Subgroups — C₄₁×#25444 #23801645

Order 10496.

934 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3155	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#472	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3062	C₂	1	2	yes	maximal, complemented
4	C₄₁×#3060	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₁×#444	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2037	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2006	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1972	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1951	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1907	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1765	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
13	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
14	C₂×#135×C₄₁	C₂²	1	1	yes	characteristic
15	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×#131×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×#131×C₄₁	C₂²	1	2	yes	—
18	C₄₁×#537	C₂²	1	2	yes	—
19	C₄₁×#535	C₂²	1	2	yes	—
20	C₄₁×#496	—	2	2	no	—
21	C₄₁×#495	—	2	1	no	—
22	C₄₁×#493	—	2	1	no	—
23	C₄₁×#472	—	2	4	no	—
24	C₄₁×#470	—	2	1	no	—
25	C₄₁×#469	—	2	1	no	—
26	C₄₁×#469	—	2	2	no	—
27	C₄₁×#468	—	2	2	no	—
28	C₄₁×#466	—	2	2	no	—
29	C₄₁×#465	—	2	2	no	—
30	C₄₁×#465	—	2	2	no	—
31	C₄₁×#444	—	2	4	no	—
32	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
33	C₄₁×#428	C₂²	1	2	yes	—
34	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
35	C₂×#104×C₄₁	C₂²	1	1	yes	characteristic
36	C₂×#104×C₄₁	C₂²	1	2	yes	—
37	C₄₁×#417	C₂²	1	2	yes	—
38	C₄₁×#416	C₂²	1	1	yes	characteristic
39	C₂×#102×C₄₁	C₂²	1	2	yes	—
40	C₂×#100×C₄₁	C₂²	1	2	yes	—
41	C₄₁×#405	C₂²	1	1	yes	characteristic
42	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
43	C₄₁×#390	C₂²	1	2	yes	—
44	C₄₁×#343	C₂²	1	2	yes	—
45	C₄₁×#341	C₂²	1	1	yes	characteristic
46	C₄₁×#339	C₂²	1	1	yes	characteristic
47	C₂×#41×C₄₁	C₂³	1	2	yes	—
48	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
49	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
50	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
51	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
52	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
53	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
54	#135×C₄₁	—	2	2	no	—
55	#135×C₄₁	—	2	2	no	—
56	C₂×Q₁₆×C₄₁	D₄	1	1	yes	characteristic
57	C₂×SD₁₆×C₄₁	—	2	2	no	—
58	C₂×SD₁₆×C₄₁	—	2	2	no	—
59	C₂×SD₁₆×C₄₁	D₄	1	2	yes	—
60	C₂×D₈×C₄₁	—	2	2	no	—
61	C₂×D₈×C₄₁	—	2	2	no	—
62	C₂×D₈×C₄₁	—	2	2	no	—
63	C₂×D₈×C₄₁	—	2	1	no	—
64	C₂×D₈×C₄₁	—	2	1	no	—
65	C₂×D₈×C₄₁	—	2	2	no	—
66	C₂×D₈×C₄₁	D₄	1	1	yes	characteristic
67	#131×C₄₁	—	2	4	no	—
68	#131×C₄₁	—	2	2	no	—
69	#131×C₄₁	—	2	2	no	—
70	#131×C₄₁	—	4	2	no	—
