Subgroups — C₄₁×#24000 #23800201

Order 10496.

1170 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3019	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#455	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2883	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2031	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2009	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1911	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1906	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1905	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1885	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1829	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×H(5,𝔽₂)×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#575	C₂²	1	2	yes	—
14	C₂×#137×C₄₁	C₂²	1	2	yes	—
15	C₄₁×#559	—	2	2	no	—
16	C₄₁×#556	—	2	2	no	—
17	C₄₁×#551	—	2	4	no	—
18	C₄₁×#542	—	2	2	no	—
19	C₄₁×#536	C₂²	1	1	yes	characteristic, complemented
20	C₄₁×#535	C₂²	1	1	yes	characteristic
21	C₄₁×#533	C₂²	1	1	yes	characteristic, complemented
22	C₄₁×#529	—	2	2	no	—
23	C₂×#124×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#519	C₂²	1	1	yes	characteristic, complemented
25	C₄₁×#518	C₂²	1	1	yes	characteristic
26	C₄₁×#518	C₂²	1	1	yes	characteristic
27	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic, complemented
28	C₄₁×#455	—	2	2	no	—
29	C₄₁×#455	—	2	2	no	—
30	C₄₁×#455	—	2	2	no	—
31	C₄₁×#453	—	2	2	no	—
32	C₄₁×#421	C₂²	1	2	yes	—
33	C₄₁×#421	C₂²	1	2	yes	complemented
34	C₂×#104×C₄₁	C₂²	1	2	yes	—
35	C₄₁×#419	C₂²	1	2	yes	complemented
36	C₂×#100×C₄₁	C₂²	1	1	yes	characteristic
37	C₂×#100×C₄₁	C₂²	1	1	yes	characteristic
38	C₄₁×#410	C₂²	1	2	yes	—
39	C₄₁×#410	C₂²	1	2	yes	—
40	C₂×#99×C₄₁	—	2	4	no	—
41	C₄₁×#344	C₂²	1	2	yes	complemented
42	C₄₁×#344	C₂²	1	2	yes	complemented
43	C₄₁×#339	C₂²	1	2	yes	complemented
44	C₄₁×H(3,ℤ/4)	C₂²	1	2	yes	complemented
45	H(5,𝔽₂)×C₄₁	—	2	2	no	—
46	H(5,𝔽₂)×C₄₁	—	2	2	no	—
47	H(5,𝔽₂)×C₄₁	—	4	2	no	—
48	C₂×#41×C₄₁	—	2	2	no	—
49	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
50	C₂×#41×C₄₁	C₂³	1	2	yes	—
51	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₄₁	—	2	2	no	—
53	C₂²×D₄×C₄₁	—	2	2	no	—
54	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
55	#137×C₄₁	—	2	4	no	—
56	#137×C₄₁	—	2	2	no	—
57	#137×C₄₁	—	2	2	no	—
58	#137×C₄₁	—	4	2	no	—
59	AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
60	AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
61	#135×C₄₁	—	4	2	no	—
62	C₂×Q₁₆×C₄₁	—	2	2	no	—
63	C₂×SD₁₆×C₄₁	—	2	2	no	—
