Subgroups — C₄₁×#20945 #23797146

Order 10496.

1862 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/8)×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#3428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#421	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₁×#418	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2309	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1883	C₂	1	8	yes	maximal, complemented
7	C₄₁×#1742	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1733	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
12	C₂×AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
13	C₂×AGL(1,ℤ/8)×C₄₁	C₄	1	8	yes	complemented
14	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	4	yes	—
15	C₂²×SD₁₆×C₄₁	—	2	1	no	—
16	C₂²×D₈×C₄₁	—	2	1	no	—
17	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
19	C₄₁×#520	—	2	2	no	—
20	C₄₁×#520	C₂²	1	4	yes	complemented
21	C₄₁×#519	—	2	1	no	—
22	C₄₁×#518	—	2	1	no	—
23	C₄₁×#518	—	2	2	no	—
24	C₄×#41×C₄₁	—	2	2	no	—
25	C₄×#41×C₄₁	—	2	2	no	—
26	C₂×#117×C₄₁	—	2	2	no	—
27	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
30	C₄₁×#421	—	2	4	no	—
31	C₂×#104×C₄₁	C₂²	1	4	yes	complemented
32	C₄₁×#418	—	2	2	no	—
33	C₄₁×#418	C₂²	1	4	yes	complemented
34	C₂×#102×C₄₁	—	2	2	no	—
35	C₄₁×#390	—	2	2	no	—
36	C₄₁×#390	C₂²	1	4	yes	complemented
37	C₄₁×#379	—	2	2	no	—
38	C₄₁×#343	C₂²	1	4	yes	complemented
39	C₄₁×#339	C₂²	1	4	yes	complemented
40	C₂⁵×C₄₁	—	2	1	no	—
41	C₂×#41×C₄₁	—	2	1	no	—
42	C₂×#41×C₄₁	—	2	1	no	—
43	C₂×#41×C₄₁	—	2	2	no	—
44	C₂×#41×C₄₁	—	2	2	no	—
45	C₂×#41×C₄₁	C₂×C₄	1	4	yes	—
46	C₂×#41×C₄₁	D₄	1	4	yes	—
47	C₂×#41×C₄₁	D₄	1	2	yes	—
48	C₂×#41×C₄₁	C₂³	1	2	yes	—
49	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
50	C₂²×D₄×C₄₁	—	2	1	no	—
51	C₂²×D₄×C₄₁	—	2	1	no	—
52	C₂²×D₄×C₄₁	—	2	1	no	—
53	C₂²×D₄×C₄₁	—	2	2	no	—
54	C₂²×D₄×C₄₁	—	2	1	no	—
55	C₂²×D₄×C₄₁	—	2	1	no	—
56	C₂²×D₄×C₄₁	—	2	1	no	—
57	C₂²×D₄×C₄₁	—	2	2	no	—
58	C₂²×D₄×C₄₁	—	2	2	no	—
59	C₂²×D₄×C₄₁	C₂×C₄	1	4	yes	complemented
60	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
61	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
62	C₂³×C₄×C₄₁	—	2	1	no	—
63	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
64	AGL(1,ℤ/8)×C₄₁	—	2	16	no	—
65	AGL(1,ℤ/8)×C₄₁	—	2	8	no	—
66	AGL(1,ℤ/8)×C₄₁	C₂×C₄	1	16	yes	complemented
67	C₂×SD₁₆×C₄₁	—	2	8	no	—
68	C₂×SD₁₆×C₄₁	—	2	4	no	—
69	C₂×D₈×C₄₁	—	2	8	no	—
70	C₂×D₈×C₄₁	—	2	4	no	—
71	C₂×M₁₆×C₄₁	—	2	2	no	—
72	C₂×M₁₆×C₄₁	—	2	1	no	—
73	C₂×M₁₆×C₄₁	—	2	1	no	—
74	C₂×M₁₆×C₄₁	—	2	2	no	—
75	C₂×M₁₆×C₄₁	—	2	2	no	—
76	C₂×M₁₆×C₄₁	C₂×C₄	1	4	yes	complemented