71	C₂×M₁₆×C₄₁	—	2	2	no	—
72	C₂×M₁₆×C₄₁	—	2	1	no	—
73	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
75	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₄₁	—	2	1	no	—
77	C₂²×C₈×C₄₁	—	2	2	no	—
78	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
80	#127×C₄₁	—	2	2	no	—
81	#124×C₄₁	—	2	2	no	—
82	C₂²⋊Q₈×C₄₁	—	2	2	no	—
83	C₂²⋊Q₈×C₄₁	—	2	2	no	—
84	#121×C₄₁	—	2	2	no	—
85	#121×C₄₁	—	2	2	no	—
86	#121×C₄₁	—	2	2	no	—
87	#121×C₄₁	—	2	2	no	—
88	#121×C₄₁	—	4	2	no	—
89	#120×C₄₁	—	4	2	no	—
90	#117×C₄₁	C₂³	1	1	yes	characteristic
91	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₄₁	—	4	2	no	—
93	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
94	#108×C₄₁	D₄	1	4	yes	complemented
95	#107×C₄₁	—	2	2	no	—
96	#106×C₄₁	—	2	1	no	—
97	#106×C₄₁	—	2	1	no	—
98	#106×C₄₁	—	2	2	no	—
99	#104×C₄₁	—	2	4	no	—
100	#104×C₄₁	—	2	2	no	—
101	#103×C₄₁	—	2	2	no	—
102	#103×C₄₁	D₄	1	2	yes	—
103	#102×C₄₁	—	2	2	no	—
104	#102×C₄₁	—	2	2	no	—
105	#102×C₄₁	—	2	2	no	—
106	#102×C₄₁	D₄	1	2	yes	—
107	#100×C₄₁	—	2	4	no	—
108	#97×C₄₁	—	2	2	no	—
109	#97×C₄₁	—	2	1	no	—
110	C₄×C₈×C₄₁	—	2	1	no	—
111	C₂⁴×C₄₁	—	2	2	no	—
112	#41×C₄₁	—	2	2	no	—
113	#41×C₄₁	—	2	1	no	—
114	#41×C₄₁	—	2	1	no	—
115	#41×C₄₁	—	2	1	no	—
116	#41×C₄₁	—	2	2	no	—
117	#41×C₄₁	—	2	2	no	—
118	#41×C₄₁	—	2	2	no	—
119	#41×C₄₁	—	2	1	no	—
120	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
122	C₂×D₄×C₄₁	—	2	2	no	—
123	C₂×D₄×C₄₁	—	2	1	no	—
124	C₂×D₄×C₄₁	—	2	2	no	—
125	C₂×D₄×C₄₁	—	2	2	no	—
126	C₂×D₄×C₄₁	—	2	2	no	—
127	C₂×D₄×C₄₁	—	2	2	no	—
128	C₂×D₄×C₄₁	—	2	2	no	—
129	C₂×D₄×C₄₁	—	2	2	no	—
130	C₂×D₄×C₄₁	—	4	2	no	—
131	C₂×D₄×C₄₁	—	4	2	no	—
132	C₂×D₄×C₄₁	—	4	2	no	—
133	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
134	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₄₁	—	2	2	no	—
136	C₂²×C₄×C₄₁	—	2	1	no	—
137	C₂²×C₄×C₄₁	—	2	1	no	—
138	C₂²×C₄×C₄₁	—	2	2	no	—
139	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
140	Q₁₆×C₄₁	—	2	2	no	—
141	SD₁₆×C₄₁	—	2	4	no	—
142	SD₁₆×C₄₁	—	4	4	no	—
143	SD₁₆×C₄₁	—	4	4	no	—
144	D₈×C₄₁	—	2	2	no	—
145	D₈×C₄₁	—	4	4	no	—
146	D₈×C₄₁	—	4	4	no	—
147	D₈×C₄₁	—	4	2	no	—
148	D₈×C₄₁	—	4	2	no	—
149	M₁₆×C₄₁	—	2	2	no	—
150	M₁₆×C₄₁	—	2	1	no	—
151	M₁₆×C₄₁	—	2	1	no	—
152	M₁₆×C₄₁	—	2	1	no	—
153	M₁₆×C₄₁	—	4	4	no	—
154	M₁₆×C₄₁	—	4	2	no	—
155	M₁₆×C₄₁	C₂×D₄	1	4	yes	—
156	M₁₆×C₄₁	#41	1	2	yes	—
157	C₂×C₈×C₄₁	—	2	1	no	—
158	C₂×C₈×C₄₁	—	2	1	no	—
159	C₂×C₈×C₄₁	—	2	2	no	—
160	C₂×C₈×C₄₁	—	2	1	no	—
161	C₂×C₈×C₄₁	—	2	1	no	—
162	C₂×C₈×C₄₁	—	4	4	no	—
163	C₂×C₈×C₄₁	—	4	2	no	—
164	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
165	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
166	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
167	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
