	1	4	yes	—
125	C₂²×C₃×D₅	C₂⁴	1	1	yes	characteristic
126	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
127	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
128	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
129	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
130	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
131	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
132	C₂×ΓL(1,𝔽₁₆)	—	2	4	no	—
133	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
134	C₂×ΓL(1,𝔽₁₆)	—	4	16	no	—
135	C₂×ΓL(1,𝔽₁₆)	C₂×D₄	1	4	yes	—
136	C₃×C₅×D₄	C₄⋊C₄	1	4	yes	complemented
137	C₂×C₃×C₄×C₅	—	2	4	no	—
138	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
139	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
140	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
142	C₃×C₂²⋊D₅	C₄⋊C₄	1	8	yes	complemented
143	C₂×C₃×Dic₅	—	2	4	no	—
144	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
145	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
146	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
147	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
148	C₃×D₂₀	C₄⋊C₄	1	4	yes	complemented
149	C₃×C₄×D₅	—	2	2	no	—
150	C₃×C₄×D₅	—	2	2	no	—
151	C₃×C₄×D₅	—	2	1	no	—
152	C₃×C₄×D₅	—	2	1	no	—
153	C₃×C₄×D₅	—	2	4	no	—
154	C₃×C₄×D₅	—	2	4	no	—
155	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
156	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
157	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
158	C₃×C₄×D₅	#41	1	2	yes	—
159	C₂²×C₃×D₄	—	5	1	no	—
160	C₂³×C₃×C₄	—	5	2	no	—
161	C₂³×Dic₃	—	5	4	no	—
162	C₃×C₄×D₄	—	5	1	no	—
163	C₃×C₄×D₄	—	5	2	no	—
164	C₃×C₄×D₄	—	5	1	no	—
165	C₃×C₄×D₄	—	5	2	no	—
166	C₃×C₄×D₄	—	5	1	no	—
167	C₃×C₄×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
168	C₂×C₃×C₄⋊C₄	—	5	1	no	—
169	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
170	C₂×C₃×C₄²	—	5	1	no	—
171	C₂×#217	—	5	4	no	—
172	D₄×Dic₃	—	10	8	no	—
173	C₂×C₄⋊Dic₃	—	5	1	no	—
174	C₂×C₄⋊Dic₃	—	5	1	no	—
175	C₂×C₄⋊Dic₃	—	5	2	no	—
176	C₂×C₄⋊Dic₃	—	5	2	no	—
177	C₂×C₄⋊Dic₃	—	5	4	no	—
178	C₂×C₄⋊Dic₃	—	5	2	no	—
179	C₂×C₄⋊Dic₃	—	10	8	no	—
180	C₂×C₄×Dic₃	—	5	2	no	—
181	C₃⋊#95	—	5	2	no	—
182	C₃⋊#95	—	5	4	no	—
183	C₂³×D₅	C₂²×D₃	1	2	yes	—
184	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
185	C₂²×AGL(1,𝔽₅)	—	3	2	no	—
186	C₂²×AGL(1,𝔽₅)	—	3	4	no	—
187	C₂²×AGL(1,𝔽₅)	—	6	8	no	—
188	C₂²×AGL(1,𝔽₅)	—	6	8	no	—
189	C₂²×AGL(1,𝔽₅)	—	12	4	no	—