64	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
65	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
66	C₄⋊Q₈×C₄₁	—	2	1	no	—
67	C₄⋊Q₈×C₄₁	—	2	1	no	—
68	#127×C₄₁	D₄	1	2	yes	—
69	#126×C₄₁	—	2	4	no	—
70	#126×C₄₁	—	2	4	no	—
71	#125×C₄₁	—	2	1	no	—
72	#124×C₄₁	—	2	4	no	—
73	#124×C₄₁	—	2	2	no	—
74	#124×C₄₁	—	2	1	no	—
75	#124×C₄₁	D₄	1	2	yes	—
76	#124×C₄₁	D₄	1	2	yes	—
77	#124×C₄₁	D₄	1	2	yes	—
78	#123×C₄₁	—	2	4	no	—
79	#123×C₄₁	—	2	4	no	—
80	#123×C₄₁	—	2	2	no	—
81	C₂²⋊Q₈×C₄₁	—	2	4	no	—
82	C₂²⋊Q₈×C₄₁	—	2	2	no	—
83	C₂²⋊Q₈×C₄₁	—	2	2	no	—
84	#121×C₄₁	—	2	2	no	—
85	#120×C₄₁	—	2	2	no	—
86	C₄×Q₈×C₄₁	—	2	1	no	—
87	C₄×Q₈×C₄₁	—	2	2	no	—
88	C₄×Q₈×C₄₁	—	2	2	no	—
89	C₄×Q₈×C₄₁	—	2	2	no	—
90	C₄×D₄×C₄₁	—	2	4	no	—
91	C₄×D₄×C₄₁	—	2	1	no	—
92	C₄×D₄×C₄₁	—	2	2	no	—
93	C₄×D₄×C₄₁	—	2	2	no	—
94	C₄×D₄×C₄₁	—	2	2	no	—
95	C₄×D₄×C₄₁	—	4	2	no	—
96	#117×C₄₁	—	2	4	no	—
97	#117×C₄₁	—	2	2	no	—
98	#117×C₄₁	—	2	1	no	—
99	#117×C₄₁	C₂³	1	2	yes	—
100	#117×C₄₁	C₂³	1	1	yes	characteristic
101	#117×C₄₁	C₂³	1	1	yes	characteristic
102	#117×C₄₁	C₂³	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
104	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
105	C₂×C₂²⋊C₄×C₄₁	D₄	1	4	yes	—
106	C₂×C₄²×C₄₁	—	2	1	no	—
107	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
108	#104×C₄₁	—	2	2	no	—
109	#104×C₄₁	—	2	2	no	—
110	#104×C₄₁	—	2	4	no	—
111	#104×C₄₁	—	4	2	no	—
112	#103×C₄₁	—	4	2	no	—
113	#100×C₄₁	—	2	2	no	—
114	#100×C₄₁	—	2	2	no	—
115	#99×C₄₁	—	2	4	no	—
116	#99×C₄₁	—	2	4	no	—
117	C₂⁴×C₄₁	—	2	2	no	—
118	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
119	#41×C₄₁	—	2	2	no	—
120	#41×C₄₁	—	2	2	no	—
121	#41×C₄₁	—	2	2	no	—
122	#41×C₄₁	—	2	2	no	—
123	#41×C₄₁	—	2	2	no	—
124	#41×C₄₁	—	2	2	no	—
125	#41×C₄₁	—	4	4	no	—
126	#41×C₄₁	—	4	4	no	—
127	C₂×Q₈×C₄₁	—	2	2	no	—
128	C₂×Q₈×C₄₁	—	2	1	no	—
129	C₂×Q₈×C₄₁	—	2	2	no	—
130	C₂×Q₈×C₄₁	—	2	2	no	—
131	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
132	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
133	C₂×Q₈×C₄₁	#41	1	2	yes	—
134	C₂×D₄×C₄₁	—	2	4	no	—
135	C₂×D₄×C₄₁	—	2	2	no	—
136	C₂×D₄×C₄₁	—	2	1	no	—
137	C₂×D₄×C₄₁	—	2	4	no	—
138	C₂×D₄×C₄₁	—	2	2	no	—
139	C₂×D₄×C₄₁	—	2	2	no	—
140	C₂×D₄×C₄₁	—	4	4	no	—
141	C₂×D₄×C₄₁	—	4	4	no	—
142	C₂×D₄×C₄₁	—	4	4	no	—
143	C₂×D₄×C₄₁	—	4	4	no	—
144	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
145	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
146	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
147	C₂×D₄×C₄₁	#41	1	2	yes	—
148	C₂²×C₄×C₄₁	—	2	2	no	—