168	C₂×C₈×C₄₁	#41	1	2	yes	—
169	C₄⋊C₄×C₄₁	—	2	2	no	—
170	C₄⋊C₄×C₄₁	—	2	2	no	—
171	C₄⋊C₄×C₄₁	—	4	2	no	—
172	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
173	C₂²⋊C₄×C₄₁	—	2	1	no	—
174	C₂²⋊C₄×C₄₁	—	4	2	no	—
175	C₂²⋊C₄×C₄₁	—	4	2	no	—
176	C₂²⋊C₄×C₄₁	—	4	2	no	—
177	C₂²⋊C₄×C₄₁	—	4	2	no	—
178	C₂²⋊C₄×C₄₁	—	8	4	no	—
179	C₄²×C₄₁	—	2	1	no	—
180	C₄²×C₄₁	—	2	1	no	—
181	C₄²×C₄₁	—	2	1	no	—
182	C₂³×C₄₁	—	2	1	no	—
183	C₂³×C₄₁	—	2	2	no	—
184	C₂³×C₄₁	—	4	2	no	—
185	C₂³×C₄₁	—	4	2	no	—
186	C₂³×C₄₁	—	4	2	no	—
187	C₂³×C₄₁	—	8	2	no	—
188	C₂³×C₄₁	—	8	2	no	—
189	C₂³×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
190	Q₈×C₄₁	—	2	2	no	—
191	Q₈×C₄₁	—	2	1	no	—
192	Q₈×C₄₁	—	2	1	no	—
193	Q₈×C₄₁	—	2	2	no	—
194	D₄×C₄₁	—	2	2	no	—
195	D₄×C₄₁	—	2	1	no	—
196	D₄×C₄₁	—	2	1	no	—
197	D₄×C₄₁	—	2	2	no	—
198	D₄×C₄₁	—	4	2	no	—
199	D₄×C₄₁	—	4	1	no	—
200	D₄×C₄₁	—	4	2	no	—
201	D₄×C₄₁	—	4	1	no	—
202	D₄×C₄₁	—	4	2	no	—
203	D₄×C₄₁	—	4	2	no	—
204	D₄×C₄₁	—	4	2	no	—
205	D₄×C₄₁	—	4	2	no	—
206	D₄×C₄₁	—	16	2	no	—
207	D₄×C₄₁	—	16	2	no	—
208	D₄×C₄₁	—	16	2	no	—
209	C₂×C₄×C₄₁	—	2	2	no	—
210	C₂×C₄×C₄₁	—	2	2	no	—
211	C₂×C₄×C₄₁	—	2	1	no	—
212	C₂×C₄×C₄₁	—	4	1	no	—
213	C₂×C₄×C₄₁	—	4	2	no	—
214	C₂×C₄×C₄₁	—	4	2	no	—
215	C₂×C₄×C₄₁	—	4	1	no	—
216	C₂×C₄×C₄₁	—	4	2	no	—
217	C₂×C₄×C₄₁	—	4	2	no	—
218	C₂×C₄×C₄₁	—	4	1	no	—
219	C₂×C₄×C₄₁	—	8	2	no	—
220	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
224	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
225	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
226	C₈×C₄₁	—	2	2	no	—
227	C₈×C₄₁	—	2	4	no	—
228	C₈×C₄₁	—	2	2	no	—
229	C₈×C₄₁	—	4	1	no	—
230	C₈×C₄₁	#121	1	2	yes	—
231	C₈×C₄₁	#121	1	2	yes	—
232	#25444	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
233	C₂²×C₄₁	—	2	2	no	—
234	C₂²×C₄₁	—	4	1	no	—
235	C₂²×C₄₁	—	4	2	no	—
236	C₂²×C₄₁	—	4	2	no	—
237	C₂²×C₄₁	—	4	2	no	—
238	C₂²×C₄₁	—	4	2	no	—
239	C₂²×C₄₁	—	8	4	no	—
240	C₂²×C₄₁	—	8	2	no	—
241	C₂²×C₄₁	—	8	2	no	—
242	C₂²×C₄₁	—	8	2	no	—
243	C₂²×C₄₁	—	16	2	no	—
244	C₂²×C₄₁	#540	1	1	yes	characteristic
245	C₂²×C₄₁	D₄²	1	1	yes	characteristic
246	C₂²×C₄₁	#550	1	1	yes	characteristic
247	C₄×C₄₁	—	4	2	no	—
248	C₄×C₄₁	—	4	1	no	—
249	C₄×C₄₁	—	4	2	no	—
250	C₄×C₄₁	—	8	1	no	—
251	C₄×C₄₁	—	8	2	no	—
252	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
253	C₄×C₄₁	#532	1	1	yes	characteristic
254	C₄×C₄₁	D₄²	1	1	yes	characteristic
255	C₄×C₄₁	#546	1	1	yes	characteristic
256	#3155	C₂×C₄₁	1	1	yes	characteristic, complemented
257	C₂×#472	C₂×C₄₁	1	1	yes	characteristic, complemented
258	#3062	C₂×C₄₁	1	2	yes	complemented
259	#3060	C₂×C₄₁	1	1	yes	characteristic, complemented
260	C₂×#444	C₂×C₄₁	1	1	yes	characteristic, complemented
261	#2037	C₂×C₄₁	1	2	yes	complemented
262	#2006	C₂×C₄₁	1	2	yes	complemented
