Subgroups — C₄₁×#23991 #23800192

Order 10496.

1164 conjugacy classes of subgroups.

The subgroup lattice has 570 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×#3027	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#3022	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2901	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2031	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2004	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1914	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1911	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1909	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1905	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1827	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#585	C₂²	1	1	yes	characteristic
13	C₂×#137×C₄₁	C₂²	1	2	yes	—
14	C₄₁×#568	C₂²	1	1	yes	characteristic
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
20	C₄₁×#535	C₂²	1	2	yes	—
21	C₄₁×#535	C₂²	1	1	yes	characteristic
22	C₄₁×#533	C₂²	1	1	yes	characteristic, complemented
23	C₄₁×#529	—	2	2	no	—
24	C₂×#124×C₄₁	C₂²	1	1	yes	characteristic, complemented
25	C₄₁×#519	C₂²	1	1	yes	characteristic, complemented
26	C₄₁×#518	C₂²	1	1	yes	characteristic, complemented
27	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
28	C₄₁×B(4,𝔽₂)	—	2	4	no	—
29	C₄₁×#456	—	2	2	no	—
30	C₄₁×#455	—	2	2	no	—
31	C₄₁×#421	—	2	2	no	—
32	C₄₁×#421	C₂²	1	2	yes	—
33	C₄₁×#421	C₂²	1	2	yes	—
34	C₂×#104×C₄₁	—	2	2	no	—
35	C₂×#104×C₄₁	C₂²	1	2	yes	—
36	C₄₁×#419	C₂²	1	2	yes	complemented
37	C₄₁×#413	C₂²	1	1	yes	characteristic
38	C₂×#100×C₄₁	C₂²	1	2	yes	complemented
39	C₄₁×#410	C₂²	1	2	yes	—
40	C₄₁×#410	C₂²	1	1	yes	characteristic
41	C₄₁×#344	C₂²	1	2	yes	complemented
42	C₄₁×#339	C₂²	1	2	yes	complemented
43	C₄₁×H(3,ℤ/4)	C₂²	1	2	yes	complemented
44	C₄₁×H(3,ℤ/4)	C₂²	1	2	yes	complemented
45	2₋¹⁺⁴×C₄₁	—	2	2	no	—
46	2₋¹⁺⁴×C₄₁	—	2	1	no	—
47	H(5,𝔽₂)×C₄₁	—	2	2	no	—
48	H(5,𝔽₂)×C₄₁	—	2	2	no	—
49	H(5,𝔽₂)×C₄₁	—	2	1	no	—
50	C₂×#41×C₄₁	—	2	2	no	—
51	C₂×#41×C₄₁	—	4	2	no	—
52	C₂×#41×C₄₁	C₂³	1	2	yes	—
53	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
54	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
55	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
56	#137×C₄₁	—	2	4	no	—
57	#137×C₄₁	—	4	2	no	—
58	C₂×Q₁₆×C₄₁	—	2	2	no	—
59	C₂×SD₁₆×C₄₁	—	2	2	no	—
60	#131×C₄₁	—	4	2	no	—
61	C₂×M₁₆×C₄₁	—	2	2	no	—
62	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
63	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₄₁	—	2	1	no	—
