Subgroups — C₄₁×#25616 #23801817

Order 10496.

1240 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3511	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3038	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2858	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2006	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1962	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1883	C₂	1	4	yes	maximal, complemented
9	C₄×#104×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	4	yes	—
11	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
12	C₄₁×#556	—	2	1	no	—
13	C₄₁×#555	—	2	1	no	—
14	C₄₁×#554	—	2	1	no	—
15	C₄₁×#553	—	2	2	no	—
16	C₄₁×#548	—	2	2	no	—
17	C₄₁×#547	—	2	1	no	—
18	C₄₁×#542	—	2	2	no	—
19	C₄₁×#537	C₂²	1	2	yes	—
20	C₄₁×#533	—	2	1	no	—
21	C₄₁×#532	—	2	1	no	—
22	C₄₁×#532	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×#124×C₄₁	—	2	2	no	—
26	C₄₁×#520	C₂²	1	2	yes	—
27	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
28	C₄₁×#496	—	2	2	no	—
29	C₄₁×#493	—	2	1	no	—
30	C₄₁×#492	—	2	1	no	—
31	C₄₁×#460	—	2	1	no	—
32	C₄₁×#460	—	2	1	no	—
33	C₄₁×#459	—	2	1	no	—
34	C₄₁×#459	—	2	1	no	—
35	C₄₁×#423	C₂²	1	1	yes	characteristic
36	C₂×#104×C₄₁	C₂²	1	2	yes	—
37	C₂×#104×C₄₁	C₂²	1	4	yes	—
38	C₄₁×#418	C₂²	1	2	yes	—
39	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
40	C₄₁×#390	—	2	2	no	—
41	C₄₁×#390	C₂²	1	4	yes	—
42	C₄³×C₄₁	C₂²	1	1	yes	characteristic
43	C₄₁×#343	C₂²	1	4	yes	—
44	C₄₁×#339	C₂²	1	2	yes	—
45	C₂×#41×C₄₁	—	2	2	no	—
46	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
47	C₂×#41×C₄₁	C₂³	1	2	yes	—
48	C₂²×Q₈×C₄₁	D₄	1	2	yes	—
49	C₂²×D₄×C₄₁	—	2	1	no	—
50	C₂²×D₄×C₄₁	—	2	1	no	—
51	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
52	AGL(1,ℤ/8)×C₄₁	—	2	8	no	—
53	AGL(1,ℤ/8)×C₄₁	—	4	4	no	—
54	AGL(1,ℤ/8)×C₄₁	—	4	4	no	—
55	C₂×SD₁₆×C₄₁	—	2	2	no	—
56	C₂×SD₁₆×C₄₁	—	2	4	no	—
57	C₂×D₈×C₄₁	—	2	2	no	—
58	C₂×D₈×C₄₁	—	2	4	no	—
59	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
60	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₄₁	—	2	1	no	—
62	C₄⋊Q₈×C₄₁	—	2	1	no	—
63	C₄⋊Q₈×C₄₁	—	2	1	no	—
64	C₄⋊Q₈×C₄₁	—	2	1	no	—
65	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
66	#127×C₄₁	—	2	1	no	—
67	#127×C₄₁	—	2	1	no	—
68	#127×C₄₁	—	2	1	no	—
69	#127×C₄₁	—	2	1	no	—
70	#127×C₄₁	D₄	1	1	yes	characteristic
71	#126×C₄₁	—	2	4	no	—
72	#125×C₄₁	—	2	2	no	—
73	#124×C₄₁	—	2	2	no	—
74	#124×C₄₁	—	2	4	no	—
75	#124×C₄₁	—	4	8	no	—
76	#124×C₄₁	D₄	1	2	yes	—
77	#123×C₄₁	—	2	4	no	—
78	C₂²⋊Q₈×C₄₁	—	2	2	no	—
79	C₂²⋊Q₈×C₄₁	—	2	4	no	—
80	C₂²⋊Q₈×C₄₁	—	2	2	no	—
81	C₂²⋊Q₈×C₄₁	—	2	2	no	—
82	#121×C₄₁	—	2	2	no	—
83	#121×C₄₁	—	2	2	no	—
84	#121×C₄₁	—	2	2	no	—
85	#121×C₄₁	—	2	1	no	—
86	#121×C₄₁	—	2	1	no	—
87	#121×C₄₁	—	2	4	no	—
88	C₄×Q₈×C₄₁	—	2	2	no	—
89	C₄×Q₈×C₄₁	—	2	2	no	—