263	#1972	C₂×C₄₁	1	1	yes	characteristic, complemented
264	#1951	C₂×C₄₁	1	1	yes	characteristic, complemented
265	#1907	C₂×C₄₁	1	2	yes	complemented
266	#1765	C₂×C₄₁	1	1	yes	characteristic, complemented
267	C₂×C₄₁	—	2	1	no	—
268	C₂×C₄₁	—	2	1	no	—
269	C₂×C₄₁	—	8	2	no	—
270	C₂×C₄₁	—	8	1	no	—
271	C₂×C₄₁	—	16	2	no	—
272	C₂×C₄₁	#2428	1	1	yes	characteristic
273	C₂×C₄₁	#3288	1	2	yes	—
274	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
275	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
276	C₂×#135	C₂²×C₄₁	1	1	yes	characteristic
277	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
278	C₂×#131	C₂²×C₄₁	1	1	yes	characteristic
279	C₂×#131	C₂²×C₄₁	1	2	yes	—
280	#537	C₂²×C₄₁	1	2	yes	—
281	#535	C₂²×C₄₁	1	2	yes	—
282	#496	—	2	2	no	—
283	#495	—	2	1	no	—
284	#493	—	2	1	no	—
285	#472	—	2	4	no	—
286	#470	—	2	1	no	—
287	#469	—	2	1	no	—
288	#469	—	2	2	no	—
289	#468	—	2	2	no	—
290	#466	—	2	2	no	—
291	#465	—	2	2	no	—
292	#465	—	2	2	no	—
293	#444	—	2	4	no	—
294	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
295	#428	C₂²×C₄₁	1	2	yes	—
296	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
297	C₂×#104	C₂²×C₄₁	1	2	yes	—
298	C₂×#104	C₂²×C₄₁	1	1	yes	characteristic
299	#417	C₂²×C₄₁	1	2	yes	—
300	#416	C₂²×C₄₁	1	1	yes	characteristic
301	C₂×#102	C₂²×C₄₁	1	2	yes	—
302	C₂×#100	C₂²×C₄₁	1	2	yes	—
303	#405	C₂²×C₄₁	1	1	yes	characteristic
304	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
305	#390	C₂²×C₄₁	1	2	yes	—
306	#343	C₂²×C₄₁	1	2	yes	—
307	#341	C₂²×C₄₁	1	1	yes	characteristic
308	#339	C₂²×C₄₁	1	1	yes	characteristic
309	C₄₁	#25444	1	1	yes	characteristic, direct product factor, Sylow
310	C₂×#41	C₂³×C₄₁	1	2	yes	—
311	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
312	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
313	AGL(1,ℤ/8)	—	2	4	no	—
314	AGL(1,ℤ/8)	—	2	4	no	—
315	AGL(1,ℤ/8)	—	4	2	no	—
316	AGL(1,ℤ/8)	—	4	2	no	—
317	#135	—	2	2	no	—
318	#135	—	2	2	no	—
319	C₂×Q₁₆	D₄×C₄₁	1	1	yes	characteristic
320	C₂×SD₁₆	—	2	2	no	—
321	C₂×SD₁₆	—	2	2	no	—
322	C₂×SD₁₆	D₄×C₄₁	1	2	yes	—
323	C₂×D₈	—	2	2	no	—
324	C₂×D₈	—	2	2	no	—
325	C₂×D₈	—	2	2	no	—
326	C₂×D₈	—	2	1	no	—
327	C₂×D₈	—	2	1	no	—
328	C₂×D₈	—	2	2	no	—
329	C₂×D₈	D₄×C₄₁	1	1	yes	characteristic
330	#131	—	2	4	no	—
331	#131	—	2	2	no	—
332	#131	—	2	2	no	—
333	#131	—	4	2	no	—
334	C₂×M₁₆	—	2	2	no	—
335	C₂×M₁₆	—	2	1	no	—
336	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
337	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
338	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
339	C₂²×C₈	—	2	1	no	—
340	C₂²×C₈	—	2	2	no	—
341	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
342	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
343	#127	—	2	2	no	—
344	#124	—	2	2	no	—
345	C₂²⋊Q₈	—	2	2	no	—