65	C₄⋊Q₈×C₄₁	D₄	1	2	yes	complemented
66	#127×C₄₁	—	2	1	no	—
67	#127×C₄₁	—	2	2	no	—
68	#126×C₄₁	—	2	4	no	—
69	#126×C₄₁	—	2	4	no	—
70	#125×C₄₁	—	2	1	no	—
71	#124×C₄₁	—	2	4	no	—
72	#124×C₄₁	—	2	2	no	—
73	#124×C₄₁	—	2	1	no	—
74	#124×C₄₁	—	2	2	no	—
75	#124×C₄₁	—	2	1	no	—
76	#124×C₄₁	—	2	1	no	—
77	#124×C₄₁	D₄	1	2	yes	complemented
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	#121×C₄₁	—	4	2	no	—
86	#120×C₄₁	—	2	4	no	—
87	#120×C₄₁	D₄	1	4	yes	—
88	C₄×Q₈×C₄₁	—	2	1	no	—
89	C₄×Q₈×C₄₁	—	2	2	no	—
90	C₄×Q₈×C₄₁	—	2	2	no	—
91	C₄×Q₈×C₄₁	—	2	2	no	—
92	C₄×D₄×C₄₁	—	2	4	no	—
93	C₄×D₄×C₄₁	—	2	2	no	—
94	C₄×D₄×C₄₁	—	2	1	no	—
95	C₄×D₄×C₄₁	—	2	2	no	—
96	C₄×D₄×C₄₁	—	2	2	no	—
97	#117×C₄₁	—	2	4	no	—
98	#117×C₄₁	—	2	2	no	—
99	#117×C₄₁	D₄	1	2	yes	complemented
100	#117×C₄₁	C₂³	1	2	yes	—
101	#117×C₄₁	C₂³	1	1	yes	characteristic
102	#117×C₄₁	C₂³	1	1	yes	characteristic
103	#117×C₄₁	C₂³	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
105	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
106	C₂×C₄²×C₄₁	D₄	1	2	yes	complemented
107	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
108	#104×C₄₁	—	2	2	no	—
109	#104×C₄₁	—	2	4	no	—
110	#104×C₄₁	—	2	2	no	—
111	#104×C₄₁	—	2	4	no	—
112	#103×C₄₁	—	4	2	no	—
113	#101×C₄₁	—	2	1	no	—
114	#100×C₄₁	—	2	1	no	—
115	#100×C₄₁	—	4	2	no	—
116	#99×C₄₁	—	2	2	no	—
117	#99×C₄₁	—	2	4	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	4	no	—
122	#41×C₄₁	—	2	2	no	—
123	#41×C₄₁	—	2	2	no	—
124	#41×C₄₁	—	2	2	no	—
125	#41×C₄₁	—	2	2	no	—
126	#41×C₄₁	—	4	4	no	—
127	#41×C₄₁	—	4	2	no	—
128	#41×C₄₁	—	4	2	no	—
129	#41×C₄₁	—	4	2	no	—
130	#41×C₄₁	—	4	2	no	—
131	C₂×Q₈×C₄₁	—	2	2	no	—
132	C₂×Q₈×C₄₁	—	2	1	no	—
133	C₂×Q₈×C₄₁	—	2	2	no	—
134	C₂×Q₈×C₄₁	—	2	2	no	—
135	C₂×Q₈×C₄₁	—	2	2	no	—
136	C₂×Q₈×C₄₁	—	4	2	no	—
137	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
138	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
139	C₂×Q₈×C₄₁	#41	1	2	yes	—
140	C₂×D₄×C₄₁	—	2	2	no	—
141	C₂×D₄×C₄₁	—	2	2	no	—
142	C₂×D₄×C₄₁	—	2	4	no	—
143	C₂×D₄×C₄₁	—	2	1	no	—
144	C₂×D₄×C₄₁	—	2	1	no	—
145	C₂×D₄×C₄₁	—	2	2	no	—
146	C₂×D₄×C₄₁	—	2	2	no	—
147	C₂×D₄×C₄₁	—	4	2	no	—
148	C₂×D₄×C₄₁	—	4	4	no	—
149	C₂×D₄×C₄₁	—	4	2	no	—
150	C₂×D₄×C₄₁	—	4	4	no	—
151	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
152	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
153	C₂×D₄×C₄₁	#41	1	2	yes	—
154	C₂²×C₄×C₄₁	—	2	1	no	—
155	C₂²×C₄×C₄₁	—	2	4	no	—
156	C₂²×C₄×C₄₁	—	2	2	no	—
157	C₂²×C₄×C₄₁	—	2	1	no	—
158	C₂²×C₄×C₄₁	—	2	2	no	—
159	C₂²×C₄×C₄₁	—	4	2	no	—
160	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
