Subgroups — C₄₁×#24226 #23800427

Order 10496.

1494 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3496	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3036	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₁×#453	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2328	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2040	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2006	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1963	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1800	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1792	C₂	1	2	yes	maximal, complemented
10	C₂×H(5,𝔽₂)×C₄₁	C₂²	1	2	yes	—
11	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	4	yes	complemented
12	C₄₁×#547	—	2	2	no	—
13	C₄₁×#546	—	2	4	no	—
14	D₄²×C₄₁	—	2	2	no	—
15	C₄₁×#537	C₂²	1	2	yes	—
16	C₄₁×#534	—	2	2	no	—
17	C₄₁×#532	C₂²	1	2	yes	—
18	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×#121×C₄₁	—	2	2	no	—
20	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic, complemented
22	C₄₁×#463	—	2	4	no	—
23	C₄₁×#461	—	2	2	no	—
24	C₄₁×#459	—	2	2	no	—
25	C₄₁×#453	—	2	8	no	—
26	C₄₁×#423	C₂²	1	2	yes	complemented
27	C₂×#104×C₄₁	C₂²	1	4	yes	complemented
28	C₂×#104×C₄₁	C₂²	1	4	yes	complemented
29	C₄₁×#417	C₂²	1	4	yes	complemented
30	C₂×#100×C₄₁	C₂²	1	2	yes	—
31	C₄₁×#390	C₂²	1	1	yes	characteristic
32	C₄₁×#390	C₂²	1	1	yes	characteristic, complemented
33	C₄₁×#386	—	2	2	no	—
34	C₄₁×#386	—	2	2	no	—
35	C₄₁×#382	C₂²	1	1	yes	characteristic, complemented
36	C₄₁×#376	C₂²	1	1	yes	characteristic, complemented
37	C₄₁×#343	C₂²	1	2	yes	complemented
38	H(5,𝔽₂)×C₄₁	—	2	4	no	—
39	H(5,𝔽₂)×C₄₁	—	2	4	no	—
40	H(5,𝔽₂)×C₄₁	—	4	4	no	—
41	C₂×#41×C₄₁	—	2	4	no	—
42	C₂×#41×C₄₁	C₂³	1	4	yes	—
43	C₂²×D₄×C₄₁	—	2	4	no	—
44	C₂²×D₄×C₄₁	—	2	2	no	—
45	C₂²×D₄×C₄₁	—	2	2	no	—
46	C₂²×D₄×C₄₁	—	2	4	no	—
47	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₄₁	—	2	1	no	—
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	4	yes	—
56	C₄⋊Q₈×C₄₁	D₄	1	2	yes	complemented
57	#127×C₄₁	—	2	2	no	—
58	#127×C₄₁	D₄	1	2	yes	complemented
59	#127×C₄₁	D₄	1	4	yes	complemented
60	#124×C₄₁	D₄	1	4	yes	complemented
61	#123×C₄₁	—	2	8	no	—
62	C₂²⋊Q₈×C₄₁	—	2	4	no	—
63	#121×C₄₁	—	2	4	no	—
64	#121×C₄₁	—	2	4	no	—
65	#121×C₄₁	—	2	4	no	—
66	#121×C₄₁	—	2	2	no	—
67	#121×C₄₁	—	2	4	no	—
68	#121×C₄₁	—	2	2	no	—
69	#120×C₄₁	—	2	8	no	—
70	C₄×D₄×C₄₁	—	2	2	no	—
71	C₄×D₄×C₄₁	—	2	2	no	—
72	C₄×D₄×C₄₁	—	2	2	no	—
73	C₄×D₄×C₄₁	—	2	2	no	—
74	C₄×D₄×C₄₁	—	2	2	no	—
75	C₄×D₄×C₄₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
80	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
81	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
82	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