77	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
78	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
79	C₂²×C₈×C₄₁	—	2	1	no	—
80	C₂²×C₈×C₄₁	—	2	1	no	—
81	C₄×Q₈×C₄₁	—	2	4	no	—
82	C₄×Q₈×C₄₁	—	2	4	no	—
83	C₄×D₄×C₄₁	—	2	8	no	—
84	C₄×D₄×C₄₁	—	2	4	no	—
85	C₄×D₄×C₄₁	—	2	4	no	—
86	C₄×D₄×C₄₁	—	2	8	no	—
87	#117×C₄₁	—	2	4	no	—
88	#117×C₄₁	—	2	4	no	—
89	#117×C₄₁	—	2	1	no	—
90	#117×C₄₁	—	2	1	no	—
91	#117×C₄₁	—	2	1	no	—
92	#117×C₄₁	—	2	1	no	—
93	#117×C₄₁	—	2	1	no	—
94	#117×C₄₁	—	2	1	no	—
95	#117×C₄₁	—	2	4	no	—
96	#117×C₄₁	—	2	2	no	—
97	#117×C₄₁	—	2	2	no	—
98	#117×C₄₁	D₄	1	2	yes	—
99	#117×C₄₁	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
101	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
102	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
103	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
105	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
106	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
108	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
109	C₂×C₄²×C₄₁	—	2	4	no	—
110	C₂×C₄²×C₄₁	—	2	1	no	—
111	C₂×C₄²×C₄₁	D₄	1	2	yes	—
112	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
113	#104×C₄₁	—	2	8	no	—
114	#102×C₄₁	—	2	8	no	—
115	C₂⁴×C₄₁	—	2	1	no	—
116	C₂⁴×C₄₁	—	2	2	no	—
117	C₂⁴×C₄₁	—	2	4	no	—
118	C₂⁴×C₄₁	—	4	1	no	—
119	C₂⁴×C₄₁	—	4	1	no	—
120	C₂⁴×C₄₁	—	4	2	no	—
121	C₂⁴×C₄₁	—	8	4	no	—
122	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
123	#41×C₄₁	—	2	4	no	—
124	#41×C₄₁	—	2	4	no	—
125	#41×C₄₁	—	2	4	no	—
126	#41×C₄₁	—	2	4	no	—
127	#41×C₄₁	—	2	4	no	—
128	#41×C₄₁	—	2	2	no	—
129	#41×C₄₁	—	2	2	no	—
130	#41×C₄₁	C₂²⋊C₄	1	8	yes	complemented
131	#41×C₄₁	C₂²×C₄	1	4	yes	—
132	#41×C₄₁	C₂×D₄	1	4	yes	—
133	C₂×Q₈×C₄₁	—	2	1	no	—
134	C₂×Q₈×C₄₁	—	2	2	no	—
135	C₂×Q₈×C₄₁	C₂²⋊C₄	1	4	yes	complemented
136	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
137	C₂×D₄×C₄₁	—	2	4	no	—
138	C₂×D₄×C₄₁	—	2	2	no	—
139	C₂×D₄×C₄₁	—	2	4	no	—
140	C₂×D₄×C₄₁	—	2	2	no	—
141	C₂×D₄×C₄₁	—	2	4	no	—
142	C₂×D₄×C₄₁	—	2	4	no	—
143	C₂×D₄×C₄₁	—	2	4	no	—
144	C₂×D₄×C₄₁	—	2	4	no	—
145	C₂×D₄×C₄₁	—	2	4	no	—
146	C₂×D₄×C₄₁	—	2	1	no	—
147	C₂×D₄×C₄₁	—	2	2	no	—
148	C₂×D₄×C₄₁	—	2	2	no	—
149	C₂×D₄×C₄₁	—	2	2	no	—
150	C₂×D₄×C₄₁	—	2	2	no	—
151	C₂×D₄×C₄₁	—	2	2	no	—
152	C₂×D₄×C₄₁	—	2	2	no	—
153	C₂×D₄×C₄₁	—	4	1	no	—
154	C₂×D₄×C₄₁	—	4	1	no	—
155	C₂×D₄×C₄₁	—	4	2	no	—
156	C₂×D₄×C₄₁	—	4	2	no	—
157	C₂×D₄×C₄₁	—	4	2	no	—
158	C₂×D₄×C₄₁	—	4	2	no	—
159	C₂×D₄×C₄₁	—	4	2	no	—
160	C₂×D₄×C₄₁	C₂²⋊C₄	1	4	yes	complemented
161	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