162	Q₁₆×C₄₁	—	4	4	no	—
163	SD₁₆×C₄₁	—	4	4	no	—
164	M₁₆×C₄₁	—	2	2	no	—
165	M₁₆×C₄₁	—	2	2	no	—
166	M₁₆×C₄₁	—	2	2	no	—
167	M₁₆×C₄₁	—	2	2	no	—
168	M₁₆×C₄₁	—	4	2	no	—
169	C₂×C₈×C₄₁	—	2	2	no	—
170	C₂×C₈×C₄₁	—	2	2	no	—
171	C₂×C₈×C₄₁	—	4	2	no	—
172	C₄⋊C₄×C₄₁	—	2	2	no	—
173	C₄⋊C₄×C₄₁	—	2	4	no	—
174	C₄⋊C₄×C₄₁	—	2	2	no	—
175	C₄⋊C₄×C₄₁	—	2	2	no	—
176	C₄⋊C₄×C₄₁	—	2	4	no	—
177	C₄⋊C₄×C₄₁	—	2	2	no	—
178	C₄⋊C₄×C₄₁	—	2	1	no	—
179	C₄⋊C₄×C₄₁	—	2	1	no	—
180	C₄⋊C₄×C₄₁	—	2	1	no	—
181	C₄⋊C₄×C₄₁	—	2	2	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	2	no	—
186	C₂²⋊C₄×C₄₁	—	2	4	no	—
187	C₂²⋊C₄×C₄₁	—	2	1	no	—
188	C₂²⋊C₄×C₄₁	—	2	1	no	—
189	C₂²⋊C₄×C₄₁	—	2	4	no	—
190	C₂²⋊C₄×C₄₁	—	2	2	no	—
191	C₂²⋊C₄×C₄₁	—	4	4	no	—
192	C₂²⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
193	C₄²×C₄₁	—	2	4	no	—
194	C₄²×C₄₁	—	2	2	no	—
195	C₄²×C₄₁	—	2	1	no	—
196	C₄²×C₄₁	—	2	1	no	—
197	C₄²×C₄₁	—	2	2	no	—
198	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₄₁	C₂×D₄	1	2	yes	—
200	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₄₁	C₂×D₄	1	2	yes	—
202	C₂³×C₄₁	—	2	2	no	—
203	C₂³×C₄₁	—	2	4	no	—
204	C₂³×C₄₁	—	2	2	no	—
205	C₂³×C₄₁	—	4	2	no	—
206	C₂³×C₄₁	—	8	2	no	—
207	C₂³×C₄₁	#120	1	2	yes	—
208	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
209	Q₈×C₄₁	—	2	2	no	—
210	Q₈×C₄₁	—	2	2	no	—
211	Q₈×C₄₁	—	2	4	no	—
212	Q₈×C₄₁	—	2	2	no	—
213	Q₈×C₄₁	—	4	2	no	—
214	Q₈×C₄₁	—	4	1	no	—
215	Q₈×C₄₁	—	4	1	no	—
216	Q₈×C₄₁	—	8	2	no	—
217	D₄×C₄₁	—	2	4	no	—
218	D₄×C₄₁	—	2	2	no	—
219	D₄×C₄₁	—	2	2	no	—
220	D₄×C₄₁	—	2	2	no	—
221	D₄×C₄₁	—	4	2	no	—
222	D₄×C₄₁	—	4	4	no	—
223	D₄×C₄₁	—	4	2	no	—
224	D₄×C₄₁	—	4	2	no	—
225	D₄×C₄₁	—	4	1	no	—
226	D₄×C₄₁	—	4	1	no	—
227	D₄×C₄₁	—	4	2	no	—
228	D₄×C₄₁	—	8	4	no	—
229	D₄×C₄₁	—	16	4	no	—
230	C₂×C₄×C₄₁	—	2	2	no	—
231	C₂×C₄×C₄₁	—	2	2	no	—
232	C₂×C₄×C₄₁	—	2	2	no	—
233	C₂×C₄×C₄₁	—	2	1	no	—
234	C₂×C₄×C₄₁	—	2	2	no	—
235	C₂×C₄×C₄₁	—	2	2	no	—
236	C₂×C₄×C₄₁	—	2	1	no	—
237	C₂×C₄×C₄₁	—	2	2	no	—
238	C₂×C₄×C₄₁	—	2	2	no	—
239	C₂×C₄×C₄₁	—	4	4	no	—
240	C₂×C₄×C₄₁	—	4	2	no	—
241	C₂×C₄×C₄₁	—	4	2	no	—
242	C₂×C₄×C₄₁	—	4	2	no	—
243	C₂×C₄×C₄₁	—	4	2	no	—
244	C₂×C₄×C₄₁	—	4	2	no	—
245	C₂×C₄×C₄₁	—	8	2	no	—
246	C₂×C₄×C₄₁	—	8	4	no	—
247	C₂×C₄×C₄₁	—	8	1	no	—
248	C₂×C₄×C₄₁	#120	1	2	yes	—
249	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
252	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
253	C₈×C₄₁	—	4	2	no	—
254	C₈×C₄₁	—	4	2	no	—
255	#23991	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
256	C₂²×C₄₁	—	2	2	no	—
257	C₂²×C₄₁	—	4	1	no	—