84	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₄₁	—	2	2	no	—
88	C₄⋊C₈×C₄₁	—	2	2	no	—
89	#104×C₄₁	—	2	8	no	—
90	#104×C₄₁	—	4	4	no	—
91	#103×C₄₁	—	2	4	no	—
92	#102×C₄₁	—	2	4	no	—
93	#100×C₄₁	—	2	4	no	—
94	#95×C₄₁	—	2	1	no	—
95	#95×C₄₁	—	2	1	no	—
96	#95×C₄₁	—	2	2	no	—
97	C₂⁴×C₄₁	—	2	4	no	—
98	C₂⁴×C₄₁	—	2	1	no	—
99	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
100	#41×C₄₁	—	2	4	no	—
101	#41×C₄₁	—	2	4	no	—
102	#41×C₄₁	—	2	4	no	—
103	#41×C₄₁	—	2	4	no	—
104	#41×C₄₁	—	4	8	no	—
105	#41×C₄₁	—	4	8	no	—
106	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
107	C₂×D₄×C₄₁	—	2	8	no	—
108	C₂×D₄×C₄₁	—	2	4	no	—
109	C₂×D₄×C₄₁	—	2	4	no	—
110	C₂×D₄×C₄₁	—	2	8	no	—
111	C₂×D₄×C₄₁	—	2	2	no	—
112	C₂×D₄×C₄₁	—	2	2	no	—
113	C₂×D₄×C₄₁	—	2	2	no	—
114	C₂×D₄×C₄₁	—	2	2	no	—
115	C₂×D₄×C₄₁	—	2	1	no	—
116	C₂×D₄×C₄₁	—	2	2	no	—
117	C₂×D₄×C₄₁	—	2	2	no	—
118	C₂×D₄×C₄₁	—	4	8	no	—
119	C₂×D₄×C₄₁	—	4	8	no	—
120	C₂×D₄×C₄₁	—	4	8	no	—
121	C₂×D₄×C₄₁	—	4	8	no	—
122	C₂×D₄×C₄₁	—	8	4	no	—
123	C₂×D₄×C₄₁	—	8	2	no	—
124	C₂×D₄×C₄₁	—	8	2	no	—
125	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
126	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
127	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	complemented
128	C₂×D₄×C₄₁	#41	1	2	yes	—
129	C₂²×C₄×C₄₁	—	2	4	no	—
130	C₂²×C₄×C₄₁	—	2	1	no	—
131	C₂²×C₄×C₄₁	—	2	2	no	—
132	C₂²×C₄×C₄₁	—	2	4	no	—
133	C₂²×C₄×C₄₁	—	2	2	no	—
134	C₂²×C₄×C₄₁	—	2	1	no	—
135	C₂²×C₄×C₄₁	—	2	1	no	—
136	C₂²×C₄×C₄₁	—	4	2	no	—
137	C₂²×C₄×C₄₁	—	8	2	no	—
138	C₂²×C₄×C₄₁	—	8	2	no	—
139	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
143	SD₁₆×C₄₁	—	4	8	no	—
144	D₈×C₄₁	—	4	8	no	—
145	M₁₆×C₄₁	—	2	4	no	—
146	M₁₆×C₄₁	—	2	4	no	—
147	C₂×C₈×C₄₁	—	2	4	no	—
148	C₄⋊C₄×C₄₁	—	2	4	no	—
149	C₄⋊C₄×C₄₁	—	2	2	no	—
150	C₄⋊C₄×C₄₁	—	2	2	no	—
151	C₄⋊C₄×C₄₁	—	2	2	no	—
152	C₄⋊C₄×C₄₁	—	2	2	no	—
153	C₂²⋊C₄×C₄₁	—	2	8	no	—
154	C₂²⋊C₄×C₄₁	—	2	4	no	—
155	C₂²⋊C₄×C₄₁	—	2	4	no	—
156	C₂²⋊C₄×C₄₁	—	2	4	no	—
157	C₂²⋊C₄×C₄₁	—	4	1	no	—
158	C₂²⋊C₄×C₄₁	—	4	1	no	—
159	C₂²⋊C₄×C₄₁	—	8	2	no	—
160	C₂²⋊C₄×C₄₁	—	8	2	no	—
161	C₄²×C₄₁	—	2	1	no	—
162	C₄²×C₄₁	—	2	2	no	—
163	C₄²×C₄₁	C₂×D₄	1	2	yes	—
164	C₄²×C₄₁	C₂×D₄	1	2	yes	—
165	C₄²×C₄₁	C₂×D₄	1	2	yes	complemented
166	C₂³×C₄₁	—	2	2	no	—
167	C₂³×C₄₁	—	2	2	no	—
168	C₂³×C₄₁	—	2	4	no	—
169	C₂³×C₄₁	—	2	2	no	—
170	C₂³×C₄₁	—	4	2	no	—
171	C₂³×C₄₁	—	4	2	no	—
172	C₂³×C₄₁	—	4	2	no	—
173	C₂³×C₄₁	—	4	2	no	—
174	C₂³×C₄₁	—	4	8	no	—
175	C₂³×C₄₁	—	4	1	no	—
176	C₂³×C₄₁	—	4	1	no	—
177	C₂³×C₄₁	—	8	4	no	—
178	C₂³×C₄₁	—	8	4	no	—
179	C₂³×C₄₁	#121	1	4	yes	—
180	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
181	Q₈×C₄₁	—	2	4	no	—
182	Q₈×C₄₁	—	2	4	no	—
183	D₄×C₄₁	—	2	4	no	—
184	D₄×C₄₁	—	2	4	no	—
185	D₄×C₄₁	—	2	4	no	—
