Subgroups — C₄₁×#19177 #23795378

Order 10496.

650 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3141	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#3103	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#462	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1702	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1700	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1682	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1680	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1547	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₄₁	—	2	1	no	—
13	C₄₁×#532	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#489	—	2	2	no	—
17	C₄₁×#487	C₂²	1	2	yes	—
18	C₄₁×#486	C₂²	1	2	yes	—
19	C₄₁×#481	—	2	1	no	—
20	C₄₁×#475	C₂²	1	1	yes	characteristic
21	C₄₁×#475	C₂²	1	1	yes	characteristic
22	C₄₁×#462	—	2	1	no	—
23	C₄₁×#462	—	2	1	no	—
24	C₄₁×#462	C₂²	1	2	yes	—
25	C₄₁×#462	C₂²	1	2	yes	—
26	C₄₁×#460	C₂²	1	1	yes	characteristic
27	C₄₁×#460	C₂²	1	1	yes	characteristic
28	C₄₁×#448	—	2	1	no	—
29	C₄₁×#446	—	2	2	no	—
30	C₄₁×#446	C₂²	1	2	yes	—
31	C₈×Q₈×C₄₁	C₂²	1	2	yes	—
32	C₄×SD₁₆×C₄₁	—	2	1	no	—
33	C₄×SD₁₆×C₄₁	—	2	1	no	—
34	C₄₁×#431	C₂²	1	1	yes	characteristic
35	C₄₁×#423	—	2	1	no	—
36	C₄₁×#423	C₂²	1	1	yes	characteristic
37	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
38	C₄₁×#417	—	2	1	no	—
39	C₄₁×#417	—	2	1	no	—
40	C₂×#103×C₄₁	—	2	1	no	—
41	C₄×M₁₆×C₄₁	—	2	1	no	—
42	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
43	C₄₁×#335	C₂²	1	2	yes	—
44	C₄₁×#334	C₂²	1	2	yes	—
45	C₄₁×#332	C₂²	1	2	yes	—
46	C₄₁×#332	C₂²	1	2	yes	—
47	C₄₁×#332	C₂²	1	2	yes	—
48	C₄₁×#331	C₂²	1	2	yes	—
49	C₂×#41×C₄₁	—	2	1	no	—
50	C₂²×Q₈×C₄₁	—	2	1	no	—
51	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
52	C₂×Q₁₆×C₄₁	—	2	2	no	—
53	C₂×Q₁₆×C₄₁	—	2	2	no	—
54	C₂×Q₁₆×C₄₁	—	4	4	no	—
55	C₂×SD₁₆×C₄₁	—	2	1	no	—
56	C₂×SD₁₆×C₄₁	—	2	1	no	—
57	C₂×M₁₆×C₄₁	—	2	1	no	—
58	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
59	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₄₁	—	2	1	no	—
61	C₄⋊Q₈×C₄₁	—	2	1	no	—
62	C₄⋊Q₈×C₄₁	—	2	1	no	—
63	C₄⋊Q₈×C₄₁	—	2	1	no	—
64	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
65	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
66	#127×C₄₁	C₂³	1	1	yes	characteristic
67	#124×C₄₁	—	2	1	no	—
68	#121×C₄₁	—	2	1	no	—
69	#121×C₄₁	—	2	1	no	—
70	C₄×Q₈×C₄₁	—	2	1	no	—
71	C₄×Q₈×C₄₁	—	2	1	no	—
72	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
73	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
74	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
75	C₄×Q₈×C₄₁	C₂³	1	1	yes	characteristic
76	C₄×D₄×C₄₁	—	2	1	no	—
77	C₄×D₄×C₄₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₄₁	—	2	1	no	—
83	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
84	#106×C₄₁	—	2	1	no	—
85	#106×C₄₁	—	2	1	no	—
86	C₄⋊C₈×C₄₁	—	2	2	no	—
87	C₄⋊C₈×C₄₁	—	2	2	no	—
88	C₄⋊C₈×C₄₁	—	2	1	no	—
89	C₄⋊C₈×C₄₁	—	2	1	no	—
90	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