258	C₂²×C₄₁	—	4	2	no	—
259	C₂²×C₄₁	—	4	4	no	—
260	C₂²×C₄₁	—	4	2	no	—
261	C₂²×C₄₁	—	4	2	no	—
262	C₂²×C₄₁	—	8	2	no	—
263	C₂²×C₄₁	—	8	4	no	—
264	C₂²×C₄₁	—	16	2	no	—
265	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
266	C₂²×C₄₁	#546	1	2	yes	—
267	C₄×C₄₁	—	2	2	no	—
268	C₄×C₄₁	—	4	2	no	—
269	C₄×C₄₁	—	4	1	no	—
270	C₄×C₄₁	—	4	2	no	—
271	C₄×C₄₁	—	4	1	no	—
272	C₄×C₄₁	—	4	2	no	—
273	C₄×C₄₁	—	4	2	no	—
274	C₄×C₄₁	—	4	2	no	—
275	C₄×C₄₁	—	8	2	no	—
276	C₄×C₄₁	#525	1	1	yes	characteristic
277	C₄×C₄₁	#535	1	1	yes	characteristic
278	C₄×C₄₁	#546	1	2	yes	—
279	#3510	C₂×C₄₁	1	1	yes	characteristic, complemented
280	#3027	C₂×C₄₁	1	1	yes	characteristic
281	#3022	C₂×C₄₁	1	1	yes	characteristic, complemented
282	#2901	C₂×C₄₁	1	1	yes	characteristic, complemented
283	#2031	C₂×C₄₁	1	2	yes	complemented
284	#2004	C₂×C₄₁	1	2	yes	complemented
285	#1914	C₂×C₄₁	1	1	yes	characteristic, complemented
286	#1911	C₂×C₄₁	1	2	yes	complemented
287	#1909	C₂×C₄₁	1	1	yes	characteristic, complemented
288	#1905	C₂×C₄₁	1	2	yes	complemented
289	#1827	C₂×C₄₁	1	1	yes	characteristic, complemented
290	C₂×C₄₁	—	2	1	no	—
291	C₂×C₄₁	—	2	2	no	—
292	C₂×C₄₁	—	4	2	no	—
293	C₂×C₄₁	—	8	2	no	—
294	C₂×C₄₁	—	8	2	no	—
295	C₂×C₄₁	#2404	1	1	yes	characteristic
296	#585	C₂²×C₄₁	1	1	yes	characteristic
297	C₂×#137	C₂²×C₄₁	1	2	yes	—
298	#568	C₂²×C₄₁	1	1	yes	characteristic
299	#559	—	2	2	no	—
300	#556	—	2	2	no	—
301	#551	—	2	4	no	—
302	#542	—	2	2	no	—
303	#536	C₂²×C₄₁	1	1	yes	characteristic
304	#535	C₂²×C₄₁	1	2	yes	—
305	#535	C₂²×C₄₁	1	1	yes	characteristic
306	#533	C₂²×C₄₁	1	1	yes	characteristic, complemented
307	#529	—	2	2	no	—
308	C₂×#124	C₂²×C₄₁	1	1	yes	characteristic, complemented
309	#519	C₂²×C₄₁	1	1	yes	characteristic, complemented
310	#518	C₂²×C₄₁	1	1	yes	characteristic, complemented
311	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
312	B(4,𝔽₂)	—	2	4	no	—
313	#456	—	2	2	no	—
314	#455	—	2	2	no	—
315	#421	—	2	2	no	—
316	#421	C₂²×C₄₁	1	2	yes	—
317	#421	C₂²×C₄₁	1	2	yes	—
318	C₂×#104	—	2	2	no	—
319	C₂×#104	C₂²×C₄₁	1	2	yes	—
320	#419	C₂²×C₄₁	1	2	yes	complemented
321	#413	C₂²×C₄₁	1	1	yes	characteristic
322	C₂×#100	C₂²×C₄₁	1	2	yes	complemented
323	#410	C₂²×C₄₁	1	1	yes	characteristic
324	#410	C₂²×C₄₁	1	2	yes	—
325	#344	C₂²×C₄₁	1	2	yes	complemented
326	#339	C₂²×C₄₁	1	2	yes	complemented
327	H(3,ℤ/4)	C₂²×C₄₁	1	2	yes	complemented
328	H(3,ℤ/4)	C₂²×C₄₁	1	2	yes	complemented
329	C₄₁	#23991	1	1	yes	characteristic, direct product factor, Sylow
330	2₋¹⁺⁴	—	2	2	no	—
331	2₋¹⁺⁴	—	2	1	no	—
332	H(5,𝔽₂)	—	2	2	no	—