190	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
191	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	complemented
192	C₂×C₂²⋊D₅	C₂×Dic₃	1	2	yes	—
193	C₂²×Dic₅	C₂×Dic₃	1	2	yes	complemented
194	D₄×D₅	Dic₆	1	4	yes	—
195	D₄×D₅	D₁₂	1	4	yes	—
196	C₂×D₂₀	C₂×Dic₃	1	1	yes	characteristic
197	C₂×C₄×D₅	—	2	4	no	—
198	C₂×C₄×D₅	—	2	4	no	—
199	C₂×C₄×D₅	C₂×Dic₃	1	2	yes	complemented
200	C₂×C₄×D₅	C₂×Dic₃	1	2	yes	complemented
201	C₂×C₄×D₅	C₂²×D₃	1	2	yes	—
202	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
203	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
204	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
205	C₂²⋊AGL(1,𝔽₅)	—	6	8	no	—
206	C₄⋊AGL(1,𝔽₅)	—	3	4	no	—
207	C₄⋊AGL(1,𝔽₅)	—	6	8	no	—
208	C₄⋊AGL(1,𝔽₅)	—	6	4	no	—
209	C₄⋊AGL(1,𝔽₅)	—	6	4	no	—
210	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
211	C₄⋊AGL(1,𝔽₅)	—	12	2	no	—
212	C₄×AGL(1,𝔽₅)	—	6	4	no	—
213	C₅×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic, complemented
214	C₅×C₂²⋊C₄	C₂×Dic₃	1	2	yes	complemented
215	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic, complemented
216	C₂²⋊Dic₅	C₂×Dic₃	1	2	yes	complemented
217	#685	C₂×Dic₃	1	2	yes	complemented
218	#685	C₂×Dic₃	1	2	yes	complemented
219	C₄⋊Dic₅	C₂×Dic₃	1	1	yes	characteristic, complemented
220	#683	C₂×Dic₃	1	2	yes	complemented
221	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic, complemented
222	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic, complemented
223	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic, complemented
224	C₂×C₄×D₄	—	5	1	no	—
225	C₂×C₄×D₄	—	15	2	no	—
226	C₂²×C₄⋊C₄	—	15	2	no	—
227	#389	—	15	2	no	—
228	#385	—	15	4	no	—
229	#384	—	15	1	no	—
230	#380	—	15	2	no	—
231	C₄×C₄⋊C₄	—	15	1	no	—
232	C₂²×C₃×C₅	—	2	2	no	—
233	C₂²×C₃×C₅	—	2	2	no	—
234	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	4	yes	complemented
235	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
236	C₂×C₃×D₅	—	2	2	no	—
237	C₂×C₃×D₅	—	2	2	no	—
238	C₂×C₃×D₅	—	2	4	no	—
239	C₂×C₃×D₅	—	2	4	no	—
240	C₂×C₃×D₅	—	2	4	no	—
241	C₂×C₃×D₅	—	2	4	no	—
242	C₂×C₃×D₅	C₂×C₄⋊C₄	1	4	yes	complemented
243	C₂×C₃×D₅	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₃×D₅	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₃×D₅	C₂²×Q₈	1	1	yes	characteristic
246	C₂×C₃×D₅	C₂×#41	1	1	yes	characteristic
247	C₂×C₃×D₅	H(5,𝔽₂)	1	1	yes	characteristic
248	C₂×C₃×D₅	2₋¹⁺⁴	1	1	yes	characteristic
249	ΓL(1,𝔽₁₆)	—	2	4	no	—