91	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
92	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
93	C₄⋊C₈×C₄₁	C₂³	1	1	yes	characteristic
94	#103×C₄₁	—	2	1	no	—
95	#103×C₄₁	—	2	1	no	—
96	#103×C₄₁	—	2	2	no	—
97	#103×C₄₁	—	2	2	no	—
98	#103×C₄₁	—	2	2	no	—
99	#103×C₄₁	—	2	2	no	—
100	#102×C₄₁	—	2	2	no	—
101	#102×C₄₁	—	2	2	no	—
102	#102×C₄₁	—	2	1	no	—
103	#102×C₄₁	—	2	1	no	—
104	C₂²⋊C₈×C₄₁	—	2	1	no	—
105	#97×C₄₁	—	2	2	no	—
106	#97×C₄₁	D₄	1	2	yes	—
107	C₄×C₈×C₄₁	—	2	2	no	—
108	C₄×C₈×C₄₁	—	2	1	no	—
109	C₄×C₈×C₄₁	C₂³	1	2	yes	—
110	#41×C₄₁	—	4	2	no	—
111	#41×C₄₁	—	4	2	no	—
112	C₂×Q₈×C₄₁	—	2	1	no	—
113	C₂×Q₈×C₄₁	—	2	2	no	—
114	C₂×Q₈×C₄₁	—	2	1	no	—
115	C₂×Q₈×C₄₁	—	2	2	no	—
116	C₂×Q₈×C₄₁	—	4	1	no	—
117	C₂×Q₈×C₄₁	—	4	2	no	—
118	C₂×Q₈×C₄₁	—	4	2	no	—
119	C₂×Q₈×C₄₁	—	4	1	no	—
120	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
122	C₂×D₄×C₄₁	—	2	1	no	—
123	C₂×D₄×C₄₁	—	2	1	no	—
124	C₂×D₄×C₄₁	—	2	1	no	—
125	C₂×D₄×C₄₁	—	4	1	no	—
126	C₂²×C₄×C₄₁	—	2	1	no	—
127	C₂²×C₄×C₄₁	—	2	1	no	—
128	C₂²×C₄×C₄₁	—	4	1	no	—
129	C₂²×C₄×C₄₁	—	4	1	no	—
130	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
133	Q₁₆×C₄₁	—	4	4	no	—
134	Q₁₆×C₄₁	—	4	4	no	—
135	SD₁₆×C₄₁	—	4	2	no	—
136	SD₁₆×C₄₁	—	4	2	no	—
137	M₁₆×C₄₁	—	2	2	no	—
138	M₁₆×C₄₁	—	2	2	no	—
139	M₁₆×C₄₁	—	4	2	no	—
140	C₂×C₈×C₄₁	—	2	2	no	—
141	C₂×C₈×C₄₁	—	2	1	no	—
142	C₂×C₈×C₄₁	—	2	1	no	—
143	C₂×C₈×C₄₁	—	2	2	no	—
144	C₂×C₈×C₄₁	—	4	2	no	—
145	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
147	C₄⋊C₄×C₄₁	—	2	1	no	—
148	C₄⋊C₄×C₄₁	—	2	1	no	—
149	C₄⋊C₄×C₄₁	—	2	1	no	—
150	C₄⋊C₄×C₄₁	—	2	1	no	—
151	C₄⋊C₄×C₄₁	—	2	1	no	—
152	C₄⋊C₄×C₄₁	—	2	2	no	—
153	C₄⋊C₄×C₄₁	—	2	1	no	—
154	C₄⋊C₄×C₄₁	—	2	2	no	—
155	C₄⋊C₄×C₄₁	—	2	1	no	—
156	C₄⋊C₄×C₄₁	—	4	1	no	—
157	C₄⋊C₄×C₄₁	—	4	1	no	—
158	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
159	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
160	C₂²⋊C₄×C₄₁	—	4	1	no	—
161	C₂²⋊C₄×C₄₁	—	4	1	no	—
162	C₄²×C₄₁	—	2	1	no	—
163	C₄²×C₄₁	—	2	1	no	—
164	C₄²×C₄₁	—	2	1	no	—
165	C₄²×C₄₁	—	2	1	no	—
166	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
167	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₄₁	#41	1	1	yes	characteristic
169	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
170	C₂³×C₄₁	—	4	1	no	—
171	C₂³×C₄₁	#121	1	1	yes	characteristic
172	Q₈×C₄₁	—	2	1	no	—
173	Q₈×C₄₁	—	2	1	no	—
174	Q₈×C₄₁	—	2	1	no	—
175	Q₈×C₄₁	—	4	2	no	—
176	Q₈×C₄₁	—	4	1	no	—
177	Q₈×C₄₁	—	4	2	no	—
178	Q₈×C₄₁	—	4	1	no	—
179	Q₈×C₄₁	—	4	2	no	—
180	Q₈×C₄₁	—	8	2	no	—
181	Q₈×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
182	D₄×C₄₁	—	4	1	no	—
183	D₄×C₄₁	—	4	1	no	—
184	D₄×C₄₁	—	4	1	no	—
185	D₄×C₄₁	—	4	1	no	—
186	D₄×C₄₁	—	8	1	no	—
187	D₄×C₄₁	—	8	2	no	—
188	C₂×C₄×C₄₁	—	2	1	no	—
189	C₂×C₄×C₄₁	—	2	1	no	—
190	C₂×C₄×C₄₁	—	2	1	no	—
191	C₂×C₄×C₄₁	—	2	1	no	—
192	C₂×C₄×C₄₁	—	2	1	no	—
193	C₂×C₄×C₄₁	—	2	1	no	—
194	C₂×C₄×C₄₁	—	4	1	no	—
195	C₂×C₄×C₄₁	—	4	2	no	—
196	C₂×C₄×C₄₁	—	4	1	no	—
197	C₂×C₄×C₄₁	—	4	2	no	—
198	C₂×C₄×C₄₁	—	8	1	no	—
199	C₂×C₄×C₄₁	—	8	2	no	—