333	H(5,𝔽₂)	—	2	2	no	—
334	H(5,𝔽₂)	—	2	1	no	—
335	C₂×#41	—	2	2	no	—
336	C₂×#41	—	4	2	no	—
337	C₂×#41	C₂³×C₄₁	1	2	yes	—
338	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
339	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
340	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
341	#137	—	2	4	no	—
342	#137	—	4	2	no	—
343	C₂×Q₁₆	—	2	2	no	—
344	C₂×SD₁₆	—	2	2	no	—
345	#131	—	4	2	no	—
346	C₂×M₁₆	—	2	2	no	—
347	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
348	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
349	C₄⋊Q₈	—	2	1	no	—
350	C₄⋊Q₈	D₄×C₄₁	1	2	yes	complemented
351	#127	—	2	1	no	—
352	#127	—	2	2	no	—
353	#126	—	2	4	no	—
354	#126	—	2	4	no	—
355	#125	—	2	1	no	—
356	#124	—	2	4	no	—
357	#124	—	2	2	no	—
358	#124	—	2	1	no	—
359	#124	—	2	2	no	—
360	#124	—	2	1	no	—
361	#124	—	2	1	no	—
362	#124	D₄×C₄₁	1	2	yes	complemented
363	#123	—	2	4	no	—
364	#123	—	2	4	no	—
365	#123	—	2	2	no	—
366	C₂²⋊Q₈	—	2	4	no	—
367	C₂²⋊Q₈	—	2	2	no	—
368	C₂²⋊Q₈	—	2	2	no	—
369	#121	—	2	2	no	—
370	#121	—	4	2	no	—
371	#120	—	2	4	no	—
372	#120	D₄×C₄₁	1	4	yes	—
373	C₄×Q₈	—	2	1	no	—
374	C₄×Q₈	—	2	2	no	—
375	C₄×Q₈	—	2	2	no	—
376	C₄×Q₈	—	2	2	no	—
377	C₄×D₄	—	2	4	no	—
378	C₄×D₄	—	2	2	no	—
379	C₄×D₄	—	2	1	no	—
380	C₄×D₄	—	2	2	no	—
381	C₄×D₄	—	2	2	no	—
382	#117	—	2	4	no	—
383	#117	—	2	2	no	—
384	#117	D₄×C₄₁	1	2	yes	complemented
385	#117	C₂³×C₄₁	1	2	yes	—
386	#117	C₂³×C₄₁	1	1	yes	characteristic
387	#117	C₂³×C₄₁	1	1	yes	characteristic
388	#117	C₂³×C₄₁	1	1	yes	characteristic
389	C₂×C₂²⋊C₄	—	2	2	no	—
390	C₂×C₂²⋊C₄	—	2	2	no	—
391	C₂×C₄²	D₄×C₄₁	1	2	yes	complemented
392	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
393	#104	—	2	2	no	—
394	#104	—	2	4	no	—
395	#104	—	2	2	no	—
396	#104	—	2	4	no	—
397	#103	—	4	2	no	—
398	#101	—	2	1	no	—
399	#100	—	2	1	no	—
400	#100	—	4	2	no	—
401	#99	—	2	2	no	—
402	#99	—	2	4	no	—
403	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
404	#41	—	2	2	no	—
405	#41	—	2	2	no	—
406	#41	—	2	4	no	—
407	#41	—	2	2	no	—
408	#41	—	2	2	no	—
409	#41	—	2	2	no	—
410	#41	—	2	2	no	—
411	#41	—	4	4	no	—
412	#41	—	4	2	no	—
413	#41	—	4	2	no	—
414	#41	—	4	2	no	—
415	#41	—	4	2	no	—
416	C₂×Q₈	—	2	2	no	—
417	C₂×Q₈	—	2	1	no	—
418	C₂×Q₈	—	2	2	no	—
419	C₂×Q₈	—	2	2	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	4	2	no	—
422	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
423	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
424	C₂×Q₈	#41×C₄₁	1	2	yes	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	2	no	—