149	C₂²×C₄×C₄₁	—	2	1	no	—
150	C₂²×C₄×C₄₁	—	2	2	no	—
151	C₂²×C₄×C₄₁	—	2	4	no	—
152	C₂²×C₄×C₄₁	—	2	1	no	—
153	C₂²×C₄×C₄₁	—	2	1	no	—
154	C₂²×C₄×C₄₁	—	2	2	no	—
155	C₂²×C₄×C₄₁	—	4	2	no	—
156	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
157	Q₁₆×C₄₁	—	4	4	no	—
158	Q₁₆×C₄₁	—	4	2	no	—
159	SD₁₆×C₄₁	—	4	4	no	—
160	SD₁₆×C₄₁	—	4	2	no	—
161	SD₁₆×C₄₁	—	4	2	no	—
162	D₈×C₄₁	—	4	2	no	—
163	M₁₆×C₄₁	—	2	2	no	—
164	M₁₆×C₄₁	—	2	2	no	—
165	M₁₆×C₄₁	—	2	2	no	—
166	M₁₆×C₄₁	—	2	2	no	—
167	C₂×C₈×C₄₁	—	2	2	no	—
168	C₂×C₈×C₄₁	—	2	2	no	—
169	C₄⋊C₄×C₄₁	—	2	2	no	—
170	C₄⋊C₄×C₄₁	—	2	2	no	—
171	C₄⋊C₄×C₄₁	—	2	4	no	—
172	C₄⋊C₄×C₄₁	—	2	4	no	—
173	C₄⋊C₄×C₄₁	—	2	2	no	—
174	C₄⋊C₄×C₄₁	—	2	2	no	—
175	C₄⋊C₄×C₄₁	—	2	1	no	—
176	C₄⋊C₄×C₄₁	—	2	1	no	—
177	C₄⋊C₄×C₄₁	—	2	1	no	—
178	C₄⋊C₄×C₄₁	—	2	2	no	—
179	C₂²⋊C₄×C₄₁	—	2	2	no	—
180	C₂²⋊C₄×C₄₁	—	2	4	no	—
181	C₂²⋊C₄×C₄₁	—	2	4	no	—
182	C₂²⋊C₄×C₄₁	—	2	4	no	—
183	C₂²⋊C₄×C₄₁	—	2	4	no	—
184	C₂²⋊C₄×C₄₁	—	2	4	no	—
185	C₂²⋊C₄×C₄₁	—	2	1	no	—
186	C₂²⋊C₄×C₄₁	—	2	2	no	—
187	C₂²⋊C₄×C₄₁	—	4	2	no	—
188	C₂²⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
189	C₂²⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
190	C₄²×C₄₁	—	2	2	no	—
191	C₄²×C₄₁	—	2	4	no	—
192	C₄²×C₄₁	—	2	1	no	—
193	C₄²×C₄₁	—	2	1	no	—
194	C₄²×C₄₁	—	2	1	no	—
195	C₄²×C₄₁	—	2	2	no	—
196	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₄₁	C₂×D₄	1	2	yes	—
199	C₂³×C₄₁	—	2	2	no	—
200	C₂³×C₄₁	—	2	2	no	—
201	C₂³×C₄₁	—	2	1	no	—
202	C₂³×C₄₁	—	2	2	no	—
203	C₂³×C₄₁	—	4	4	no	—
204	C₂³×C₄₁	—	8	2	no	—
205	C₂³×C₄₁	—	16	2	no	—
206	C₂³×C₄₁	#120	1	4	yes	—
207	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
208	Q₈×C₄₁	—	2	4	no	—
209	Q₈×C₄₁	—	2	2	no	—
210	Q₈×C₄₁	—	2	2	no	—
211	Q₈×C₄₁	—	2	2	no	—
212	Q₈×C₄₁	—	4	2	no	—
213	Q₈×C₄₁	—	8	2	no	—
214	D₄×C₄₁	—	2	4	no	—
215	D₄×C₄₁	—	2	4	no	—
216	D₄×C₄₁	—	2	2	no	—
217	D₄×C₄₁	—	2	2	no	—
218	D₄×C₄₁	—	2	2	no	—
219	D₄×C₄₁	—	4	8	no	—
220	D₄×C₄₁	—	4	2	no	—
221	D₄×C₄₁	—	4	1	no	—
222	D₄×C₄₁	—	4	2	no	—
223	D₄×C₄₁	—	4	1	no	—
224	D₄×C₄₁	—	8	4	no	—
225	C₂×C₄×C₄₁	—	2	2	no	—
226	C₂×C₄×C₄₁	—	2	1	no	—
227	C₂×C₄×C₄₁	—	2	2	no	—
228	C₂×C₄×C₄₁	—	2	2	no	—
229	C₂×C₄×C₄₁	—	2	1	no	—
230	C₂×C₄×C₄₁	—	2	1	no	—
231	C₂×C₄×C₄₁	—	2	2	no	—
232	C₂×C₄×C₄₁	—	2	2	no	—
233	C₂×C₄×C₄₁	—	4	4	no	—
234	C₂×C₄×C₄₁	—	4	1	no	—
235	C₂×C₄×C₄₁	—	4	1	no	—
236	C₂×C₄×C₄₁	—	4	2	no	—
237	C₂×C₄×C₄₁	—	4	2	no	—
238	C₂×C₄×C₄₁	—	4	2	no	—
239	C₂×C₄×C₄₁	—	4	2	no	—
240	C₂×C₄×C₄₁	—	8	2	no	—
241	C₂×C₄×C₄₁	—	8	4	no	—
242	C₂×C₄×C₄₁	—	8	1	no	—
243	C₂×C₄×C₄₁	—	16	2	no	—
244	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