90	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
91	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
92	C₄×D₄×C₄₁	—	2	2	no	—
93	C₄×D₄×C₄₁	—	2	2	no	—
94	C₄×D₄×C₄₁	—	2	1	no	—
95	C₄×D₄×C₄₁	—	2	1	no	—
96	C₄×D₄×C₄₁	—	2	4	no	—
97	C₄×D₄×C₄₁	—	2	1	no	—
98	C₄×D₄×C₄₁	—	2	1	no	—
99	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
100	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
101	#117×C₄₁	—	2	2	no	—
102	#117×C₄₁	—	2	2	no	—
103	#117×C₄₁	—	2	1	no	—
104	#117×C₄₁	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
107	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
108	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
109	C₂×C₄²×C₄₁	—	2	1	no	—
110	C₂×C₄²×C₄₁	—	2	1	no	—
111	C₂×C₄²×C₄₁	—	2	1	no	—
112	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
113	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
114	C₄⋊C₈×C₄₁	—	2	1	no	—
115	C₄⋊C₈×C₄₁	—	2	1	no	—
116	#104×C₄₁	—	2	4	no	—
117	#104×C₄₁	—	4	4	no	—
118	#102×C₄₁	—	2	2	no	—
119	#102×C₄₁	—	2	2	no	—
120	#97×C₄₁	—	2	1	no	—
121	C₄×C₈×C₄₁	—	2	1	no	—
122	C₂⁴×C₄₁	—	2	2	no	—
123	#41×C₄₁	—	2	2	no	—
124	#41×C₄₁	—	2	4	no	—
125	#41×C₄₁	—	2	4	no	—
126	#41×C₄₁	—	2	2	no	—
127	#41×C₄₁	—	8	4	no	—
128	C₂×Q₈×C₄₁	—	2	2	no	—
129	C₂×Q₈×C₄₁	—	2	2	no	—
130	C₂×Q₈×C₄₁	—	4	4	no	—
131	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
132	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
133	C₂×D₄×C₄₁	—	2	1	no	—
134	C₂×D₄×C₄₁	—	2	2	no	—
135	C₂×D₄×C₄₁	—	2	4	no	—
136	C₂×D₄×C₄₁	—	2	1	no	—
137	C₂×D₄×C₄₁	—	2	1	no	—
138	C₂×D₄×C₄₁	—	2	2	no	—
139	C₂×D₄×C₄₁	—	2	2	no	—
140	C₂×D₄×C₄₁	—	2	4	no	—
141	C₂×D₄×C₄₁	—	2	4	no	—
142	C₂×D₄×C₄₁	—	4	4	no	—
143	C₂×D₄×C₄₁	—	4	4	no	—
144	C₂×D₄×C₄₁	—	4	1	no	—
145	C₂×D₄×C₄₁	—	4	1	no	—
146	C₂×D₄×C₄₁	—	4	1	no	—
147	C₂×D₄×C₄₁	—	4	1	no	—
148	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
149	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
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	1	no	—
155	C₂²×C₄×C₄₁	—	2	2	no	—
156	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
159	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
160	SD₁₆×C₄₁	—	4	8	no	—
161	SD₁₆×C₄₁	—	4	4	no	—
162	D₈×C₄₁	—	4	8	no	—
163	D₈×C₄₁	—	4	4	no	—
164	M₁₆×C₄₁	—	2	2	no	—
165	M₁₆×C₄₁	—	2	4	no	—
166	M₁₆×C₄₁	#41	1	4	yes	complemented
167	C₂×C₈×C₄₁	—	2	2	no	—
168	C₂×C₈×C₄₁	—	2	2	no	—
169	C₄⋊C₄×C₄₁	—	2	4	no	—
170	C₄⋊C₄×C₄₁	—	2	4	no	—
171	C₄⋊C₄×C₄₁	—	2	2	no	—
172	C₄⋊C₄×C₄₁	—	2	2	no	—
173	C₄⋊C₄×C₄₁	—	2	1	no	—
174	C₄⋊C₄×C₄₁	—	2	1	no	—
175	C₄⋊C₄×C₄₁	—	2	1	no	—
176	C₄⋊C₄×C₄₁	—	2	1	no	—
177	C₄⋊C₄×C₄₁	—	2	2	no	—
178	C₄⋊C₄×C₄₁	—	2	2	no	—
179	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
180	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
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	1	no	—
185	C₂²⋊C₄×C₄₁	—	2	1	no	—
186	C₂²⋊C₄×C₄₁	—	2	1	no	—
