Subgroups — C₄₁×#24154 #23800355

Order 10496.

1394 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂²⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2034	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2000	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1988	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1982	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1877	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1864	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₄₁	—	2	1	no	—
13	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
14	C₄₁×#532	—	2	4	no	—
15	C₄₁×#532	—	2	4	no	—
16	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
18	C₂×#124×C₄₁	—	2	1	no	—
19	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×#121×C₄₁	—	2	1	no	—
21	C₂×#121×C₄₁	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₄₁	—	2	1	no	—
24	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
25	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
26	C₄₁×#475	—	2	4	no	—
27	C₄₁×#460	—	2	4	no	—
28	C₂×#106×C₄₁	C₂²	1	2	yes	—
29	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×#103×C₄₁	C₂²	1	2	yes	—
32	C₂×#102×C₄₁	C₂²	1	2	yes	—
33	C₂×#102×C₄₁	C₂²	1	2	yes	—
34	C₂×#102×C₄₁	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
36	C₄₁×#391	C₂²	1	1	yes	characteristic
37	C₄₁×#389	C₂²	1	1	yes	characteristic
38	C₄₁×#387	—	2	1	no	—
39	C₄₁×#385	—	2	1	no	—
40	C₄₁×#380	C₂²	1	1	yes	characteristic
41	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
42	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
44	C₄₁×#340	C₂²	1	2	yes	—
45	C₄₁×#340	C₂²	1	2	yes	—
46	C₄₁×#336	C₂²	1	2	yes	—
47	C₂×#41×C₄₁	—	2	8	no	—
48	C₂×#41×C₄₁	—	4	4	no	—
49	C₂×#41×C₄₁	—	4	4	no	—
50	C₂²×Q₈×C₄₁	—	2	1	no	—
51	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₄₁	—	2	2	no	—
53	C₂²×D₄×C₄₁	—	2	1	no	—
54	C₂²×D₄×C₄₁	—	2	1	no	—
55	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
56	C₂³×C₄×C₄₁	—	2	2	no	—
57	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
58	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
60	C₂×SD₁₆×C₄₁	—	2	4	no	—
61	C₂×SD₁₆×C₄₁	—	2	4	no	—
62	C₂×SD₁₆×C₄₁	—	4	2	no	—
63	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
64	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₄₁	—	2	2	no	—
66	C₄⋊Q₈×C₄₁	D₄	1	4	yes	—
67	#127×C₄₁	—	2	2	no	—
68	#127×C₄₁	D₄	1	4	yes	—
69	#124×C₄₁	—	2	8	no	—
70	C₂²⋊Q₈×C₄₁	—	2	2	no	—
71	C₂²⋊Q₈×C₄₁	—	2	2	no	—
72	#121×C₄₁	—	2	8	no	—
73	#121×C₄₁	—	2	4	no	—
74	#121×C₄₁	—	2	4	no	—
75	C₄×Q₈×C₄₁	—	2	2	no	—
76	C₄×Q₈×C₄₁	—	2	2	no	—
77	C₄×D₄×C₄₁	—	2	8	no	—
78	C₄×D₄×C₄₁	—	2	4	no	—
79	C₄×D₄×C₄₁	—	2	4	no	—