250	ΓL(1,𝔽₁₆)	—	4	4	no	—
251	ΓL(1,𝔽₁₆)	—	4	2	no	—
252	C₃×C₄×C₅	—	2	2	no	—
253	C₃×C₄×C₅	—	2	1	no	—
254	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	complemented
255	C₃×C₄×C₅	C₄×D₄	1	2	yes	complemented
256	C₃×Dic₅	—	2	2	no	—
257	C₃×Dic₅	—	2	1	no	—
258	C₃×Dic₅	C₂×C₄⋊C₄	1	2	yes	complemented
259	C₃×Dic₅	C₄×D₄	1	2	yes	complemented
260	C₂⁴×C₃	—	5	2	no	—
261	C₂×C₃×D₄	—	5	2	no	—
262	C₂×C₃×D₄	—	5	4	no	—
263	C₂×C₃×D₄	—	5	4	no	—
264	C₂×C₃×D₄	—	5	1	no	—
265	C₂×C₃×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
266	C₂²×C₃×C₄	—	5	1	no	—
267	C₂²×C₃×C₄	—	5	1	no	—
268	C₂²×C₃×C₄	—	5	2	no	—
269	C₂²×C₃×C₄	—	5	1	no	—
270	C₂²×C₃×C₄	—	5	2	no	—
271	C₂²×C₃×C₄	—	5	2	no	—
272	C₂²×C₃×C₄	—	5	2	no	—
273	C₂²×C₃×C₄	—	10	4	no	—
274	C₂²×C₃×C₄	—	10	4	no	—
275	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
276	C₂²×Dic₃	—	5	2	no	—
277	C₂²×Dic₃	—	5	2	no	—
278	C₂²×Dic₃	—	5	4	no	—
279	C₂²×Dic₃	—	10	8	no	—
280	C₂²×Dic₃	—	10	8	no	—
281	C₂²×Dic₃	—	20	4	no	—
282	C₃×C₄⋊C₄	—	5	2	no	—
283	C₃×C₄⋊C₄	—	5	1	no	—
284	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
285	C₃×C₂²⋊C₄	—	5	2	no	—
286	C₃×C₂²⋊C₄	—	5	2	no	—
287	C₃×C₂²⋊C₄	—	5	2	no	—
288	C₃×C₂²⋊C₄	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
289	C₃×C₄²	—	5	1	no	—
290	C₃×C₄²	—	5	1	no	—
291	C₃×C₄²	—	5	1	no	—
292	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
293	#217	—	10	8	no	—
294	C₄⋊Dic₃	—	5	4	no	—
295	C₄⋊Dic₃	—	10	8	no	—
296	C₄⋊Dic₃	—	10	2	no	—
297	C₄⋊Dic₃	—	10	4	no	—
298	C₄⋊Dic₃	—	10	4	no	—
299	C₄⋊Dic₃	—	20	2	no	—
300	C₄×Dic₃	—	10	4	no	—
301	C₂³×C₅	C₂²×Dic₃	1	2	yes	—
302	C₂²×D₅	—	2	2	no	—
303	C₂²×D₅	—	2	2	no	—
304	C₂²×D₅	—	2	2	no	—
305	C₂²×D₅	—	2	2	no	—
306	C₂²×D₅	C₂×Dic₆	1	4	yes	—
307	C₂²×D₅	C₂×D₁₂	1	4	yes	—
308	C₂²×D₅	C₂²×Dic₃	1	2	yes	—
309	C₂²×D₅	C₂³×D₃	1	1	yes	characteristic
310	C₂×AGL(1,𝔽₅)	—	3	4	no	—
311	C₂×AGL(1,𝔽₅)	—	6	2	no	—
312	C₂×AGL(1,𝔽₅)	—	6	2	no	—
313	C₂×AGL(1,𝔽₅)	—	6	2	no	—
314	C₂×AGL(1,𝔽₅)	—	6	4	no	—
315	C₂×AGL(1,𝔽₅)	—	6	4	no	—
316	C₂×AGL(1,𝔽₅)	—	6	2	no	—
317	C₂×AGL(1,𝔽₅)	—	6	4	no	—
318	C₂×AGL(1,𝔽₅)	—	6	2	no	—
319	C₂×AGL(1,𝔽₅)	—	12	16	no	—
320	C₅×D₄	C₄⋊Dic₃	1	4	yes	complemented
321	C₂×C₄×C₅	—	2	4	no	—
322	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
323	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
324	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
325	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
326	C₂²⋊D₅	C₄⋊Dic₃	1	8	yes	complemented
