Subgroups — C₄₁×#24806 #23801007

Order 10496.

896 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#472	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3055	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₁×#469	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2930	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₁×#430	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2063	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2051	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1951	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1900	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1815	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
13	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×#131×C₄₁	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#525	C₂²	1	1	yes	characteristic
18	C₂×#121×C₄₁	C₂²	1	2	yes	—
19	C₄₁×#472	—	2	4	no	—
20	C₄₁×#469	—	2	2	no	—
21	C₄₁×#469	—	2	2	no	—
22	C₄₁×#469	—	2	2	no	—
23	C₄₁×#468	—	2	2	no	—
24	C₄₁×#466	—	2	2	no	—
25	C₄₁×#465	—	2	2	no	—
26	C₄₁×#436	—	2	2	no	—
27	C₄₁×#435	—	2	2	no	—
28	C₄₁×#430	—	2	4	no	—
29	C₂×#108×C₄₁	C₂²	1	2	yes	—
30	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
32	C₄₁×#428	C₂²	1	2	yes	—
33	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
34	C₄₁×#424	C₂²	1	1	yes	characteristic
35	C₄₁×#418	C₂²	1	2	yes	—
36	C₄₁×#417	C₂²	1	2	yes	—
37	C₂×#102×C₄₁	C₂²	1	2	yes	—
38	C₂×#100×C₄₁	C₂²	1	2	yes	—
39	C₂×#100×C₄₁	C₂²	1	2	yes	—
40	C₄₁×#408	C₂²	1	1	yes	characteristic
41	C₄₁×#407	C₂²	1	1	yes	characteristic
42	C₄₁×#405	C₂²	1	1	yes	characteristic
43	C₄₁×#343	C₂²	1	2	yes	—
44	C₄₁×#343	C₂²	1	2	yes	—
45	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
46	C₂²×D₄×C₄₁	—	2	2	no	—
47	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
48	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
49	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
50	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
51	C₂×SD₁₆×C₄₁	—	2	2	no	—
52	C₂×D₈×C₄₁	—	2	2	no	—
53	C₂×D₈×C₄₁	—	2	1	no	—
54	C₂×D₈×C₄₁	—	2	1	no	—
55	C₂×D₈×C₄₁	—	2	2	no	—
56	C₂×D₈×C₄₁	—	2	2	no	—
57	#131×C₄₁	—	2	2	no	—
58	#131×C₄₁	—	2	2	no	—
59	C₂×M₁₆×C₄₁	—	2	2	no	—
60	C₂×M₁₆×C₄₁	—	2	2	no	—
61	C₂×M₁₆×C₄₁	—	2	1	no	—
62	C₂×M₁₆×C₄₁	—	2	2	no	—
63	C₂×M₁₆×C₄₁	—	2	4	no	—
64	C₂×M₁₆×C₄₁	—	4	2	no	—
65	C₂×M₁₆×C₄₁	D₄	1	2	yes	—
66	C₂×M₁₆×C₄₁	D₄	1	2	yes	—
67	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
68	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
69	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
70	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
71	C₂²×C₈×C₄₁	—	2	2	no	—
