Subgroups — C₄₁×#25404 #23801605

Order 10496.

968 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×#3151	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#472	C₂	1	2	yes	maximal, complemented
3	C₄₁×#3036	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₁×#459	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2935	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2051	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2018	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1965	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1907	C₂	1	2	yes	maximal, complemented
11	C₄×#100×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
13	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
14	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
15	C₂²×D₈×C₄₁	C₂²	1	2	yes	—
16	C₂×#131×C₄₁	C₂²	1	2	yes	—
17	C₄₁×#532	C₂²	1	1	yes	characteristic
18	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
20	C₄₁×#496	—	2	2	no	—
21	C₄₁×#493	—	2	1	no	—
22	C₄₁×#492	—	2	1	no	—
23	C₄₁×#472	—	2	8	no	—
24	C₄₁×#463	—	2	2	no	—
25	C₄₁×#461	—	2	1	no	—
26	C₄₁×#459	—	2	1	no	—
27	C₄₁×#459	—	2	1	no	—
28	C₄₁×#459	—	2	2	no	—
29	C₄₁×#459	—	2	1	no	—
30	C₄₁×#436	—	2	1	no	—
31	C₄₁×#435	—	2	2	no	—
32	C₈×D₄×C₄₁	—	2	1	no	—
33	C₂×#108×C₄₁	C₂²	1	2	yes	—
34	C₄₁×#423	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
36	C₄₁×#417	C₂²	1	2	yes	—
37	C₂×#102×C₄₁	C₂²	1	2	yes	—
38	C₂×#100×C₄₁	C₂²	1	1	yes	characteristic
39	C₂×#100×C₄₁	C₂²	1	2	yes	—
40	C₂×#100×C₄₁	C₂²	1	2	yes	—
41	C₂×#100×C₄₁	C₂²	1	1	yes	characteristic
42	C₄₁×#408	C₂²	1	2	yes	—
43	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
45	C₄₁×#343	C₂²	1	1	yes	characteristic
46	C₄₁×#343	C₂²	1	1	yes	characteristic
47	C₄₁×#341	C₂²	1	2	yes	—
48	C₂×#41×C₄₁	C₂³	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	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₄₁	—	2	1	no	—
54	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
55	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
56	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
57	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
58	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
59	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
60	C₂×SD₁₆×C₄₁	—	2	2	no	—
61	C₂×SD₁₆×C₄₁	—	2	2	no	—
62	C₂×SD₁₆×C₄₁	—	2	2	no	—
63	C₂×D₈×C₄₁	—	2	4	no	—
64	C₂×D₈×C₄₁	—	2	2	no	—
65	C₂×D₈×C₄₁	—	2	2	no	—
66	C₂×D₈×C₄₁	—	2	2	no	—
67	C₂×D₈×C₄₁	—	2	2	no	—
68	C₂×D₈×C₄₁	—	2	2	no	—
69	C₂×D₈×C₄₁	—	4	2	no	—
70	#131×C₄₁	—	2	4	no	—
71	#131×C₄₁	—	4	2	no	—
72	C₂×M₁₆×C₄₁	—	2	2	no	—
73	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
74	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
76	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
77	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₄₁	—	2	2	no	—
79	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
81	#127×C₄₁	—	2	1	no	—
82	#127×C₄₁	—	2	1	no	—
83	#127×C₄₁	—	2	1	no	—
84	#127×C₄₁	—	2	1	no	—