162	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₄₁	—	2	1	no	—
164	C₂²×C₄×C₄₁	—	2	4	no	—
165	C₂²×C₄×C₄₁	—	2	2	no	—
166	C₂²×C₄×C₄₁	—	2	4	no	—
167	C₂²×C₄×C₄₁	—	2	2	no	—
168	C₂²×C₄×C₄₁	—	2	8	no	—
169	C₂²×C₄×C₄₁	—	2	2	no	—
170	C₂²×C₄×C₄₁	—	2	2	no	—
171	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
172	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
173	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
176	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₄₁	#41	1	2	yes	—
180	C₂²×C₄×C₄₁	#41	1	2	yes	—
181	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
182	SD₁₆×C₄₁	—	2	16	no	—
183	D₈×C₄₁	—	2	16	no	—
184	M₁₆×C₄₁	—	2	4	no	—
185	M₁₆×C₄₁	—	2	2	no	—
186	M₁₆×C₄₁	—	2	2	no	—
187	M₁₆×C₄₁	—	2	2	no	—
188	M₁₆×C₄₁	—	4	2	no	—
189	M₁₆×C₄₁	C₂²×C₄	1	4	yes	—
190	C₂×C₈×C₄₁	—	2	4	no	—
191	C₂×C₈×C₄₁	—	2	2	no	—
192	C₂×C₈×C₄₁	—	2	2	no	—
193	C₂×C₈×C₄₁	—	4	2	no	—
194	C₄⋊C₄×C₄₁	—	2	8	no	—
195	C₄⋊C₄×C₄₁	—	2	2	no	—
196	C₄⋊C₄×C₄₁	—	2	2	no	—
197	C₄⋊C₄×C₄₁	—	2	2	no	—
198	C₄⋊C₄×C₄₁	—	4	2	no	—
199	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
200	C₂²⋊C₄×C₄₁	—	2	8	no	—
201	C₂²⋊C₄×C₄₁	—	2	2	no	—
202	C₂²⋊C₄×C₄₁	—	2	2	no	—
203	C₂²⋊C₄×C₄₁	—	2	2	no	—
204	C₂²⋊C₄×C₄₁	—	2	2	no	—
205	C₂²⋊C₄×C₄₁	—	4	2	no	—
206	C₂²⋊C₄×C₄₁	—	8	8	no	—
207	C₄²×C₄₁	—	2	8	no	—
208	C₄²×C₄₁	—	2	2	no	—
209	C₄²×C₄₁	—	2	2	no	—
210	C₄²×C₄₁	—	2	2	no	—
211	C₄²×C₄₁	—	2	2	no	—
212	C₄²×C₄₁	—	2	2	no	—
213	C₄²×C₄₁	—	2	2	no	—
214	C₂³×C₄₁	—	2	2	no	—
215	C₂³×C₄₁	—	2	4	no	—
216	C₂³×C₄₁	—	2	4	no	—
217	C₂³×C₄₁	—	4	2	no	—
218	C₂³×C₄₁	—	4	8	no	—
219	C₂³×C₄₁	—	4	2	no	—
220	C₂³×C₄₁	—	4	2	no	—
221	C₂³×C₄₁	—	4	4	no	—
222	C₂³×C₄₁	—	4	4	no	—
223	C₂³×C₄₁	—	8	2	no	—
224	C₂³×C₄₁	—	8	4	no	—
225	C₂³×C₄₁	—	8	4	no	—
226	C₂³×C₄₁	—	8	8	no	—
227	C₂³×C₄₁	—	8	8	no	—
228	C₂³×C₄₁	C₄×D₄	1	2	yes	—
229	C₂³×C₄₁	#120	1	1	yes	characteristic
230	C₂³×C₄₁	#120	1	1	yes	characteristic
231	C₂³×C₄₁	#121	1	2	yes	—
232	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
233	Q₈×C₄₁	—	2	2	no	—
234	Q₈×C₄₁	—	2	4	no	—
235	Q₈×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
236	D₄×C₄₁	—	2	4	no	—
237	D₄×C₄₁	—	2	4	no	—
238	D₄×C₄₁	—	2	8	no	—
239	D₄×C₄₁	—	2	4	no	—
240	D₄×C₄₁	—	2	4	no	—
241	D₄×C₄₁	—	2	2	no	—
242	D₄×C₄₁	—	2	4	no	—
243	D₄×C₄₁	—	4	2	no	—
244	D₄×C₄₁	—	4	2	no	—
245	D₄×C₄₁	—	4	4	no	—
246	D₄×C₄₁	—	4	4	no	—
247	D₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
248	C₂×C₄×C₄₁	—	2	2	no	—
249	C₂×C₄×C₄₁	—	2	4	no	—
250	C₂×C₄×C₄₁	—	2	4	no	—
251	C₂×C₄×C₄₁	—	2	4	no	—
252	C₂×C₄×C₄₁	—	2	8	no	—
253	C₂×C₄×C₄₁	—	2	4	no	—
254	C₂×C₄×C₄₁	—	2	4	no	—
255	C₂×C₄×C₄₁	—	4	4	no	—
