Subgroups — C₄₁×#24146 #23800347

Order 10496.

1142 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3508	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#3078	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#137×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2037	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2004	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1986	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1885	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1856	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1806	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#559	—	2	1	no	—
14	C₄₁×#559	—	2	1	no	—
15	C₄₁×#550	—	2	1	no	—
16	C₄₁×#550	—	2	1	no	—
17	C₄₁×#546	—	2	2	no	—
18	C₄₁×#546	—	2	2	no	—
19	C₄₁×#540	—	2	2	no	—
20	C₄₁×#535	C₂²	1	2	yes	—
21	C₄₁×#533	C₂²	1	1	yes	characteristic
22	C₄₁×#532	—	2	1	no	—
23	C₄₁×#532	—	2	1	no	—
24	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×#124×C₄₁	C₂²	1	2	yes	—
26	C₄₁×#518	C₂²	1	2	yes	—
27	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
29	C₄₁×#479	—	2	2	no	—
30	C₄₁×#477	—	2	1	no	—
31	C₄₁×#475	—	2	1	no	—
32	C₄₁×#441	—	2	1	no	—
33	C₄₁×#440	—	2	1	no	—
34	C₄×Q₁₆×C₄₁	—	2	1	no	—
35	C₄×SD₁₆×C₄₁	—	2	1	no	—
36	C₄₁×#428	C₂²	1	2	yes	—
37	C₄₁×#428	C₂²	1	1	yes	characteristic
38	C₄₁×#423	C₂²	1	1	yes	characteristic
39	C₂×#104×C₄₁	C₂²	1	2	yes	—
40	C₄₁×#419	C₂²	1	2	yes	—
41	C₄₁×#419	C₂²	1	1	yes	characteristic
42	C₄₁×#417	C₂²	1	1	yes	characteristic
43	C₂×#100×C₄₁	C₂²	1	2	yes	—
44	C₂×#100×C₄₁	C₂²	1	2	yes	—
45	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
46	C₄₁×#394	—	2	2	no	—
47	C₄₁×#390	—	2	2	no	—
48	C₄₁×#390	C₂²	1	2	yes	—
49	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
50	C₄₁×#339	C₂²	1	2	yes	—
51	C₄₁×#339	C₂²	1	2	yes	—
52	C₄₁×#339	C₂²	1	2	yes	—
53	C₂×#41×C₄₁	—	2	2	no	—
54	C₂×#41×C₄₁	—	2	2	no	—
55	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
56	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
57	C₂²×D₄×C₄₁	D₄	1	2	yes	—
58	C₂²×D₄×C₄₁	D₄	1	2	yes	—
59	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
60	#137×C₄₁	—	2	2	no	—
61	#137×C₄₁	—	2	2	no	—
62	C₂×Q₁₆×C₄₁	—	2	1	no	—
63	C₂×SD₁₆×C₄₁	—	2	1	no	—
64	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
65	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
66	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
68	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
69	#127×C₄₁	—	2	2	no	—
70	#127×C₄₁	—	2	1	no	—
71	#127×C₄₁	—	2	1	no	—
72	#127×C₄₁	—	2	1	no	—
73	#127×C₄₁	—	2	1	no	—
