Subgroups — C₄₁×#25622 #23801823

Order 10496.

1294 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3506	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2541	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2002	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1986	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1980	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1883	C₂	1	4	yes	maximal, complemented
9	C₄×#100×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	4	yes	—
11	C₄₁×#559	—	2	2	no	—
12	C₄₁×#556	—	2	1	no	—
13	C₄₁×#555	—	2	1	no	—
14	C₄₁×#553	—	2	2	no	—
15	C₄₁×#550	—	2	1	no	—
16	C₄₁×#547	—	2	1	no	—
17	C₄₁×#538	—	2	2	no	—
18	C₄₁×#535	C₂²	1	2	yes	—
19	C₄₁×#533	C₂²	1	1	yes	characteristic
20	C₄₁×#532	C₂²	1	1	yes	characteristic
21	C₂×#127×C₄₁	C₂²	1	2	yes	—
22	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
23	C₄₁×#525	—	2	2	no	—
24	C₂×#121×C₄₁	—	2	2	no	—
25	C₄₁×#520	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#496	—	2	2	no	—
28	C₄₁×#493	—	2	1	no	—
29	C₄₁×#492	—	2	1	no	—
30	C₄₁×#490	—	2	2	no	—
31	C₄₁×#486	—	2	1	no	—
32	C₄₁×#486	—	2	1	no	—
33	C₂×#104×C₄₁	C₂²	1	4	yes	—
34	C₄₁×#418	C₂²	1	4	yes	—
35	C₂×#100×C₄₁	C₂²	1	2	yes	—
36	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
38	C₄₁×#392	—	2	2	no	—
39	C₄₁×#390	C₂²	1	2	yes	—
40	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
41	C₄₁×#343	C₂²	1	2	yes	—
42	C₄₁×#339	C₂²	1	4	yes	—
43	C₂×#41×C₄₁	—	2	2	no	—
44	C₂×#41×C₄₁	C₂³	1	2	yes	—
45	C₂²×D₄×C₄₁	—	2	4	no	—
46	C₂²×D₄×C₄₁	—	2	1	no	—
47	C₂²×D₄×C₄₁	—	2	1	no	—
48	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
49	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₄₁	D₄	1	2	yes	—
51	AGL(1,ℤ/8)×C₄₁	—	2	8	no	—
52	AGL(1,ℤ/8)×C₄₁	—	4	4	no	—
53	C₂×SD₁₆×C₄₁	—	2	4	no	—
54	C₂×D₈×C₄₁	—	2	4	no	—
55	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
56	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
57	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic, complemented
58	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic, complemented
59	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic, complemented
60	#127×C₄₁	—	2	2	no	—
61	#127×C₄₁	—	2	1	no	—
62	#127×C₄₁	—	2	1	no	—
63	#127×C₄₁	—	2	1	no	—
64	#127×C₄₁	—	2	1	no	—
65	#127×C₄₁	—	4	4	no	—
66	#127×C₄₁	D₄	1	1	yes	characteristic, complemented
67	#126×C₄₁	—	2	4	no	—
68	#125×C₄₁	D₄	1	2	yes	complemented
69	#124×C₄₁	—	2	4	no	—
70	#124×C₄₁	—	2	2	no	—
71	#124×C₄₁	D₄	1	2	yes	complemented
