Subgroups — C₄₁×#32678 #23808879

Order 10496.

758 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#2592	C₂	1	2	yes	maximal, complemented
2	C₄₁×#2055	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2015	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2012	C₂	1	2	yes	maximal
5	C₄₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1972	C₂	1	1	yes	maximal, characteristic
7	C₄₁×#1971	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1927	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1926	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
11	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
12	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×#124×C₄₁	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×#103×C₄₁	C₂²	1	2	yes	—
19	C₂×#103×C₄₁	C₂²	1	2	yes	—
20	C₂×#102×C₄₁	C₂²	1	2	yes	—
21	C₂×#102×C₄₁	C₂²	1	2	yes	—
22	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
23	C₄₁×#397	—	2	2	no	—
24	C₄₁×#393	—	2	2	no	—
25	C₄₁×#391	C₂²	1	2	yes	—
26	C₄₁×#391	C₂²	1	2	yes	—
27	C₄₁×#390	C₂²	1	2	yes	—
28	C₄₁×#390	C₂²	1	2	yes	—
29	C₄₁×#388	—	2	2	no	—
30	C₄₁×#388	—	2	2	no	—
31	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
32	C₄₁×#340	C₂²	1	2	yes	—
33	C₄₁×#336	C₂²	1	2	yes	—
34	C₄₁×#336	C₂²	1	2	yes	—
35	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
37	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₄₁	—	2	2	no	—
39	C₂×Q₁₆×C₄₁	—	4	1	no	—
40	C₂×Q₁₆×C₄₁	—	4	1	no	—
41	C₂×SD₁₆×C₄₁	—	4	2	no	—
42	C₂×SD₁₆×C₄₁	—	4	2	no	—
43	C₂×SD₁₆×C₄₁	—	4	2	no	—
44	C₂×D₈×C₄₁	—	2	2	no	—
45	C₂×D₈×C₄₁	—	4	1	no	—
46	C₂×D₈×C₄₁	—	4	1	no	—
47	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
50	#124×C₄₁	—	4	2	no	—
51	#124×C₄₁	—	4	2	no	—
52	C₄×Q₈×C₄₁	—	4	1	no	—
53	C₄×Q₈×C₄₁	—	4	1	no	—
54	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
64	C₂×C₄²×C₄₁	—	2	2	no	—
65	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
66	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
67	#106×C₄₁	—	4	1	no	—
68	#106×C₄₁	—	4	1	no	—
69	C₄⋊C₈×C₄₁	—	4	1	no	—
70	#103×C₄₁	—	2	4	no	—
71	#103×C₄₁	—	4	2	no	—
72	#102×C₄₁	—	2	4	no	—
73	#102×C₄₁	—	4	2	no	—
74	#97×C₄₁	—	4	1	no	—
75	#95×C₄₁	—	2	2	no	—
76	#95×C₄₁	—	2	2	no	—
77	#95×C₄₁	—	2	2	no	—
78	#95×C₄₁	—	2	2	no	—
79	#95×C₄₁	—	2	2	no	—
80	C₂⁴×C₄₁	—	2	2	no	—
81	C₂×Q₈×C₄₁	—	2	2	no	—
82	C₂×Q₈×C₄₁	—	2	2	no	—
83	C₂×Q₈×C₄₁	—	2	2	no	—
84	C₂×Q₈×C₄₁	—	2	2	no	—
85	C₂×Q₈×C₄₁	—	2	2	no	—
86	C₂×Q₈×C₄₁	—	2	2	no	—
87	C₂×Q₈×C₄₁	—	4	1	no	—
88	C₂×Q₈×C₄₁	—	4	2	no	—
89	C₂×D₄×C₄₁	—	2	2	no	—
90	C₂×D₄×C₄₁	—	2	2	no	—
91	C₂×D₄×C₄₁	—	2	2	no	—
92	C₂×D₄×C₄₁	—	2	2	no	—
93	C₂×D₄×C₄₁	—	2	2	no	—
94	C₂×D₄×C₄₁	—	2	2	no	—
95	C₂²×C₄×C₄₁	—	2	2	no	—
96	C₂²×C₄×C₄₁	—	2	2	no	—
97	C₂²×C₄×C₄₁	—	2	2	no	—
98	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
103	Q₁₆×C₄₁	—	4	4	no	—
104	SD₁₆×C₄₁	—	4	8	no	—
105	D₈×C₄₁	—	4	4	no	—
106	C₂×C₈×C₄₁	—	2	1	no	—
107	C₂×C₈×C₄₁	—	2	1	no	—
108	C₂×C₈×C₄₁	—	2	1	no	—