74	#125×C₄₁	D₄	1	2	yes	—
75	#124×C₄₁	—	2	2	no	—
76	#124×C₄₁	—	2	2	no	—
77	#124×C₄₁	—	2	2	no	—
78	#124×C₄₁	—	2	4	no	—
79	#124×C₄₁	—	4	2	no	—
80	#123×C₄₁	—	2	4	no	—
81	#123×C₄₁	—	2	4	no	—
82	C₂²⋊Q₈×C₄₁	—	2	1	no	—
83	C₂²⋊Q₈×C₄₁	—	2	1	no	—
84	#121×C₄₁	—	2	2	no	—
85	#121×C₄₁	—	2	2	no	—
86	#121×C₄₁	—	2	4	no	—
87	#121×C₄₁	—	2	4	no	—
88	#121×C₄₁	—	2	2	no	—
89	#121×C₄₁	—	2	2	no	—
90	#120×C₄₁	—	2	4	no	—
91	C₄×Q₈×C₄₁	—	2	1	no	—
92	C₄×Q₈×C₄₁	—	2	1	no	—
93	C₄×Q₈×C₄₁	—	2	1	no	—
94	C₄×Q₈×C₄₁	—	2	1	no	—
95	C₄×Q₈×C₄₁	—	2	1	no	—
96	C₄×Q₈×C₄₁	—	2	1	no	—
97	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
98	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
99	C₄×D₄×C₄₁	—	2	2	no	—
100	C₄×D₄×C₄₁	—	2	2	no	—
101	C₄×D₄×C₄₁	—	2	1	no	—
102	C₄×D₄×C₄₁	—	2	1	no	—
103	C₄×D₄×C₄₁	—	2	2	no	—
104	C₄×D₄×C₄₁	—	2	2	no	—
105	C₄×D₄×C₄₁	—	2	2	no	—
106	C₄×D₄×C₄₁	—	2	2	no	—
107	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
108	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
109	#117×C₄₁	—	2	1	no	—
110	#117×C₄₁	C₂³	1	1	yes	characteristic
111	#117×C₄₁	C₂³	1	2	yes	—
112	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
113	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
114	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
115	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
116	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
117	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
118	C₂×C₄²×C₄₁	—	2	1	no	—
119	C₂×C₄²×C₄₁	—	2	1	no	—
120	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
121	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
122	#107×C₄₁	—	2	1	no	—
123	#107×C₄₁	—	2	2	no	—
124	#106×C₄₁	—	2	1	no	—
125	#106×C₄₁	—	2	2	no	—
126	C₄⋊C₈×C₄₁	—	2	1	no	—
127	C₄⋊C₈×C₄₁	—	2	1	no	—
128	#104×C₄₁	—	4	2	no	—
129	#103×C₄₁	—	2	1	no	—
130	#103×C₄₁	—	2	1	no	—
131	#103×C₄₁	—	2	1	no	—
132	#103×C₄₁	—	2	2	no	—
133	#103×C₄₁	—	2	2	no	—
134	#102×C₄₁	—	2	1	no	—
135	#100×C₄₁	—	2	4	no	—
136	#100×C₄₁	—	4	2	no	—
137	#97×C₄₁	—	2	1	no	—
138	C₄×C₈×C₄₁	—	2	1	no	—
139	#95×C₄₁	—	2	1	no	—
140	C₂⁴×C₄₁	C₂×D₄	1	4	yes	—
141	#41×C₄₁	—	2	1	no	—
142	#41×C₄₁	—	2	1	no	—
143	#41×C₄₁	—	2	1	no	—
144	#41×C₄₁	—	2	1	no	—
145	#41×C₄₁	—	8	4	no	—
146	#41×C₄₁	—	8	4	no	—
147	C₂×Q₈×C₄₁	—	2	1	no	—
148	C₂×Q₈×C₄₁	—	2	1	no	—
149	C₂×Q₈×C₄₁	—	2	1	no	—
150	C₂×Q₈×C₄₁	—	2	1	no	—
151	C₂×Q₈×C₄₁	—	2	1	no	—
152	C₂×Q₈×C₄₁	—	2	1	no	—
153	C₂×Q₈×C₄₁	—	2	1	no	—
154	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
155	C₂×Q₈×C₄₁	#41	1	2	yes	complemented
156	C₂×D₄×C₄₁	—	2	1	no	—
157	C₂×D₄×C₄₁	—	2	4	no	—