327	C₂×Dic₅	—	2	4	no	—
328	C₂×Dic₅	C₂²×Dic₃	1	1	yes	characteristic
329	C₂×Dic₅	C₂²×Dic₃	1	2	yes	—
330	C₂×Dic₅	C₂²×Dic₃	1	1	yes	characteristic
331	C₂×Dic₅	C₂²×Dic₃	1	1	yes	characteristic
332	D₂₀	C₄⋊Dic₃	1	4	yes	complemented
333	C₄×D₅	—	2	2	no	—
334	C₄×D₅	—	2	2	no	—
335	C₄×D₅	—	2	1	no	—
336	C₄×D₅	—	2	1	no	—
337	C₄×D₅	—	2	4	no	—
338	C₄×D₅	—	2	4	no	—
339	C₄×D₅	C₂×Dic₆	1	2	yes	—
340	C₄×D₅	C₂×D₁₂	1	2	yes	—
341	C₄×D₅	D₃×D₄	1	2	yes	—
342	C₄×D₅	#237	1	2	yes	—
343	C₂²×D₄	—	5	1	no	—
344	C₂³×C₄	—	5	2	no	—
345	C₂³×C₄	—	15	4	no	—
346	C₄×D₄	—	5	1	no	—
347	C₄×D₄	—	5	2	no	—
348	C₄×D₄	—	5	1	no	—
349	C₄×D₄	—	5	2	no	—
350	C₄×D₄	—	5	1	no	—
351	C₄×D₄	—	30	8	no	—
352	C₄×D₄	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
353	C₂×C₄⋊C₄	—	5	1	no	—
354	C₂×C₄⋊C₄	—	15	1	no	—
355	C₂×C₄⋊C₄	—	15	2	no	—
356	C₂×C₄⋊C₄	—	15	4	no	—
357	C₂×C₄⋊C₄	—	15	2	no	—
358	C₂×C₄⋊C₄	—	15	1	no	—
359	C₂×C₄⋊C₄	—	15	2	no	—
360	C₂×C₄⋊C₄	—	30	8	no	—
361	C₂×C₂²⋊C₄	—	5	2	no	—
362	C₂×C₂²⋊C₄	—	15	4	no	—
363	C₂×C₄²	—	5	1	no	—
364	C₂×C₄²	—	15	2	no	—
365	#95	—	15	2	no	—
366	#95	—	15	4	no	—
367	C₂×C₃×C₅	—	2	4	no	—
368	C₂×C₃×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
369	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
370	C₂×C₃×C₅	#520	1	1	yes	characteristic
371	C₃×D₅	—	2	4	no	—
372	C₃×D₅	D₄²	1	1	yes	characteristic
373	C₃×D₅	#547	1	1	yes	characteristic
374	C₃×D₅	D₄×Q₈	1	1	yes	characteristic
375	C₃×D₅	#554	1	1	yes	characteristic
376	C₂³×C₃	—	5	1	no	—
377	C₂³×C₃	—	5	4	no	—
378	C₂³×C₃	—	5	4	no	—
379	C₂³×C₃	—	5	2	no	—
380	C₂³×C₃	—	10	2	no	—
381	C₂³×C₃	—	10	2	no	—
382	C₂³×C₃	—	10	2	no	—
383	C₂³×C₃	—	10	2	no	—
384	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	2	yes	—
385	C₃×D₄	—	5	8	no	—
386	C₃×D₄	—	5	4	no	—
387	C₃×D₄	C₄⋊AGL(1,𝔽₅)	1	4	yes	complemented
388	C₂×C₃×C₄	—	2	4	no	—
389	C₂×C₃×C₄	—	5	1	no	—
390	C₂×C₃×C₄	—	5	1	no	—
391	C₂×C₃×C₄	—	5	2	no	—
392	C₂×C₃×C₄	—	5	2	no	—
393	C₂×C₃×C₄	—	5	2	no	—
394	C₂×C₃×C₄	—	5	2	no	—
395	C₂×C₃×C₄	—	5	1	no	—
396	C₂×C₃×C₄	—	5	2	no	—
397	C₂×C₃×C₄	—	10	4	no	—
398	C₂×C₃×C₄	—	10	2	no	—
399	C₂×C₃×C₄	—	10	1	no	—
400	C₂×C₃×C₄	—	10	1	no	—
401	C₂×C₃×C₄	—	10	2	no	—
402	C₂×C₃×C₄	—	10	4	no	—
403	C₂×C₃×C₄	—	10	4	no	—
404	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
405	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
406	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
407	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
408	C₂×Dic₃	—	5	4	no	—