72	C₂²×C₈×C₄₁	—	2	1	no	—
73	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
75	#123×C₄₁	—	2	1	no	—
76	C₂²⋊Q₈×C₄₁	D₄	1	2	yes	—
77	#121×C₄₁	—	2	2	no	—
78	#121×C₄₁	—	2	2	no	—
79	#121×C₄₁	—	2	2	no	—
80	#121×C₄₁	—	2	2	no	—
81	#121×C₄₁	D₄	1	2	yes	—
82	#120×C₄₁	—	2	1	no	—
83	C₄×D₄×C₄₁	—	2	2	no	—
84	#117×C₄₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
86	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
87	#108×C₄₁	—	2	4	no	—
88	#108×C₄₁	—	2	2	no	—
89	#108×C₄₁	D₄	1	4	yes	complemented
90	#107×C₄₁	—	2	2	no	—
91	#106×C₄₁	—	2	1	no	—
92	#106×C₄₁	—	2	1	no	—
93	#106×C₄₁	—	2	2	no	—
94	C₄⋊C₈×C₄₁	—	2	2	no	—
95	#103×C₄₁	—	2	2	no	—
96	#102×C₄₁	—	2	2	no	—
97	#102×C₄₁	—	2	4	no	—
98	#102×C₄₁	—	2	2	no	—
99	#102×C₄₁	—	2	2	no	—
100	#100×C₄₁	—	2	4	no	—
101	#100×C₄₁	—	4	2	no	—
102	C₂²⋊C₈×C₄₁	—	2	2	no	—
103	C₂²⋊C₈×C₄₁	—	2	2	no	—
104	#97×C₄₁	—	2	1	no	—
105	C₄×C₈×C₄₁	—	2	1	no	—
106	C₂⁴×C₄₁	—	2	2	no	—
107	C₂⁴×C₄₁	#41	1	1	yes	characteristic
108	#41×C₄₁	—	2	1	no	—
109	#41×C₄₁	—	2	1	no	—
110	#41×C₄₁	—	2	1	no	—
111	#41×C₄₁	—	2	1	no	—
112	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
113	C₂×D₄×C₄₁	—	2	1	no	—
114	C₂×D₄×C₄₁	—	2	2	no	—
115	C₂×D₄×C₄₁	—	2	2	no	—
116	C₂×D₄×C₄₁	—	2	2	no	—
117	C₂×D₄×C₄₁	—	2	2	no	—
118	C₂×D₄×C₄₁	—	2	2	no	—
119	C₂×D₄×C₄₁	—	2	2	no	—
120	C₂×D₄×C₄₁	—	2	2	no	—
121	C₂×D₄×C₄₁	—	4	2	no	—
122	C₂×D₄×C₄₁	—	4	2	no	—
123	C₂×D₄×C₄₁	—	8	4	no	—
124	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₄₁	—	2	1	no	—
126	C₂²×C₄×C₄₁	—	2	1	no	—
127	C₂²×C₄×C₄₁	—	2	1	no	—
128	C₂²×C₄×C₄₁	—	2	1	no	—
129	C₂²×C₄×C₄₁	—	2	1	no	—
130	C₂²×C₄×C₄₁	—	2	2	no	—
131	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
132	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
134	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
135	SD₁₆×C₄₁	—	4	4	no	—
136	D₈×C₄₁	—	4	4	no	—
137	D₈×C₄₁	—	4	2	no	—
138	D₈×C₄₁	—	4	2	no	—
139	M₁₆×C₄₁	—	2	2	no	—
140	M₁₆×C₄₁	—	2	4	no	—
141	M₁₆×C₄₁	—	2	4	no	—
142	M₁₆×C₄₁	—	2	1	no	—
143	M₁₆×C₄₁	—	2	2	no	—
144	M₁₆×C₄₁	—	2	1	no	—
145	M₁₆×C₄₁	—	2	1	no	—
146	M₁₆×C₄₁	—	2	2	no	—
147	M₁₆×C₄₁	—	2	1	no	—
148	M₁₆×C₄₁	—	4	4	no	—
149	M₁₆×C₄₁	—	4	2	no	—
150	M₁₆×C₄₁	C₂×D₄	1	4	yes	—
151	C₂×C₈×C₄₁	—	2	2	no	—
152	C₂×C₈×C₄₁	—	2	1	no	—
153	C₂×C₈×C₄₁	—	2	1	no	—
154	C₂×C₈×C₄₁	—	2	1	no	—
155	C₂×C₈×C₄₁	—	2	1	no	—
156	C₂×C₈×C₄₁	—	2	2	no	—
157	C₂×C₈×C₄₁	—	4	4	no	—
158	C₂×C₈×C₄₁	—	4	2	no	—
159	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
160	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
162	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
163	C₂×C₈×C₄₁	#41	1	2	yes	—
164	C₄⋊C₄×C₄₁	—	2	2	no	—
165	C₄⋊C₄×C₄₁	—	2	2	no	—
166	C₄⋊C₄×C₄₁	—	2	2	no	—