186	D₄×C₄₁	—	2	4	no	—
187	D₄×C₄₁	—	2	4	no	—
188	D₄×C₄₁	—	4	4	no	—
189	D₄×C₄₁	—	4	16	no	—
190	D₄×C₄₁	—	4	4	no	—
191	D₄×C₄₁	—	4	4	no	—
192	D₄×C₄₁	—	4	2	no	—
193	D₄×C₄₁	—	8	2	no	—
194	D₄×C₄₁	—	8	8	no	—
195	D₄×C₄₁	—	8	4	no	—
196	C₂×C₄×C₄₁	—	2	2	no	—
197	C₂×C₄×C₄₁	—	2	4	no	—
198	C₂×C₄×C₄₁	—	2	2	no	—
199	C₂×C₄×C₄₁	—	4	1	no	—
200	C₂×C₄×C₄₁	—	4	1	no	—
201	C₂×C₄×C₄₁	—	4	4	no	—
202	C₂×C₄×C₄₁	—	4	8	no	—
203	C₂×C₄×C₄₁	—	4	1	no	—
204	C₂×C₄×C₄₁	—	4	1	no	—
205	C₂×C₄×C₄₁	—	4	4	no	—
206	C₂×C₄×C₄₁	—	4	4	no	—
207	C₂×C₄×C₄₁	—	4	2	no	—
208	C₂×C₄×C₄₁	—	4	1	no	—
209	C₂×C₄×C₄₁	—	4	1	no	—
210	C₂×C₄×C₄₁	—	4	2	no	—
211	C₂×C₄×C₄₁	—	4	1	no	—
212	C₂×C₄×C₄₁	—	4	1	no	—
213	C₂×C₄×C₄₁	—	8	2	no	—
214	C₂×C₄×C₄₁	—	8	4	no	—
215	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
216	C₂×C₄×C₄₁	#120	1	2	yes	—
217	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
218	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
219	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
220	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
222	C₈×C₄₁	—	4	4	no	—
223	#24226	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂²×C₄₁	—	2	1	no	—
225	C₂²×C₄₁	—	2	1	no	—
226	C₂²×C₄₁	—	2	4	no	—
227	C₂²×C₄₁	—	4	4	no	—
228	C₂²×C₄₁	—	4	4	no	—
229	C₂²×C₄₁	—	4	2	no	—
230	C₂²×C₄₁	—	4	2	no	—
231	C₂²×C₄₁	—	4	4	no	—
232	C₂²×C₄₁	—	4	2	no	—
233	C₂²×C₄₁	—	4	4	no	—
234	C₂²×C₄₁	—	4	4	no	—
235	C₂²×C₄₁	—	8	2	no	—
236	C₂²×C₄₁	—	8	2	no	—
237	C₂²×C₄₁	—	8	1	no	—
238	C₂²×C₄₁	—	8	1	no	—
239	C₂²×C₄₁	—	16	8	no	—
240	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
241	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
242	C₂²×C₄₁	#535	1	1	yes	characteristic
243	C₄×C₄₁	—	2	1	no	—
244	C₄×C₄₁	—	2	1	no	—
245	C₄×C₄₁	—	2	2	no	—
246	C₄×C₄₁	—	4	2	no	—
247	C₄×C₄₁	—	4	4	no	—
248	C₄×C₄₁	—	4	2	no	—
249	C₄×C₄₁	—	8	1	no	—
250	C₄×C₄₁	—	8	1	no	—
251	C₄×C₄₁	D₄²	1	2	yes	—
252	C₄×C₄₁	#546	1	2	yes	—
253	#3496	C₂×C₄₁	1	1	yes	characteristic, complemented
254	#3036	C₂×C₄₁	1	2	yes	complemented
255	C₂×#453	C₂×C₄₁	1	2	yes	complemented
256	#2328	C₂×C₄₁	1	1	yes	characteristic, complemented
257	#2040	C₂×C₄₁	1	2	yes	complemented
258	#2006	C₂×C₄₁	1	2	yes	complemented
259	#1963	C₂×C₄₁	1	2	yes	complemented
260	#1800	C₂×C₄₁	1	1	yes	characteristic, complemented
261	#1792	C₂×C₄₁	1	2	yes	complemented
262	C₂×C₄₁	—	2	1	no	—
263	C₂×C₄₁	—	2	1	no	—
264	C₂×C₄₁	—	4	4	no	—
265	C₂×C₄₁	—	8	2	no	—
266	C₂×C₄₁	—	8	4	no	—
267	C₂×C₄₁	#2408	1	1	yes	characteristic
268	C₂×C₄₁	#3019	1	1	yes	characteristic
269	C₂×C₄₁	#3021	1	1	yes	characteristic
270	C₂×H(5,𝔽₂)	C₂²×C₄₁	1	2	yes	—
271	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	4	yes	complemented
272	#547	—	2	2	no	—
273	#546	—	2	4	no	—
274	D₄²	—	2	2	no	—
275	#537	C₂²×C₄₁	1	2	yes	—