80	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
82	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
92	C₂×C₄²×C₄₁	—	2	2	no	—
93	C₂×C₄²×C₄₁	—	2	2	no	—
94	C₂×C₄²×C₄₁	—	2	1	no	—
95	C₂×C₄²×C₄₁	—	2	1	no	—
96	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
100	#106×C₄₁	—	2	4	no	—
101	C₄⋊C₈×C₄₁	—	2	2	no	—
102	C₄⋊C₈×C₄₁	—	2	2	no	—
103	#103×C₄₁	—	2	4	no	—
104	#102×C₄₁	—	2	4	no	—
105	#102×C₄₁	—	2	4	no	—
106	#102×C₄₁	—	4	2	no	—
107	C₂²⋊C₈×C₄₁	—	4	2	no	—
108	#95×C₄₁	—	2	1	no	—
109	#95×C₄₁	—	2	1	no	—
110	#95×C₄₁	—	2	1	no	—
111	#95×C₄₁	—	2	1	no	—
112	#95×C₄₁	—	2	1	no	—
113	C₂⁴×C₄₁	—	2	2	no	—
114	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
115	#41×C₄₁	—	4	32	no	—
116	C₂×Q₈×C₄₁	—	2	4	no	—
117	C₂×Q₈×C₄₁	—	2	1	no	—
118	C₂×Q₈×C₄₁	—	2	1	no	—
119	C₂×Q₈×C₄₁	—	2	1	no	—
120	C₂×Q₈×C₄₁	—	2	1	no	—
121	C₂×Q₈×C₄₁	—	2	1	no	—
122	C₂×Q₈×C₄₁	—	4	2	no	—
123	C₂×Q₈×C₄₁	—	4	2	no	—
124	C₂×Q₈×C₄₁	#41	1	2	yes	complemented
125	C₂×D₄×C₄₁	—	2	8	no	—
126	C₂×D₄×C₄₁	—	2	4	no	—
127	C₂×D₄×C₄₁	—	2	4	no	—
128	C₂×D₄×C₄₁	—	2	2	no	—
129	C₂×D₄×C₄₁	—	2	2	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	—	4	2	no	—
134	C₂×D₄×C₄₁	—	4	2	no	—
135	C₂×D₄×C₄₁	—	4	4	no	—
136	C₂×D₄×C₄₁	—	4	4	no	—
137	C₂×D₄×C₄₁	—	4	2	no	—
138	C₂×D₄×C₄₁	—	4	2	no	—
139	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
140	C₂²×C₄×C₄₁	—	2	2	no	—
141	C₂²×C₄×C₄₁	—	2	2	no	—
142	C₂²×C₄×C₄₁	—	2	8	no	—
143	C₂²×C₄×C₄₁	—	2	1	no	—
144	C₂²×C₄×C₄₁	—	2	1	no	—
145	C₂²×C₄×C₄₁	—	2	2	no	—
146	C₂²×C₄×C₄₁	—	2	2	no	—
147	C₂²×C₄×C₄₁	—	2	1	no	—
148	C₂²×C₄×C₄₁	—	2	1	no	—
149	C₂²×C₄×C₄₁	—	2	1	no	—
150	C₂²×C₄×C₄₁	—	2	1	no	—
151	C₂²×C₄×C₄₁	—	2	1	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	—	4	1	no	—
156	C₂²×C₄×C₄₁	—	4	1	no	—
157	C₂²×C₄×C₄₁	—	4	4	no	—
158	C₂²×C₄×C₄₁	—	4	4	no	—
159	C₂²×C₄×C₄₁	SD₁₆	1	4	yes	complemented
160	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
166	SD₁₆×C₄₁	—	4	8	no	—
167	C₂×C₈×C₄₁	—	2	2	no	—
168	C₂×C₈×C₄₁	—	2	2	no	—
169	C₂×C₈×C₄₁	—	2	2	no	—
170	C₂×C₈×C₄₁	—	2	2	no	—
171	C₂×C₈×C₄₁	—	2	2	no	—
172	C₂×C₈×C₄₁	—	2	2	no	—
173	C₄⋊C₄×C₄₁	—	2	4	no	—
174	C₄⋊C₄×C₄₁	—	2	4	no	—
175	C₄⋊C₄×C₄₁	—	2	2	no	—
176	C₄⋊C₄×C₄₁	—	2	1	no	—
177	C₄⋊C₄×C₄₁	—	2	1	no	—
178	C₄⋊C₄×C₄₁	—	2	1	no	—
179	C₄⋊C₄×C₄₁	—	2	1	no	—
180	C₄⋊C₄×C₄₁	—	2	1	no	—
181	C₄⋊C₄×C₄₁	—	4	2	no	—
182	C₄⋊C₄×C₄₁	—	4	2	no	—
183	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
184	C₄⋊C₄×C₄₁	#41	1	2	yes	complemented
185	C₂²⋊C₄×C₄₁	—	2	8	no	—
186	C₂²⋊C₄×C₄₁	—	2	8	no	—
187	C₂²⋊C₄×C₄₁	—	2	4	no	—
188	C₂²⋊C₄×C₄₁	—	4	2	no	—
189	C₄²×C₄₁	—	2	2	no	—
190	C₄²×C₄₁	—	2	2	no	—
191	C₄²×C₄₁	—	2	1	no	—
192	C₄²×C₄₁	—	2	1	no	—
193	C₄²×C₄₁	—	4	2	no	—
194	C₄²×C₄₁	—	4	2	no	—