200	C₂×C₄×C₄₁	—	8	2	no	—
201	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
202	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
203	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
204	C₂×C₄×C₄₁	#135	1	2	yes	—
205	C₂×C₄×C₄₁	#137	1	2	yes	—
206	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
209	C₈×C₄₁	—	2	2	no	—
210	C₈×C₄₁	—	2	2	no	—
211	C₈×C₄₁	—	4	2	no	—
212	C₈×C₄₁	—	4	2	no	—
213	C₈×C₄₁	—	4	1	no	—
214	C₈×C₄₁	—	4	1	no	—
215	#19177	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂²×C₄₁	—	2	1	no	—
217	C₂²×C₄₁	—	2	1	no	—
218	C₂²×C₄₁	—	8	1	no	—
219	C₂²×C₄₁	—	8	1	no	—
220	C₂²×C₄₁	—	8	1	no	—
221	C₂²×C₄₁	#464	1	2	yes	—
222	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
223	C₄×C₄₁	—	2	1	no	—
224	C₄×C₄₁	—	2	2	no	—
225	C₄×C₄₁	—	2	1	no	—
226	C₄×C₄₁	—	2	1	no	—
227	C₄×C₄₁	—	4	1	no	—
228	C₄×C₄₁	—	4	1	no	—
229	C₄×C₄₁	—	4	1	no	—
230	C₄×C₄₁	—	8	1	no	—
231	C₄×C₄₁	—	8	2	no	—
232	C₄×C₄₁	#464	1	2	yes	—
233	C₄×C₄₁	C₂×#135	1	1	yes	characteristic
234	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
235	C₄×C₄₁	C₂×#137	1	1	yes	characteristic
236	C₄×C₄₁	#575	1	1	yes	characteristic
237	#3141	C₂×C₄₁	1	1	yes	characteristic
238	#3103	C₂×C₄₁	1	1	yes	characteristic
239	C₂×#462	C₂×C₄₁	1	1	yes	characteristic, complemented
240	Q₈×M₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
241	#1702	C₂×C₄₁	1	2	yes	complemented
242	#1700	C₂×C₄₁	1	2	yes	complemented
243	#1682	C₂×C₄₁	1	2	yes	complemented
244	#1680	C₂×C₄₁	1	2	yes	complemented
245	#1565	C₂×C₄₁	1	1	yes	characteristic, complemented
246	#1549	C₂×C₄₁	1	1	yes	characteristic, complemented
247	#1547	C₂×C₄₁	1	1	yes	characteristic
248	C₂×C₄₁	—	2	2	no	—
249	C₂×C₄₁	—	16	1	no	—
250	C₂×C₄₁	C₂×#464	1	1	yes	characteristic
251	C₂×C₄₁	#3055	1	1	yes	characteristic
252	C₂×C₄₁	#3069	1	1	yes	characteristic
253	C₂²×Q₁₆	—	2	1	no	—
254	#532	C₂²×C₄₁	1	1	yes	characteristic
255	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
256	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
257	#489	—	2	2	no	—
258	#487	C₂²×C₄₁	1	2	yes	—
259	#486	C₂²×C₄₁	1	2	yes	—
260	#481	—	2	1	no	—
261	#475	C₂²×C₄₁	1	1	yes	characteristic
262	#475	C₂²×C₄₁	1	1	yes	characteristic
263	#462	—	2	1	no	—
264	#462	—	2	1	no	—
265	#462	C₂²×C₄₁	1	2	yes	—
266	#462	C₂²×C₄₁	1	2	yes	—
267	#460	C₂²×C₄₁	1	1	yes	characteristic
268	#460	C₂²×C₄₁	1	1	yes	characteristic
269	#448	—	2	1	no	—
270	#446	—	2	2	no	—
271	#446	C₂²×C₄₁	1	2	yes	—
272	C₈×Q₈	C₂²×C₄₁	1	2	yes	—
273	C₄×SD₁₆	—	2	1	no	—
274	C₄×SD₁₆	—	2	1	no	—
275	#431	C₂²×C₄₁	1	1	yes	characteristic
276	#423	—	2	1	no	—
277	#423	C₂²×C₄₁	1	1	yes	characteristic
278	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
279	#417	—	2	1	no	—
280	#417	—	2	1	no	—
281	C₂×#103	—	2	1	no	—
282	C₄×M₁₆	—	2	1	no	—
283	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
284	#335	C₂²×C₄₁	1	2	yes	—
285	#334	C₂²×C₄₁	1	2	yes	—
286	#332	C₂²×C₄₁	1	2	yes	—
287	#332	C₂²×C₄₁	1	2	yes	—
288	#332	C₂²×C₄₁	1	2	yes	—
289	#331	C₂²×C₄₁	1	2	yes	—