256	C₂×C₄×C₄₁	—	4	4	no	—
257	C₂×C₄×C₄₁	—	4	2	no	—
258	C₂×C₄×C₄₁	—	8	2	no	—
259	C₂×C₄×C₄₁	—	8	8	no	—
260	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
261	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
262	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
263	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
264	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
265	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
266	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
267	C₂×C₄×C₄₁	#121	1	2	yes	—
268	C₂×C₄×C₄₁	#121	1	2	yes	—
269	C₂×C₄×C₄₁	#121	1	2	yes	—
270	C₂×C₄×C₄₁	#123	1	2	yes	—
271	C₂×C₄×C₄₁	#123	1	2	yes	—
272	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
273	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
274	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
277	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
278	C₈×C₄₁	—	2	4	no	—
279	C₈×C₄₁	—	4	2	no	—
280	#20945	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
281	C₂²×C₄₁	—	2	2	no	—
282	C₂²×C₄₁	—	2	2	no	—
283	C₂²×C₄₁	—	2	2	no	—
284	C₂²×C₄₁	—	2	4	no	—
285	C₂²×C₄₁	—	4	2	no	—
286	C₂²×C₄₁	—	4	2	no	—
287	C₂²×C₄₁	—	4	8	no	—
288	C₂²×C₄₁	—	4	4	no	—
289	C₂²×C₄₁	—	4	4	no	—
290	C₂²×C₄₁	—	8	4	no	—
291	C₂²×C₄₁	—	8	4	no	—
292	C₂²×C₄₁	—	8	8	no	—
293	C₂²×C₄₁	—	8	8	no	—
294	C₂²×C₄₁	—	8	4	no	—
295	C₂²×C₄₁	#386	1	2	yes	—
296	C₂²×C₄₁	#386	1	2	yes	—
297	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
298	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
299	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
300	C₄×C₄₁	—	2	4	no	—
301	C₄×C₄₁	—	4	4	no	—
302	C₄×C₄₁	—	4	4	no	—
303	C₄×C₄₁	—	8	2	no	—
304	C₄×C₄₁	#386	1	2	yes	—
305	C₄×C₄₁	#386	1	2	yes	—
306	C₄×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
307	C₄×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
308	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
309	C₄×C₄₁	C₂×#123	1	1	yes	characteristic
310	C₂²×AGL(1,ℤ/8)	C₂×C₄₁	1	1	yes	characteristic
311	#3428	C₂×C₄₁	1	1	yes	characteristic, complemented
312	C₂×#421	C₂×C₄₁	1	1	yes	characteristic, complemented
313	C₂×#418	C₂×C₄₁	1	1	yes	characteristic, complemented
314	#2309	C₂×C₄₁	1	1	yes	characteristic, complemented
315	#1883	C₂×C₄₁	1	8	yes	complemented
316	#1742	C₂×C₄₁	1	1	yes	characteristic, complemented
317	#1733	C₂×C₄₁	1	1	yes	characteristic, complemented
318	C₂×C₄₁	—	2	1	no	—
319	C₂×C₄₁	—	2	1	no	—
320	C₂×C₄₁	—	2	2	no	—
321	C₂×C₄₁	—	2	2	no	—
322	C₂×C₄₁	—	4	4	no	—
323	C₂×C₄₁	—	8	4	no	—
324	C₂×C₄₁	C₂×#386	1	1	yes	characteristic