276	#534	—	2	2	no	—
277	#532	C₂²×C₄₁	1	2	yes	—
278	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
279	C₂×#121	—	2	2	no	—
280	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
281	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic, complemented
282	#463	—	2	4	no	—
283	#461	—	2	2	no	—
284	#459	—	2	2	no	—
285	#453	—	2	8	no	—
286	#423	C₂²×C₄₁	1	2	yes	complemented
287	C₂×#104	C₂²×C₄₁	1	4	yes	complemented
288	C₂×#104	C₂²×C₄₁	1	4	yes	complemented
289	#417	C₂²×C₄₁	1	4	yes	complemented
290	C₂×#100	C₂²×C₄₁	1	2	yes	—
291	#390	C₂²×C₄₁	1	1	yes	characteristic
292	#390	C₂²×C₄₁	1	1	yes	characteristic, complemented
293	#386	—	2	2	no	—
294	#386	—	2	2	no	—
295	#382	C₂²×C₄₁	1	1	yes	characteristic, complemented
296	#376	C₂²×C₄₁	1	1	yes	characteristic, complemented
297	#343	C₂²×C₄₁	1	2	yes	complemented
298	C₄₁	#24226	1	1	yes	characteristic, direct product factor, Sylow
299	H(5,𝔽₂)	—	2	4	no	—
300	H(5,𝔽₂)	—	2	4	no	—
301	H(5,𝔽₂)	—	4	4	no	—
302	C₂×#41	—	2	4	no	—
303	C₂×#41	C₂³×C₄₁	1	4	yes	—
304	C₂²×D₄	—	2	4	no	—
305	C₂²×D₄	—	2	2	no	—
306	C₂²×D₄	—	2	2	no	—
307	C₂²×D₄	—	2	4	no	—
308	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
309	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
310	C₂³×C₄	—	2	1	no	—
311	C₂³×C₄	D₄×C₄₁	1	2	yes	—
312	AGL(1,ℤ/8)	—	2	8	no	—
313	AGL(1,ℤ/8)	—	4	4	no	—
314	C₂×SD₁₆	—	2	4	no	—
315	C₂×D₈	—	2	4	no	—
316	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
317	C₄⋊Q₈	D₄×C₄₁	1	2	yes	complemented
318	#127	—	2	2	no	—
319	#127	D₄×C₄₁	1	2	yes	complemented
320	#127	D₄×C₄₁	1	4	yes	complemented
321	#124	D₄×C₄₁	1	4	yes	complemented
322	#123	—	2	8	no	—
323	C₂²⋊Q₈	—	2	4	no	—
324	#121	—	2	4	no	—
325	#121	—	2	4	no	—
326	#121	—	2	4	no	—
327	#121	—	2	2	no	—
328	#121	—	2	4	no	—
329	#121	—	2	2	no	—
330	#120	—	2	8	no	—
331	C₄×D₄	—	2	2	no	—
332	C₄×D₄	—	2	2	no	—
333	C₄×D₄	—	2	2	no	—
334	C₄×D₄	—	2	2	no	—
335	C₄×D₄	—	2	2	no	—
336	C₄×D₄	—	2	2	no	—
337	C₂×C₄⋊C₄	—	2	2	no	—
338	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
339	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
340	C₂×C₂²⋊C₄	—	2	4	no	—
341	C₂×C₂²⋊C₄	—	2	1	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
345	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
346	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
347	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
348	C₄⋊C₈	—	2	2	no	—
349	C₄⋊C₈	—	2	2	no	—
350	#104	—	2	8	no	—
351	#104	—	4	4	no	—
352	#103	—	2	4	no	—
353	#102	—	2	4	no	—
354	#100	—	2	4	no	—
355	#95	—	2	1	no	—
356	#95	—	2	1	no	—
357	#95	—	2	2	no	—
358	C₂⁴	—	2	4	no	—
359	C₂⁴	—	2	1	no	—
360	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
361	#41	—	2	4	no	—
362	#41	—	2	4	no	—
363	#41	—	2	4	no	—