109	C₂×C₈×C₄₁	—	2	1	no	—
110	C₂×C₈×C₄₁	—	2	2	no	—
111	C₂×C₈×C₄₁	—	2	2	no	—
112	C₂×C₈×C₄₁	—	2	2	no	—
113	C₂×C₈×C₄₁	#41	1	2	yes	—
114	C₂×C₈×C₄₁	#41	1	2	yes	—
115	C₄⋊C₄×C₄₁	—	2	2	no	—
116	C₄⋊C₄×C₄₁	—	2	2	no	—
117	C₄⋊C₄×C₄₁	—	2	2	no	—
118	C₄⋊C₄×C₄₁	—	2	2	no	—
119	C₄⋊C₄×C₄₁	—	4	1	no	—
120	C₄⋊C₄×C₄₁	—	4	2	no	—
121	C₂²⋊C₄×C₄₁	—	4	4	no	—
122	C₂²⋊C₄×C₄₁	—	4	4	no	—
123	C₂²⋊C₄×C₄₁	—	4	4	no	—
124	C₂²⋊C₄×C₄₁	—	4	4	no	—
125	C₂²⋊C₄×C₄₁	—	8	2	no	—
126	C₂²⋊C₄×C₄₁	—	8	2	no	—
127	C₄²×C₄₁	—	2	1	no	—
128	C₄²×C₄₁	—	2	1	no	—
129	C₄²×C₄₁	—	2	2	no	—
130	C₄²×C₄₁	—	2	2	no	—
131	C₄²×C₄₁	—	8	2	no	—
132	C₂³×C₄₁	—	4	2	no	—
133	C₂³×C₄₁	—	4	2	no	—
134	C₂³×C₄₁	—	4	2	no	—
135	C₂³×C₄₁	—	4	2	no	—
136	C₂³×C₄₁	—	4	2	no	—
137	C₂³×C₄₁	—	4	2	no	—
138	C₂³×C₄₁	—	4	2	no	—
139	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
140	Q₈×C₄₁	—	4	4	no	—
141	Q₈×C₄₁	—	4	2	no	—
142	Q₈×C₄₁	—	4	2	no	—
143	Q₈×C₄₁	—	4	2	no	—
144	Q₈×C₄₁	—	4	2	no	—
145	D₄×C₄₁	—	4	2	no	—
146	D₄×C₄₁	—	4	2	no	—
147	D₄×C₄₁	—	4	2	no	—
148	D₄×C₄₁	—	4	2	no	—
149	C₂×C₄×C₄₁	—	2	2	no	—
150	C₂×C₄×C₄₁	—	2	2	no	—
151	C₂×C₄×C₄₁	—	2	1	no	—
152	C₂×C₄×C₄₁	—	2	1	no	—
153	C₂×C₄×C₄₁	—	2	2	no	—
154	C₂×C₄×C₄₁	—	2	2	no	—
155	C₂×C₄×C₄₁	—	2	2	no	—
156	C₂×C₄×C₄₁	—	2	2	no	—
157	C₂×C₄×C₄₁	—	2	2	no	—
158	C₂×C₄×C₄₁	—	2	2	no	—
159	C₂×C₄×C₄₁	—	4	2	no	—
160	C₂×C₄×C₄₁	—	4	2	no	—
161	C₂×C₄×C₄₁	—	4	2	no	—
162	C₂×C₄×C₄₁	—	4	2	no	—
163	C₂×C₄×C₄₁	—	4	2	no	—
164	C₂×C₄×C₄₁	—	4	2	no	—
165	C₂×C₄×C₄₁	—	4	2	no	—
166	C₂×C₄×C₄₁	—	4	2	no	—
167	C₂×C₄×C₄₁	—	4	2	no	—
168	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
173	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
174	C₈×C₄₁	—	2	2	no	—
175	C₈×C₄₁	—	2	2	no	—
176	C₈×C₄₁	—	4	2	no	—
177	#32678	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂²×C₄₁	—	8	2	no	—
179	C₂²×C₄₁	—	8	2	no	—
180	C₂²×C₄₁	—	8	2	no	—
181	C₂²×C₄₁	—	8	2	no	—
182	C₂²×C₄₁	—	8	2	no	—
183	C₂²×C₄₁	—	8	2	no	—
184	C₂²×C₄₁	—	8	2	no	—
185	C₂²×C₄₁	#534	1	1	yes	characteristic
186	C₂²×C₄₁	#550	1	1	yes	characteristic
187	C₂²×C₄₁	#550	1	1	yes	characteristic
188	C₂²×C₄₁	#575	1	2	yes	—
189	C₂²×C₄₁	#577	1	2	yes	—
190	C₄×C₄₁	—	4	2	no	—
191	C₄×C₄₁	—	4	1	no	—
192	C₄×C₄₁	—	4	2	no	—
193	C₄×C₄₁	—	4	1	no	—
194	C₄×C₄₁	—	8	2	no	—
195	C₄×C₄₁	—	8	2	no	—
196	C₄×C₄₁	—	8	2	no	—
197	C₄×C₄₁	#538	1	1	yes	characteristic
198	C₄×C₄₁	#538	1	1	yes	characteristic
199	C₄×C₄₁	#555	1	1	yes	characteristic
200	C₄×C₄₁	H(3,𝔽₄)	1	1	yes	characteristic
201	#2592	C₂×C₄₁	1	2	yes	complemented
202	#2055	C₂×C₄₁	1	2	yes	complemented
203	#2015	C₂×C₄₁	1	2	yes	complemented
204	#2012	C₂×C₄₁	1	2	yes	—
205	#1985	C₂×C₄₁	1	1	yes	characteristic, complemented
206	#1972	C₂×C₄₁	1	1	yes	characteristic
207	#1971	C₂×C₄₁	1	1	yes	characteristic, complemented
208	#1927	C₂×C₄₁	1	2	yes	complemented
209	#1926	C₂×C₄₁	1	2	yes	complemented
210	C₂×C₄₁	—	16	2	no	—
211	C₂×C₄₁	#2733	1	1	yes	characteristic
212	C₂×C₄₁	#3197	1	1	yes	characteristic
213	C₂×C₄₁	#3204	1	1	yes	characteristic