187	C₂²⋊C₄×C₄₁	—	2	2	no	—
188	C₂²⋊C₄×C₄₁	—	2	2	no	—
189	C₂²⋊C₄×C₄₁	—	4	16	no	—
190	C₄²×C₄₁	—	2	4	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	—	2	2	no	—
197	C₄²×C₄₁	—	4	2	no	—
198	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
202	C₂³×C₄₁	—	2	1	no	—
203	C₂³×C₄₁	—	2	2	no	—
204	C₂³×C₄₁	—	4	4	no	—
205	C₂³×C₄₁	—	4	4	no	—
206	C₂³×C₄₁	—	4	2	no	—
207	C₂³×C₄₁	—	4	4	no	—
208	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
209	Q₈×C₄₁	—	2	1	no	—
210	Q₈×C₄₁	—	2	1	no	—
211	Q₈×C₄₁	—	2	4	no	—
212	Q₈×C₄₁	—	4	4	no	—
213	Q₈×C₄₁	—	8	2	no	—
214	D₄×C₄₁	—	2	1	no	—
215	D₄×C₄₁	—	2	1	no	—
216	D₄×C₄₁	—	2	4	no	—
217	D₄×C₄₁	—	4	2	no	—
218	D₄×C₄₁	—	4	4	no	—
219	D₄×C₄₁	—	4	2	no	—
220	D₄×C₄₁	—	4	2	no	—
221	D₄×C₄₁	—	4	4	no	—
222	D₄×C₄₁	—	4	2	no	—
223	D₄×C₄₁	—	4	2	no	—
224	D₄×C₄₁	—	8	2	no	—
225	D₄×C₄₁	—	8	2	no	—
226	D₄×C₄₁	—	8	4	no	—
227	D₄×C₄₁	—	8	2	no	—
228	D₄×C₄₁	—	8	2	no	—
229	C₂×C₄×C₄₁	—	2	2	no	—
230	C₂×C₄×C₄₁	—	2	2	no	—
231	C₂×C₄×C₄₁	—	2	4	no	—
232	C₂×C₄×C₄₁	—	2	1	no	—
233	C₂×C₄×C₄₁	—	2	2	no	—
234	C₂×C₄×C₄₁	—	2	1	no	—
235	C₂×C₄×C₄₁	—	2	1	no	—
236	C₂×C₄×C₄₁	—	2	4	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	—	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	4	no	—
245	C₂×C₄×C₄₁	—	8	4	no	—
246	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
247	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
248	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
249	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
250	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
252	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
253	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
254	C₈×C₄₁	—	2	4	no	—
255	C₈×C₄₁	—	4	2	no	—
256	#25616	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
257	C₂²×C₄₁	—	2	1	no	—
258	C₂²×C₄₁	—	2	1	no	—
259	C₂²×C₄₁	—	4	2	no	—
260	C₂²×C₄₁	—	4	1	no	—
261	C₂²×C₄₁	—	4	1	no	—
262	C₂²×C₄₁	—	4	2	no	—
263	C₂²×C₄₁	—	4	1	no	—
264	C₂²×C₄₁	—	4	2	no	—
265	C₂²×C₄₁	—	8	8	no	—
266	C₂²×C₄₁	—	8	2	no	—
267	C₂²×C₄₁	—	8	2	no	—
268	C₂²×C₄₁	—	8	2	no	—
269	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
270	C₂²×C₄₁	#550	1	2	yes	—
271	C₄×C₄₁	—	2	1	no	—
272	C₄×C₄₁	—	2	1	no	—
273	C₄×C₄₁	—	4	4	no	—
274	C₄×C₄₁	—	4	2	no	—
275	C₄×C₄₁	—	4	1	no	—
276	C₄×C₄₁	—	4	1	no	—
277	C₄×C₄₁	—	4	4	no	—
278	C₄×C₄₁	—	4	1	no	—
279	C₄×C₄₁	#453	1	2	yes	—
280	C₄×C₄₁	#454	1	2	yes	—
281	C₄×C₄₁	#532	1	1	yes	characteristic
282	C₄×C₄₁	#535	1	1	yes	characteristic
283	C₄×C₄₁	#546	1	2	yes	—
284	#3511	C₂×C₄₁	1	1	yes	characteristic, complemented
285	#3151	C₂×C₄₁	1	1	yes	characteristic, complemented
286	#3038	C₂×C₄₁	1	1	yes	characteristic, complemented
287	#2858	C₂×C₄₁	1	1	yes	characteristic, complemented
288	#2006	C₂×C₄₁	1	4	yes	complemented
289	#1997	C₂×C₄₁	1	1	yes	characteristic, complemented