195	C₄²×C₄₁	C₂×D₄	1	2	yes	—
196	C₄²×C₄₁	C₂×D₄	1	2	yes	—
197	C₂³×C₄₁	—	2	1	no	—
198	C₂³×C₄₁	—	2	1	no	—
199	C₂³×C₄₁	—	2	1	no	—
200	C₂³×C₄₁	—	2	1	no	—
201	C₂³×C₄₁	—	2	1	no	—
202	C₂³×C₄₁	—	2	1	no	—
203	C₂³×C₄₁	—	2	4	no	—
204	C₂³×C₄₁	—	4	2	no	—
205	C₂³×C₄₁	—	4	2	no	—
206	C₂³×C₄₁	—	4	2	no	—
207	C₂³×C₄₁	—	4	2	no	—
208	C₂³×C₄₁	—	4	2	no	—
209	C₂³×C₄₁	—	4	2	no	—
210	C₂³×C₄₁	C₂×SD₁₆	1	2	yes	—
211	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
212	Q₈×C₄₁	—	2	2	no	—
213	Q₈×C₄₁	—	2	2	no	—
214	Q₈×C₄₁	—	4	1	no	—
215	Q₈×C₄₁	—	4	1	no	—
216	Q₈×C₄₁	—	4	8	no	—
217	D₄×C₄₁	—	2	4	no	—
218	D₄×C₄₁	—	2	4	no	—
219	D₄×C₄₁	—	4	8	no	—
220	D₄×C₄₁	—	4	16	no	—
221	D₄×C₄₁	—	4	2	no	—
222	D₄×C₄₁	—	4	2	no	—
223	D₄×C₄₁	—	4	8	no	—
224	C₂×C₄×C₄₁	—	2	1	no	—
225	C₂×C₄×C₄₁	—	2	1	no	—
226	C₂×C₄×C₄₁	—	2	1	no	—
227	C₂×C₄×C₄₁	—	2	1	no	—
228	C₂×C₄×C₄₁	—	2	8	no	—
229	C₂×C₄×C₄₁	—	2	1	no	—
230	C₂×C₄×C₄₁	—	2	1	no	—
231	C₂×C₄×C₄₁	—	2	4	no	—
232	C₂×C₄×C₄₁	—	2	2	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	1	no	—
237	C₂×C₄×C₄₁	—	2	1	no	—
238	C₂×C₄×C₄₁	—	4	4	no	—
239	C₂×C₄×C₄₁	—	4	16	no	—
240	C₂×C₄×C₄₁	—	4	1	no	—
241	C₂×C₄×C₄₁	—	4	1	no	—
242	C₂×C₄×C₄₁	—	4	2	no	—
243	C₂×C₄×C₄₁	—	4	1	no	—
244	C₂×C₄×C₄₁	—	4	2	no	—
245	C₂×C₄×C₄₁	—	4	1	no	—
246	C₂×C₄×C₄₁	—	4	1	no	—
247	C₂×C₄×C₄₁	—	4	1	no	—
248	C₂×C₄×C₄₁	—	4	1	no	—
249	C₂×C₄×C₄₁	—	4	1	no	—
250	C₂×C₄×C₄₁	—	4	1	no	—
251	C₂×C₄×C₄₁	—	4	1	no	—
252	C₂×C₄×C₄₁	—	4	1	no	—
253	C₂×C₄×C₄₁	—	4	1	no	—
254	C₂×C₄×C₄₁	—	4	1	no	—
255	C₂×C₄×C₄₁	—	4	1	no	—
256	C₂×C₄×C₄₁	#120	1	2	yes	—
257	C₂×C₄×C₄₁	#120	1	2	yes	—
258	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
259	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
260	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
261	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
264	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
265	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
266	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
267	C₈×C₄₁	—	4	2	no	—
268	C₈×C₄₁	—	4	2	no	—
269	#24154	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
270	C₂²×C₄₁	—	2	2	no	—
271	C₂²×C₄₁	—	2	2	no	—
272	C₂²×C₄₁	—	2	2	no	—
273	C₂²×C₄₁	—	4	1	no	—
274	C₂²×C₄₁	—	4	1	no	—
275	C₂²×C₄₁	—	4	1	no	—
276	C₂²×C₄₁	—	4	1	no	—
277	C₂²×C₄₁	—	4	4	no	—
278	C₂²×C₄₁	—	4	4	no	—
279	C₂²×C₄₁	—	4	4	no	—
280	C₂²×C₄₁	—	8	2	no	—
281	C₂²×C₄₁	—	8	2	no	—
282	C₂²×C₄₁	—	8	2	no	—
283	C₂²×C₄₁	—	8	2	no	—
284	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
285	C₂²×C₄₁	#525	1	1	yes	characteristic
286	C₂²×C₄₁	#537	1	1	yes	characteristic
287	C₂²×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
288	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