247	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
248	C₈×C₄₁	—	4	2	no	—
249	C₈×C₄₁	—	4	2	no	—
250	#24000	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂²×C₄₁	—	2	4	no	—
252	C₂²×C₄₁	—	4	1	no	—
253	C₂²×C₄₁	—	4	4	no	—
254	C₂²×C₄₁	—	4	2	no	—
255	C₂²×C₄₁	—	4	2	no	—
256	C₂²×C₄₁	—	4	1	no	—
257	C₂²×C₄₁	—	8	2	no	—
258	C₂²×C₄₁	—	8	2	no	—
259	C₂²×C₄₁	—	8	2	no	—
260	C₂²×C₄₁	—	16	4	no	—
261	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
262	C₂²×C₄₁	#525	1	1	yes	characteristic
263	C₂²×C₄₁	#535	1	1	yes	characteristic
264	C₄×C₄₁	—	4	2	no	—
265	C₄×C₄₁	—	4	1	no	—
266	C₄×C₄₁	—	4	2	no	—
267	C₄×C₄₁	—	4	1	no	—
268	C₄×C₄₁	—	4	2	no	—
269	C₄×C₄₁	—	4	2	no	—
270	C₄×C₄₁	—	4	1	no	—
271	C₄×C₄₁	—	4	2	no	—
272	C₄×C₄₁	—	8	2	no	—
273	C₄×C₄₁	#546	1	2	yes	—
274	C₄×C₄₁	#546	1	2	yes	—
275	#3510	C₂×C₄₁	1	1	yes	characteristic, complemented
276	#3019	C₂×C₄₁	1	1	yes	characteristic
277	C₂×#455	C₂×C₄₁	1	1	yes	characteristic, complemented
278	#2883	C₂×C₄₁	1	1	yes	characteristic, complemented
279	#2031	C₂×C₄₁	1	2	yes	complemented
280	#2009	C₂×C₄₁	1	2	yes	complemented
281	#1911	C₂×C₄₁	1	1	yes	characteristic, complemented
282	#1906	C₂×C₄₁	1	2	yes	complemented
283	#1905	C₂×C₄₁	1	2	yes	complemented
284	#1885	C₂×C₄₁	1	1	yes	characteristic, complemented
285	#1829	C₂×C₄₁	1	1	yes	characteristic, complemented
286	C₂×C₄₁	—	2	1	no	—
287	C₂×C₄₁	—	2	1	no	—
288	C₂×C₄₁	—	2	1	no	—
289	C₂×C₄₁	—	4	4	no	—
290	C₂×C₄₁	—	8	1	no	—
291	C₂×C₄₁	—	8	2	no	—
292	C₂×C₄₁	#2404	1	1	yes	characteristic
293	C₂×H(5,𝔽₂)	C₂²×C₄₁	1	1	yes	characteristic
294	#575	C₂²×C₄₁	1	2	yes	—
295	C₂×#137	C₂²×C₄₁	1	2	yes	—
296	#559	—	2	2	no	—
297	#556	—	2	2	no	—
298	#551	—	2	4	no	—
299	#542	—	2	2	no	—
300	#536	C₂²×C₄₁	1	1	yes	characteristic, complemented
301	#535	C₂²×C₄₁	1	1	yes	characteristic
302	#533	C₂²×C₄₁	1	1	yes	characteristic, complemented
303	#529	—	2	2	no	—
304	C₂×#124	C₂²×C₄₁	1	1	yes	characteristic
305	#519	C₂²×C₄₁	1	1	yes	characteristic, complemented
306	#518	C₂²×C₄₁	1	1	yes	characteristic
307	#518	C₂²×C₄₁	1	1	yes	characteristic
308	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic, complemented
309	#455	—	2	2	no	—
310	#455	—	2	2	no	—
311	#455	—	2	2	no	—
312	#453	—	2	2	no	—
313	#421	C₂²×C₄₁	1	2	yes	—
314	#421	C₂²×C₄₁	1	2	yes	complemented
315	C₂×#104	C₂²×C₄₁	1	2	yes	—
316	#419	C₂²×C₄₁	1	2	yes	complemented
317	C₂×#100	C₂²×C₄₁	1	1	yes	characteristic
318	C₂×#100	C₂²×C₄₁	1	1	yes	characteristic
319	#410	C₂²×C₄₁	1	2	yes	—
320	#410	C₂²×C₄₁	1	2	yes	—
321	C₂×#99	—	2	4	no	—