158	C₂×D₄×C₄₁	—	2	4	no	—
159	C₂×D₄×C₄₁	—	2	2	no	—
160	C₂×D₄×C₄₁	—	2	2	no	—
161	C₂×D₄×C₄₁	—	2	2	no	—
162	C₂×D₄×C₄₁	—	2	2	no	—
163	C₂×D₄×C₄₁	—	2	2	no	—
164	C₂×D₄×C₄₁	—	2	2	no	—
165	C₂×D₄×C₄₁	—	2	2	no	—
166	C₂×D₄×C₄₁	—	2	2	no	—
167	C₂×D₄×C₄₁	—	2	2	no	—
168	C₂×D₄×C₄₁	—	2	2	no	—
169	C₂×D₄×C₄₁	—	4	4	no	—
170	C₂×D₄×C₄₁	—	8	2	no	—
171	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₄₁	—	2	1	no	—
173	C₂²×C₄×C₄₁	—	2	4	no	—
174	C₂²×C₄×C₄₁	—	2	4	no	—
175	C₂²×C₄×C₄₁	—	2	1	no	—
176	C₂²×C₄×C₄₁	—	2	1	no	—
177	C₂²×C₄×C₄₁	—	2	1	no	—
178	C₂²×C₄×C₄₁	—	2	2	no	—
179	C₂²×C₄×C₄₁	—	2	1	no	—
180	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
183	Q₁₆×C₄₁	—	2	4	no	—
184	SD₁₆×C₄₁	—	2	4	no	—
185	M₁₆×C₄₁	—	2	1	no	—
186	M₁₆×C₄₁	—	2	2	no	—
187	M₁₆×C₄₁	—	2	1	no	—
188	M₁₆×C₄₁	—	2	2	no	—
189	M₁₆×C₄₁	—	2	1	no	—
190	M₁₆×C₄₁	#41	1	2	yes	complemented
191	C₂×C₈×C₄₁	—	2	1	no	—
192	C₂×C₈×C₄₁	—	2	1	no	—
193	C₂×C₈×C₄₁	—	2	2	no	—
194	C₄⋊C₄×C₄₁	—	2	1	no	—
195	C₄⋊C₄×C₄₁	—	2	1	no	—
196	C₄⋊C₄×C₄₁	—	2	1	no	—
197	C₄⋊C₄×C₄₁	—	2	1	no	—
198	C₄⋊C₄×C₄₁	—	2	2	no	—
199	C₄⋊C₄×C₄₁	—	2	2	no	—
200	C₄⋊C₄×C₄₁	—	2	1	no	—
201	C₄⋊C₄×C₄₁	—	2	1	no	—
202	C₄⋊C₄×C₄₁	—	4	2	no	—
203	C₄⋊C₄×C₄₁	—	4	1	no	—
204	C₄⋊C₄×C₄₁	—	4	1	no	—
205	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
206	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
207	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
208	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
209	C₂²⋊C₄×C₄₁	—	2	4	no	—
210	C₂²⋊C₄×C₄₁	—	2	4	no	—
211	C₂²⋊C₄×C₄₁	—	2	4	no	—
212	C₂²⋊C₄×C₄₁	—	2	4	no	—
213	C₂²⋊C₄×C₄₁	—	2	1	no	—
214	C₂²⋊C₄×C₄₁	—	2	2	no	—
215	C₂²⋊C₄×C₄₁	—	2	1	no	—
216	C₂²⋊C₄×C₄₁	—	2	1	no	—
217	C₂²⋊C₄×C₄₁	—	4	8	no	—
218	C₂²⋊C₄×C₄₁	—	8	4	no	—
219	C₄²×C₄₁	—	2	1	no	—
220	C₄²×C₄₁	—	2	1	no	—
221	C₄²×C₄₁	—	2	1	no	—
222	C₄²×C₄₁	—	2	2	no	—
223	C₄²×C₄₁	—	2	2	no	—
224	C₄²×C₄₁	—	2	2	no	—
225	C₄²×C₄₁	—	4	1	no	—
226	C₄²×C₄₁	—	4	1	no	—
227	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
228	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
229	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
230	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
231	C₂³×C₄₁	—	2	4	no	—
232	C₂³×C₄₁	—	2	1	no	—
233	C₂³×C₄₁	—	2	4	no	—
234	C₂³×C₄₁	—	2	4	no	—
235	C₂³×C₄₁	—	4	4	no	—
236	C₂³×C₄₁	—	4	4	no	—
237	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
238	Q₈×C₄₁	—	2	2	no	—
239	Q₈×C₄₁	—	2	1	no	—
240	Q₈×C₄₁	—	2	1	no	—
241	Q₈×C₄₁	—	2	2	no	—
242	Q₈×C₄₁	—	4	2	no	—
243	Q₈×C₄₁	—	4	1	no	—
244	Q₈×C₄₁	—	4	1	no	—
245	Q₈×C₄₁	—	4	2	no	—
246	D₄×C₄₁	—	2	1	no	—