289	C₂²×C₄₁	#575	1	1	yes	characteristic
290	C₂²×C₄₁	#575	1	1	yes	characteristic
291	C₄×C₄₁	—	2	2	no	—
292	C₄×C₄₁	—	2	2	no	—
293	C₄×C₄₁	—	4	2	no	—
294	C₄×C₄₁	—	4	1	no	—
295	C₄×C₄₁	—	4	2	no	—
296	C₄×C₄₁	—	4	4	no	—
297	C₄×C₄₁	—	4	1	no	—
298	C₄×C₄₁	—	8	1	no	—
299	C₄×C₄₁	—	8	1	no	—
300	C₄×C₄₁	#450	1	4	yes	—
301	C₄×C₄₁	#546	1	1	yes	characteristic
302	C₄×C₄₁	#547	1	1	yes	characteristic
303	C₄×C₄₁	#548	1	1	yes	characteristic
304	C₄×C₄₁	#550	1	1	yes	characteristic
305	C₂×#532	C₂×C₄₁	1	1	yes	characteristic
306	C₂×#475	C₂×C₄₁	1	1	yes	characteristic, complemented
307	C₂×#460	C₂×C₄₁	1	1	yes	characteristic, complemented
308	C₄×C₂²⋊Q₈	C₂×C₄₁	1	1	yes	characteristic, complemented
309	#2034	C₂×C₄₁	1	2	yes	complemented
310	#2000	C₂×C₄₁	1	2	yes	complemented
311	#1988	C₂×C₄₁	1	1	yes	characteristic, complemented
312	#1982	C₂×C₄₁	1	1	yes	characteristic, complemented
313	#1877	C₂×C₄₁	1	2	yes	complemented
314	#1864	C₂×C₄₁	1	2	yes	complemented
315	#1797	C₂×C₄₁	1	1	yes	characteristic, complemented
316	C₂×C₄₁	—	4	2	no	—
317	C₂×C₄₁	—	8	4	no	—
318	C₂×C₄₁	#2407	1	1	yes	characteristic
319	C₂×C₄₁	C₂×#450	1	1	yes	characteristic
320	C₂×C₄₁	#3005	1	1	yes	characteristic
321	C₂×C₄₁	#3103	1	1	yes	characteristic
322	C₂×C₄₁	#3115	1	1	yes	characteristic
323	C₂×C₄₁	#3246	1	1	yes	characteristic
324	C₂×C₄₁	#3251	1	1	yes	characteristic
325	C₂²×#41	—	2	1	no	—
326	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
327	#532	—	2	4	no	—
328	#532	—	2	4	no	—
329	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
330	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
331	C₂×#124	—	2	1	no	—
332	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
333	C₂×#121	—	2	1	no	—
334	C₂×#121	C₂²×C₄₁	1	2	yes	—
335	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
336	C₂×C₄×D₄	—	2	1	no	—
337	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
338	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
339	#475	—	2	4	no	—
340	#460	—	2	4	no	—
341	C₂×#106	C₂²×C₄₁	1	2	yes	—
342	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
343	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
344	C₂×#103	C₂²×C₄₁	1	2	yes	—
345	C₂×#102	C₂²×C₄₁	1	2	yes	—
346	C₂×#102	C₂²×C₄₁	1	2	yes	—
347	C₂×#102	C₂²×C₄₁	1	2	yes	—
348	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
349	#391	C₂²×C₄₁	1	1	yes	characteristic
350	#389	C₂²×C₄₁	1	1	yes	characteristic
351	#387	—	2	1	no	—
352	#385	—	2	1	no	—
353	#380	C₂²×C₄₁	1	1	yes	characteristic
354	C₄×C₄⋊C₄	—	2	1	no	—
355	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
356	C₄×C₂²⋊C₄	—	2	1	no	—
357	#340	C₂²×C₄₁	1	2	yes	—
358	#340	C₂²×C₄₁	1	2	yes	—