322	#344	C₂²×C₄₁	1	2	yes	complemented
323	#344	C₂²×C₄₁	1	2	yes	complemented
324	#339	C₂²×C₄₁	1	2	yes	complemented
325	H(3,ℤ/4)	C₂²×C₄₁	1	2	yes	complemented
326	C₄₁	#24000	1	1	yes	characteristic, direct product factor, Sylow
327	H(5,𝔽₂)	—	2	2	no	—
328	H(5,𝔽₂)	—	2	2	no	—
329	H(5,𝔽₂)	—	4	2	no	—
330	C₂×#41	—	2	2	no	—
331	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
332	C₂×#41	C₂³×C₄₁	1	2	yes	—
333	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
334	C₂²×D₄	—	2	2	no	—
335	C₂²×D₄	—	2	2	no	—
336	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
337	#137	—	2	4	no	—
338	#137	—	2	2	no	—
339	#137	—	2	2	no	—
340	#137	—	4	2	no	—
341	AGL(1,ℤ/8)	—	2	2	no	—
342	AGL(1,ℤ/8)	—	2	2	no	—
343	#135	—	4	2	no	—
344	C₂×Q₁₆	—	2	2	no	—
345	C₂×SD₁₆	—	2	2	no	—
346	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
347	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
348	C₄⋊Q₈	—	2	1	no	—
349	C₄⋊Q₈	—	2	1	no	—
350	#127	D₄×C₄₁	1	2	yes	—
351	#126	—	2	4	no	—
352	#126	—	2	4	no	—
353	#125	—	2	1	no	—
354	#124	—	2	4	no	—
355	#124	—	2	2	no	—
356	#124	—	2	1	no	—
357	#124	D₄×C₄₁	1	2	yes	—
358	#124	D₄×C₄₁	1	2	yes	—
359	#124	D₄×C₄₁	1	2	yes	—
360	#123	—	2	4	no	—
361	#123	—	2	4	no	—
362	#123	—	2	2	no	—
363	C₂²⋊Q₈	—	2	4	no	—
364	C₂²⋊Q₈	—	2	2	no	—
365	C₂²⋊Q₈	—	2	2	no	—
366	#121	—	2	2	no	—
367	#120	—	2	2	no	—
368	C₄×Q₈	—	2	1	no	—
369	C₄×Q₈	—	2	2	no	—
370	C₄×Q₈	—	2	2	no	—
371	C₄×Q₈	—	2	2	no	—
372	C₄×D₄	—	2	4	no	—
373	C₄×D₄	—	2	1	no	—
374	C₄×D₄	—	2	2	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	2	2	no	—
377	C₄×D₄	—	4	2	no	—
378	#117	—	2	4	no	—
379	#117	—	2	2	no	—
380	#117	—	2	1	no	—
381	#117	C₂³×C₄₁	1	2	yes	—
382	#117	C₂³×C₄₁	1	1	yes	characteristic
383	#117	C₂³×C₄₁	1	1	yes	characteristic
384	#117	C₂³×C₄₁	1	1	yes	characteristic
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	D₄×C₄₁	1	4	yes	—
388	C₂×C₄²	—	2	1	no	—
389	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
390	#104	—	2	2	no	—
391	#104	—	2	2	no	—
392	#104	—	2	4	no	—
393	#104	—	4	2	no	—
394	#103	—	4	2	no	—
395	#100	—	2	2	no	—
396	#100	—	2	2	no	—
397	#99	—	2	4	no	—
398	#99	—	2	4	no	—
399	C₂⁴	—	2	2	no	—
400	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
401	#41	—	2	2	no	—
402	#41	—	2	2	no	—
403	#41	—	2	2	no	—
404	#41	—	2	2	no	—
405	#41	—	2	2	no	—
406	#41	—	2	2	no	—
407	#41	—	4	4	no	—
408	#41	—	4	4	no	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	—	2	2	no	—
413	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
414	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