247	D₄×C₄₁	—	2	1	no	—
248	D₄×C₄₁	—	4	4	no	—
249	D₄×C₄₁	—	4	1	no	—
250	D₄×C₄₁	—	4	1	no	—
251	D₄×C₄₁	—	4	2	no	—
252	D₄×C₄₁	—	4	2	no	—
253	D₄×C₄₁	—	4	2	no	—
254	D₄×C₄₁	—	4	2	no	—
255	D₄×C₄₁	—	4	4	no	—
256	D₄×C₄₁	—	8	2	no	—
257	D₄×C₄₁	—	8	2	no	—
258	D₄×C₄₁	—	8	4	no	—
259	D₄×C₄₁	—	8	4	no	—
260	C₂×C₄×C₄₁	—	2	2	no	—
261	C₂×C₄×C₄₁	—	2	1	no	—
262	C₂×C₄×C₄₁	—	2	1	no	—
263	C₂×C₄×C₄₁	—	2	1	no	—
264	C₂×C₄×C₄₁	—	2	2	no	—
265	C₂×C₄×C₄₁	—	2	2	no	—
266	C₂×C₄×C₄₁	—	2	2	no	—
267	C₂×C₄×C₄₁	—	4	1	no	—
268	C₂×C₄×C₄₁	—	4	2	no	—
269	C₂×C₄×C₄₁	—	4	1	no	—
270	C₂×C₄×C₄₁	—	4	1	no	—
271	C₂×C₄×C₄₁	—	4	1	no	—
272	C₂×C₄×C₄₁	—	4	1	no	—
273	C₂×C₄×C₄₁	—	4	2	no	—
274	C₂×C₄×C₄₁	—	4	2	no	—
275	C₂×C₄×C₄₁	—	4	2	no	—
276	C₂×C₄×C₄₁	—	4	2	no	—
277	C₂×C₄×C₄₁	—	4	1	no	—
278	C₂×C₄×C₄₁	—	4	2	no	—
279	C₂×C₄×C₄₁	—	8	4	no	—
280	C₂×C₄×C₄₁	—	8	4	no	—
281	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
282	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
283	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
284	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
285	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
287	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
288	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
289	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
290	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
291	C₈×C₄₁	—	2	2	no	—
292	C₈×C₄₁	—	4	1	no	—
293	C₈×C₄₁	—	4	2	no	—
294	#24146	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
295	C₂²×C₄₁	—	2	1	no	—
296	C₂²×C₄₁	—	2	1	no	—
297	C₂²×C₄₁	—	4	1	no	—
298	C₂²×C₄₁	—	4	4	no	—
299	C₂²×C₄₁	—	4	1	no	—
300	C₂²×C₄₁	—	4	4	no	—
301	C₂²×C₄₁	—	4	1	no	—
302	C₂²×C₄₁	—	4	4	no	—
303	C₂²×C₄₁	—	8	4	no	—
304	C₂²×C₄₁	—	8	4	no	—
305	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
306	C₂²×C₄₁	#546	1	1	yes	characteristic
307	C₂²×C₄₁	#547	1	1	yes	characteristic
308	C₄×C₄₁	—	2	1	no	—
309	C₄×C₄₁	—	2	1	no	—
310	C₄×C₄₁	—	2	1	no	—
311	C₄×C₄₁	—	4	1	no	—
312	C₄×C₄₁	—	4	2	no	—
313	C₄×C₄₁	—	4	1	no	—
314	C₄×C₄₁	—	4	2	no	—
315	C₄×C₄₁	—	4	1	no	—
316	C₄×C₄₁	—	4	2	no	—
317	C₄×C₄₁	—	8	2	no	—
318	C₄×C₄₁	—	8	1	no	—
319	C₄×C₄₁	#455	1	2	yes	—
320	C₄×C₄₁	#525	1	1	yes	characteristic
321	C₄×C₄₁	#537	1	1	yes	characteristic
322	C₄×C₄₁	#548	1	1	yes	characteristic
323	C₄×C₄₁	#550	1	1	yes	characteristic
324	#3508	C₂×C₄₁	1	1	yes	characteristic
325	#3078	C₂×C₄₁	1	1	yes	characteristic, complemented
326	C₄×#137	C₂×C₄₁	1	1	yes	characteristic, complemented
327	#2548	C₂×C₄₁	1	1	yes	characteristic, complemented
328	#2037	C₂×C₄₁	1	2	yes	complemented
