Subgroups — C₄₁×#24286 #23800487

Order 10496.

1322 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3584	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3054	C₂	1	2	yes	maximal, complemented
3	C₄₁×#3025	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2930	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2053	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2010	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1976	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1812	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1794	C₂	1	2	yes	maximal, complemented
10	C₄₁×#585	C₂²	1	2	yes	—
11	C₄₁×#578	—	2	2	no	—
12	C₄₁×#577	—	2	4	no	—
13	C₄₁×#576	—	2	2	no	—
14	C₄₁×#575	—	2	2	no	—
15	C₄₁×#575	C₂²	1	2	yes	—
16	C₂×#135×C₄₁	—	2	2	no	—
17	C₂×#135×C₄₁	C₂²	1	2	yes	—
18	C₂×#131×C₄₁	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic, complemented
20	C₄₁×#525	C₂²	1	1	yes	characteristic
21	C₄₁×#525	C₂²	1	4	yes	complemented
22	C₄₁×#470	—	2	2	no	—
23	C₄₁×#469	—	2	2	no	—
24	C₄₁×#468	—	2	4	no	—
25	C₄₁×#458	—	2	4	no	—
26	C₄₁×B(4,𝔽₂)	—	2	4	no	—
27	C₄₁×#436	—	2	2	no	—
28	C₄₁×#435	—	2	2	no	—
29	C₄₁×#427	C₂²	1	2	yes	complemented
30	C₄₁×#424	C₂²	1	1	yes	characteristic, complemented
31	C₄₁×#421	C₂²	1	4	yes	complemented
32	C₄₁×#416	C₂²	1	4	yes	complemented
33	C₄₁×#410	C₂²	1	2	yes	—
34	C₄₁×#410	C₂²	1	4	yes	complemented
35	C₄₁×#408	C₂²	1	1	yes	characteristic
36	C₄₁×#407	C₂²	1	1	yes	characteristic, complemented
37	C₄₁×#405	C₂²	1	1	yes	characteristic, complemented
38	C₄₁×H(3,ℤ/4)	C₂²	1	2	yes	complemented
39	2₋¹⁺⁴×C₄₁	—	2	4	no	—
40	2₋¹⁺⁴×C₄₁	—	2	2	no	—
41	H(5,𝔽₂)×C₄₁	—	2	4	no	—
42	H(5,𝔽₂)×C₄₁	—	2	4	no	—
43	H(5,𝔽₂)×C₄₁	—	2	2	no	—
44	C₂×#41×C₄₁	—	2	4	no	—
45	C₂×#41×C₄₁	—	4	4	no	—
46	C₂×#41×C₄₁	C₂³	1	4	yes	—
47	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
48	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
49	#137×C₄₁	—	2	8	no	—
50	#137×C₄₁	—	2	4	no	—
51	AGL(1,ℤ/8)×C₄₁	—	2	8	no	—
52	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
53	#135×C₄₁	—	2	4	no	—
54	#135×C₄₁	—	2	4	no	—
55	#135×C₄₁	—	2	2	no	—
56	#135×C₄₁	—	2	4	no	—
57	#135×C₄₁	—	2	2	no	—
58	#135×C₄₁	—	2	2	no	—
59	#135×C₄₁	—	2	2	no	—
60	C₂×Q₁₆×C₄₁	—	2	2	no	—
61	C₂×Q₁₆×C₄₁	D₄	1	2	yes	complemented
62	C₂×SD₁₆×C₄₁	—	2	4	no	—
63	C₂×SD₁₆×C₄₁	D₄	1	4	yes	complemented