72	#123×C₄₁	—	2	4	no	—
73	#123×C₄₁	—	2	4	no	—
74	C₂²⋊Q₈×C₄₁	—	2	2	no	—
75	C₂²⋊Q₈×C₄₁	—	2	1	no	—
76	C₂²⋊Q₈×C₄₁	—	2	1	no	—
77	#121×C₄₁	—	2	2	no	—
78	#121×C₄₁	—	2	4	no	—
79	#121×C₄₁	—	2	1	no	—
80	#121×C₄₁	—	2	1	no	—
81	#121×C₄₁	—	2	2	no	—
82	#121×C₄₁	—	2	2	no	—
83	#121×C₄₁	—	4	8	no	—
84	#120×C₄₁	—	2	4	no	—
85	C₄×Q₈×C₄₁	—	2	2	no	—
86	C₄×Q₈×C₄₁	—	2	1	no	—
87	C₄×Q₈×C₄₁	—	2	1	no	—
88	C₄×D₄×C₄₁	—	2	4	no	—
89	C₄×D₄×C₄₁	—	2	2	no	—
90	C₄×D₄×C₄₁	—	2	4	no	—
91	C₄×D₄×C₄₁	—	2	1	no	—
92	C₄×D₄×C₄₁	—	2	1	no	—
93	C₄×D₄×C₄₁	—	2	1	no	—
94	C₄×D₄×C₄₁	—	2	1	no	—
95	C₄×D₄×C₄₁	—	2	1	no	—
96	C₄×D₄×C₄₁	—	2	1	no	—
97	#117×C₄₁	—	2	2	no	—
98	#117×C₄₁	—	2	2	no	—
99	#117×C₄₁	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
101	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
102	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
103	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
104	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
105	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
106	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
107	#104×C₄₁	—	4	4	no	—
108	#102×C₄₁	—	2	4	no	—
109	#102×C₄₁	—	2	4	no	—
110	#100×C₄₁	—	2	4	no	—
111	#97×C₄₁	—	2	1	no	—
112	#97×C₄₁	—	2	1	no	—
113	C₄×C₈×C₄₁	—	2	1	no	—
114	C₄×C₈×C₄₁	—	2	1	no	—
115	#95×C₄₁	—	2	1	no	—
116	C₂⁴×C₄₁	—	2	2	no	—
117	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
118	#41×C₄₁	—	2	4	no	—
119	#41×C₄₁	—	2	4	no	—
120	#41×C₄₁	—	8	4	no	—
121	C₂×Q₈×C₄₁	—	2	1	no	—
122	C₂×Q₈×C₄₁	—	2	1	no	—
123	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
124	C₂×D₄×C₄₁	—	2	2	no	—
125	C₂×D₄×C₄₁	—	2	4	no	—
126	C₂×D₄×C₄₁	—	2	4	no	—
127	C₂×D₄×C₄₁	—	2	1	no	—
128	C₂×D₄×C₄₁	—	2	1	no	—
129	C₂×D₄×C₄₁	—	2	1	no	—
130	C₂×D₄×C₄₁	—	2	1	no	—
131	C₂×D₄×C₄₁	—	2	2	no	—
132	C₂×D₄×C₄₁	—	2	2	no	—
133	C₂×D₄×C₄₁	—	2	2	no	—
134	C₂×D₄×C₄₁	—	2	4	no	—
135	C₂×D₄×C₄₁	—	2	2	no	—
136	C₂×D₄×C₄₁	—	2	4	no	—
137	C₂×D₄×C₄₁	—	4	4	no	—
138	C₂×D₄×C₄₁	—	4	8	no	—
139	C₂×D₄×C₄₁	—	4	8	no	—
140	C₂×D₄×C₄₁	—	4	1	no	—
141	C₂×D₄×C₄₁	—	4	1	no	—
142	C₂×D₄×C₄₁	—	4	1	no	—
143	C₂×D₄×C₄₁	—	4	1	no	—
144	C₂×D₄×C₄₁	—	8	2	no	—
145	C₂×D₄×C₄₁	—	8	4	no	—
146	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₄₁	—	2	2	no	—
148	C₂²×C₄×C₄₁	—	2	4	no	—
149	C₂²×C₄×C₄₁	—	2	2	no	—
150	C₂²×C₄×C₄₁	—	2	2	no	—
151	C₂²×C₄×C₄₁	—	2	2	no	—
152	C₂²×C₄×C₄₁	—	4	4	no	—
153	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
157	SD₁₆×C₄₁	—	4	8	no	—
158	D₈×C₄₁	—	4	8	no	—
159	M₁₆×C₄₁	—	2	2	no	—