329	#2004	C₂×C₄₁	1	2	yes	complemented
330	#1986	C₂×C₄₁	1	1	yes	characteristic, complemented
331	#1965	C₂×C₄₁	1	1	yes	characteristic, complemented
332	#1885	C₂×C₄₁	1	2	yes	complemented
333	#1856	C₂×C₄₁	1	2	yes	complemented
334	#1806	C₂×C₄₁	1	1	yes	characteristic, complemented
335	C₂×C₄₁	—	2	1	no	—
336	C₂×C₄₁	—	2	1	no	—
337	C₂×C₄₁	—	8	1	no	—
338	C₂×C₄₁	—	8	4	no	—
339	C₂×C₄₁	#2407	1	1	yes	characteristic
340	C₂×C₄₁	C₂×#455	1	1	yes	characteristic
341	C₂×C₄₁	#3019	1	1	yes	characteristic
342	C₂×#137	C₂²×C₄₁	1	1	yes	characteristic
343	#559	—	2	1	no	—
344	#559	—	2	1	no	—
345	#550	—	2	1	no	—
346	#550	—	2	1	no	—
347	#546	—	2	2	no	—
348	#546	—	2	2	no	—
349	#540	—	2	2	no	—
350	#535	C₂²×C₄₁	1	2	yes	—
351	#533	C₂²×C₄₁	1	1	yes	characteristic
352	#532	—	2	1	no	—
353	#532	—	2	1	no	—
354	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
355	C₂×#124	C₂²×C₄₁	1	2	yes	—
356	#518	C₂²×C₄₁	1	2	yes	—
357	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
358	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
359	#479	—	2	2	no	—
360	#477	—	2	1	no	—
361	#475	—	2	1	no	—
362	#441	—	2	1	no	—
363	#440	—	2	1	no	—
364	C₄×Q₁₆	—	2	1	no	—
365	C₄×SD₁₆	—	2	1	no	—
366	#428	C₂²×C₄₁	1	2	yes	—
367	#428	C₂²×C₄₁	1	1	yes	characteristic
368	#423	C₂²×C₄₁	1	1	yes	characteristic
369	C₂×#104	C₂²×C₄₁	1	2	yes	—
370	#419	C₂²×C₄₁	1	2	yes	—
371	#419	C₂²×C₄₁	1	1	yes	characteristic
372	#417	C₂²×C₄₁	1	1	yes	characteristic
373	C₂×#100	C₂²×C₄₁	1	2	yes	—
374	C₂×#100	C₂²×C₄₁	1	2	yes	—
375	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
376	#394	—	2	2	no	—
377	#390	—	2	2	no	—
378	#390	C₂²×C₄₁	1	2	yes	—
379	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
380	#339	C₂²×C₄₁	1	2	yes	—
381	#339	C₂²×C₄₁	1	2	yes	—
382	#339	C₂²×C₄₁	1	2	yes	—
383	C₄₁	#24146	1	1	yes	characteristic, direct product factor, Sylow
384	C₂×#41	—	2	2	no	—
385	C₂×#41	—	2	2	no	—
386	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
387	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
388	C₂²×D₄	D₄×C₄₁	1	2	yes	—
389	C₂²×D₄	D₄×C₄₁	1	2	yes	—
390	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
391	#137	—	2	2	no	—
392	#137	—	2	2	no	—
393	C₂×Q₁₆	—	2	1	no	—
394	C₂×SD₁₆	—	2	1	no	—
395	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
396	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
397	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
398	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
399	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
400	#127	—	2	2	no	—
401	#127	—	2	1	no	—
402	#127	—	2	1	no	—