346	C₂²⋊Q₈	—	2	2	no	—
347	#121	—	2	2	no	—
348	#121	—	2	2	no	—
349	#121	—	2	2	no	—
350	#121	—	2	2	no	—
351	#121	—	4	2	no	—
352	#120	—	4	2	no	—
353	#117	C₂³×C₄₁	1	1	yes	characteristic
354	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
355	C₂×C₂²⋊C₄	—	4	2	no	—
356	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
357	#108	D₄×C₄₁	1	4	yes	complemented
358	#107	—	2	2	no	—
359	#106	—	2	1	no	—
360	#106	—	2	1	no	—
361	#106	—	2	2	no	—
362	#104	—	2	4	no	—
363	#104	—	2	2	no	—
364	#103	—	2	2	no	—
365	#103	D₄×C₄₁	1	2	yes	—
366	#102	—	2	2	no	—
367	#102	—	2	2	no	—
368	#102	—	2	2	no	—
369	#102	D₄×C₄₁	1	2	yes	—
370	#100	—	2	4	no	—
371	#97	—	2	2	no	—
372	#97	—	2	1	no	—
373	C₄×C₈	—	2	1	no	—
374	C₂⁴	—	2	2	no	—
375	#41	—	2	2	no	—
376	#41	—	2	1	no	—
377	#41	—	2	1	no	—
378	#41	—	2	1	no	—
379	#41	—	2	2	no	—
380	#41	—	2	2	no	—
381	#41	—	2	2	no	—
382	#41	—	2	1	no	—
383	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
384	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
385	C₂×D₄	—	2	2	no	—
386	C₂×D₄	—	2	1	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	2	no	—
390	C₂×D₄	—	2	2	no	—
391	C₂×D₄	—	2	2	no	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	4	2	no	—
394	C₂×D₄	—	4	2	no	—
395	C₂×D₄	—	4	2	no	—
396	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
397	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	1	no	—
400	C₂²×C₄	—	2	1	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
403	Q₁₆	—	2	2	no	—
404	SD₁₆	—	2	4	no	—
405	SD₁₆	—	4	4	no	—
406	SD₁₆	—	4	4	no	—
407	D₈	—	2	2	no	—
408	D₈	—	4	4	no	—
409	D₈	—	4	4	no	—
410	D₈	—	4	2	no	—
411	D₈	—	4	2	no	—
412	M₁₆	—	2	2	no	—
413	M₁₆	—	2	1	no	—
414	M₁₆	—	2	1	no	—
415	M₁₆	—	2	1	no	—
416	M₁₆	—	4	4	no	—
417	M₁₆	—	4	2	no	—
418	M₁₆	C₂×D₄×C₄₁	1	4	yes	—
419	M₁₆	#41×C₄₁	1	2	yes	—
420	C₂×C₈	—	2	1	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	1	no	—
424	C₂×C₈	—	2	1	no	—
425	C₂×C₈	—	4	4	no	—
426	C₂×C₈	—	4	2	no	—
427	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
428	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
429	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
430	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
431	C₂×C₈	#41×C₄₁	1	2	yes	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	2	no	—
434	C₄⋊C₄	—	4	2	no	—
435	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
436	C₂²⋊C₄	—	2	1	no	—
437	C₂²⋊C₄	—	4	2	no	—
438	C₂²⋊C₄	—	4	2	no	—
439	C₂²⋊C₄	—	4	2	no	—
440	C₂²⋊C₄	—	4	2	no	—
441	C₂²⋊C₄	—	8	4	no	—
442	C₄²	—	2	1	no	—
443	C₄²	—	2	1	no	—
444	C₄²	—	2	1	no	—
445	C₂³	—	2	1	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	4	2	no	—
448	C₂³	—	4	2	no	—
449	C₂³	—	4	2	no	—
450	C₂³	—	8	2	no	—
451	C₂³	—	8	2	no	—
452	C₂³	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