432	C₂×D₄	—	4	2	no	—
433	C₂×D₄	—	4	4	no	—
434	C₂×D₄	—	4	2	no	—
435	C₂×D₄	—	4	4	no	—
436	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
437	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
438	C₂×D₄	#41×C₄₁	1	2	yes	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	2	4	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	4	2	no	—
445	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
446	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
447	Q₁₆	—	4	4	no	—
448	SD₁₆	—	4	4	no	—
449	M₁₆	—	2	2	no	—
450	M₁₆	—	2	2	no	—
451	M₁₆	—	2	2	no	—
452	M₁₆	—	2	2	no	—
453	M₁₆	—	4	2	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	2	no	—
456	C₂×C₈	—	4	2	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	4	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	—	2	4	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	2	no	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	2	4	no	—
470	C₂²⋊C₄	—	2	2	no	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	—	2	1	no	—
473	C₂²⋊C₄	—	2	1	no	—
474	C₂²⋊C₄	—	2	4	no	—
475	C₂²⋊C₄	—	2	2	no	—
476	C₂²⋊C₄	—	4	4	no	—
477	C₂²⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
478	C₄²	—	2	4	no	—
479	C₄²	—	2	2	no	—
480	C₄²	—	2	1	no	—
481	C₄²	—	2	1	no	—
482	C₄²	—	2	2	no	—
483	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
484	C₄²	C₂×D₄×C₄₁	1	2	yes	—
485	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
486	C₄²	C₂×D₄×C₄₁	1	2	yes	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	4	no	—
489	C₂³	—	2	2	no	—
490	C₂³	—	4	2	no	—
491	C₂³	—	8	2	no	—
492	C₂³	#120×C₄₁	1	2	yes	—
493	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
494	Q₈	—	2	2	no	—
495	Q₈	—	2	2	no	—
496	Q₈	—	2	4	no	—
497	Q₈	—	2	2	no	—
498	Q₈	—	4	2	no	—
499	Q₈	—	4	1	no	—
500	Q₈	—	4	1	no	—
501	Q₈	—	8	2	no	—
502	D₄	—	2	4	no	—
503	D₄	—	2	2	no	—
504	D₄	—	2	2	no	—
505	D₄	—	2	2	no	—
506	D₄	—	4	2	no	—
507	D₄	—	4	4	no	—
508	D₄	—	4	2	no	—
509	D₄	—	4	2	no	—
510	D₄	—	4	1	no	—
511	D₄	—	4	1	no	—
512	D₄	—	4	2	no	—
513	D₄	—	8	4	no	—
514	D₄	—	16	4	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	1	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	4	4	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	8	2	no	—
531	C₂×C₄	—	8	4	no	—
532	C₂×C₄	—	8	1	no	—
533	C₂×C₄	#120×C₄₁	1	2	yes	—
534	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
535	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
536	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
537	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
538	C₈	—	4	2	no	—