409	C₂×Dic₃	—	10	2	no	—
410	C₂×Dic₃	—	10	2	no	—
411	C₂×Dic₃	—	10	2	no	—
412	C₂×Dic₃	—	10	4	no	—
413	C₂×Dic₃	—	10	4	no	—
414	C₂×Dic₃	—	10	2	no	—
415	C₂×Dic₃	—	10	4	no	—
416	C₂×Dic₃	—	10	2	no	—
417	C₂×Dic₃	—	20	16	no	—
418	C₂²×C₅	—	2	2	no	—
419	C₂²×C₅	—	2	2	no	—
420	C₂²×C₅	C₂×C₄⋊Dic₃	1	4	yes	complemented
421	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
422	C₂×D₅	—	2	2	no	—
423	C₂×D₅	—	2	2	no	—
424	C₂×D₅	—	2	4	no	—
425	C₂×D₅	—	2	4	no	—
426	C₂×D₅	—	2	4	no	—
427	C₂×D₅	—	2	4	no	—
428	C₂×D₅	C₂×C₄⋊Dic₃	1	4	yes	complemented
429	C₂×D₅	C₂²×Dic₆	1	1	yes	characteristic
430	C₂×D₅	C₂²×D₁₂	1	1	yes	characteristic
431	C₂×D₅	C₂×D₃×D₄	1	1	yes	characteristic
432	C₂×D₅	C₂×#237	1	1	yes	characteristic
433	C₂×D₅	#1009	1	1	yes	characteristic
434	C₂×D₅	#1010	1	1	yes	characteristic
435	AGL(1,𝔽₅)	—	6	4	no	—
436	AGL(1,𝔽₅)	—	12	4	no	—
437	AGL(1,𝔽₅)	—	12	2	no	—
438	C₄×C₅	—	2	2	no	—
439	C₄×C₅	—	2	1	no	—
440	C₄×C₅	C₂×C₄⋊Dic₃	1	2	yes	complemented
441	C₄×C₅	D₄×Dic₃	1	2	yes	complemented
442	Dic₅	—	2	2	no	—
443	Dic₅	—	2	1	no	—
444	Dic₅	C₂×C₄⋊Dic₃	1	2	yes	complemented
445	Dic₅	D₄×Dic₃	1	2	yes	complemented
446	C₂⁴	—	5	2	no	—
447	C₂×D₄	—	5	2	no	—
448	C₂×D₄	—	5	4	no	—
449	C₂×D₄	—	5	4	no	—
450	C₂×D₄	—	5	1	no	—
451	C₂×D₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
452	C₂²×C₄	—	5	1	no	—
453	C₂²×C₄	—	5	1	no	—
454	C₂²×C₄	—	5	2	no	—
455	C₂²×C₄	—	5	1	no	—
456	C₂²×C₄	—	5	2	no	—
457	C₂²×C₄	—	5	2	no	—
458	C₂²×C₄	—	5	2	no	—
459	C₂²×C₄	—	10	4	no	—
460	C₂²×C₄	—	10	4	no	—
461	C₂²×C₄	—	15	2	no	—
462	C₂²×C₄	—	15	2	no	—
463	C₂²×C₄	—	15	4	no	—
464	C₂²×C₄	—	30	8	no	—
465	C₂²×C₄	—	30	8	no	—
466	C₂²×C₄	—	60	4	no	—
467	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	complemented
468	C₄⋊C₄	—	5	2	no	—
469	C₄⋊C₄	—	5	1	no	—
470	C₄⋊C₄	—	15	4	no	—
471	C₄⋊C₄	—	30	4	no	—
472	C₄⋊C₄	—	30	4	no	—
473	C₄⋊C₄	—	30	2	no	—
474	C₄⋊C₄	—	30	8	no	—
475	C₄⋊C₄	—	60	2	no	—
476	C₄⋊C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
477	C₂²⋊C₄	—	5	2	no	—
478	C₂²⋊C₄	—	5	2	no	—
479	C₂²⋊C₄	—	5	2	no	—
480	C₂²⋊C₄	—	30	8	no	—
481	C₂²⋊C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	complemented
482	C₄²	—	5	1	no	—
483	C₄²	—	5	1	no	—
484	C₄²	—	5	1	no	—
485	C₄²	—	30	4	no	—
486	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
487	C₃×C₅	D₄×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
488	C₂²×C₃	—	2	2	no	—
489	C₂²×C₃	—	2	2	no	—
490	C₂²×C₃	—	5	1	no	—
491	C₂²×C₃	—	5	1	no	—
492	C₂²×C₃	—	5	1	no	—