359	#336	C₂²×C₄₁	1	2	yes	—
360	C₄₁	#24154	1	1	yes	characteristic, direct product factor, Sylow
361	C₂×#41	—	2	8	no	—
362	C₂×#41	—	4	4	no	—
363	C₂×#41	—	4	4	no	—
364	C₂²×Q₈	—	2	1	no	—
365	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
366	C₂²×D₄	—	2	2	no	—
367	C₂²×D₄	—	2	1	no	—
368	C₂²×D₄	—	2	1	no	—
369	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
370	C₂³×C₄	—	2	2	no	—
371	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
372	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
373	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
374	C₂×SD₁₆	—	2	4	no	—
375	C₂×SD₁₆	—	2	4	no	—
376	C₂×SD₁₆	—	4	2	no	—
377	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
378	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
379	C₄⋊Q₈	—	2	2	no	—
380	C₄⋊Q₈	D₄×C₄₁	1	4	yes	—
381	#127	—	2	2	no	—
382	#127	D₄×C₄₁	1	4	yes	—
383	#124	—	2	8	no	—
384	C₂²⋊Q₈	—	2	2	no	—
385	C₂²⋊Q₈	—	2	2	no	—
386	#121	—	2	8	no	—
387	#121	—	2	4	no	—
388	#121	—	2	4	no	—
389	C₄×Q₈	—	2	2	no	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×D₄	—	2	8	no	—
392	C₄×D₄	—	2	4	no	—
393	C₄×D₄	—	2	4	no	—
394	C₂×C₄⋊C₄	—	2	1	no	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	—	2	1	no	—
398	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
400	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
401	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
402	C₂×C₂²⋊C₄	—	2	1	no	—
403	C₂×C₂²⋊C₄	—	2	2	no	—
404	C₂×C₂²⋊C₄	—	2	2	no	—
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₄²	—	2	2	no	—
407	C₂×C₄²	—	2	2	no	—
408	C₂×C₄²	—	2	1	no	—
409	C₂×C₄²	—	2	1	no	—
410	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
411	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
412	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
413	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
414	#106	—	2	4	no	—
415	C₄⋊C₈	—	2	2	no	—
416	C₄⋊C₈	—	2	2	no	—
417	#103	—	2	4	no	—
418	#102	—	2	4	no	—
419	#102	—	2	4	no	—
420	#102	—	4	2	no	—
421	C₂²⋊C₈	—	4	2	no	—
422	#95	—	2	1	no	—
423	#95	—	2	1	no	—
424	#95	—	2	1	no	—
425	#95	—	2	1	no	—
426	#95	—	2	1	no	—
427	C₂⁴	—	2	2	no	—
428	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
429	#41	—	4	32	no	—
430	C₂×Q₈	—	2	4	no	—
431	C₂×Q₈	—	2	1	no	—
432	C₂×Q₈	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	4	2	no	—
437	C₂×Q₈	—	4	2	no	—
438	C₂×Q₈	#41×C₄₁	1	2	yes	complemented
439	C₂×D₄	—	2	8	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	2	no	—
447	C₂×D₄	—	4	2	no	—
448	C₂×D₄	—	4	2	no	—
449	C₂×D₄	—	4	4	no	—
450	C₂×D₄	—	4	4	no	—
451	C₂×D₄	—	4	2	no	—
452	C₂×D₄	—	4	2	no	—
453	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	8	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	4	1	no	—