403	#127	—	2	1	no	—
404	#127	—	2	1	no	—
405	#125	D₄×C₄₁	1	2	yes	—
406	#124	—	2	2	no	—
407	#124	—	2	2	no	—
408	#124	—	2	2	no	—
409	#124	—	2	4	no	—
410	#124	—	4	2	no	—
411	#123	—	2	4	no	—
412	#123	—	2	4	no	—
413	C₂²⋊Q₈	—	2	1	no	—
414	C₂²⋊Q₈	—	2	1	no	—
415	#121	—	2	2	no	—
416	#121	—	2	2	no	—
417	#121	—	2	4	no	—
418	#121	—	2	4	no	—
419	#121	—	2	2	no	—
420	#121	—	2	2	no	—
421	#120	—	2	4	no	—
422	C₄×Q₈	—	2	1	no	—
423	C₄×Q₈	—	2	1	no	—
424	C₄×Q₈	—	2	1	no	—
425	C₄×Q₈	—	2	1	no	—
426	C₄×Q₈	—	2	1	no	—
427	C₄×Q₈	—	2	1	no	—
428	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
429	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
430	C₄×D₄	—	2	2	no	—
431	C₄×D₄	—	2	2	no	—
432	C₄×D₄	—	2	1	no	—
433	C₄×D₄	—	2	1	no	—
434	C₄×D₄	—	2	2	no	—
435	C₄×D₄	—	2	2	no	—
436	C₄×D₄	—	2	2	no	—
437	C₄×D₄	—	2	2	no	—
438	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
439	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
440	#117	—	2	1	no	—
441	#117	C₂³×C₄₁	1	1	yes	characteristic
442	#117	C₂³×C₄₁	1	2	yes	—
443	C₂×C₄⋊C₄	—	2	1	no	—
444	C₂×C₄⋊C₄	—	2	1	no	—
445	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
446	C₂×C₂²⋊C₄	—	2	4	no	—
447	C₂×C₂²⋊C₄	—	2	4	no	—
448	C₂×C₂²⋊C₄	—	2	4	no	—
449	C₂×C₄²	—	2	1	no	—
450	C₂×C₄²	—	2	1	no	—
451	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
452	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
453	#107	—	2	1	no	—
454	#107	—	2	2	no	—
455	#106	—	2	1	no	—
456	#106	—	2	2	no	—
457	C₄⋊C₈	—	2	1	no	—
458	C₄⋊C₈	—	2	1	no	—
459	#104	—	4	2	no	—
460	#103	—	2	1	no	—
461	#103	—	2	1	no	—
462	#103	—	2	1	no	—
463	#103	—	2	2	no	—
464	#103	—	2	2	no	—
465	#102	—	2	1	no	—
466	#100	—	2	4	no	—
467	#100	—	4	2	no	—
468	#97	—	2	1	no	—
469	C₄×C₈	—	2	1	no	—
470	#95	—	2	1	no	—
471	C₂⁴	C₂×D₄×C₄₁	1	4	yes	—
472	#41	—	2	1	no	—
473	#41	—	2	1	no	—
474	#41	—	2	1	no	—
475	#41	—	2	1	no	—
476	#41	—	8	4	no	—
477	#41	—	8	4	no	—
478	C₂×Q₈	—	2	1	no	—
479	C₂×Q₈	—	2	1	no	—
480	C₂×Q₈	—	2	1	no	—
481	C₂×Q₈	—	2	1	no	—
482	C₂×Q₈	—	2	1	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
486	C₂×Q₈	#41×C₄₁	1	2	yes	complemented
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	—	2	4	no	—
489	C₂×D₄	—	2	4	no	—
490	C₂×D₄	—	2	2	no	—
491	C₂×D₄	—	2	2	no	—
492	C₂×D₄	—	2	2	no	—
493	C₂×D₄	—	2	2	no	—
494	C₂×D₄	—	2	2	no	—
495	C₂×D₄	—	2	2	no	—
496	C₂×D₄	—	2	2	no	—
497	C₂×D₄	—	2	2	no	—
498	C₂×D₄	—	2	2	no	—
499	C₂×D₄	—	2	2	no	—
500	C₂×D₄	—	4	4	no	—
501	C₂×D₄	—	8	2	no	—
502	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
503	C₂²×C₄	—	2	1	no	—
504	C₂²×C₄	—	2	4	no	—