539	C₈	—	4	2	no	—
540	C₂²	—	2	2	no	—
541	C₂²	—	4	1	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	4	4	no	—
544	C₂²	—	4	2	no	—
545	C₂²	—	4	2	no	—
546	C₂²	—	8	2	no	—
547	C₂²	—	8	4	no	—
548	C₂²	—	16	2	no	—
549	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
550	C₂²	C₄₁×#546	1	2	yes	—
551	C₄	—	2	2	no	—
552	C₄	—	4	2	no	—
553	C₄	—	4	1	no	—
554	C₄	—	4	2	no	—
555	C₄	—	4	1	no	—
556	C₄	—	4	2	no	—
557	C₄	—	4	2	no	—
558	C₄	—	4	2	no	—
559	C₄	—	8	2	no	—
560	C₄	C₄₁×#525	1	1	yes	characteristic
561	C₄	C₄₁×#535	1	1	yes	characteristic
562	C₄	C₄₁×#546	1	2	yes	—
563	C₂	—	2	1	no	—
564	C₂	—	2	2	no	—
565	C₂	—	4	2	no	—
566	C₂	—	8	2	no	—
567	C₂	—	8	2	no	—
568	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×#3027	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. C₄₁×#3022	C₂	semidirect
4. C₄₁×#2901	C₂	semidirect
5. C₄₁×#2031	C₂	semidirect
6. C₄₁×#2004	C₂	semidirect
7. C₄₁×#1914	C₂	semidirect
8. C₄₁×#1911	C₂	semidirect
9. C₄₁×#1909	C₂	semidirect
10. C₄₁×#1905	C₂	semidirect
11. C₄₁×#1827	C₂	semidirect
22. C₄₁×#533	C₂²	semidirect
24. C₂×#124×C₄₁	C₂²	semidirect
25. C₄₁×#519	C₂²	semidirect
26. C₄₁×#518	C₂²	semidirect
36. C₄₁×#419	C₂²	semidirect
38. C₂×#100×C₄₁	C₂²	semidirect
41. C₄₁×#344	C₂²	semidirect
42. C₄₁×#339	C₂²	semidirect
43. C₄₁×H(3,ℤ/4)	C₂²	semidirect
44. C₄₁×H(3,ℤ/4)	C₂²	semidirect
65. C₄⋊Q₈×C₄₁	D₄	semidirect
77. #124×C₄₁	D₄	semidirect
99. #117×C₄₁	D₄	semidirect
106. C₂×C₄²×C₄₁	D₄	semidirect
255. #23991	C₄₁	direct (N × H)
279. #3510	C₂×C₄₁	semidirect
281. #3022	C₂×C₄₁	semidirect
282. #2901	C₂×C₄₁	semidirect
283. #2031	C₂×C₄₁	semidirect
284. #2004	C₂×C₄₁	semidirect
285. #1914	C₂×C₄₁	semidirect
286. #1911	C₂×C₄₁	semidirect
287. #1909	C₂×C₄₁	semidirect
288. #1905	C₂×C₄₁	semidirect
289. #1827	C₂×C₄₁	semidirect
306. #533	C₂²×C₄₁	semidirect
308. C₂×#124	C₂²×C₄₁	semidirect
309. #519	C₂²×C₄₁	semidirect
310. #518	C₂²×C₄₁	semidirect
320. #419	C₂²×C₄₁	semidirect
322. C₂×#100	C₂²×C₄₁	semidirect
325. #344	C₂²×C₄₁	semidirect
326. #339	C₂²×C₄₁	semidirect
327. H(3,ℤ/4)	C₂²×C₄₁	semidirect
328. H(3,ℤ/4)	C₂²×C₄₁	semidirect
329. C₄₁	#23991	direct (N × H)
350. C₄⋊Q₈	D₄×C₄₁	semidirect
362. #124	D₄×C₄₁	semidirect
384. #117	D₄×C₄₁	semidirect
391. C₂×C₄²	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×#3510	2
C₄₁×#3027	2
C₄₁×#3022	2
C₄₁×#2901	2
C₄₁×#2031	2
C₄₁×#2004	2
C₄₁×#1914	2
C₄₁×#1911	2
C₄₁×#1909	2
C₄₁×#1905	2
C₄₁×#1827	2
#23991	41

Subgroups — C41×#23991 #23800192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23991 #23800192