290	C₄₁	#19177	1	1	yes	characteristic, direct product factor, Sylow
291	C₂×#41	—	2	1	no	—
292	C₂²×Q₈	—	2	1	no	—
293	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
294	C₂×Q₁₆	—	2	2	no	—
295	C₂×Q₁₆	—	2	2	no	—
296	C₂×Q₁₆	—	4	4	no	—
297	C₂×SD₁₆	—	2	1	no	—
298	C₂×SD₁₆	—	2	1	no	—
299	C₂×M₁₆	—	2	1	no	—
300	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
301	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
302	C₂²×C₈	—	2	1	no	—
303	C₄⋊Q₈	—	2	1	no	—
304	C₄⋊Q₈	—	2	1	no	—
305	C₄⋊Q₈	—	2	1	no	—
306	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
307	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
308	#127	C₂³×C₄₁	1	1	yes	characteristic
309	#124	—	2	1	no	—
310	#121	—	2	1	no	—
311	#121	—	2	1	no	—
312	C₄×Q₈	—	2	1	no	—
313	C₄×Q₈	—	2	1	no	—
314	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
315	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
316	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
317	C₄×Q₈	C₂³×C₄₁	1	1	yes	characteristic
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
324	C₂×C₄²	—	2	1	no	—
325	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
326	#106	—	2	1	no	—
327	#106	—	2	1	no	—
328	C₄⋊C₈	—	2	2	no	—
329	C₄⋊C₈	—	2	2	no	—
330	C₄⋊C₈	—	2	1	no	—
331	C₄⋊C₈	—	2	1	no	—
332	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
333	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
334	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
335	C₄⋊C₈	C₂³×C₄₁	1	1	yes	characteristic
336	#103	—	2	1	no	—
337	#103	—	2	1	no	—
338	#103	—	2	2	no	—
339	#103	—	2	2	no	—
340	#103	—	2	2	no	—
341	#103	—	2	2	no	—
342	#102	—	2	2	no	—
343	#102	—	2	2	no	—
344	#102	—	2	1	no	—
345	#102	—	2	1	no	—
346	C₂²⋊C₈	—	2	1	no	—
347	#97	—	2	2	no	—
348	#97	D₄×C₄₁	1	2	yes	—
349	C₄×C₈	—	2	2	no	—
350	C₄×C₈	—	2	1	no	—
351	C₄×C₈	C₂³×C₄₁	1	2	yes	—
352	#41	—	4	2	no	—
353	#41	—	4	2	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	4	1	no	—
359	C₂×Q₈	—	4	2	no	—
360	C₂×Q₈	—	4	2	no	—
361	C₂×Q₈	—	4	1	no	—
362	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
363	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	4	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	4	1	no	—
371	C₂²×C₄	—	4	1	no	—
372	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
373	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
374	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
375	Q₁₆	—	4	4	no	—
376	Q₁₆	—	4	4	no	—
377	SD₁₆	—	4	2	no	—
378	SD₁₆	—	4	2	no	—
379	M₁₆	—	2	2	no	—
380	M₁₆	—	2	2	no	—
381	M₁₆	—	4	2	no	—
382	C₂×C₈	—	2	2	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	4	2	no	—
387	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
388	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	1	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	4	1	no	—
399	C₄⋊C₄	—	4	1	no	—
400	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
401	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
402	C₂²⋊C₄	—	4	1	no	—
403	C₂²⋊C₄	—	4	1	no	—
404	C₄²	—	2	1	no	—