167	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
168	C₂²⋊C₄×C₄₁	—	2	2	no	—
169	C₂²⋊C₄×C₄₁	—	2	1	no	—
170	C₂²⋊C₄×C₄₁	—	2	2	no	—
171	C₂²⋊C₄×C₄₁	—	4	2	no	—
172	C₂²⋊C₄×C₄₁	—	4	2	no	—
173	C₄²×C₄₁	—	2	1	no	—
174	C₂³×C₄₁	—	2	1	no	—
175	C₂³×C₄₁	—	2	1	no	—
176	C₂³×C₄₁	—	2	1	no	—
177	C₂³×C₄₁	—	4	2	no	—
178	C₂³×C₄₁	—	4	2	no	—
179	C₂³×C₄₁	—	4	2	no	—
180	C₂³×C₄₁	—	4	2	no	—
181	C₂³×C₄₁	—	8	2	no	—
182	C₂³×C₄₁	—	8	2	no	—
183	C₂³×C₄₁	#121	1	2	yes	—
184	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
185	Q₈×C₄₁	—	2	1	no	—
186	Q₈×C₄₁	—	2	1	no	—
187	D₄×C₄₁	—	2	1	no	—
188	D₄×C₄₁	—	2	1	no	—
189	D₄×C₄₁	—	4	1	no	—
190	D₄×C₄₁	—	4	1	no	—
191	D₄×C₄₁	—	4	2	no	—
192	D₄×C₄₁	—	4	2	no	—
193	D₄×C₄₁	—	4	2	no	—
194	D₄×C₄₁	—	4	2	no	—
195	D₄×C₄₁	—	8	2	no	—
196	D₄×C₄₁	—	8	4	no	—
197	D₄×C₄₁	—	8	4	no	—
198	C₂×C₄×C₄₁	—	2	1	no	—
199	C₂×C₄×C₄₁	—	2	1	no	—
200	C₂×C₄×C₄₁	—	2	2	no	—
201	C₂×C₄×C₄₁	—	2	2	no	—
202	C₂×C₄×C₄₁	—	2	2	no	—
203	C₂×C₄×C₄₁	—	2	2	no	—
204	C₂×C₄×C₄₁	—	2	2	no	—
205	C₂×C₄×C₄₁	—	2	1	no	—
206	C₂×C₄×C₄₁	—	4	1	no	—
207	C₂×C₄×C₄₁	—	4	1	no	—
208	C₂×C₄×C₄₁	—	4	2	no	—
209	C₂×C₄×C₄₁	—	8	2	no	—
210	C₂×C₄×C₄₁	#121	1	2	yes	—
211	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
215	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
216	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
217	C₈×C₄₁	—	2	2	no	—
218	C₈×C₄₁	—	2	4	no	—
219	C₈×C₄₁	—	4	1	no	—
220	C₈×C₄₁	—	4	1	no	—
221	C₈×C₄₁	—	4	2	no	—
222	C₈×C₄₁	#121	1	2	yes	—
223	C₈×C₄₁	#121	1	2	yes	—
224	#24806	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂²×C₄₁	—	2	2	no	—
226	C₂²×C₄₁	—	2	2	no	—
227	C₂²×C₄₁	—	2	2	no	—
228	C₂²×C₄₁	—	2	2	no	—
229	C₂²×C₄₁	—	4	1	no	—
230	C₂²×C₄₁	—	4	2	no	—
231	C₂²×C₄₁	—	8	1	no	—
232	C₂²×C₄₁	—	8	1	no	—
233	C₂²×C₄₁	—	8	2	no	—
234	C₂²×C₄₁	—	8	2	no	—
235	C₂²×C₄₁	—	8	2	no	—
236	C₂²×C₄₁	—	16	2	no	—
237	C₂²×C₄₁	—	16	2	no	—
238	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
239	C₂²×C₄₁	#548	1	1	yes	characteristic
240	C₂²×C₄₁	#553	1	1	yes	characteristic
241	C₄×C₄₁	—	2	2	no	—
242	C₄×C₄₁	—	4	1	no	—
243	C₄×C₄₁	—	4	2	no	—
244	C₄×C₄₁	—	8	2	no	—
245	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
246	C₄×C₄₁	#537	1	1	yes	characteristic
247	C₄×C₄₁	D₄²	1	1	yes	characteristic
248	C₄×C₄₁	#547	1	1	yes	characteristic
249	C₂×#472	C₂×C₄₁	1	1	yes	characteristic, complemented
250	#3055	C₂×C₄₁	1	2	yes	complemented
251	C₂×#469	C₂×C₄₁	1	1	yes	characteristic, complemented
252	#2930	C₂×C₄₁	1	1	yes	characteristic, complemented
253	C₂×#430	C₂×C₄₁	1	1	yes	characteristic, complemented
254	#2063	C₂×C₄₁	1	2	yes	complemented
255	#2051	C₂×C₄₁	1	2	yes	complemented
256	#1978	C₂×C₄₁	1	1	yes	characteristic, complemented