493	C₂²×C₃	—	5	1	no	—
494	C₂²×C₃	—	5	1	no	—
495	C₂²×C₃	—	5	1	no	—
496	C₂²×C₃	—	5	4	no	—
497	C₂²×C₃	—	10	4	no	—
498	C₂²×C₃	—	10	4	no	—
499	C₂²×C₃	—	10	4	no	—
500	C₂²×C₃	—	10	4	no	—
501	C₂²×C₃	—	10	2	no	—
502	C₂²×C₃	—	10	2	no	—
503	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	4	yes	complemented
504	C₂²×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
505	C₃×C₄	—	2	2	no	—
506	C₃×C₄	—	2	1	no	—
507	C₃×C₄	—	5	2	no	—
508	C₃×C₄	—	5	2	no	—
509	C₃×C₄	—	10	2	no	—
510	C₃×C₄	—	10	1	no	—
511	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	complemented
512	C₃×C₄	D₄×AGL(1,𝔽₅)	1	2	yes	complemented
513	Dic₃	—	10	4	no	—
514	Dic₃	—	20	2	no	—
515	Dic₃	—	20	4	no	—
516	C₂×C₅	—	2	4	no	—
517	C₂×C₅	C₂²×C₄⋊Dic₃	1	1	yes	characteristic
518	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
519	C₂×C₅	#5827	1	1	yes	characteristic
520	D₅	—	2	4	no	—
521	D₅	D₄×Dic₆	1	1	yes	characteristic
522	D₅	#5543	1	1	yes	characteristic
523	D₅	D₄×D₁₂	1	1	yes	characteristic
524	D₅	#5555	1	1	yes	characteristic
525	C₂³	—	5	1	no	—
526	C₂³	—	5	4	no	—
527	C₂³	—	5	4	no	—
528	C₂³	—	5	2	no	—
529	C₂³	—	10	2	no	—
530	C₂³	—	10	2	no	—
531	C₂³	—	10	2	no	—
532	C₂³	—	10	2	no	—
533	C₂³	C₂²×ΓL(1,𝔽₁₆)	1	2	yes	—
534	D₄	—	5	8	no	—
535	D₄	—	5	4	no	—
536	D₄	C₄⋊ΓL(1,𝔽₁₆)	1	4	yes	direct product factor
537	C₂×C₄	—	2	4	no	—
538	C₂×C₄	—	5	1	no	—
539	C₂×C₄	—	5	1	no	—
540	C₂×C₄	—	5	2	no	—
541	C₂×C₄	—	5	2	no	—
542	C₂×C₄	—	5	2	no	—
543	C₂×C₄	—	5	2	no	—
544	C₂×C₄	—	5	1	no	—
545	C₂×C₄	—	5	2	no	—
546	C₂×C₄	—	10	4	no	—
547	C₂×C₄	—	10	2	no	—
548	C₂×C₄	—	10	1	no	—
549	C₂×C₄	—	10	2	no	—
550	C₂×C₄	—	10	1	no	—
551	C₂×C₄	—	10	4	no	—
552	C₂×C₄	—	10	4	no	—
553	C₂×C₄	—	15	4	no	—
554	C₂×C₄	—	30	2	no	—
555	C₂×C₄	—	30	2	no	—
556	C₂×C₄	—	30	2	no	—
557	C₂×C₄	—	30	4	no	—
558	C₂×C₄	—	30	4	no	—
559	C₂×C₄	—	30	2	no	—
560	C₂×C₄	—	30	4	no	—
561	C₂×C₄	—	30	2	no	—
562	C₂×C₄	—	60	16	no	—
563	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
564	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	2	yes	—
565	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
566	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
567	C₂×C₃	—	2	4	no	—
568	C₂×C₃	—	5	1	no	—
569	C₂×C₃	—	5	1	no	—
570	C₂×C₃	—	5	1	no	—
571	C₂×C₃	—	5	1	no	—
572	C₂×C₃	—	10	4	no	—
573	C₂×C₃	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
574	C₂×C₃	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
575	C₂×C₃	#66266	1	1	yes	characteristic