364	#41	—	2	4	no	—
365	#41	—	4	8	no	—
366	#41	—	4	8	no	—
367	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
368	C₂×D₄	—	2	8	no	—
369	C₂×D₄	—	2	4	no	—
370	C₂×D₄	—	2	4	no	—
371	C₂×D₄	—	2	8	no	—
372	C₂×D₄	—	2	2	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	4	8	no	—
380	C₂×D₄	—	4	8	no	—
381	C₂×D₄	—	4	8	no	—
382	C₂×D₄	—	4	8	no	—
383	C₂×D₄	—	8	4	no	—
384	C₂×D₄	—	8	2	no	—
385	C₂×D₄	—	8	2	no	—
386	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
387	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
388	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	complemented
389	C₂×D₄	#41×C₄₁	1	2	yes	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	8	2	no	—
399	C₂²×C₄	—	8	2	no	—
400	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
401	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
403	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
404	SD₁₆	—	4	8	no	—
405	D₈	—	4	8	no	—
406	M₁₆	—	2	4	no	—
407	M₁₆	—	2	4	no	—
408	C₂×C₈	—	2	4	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₂²⋊C₄	—	2	8	no	—
415	C₂²⋊C₄	—	2	4	no	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	4	1	no	—
419	C₂²⋊C₄	—	4	1	no	—
420	C₂²⋊C₄	—	8	2	no	—
421	C₂²⋊C₄	—	8	2	no	—
422	C₄²	—	2	1	no	—
423	C₄²	—	2	2	no	—
424	C₄²	C₂×D₄×C₄₁	1	2	yes	—
425	C₄²	C₂×D₄×C₄₁	1	2	yes	—
426	C₄²	C₂×D₄×C₄₁	1	2	yes	complemented
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	4	no	—
430	C₂³	—	2	2	no	—
431	C₂³	—	4	2	no	—
432	C₂³	—	4	2	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	—	4	8	no	—
436	C₂³	—	4	1	no	—
437	C₂³	—	4	1	no	—
438	C₂³	—	8	4	no	—
439	C₂³	—	8	4	no	—
440	C₂³	#121×C₄₁	1	4	yes	—
441	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
442	Q₈	—	2	4	no	—
443	Q₈	—	2	4	no	—
444	D₄	—	2	4	no	—
445	D₄	—	2	4	no	—
446	D₄	—	2	4	no	—
447	D₄	—	2	4	no	—
448	D₄	—	2	4	no	—
449	D₄	—	4	4	no	—
450	D₄	—	4	16	no	—
451	D₄	—	4	4	no	—
452	D₄	—	4	4	no	—
453	D₄	—	4	2	no	—
454	D₄	—	8	2	no	—
455	D₄	—	8	8	no	—
456	D₄	—	8	4	no	—
457	C₂×C₄	—	2	2	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	4	no	—
463	C₂×C₄	—	4	8	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	4	no	—
467	C₂×C₄	—	4	4	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	8	2	no	—
475	C₂×C₄	—	8	4	no	—
476	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
477	C₂×C₄	#120×C₄₁	1	2	yes	—
478	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
479	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
480	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
481	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
482	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
483	C₈	—	4	4	no	—
484	C₂²	—	2	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	2	4	no	—