64	C₂×D₈×C₄₁	—	2	2	no	—
65	C₂×D₈×C₄₁	D₄	1	2	yes	complemented
66	#131×C₄₁	—	2	2	no	—
67	#131×C₄₁	—	2	2	no	—
68	C₂×M₁₆×C₄₁	—	2	1	no	—
69	C₂×M₁₆×C₄₁	—	2	1	no	—
70	C₂×M₁₆×C₄₁	—	2	2	no	—
71	C₂×M₁₆×C₄₁	—	2	2	no	—
72	C₂×M₁₆×C₄₁	D₄	1	2	yes	complemented
73	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
74	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₄₁	D₄	1	2	yes	complemented
76	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
78	#123×C₄₁	—	2	4	no	—
79	#123×C₄₁	D₄	1	2	yes	—
80	C₂²⋊Q₈×C₄₁	—	2	1	no	—
81	C₂²⋊Q₈×C₄₁	—	2	4	no	—
82	#121×C₄₁	—	2	1	no	—
83	#121×C₄₁	—	2	4	no	—
84	#120×C₄₁	—	2	4	no	—
85	#120×C₄₁	D₄	1	2	yes	—
86	C₄×D₄×C₄₁	—	2	2	no	—
87	C₄×D₄×C₄₁	—	4	4	no	—
88	#117×C₄₁	C₂³	1	4	yes	—
89	#117×C₄₁	C₂³	1	1	yes	characteristic
90	#107×C₄₁	—	2	2	no	—
91	#106×C₄₁	—	2	2	no	—
92	C₄⋊C₈×C₄₁	—	2	2	no	—
93	#104×C₄₁	—	4	4	no	—
94	#103×C₄₁	—	2	4	no	—
95	#102×C₄₁	—	2	4	no	—
96	#99×C₄₁	—	2	4	no	—
97	#99×C₄₁	—	2	8	no	—
98	C₂²⋊C₈×C₄₁	—	2	2	no	—
99	C₂²⋊C₈×C₄₁	—	2	2	no	—
100	#97×C₄₁	—	2	1	no	—
101	C₄×C₈×C₄₁	—	2	1	no	—
102	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
103	#41×C₄₁	—	2	2	no	—
104	#41×C₄₁	—	2	2	no	—
105	#41×C₄₁	—	2	8	no	—
106	#41×C₄₁	—	2	4	no	—
107	#41×C₄₁	—	2	4	no	—
108	#41×C₄₁	—	2	4	no	—
109	#41×C₄₁	—	2	4	no	—
110	#41×C₄₁	—	4	8	no	—
111	#41×C₄₁	—	4	4	no	—
112	#41×C₄₁	—	4	4	no	—
113	#41×C₄₁	—	4	4	no	—
114	#41×C₄₁	—	4	4	no	—
115	C₂×Q₈×C₄₁	—	2	4	no	—
116	C₂×Q₈×C₄₁	—	4	4	no	—
117	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
118	C₂×Q₈×C₄₁	#41	1	1	yes	characteristic
119	C₂×D₄×C₄₁	—	2	4	no	—
120	C₂×D₄×C₄₁	—	2	4	no	—
121	C₂×D₄×C₄₁	—	2	2	no	—
122	C₂×D₄×C₄₁	—	4	4	no	—
123	C₂×D₄×C₄₁	—	4	8	no	—
124	C₂×D₄×C₄₁	—	4	4	no	—
125	C₂×D₄×C₄₁	—	4	4	no	—
126	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
127	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
128	C₂×D₄×C₄₁	#41	1	1	yes	characteristic
129	C₂²×C₄×C₄₁	—	2	1	no	—
130	C₂²×C₄×C₄₁	—	2	2	no	—
131	C₂²×C₄×C₄₁	—	2	1	no	—
132	C₂²×C₄×C₄₁	—	2	2	no	—
133	C₂²×C₄×C₄₁	—	2	4	no	—
134	C₂²×C₄×C₄₁	—	4	4	no	—
135	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
137	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
138	Q₁₆×C₄₁	—	2	4	no	—
139	Q₁₆×C₄₁	—	2	2	no	—
140	Q₁₆×C₄₁	—	2	2	no	—
141	SD₁₆×C₄₁	—	2	8	no	—
142	SD₁₆×C₄₁	—	2	4	no	—