470	C₂²×C₄	—	4	1	no	—
471	C₂²×C₄	—	4	4	no	—
472	C₂²×C₄	—	4	4	no	—
473	C₂²×C₄	SD₁₆×C₄₁	1	4	yes	complemented
474	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
476	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
479	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
480	SD₁₆	—	4	8	no	—
481	C₂×C₈	—	2	2	no	—
482	C₂×C₈	—	2	2	no	—
483	C₂×C₈	—	2	2	no	—
484	C₂×C₈	—	2	2	no	—
485	C₂×C₈	—	2	2	no	—
486	C₂×C₈	—	2	2	no	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	4	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	4	2	no	—
496	C₄⋊C₄	—	4	2	no	—
497	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
498	C₄⋊C₄	#41×C₄₁	1	2	yes	complemented
499	C₂²⋊C₄	—	2	8	no	—
500	C₂²⋊C₄	—	2	8	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	4	2	no	—
503	C₄²	—	2	2	no	—
504	C₄²	—	2	2	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	2	1	no	—
507	C₄²	—	4	2	no	—
508	C₄²	—	4	2	no	—
509	C₄²	C₂×D₄×C₄₁	1	2	yes	—
510	C₄²	C₂×D₄×C₄₁	1	2	yes	—
511	C₂³	—	2	1	no	—
512	C₂³	—	2	1	no	—
513	C₂³	—	2	1	no	—
514	C₂³	—	2	1	no	—
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	2	4	no	—
518	C₂³	—	4	2	no	—
519	C₂³	—	4	2	no	—
520	C₂³	—	4	2	no	—
521	C₂³	—	4	2	no	—
522	C₂³	—	4	2	no	—
523	C₂³	—	4	2	no	—
524	C₂³	C₂×SD₁₆×C₄₁	1	2	yes	—
525	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
526	Q₈	—	2	2	no	—
527	Q₈	—	2	2	no	—
528	Q₈	—	4	1	no	—
529	Q₈	—	4	1	no	—
530	Q₈	—	4	8	no	—
531	D₄	—	2	4	no	—
532	D₄	—	2	4	no	—
533	D₄	—	4	8	no	—
534	D₄	—	4	16	no	—
535	D₄	—	4	2	no	—
536	D₄	—	4	2	no	—
537	D₄	—	4	8	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	8	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	1	no	—
550	C₂×C₄	—	2	1	no	—
551	C₂×C₄	—	2	1	no	—
552	C₂×C₄	—	4	16	no	—
553	C₂×C₄	—	4	4	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	2	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	2	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	1	no	—
564	C₂×C₄	—	4	1	no	—
565	C₂×C₄	—	4	1	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	1	no	—
568	C₂×C₄	—	4	1	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	#120×C₄₁	1	2	yes	—
571	C₂×C₄	#120×C₄₁	1	2	yes	—
572	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
573	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
574	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
575	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
576	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
577	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
578	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
579	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