576	C₅	D₄×C₄⋊Dic₃	1	1	yes	characteristic, complemented, Sylow
577	C₂²	—	2	2	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	5	1	no	—
580	C₂²	—	5	1	no	—
581	C₂²	—	5	1	no	—
582	C₂²	—	5	1	no	—
583	C₂²	—	5	1	no	—
584	C₂²	—	5	1	no	—
585	C₂²	—	5	4	no	—
586	C₂²	—	10	4	no	—
587	C₂²	—	10	4	no	—
588	C₂²	—	10	4	no	—
589	C₂²	—	10	4	no	—
590	C₂²	—	10	2	no	—
591	C₂²	—	10	2	no	—
592	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	4	yes	complemented
593	C₂²	C₂³×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
594	C₄	—	2	2	no	—
595	C₄	—	2	1	no	—
596	C₄	—	5	2	no	—
597	C₄	—	5	2	no	—
598	C₄	—	10	2	no	—
599	C₄	—	10	1	no	—
600	C₄	—	30	4	no	—
601	C₄	—	60	4	no	—
602	C₄	—	60	2	no	—
603	C₄	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	2	yes	complemented
604	C₄	D₄×ΓL(1,𝔽₁₆)	1	2	yes	complemented
605	C₃	D₄×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic, complemented, Sylow
606	C₂	—	2	4	no	—
607	C₂	—	5	1	no	—
608	C₂	—	5	1	no	—
609	C₂	—	5	1	no	—
610	C₂	—	5	1	no	—
611	C₂	—	10	4	no	—
612	C₂	C₂²×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
613	C₂	C₂×D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
614	C₂	#11756995	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×ΓL(1,𝔽₁₆)
commutator, socle	C₂²×C₃×C₅	C₂³×C₄
Fitting	C₃×C₄×C₅×D₄	C₄
prime radical	C₂×C₃×D₄×D₅	C₂²
cosocle	C₂²×C₃×D₅	C₂⁴
hypercenter	C₄×D₄	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	D₄×C₄⋊C₄
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄⋊ΓL(1,𝔽₁₆)	C₂	semidirect
4. #11752424	C₂	semidirect
5. #11752371	C₂	semidirect
6. #11752358	C₂	semidirect
7. #11752347	C₂	semidirect
8. C₄×C₄⋊ΓL(1,𝔽₁₆)	C₂	semidirect
23. C₃×C₄×C₅×D₄	C₄	semidirect
24. C₃×D₄×Dic₅	C₄	semidirect
25. C₃×C₄×C₂²⋊D₅	C₄	semidirect
26. C₃×#4039	C₄	semidirect
28. C₃×#4027	C₄	semidirect
30. C₃×C₄×D₂₀	C₄	semidirect
51. C₂²×C₃×C₄×C₅	C₂×C₄	semidirect
53. C₂²×C₃×Dic₅	C₂×C₄	semidirect
59. C₂×C₃×C₄×D₅	C₂×C₄	semidirect
60. C₂×C₃×C₄×D₅	C₂×C₄	semidirect
71. C₄⋊ΓL(1,𝔽₁₆)	D₄	direct (N × H)
73. C₃×C₅×C₄⋊C₄	C₂×C₄	semidirect
74. C₃×C₅×C₂²⋊C₄	C₂×C₄	semidirect
75. C₃×C₄²×C₅	C₂×C₄	semidirect
76. C₃×C₂²⋊Dic₅	C₂×C₄	semidirect
77. C₃×#685	C₂×C₄	semidirect
78. C₃×#685	C₂×C₄	semidirect
79. C₃×C₄⋊Dic₅	C₂×C₄	semidirect
80. C₃×#683	C₂×C₄	semidirect
81. C₃×C₄×Dic₅	C₂×C₄	semidirect
82. C₃×C₄×Dic₅	C₂×C₄	semidirect
83. C₃×C₄×Dic₅	C₂×C₄	semidirect
105. C₄×C₅×D₄	Dic₃	semidirect
106. D₄×Dic₅	Dic₃	semidirect
107. C₄×C₂²⋊D₅	Dic₃	semidirect