143	SD₁₆×C₄₁	—	2	4	no	—
144	D₈×C₄₁	—	2	4	no	—
145	D₈×C₄₁	—	2	2	no	—
146	D₈×C₄₁	—	2	2	no	—
147	M₁₆×C₄₁	—	2	2	no	—
148	M₁₆×C₄₁	—	2	4	no	—
149	M₁₆×C₄₁	—	2	2	no	—
150	M₁₆×C₄₁	—	4	1	no	—
151	M₁₆×C₄₁	—	4	1	no	—
152	M₁₆×C₄₁	—	4	2	no	—
153	C₂×C₈×C₄₁	—	2	2	no	—
154	C₂×C₈×C₄₁	—	2	2	no	—
155	C₂×C₈×C₄₁	—	2	1	no	—
156	C₂×C₈×C₄₁	—	2	1	no	—
157	C₂×C₈×C₄₁	—	2	2	no	—
158	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
159	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
160	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
161	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
162	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
163	C₄⋊C₄×C₄₁	—	2	2	no	—
164	C₄⋊C₄×C₄₁	—	2	1	no	—
165	C₄⋊C₄×C₄₁	—	2	4	no	—
166	C₄⋊C₄×C₄₁	—	4	4	no	—
167	C₂²⋊C₄×C₄₁	—	2	2	no	—
168	C₂²⋊C₄×C₄₁	—	2	2	no	—
169	C₂²⋊C₄×C₄₁	—	2	4	no	—
170	C₂²⋊C₄×C₄₁	—	4	4	no	—
171	C₂²⋊C₄×C₄₁	—	4	4	no	—
172	C₂²⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
173	C₄²×C₄₁	—	2	1	no	—
174	C₄²×C₄₁	—	2	4	no	—
175	C₂³×C₄₁	—	2	1	no	—
176	C₂³×C₄₁	—	2	2	no	—
177	C₂³×C₄₁	—	2	4	no	—
178	C₂³×C₄₁	—	4	4	no	—
179	C₂³×C₄₁	—	8	1	no	—
180	C₂³×C₄₁	#121	1	2	yes	—
181	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₄₁	—	2	2	no	—
183	Q₈×C₄₁	—	2	4	no	—
184	Q₈×C₄₁	—	2	4	no	—
185	Q₈×C₄₁	—	4	4	no	—
186	D₄×C₄₁	—	2	4	no	—
187	D₄×C₄₁	—	2	4	no	—
188	D₄×C₄₁	—	2	2	no	—
189	D₄×C₄₁	—	2	4	no	—
190	D₄×C₄₁	—	4	4	no	—
191	D₄×C₄₁	—	4	8	no	—
192	D₄×C₄₁	—	4	4	no	—
193	D₄×C₄₁	—	4	4	no	—
194	D₄×C₄₁	—	8	2	no	—
195	D₄×C₄₁	—	8	8	no	—
196	C₂×C₄×C₄₁	—	2	2	no	—
197	C₂×C₄×C₄₁	—	2	2	no	—
198	C₂×C₄×C₄₁	—	2	2	no	—
199	C₂×C₄×C₄₁	—	2	1	no	—
200	C₂×C₄×C₄₁	—	2	2	no	—
201	C₂×C₄×C₄₁	—	2	4	no	—
202	C₂×C₄×C₄₁	—	4	8	no	—
203	C₂×C₄×C₄₁	—	4	4	no	—
204	C₂×C₄×C₄₁	—	4	1	no	—
205	C₂×C₄×C₄₁	—	4	1	no	—
206	C₂×C₄×C₄₁	—	4	4	no	—
207	C₂×C₄×C₄₁	—	4	2	no	—
208	C₂×C₄×C₄₁	—	4	4	no	—
209	C₂×C₄×C₄₁	—	4	4	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	2	yes	—
213	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
215	C₈×C₄₁	—	2	2	no	—
216	C₈×C₄₁	—	4	1	no	—
217	C₈×C₄₁	—	4	1	no	—
218	C₈×C₄₁	#120	1	1	yes	characteristic
219	C₈×C₄₁	#120	1	1	yes	characteristic
220	C₈×C₄₁	#120	1	2	yes	—
221	#24286	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂²×C₄₁	—	2	2	no	—
223	C₂²×C₄₁	—	4	1	no	—
224	C₂²×C₄₁	—	4	2	no	—
225	C₂²×C₄₁	—	4	1	no	—