85	#127×C₄₁	D₄	1	1	yes	characteristic
86	#124×C₄₁	D₄	1	2	yes	—
87	#121×C₄₁	—	2	1	no	—
88	#121×C₄₁	—	2	1	no	—
89	C₄×D₄×C₄₁	—	2	1	no	—
90	C₄×D₄×C₄₁	—	2	1	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	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
96	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
97	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
98	#108×C₄₁	—	2	4	no	—
99	C₄⋊C₈×C₄₁	—	2	1	no	—
100	C₄⋊C₈×C₄₁	—	2	1	no	—
101	C₄⋊C₈×C₄₁	D₄	1	1	yes	characteristic
102	C₄⋊C₈×C₄₁	D₄	1	2	yes	—
103	C₄⋊C₈×C₄₁	D₄	1	1	yes	characteristic
104	#103×C₄₁	—	2	2	no	—
105	#102×C₄₁	—	2	2	no	—
106	#102×C₄₁	—	2	2	no	—
107	#102×C₄₁	—	2	2	no	—
108	#100×C₄₁	—	2	4	no	—
109	#100×C₄₁	—	2	2	no	—
110	#100×C₄₁	—	4	2	no	—
111	#100×C₄₁	D₄	1	4	yes	complemented
112	C₂²⋊C₈×C₄₁	—	2	2	no	—
113	C₂²⋊C₈×C₄₁	—	2	2	no	—
114	#97×C₄₁	—	2	1	no	—
115	#97×C₄₁	—	2	1	no	—
116	C₄×C₈×C₄₁	—	2	1	no	—
117	C₄×C₈×C₄₁	—	2	1	no	—
118	C₂⁴×C₄₁	—	2	1	no	—
119	C₂⁴×C₄₁	—	2	2	no	—
120	#41×C₄₁	—	2	2	no	—
121	#41×C₄₁	—	2	2	no	—
122	#41×C₄₁	—	2	2	no	—
123	#41×C₄₁	—	2	2	no	—
124	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
125	C₂×D₄×C₄₁	—	2	2	no	—
126	C₂×D₄×C₄₁	—	2	1	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	2	no	—
131	C₂×D₄×C₄₁	—	2	2	no	—
132	C₂×D₄×C₄₁	—	2	2	no	—
133	C₂×D₄×C₄₁	—	2	1	no	—
134	C₂×D₄×C₄₁	—	2	2	no	—
135	C₂×D₄×C₄₁	—	2	1	no	—
136	C₂×D₄×C₄₁	—	2	1	no	—
137	C₂×D₄×C₄₁	—	2	2	no	—
138	C₂×D₄×C₄₁	—	2	2	no	—
139	C₂×D₄×C₄₁	—	4	1	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	—	8	2	no	—
144	C₂×D₄×C₄₁	—	8	2	no	—
145	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
146	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
147	C₂²×C₄×C₄₁	—	2	2	no	—
148	C₂²×C₄×C₄₁	—	2	1	no	—
149	C₂²×C₄×C₄₁	—	4	1	no	—
150	C₂²×C₄×C₄₁	—	4	1	no	—
151	C₂²×C₄×C₄₁	—	8	2	no	—
152	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
153	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
154	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
155	SD₁₆×C₄₁	—	4	4	no	—
156	SD₁₆×C₄₁	—	4	4	no	—
157	SD₁₆×C₄₁	—	4	4	no	—
158	D₈×C₄₁	—	4	4	no	—
159	D₈×C₄₁	—	4	4	no	—
160	D₈×C₄₁	—	4	4	no	—
161	D₈×C₄₁	—	4	4	no	—
162	D₈×C₄₁	—	4	4	no	—
163	M₁₆×C₄₁	—	2	1	no	—
164	M₁₆×C₄₁	—	2	1	no	—
165	M₁₆×C₄₁	—	2	1	no	—
166	M₁₆×C₄₁	—	2	2	no	—
167	M₁₆×C₄₁	—	2	1	no	—
168	M₁₆×C₄₁	—	2	1	no	—
169	M₁₆×C₄₁	—	2	2	no	—
170	M₁₆×C₄₁	—	4	4	no	—
171	M₁₆×C₄₁	C₂×D₄	1	2	yes	—
172	M₁₆×C₄₁	C₂×D₄	1	2	yes	—
173	M₁₆×C₄₁	#41	1	2	yes	—
174	C₂×C₈×C₄₁	—	2	1	no	—
175	C₂×C₈×C₄₁	—	2	1	no	—
176	C₂×C₈×C₄₁	—	2	1	no	—
177	C₂×C₈×C₄₁	—	2	2	no	—
178	C₂×C₈×C₄₁	—	2	2	no	—
179	C₂×C₈×C₄₁	—	2	1	no	—
180	C₂×C₈×C₄₁	—	2	2	no	—
181	C₂×C₈×C₄₁	—	4	4	no	—
182	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
183	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
184	C₄⋊C₄×C₄₁	—	2	1	no	—
185	C₄⋊C₄×C₄₁	—	2	1	no	—
186	C₄⋊C₄×C₄₁	—	2	1	no	—
187	C₄⋊C₄×C₄₁	—	2	1	no	—
188	C₂²⋊C₄×C₄₁	—	2	2	no	—
189	C₂²⋊C₄×C₄₁	—	2	2	no	—
190	C₂²⋊C₄×C₄₁	—	4	1	no	—
191	C₂²⋊C₄×C₄₁	—	4	1	no	—