325	C₂×C₄₁	#3021	1	2	yes	—
326	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
327	C₂³×D₄	C₂²×C₄₁	1	1	yes	characteristic
328	C₂×AGL(1,ℤ/8)	—	2	4	no	—
329	C₂×AGL(1,ℤ/8)	—	2	2	no	—
330	C₂×AGL(1,ℤ/8)	C₄×C₄₁	1	8	yes	complemented
331	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	4	yes	—
332	C₂²×SD₁₆	—	2	1	no	—
333	C₂²×D₈	—	2	1	no	—
334	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
335	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
336	#520	—	2	2	no	—
337	#520	C₂²×C₄₁	1	4	yes	complemented
338	#519	—	2	1	no	—
339	#518	—	2	1	no	—
340	#518	—	2	2	no	—
341	C₄×#41	—	2	2	no	—
342	C₄×#41	—	2	2	no	—
343	C₂×#117	—	2	2	no	—
344	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
345	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
346	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
347	#421	—	2	4	no	—
348	C₂×#104	C₂²×C₄₁	1	4	yes	complemented
349	#418	—	2	2	no	—
350	#418	C₂²×C₄₁	1	4	yes	complemented
351	C₂×#102	—	2	2	no	—
352	#390	—	2	2	no	—
353	#390	C₂²×C₄₁	1	4	yes	complemented
354	#379	—	2	2	no	—
355	#343	C₂²×C₄₁	1	4	yes	complemented
356	#339	C₂²×C₄₁	1	4	yes	complemented
357	C₄₁	#20945	1	1	yes	characteristic, direct product factor, Sylow
358	C₂⁵	—	2	1	no	—
359	C₂×#41	—	2	1	no	—
360	C₂×#41	—	2	1	no	—
361	C₂×#41	—	2	2	no	—
362	C₂×#41	—	2	2	no	—
363	C₂×#41	C₂×C₄×C₄₁	1	4	yes	—
364	C₂×#41	D₄×C₄₁	1	4	yes	—
365	C₂×#41	D₄×C₄₁	1	2	yes	—
366	C₂×#41	C₂³×C₄₁	1	2	yes	—
367	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
368	C₂²×D₄	—	2	1	no	—
369	C₂²×D₄	—	2	1	no	—
370	C₂²×D₄	—	2	1	no	—
371	C₂²×D₄	—	2	2	no	—
372	C₂²×D₄	—	2	1	no	—
373	C₂²×D₄	—	2	1	no	—
374	C₂²×D₄	—	2	1	no	—
375	C₂²×D₄	—	2	2	no	—
376	C₂²×D₄	—	2	2	no	—
377	C₂²×D₄	C₂×C₄×C₄₁	1	4	yes	complemented
378	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
379	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
380	C₂³×C₄	—	2	1	no	—
381	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
382	AGL(1,ℤ/8)	—	2	16	no	—
383	AGL(1,ℤ/8)	—	2	8	no	—
384	AGL(1,ℤ/8)	C₂×C₄×C₄₁	1	16	yes	complemented
385	C₂×SD₁₆	—	2	8	no	—
386	C₂×SD₁₆	—	2	4	no	—
387	C₂×D₈	—	2	8	no	—
388	C₂×D₈	—	2	4	no	—
389	C₂×M₁₆	—	2	2	no	—
390	C₂×M₁₆	—	2	1	no	—
391	C₂×M₁₆	—	2	1	no	—
392	C₂×M₁₆	—	2	2	no	—
393	C₂×M₁₆	—	2	2	no	—
394	C₂×M₁₆	C₂×C₄×C₄₁	1	4	yes	complemented
395	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
396	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
397	C₂²×C₈	—	2	1	no	—
398	C₂²×C₈	—	2	1	no	—
399	C₄×Q₈	—	2	4	no	—
400	C₄×Q₈	—	2	4	no	—
401	C₄×D₄	—	2	8	no	—
402	C₄×D₄	—	2	4	no	—
403	C₄×D₄	—	2	4	no	—
404	C₄×D₄	—	2	8	no	—