226	C₂²×C₄₁	—	4	4	no	—
227	C₂²×C₄₁	—	4	1	no	—
228	C₂²×C₄₁	—	4	4	no	—
229	C₂²×C₄₁	—	4	1	no	—
230	C₂²×C₄₁	—	8	4	no	—
231	C₂²×C₄₁	—	16	4	no	—
232	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
233	C₂²×C₄₁	D₄²	1	2	yes	—
234	C₄×C₄₁	—	2	2	no	—
235	C₄×C₄₁	—	4	2	no	—
236	C₄×C₄₁	—	4	1	no	—
237	C₄×C₄₁	—	4	4	no	—
238	C₄×C₄₁	—	8	4	no	—
239	C₄×C₄₁	C₂×#120	1	1	yes	characteristic
240	C₄×C₄₁	#535	1	1	yes	characteristic
241	C₄×C₄₁	#546	1	2	yes	—
242	#3584	C₂×C₄₁	1	1	yes	characteristic, complemented
243	#3054	C₂×C₄₁	1	2	yes	complemented
244	#3025	C₂×C₄₁	1	2	yes	complemented
245	#2930	C₂×C₄₁	1	1	yes	characteristic, complemented
246	#2053	C₂×C₄₁	1	2	yes	complemented
247	#2010	C₂×C₄₁	1	2	yes	complemented
248	#1976	C₂×C₄₁	1	2	yes	complemented
249	#1812	C₂×C₄₁	1	1	yes	characteristic, complemented
250	#1794	C₂×C₄₁	1	2	yes	complemented
251	C₂×C₄₁	—	2	1	no	—
252	C₂×C₄₁	—	2	2	no	—
253	C₂×C₄₁	—	4	2	no	—
254	C₂×C₄₁	—	8	1	no	—
255	C₂×C₄₁	—	8	4	no	—
256	C₂×C₄₁	#2408	1	1	yes	characteristic
257	#585	C₂²×C₄₁	1	2	yes	—
258	#578	—	2	2	no	—
259	#577	—	2	4	no	—
260	#576	—	2	2	no	—
261	#575	—	2	2	no	—
262	#575	C₂²×C₄₁	1	2	yes	—
263	C₂×#135	—	2	2	no	—
264	C₂×#135	C₂²×C₄₁	1	2	yes	—
265	C₂×#131	C₂²×C₄₁	1	1	yes	characteristic
266	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic, complemented
267	#525	C₂²×C₄₁	1	1	yes	characteristic
268	#525	C₂²×C₄₁	1	4	yes	complemented
269	#470	—	2	2	no	—
270	#469	—	2	2	no	—
271	#468	—	2	4	no	—
272	#458	—	2	4	no	—
273	B(4,𝔽₂)	—	2	4	no	—
274	#436	—	2	2	no	—
275	#435	—	2	2	no	—
276	#427	C₂²×C₄₁	1	2	yes	complemented
277	#424	C₂²×C₄₁	1	1	yes	characteristic, complemented
278	#421	C₂²×C₄₁	1	4	yes	complemented
279	#416	C₂²×C₄₁	1	4	yes	complemented
280	#410	C₂²×C₄₁	1	2	yes	—
281	#410	C₂²×C₄₁	1	4	yes	complemented
282	#408	C₂²×C₄₁	1	1	yes	characteristic
283	#407	C₂²×C₄₁	1	1	yes	characteristic, complemented
284	#405	C₂²×C₄₁	1	1	yes	characteristic, complemented
285	H(3,ℤ/4)	C₂²×C₄₁	1	2	yes	complemented
286	C₄₁	#24286	1	1	yes	characteristic, direct product factor, Sylow
287	2₋¹⁺⁴	—	2	4	no	—
288	2₋¹⁺⁴	—	2	2	no	—
289	H(5,𝔽₂)	—	2	4	no	—
290	H(5,𝔽₂)	—	2	4	no	—
291	H(5,𝔽₂)	—	2	2	no	—
292	C₂×#41	—	2	4	no	—
293	C₂×#41	—	4	4	no	—
294	C₂×#41	C₂³×C₄₁	1	4	yes	—
295	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