487	C₂²	—	4	4	no	—
488	C₂²	—	4	4	no	—
489	C₂²	—	4	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	4	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	4	no	—
494	C₂²	—	4	4	no	—
495	C₂²	—	8	2	no	—
496	C₂²	—	8	2	no	—
497	C₂²	—	8	1	no	—
498	C₂²	—	8	1	no	—
499	C₂²	—	16	8	no	—
500	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
501	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
502	C₂²	C₄₁×#535	1	1	yes	characteristic
503	C₄	—	2	1	no	—
504	C₄	—	2	1	no	—
505	C₄	—	2	2	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	4	no	—
508	C₄	—	4	2	no	—
509	C₄	—	8	1	no	—
510	C₄	—	8	1	no	—
511	C₄	D₄²×C₄₁	1	2	yes	—
512	C₄	C₄₁×#546	1	2	yes	—
513	C₂	—	2	1	no	—
514	C₂	—	2	1	no	—
515	C₂	—	4	4	no	—
516	C₂	—	8	2	no	—
517	C₂	—	8	4	no	—
518	C₂	C₄₁×#2408	1	1	yes	characteristic
519	C₂	C₄₁×#3019	1	1	yes	characteristic
520	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×#3496	C₂	semidirect
2. C₄₁×#3036	C₂	semidirect
3. C₂×C₄₁×#453	C₂	semidirect
4. C₄₁×#2328	C₂	semidirect
5. C₄₁×#2040	C₂	semidirect
6. C₄₁×#2006	C₂	semidirect
7. C₄₁×#1963	C₂	semidirect
8. C₄₁×#1800	C₂	semidirect
9. C₄₁×#1792	C₂	semidirect
11. C₂×AGL(1,ℤ/8)×C₄₁	C₂²	semidirect
21. C₂×C₄×D₄×C₄₁	C₂²	semidirect
26. C₄₁×#423	C₂²	semidirect
27. C₂×#104×C₄₁	C₂²	semidirect
28. C₂×#104×C₄₁	C₂²	semidirect
29. C₄₁×#417	C₂²	semidirect
32. C₄₁×#390	C₂²	semidirect
35. C₄₁×#382	C₂²	semidirect
36. C₄₁×#376	C₂²	semidirect
37. C₄₁×#343	C₂²	semidirect
56. C₄⋊Q₈×C₄₁	D₄	semidirect
58. #127×C₄₁	D₄	semidirect
59. #127×C₄₁	D₄	semidirect
60. #124×C₄₁	D₄	semidirect
127. C₂×D₄×C₄₁	C₂×D₄	semidirect
165. C₄²×C₄₁	C₂×D₄	semidirect
223. #24226	C₄₁	direct (N × H)
253. #3496	C₂×C₄₁	semidirect
254. #3036	C₂×C₄₁	semidirect
255. C₂×#453	C₂×C₄₁	semidirect
256. #2328	C₂×C₄₁	semidirect
257. #2040	C₂×C₄₁	semidirect
258. #2006	C₂×C₄₁	semidirect
259. #1963	C₂×C₄₁	semidirect
260. #1800	C₂×C₄₁	semidirect
261. #1792	C₂×C₄₁	semidirect
271. C₂×AGL(1,ℤ/8)	C₂²×C₄₁	semidirect
281. C₂×C₄×D₄	C₂²×C₄₁	semidirect
286. #423	C₂²×C₄₁	semidirect
287. C₂×#104	C₂²×C₄₁	semidirect
288. C₂×#104	C₂²×C₄₁	semidirect
289. #417	C₂²×C₄₁	semidirect
292. #390	C₂²×C₄₁	semidirect
295. #382	C₂²×C₄₁	semidirect
296. #376	C₂²×C₄₁	semidirect
297. #343	C₂²×C₄₁	semidirect
298. C₄₁	#24226	direct (N × H)
317. C₄⋊Q₈	D₄×C₄₁	semidirect
319. #127	D₄×C₄₁	semidirect
320. #127	D₄×C₄₁	semidirect
321. #124	D₄×C₄₁	semidirect
388. C₂×D₄	C₂×D₄×C₄₁	semidirect
426. C₄²	C₂×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×#3496	2
C₄₁×#3036	2
C₂×C₄₁×#453	2
C₄₁×#2328	2
C₄₁×#2040	2
C₄₁×#2006	2
C₄₁×#1963	2
C₄₁×#1800	2
C₄₁×#1792	2
#24226	41

Subgroups — C41×#24226 #23800427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24226 #23800427