415	C₂×Q₈	#41×C₄₁	1	2	yes	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	4	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	4	4	no	—
423	C₂×D₄	—	4	4	no	—
424	C₂×D₄	—	4	4	no	—
425	C₂×D₄	—	4	4	no	—
426	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
427	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
428	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
429	C₂×D₄	#41×C₄₁	1	2	yes	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	2	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	1	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	4	2	no	—
438	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
439	Q₁₆	—	4	4	no	—
440	Q₁₆	—	4	2	no	—
441	SD₁₆	—	4	4	no	—
442	SD₁₆	—	4	2	no	—
443	SD₁₆	—	4	2	no	—
444	D₈	—	4	2	no	—
445	M₁₆	—	2	2	no	—
446	M₁₆	—	2	2	no	—
447	M₁₆	—	2	2	no	—
448	M₁₆	—	2	2	no	—
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	2	2	no	—
453	C₄⋊C₄	—	2	4	no	—
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	2	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₂²⋊C₄	—	2	2	no	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₂²⋊C₄	—	2	4	no	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	1	no	—
468	C₂²⋊C₄	—	2	2	no	—
469	C₂²⋊C₄	—	4	2	no	—
470	C₂²⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
471	C₂²⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
472	C₄²	—	2	2	no	—
473	C₄²	—	2	4	no	—
474	C₄²	—	2	1	no	—
475	C₄²	—	2	1	no	—
476	C₄²	—	2	1	no	—
477	C₄²	—	2	2	no	—
478	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
479	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
480	C₄²	C₂×D₄×C₄₁	1	2	yes	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	1	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	4	4	no	—
486	C₂³	—	8	2	no	—
487	C₂³	—	16	2	no	—
488	C₂³	#120×C₄₁	1	4	yes	—
489	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
490	Q₈	—	2	4	no	—
491	Q₈	—	2	2	no	—
492	Q₈	—	2	2	no	—
493	Q₈	—	2	2	no	—
494	Q₈	—	4	2	no	—
495	Q₈	—	8	2	no	—
496	D₄	—	2	4	no	—
497	D₄	—	2	4	no	—
498	D₄	—	2	2	no	—
499	D₄	—	2	2	no	—
500	D₄	—	2	2	no	—
501	D₄	—	4	8	no	—
502	D₄	—	4	2	no	—
503	D₄	—	4	1	no	—
504	D₄	—	4	2	no	—
505	D₄	—	4	1	no	—
506	D₄	—	8	4	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	4	4	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	2	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	8	2	no	—
523	C₂×C₄	—	8	4	no	—
524	C₂×C₄	—	8	1	no	—
525	C₂×C₄	—	16	2	no	—
526	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