290	#1962	C₂×C₄₁	1	1	yes	characteristic, complemented
291	#1883	C₂×C₄₁	1	4	yes	complemented
292	C₄×#104	C₂×C₄₁	1	1	yes	characteristic, complemented
293	C₂×C₄₁	—	2	2	no	—
294	C₂×C₄₁	—	8	1	no	—
295	C₂×C₄₁	—	8	2	no	—
296	C₂×C₄₁	—	16	2	no	—
297	C₂×C₄₁	#2431	1	1	yes	characteristic
298	C₂×C₄₁	C₂×#453	1	1	yes	characteristic
299	C₂×C₄₁	C₂×#454	1	1	yes	characteristic
300	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	4	yes	—
301	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
302	#556	—	2	1	no	—
303	#555	—	2	1	no	—
304	#554	—	2	1	no	—
305	#553	—	2	2	no	—
306	#548	—	2	2	no	—
307	#547	—	2	1	no	—
308	#542	—	2	2	no	—
309	#537	C₂²×C₄₁	1	2	yes	—
310	#533	—	2	1	no	—
311	#532	—	2	1	no	—
312	#532	C₂²×C₄₁	1	1	yes	characteristic
313	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
314	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
315	C₂×#124	—	2	2	no	—
316	#520	C₂²×C₄₁	1	2	yes	—
317	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
318	#496	—	2	2	no	—
319	#493	—	2	1	no	—
320	#492	—	2	1	no	—
321	#460	—	2	1	no	—
322	#460	—	2	1	no	—
323	#459	—	2	1	no	—
324	#459	—	2	1	no	—
325	#423	C₂²×C₄₁	1	1	yes	characteristic
326	C₂×#104	C₂²×C₄₁	1	2	yes	—
327	C₂×#104	C₂²×C₄₁	1	4	yes	—
328	#418	C₂²×C₄₁	1	2	yes	—
329	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
330	#390	—	2	2	no	—
331	#390	C₂²×C₄₁	1	4	yes	—
332	C₄³	C₂²×C₄₁	1	1	yes	characteristic
333	#343	C₂²×C₄₁	1	4	yes	—
334	#339	C₂²×C₄₁	1	2	yes	—
335	C₄₁	#25616	1	1	yes	characteristic, direct product factor, Sylow
336	C₂×#41	—	2	2	no	—
337	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
338	C₂×#41	C₂³×C₄₁	1	2	yes	—
339	C₂²×Q₈	D₄×C₄₁	1	2	yes	—
340	C₂²×D₄	—	2	1	no	—
341	C₂²×D₄	—	2	1	no	—
342	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
343	AGL(1,ℤ/8)	—	2	8	no	—
344	AGL(1,ℤ/8)	—	4	4	no	—
345	AGL(1,ℤ/8)	—	4	4	no	—
346	C₂×SD₁₆	—	2	2	no	—
347	C₂×SD₁₆	—	2	4	no	—
348	C₂×D₈	—	2	2	no	—
349	C₂×D₈	—	2	4	no	—
350	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
351	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
352	C₄⋊Q₈	—	2	1	no	—
353	C₄⋊Q₈	—	2	1	no	—
354	C₄⋊Q₈	—	2	1	no	—
355	C₄⋊Q₈	—	2	1	no	—
356	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
357	#127	—	2	1	no	—
358	#127	—	2	1	no	—
359	#127	—	2	1	no	—
360	#127	—	2	1	no	—
361	#127	D₄×C₄₁	1	1	yes	characteristic
362	#126	—	2	4	no	—
363	#125	—	2	2	no	—
364	#124	—	2	2	no	—
365	#124	—	2	4	no	—
366	#124	—	4	8	no	—
367	#124	D₄×C₄₁	1	2	yes	—
368	#123	—	2	4	no	—
369	C₂²⋊Q₈	—	2	2	no	—
370	C₂²⋊Q₈	—	2	4	no	—
371	C₂²⋊Q₈	—	2	2	no	—
372	C₂²⋊Q₈	—	2	2	no	—
373	#121	—	2	2	no	—
374	#121	—	2	2	no	—
375	#121	—	2	2	no	—
376	#121	—	2	1	no	—
377	#121	—	2	1	no	—
378	#121	—	2	4	no	—
379	C₄×Q₈	—	2	2	no	—
380	C₄×Q₈	—	2	2	no	—
381	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