296	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
297	#137	—	2	8	no	—
298	#137	—	2	4	no	—
299	AGL(1,ℤ/8)	—	2	8	no	—
300	AGL(1,ℤ/8)	—	2	4	no	—
301	#135	—	2	4	no	—
302	#135	—	2	4	no	—
303	#135	—	2	2	no	—
304	#135	—	2	4	no	—
305	#135	—	2	2	no	—
306	#135	—	2	2	no	—
307	#135	—	2	2	no	—
308	C₂×Q₁₆	—	2	2	no	—
309	C₂×Q₁₆	D₄×C₄₁	1	2	yes	complemented
310	C₂×SD₁₆	—	2	4	no	—
311	C₂×SD₁₆	D₄×C₄₁	1	4	yes	complemented
312	C₂×D₈	—	2	2	no	—
313	C₂×D₈	D₄×C₄₁	1	2	yes	complemented
314	#131	—	2	2	no	—
315	#131	—	2	2	no	—
316	C₂×M₁₆	—	2	1	no	—
317	C₂×M₁₆	—	2	1	no	—
318	C₂×M₁₆	—	2	2	no	—
319	C₂×M₁₆	—	2	2	no	—
320	C₂×M₁₆	D₄×C₄₁	1	2	yes	complemented
321	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
322	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
323	C₂²×C₈	D₄×C₄₁	1	2	yes	complemented
324	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
325	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
326	#123	—	2	4	no	—
327	#123	D₄×C₄₁	1	2	yes	—
328	C₂²⋊Q₈	—	2	1	no	—
329	C₂²⋊Q₈	—	2	4	no	—
330	#121	—	2	1	no	—
331	#121	—	2	4	no	—
332	#120	—	2	4	no	—
333	#120	D₄×C₄₁	1	2	yes	—
334	C₄×D₄	—	2	2	no	—
335	C₄×D₄	—	4	4	no	—
336	#117	C₂³×C₄₁	1	4	yes	—
337	#117	C₂³×C₄₁	1	1	yes	characteristic
338	#107	—	2	2	no	—
339	#106	—	2	2	no	—
340	C₄⋊C₈	—	2	2	no	—
341	#104	—	4	4	no	—
342	#103	—	2	4	no	—
343	#102	—	2	4	no	—
344	#99	—	2	4	no	—
345	#99	—	2	8	no	—
346	C₂²⋊C₈	—	2	2	no	—
347	C₂²⋊C₈	—	2	2	no	—
348	#97	—	2	1	no	—
349	C₄×C₈	—	2	1	no	—
350	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
351	#41	—	2	2	no	—
352	#41	—	2	2	no	—
353	#41	—	2	8	no	—
354	#41	—	2	4	no	—
355	#41	—	2	4	no	—
356	#41	—	2	4	no	—
357	#41	—	2	4	no	—
358	#41	—	4	8	no	—
359	#41	—	4	4	no	—
360	#41	—	4	4	no	—
361	#41	—	4	4	no	—
362	#41	—	4	4	no	—
363	C₂×Q₈	—	2	4	no	—
364	C₂×Q₈	—	4	4	no	—
365	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
366	C₂×Q₈	#41×C₄₁	1	1	yes	characteristic
367	C₂×D₄	—	2	4	no	—
368	C₂×D₄	—	2	4	no	—
369	C₂×D₄	—	2	2	no	—
370	C₂×D₄	—	4	4	no	—
371	C₂×D₄	—	4	8	no	—
372	C₂×D₄	—	4	4	no	—
373	C₂×D₄	—	4	4	no	—
374	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
375	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
376	C₂×D₄	#41×C₄₁	1	1	yes	characteristic
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	1	no	—