108. #4039	Dic₃	semidirect
110. #4027	Dic₃	semidirect
112. C₄×D₂₀	Dic₃	semidirect
136. C₃×C₅×D₄	C₄⋊C₄	semidirect
142. C₃×C₂²⋊D₅	C₄⋊C₄	semidirect
148. C₃×D₂₀	C₄⋊C₄	semidirect
167. C₃×C₄×D₄	AGL(1,𝔽₅)	semidirect
191. C₂²×C₄×C₅	C₂×Dic₃	semidirect
193. C₂²×Dic₅	C₂×Dic₃	semidirect
199. C₂×C₄×D₅	C₂×Dic₃	semidirect
200. C₂×C₄×D₅	C₂×Dic₃	semidirect
213. C₅×C₄⋊C₄	C₂×Dic₃	semidirect
214. C₅×C₂²⋊C₄	C₂×Dic₃	semidirect
215. C₄²×C₅	C₂×Dic₃	semidirect
216. C₂²⋊Dic₅	C₂×Dic₃	semidirect
217. #685	C₂×Dic₃	semidirect
218. #685	C₂×Dic₃	semidirect
219. C₄⋊Dic₅	C₂×Dic₃	semidirect
220. #683	C₂×Dic₃	semidirect
221. C₄×Dic₅	C₂×Dic₃	semidirect
222. C₄×Dic₅	C₂×Dic₃	semidirect
223. C₄×Dic₅	C₂×Dic₃	semidirect
234. C₂²×C₃×C₅	C₂×C₄⋊C₄	semidirect
242. C₂×C₃×D₅	C₂×C₄⋊C₄	semidirect
254. C₃×C₄×C₅	C₂×C₄⋊C₄	semidirect
255. C₃×C₄×C₅	C₄×D₄	semidirect
258. C₃×Dic₅	C₂×C₄⋊C₄	semidirect
259. C₃×Dic₅	C₄×D₄	semidirect
275. C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	semidirect
284. C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	semidirect
288. C₃×C₂²⋊C₄	C₂×AGL(1,𝔽₅)	semidirect
292. C₃×C₄²	C₂×AGL(1,𝔽₅)	semidirect
320. C₅×D₄	C₄⋊Dic₃	semidirect
326. C₂²⋊D₅	C₄⋊Dic₃	semidirect
332. D₂₀	C₄⋊Dic₃	semidirect
352. C₄×D₄	ΓL(1,𝔽₁₆)	semidirect
387. C₃×D₄	C₄⋊AGL(1,𝔽₅)	semidirect
420. C₂²×C₅	C₂×C₄⋊Dic₃	semidirect
428. C₂×D₅	C₂×C₄⋊Dic₃	semidirect
440. C₄×C₅	C₂×C₄⋊Dic₃	semidirect
441. C₄×C₅	D₄×Dic₃	semidirect
444. Dic₅	C₂×C₄⋊Dic₃	semidirect
445. Dic₅	D₄×Dic₃	semidirect
467. C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	semidirect
476. C₄⋊C₄	C₂×ΓL(1,𝔽₁₆)	semidirect
481. C₂²⋊C₄	C₂×ΓL(1,𝔽₁₆)	semidirect
486. C₄²	C₂×ΓL(1,𝔽₁₆)	semidirect
487. C₃×C₅	D₄×C₄⋊C₄	semidirect
503. C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	semidirect
511. C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	semidirect
512. C₃×C₄	D₄×AGL(1,𝔽₅)	semidirect
536. D₄	C₄⋊ΓL(1,𝔽₁₆)	direct (N × H)
576. C₅	D₄×C₄⋊Dic₃	semidirect
592. C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)	semidirect
603. C₄	C₂×C₄⋊ΓL(1,𝔽₁₆)	semidirect
604. C₄	D₄×ΓL(1,𝔽₁₆)	semidirect
605. C₃	D₄×C₄⋊AGL(1,𝔽₅)	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×ΓL(1,𝔽₁₆)	2
C₂²×C₄⋊ΓL(1,𝔽₁₆)	2
C₃×C₄×D₄×D₅	2
#11752424	2
#11752371	2
#11752358	2
#11752347	2
C₄×C₄⋊ΓL(1,𝔽₁₆)	2
D₄×C₄⋊AGL(1,𝔽₅)	3
D₄×C₄⋊Dic₃	5

Subgroups — D4×C4⋊ΓL(1,𝔽16) #14327017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₄⋊ΓL(1,𝔽₁₆) #14327017