214	C₂×C₄₁	#3401	1	2	yes	—
215	C₂×C₄₁	#3401	1	2	yes	—
216	C₂²×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
217	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
218	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
219	C₂×#124	C₂²×C₄₁	1	2	yes	—
220	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
221	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
222	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
223	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
224	C₂×#103	C₂²×C₄₁	1	2	yes	—
225	C₂×#103	C₂²×C₄₁	1	2	yes	—
226	C₂×#102	C₂²×C₄₁	1	2	yes	—
227	C₂×#102	C₂²×C₄₁	1	2	yes	—
228	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
229	#397	—	2	2	no	—
230	#393	—	2	2	no	—
231	#391	C₂²×C₄₁	1	2	yes	—
232	#391	C₂²×C₄₁	1	2	yes	—
233	#390	C₂²×C₄₁	1	2	yes	—
234	#390	C₂²×C₄₁	1	2	yes	—
235	#388	—	2	2	no	—
236	#388	—	2	2	no	—
237	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
238	#340	C₂²×C₄₁	1	2	yes	—
239	#336	C₂²×C₄₁	1	2	yes	—
240	#336	C₂²×C₄₁	1	2	yes	—
241	C₄₁	#32678	1	1	yes	characteristic, direct product factor, Sylow
242	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
243	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
244	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
245	C₂×Q₁₆	—	2	2	no	—
246	C₂×Q₁₆	—	4	1	no	—
247	C₂×Q₁₆	—	4	1	no	—
248	C₂×SD₁₆	—	4	2	no	—
249	C₂×SD₁₆	—	4	2	no	—
250	C₂×SD₁₆	—	4	2	no	—
251	C₂×D₈	—	2	2	no	—
252	C₂×D₈	—	4	1	no	—
253	C₂×D₈	—	4	1	no	—
254	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
255	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
256	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
257	#124	—	4	2	no	—
258	#124	—	4	2	no	—
259	C₄×Q₈	—	4	1	no	—
260	C₄×Q₈	—	4	1	no	—
261	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
262	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
263	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
264	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
265	C₂×C₂²⋊C₄	—	2	2	no	—
266	C₂×C₂²⋊C₄	—	2	2	no	—
267	C₂×C₂²⋊C₄	—	2	2	no	—
268	C₂×C₂²⋊C₄	—	2	2	no	—
269	C₂×C₂²⋊C₄	—	2	2	no	—
270	C₂×C₂²⋊C₄	—	2	2	no	—
271	C₂×C₄²	—	2	2	no	—
272	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
273	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
274	#106	—	4	1	no	—
275	#106	—	4	1	no	—
276	C₄⋊C₈	—	4	1	no	—
277	#103	—	2	4	no	—
278	#103	—	4	2	no	—
279	#102	—	2	4	no	—
280	#102	—	4	2	no	—
281	#97	—	4	1	no	—
282	#95	—	2	2	no	—
283	#95	—	2	2	no	—
284	#95	—	2	2	no	—
285	#95	—	2	2	no	—
286	#95	—	2	2	no	—
287	C₂⁴	—	2	2	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	4	1	no	—
295	C₂×Q₈	—	4	2	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	2	no	—
298	C₂×D₄	—	2	2	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	2	2	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
307	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
309	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