257	#1951	C₂×C₄₁	1	1	yes	characteristic, complemented
258	#1900	C₂×C₄₁	1	2	yes	complemented
259	#1815	C₂×C₄₁	1	1	yes	characteristic, complemented
260	C₂×C₄₁	—	2	1	no	—
261	C₂×C₄₁	—	2	1	no	—
262	C₂×C₄₁	—	4	2	no	—
263	C₂×C₄₁	—	8	1	no	—
264	C₂×C₄₁	—	16	2	no	—
265	C₂×C₄₁	#2416	1	1	yes	characteristic
266	C₂×C₄₁	#3068	1	2	yes	—
267	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
268	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
269	C₂×#131	C₂²×C₄₁	1	1	yes	characteristic
270	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
271	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
272	#525	C₂²×C₄₁	1	1	yes	characteristic
273	C₂×#121	C₂²×C₄₁	1	2	yes	—
274	#472	—	2	4	no	—
275	#469	—	2	2	no	—
276	#469	—	2	2	no	—
277	#469	—	2	2	no	—
278	#468	—	2	2	no	—
279	#466	—	2	2	no	—
280	#465	—	2	2	no	—
281	#436	—	2	2	no	—
282	#435	—	2	2	no	—
283	#430	—	2	4	no	—
284	C₂×#108	C₂²×C₄₁	1	2	yes	—
285	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
286	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
287	#428	C₂²×C₄₁	1	2	yes	—
288	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
289	#424	C₂²×C₄₁	1	1	yes	characteristic
290	#418	C₂²×C₄₁	1	2	yes	—
291	#417	C₂²×C₄₁	1	2	yes	—
292	C₂×#102	C₂²×C₄₁	1	2	yes	—
293	C₂×#100	C₂²×C₄₁	1	2	yes	—
294	C₂×#100	C₂²×C₄₁	1	2	yes	—
295	#408	C₂²×C₄₁	1	1	yes	characteristic
296	#407	C₂²×C₄₁	1	1	yes	characteristic
297	#405	C₂²×C₄₁	1	1	yes	characteristic
298	#343	C₂²×C₄₁	1	2	yes	—
299	#343	C₂²×C₄₁	1	2	yes	—
300	C₄₁	#24806	1	1	yes	characteristic, direct product factor, Sylow
301	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
302	C₂²×D₄	—	2	2	no	—
303	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
304	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
305	AGL(1,ℤ/8)	—	2	4	no	—
306	AGL(1,ℤ/8)	—	4	2	no	—
307	C₂×SD₁₆	—	2	2	no	—
308	C₂×D₈	—	2	2	no	—
309	C₂×D₈	—	2	1	no	—
310	C₂×D₈	—	2	1	no	—
311	C₂×D₈	—	2	2	no	—
312	C₂×D₈	—	2	2	no	—
313	#131	—	2	2	no	—
314	#131	—	2	2	no	—
315	C₂×M₁₆	—	2	2	no	—
316	C₂×M₁₆	—	2	2	no	—
317	C₂×M₁₆	—	2	1	no	—
318	C₂×M₁₆	—	2	2	no	—
319	C₂×M₁₆	—	2	4	no	—
320	C₂×M₁₆	—	4	2	no	—
321	C₂×M₁₆	D₄×C₄₁	1	2	yes	—
322	C₂×M₁₆	D₄×C₄₁	1	2	yes	—
323	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
324	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
325	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
326	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
327	C₂²×C₈	—	2	2	no	—
328	C₂²×C₈	—	2	1	no	—
329	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
330	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
331	#123	—	2	1	no	—
332	C₂²⋊Q₈	D₄×C₄₁	1	2	yes	—