382	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
383	C₄×D₄	—	2	2	no	—
384	C₄×D₄	—	2	2	no	—
385	C₄×D₄	—	2	1	no	—
386	C₄×D₄	—	2	1	no	—
387	C₄×D₄	—	2	4	no	—
388	C₄×D₄	—	2	1	no	—
389	C₄×D₄	—	2	1	no	—
390	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
391	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
392	#117	—	2	2	no	—
393	#117	—	2	2	no	—
394	#117	—	2	1	no	—
395	#117	C₂³×C₄₁	1	1	yes	characteristic
396	C₂×C₄⋊C₄	—	2	2	no	—
397	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
398	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
399	C₂×C₂²⋊C₄	—	2	8	no	—
400	C₂×C₄²	—	2	1	no	—
401	C₂×C₄²	—	2	1	no	—
402	C₂×C₄²	—	2	1	no	—
403	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
404	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
405	C₄⋊C₈	—	2	1	no	—
406	C₄⋊C₈	—	2	1	no	—
407	#104	—	2	4	no	—
408	#104	—	4	4	no	—
409	#102	—	2	2	no	—
410	#102	—	2	2	no	—
411	#97	—	2	1	no	—
412	C₄×C₈	—	2	1	no	—
413	C₂⁴	—	2	2	no	—
414	#41	—	2	2	no	—
415	#41	—	2	4	no	—
416	#41	—	2	4	no	—
417	#41	—	2	2	no	—
418	#41	—	8	4	no	—
419	C₂×Q₈	—	2	2	no	—
420	C₂×Q₈	—	2	2	no	—
421	C₂×Q₈	—	4	4	no	—
422	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
423	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	2	4	no	—
433	C₂×D₄	—	4	4	no	—
434	C₂×D₄	—	4	4	no	—
435	C₂×D₄	—	4	1	no	—
436	C₂×D₄	—	4	1	no	—
437	C₂×D₄	—	4	1	no	—
438	C₂×D₄	—	4	1	no	—
439	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
440	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	1	no	—
444	C₂²×C₄	—	2	1	no	—
445	C₂²×C₄	—	2	1	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
450	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
451	SD₁₆	—	4	8	no	—
452	SD₁₆	—	4	4	no	—
453	D₈	—	4	8	no	—
454	D₈	—	4	4	no	—
455	M₁₆	—	2	2	no	—
456	M₁₆	—	2	4	no	—
457	M₁₆	#41×C₄₁	1	4	yes	complemented
458	C₂×C₈	—	2	2	no	—
459	C₂×C₈	—	2	2	no	—
460	C₄⋊C₄	—	2	4	no	—
461	C₄⋊C₄	—	2	4	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
471	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
472	C₂²⋊C₄	—	2	4	no	—
473	C₂²⋊C₄	—	2	4	no	—
474	C₂²⋊C₄	—	2	4	no	—
475	C₂²⋊C₄	—	2	1	no	—
476	C₂²⋊C₄	—	2	1	no	—
477	C₂²⋊C₄	—	2	1	no	—
478	C₂²⋊C₄	—	2	2	no	—
479	C₂²⋊C₄	—	2	2	no	—
480	C₂²⋊C₄	—	4	16	no	—
481	C₄²	—	2	4	no	—
482	C₄²	—	2	4	no	—
483	C₄²	—	2	1	no	—
484	C₄²	—	2	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	2	2	no	—
487	C₄²	—	2	2	no	—
488	C₄²	—	4	2	no	—
489	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
490	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
491	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
492	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
493	C₂³	—	2	1	no	—
494	C₂³	—	2	2	no	—
495	C₂³	—	4	4	no	—
496	C₂³	—	4	4	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	4	4	no	—
499	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