160	M₁₆×C₄₁	—	2	2	no	—
161	M₁₆×C₄₁	—	2	2	no	—
162	M₁₆×C₄₁	#41	1	4	yes	complemented
163	C₂×C₈×C₄₁	—	2	2	no	—
164	C₂×C₈×C₄₁	—	2	2	no	—
165	C₄⋊C₄×C₄₁	—	2	4	no	—
166	C₄⋊C₄×C₄₁	—	2	4	no	—
167	C₄⋊C₄×C₄₁	—	2	1	no	—
168	C₄⋊C₄×C₄₁	—	2	1	no	—
169	C₄⋊C₄×C₄₁	—	2	2	no	—
170	C₄⋊C₄×C₄₁	—	2	1	no	—
171	C₄⋊C₄×C₄₁	—	2	2	no	—
172	C₄⋊C₄×C₄₁	—	2	1	no	—
173	C₄⋊C₄×C₄₁	—	2	1	no	—
174	C₄⋊C₄×C₄₁	—	2	1	no	—
175	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
176	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
177	C₂²⋊C₄×C₄₁	—	2	4	no	—
178	C₂²⋊C₄×C₄₁	—	2	4	no	—
179	C₂²⋊C₄×C₄₁	—	2	4	no	—
180	C₂²⋊C₄×C₄₁	—	2	2	no	—
181	C₂²⋊C₄×C₄₁	—	2	2	no	—
182	C₂²⋊C₄×C₄₁	—	2	2	no	—
183	C₂²⋊C₄×C₄₁	—	2	2	no	—
184	C₂²⋊C₄×C₄₁	—	2	2	no	—
185	C₂²⋊C₄×C₄₁	—	4	8	no	—
186	C₄²×C₄₁	—	2	4	no	—
187	C₄²×C₄₁	—	2	2	no	—
188	C₄²×C₄₁	—	2	4	no	—
189	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
193	C₂³×C₄₁	—	2	2	no	—
194	C₂³×C₄₁	—	2	2	no	—
195	C₂³×C₄₁	—	2	2	no	—
196	C₂³×C₄₁	—	2	2	no	—
197	C₂³×C₄₁	—	4	2	no	—
198	C₂³×C₄₁	—	4	2	no	—
199	C₂³×C₄₁	—	4	4	no	—
200	C₂³×C₄₁	—	4	4	no	—
201	C₂³×C₄₁	—	4	2	no	—
202	C₂³×C₄₁	—	4	2	no	—
203	C₂³×C₄₁	—	4	2	no	—
204	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₄₁	—	2	4	no	—
206	Q₈×C₄₁	—	4	2	no	—
207	Q₈×C₄₁	—	8	1	no	—
208	D₄×C₄₁	—	2	4	no	—
209	D₄×C₄₁	—	4	2	no	—
210	D₄×C₄₁	—	4	2	no	—
211	D₄×C₄₁	—	4	2	no	—
212	D₄×C₄₁	—	4	4	no	—
213	D₄×C₄₁	—	4	2	no	—
214	D₄×C₄₁	—	4	4	no	—
215	D₄×C₄₁	—	4	2	no	—
216	D₄×C₄₁	—	8	1	no	—
217	D₄×C₄₁	—	8	2	no	—
218	D₄×C₄₁	—	8	2	no	—
219	D₄×C₄₁	—	8	2	no	—
220	D₄×C₄₁	—	8	4	no	—
221	D₄×C₄₁	—	8	4	no	—
222	D₄×C₄₁	—	8	16	no	—
223	D₄×C₄₁	—	8	2	no	—
224	C₂×C₄×C₄₁	—	2	2	no	—
225	C₂×C₄×C₄₁	—	2	4	no	—
226	C₂×C₄×C₄₁	—	2	2	no	—
227	C₂×C₄×C₄₁	—	2	2	no	—
228	C₂×C₄×C₄₁	—	2	2	no	—
229	C₂×C₄×C₄₁	—	2	2	no	—
230	C₂×C₄×C₄₁	—	4	1	no	—
231	C₂×C₄×C₄₁	—	4	2	no	—
232	C₂×C₄×C₄₁	—	4	2	no	—
233	C₂×C₄×C₄₁	—	4	4	no	—
234	C₂×C₄×C₄₁	—	4	4	no	—
235	C₂×C₄×C₄₁	—	4	2	no	—
236	C₂×C₄×C₄₁	—	8	2	no	—
237	C₂×C₄×C₄₁	—	8	4	no	—
238	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
239	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
240	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
241	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
242	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
245	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
246	C₈×C₄₁	—	2	4	no	—
247	C₈×C₄₁	—	4	2	no	—
248	#25622	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
249	C₂²×C₄₁	—	2	1	no	—