333	#121	—	2	2	no	—
334	#121	—	2	2	no	—
335	#121	—	2	2	no	—
336	#121	—	2	2	no	—
337	#121	D₄×C₄₁	1	2	yes	—
338	#120	—	2	1	no	—
339	C₄×D₄	—	2	2	no	—
340	#117	C₂³×C₄₁	1	1	yes	characteristic
341	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	#108	—	2	4	no	—
344	#108	—	2	2	no	—
345	#108	D₄×C₄₁	1	4	yes	complemented
346	#107	—	2	2	no	—
347	#106	—	2	1	no	—
348	#106	—	2	1	no	—
349	#106	—	2	2	no	—
350	C₄⋊C₈	—	2	2	no	—
351	#103	—	2	2	no	—
352	#102	—	2	2	no	—
353	#102	—	2	4	no	—
354	#102	—	2	2	no	—
355	#102	—	2	2	no	—
356	#100	—	2	4	no	—
357	#100	—	4	2	no	—
358	C₂²⋊C₈	—	2	2	no	—
359	C₂²⋊C₈	—	2	2	no	—
360	#97	—	2	1	no	—
361	C₄×C₈	—	2	1	no	—
362	C₂⁴	—	2	2	no	—
363	C₂⁴	#41×C₄₁	1	1	yes	characteristic
364	#41	—	2	1	no	—
365	#41	—	2	1	no	—
366	#41	—	2	1	no	—
367	#41	—	2	1	no	—
368	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	2	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	2	no	—
377	C₂×D₄	—	4	2	no	—
378	C₂×D₄	—	4	2	no	—
379	C₂×D₄	—	8	4	no	—
380	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	2	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
388	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
389	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
390	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
391	SD₁₆	—	4	4	no	—
392	D₈	—	4	4	no	—
393	D₈	—	4	2	no	—
394	D₈	—	4	2	no	—
395	M₁₆	—	2	2	no	—
396	M₁₆	—	2	4	no	—
397	M₁₆	—	2	4	no	—
398	M₁₆	—	2	1	no	—
399	M₁₆	—	2	2	no	—
400	M₁₆	—	2	1	no	—
401	M₁₆	—	2	1	no	—
402	M₁₆	—	2	2	no	—
403	M₁₆	—	2	1	no	—
404	M₁₆	—	4	4	no	—
405	M₁₆	—	4	2	no	—
406	M₁₆	C₂×D₄×C₄₁	1	4	yes	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	1	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	4	4	no	—
414	C₂×C₈	—	4	2	no	—
415	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
416	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
417	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
418	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
419	C₂×C₈	#41×C₄₁	1	2	yes	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
424	C₂²⋊C₄	—	2	2	no	—
425	C₂²⋊C₄	—	2	1	no	—
426	C₂²⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	4	2	no	—
428	C₂²⋊C₄	—	4	2	no	—
429	C₄²	—	2	1	no	—
430	C₂³	—	2	1	no	—
431	C₂³	—	2	1	no	—
432	C₂³	—	2	1	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	2	no	—
436	C₂³	—	4	2	no	—
437	C₂³	—	8	2	no	—
438	C₂³	—	8	2	no	—
439	C₂³	#121×C₄₁	1	2	yes	—
440	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
441	Q₈	—	2	1	no	—
442	Q₈	—	2	1	no	—
443	D₄	—	2	1	no	—
444	D₄	—	2	1	no	—
445	D₄	—	4	1	no	—
446	D₄	—	4	1	no	—
447	D₄	—	4	2	no	—
448	D₄	—	4	2	no	—
449	D₄	—	4	2	no	—