580	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
581	C₈	—	4	2	no	—
582	C₈	—	4	2	no	—
583	C₂²	—	2	2	no	—
584	C₂²	—	2	2	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	4	1	no	—
587	C₂²	—	4	1	no	—
588	C₂²	—	4	1	no	—
589	C₂²	—	4	1	no	—
590	C₂²	—	4	4	no	—
591	C₂²	—	4	4	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	8	2	no	—
594	C₂²	—	8	2	no	—
595	C₂²	—	8	2	no	—
596	C₂²	—	8	2	no	—
597	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
598	C₂²	C₄₁×#525	1	1	yes	characteristic
599	C₂²	C₄₁×#537	1	1	yes	characteristic
600	C₂²	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
601	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
602	C₂²	C₄₁×#575	1	1	yes	characteristic
603	C₂²	C₄₁×#575	1	1	yes	characteristic
604	C₄	—	2	2	no	—
605	C₄	—	2	2	no	—
606	C₄	—	4	2	no	—
607	C₄	—	4	1	no	—
608	C₄	—	4	2	no	—
609	C₄	—	4	4	no	—
610	C₄	—	4	1	no	—
611	C₄	—	8	1	no	—
612	C₄	—	8	1	no	—
613	C₄	C₄₁×#450	1	4	yes	—
614	C₄	C₄₁×#546	1	1	yes	characteristic
615	C₄	C₄₁×#547	1	1	yes	characteristic
616	C₄	C₄₁×#548	1	1	yes	characteristic
617	C₄	C₄₁×#550	1	1	yes	characteristic
618	C₂	—	4	2	no	—
619	C₂	—	8	4	no	—
620	C₂	C₄₁×#2407	1	1	yes	characteristic
621	C₂	C₂×C₄₁×#450	1	1	yes	characteristic
622	C₂	C₄₁×#3005	1	1	yes	characteristic
623	C₂	C₄₁×#3103	1	1	yes	characteristic
624	C₂	C₄₁×#3115	1	1	yes	characteristic
625	C₂	C₄₁×#3246	1	1	yes	characteristic
626	C₂	C₄₁×#3251	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#532	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#475	C₂	semidirect
3. C₂×C₄₁×#460	C₂	semidirect
4. C₄×C₂²⋊Q₈×C₄₁	C₂	semidirect
5. C₄₁×#2034	C₂	semidirect
6. C₄₁×#2000	C₂	semidirect
7. C₄₁×#1988	C₂	semidirect
8. C₄₁×#1982	C₂	semidirect
9. C₄₁×#1877	C₂	semidirect
10. C₄₁×#1864	C₂	semidirect
11. C₄₁×#1797	C₂	semidirect
124. C₂×Q₈×C₄₁	#41	semidirect
159. C₂²×C₄×C₄₁	SD₁₆	semidirect
184. C₄⋊C₄×C₄₁	#41	semidirect
269. #24154	C₄₁	direct (N × H)
306. C₂×#475	C₂×C₄₁	semidirect
307. C₂×#460	C₂×C₄₁	semidirect
308. C₄×C₂²⋊Q₈	C₂×C₄₁	semidirect
309. #2034	C₂×C₄₁	semidirect
310. #2000	C₂×C₄₁	semidirect
311. #1988	C₂×C₄₁	semidirect
312. #1982	C₂×C₄₁	semidirect
313. #1877	C₂×C₄₁	semidirect
314. #1864	C₂×C₄₁	semidirect
315. #1797	C₂×C₄₁	semidirect
360. C₄₁	#24154	direct (N × H)
438. C₂×Q₈	#41×C₄₁	semidirect
473. C₂²×C₄	SD₁₆×C₄₁	semidirect
498. C₄⋊C₄	#41×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#532	2
C₂×C₄₁×#475	2
C₂×C₄₁×#460	2
C₄×C₂²⋊Q₈×C₄₁	2
C₄₁×#2034	2
C₄₁×#2000	2
C₄₁×#1988	2
C₄₁×#1982	2
C₄₁×#1877	2
C₄₁×#1864	2
C₄₁×#1797	2
#24154	41

Subgroups — C41×#24154 #23800355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24154 #23800355