250	C₂²×C₄₁	—	2	1	no	—
251	C₂²×C₄₁	—	4	4	no	—
252	C₂²×C₄₁	—	4	2	no	—
253	C₂²×C₄₁	—	4	2	no	—
254	C₂²×C₄₁	—	4	2	no	—
255	C₂²×C₄₁	—	4	2	no	—
256	C₂²×C₄₁	—	4	2	no	—
257	C₂²×C₄₁	—	4	2	no	—
258	C₂²×C₄₁	—	8	2	no	—
259	C₂²×C₄₁	—	8	4	no	—
260	C₂²×C₄₁	—	8	4	no	—
261	C₂²×C₄₁	—	8	2	no	—
262	C₂²×C₄₁	—	8	2	no	—
263	C₂²×C₄₁	—	8	2	no	—
264	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
265	C₂²×C₄₁	#532	1	1	yes	characteristic
266	C₂²×C₄₁	#535	1	1	yes	characteristic
267	C₄×C₄₁	—	2	1	no	—
268	C₄×C₄₁	—	2	1	no	—
269	C₄×C₄₁	—	2	1	no	—
270	C₄×C₄₁	—	4	4	no	—
271	C₄×C₄₁	—	4	2	no	—
272	C₄×C₄₁	—	4	2	no	—
273	C₄×C₄₁	—	4	2	no	—
274	C₄×C₄₁	—	4	2	no	—
275	C₄×C₄₁	#453	1	2	yes	—
276	C₄×C₄₁	#546	1	2	yes	—
277	C₄×C₄₁	#550	1	2	yes	—
278	#3506	C₂×C₄₁	1	1	yes	characteristic, complemented
279	#3151	C₂×C₄₁	1	1	yes	characteristic, complemented
280	#3138	C₂×C₄₁	1	1	yes	characteristic, complemented
281	#2541	C₂×C₄₁	1	1	yes	characteristic, complemented
282	#2002	C₂×C₄₁	1	4	yes	complemented
283	#1986	C₂×C₄₁	1	1	yes	characteristic, complemented
284	#1980	C₂×C₄₁	1	1	yes	characteristic, complemented
285	#1883	C₂×C₄₁	1	4	yes	complemented
286	C₄×#100	C₂×C₄₁	1	1	yes	characteristic, complemented
287	C₂×C₄₁	—	2	1	no	—
288	C₂×C₄₁	—	2	1	no	—
289	C₂×C₄₁	—	8	2	no	—
290	C₂×C₄₁	—	8	2	no	—
291	C₂×C₄₁	—	16	2	no	—
292	C₂×C₄₁	#2431	1	1	yes	characteristic
293	C₂×C₄₁	C₂×#453	1	1	yes	characteristic
294	C₂×C₄₁	#3021	1	1	yes	characteristic
295	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	4	yes	—
296	#559	—	2	2	no	—
297	#556	—	2	1	no	—
298	#555	—	2	1	no	—
299	#553	—	2	2	no	—
300	#550	—	2	1	no	—
301	#547	—	2	1	no	—
302	#538	—	2	2	no	—
303	#535	C₂²×C₄₁	1	2	yes	—
304	#533	C₂²×C₄₁	1	1	yes	characteristic
305	#532	C₂²×C₄₁	1	1	yes	characteristic
306	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
307	C₂×#127	C₂²×C₄₁	1	2	yes	—
308	#525	—	2	2	no	—
309	C₂×#121	—	2	2	no	—
310	#520	C₂²×C₄₁	1	2	yes	—
311	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
312	#496	—	2	2	no	—
313	#493	—	2	1	no	—
314	#492	—	2	1	no	—
315	#490	—	2	2	no	—
316	#486	—	2	1	no	—
317	#486	—	2	1	no	—
318	C₂×#104	C₂²×C₄₁	1	4	yes	—
319	#418	C₂²×C₄₁	1	4	yes	—
320	C₂×#100	C₂²×C₄₁	1	2	yes	—
321	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
322	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
323	#392	—	2	2	no	—
324	#390	C₂²×C₄₁	1	2	yes	—
325	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
326	#343	C₂²×C₄₁	1	2	yes	—
327	#339	C₂²×C₄₁	1	4	yes	—