192	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
196	C₂³×C₄₁	—	2	2	no	—
197	C₂³×C₄₁	—	2	2	no	—
198	C₂³×C₄₁	—	4	1	no	—
199	C₂³×C₄₁	—	4	1	no	—
200	C₂³×C₄₁	—	4	2	no	—
201	C₂³×C₄₁	—	4	2	no	—
202	C₂³×C₄₁	—	4	2	no	—
203	C₂³×C₄₁	—	8	1	no	—
204	C₂³×C₄₁	—	8	1	no	—
205	C₂³×C₄₁	—	8	2	no	—
206	C₂³×C₄₁	—	8	2	no	—
207	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
208	Q₈×C₄₁	—	2	2	no	—
209	Q₈×C₄₁	—	2	2	no	—
210	D₄×C₄₁	—	2	2	no	—
211	D₄×C₄₁	—	2	2	no	—
212	D₄×C₄₁	—	2	2	no	—
213	D₄×C₄₁	—	2	2	no	—
214	D₄×C₄₁	—	4	2	no	—
215	D₄×C₄₁	—	4	2	no	—
216	D₄×C₄₁	—	4	2	no	—
217	D₄×C₄₁	—	4	2	no	—
218	D₄×C₄₁	—	4	1	no	—
219	D₄×C₄₁	—	4	2	no	—
220	D₄×C₄₁	—	4	1	no	—
221	D₄×C₄₁	—	4	2	no	—
222	D₄×C₄₁	—	4	2	no	—
223	D₄×C₄₁	—	8	1	no	—
224	D₄×C₄₁	—	8	2	no	—
225	D₄×C₄₁	—	8	2	no	—
226	D₄×C₄₁	—	8	2	no	—
227	D₄×C₄₁	—	8	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	1	no	—
234	C₂×C₄×C₄₁	—	8	2	no	—
235	C₂×C₄×C₄₁	—	8	2	no	—
236	C₂×C₄×C₄₁	—	8	1	no	—
237	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
238	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
239	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
240	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
241	C₂×C₄×C₄₁	C₂²×D₄	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	1	yes	characteristic
245	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
246	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
247	C₈×C₄₁	—	2	2	no	—
248	C₈×C₄₁	—	2	2	no	—
249	C₈×C₄₁	—	2	2	no	—
250	C₈×C₄₁	—	2	2	no	—
251	C₈×C₄₁	—	2	2	no	—
252	C₈×C₄₁	—	4	2	no	—
253	C₈×C₄₁	—	4	1	no	—
254	#25404	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂²×C₄₁	—	2	1	no	—
256	C₂²×C₄₁	—	2	1	no	—
257	C₂²×C₄₁	—	4	2	no	—
258	C₂²×C₄₁	—	4	2	no	—
259	C₂²×C₄₁	—	4	2	no	—
260	C₂²×C₄₁	—	4	2	no	—
261	C₂²×C₄₁	—	8	2	no	—
262	C₂²×C₄₁	—	8	2	no	—
263	C₂²×C₄₁	—	8	1	no	—
264	C₂²×C₄₁	—	8	1	no	—
265	C₂²×C₄₁	—	8	2	no	—
266	C₂²×C₄₁	—	8	2	no	—
267	C₂²×C₄₁	—	8	2	no	—
268	C₂²×C₄₁	—	16	2	no	—
269	C₂²×C₄₁	—	16	2	no	—
270	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
271	C₂²×C₄₁	#532	1	1	yes	characteristic
272	C₂²×C₄₁	#540	1	1	yes	characteristic
273	C₄×C₄₁	—	2	1	no	—
274	C₄×C₄₁	—	2	1	no	—
275	C₄×C₄₁	—	2	1	no	—
276	C₄×C₄₁	—	4	2	no	—
277	C₄×C₄₁	—	8	1	no	—
278	C₄×C₄₁	#472	1	2	yes	complemented
279	C₄×C₄₁	D₄²	1	1	yes	characteristic
280	C₄×C₄₁	D₄²	1	1	yes	characteristic
281	C₄×C₄₁	#546	1	1	yes	characteristic
282	C₄×C₄₁	#550	1	1	yes	characteristic
283	#3151	C₂×C₄₁	1	1	yes	characteristic, complemented
284	C₂×#472	C₂×C₄₁	1	2	yes	complemented
285	#3036	C₂×C₄₁	1	1	yes	characteristic, complemented
286	C₂×#459	C₂×C₄₁	1	1	yes	characteristic, complemented
287	#2935	C₂×C₄₁	1	1	yes	characteristic, complemented
288	#2051	C₂×C₄₁	1	2	yes	complemented
289	#2018	C₂×C₄₁	1	2	yes	complemented
290	#1965	C₂×C₄₁	1	1	yes	characteristic, complemented
291	#1950	C₂×C₄₁	1	1	yes	characteristic, complemented
292	#1907	C₂×C₄₁	1	2	yes	complemented
293	C₄×#100	C₂×C₄₁	1	1	yes	characteristic, complemented
294	C₂×C₄₁	—	2	1	no	—