450	D₄	—	4	2	no	—
451	D₄	—	8	2	no	—
452	D₄	—	8	4	no	—
453	D₄	—	8	4	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	1	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	8	2	no	—
466	C₂×C₄	#121×C₄₁	1	2	yes	—
467	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
468	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
469	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
470	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
471	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
472	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
473	C₈	—	2	2	no	—
474	C₈	—	2	4	no	—
475	C₈	—	4	1	no	—
476	C₈	—	4	1	no	—
477	C₈	—	4	2	no	—
478	C₈	#121×C₄₁	1	2	yes	—
479	C₈	#121×C₄₁	1	2	yes	—
480	C₂²	—	2	2	no	—
481	C₂²	—	2	2	no	—
482	C₂²	—	2	2	no	—
483	C₂²	—	2	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	4	1	no	—
486	C₂²	—	8	1	no	—
487	C₂²	—	8	1	no	—
488	C₂²	—	8	2	no	—
489	C₂²	—	8	2	no	—
490	C₂²	—	8	2	no	—
491	C₂²	—	16	2	no	—
492	C₂²	—	16	2	no	—
493	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
494	C₂²	C₄₁×#548	1	1	yes	characteristic
495	C₂²	C₄₁×#553	1	1	yes	characteristic
496	C₄	—	2	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	4	1	no	—
499	C₄	—	8	2	no	—
500	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
501	C₄	C₄₁×#537	1	1	yes	characteristic
502	C₄	D₄²×C₄₁	1	1	yes	characteristic
503	C₄	C₄₁×#547	1	1	yes	characteristic
504	C₂	—	2	1	no	—
505	C₂	—	2	1	no	—
506	C₂	—	4	2	no	—
507	C₂	—	8	1	no	—
508	C₂	—	16	2	no	—
509	C₂	C₄₁×#2416	1	1	yes	characteristic
510	C₂	C₄₁×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#472	C₂	semidirect
2. C₄₁×#3055	C₂	semidirect
3. C₂×C₄₁×#469	C₂	semidirect
4. C₄₁×#2930	C₂	semidirect
5. C₂×C₄₁×#430	C₂	semidirect
6. C₄₁×#2063	C₂	semidirect
7. C₄₁×#2051	C₂	semidirect
8. C₄₁×#1978	C₂	semidirect
9. C₄₁×#1951	C₂	semidirect
10. C₄₁×#1900	C₂	semidirect
11. C₄₁×#1815	C₂	semidirect
89. #108×C₄₁	D₄	semidirect
224. #24806	C₄₁	direct (N × H)
249. C₂×#472	C₂×C₄₁	semidirect
250. #3055	C₂×C₄₁	semidirect
251. C₂×#469	C₂×C₄₁	semidirect
252. #2930	C₂×C₄₁	semidirect
253. C₂×#430	C₂×C₄₁	semidirect
254. #2063	C₂×C₄₁	semidirect
255. #2051	C₂×C₄₁	semidirect
256. #1978	C₂×C₄₁	semidirect
257. #1951	C₂×C₄₁	semidirect
258. #1900	C₂×C₄₁	semidirect
259. #1815	C₂×C₄₁	semidirect
300. C₄₁	#24806	direct (N × H)
345. #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
C₂×C₄₁×#472	2
C₄₁×#3055	2
C₂×C₄₁×#469	2
C₄₁×#2930	2
C₂×C₄₁×#430	2
C₄₁×#2063	2
C₄₁×#2051	2
C₄₁×#1978	2
C₄₁×#1951	2
C₄₁×#1900	2
C₄₁×#1815	2
#24806	41

Subgroups — C41×#24806 #23801007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24806 #23801007