405	#117	—	2	4	no	—
406	#117	—	2	4	no	—
407	#117	—	2	1	no	—
408	#117	—	2	1	no	—
409	#117	—	2	1	no	—
410	#117	—	2	1	no	—
411	#117	—	2	1	no	—
412	#117	—	2	1	no	—
413	#117	—	2	4	no	—
414	#117	—	2	2	no	—
415	#117	—	2	2	no	—
416	#117	D₄×C₄₁	1	2	yes	—
417	#117	C₂³×C₄₁	1	2	yes	—
418	C₂×C₄⋊C₄	—	2	4	no	—
419	C₂×C₄⋊C₄	—	2	1	no	—
420	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
421	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
422	C₂×C₂²⋊C₄	—	2	4	no	—
423	C₂×C₂²⋊C₄	—	2	1	no	—
424	C₂×C₂²⋊C₄	—	2	1	no	—
425	C₂×C₂²⋊C₄	—	2	8	no	—
426	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
427	C₂×C₄²	—	2	4	no	—
428	C₂×C₄²	—	2	1	no	—
429	C₂×C₄²	D₄×C₄₁	1	2	yes	—
430	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
431	#104	—	2	8	no	—
432	#102	—	2	8	no	—
433	C₂⁴	—	2	1	no	—
434	C₂⁴	—	2	2	no	—
435	C₂⁴	—	2	4	no	—
436	C₂⁴	—	4	1	no	—
437	C₂⁴	—	4	1	no	—
438	C₂⁴	—	4	2	no	—
439	C₂⁴	—	8	4	no	—
440	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
441	#41	—	2	4	no	—
442	#41	—	2	4	no	—
443	#41	—	2	4	no	—
444	#41	—	2	4	no	—
445	#41	—	2	4	no	—
446	#41	—	2	2	no	—
447	#41	—	2	2	no	—
448	#41	C₂²⋊C₄×C₄₁	1	8	yes	complemented
449	#41	C₂²×C₄×C₄₁	1	4	yes	—
450	#41	C₂×D₄×C₄₁	1	4	yes	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	—	2	2	no	—
453	C₂×Q₈	C₂²⋊C₄×C₄₁	1	4	yes	complemented
454	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
455	C₂×D₄	—	2	4	no	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	4	no	—
460	C₂×D₄	—	2	4	no	—
461	C₂×D₄	—	2	4	no	—
462	C₂×D₄	—	2	4	no	—
463	C₂×D₄	—	2	4	no	—
464	C₂×D₄	—	2	1	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	2	no	—
470	C₂×D₄	—	2	2	no	—
471	C₂×D₄	—	4	1	no	—
472	C₂×D₄	—	4	1	no	—
473	C₂×D₄	—	4	2	no	—
474	C₂×D₄	—	4	2	no	—
475	C₂×D₄	—	4	2	no	—
476	C₂×D₄	—	4	2	no	—
477	C₂×D₄	—	4	2	no	—
478	C₂×D₄	C₂²⋊C₄×C₄₁	1	4	yes	complemented
479	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
480	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	8	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
490	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
491	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
494	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
497	C₂²×C₄	#41×C₄₁	1	2	yes	—
498	C₂²×C₄	#41×C₄₁	1	2	yes	—
499	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
500	SD₁₆	—	2	16	no	—
501	D₈	—	2	16	no	—
502	M₁₆	—	2	4	no	—
503	M₁₆	—	2	2	no	—
504	M₁₆	—	2	2	no	—
505	M₁₆	—	2	2	no	—
506	M₁₆	—	4	2	no	—
507	M₁₆	C₂²×C₄×C₄₁	1	4	yes	—
508	C₂×C₈	—	2	4	no	—
509	C₂×C₈	—	2	2	no	—
510	C₂×C₈	—	2	2	no	—
511	C₂×C₈	—	4	2	no	—