295	C₂×C₄₁	—	2	1	no	—
296	C₂×C₄₁	—	8	2	no	—
297	C₂×C₄₁	—	8	2	no	—
298	C₂×C₄₁	—	16	2	no	—
299	C₂×C₄₁	#2428	1	1	yes	characteristic
300	C₂×C₄₁	C₂×#472	1	1	yes	characteristic
301	C₂×C₄₁	#3068	1	1	yes	characteristic
302	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
303	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
304	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
305	C₂²×D₈	C₂²×C₄₁	1	2	yes	—
306	C₂×#131	C₂²×C₄₁	1	2	yes	—
307	#532	C₂²×C₄₁	1	1	yes	characteristic
308	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
309	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
310	#496	—	2	2	no	—
311	#493	—	2	1	no	—
312	#492	—	2	1	no	—
313	#472	—	2	8	no	—
314	#463	—	2	2	no	—
315	#461	—	2	1	no	—
316	#459	—	2	1	no	—
317	#459	—	2	1	no	—
318	#459	—	2	2	no	—
319	#459	—	2	1	no	—
320	#436	—	2	1	no	—
321	#435	—	2	2	no	—
322	C₈×D₄	—	2	1	no	—
323	C₂×#108	C₂²×C₄₁	1	2	yes	—
324	#423	C₂²×C₄₁	1	1	yes	characteristic
325	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
326	#417	C₂²×C₄₁	1	2	yes	—
327	C₂×#102	C₂²×C₄₁	1	2	yes	—
328	C₂×#100	C₂²×C₄₁	1	2	yes	—
329	C₂×#100	C₂²×C₄₁	1	1	yes	characteristic
330	C₂×#100	C₂²×C₄₁	1	2	yes	—
331	C₂×#100	C₂²×C₄₁	1	1	yes	characteristic
332	#408	C₂²×C₄₁	1	2	yes	—
333	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
334	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
335	#343	C₂²×C₄₁	1	1	yes	characteristic
336	#343	C₂²×C₄₁	1	1	yes	characteristic
337	#341	C₂²×C₄₁	1	2	yes	—
338	C₄₁	#25404	1	1	yes	characteristic, direct product factor, Sylow
339	C₂×#41	C₂³×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	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
344	C₂³×C₄	—	2	1	no	—
345	AGL(1,ℤ/8)	—	2	4	no	—
346	AGL(1,ℤ/8)	—	2	4	no	—
347	AGL(1,ℤ/8)	—	2	4	no	—
348	AGL(1,ℤ/8)	—	4	2	no	—
349	AGL(1,ℤ/8)	—	4	2	no	—
350	AGL(1,ℤ/8)	—	4	2	no	—
351	C₂×SD₁₆	—	2	2	no	—
352	C₂×SD₁₆	—	2	2	no	—
353	C₂×SD₁₆	—	2	2	no	—
354	C₂×D₈	—	2	4	no	—
355	C₂×D₈	—	2	2	no	—
356	C₂×D₈	—	2	2	no	—
357	C₂×D₈	—	2	2	no	—
358	C₂×D₈	—	2	2	no	—
359	C₂×D₈	—	2	2	no	—
360	C₂×D₈	—	4	2	no	—
361	#131	—	2	4	no	—
362	#131	—	4	2	no	—
363	C₂×M₁₆	—	2	2	no	—
364	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
365	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
366	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
367	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
368	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
369	C₂²×C₈	—	2	2	no	—
370	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
371	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
372	#127	—	2	1	no	—
373	#127	—	2	1	no	—
374	#127	—	2	1	no	—
375	#127	—	2	1	no	—
376	#127	D₄×C₄₁	1	1	yes	characteristic
377	#124	D₄×C₄₁	1	2	yes	—
378	#121	—	2	1	no	—
379	#121	—	2	1	no	—
380	C₄×D₄	—	2	1	no	—
381	C₄×D₄	—	2	1	no	—
382	C₄×D₄	—	2	1	no	—
383	C₄×D₄	—	2	1	no	—
384	C₄×D₄	—	2	1	no	—
385	C₄×D₄	—	2	1	no	—
386	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
387	C₂×C₂²⋊C₄	—	2	1	no	—