328	C₄₁	#25622	1	1	yes	characteristic, direct product factor, Sylow
329	C₂×#41	—	2	2	no	—
330	C₂×#41	C₂³×C₄₁	1	2	yes	—
331	C₂²×D₄	—	2	4	no	—
332	C₂²×D₄	—	2	1	no	—
333	C₂²×D₄	—	2	1	no	—
334	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
335	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
336	C₂³×C₄	D₄×C₄₁	1	2	yes	—
337	AGL(1,ℤ/8)	—	2	8	no	—
338	AGL(1,ℤ/8)	—	4	4	no	—
339	C₂×SD₁₆	—	2	4	no	—
340	C₂×D₈	—	2	4	no	—
341	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
342	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
343	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic, complemented
344	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic, complemented
345	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic, complemented
346	#127	—	2	2	no	—
347	#127	—	2	1	no	—
348	#127	—	2	1	no	—
349	#127	—	2	1	no	—
350	#127	—	2	1	no	—
351	#127	—	4	4	no	—
352	#127	D₄×C₄₁	1	1	yes	characteristic, complemented
353	#126	—	2	4	no	—
354	#125	D₄×C₄₁	1	2	yes	complemented
355	#124	—	2	4	no	—
356	#124	—	2	2	no	—
357	#124	D₄×C₄₁	1	2	yes	complemented
358	#123	—	2	4	no	—
359	#123	—	2	4	no	—
360	C₂²⋊Q₈	—	2	2	no	—
361	C₂²⋊Q₈	—	2	1	no	—
362	C₂²⋊Q₈	—	2	1	no	—
363	#121	—	2	2	no	—
364	#121	—	2	4	no	—
365	#121	—	2	1	no	—
366	#121	—	2	1	no	—
367	#121	—	2	2	no	—
368	#121	—	2	2	no	—
369	#121	—	4	8	no	—
370	#120	—	2	4	no	—
371	C₄×Q₈	—	2	2	no	—
372	C₄×Q₈	—	2	1	no	—
373	C₄×Q₈	—	2	1	no	—
374	C₄×D₄	—	2	4	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	2	4	no	—
377	C₄×D₄	—	2	1	no	—
378	C₄×D₄	—	2	1	no	—
379	C₄×D₄	—	2	1	no	—
380	C₄×D₄	—	2	1	no	—
381	C₄×D₄	—	2	1	no	—
382	C₄×D₄	—	2	1	no	—
383	#117	—	2	2	no	—
384	#117	—	2	2	no	—
385	#117	C₂³×C₄₁	1	2	yes	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	2	2	no	—
388	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
389	C₂×C₂²⋊C₄	—	2	4	no	—
390	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
391	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
392	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
393	#104	—	4	4	no	—
394	#102	—	2	4	no	—
395	#102	—	2	4	no	—
396	#100	—	2	4	no	—
397	#97	—	2	1	no	—
398	#97	—	2	1	no	—
399	C₄×C₈	—	2	1	no	—
400	C₄×C₈	—	2	1	no	—
401	#95	—	2	1	no	—
402	C₂⁴	—	2	2	no	—
403	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
404	#41	—	2	4	no	—
405	#41	—	2	4	no	—
406	#41	—	8	4	no	—
407	C₂×Q₈	—	2	1	no	—
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
410	C₂×D₄	—	2	2	no	—