512	C₄⋊C₄	—	2	8	no	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	2	2	no	—
516	C₄⋊C₄	—	4	2	no	—
517	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
518	C₂²⋊C₄	—	2	8	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	2	no	—
523	C₂²⋊C₄	—	4	2	no	—
524	C₂²⋊C₄	—	8	8	no	—
525	C₄²	—	2	8	no	—
526	C₄²	—	2	2	no	—
527	C₄²	—	2	2	no	—
528	C₄²	—	2	2	no	—
529	C₄²	—	2	2	no	—
530	C₄²	—	2	2	no	—
531	C₄²	—	2	2	no	—
532	C₂³	—	2	2	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	2	4	no	—
535	C₂³	—	4	2	no	—
536	C₂³	—	4	8	no	—
537	C₂³	—	4	2	no	—
538	C₂³	—	4	2	no	—
539	C₂³	—	4	4	no	—
540	C₂³	—	4	4	no	—
541	C₂³	—	8	2	no	—
542	C₂³	—	8	4	no	—
543	C₂³	—	8	4	no	—
544	C₂³	—	8	8	no	—
545	C₂³	—	8	8	no	—
546	C₂³	C₄×D₄×C₄₁	1	2	yes	—
547	C₂³	#120×C₄₁	1	1	yes	characteristic
548	C₂³	#120×C₄₁	1	1	yes	characteristic
549	C₂³	#121×C₄₁	1	2	yes	—
550	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
551	Q₈	—	2	2	no	—
552	Q₈	—	2	4	no	—
553	Q₈	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
554	D₄	—	2	4	no	—
555	D₄	—	2	4	no	—
556	D₄	—	2	8	no	—
557	D₄	—	2	4	no	—
558	D₄	—	2	4	no	—
559	D₄	—	2	2	no	—
560	D₄	—	2	4	no	—
561	D₄	—	4	2	no	—
562	D₄	—	4	2	no	—
563	D₄	—	4	4	no	—
564	D₄	—	4	4	no	—
565	D₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	8	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	4	no	—
573	C₂×C₄	—	4	4	no	—
574	C₂×C₄	—	4	4	no	—
575	C₂×C₄	—	4	2	no	—
576	C₂×C₄	—	8	2	no	—
577	C₂×C₄	—	8	8	no	—
578	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
579	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
580	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
581	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
582	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
583	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
584	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
585	C₂×C₄	#121×C₄₁	1	2	yes	—
586	C₂×C₄	#121×C₄₁	1	2	yes	—
587	C₂×C₄	#121×C₄₁	1	2	yes	—
588	C₂×C₄	#123×C₄₁	1	2	yes	—
589	C₂×C₄	#123×C₄₁	1	2	yes	—
590	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
591	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
592	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
593	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
594	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
595	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
596	C₈	—	2	4	no	—
597	C₈	—	4	2	no	—
598	C₂²	—	2	2	no	—
599	C₂²	—	2	2	no	—
600	C₂²	—	2	2	no	—
601	C₂²	—	2	4	no	—
602	C₂²	—	4	2	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	8	no	—
605	C₂²	—	4	4	no	—