505	C₂²×C₄	—	2	4	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	2	2	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
512	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
513	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
514	Q₁₆	—	2	4	no	—
515	SD₁₆	—	2	4	no	—
516	M₁₆	—	2	1	no	—
517	M₁₆	—	2	2	no	—
518	M₁₆	—	2	1	no	—
519	M₁₆	—	2	2	no	—
520	M₁₆	—	2	1	no	—
521	M₁₆	#41×C₄₁	1	2	yes	complemented
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	2	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	—	2	2	no	—
530	C₄⋊C₄	—	2	2	no	—
531	C₄⋊C₄	—	2	1	no	—
532	C₄⋊C₄	—	2	1	no	—
533	C₄⋊C₄	—	4	2	no	—
534	C₄⋊C₄	—	4	1	no	—
535	C₄⋊C₄	—	4	1	no	—
536	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
537	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
538	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
539	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
540	C₂²⋊C₄	—	2	4	no	—
541	C₂²⋊C₄	—	2	4	no	—
542	C₂²⋊C₄	—	2	4	no	—
543	C₂²⋊C₄	—	2	4	no	—
544	C₂²⋊C₄	—	2	1	no	—
545	C₂²⋊C₄	—	2	2	no	—
546	C₂²⋊C₄	—	2	1	no	—
547	C₂²⋊C₄	—	2	1	no	—
548	C₂²⋊C₄	—	4	8	no	—
549	C₂²⋊C₄	—	8	4	no	—
550	C₄²	—	2	1	no	—
551	C₄²	—	2	1	no	—
552	C₄²	—	2	1	no	—
553	C₄²	—	2	2	no	—
554	C₄²	—	2	2	no	—
555	C₄²	—	2	2	no	—
556	C₄²	—	4	1	no	—
557	C₄²	—	4	1	no	—
558	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
559	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
560	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
561	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
562	C₂³	—	2	4	no	—
563	C₂³	—	2	1	no	—
564	C₂³	—	2	4	no	—
565	C₂³	—	2	4	no	—
566	C₂³	—	4	4	no	—
567	C₂³	—	4	4	no	—
568	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
569	Q₈	—	2	2	no	—
570	Q₈	—	2	1	no	—
571	Q₈	—	2	1	no	—
572	Q₈	—	2	2	no	—
573	Q₈	—	4	2	no	—
574	Q₈	—	4	1	no	—
575	Q₈	—	4	1	no	—
576	Q₈	—	4	2	no	—
577	D₄	—	2	1	no	—
578	D₄	—	2	1	no	—
579	D₄	—	4	4	no	—
580	D₄	—	4	1	no	—
581	D₄	—	4	1	no	—
582	D₄	—	4	2	no	—
583	D₄	—	4	2	no	—
584	D₄	—	4	2	no	—
585	D₄	—	4	2	no	—
586	D₄	—	4	4	no	—
587	D₄	—	8	2	no	—
588	D₄	—	8	2	no	—
589	D₄	—	8	4	no	—
590	D₄	—	8	4	no	—
591	C₂×C₄	—	2	2	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	4	1	no	—
599	C₂×C₄	—	4	2	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	2	no	—
605	C₂×C₄	—	4	2	no	—
606	C₂×C₄	—	4	2	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	2	no	—
610	C₂×C₄	—	8	4	no	—
611	C₂×C₄	—	8	4	no	—
612	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
613	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