388	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
389	#108	—	2	4	no	—
390	C₄⋊C₈	—	2	1	no	—
391	C₄⋊C₈	—	2	1	no	—
392	C₄⋊C₈	D₄×C₄₁	1	1	yes	characteristic
393	C₄⋊C₈	D₄×C₄₁	1	2	yes	—
394	C₄⋊C₈	D₄×C₄₁	1	1	yes	characteristic
395	#103	—	2	2	no	—
396	#102	—	2	2	no	—
397	#102	—	2	2	no	—
398	#102	—	2	2	no	—
399	#100	—	2	4	no	—
400	#100	—	2	2	no	—
401	#100	—	4	2	no	—
402	#100	D₄×C₄₁	1	4	yes	complemented
403	C₂²⋊C₈	—	2	2	no	—
404	C₂²⋊C₈	—	2	2	no	—
405	#97	—	2	1	no	—
406	#97	—	2	1	no	—
407	C₄×C₈	—	2	1	no	—
408	C₄×C₈	—	2	1	no	—
409	C₂⁴	—	2	1	no	—
410	C₂⁴	—	2	2	no	—
411	#41	—	2	2	no	—
412	#41	—	2	2	no	—
413	#41	—	2	2	no	—
414	#41	—	2	2	no	—
415	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	1	no	—
418	C₂×D₄	—	2	1	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	1	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	4	1	no	—
431	C₂×D₄	—	4	1	no	—
432	C₂×D₄	—	4	1	no	—
433	C₂×D₄	—	4	1	no	—
434	C₂×D₄	—	8	2	no	—
435	C₂×D₄	—	8	2	no	—
436	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
437	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	1	no	—
440	C₂²×C₄	—	4	1	no	—
441	C₂²×C₄	—	4	1	no	—
442	C₂²×C₄	—	8	2	no	—
443	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
444	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
446	SD₁₆	—	4	4	no	—
447	SD₁₆	—	4	4	no	—
448	SD₁₆	—	4	4	no	—
449	D₈	—	4	4	no	—
450	D₈	—	4	4	no	—
451	D₈	—	4	4	no	—
452	D₈	—	4	4	no	—
453	D₈	—	4	4	no	—
454	M₁₆	—	2	1	no	—
455	M₁₆	—	2	1	no	—
456	M₁₆	—	2	1	no	—
457	M₁₆	—	2	2	no	—
458	M₁₆	—	2	1	no	—
459	M₁₆	—	2	1	no	—
460	M₁₆	—	2	2	no	—
461	M₁₆	—	4	4	no	—
462	M₁₆	C₂×D₄×C₄₁	1	2	yes	—
463	M₁₆	C₂×D₄×C₄₁	1	2	yes	—
464	M₁₆	#41×C₄₁	1	2	yes	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	2	2	no	—
469	C₂×C₈	—	2	2	no	—
470	C₂×C₈	—	2	1	no	—
471	C₂×C₈	—	2	2	no	—
472	C₂×C₈	—	4	4	no	—
473	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
474	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₂²⋊C₄	—	2	2	no	—
480	C₂²⋊C₄	—	2	2	no	—
481	C₂²⋊C₄	—	4	1	no	—
482	C₂²⋊C₄	—	4	1	no	—
483	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
484	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
485	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
486	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
487	C₂³	—	2	2	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	4	1	no	—
490	C₂³	—	4	1	no	—
491	C₂³	—	4	2	no	—
492	C₂³	—	4	2	no	—
493	C₂³	—	4	2	no	—
494	C₂³	—	8	1	no	—
495	C₂³	—	8	1	no	—
496	C₂³	—	8	2	no	—
497	C₂³	—	8	2	no	—
498	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
499	Q₈	—	2	2	no	—
500	Q₈	—	2	2	no	—
501	D₄	—	2	2	no	—
502	D₄	—	2	2	no	—
503	D₄	—	2	2	no	—
504	D₄	—	2	2	no	—
505	D₄	—	4	2	no	—
506	D₄	—	4	2	no	—
507	D₄	—	4	2	no	—
508	D₄	—	4	2	no	—
509	D₄	—	4	1	no	—
510	D₄	—	4	2	no	—
511	D₄	—	4	1	no	—
512	D₄	—	4	2	no	—
513	D₄	—	4	2	no	—
514	D₄	—	8	1	no	—
515	D₄	—	8	2	no	—
516	D₄	—	8	2	no	—
517	D₄	—	8	2	no	—