380	C₂²×C₄	—	2	2	no	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	4	4	no	—
383	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
385	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
386	Q₁₆	—	2	4	no	—
387	Q₁₆	—	2	2	no	—
388	Q₁₆	—	2	2	no	—
389	SD₁₆	—	2	8	no	—
390	SD₁₆	—	2	4	no	—
391	SD₁₆	—	2	4	no	—
392	D₈	—	2	4	no	—
393	D₈	—	2	2	no	—
394	D₈	—	2	2	no	—
395	M₁₆	—	2	2	no	—
396	M₁₆	—	2	4	no	—
397	M₁₆	—	2	2	no	—
398	M₁₆	—	4	1	no	—
399	M₁₆	—	4	1	no	—
400	M₁₆	—	4	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
407	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
408	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
409	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
410	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	1	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	4	4	no	—
415	C₂²⋊C₄	—	2	2	no	—
416	C₂²⋊C₄	—	2	2	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	4	4	no	—
419	C₂²⋊C₄	—	4	4	no	—
420	C₂²⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
421	C₄²	—	2	1	no	—
422	C₄²	—	2	4	no	—
423	C₂³	—	2	1	no	—
424	C₂³	—	2	2	no	—
425	C₂³	—	2	4	no	—
426	C₂³	—	4	4	no	—
427	C₂³	—	8	1	no	—
428	C₂³	#121×C₄₁	1	2	yes	—
429	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
430	Q₈	—	2	2	no	—
431	Q₈	—	2	4	no	—
432	Q₈	—	2	4	no	—
433	Q₈	—	4	4	no	—
434	D₄	—	2	4	no	—
435	D₄	—	2	4	no	—
436	D₄	—	2	2	no	—
437	D₄	—	2	4	no	—
438	D₄	—	4	4	no	—
439	D₄	—	4	8	no	—
440	D₄	—	4	4	no	—
441	D₄	—	4	4	no	—
442	D₄	—	8	2	no	—
443	D₄	—	8	8	no	—
444	C₂×C₄	—	2	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	4	8	no	—
451	C₂×C₄	—	4	4	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	—	4	1	no	—
454	C₂×C₄	—	4	4	no	—
455	C₂×C₄	—	4	2	no	—
456	C₂×C₄	—	4	4	no	—
457	C₂×C₄	—	4	4	no	—
458	C₂×C₄	#121×C₄₁	1	2	yes	—
459	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
461	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
462	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
463	C₈	—	2	2	no	—
464	C₈	—	4	1	no	—
465	C₈	—	4	1	no	—
466	C₈	#120×C₄₁	1	1	yes	characteristic
467	C₈	#120×C₄₁	1	1	yes	characteristic
468	C₈	#120×C₄₁	1	2	yes	—
469	C₂²	—	2	2	no	—
470	C₂²	—	4	1	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	4	1	no	—
473	C₂²	—	4	4	no	—
474	C₂²	—	4	1	no	—
475	C₂²	—	4	4	no	—
476	C₂²	—	4	1	no	—
477	C₂²	—	8	4	no	—
478	C₂²	—	16	4	no	—