310	Q₁₆	—	4	4	no	—
311	SD₁₆	—	4	8	no	—
312	D₈	—	4	4	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	#41×C₄₁	1	2	yes	—
321	C₂×C₈	#41×C₄₁	1	2	yes	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₂²⋊C₄	—	4	4	no	—
329	C₂²⋊C₄	—	4	4	no	—
330	C₂²⋊C₄	—	4	4	no	—
331	C₂²⋊C₄	—	4	4	no	—
332	C₂²⋊C₄	—	8	2	no	—
333	C₂²⋊C₄	—	8	2	no	—
334	C₄²	—	2	1	no	—
335	C₄²	—	2	1	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	8	2	no	—
339	C₂³	—	4	2	no	—
340	C₂³	—	4	2	no	—
341	C₂³	—	4	2	no	—
342	C₂³	—	4	2	no	—
343	C₂³	—	4	2	no	—
344	C₂³	—	4	2	no	—
345	C₂³	—	4	2	no	—
346	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
347	Q₈	—	4	4	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	4	2	no	—
351	Q₈	—	4	2	no	—
352	D₄	—	4	2	no	—
353	D₄	—	4	2	no	—
354	D₄	—	4	2	no	—
355	D₄	—	4	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
376	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
377	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
378	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
379	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
380	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
381	C₈	—	2	2	no	—
382	C₈	—	2	2	no	—
383	C₈	—	4	2	no	—
384	C₂²	—	8	2	no	—
385	C₂²	—	8	2	no	—
386	C₂²	—	8	2	no	—
387	C₂²	—	8	2	no	—
388	C₂²	—	8	2	no	—
389	C₂²	—	8	2	no	—
390	C₂²	—	8	2	no	—
391	C₂²	C₄₁×#534	1	1	yes	characteristic
392	C₂²	C₄₁×#550	1	1	yes	characteristic
393	C₂²	C₄₁×#550	1	1	yes	characteristic
394	C₂²	C₄₁×#575	1	2	yes	—
395	C₂²	C₄₁×#577	1	2	yes	—
396	C₄	—	4	2	no	—
397	C₄	—	4	1	no	—
398	C₄	—	4	2	no	—
399	C₄	—	4	1	no	—
400	C₄	—	8	2	no	—
401	C₄	—	8	2	no	—
402	C₄	—	8	2	no	—
403	C₄	C₄₁×#538	1	1	yes	characteristic
404	C₄	C₄₁×#538	1	1	yes	characteristic
405	C₄	C₄₁×#555	1	1	yes	characteristic
406	C₄	C₄₁×H(3,𝔽₄)	1	1	yes	characteristic
407	C₂	—	16	2	no	—
408	C₂	C₄₁×#2733	1	1	yes	characteristic
409	C₂	C₄₁×#3197	1	1	yes	characteristic
410	C₂	C₄₁×#3204	1	1	yes	characteristic
411	C₂	C₄₁×#3401	1	2	yes	—
412	C₂	C₄₁×#3401	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂²×D₈×C₄₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄×Q₈×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#2592	C₂	semidirect
2. C₄₁×#2055	C₂	semidirect
3. C₄₁×#2015	C₂	semidirect
5. C₄₁×#1985	C₂	semidirect
7. C₄₁×#1971	C₂	semidirect
8. C₄₁×#1927	C₂	semidirect
9. C₄₁×#1926	C₂	semidirect
177. #32678	C₄₁	direct (N × H)
201. #2592	C₂×C₄₁	semidirect
202. #2055	C₂×C₄₁	semidirect
203. #2015	C₂×C₄₁	semidirect
205. #1985	C₂×C₄₁	semidirect
207. #1971	C₂×C₄₁	semidirect
208. #1927	C₂×C₄₁	semidirect
209. #1926	C₂×C₄₁	semidirect
241. C₄₁	#32678	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×#2592	2
C₄₁×#2055	2
C₄₁×#2015	2
C₄₁×#2012	2
C₄₁×#1985	2
C₄₁×#1972	2
C₄₁×#1971	2
C₄₁×#1927	2
C₄₁×#1926	2
#32678	41

Subgroups — C41×#32678 #23808879

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#32678 #23808879