518	D₄	—	8	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	4	1	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	4	2	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	8	2	no	—
526	C₂×C₄	—	8	2	no	—
527	C₂×C₄	—	8	1	no	—
528	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
529	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
530	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
531	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
532	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
533	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
534	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
535	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
536	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
537	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
538	C₈	—	2	2	no	—
539	C₈	—	2	2	no	—
540	C₈	—	2	2	no	—
541	C₈	—	2	2	no	—
542	C₈	—	2	2	no	—
543	C₈	—	4	2	no	—
544	C₈	—	4	1	no	—
545	C₂²	—	2	1	no	—
546	C₂²	—	2	1	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	8	2	no	—
552	C₂²	—	8	2	no	—
553	C₂²	—	8	1	no	—
554	C₂²	—	8	1	no	—
555	C₂²	—	8	2	no	—
556	C₂²	—	8	2	no	—
557	C₂²	—	8	2	no	—
558	C₂²	—	16	2	no	—
559	C₂²	—	16	2	no	—
560	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
561	C₂²	C₄₁×#532	1	1	yes	characteristic
562	C₂²	C₄₁×#540	1	1	yes	characteristic
563	C₄	—	2	1	no	—
564	C₄	—	2	1	no	—
565	C₄	—	2	1	no	—
566	C₄	—	4	2	no	—
567	C₄	—	8	1	no	—
568	C₄	C₄₁×#472	1	2	yes	complemented
569	C₄	D₄²×C₄₁	1	1	yes	characteristic
570	C₄	D₄²×C₄₁	1	1	yes	characteristic
571	C₄	C₄₁×#546	1	1	yes	characteristic
572	C₄	C₄₁×#550	1	1	yes	characteristic
573	C₂	—	2	1	no	—
574	C₂	—	2	1	no	—
575	C₂	—	8	2	no	—
576	C₂	—	8	2	no	—
577	C₂	—	16	2	no	—
578	C₂	C₄₁×#2428	1	1	yes	characteristic
579	C₂	C₂×C₄₁×#472	1	1	yes	characteristic
580	C₂	C₄₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#121
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×#3151	C₂	semidirect
2. C₂×C₄₁×#472	C₂	semidirect
3. C₄₁×#3036	C₂	semidirect
4. C₂×C₄₁×#459	C₂	semidirect
5. C₄₁×#2935	C₂	semidirect
6. C₄₁×#2051	C₂	semidirect
7. C₄₁×#2018	C₂	semidirect
8. C₄₁×#1965	C₂	semidirect
9. C₄₁×#1950	C₂	semidirect
10. C₄₁×#1907	C₂	semidirect
11. C₄×#100×C₄₁	C₂	semidirect
111. #100×C₄₁	D₄	semidirect
254. #25404	C₄₁	direct (N × H)
278. C₄×C₄₁	#472	semidirect
283. #3151	C₂×C₄₁	semidirect
284. C₂×#472	C₂×C₄₁	semidirect
285. #3036	C₂×C₄₁	semidirect
286. C₂×#459	C₂×C₄₁	semidirect
287. #2935	C₂×C₄₁	semidirect
288. #2051	C₂×C₄₁	semidirect
289. #2018	C₂×C₄₁	semidirect
290. #1965	C₂×C₄₁	semidirect
291. #1950	C₂×C₄₁	semidirect
292. #1907	C₂×C₄₁	semidirect
293. C₄×#100	C₂×C₄₁	semidirect
338. C₄₁	#25404	direct (N × H)
402. #100	D₄×C₄₁	semidirect
568. C₄	C₄₁×#472	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3151	2
C₂×C₄₁×#472	2
C₄₁×#3036	2
C₂×C₄₁×#459	2
C₄₁×#2935	2
C₄₁×#2051	2
C₄₁×#2018	2
C₄₁×#1965	2
C₄₁×#1950	2
C₄₁×#1907	2
C₄×#100×C₄₁	2
#25404	41

Subgroups — C41×#25404 #23801605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#25404 #23801605