614	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
615	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
616	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
617	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
618	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
619	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
620	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
621	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
622	C₈	—	2	2	no	—
623	C₈	—	4	1	no	—
624	C₈	—	4	2	no	—
625	C₂²	—	2	1	no	—
626	C₂²	—	2	1	no	—
627	C₂²	—	4	1	no	—
628	C₂²	—	4	4	no	—
629	C₂²	—	4	1	no	—
630	C₂²	—	4	4	no	—
631	C₂²	—	4	1	no	—
632	C₂²	—	4	4	no	—
633	C₂²	—	8	4	no	—
634	C₂²	—	8	4	no	—
635	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
636	C₂²	C₄₁×#546	1	1	yes	characteristic
637	C₂²	C₄₁×#547	1	1	yes	characteristic
638	C₄	—	2	1	no	—
639	C₄	—	2	1	no	—
640	C₄	—	2	1	no	—
641	C₄	—	4	1	no	—
642	C₄	—	4	2	no	—
643	C₄	—	4	1	no	—
644	C₄	—	4	2	no	—
645	C₄	—	4	1	no	—
646	C₄	—	4	2	no	—
647	C₄	—	8	2	no	—
648	C₄	—	8	1	no	—
649	C₄	C₄₁×#455	1	2	yes	—
650	C₄	C₄₁×#525	1	1	yes	characteristic
651	C₄	C₄₁×#537	1	1	yes	characteristic
652	C₄	C₄₁×#548	1	1	yes	characteristic
653	C₄	C₄₁×#550	1	1	yes	characteristic
654	C₂	—	2	1	no	—
655	C₂	—	2	1	no	—
656	C₂	—	8	4	no	—
657	C₂	—	8	1	no	—
658	C₂	C₄₁×#2407	1	1	yes	characteristic
659	C₂	C₂×C₄₁×#455	1	1	yes	characteristic
660	C₂	C₄₁×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#3508	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#3078	C₂	semidirect
3. C₄×#137×C₄₁	C₂	semidirect
4. C₄₁×#2548	C₂	semidirect
5. C₄₁×#2037	C₂	semidirect
6. C₄₁×#2004	C₂	semidirect
7. C₄₁×#1986	C₂	semidirect
8. C₄₁×#1965	C₂	semidirect
9. C₄₁×#1885	C₂	semidirect
10. C₄₁×#1856	C₂	semidirect
11. C₄₁×#1806	C₂	semidirect
155. C₂×Q₈×C₄₁	#41	semidirect
190. M₁₆×C₄₁	#41	semidirect
294. #24146	C₄₁	direct (N × H)
325. #3078	C₂×C₄₁	semidirect
326. C₄×#137	C₂×C₄₁	semidirect
327. #2548	C₂×C₄₁	semidirect
328. #2037	C₂×C₄₁	semidirect
329. #2004	C₂×C₄₁	semidirect
330. #1986	C₂×C₄₁	semidirect
331. #1965	C₂×C₄₁	semidirect
332. #1885	C₂×C₄₁	semidirect
333. #1856	C₂×C₄₁	semidirect
334. #1806	C₂×C₄₁	semidirect
383. C₄₁	#24146	direct (N × H)
486. C₂×Q₈	#41×C₄₁	semidirect
521. M₁₆	#41×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3508	2
C₄₁×#3078	2
C₄×#137×C₄₁	2
C₄₁×#2548	2
C₄₁×#2037	2
C₄₁×#2004	2
C₄₁×#1986	2
C₄₁×#1965	2
C₄₁×#1885	2
C₄₁×#1856	2
C₄₁×#1806	2
#24146	41

Subgroups — C41×#24146 #23800347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24146 #23800347