405	C₄²	—	2	1	no	—
406	C₄²	—	2	1	no	—
407	C₄²	—	2	1	no	—
408	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
409	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
410	C₄²	#41×C₄₁	1	1	yes	characteristic
411	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
412	C₂³	—	4	1	no	—
413	C₂³	#121×C₄₁	1	1	yes	characteristic
414	Q₈	—	2	1	no	—
415	Q₈	—	2	1	no	—
416	Q₈	—	2	1	no	—
417	Q₈	—	4	2	no	—
418	Q₈	—	4	1	no	—
419	Q₈	—	4	2	no	—
420	Q₈	—	4	1	no	—
421	Q₈	—	4	2	no	—
422	Q₈	—	8	2	no	—
423	Q₈	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
424	D₄	—	4	1	no	—
425	D₄	—	4	1	no	—
426	D₄	—	4	1	no	—
427	D₄	—	4	1	no	—
428	D₄	—	8	1	no	—
429	D₄	—	8	2	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	8	1	no	—
441	C₂×C₄	—	8	2	no	—
442	C₂×C₄	—	8	2	no	—
443	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
444	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
445	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
446	C₂×C₄	#135×C₄₁	1	2	yes	—
447	C₂×C₄	#137×C₄₁	1	2	yes	—
448	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
449	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
450	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
451	C₈	—	2	2	no	—
452	C₈	—	2	2	no	—
453	C₈	—	4	2	no	—
454	C₈	—	4	2	no	—
455	C₈	—	4	1	no	—
456	C₈	—	4	1	no	—
457	C₂²	—	2	1	no	—
458	C₂²	—	2	1	no	—
459	C₂²	—	8	1	no	—
460	C₂²	—	8	1	no	—
461	C₂²	—	8	1	no	—
462	C₂²	C₄₁×#464	1	2	yes	—
463	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
464	C₄	—	2	1	no	—
465	C₄	—	2	2	no	—
466	C₄	—	2	1	no	—
467	C₄	—	2	1	no	—
468	C₄	—	4	1	no	—
469	C₄	—	4	1	no	—
470	C₄	—	4	1	no	—
471	C₄	—	8	2	no	—
472	C₄	—	8	1	no	—
473	C₄	C₄₁×#464	1	2	yes	—
474	C₄	C₂×#135×C₄₁	1	1	yes	characteristic
475	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
476	C₄	C₂×#137×C₄₁	1	1	yes	characteristic
477	C₄	C₄₁×#575	1	1	yes	characteristic
478	C₂	—	2	2	no	—
479	C₂	—	16	1	no	—
480	C₂	C₂×C₄₁×#464	1	1	yes	characteristic
481	C₂	C₄₁×#3055	1	1	yes	characteristic
482	C₂	C₄₁×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
prime radical	C₄₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ 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
3. C₂×C₄₁×#462	C₂	semidirect
4. Q₈×M₁₆×C₄₁	C₂	semidirect
5. C₄₁×#1702	C₂	semidirect
6. C₄₁×#1700	C₂	semidirect
7. C₄₁×#1682	C₂	semidirect
8. C₄₁×#1680	C₂	semidirect
9. C₄₁×#1565	C₂	semidirect
10. C₄₁×#1549	C₂	semidirect
215. #19177	C₄₁	direct (N × H)
239. C₂×#462	C₂×C₄₁	semidirect
240. Q₈×M₁₆	C₂×C₄₁	semidirect
241. #1702	C₂×C₄₁	semidirect
242. #1700	C₂×C₄₁	semidirect
243. #1682	C₂×C₄₁	semidirect
244. #1680	C₂×C₄₁	semidirect
245. #1565	C₂×C₄₁	semidirect
246. #1549	C₂×C₄₁	semidirect
290. C₄₁	#19177	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×#3141	2
C₄₁×#3103	2
C₂×C₄₁×#462	2
Q₈×M₁₆×C₄₁	2
C₄₁×#1702	2
C₄₁×#1700	2
C₄₁×#1682	2
C₄₁×#1680	2
C₄₁×#1565	2
C₄₁×#1549	2
C₄₁×#1547	2
#19177	41

Subgroups — C41×#19177 #23795378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#19177 #23795378