500	Q₈	—	2	1	no	—
501	Q₈	—	2	1	no	—
502	Q₈	—	2	4	no	—
503	Q₈	—	4	4	no	—
504	Q₈	—	8	2	no	—
505	D₄	—	2	1	no	—
506	D₄	—	2	1	no	—
507	D₄	—	2	4	no	—
508	D₄	—	4	2	no	—
509	D₄	—	4	4	no	—
510	D₄	—	4	2	no	—
511	D₄	—	4	2	no	—
512	D₄	—	4	4	no	—
513	D₄	—	4	2	no	—
514	D₄	—	4	2	no	—
515	D₄	—	8	2	no	—
516	D₄	—	8	2	no	—
517	D₄	—	8	4	no	—
518	D₄	—	8	2	no	—
519	D₄	—	8	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	1	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	4	no	—
536	C₂×C₄	—	8	4	no	—
537	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
538	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
539	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
540	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
541	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
542	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
543	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
544	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
545	C₈	—	2	4	no	—
546	C₈	—	4	2	no	—
547	C₂²	—	2	1	no	—
548	C₂²	—	2	1	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	1	no	—
551	C₂²	—	4	1	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	2	no	—
555	C₂²	—	8	8	no	—
556	C₂²	—	8	2	no	—
557	C₂²	—	8	2	no	—
558	C₂²	—	8	2	no	—
559	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
560	C₂²	C₄₁×#550	1	2	yes	—
561	C₄	—	2	1	no	—
562	C₄	—	2	1	no	—
563	C₄	—	4	4	no	—
564	C₄	—	4	2	no	—
565	C₄	—	4	1	no	—
566	C₄	—	4	1	no	—
567	C₄	—	4	4	no	—
568	C₄	—	4	1	no	—
569	C₄	C₄₁×#453	1	2	yes	—
570	C₄	C₄₁×#454	1	2	yes	—
571	C₄	C₄₁×#532	1	1	yes	characteristic
572	C₄	C₄₁×#535	1	1	yes	characteristic
573	C₄	C₄₁×#546	1	2	yes	—
574	C₂	—	2	2	no	—
575	C₂	—	8	1	no	—
576	C₂	—	8	2	no	—
577	C₂	—	16	2	no	—
578	C₂	C₄₁×#2431	1	1	yes	characteristic
579	C₂	C₂×C₄₁×#453	1	1	yes	characteristic
580	C₂	C₂×C₄₁×#454	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×#3511	C₂	semidirect
2. C₄₁×#3151	C₂	semidirect
3. C₄₁×#3038	C₂	semidirect
4. C₄₁×#2858	C₂	semidirect
5. C₄₁×#2006	C₂	semidirect
6. C₄₁×#1997	C₂	semidirect
7. C₄₁×#1962	C₂	semidirect
8. C₄₁×#1883	C₂	semidirect
9. C₄×#104×C₄₁	C₂	semidirect
166. M₁₆×C₄₁	#41	semidirect
256. #25616	C₄₁	direct (N × H)
284. #3511	C₂×C₄₁	semidirect
285. #3151	C₂×C₄₁	semidirect
286. #3038	C₂×C₄₁	semidirect
287. #2858	C₂×C₄₁	semidirect
288. #2006	C₂×C₄₁	semidirect
289. #1997	C₂×C₄₁	semidirect
290. #1962	C₂×C₄₁	semidirect
291. #1883	C₂×C₄₁	semidirect
292. C₄×#104	C₂×C₄₁	semidirect
335. C₄₁	#25616	direct (N × H)
457. 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×#3511	2
C₄₁×#3151	2
C₄₁×#3038	2
C₄₁×#2858	2
C₄₁×#2006	2
C₄₁×#1997	2
C₄₁×#1962	2
C₄₁×#1883	2
C₄×#104×C₄₁	2
#25616	41

Subgroups — C41×#25616 #23801817

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#25616 #23801817