411	C₂×D₄	—	2	4	no	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	1	no	—
414	C₂×D₄	—	2	1	no	—
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	4	4	no	—
424	C₂×D₄	—	4	8	no	—
425	C₂×D₄	—	4	8	no	—
426	C₂×D₄	—	4	1	no	—
427	C₂×D₄	—	4	1	no	—
428	C₂×D₄	—	4	1	no	—
429	C₂×D₄	—	4	1	no	—
430	C₂×D₄	—	8	2	no	—
431	C₂×D₄	—	8	4	no	—
432	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	4	4	no	—
439	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
440	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
441	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
443	SD₁₆	—	4	8	no	—
444	D₈	—	4	8	no	—
445	M₁₆	—	2	2	no	—
446	M₁₆	—	2	2	no	—
447	M₁₆	—	2	2	no	—
448	M₁₆	#41×C₄₁	1	4	yes	complemented
449	C₂×C₈	—	2	2	no	—
450	C₂×C₈	—	2	2	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	4	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	2	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
462	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
463	C₂²⋊C₄	—	2	4	no	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	2	no	—
467	C₂²⋊C₄	—	2	2	no	—
468	C₂²⋊C₄	—	2	2	no	—
469	C₂²⋊C₄	—	2	2	no	—
470	C₂²⋊C₄	—	2	2	no	—
471	C₂²⋊C₄	—	4	8	no	—
472	C₄²	—	2	4	no	—
473	C₄²	—	2	2	no	—
474	C₄²	—	2	4	no	—
475	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
476	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
477	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
478	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	4	2	no	—
484	C₂³	—	4	2	no	—
485	C₂³	—	4	4	no	—
486	C₂³	—	4	4	no	—
487	C₂³	—	4	2	no	—
488	C₂³	—	4	2	no	—
489	C₂³	—	4	2	no	—
490	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
491	Q₈	—	2	4	no	—
492	Q₈	—	4	2	no	—
493	Q₈	—	8	1	no	—
494	D₄	—	2	4	no	—
495	D₄	—	4	2	no	—
496	D₄	—	4	2	no	—
497	D₄	—	4	2	no	—
498	D₄	—	4	4	no	—
499	D₄	—	4	2	no	—
500	D₄	—	4	4	no	—
501	D₄	—	4	2	no	—
502	D₄	—	8	1	no	—
503	D₄	—	8	2	no	—
504	D₄	—	8	2	no	—
505	D₄	—	8	2	no	—
506	D₄	—	8	4	no	—
507	D₄	—	8	4	no	—
508	D₄	—	8	16	no	—
509	D₄	—	8	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	4	1	no	—
517	C₂×C₄	—	4	2	no	—
518	C₂×C₄	—	4	2	no	—
519	C₂×C₄	—	4	4	no	—
520	C₂×C₄	—	4	4	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	8	2	no	—
523	C₂×C₄	—	8	4	no	—
524	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
525	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
526	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
527	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