606	C₂²	—	4	4	no	—
607	C₂²	—	8	4	no	—
608	C₂²	—	8	4	no	—
609	C₂²	—	8	8	no	—
610	C₂²	—	8	8	no	—
611	C₂²	—	8	4	no	—
612	C₂²	C₄₁×#386	1	2	yes	—
613	C₂²	C₄₁×#386	1	2	yes	—
614	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
615	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
616	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
617	C₄	—	2	4	no	—
618	C₄	—	4	4	no	—
619	C₄	—	4	4	no	—
620	C₄	—	8	2	no	—
621	C₄	C₄₁×#386	1	2	yes	—
622	C₄	C₄₁×#386	1	2	yes	—
623	C₄	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
624	C₄	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
625	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
626	C₄	C₂×#123×C₄₁	1	1	yes	characteristic
627	C₂	—	2	1	no	—
628	C₂	—	2	1	no	—
629	C₂	—	2	2	no	—
630	C₂	—	2	2	no	—
631	C₂	—	4	4	no	—
632	C₂	—	8	4	no	—
633	C₂	C₂×C₄₁×#386	1	1	yes	characteristic
634	C₂	C₄₁×#3021	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂²×AGL(1,ℤ/8)×C₄₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#3428	C₂	semidirect
3. C₂×C₄₁×#421	C₂	semidirect
4. C₂×C₄₁×#418	C₂	semidirect
5. C₄₁×#2309	C₂	semidirect
6. C₄₁×#1883	C₂	semidirect
7. C₄₁×#1742	C₂	semidirect
8. C₄₁×#1733	C₂	semidirect
13. C₂×AGL(1,ℤ/8)×C₄₁	C₄	semidirect
20. C₄₁×#520	C₂²	semidirect
31. C₂×#104×C₄₁	C₂²	semidirect
33. C₄₁×#418	C₂²	semidirect
36. C₄₁×#390	C₂²	semidirect
38. C₄₁×#343	C₂²	semidirect
39. C₄₁×#339	C₂²	semidirect
59. C₂²×D₄×C₄₁	C₂×C₄	semidirect
66. AGL(1,ℤ/8)×C₄₁	C₂×C₄	semidirect
76. C₂×M₁₆×C₄₁	C₂×C₄	semidirect
130. #41×C₄₁	C₂²⋊C₄	semidirect
135. C₂×Q₈×C₄₁	C₂²⋊C₄	semidirect
160. C₂×D₄×C₄₁	C₂²⋊C₄	semidirect
280. #20945	C₄₁	direct (N × H)
311. #3428	C₂×C₄₁	semidirect
312. C₂×#421	C₂×C₄₁	semidirect
313. C₂×#418	C₂×C₄₁	semidirect
314. #2309	C₂×C₄₁	semidirect
315. #1883	C₂×C₄₁	semidirect
316. #1742	C₂×C₄₁	semidirect
317. #1733	C₂×C₄₁	semidirect
330. C₂×AGL(1,ℤ/8)	C₄×C₄₁	semidirect
337. #520	C₂²×C₄₁	semidirect
348. C₂×#104	C₂²×C₄₁	semidirect
350. #418	C₂²×C₄₁	semidirect
353. #390	C₂²×C₄₁	semidirect
355. #343	C₂²×C₄₁	semidirect
356. #339	C₂²×C₄₁	semidirect
357. C₄₁	#20945	direct (N × H)
377. C₂²×D₄	C₂×C₄×C₄₁	semidirect
384. AGL(1,ℤ/8)	C₂×C₄×C₄₁	semidirect
394. C₂×M₁₆	C₂×C₄×C₄₁	semidirect
448. #41	C₂²⋊C₄×C₄₁	semidirect
453. C₂×Q₈	C₂²⋊C₄×C₄₁	semidirect
478. C₂×D₄	C₂²⋊C₄×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×AGL(1,ℤ/8)×C₄₁	2
C₄₁×#3428	2
C₂×C₄₁×#421	2
C₂×C₄₁×#418	2
C₄₁×#2309	2
C₄₁×#1883	2
C₄₁×#1742	2
C₄₁×#1733	2
#20945	41

Subgroups — C41×#20945 #23797146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20945 #23797146