528	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
529	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
530	C₈	—	4	2	no	—
531	C₈	—	4	2	no	—
532	C₂²	—	2	4	no	—
533	C₂²	—	4	1	no	—
534	C₂²	—	4	4	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	1	no	—
538	C₂²	—	8	2	no	—
539	C₂²	—	8	2	no	—
540	C₂²	—	8	2	no	—
541	C₂²	—	16	4	no	—
542	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
543	C₂²	C₄₁×#525	1	1	yes	characteristic
544	C₂²	C₄₁×#535	1	1	yes	characteristic
545	C₄	—	4	2	no	—
546	C₄	—	4	1	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	1	no	—
549	C₄	—	4	2	no	—
550	C₄	—	4	2	no	—
551	C₄	—	4	1	no	—
552	C₄	—	4	2	no	—
553	C₄	—	8	2	no	—
554	C₄	C₄₁×#546	1	2	yes	—
555	C₄	C₄₁×#546	1	2	yes	—
556	C₂	—	2	1	no	—
557	C₂	—	2	1	no	—
558	C₂	—	2	1	no	—
559	C₂	—	4	4	no	—
560	C₂	—	8	1	no	—
561	C₂	—	8	2	no	—
562	C₂	C₄₁×#2404	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁	#2404
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#3019	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3510	C₂	semidirect
3. C₂×C₄₁×#455	C₂	semidirect
4. C₄₁×#2883	C₂	semidirect
5. C₄₁×#2031	C₂	semidirect
6. C₄₁×#2009	C₂	semidirect
7. C₄₁×#1911	C₂	semidirect
8. C₄₁×#1906	C₂	semidirect
9. C₄₁×#1905	C₂	semidirect
10. C₄₁×#1885	C₂	semidirect
11. C₄₁×#1829	C₂	semidirect
19. C₄₁×#536	C₂²	semidirect
21. C₄₁×#533	C₂²	semidirect
24. C₄₁×#519	C₂²	semidirect
27. C₄×#41×C₄₁	C₂²	semidirect
33. C₄₁×#421	C₂²	semidirect
35. C₄₁×#419	C₂²	semidirect
41. C₄₁×#344	C₂²	semidirect
42. C₄₁×#344	C₂²	semidirect
43. C₄₁×#339	C₂²	semidirect
44. C₄₁×H(3,ℤ/4)	C₂²	semidirect
250. #24000	C₄₁	direct (N × H)
275. #3510	C₂×C₄₁	semidirect
277. C₂×#455	C₂×C₄₁	semidirect
278. #2883	C₂×C₄₁	semidirect
279. #2031	C₂×C₄₁	semidirect
280. #2009	C₂×C₄₁	semidirect
281. #1911	C₂×C₄₁	semidirect
282. #1906	C₂×C₄₁	semidirect
283. #1905	C₂×C₄₁	semidirect
284. #1885	C₂×C₄₁	semidirect
285. #1829	C₂×C₄₁	semidirect
300. #536	C₂²×C₄₁	semidirect
302. #533	C₂²×C₄₁	semidirect
305. #519	C₂²×C₄₁	semidirect
308. C₄×#41	C₂²×C₄₁	semidirect
314. #421	C₂²×C₄₁	semidirect
316. #419	C₂²×C₄₁	semidirect
322. #344	C₂²×C₄₁	semidirect
323. #344	C₂²×C₄₁	semidirect
324. #339	C₂²×C₄₁	semidirect
325. H(3,ℤ/4)	C₂²×C₄₁	semidirect
326. C₄₁	#24000	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#3510	2
C₄₁×#3019	2
C₂×C₄₁×#455	2
C₄₁×#2883	2
C₄₁×#2031	2
C₄₁×#2009	2
C₄₁×#1911	2
C₄₁×#1906	2
C₄₁×#1905	2
C₄₁×#1885	2
C₄₁×#1829	2
#24000	41

Subgroups — C41×#24000 #23800201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24000 #23800201