479	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
480	C₂²	D₄²×C₄₁	1	2	yes	—
481	C₄	—	2	2	no	—
482	C₄	—	4	2	no	—
483	C₄	—	4	1	no	—
484	C₄	—	4	4	no	—
485	C₄	—	8	4	no	—
486	C₄	C₂×#120×C₄₁	1	1	yes	characteristic
487	C₄	C₄₁×#535	1	1	yes	characteristic
488	C₄	C₄₁×#546	1	2	yes	—
489	C₂	—	2	1	no	—
490	C₂	—	2	2	no	—
491	C₂	—	4	2	no	—
492	C₂	—	8	1	no	—
493	C₂	—	8	4	no	—
494	C₂	C₄₁×#2408	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₄₁	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂×C₄₁	#2408
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3584	C₂	semidirect
2. C₄₁×#3054	C₂	semidirect
3. C₄₁×#3025	C₂	semidirect
4. C₄₁×#2930	C₂	semidirect
5. C₄₁×#2053	C₂	semidirect
6. C₄₁×#2010	C₂	semidirect
7. C₄₁×#1976	C₂	semidirect
8. C₄₁×#1812	C₂	semidirect
9. C₄₁×#1794	C₂	semidirect
19. C₂²×M₁₆×C₄₁	C₂²	semidirect
21. C₄₁×#525	C₂²	semidirect
29. C₄₁×#427	C₂²	semidirect
30. C₄₁×#424	C₂²	semidirect
31. C₄₁×#421	C₂²	semidirect
32. C₄₁×#416	C₂²	semidirect
34. C₄₁×#410	C₂²	semidirect
36. C₄₁×#407	C₂²	semidirect
37. C₄₁×#405	C₂²	semidirect
38. C₄₁×H(3,ℤ/4)	C₂²	semidirect
61. C₂×Q₁₆×C₄₁	D₄	semidirect
63. C₂×SD₁₆×C₄₁	D₄	semidirect
65. C₂×D₈×C₄₁	D₄	semidirect
72. C₂×M₁₆×C₄₁	D₄	semidirect
75. C₂²×C₈×C₄₁	D₄	semidirect
221. #24286	C₄₁	direct (N × H)
242. #3584	C₂×C₄₁	semidirect
243. #3054	C₂×C₄₁	semidirect
244. #3025	C₂×C₄₁	semidirect
245. #2930	C₂×C₄₁	semidirect
246. #2053	C₂×C₄₁	semidirect
247. #2010	C₂×C₄₁	semidirect
248. #1976	C₂×C₄₁	semidirect
249. #1812	C₂×C₄₁	semidirect
250. #1794	C₂×C₄₁	semidirect
266. C₂²×M₁₆	C₂²×C₄₁	semidirect
268. #525	C₂²×C₄₁	semidirect
276. #427	C₂²×C₄₁	semidirect
277. #424	C₂²×C₄₁	semidirect
278. #421	C₂²×C₄₁	semidirect
279. #416	C₂²×C₄₁	semidirect
281. #410	C₂²×C₄₁	semidirect
283. #407	C₂²×C₄₁	semidirect
284. #405	C₂²×C₄₁	semidirect
285. H(3,ℤ/4)	C₂²×C₄₁	semidirect
286. C₄₁	#24286	direct (N × H)
309. C₂×Q₁₆	D₄×C₄₁	semidirect
311. C₂×SD₁₆	D₄×C₄₁	semidirect
313. C₂×D₈	D₄×C₄₁	semidirect
320. C₂×M₁₆	D₄×C₄₁	semidirect
323. C₂²×C₈	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×#3584	2
C₄₁×#3054	2
C₄₁×#3025	2
C₄₁×#2930	2
C₄₁×#2053	2
C₄₁×#2010	2
C₄₁×#1976	2
C₄₁×#1812	2
C₄₁×#1794	2
#24286	41

Subgroups — C41×#24286 #23800487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24286 #23800487