528	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
529	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
530	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
531	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
532	C₈	—	2	4	no	—
533	C₈	—	4	2	no	—
534	C₂²	—	2	1	no	—
535	C₂²	—	2	1	no	—
536	C₂²	—	4	4	no	—
537	C₂²	—	4	2	no	—
538	C₂²	—	4	2	no	—
539	C₂²	—	4	2	no	—
540	C₂²	—	4	2	no	—
541	C₂²	—	4	2	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	8	2	no	—
544	C₂²	—	8	4	no	—
545	C₂²	—	8	4	no	—
546	C₂²	—	8	2	no	—
547	C₂²	—	8	2	no	—
548	C₂²	—	8	2	no	—
549	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
550	C₂²	C₄₁×#532	1	1	yes	characteristic
551	C₂²	C₄₁×#535	1	1	yes	characteristic
552	C₄	—	2	1	no	—
553	C₄	—	2	1	no	—
554	C₄	—	2	1	no	—
555	C₄	—	4	4	no	—
556	C₄	—	4	2	no	—
557	C₄	—	4	2	no	—
558	C₄	—	4	2	no	—
559	C₄	—	4	2	no	—
560	C₄	C₄₁×#453	1	2	yes	—
561	C₄	C₄₁×#546	1	2	yes	—
562	C₄	C₄₁×#550	1	2	yes	—
563	C₂	—	2	1	no	—
564	C₂	—	2	1	no	—
565	C₂	—	8	2	no	—
566	C₂	—	8	2	no	—
567	C₂	—	16	2	no	—
568	C₂	C₄₁×#2431	1	1	yes	characteristic
569	C₂	C₂×C₄₁×#453	1	1	yes	characteristic
570	C₂	C₄₁×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
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
1. C₄₁×#3506	C₂	semidirect
2. C₄₁×#3151	C₂	semidirect
3. C₄₁×#3138	C₂	semidirect
4. C₄₁×#2541	C₂	semidirect
5. C₄₁×#2002	C₂	semidirect
6. C₄₁×#1986	C₂	semidirect
7. C₄₁×#1980	C₂	semidirect
8. C₄₁×#1883	C₂	semidirect
9. C₄×#100×C₄₁	C₂	semidirect
57. C₄⋊Q₈×C₄₁	D₄	semidirect
58. C₄⋊Q₈×C₄₁	D₄	semidirect
59. C₄⋊Q₈×C₄₁	D₄	semidirect
66. #127×C₄₁	D₄	semidirect
68. #125×C₄₁	D₄	semidirect
71. #124×C₄₁	D₄	semidirect
162. M₁₆×C₄₁	#41	semidirect
248. #25622	C₄₁	direct (N × H)
278. #3506	C₂×C₄₁	semidirect
279. #3151	C₂×C₄₁	semidirect
280. #3138	C₂×C₄₁	semidirect
281. #2541	C₂×C₄₁	semidirect
282. #2002	C₂×C₄₁	semidirect
283. #1986	C₂×C₄₁	semidirect
284. #1980	C₂×C₄₁	semidirect
285. #1883	C₂×C₄₁	semidirect
286. C₄×#100	C₂×C₄₁	semidirect
328. C₄₁	#25622	direct (N × H)
343. C₄⋊Q₈	D₄×C₄₁	semidirect
344. C₄⋊Q₈	D₄×C₄₁	semidirect
345. C₄⋊Q₈	D₄×C₄₁	semidirect
352. #127	D₄×C₄₁	semidirect
354. #125	D₄×C₄₁	semidirect
357. #124	D₄×C₄₁	semidirect
448. 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×#3506	2
C₄₁×#3151	2
C₄₁×#3138	2
C₄₁×#2541	2
C₄₁×#2002	2
C₄₁×#1986	2
C₄₁×#1980	2
C₄₁×#1883	2
C₄×#100×C₄₁	2
#25622	41

Subgroups — C41×#25622 #23801823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#25622 #23801823