453	Q₈	—	2	2	no	—
454	Q₈	—	2	1	no	—
455	Q₈	—	2	1	no	—
456	Q₈	—	2	2	no	—
457	D₄	—	2	2	no	—
458	D₄	—	2	1	no	—
459	D₄	—	2	1	no	—
460	D₄	—	2	2	no	—
461	D₄	—	4	2	no	—
462	D₄	—	4	1	no	—
463	D₄	—	4	2	no	—
464	D₄	—	4	1	no	—
465	D₄	—	4	2	no	—
466	D₄	—	4	2	no	—
467	D₄	—	4	2	no	—
468	D₄	—	4	2	no	—
469	D₄	—	16	2	no	—
470	D₄	—	16	2	no	—
471	D₄	—	16	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	8	2	no	—
483	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
485	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
486	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
487	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
488	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
489	C₈	—	2	2	no	—
490	C₈	—	2	4	no	—
491	C₈	—	2	2	no	—
492	C₈	—	4	1	no	—
493	C₈	#121×C₄₁	1	2	yes	—
494	C₈	#121×C₄₁	1	2	yes	—
495	C₂²	—	2	2	no	—
496	C₂²	—	4	1	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	4	2	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	2	no	—
501	C₂²	—	8	4	no	—
502	C₂²	—	8	2	no	—
503	C₂²	—	8	2	no	—
504	C₂²	—	8	2	no	—
505	C₂²	—	16	2	no	—
506	C₂²	C₄₁×#540	1	1	yes	characteristic
507	C₂²	D₄²×C₄₁	1	1	yes	characteristic
508	C₂²	C₄₁×#550	1	1	yes	characteristic
509	C₄	—	4	2	no	—
510	C₄	—	4	1	no	—
511	C₄	—	4	2	no	—
512	C₄	—	8	1	no	—
513	C₄	—	8	2	no	—
514	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
515	C₄	C₄₁×#532	1	1	yes	characteristic
516	C₄	D₄²×C₄₁	1	1	yes	characteristic
517	C₄	C₄₁×#546	1	1	yes	characteristic
518	C₂	—	2	1	no	—
519	C₂	—	2	1	no	—
520	C₂	—	8	2	no	—
521	C₂	—	8	1	no	—
522	C₂	—	16	2	no	—
523	C₂	C₄₁×#2428	1	1	yes	characteristic
524	C₂	C₄₁×#3288	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3155	C₂	semidirect
2. C₂×C₄₁×#472	C₂	semidirect
3. C₄₁×#3062	C₂	semidirect
4. C₄₁×#3060	C₂	semidirect
5. C₂×C₄₁×#444	C₂	semidirect
6. C₄₁×#2037	C₂	semidirect
7. C₄₁×#2006	C₂	semidirect
8. C₄₁×#1972	C₂	semidirect
9. C₄₁×#1951	C₂	semidirect
10. C₄₁×#1907	C₂	semidirect
11. C₄₁×#1765	C₂	semidirect
94. #108×C₄₁	D₄	semidirect
232. #25444	C₄₁	direct (N × H)
256. #3155	C₂×C₄₁	semidirect
257. C₂×#472	C₂×C₄₁	semidirect
258. #3062	C₂×C₄₁	semidirect
259. #3060	C₂×C₄₁	semidirect
260. C₂×#444	C₂×C₄₁	semidirect
261. #2037	C₂×C₄₁	semidirect
262. #2006	C₂×C₄₁	semidirect
263. #1972	C₂×C₄₁	semidirect
264. #1951	C₂×C₄₁	semidirect
265. #1907	C₂×C₄₁	semidirect
266. #1765	C₂×C₄₁	semidirect
309. C₄₁	#25444	direct (N × H)
357. #108	D₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3155	2
C₂×C₄₁×#472	2
C₄₁×#3062	2
C₄₁×#3060	2
C₂×C₄₁×#444	2
C₄₁×#2037	2
C₄₁×#2006	2
C₄₁×#1972	2
C₄₁×#1951	2
C₄₁×#1907	2
C₄₁×#1765	2
#25444	41

Subgroups — C41×#25444 #23801645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#25444 #23801645