Subgroups — C₄₁×#18772 #23794973

Order 10496.

1040 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	C₂×C₄₁×#555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3206	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#3036	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1646	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1624	C₂	1	4	yes	maximal, complemented
7	C₄₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1486	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1478	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₄₁	—	2	1	no	—
11	C₄₁×#555	—	2	2	no	—
12	C₄₁×#555	—	2	2	no	—
13	C₄₁×#555	—	2	2	no	—
14	C₄₁×#555	C₂²	1	4	yes	—
15	C₄₁×#532	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×#124×C₄₁	—	2	2	no	—
18	C₂×#121×C₄₁	—	2	1	no	—
19	C₂×#121×C₄₁	—	2	1	no	—
20	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×#117×C₄₁	—	2	1	no	—
23	C₂×#117×C₄₁	—	2	1	no	—
24	C₄₁×#492	C₂²	1	2	yes	complemented
25	C₄₁×#487	C₂²	1	2	yes	complemented
26	C₄₁×#480	—	2	1	no	—
27	C₄₁×#474	C₂²	1	2	yes	complemented
28	C₄₁×#463	—	2	2	no	—
29	C₄₁×#461	—	2	2	no	—
30	C₄₁×#461	C₂²	1	2	yes	complemented
31	C₄₁×#459	C₂²	1	2	yes	complemented
32	C₄₁×#459	C₂²	1	2	yes	complemented
33	C₄₁×#448	—	2	2	no	—
34	C₄₁×#447	—	2	1	no	—
35	C₄×D₈×C₄₁	—	2	2	no	—
36	C₄₁×#431	C₂²	1	1	yes	characteristic
37	C₄₁×#431	C₂²	1	1	yes	characteristic
38	C₄₁×#423	C₂²	1	1	yes	characteristic
39	C₄₁×#417	—	2	1	no	—
40	C₄₁×#333	C₂²	1	4	yes	complemented
41	C₄₁×D₄⋊C₈	C₂²	1	4	yes	complemented
42	C₄₁×D₄⋊C₈	C₂²	1	4	yes	complemented
43	C₂×#41×C₄₁	—	2	1	no	—
44	C₂²×Q₈×C₄₁	D₄	1	2	yes	—
45	C₂²×D₄×C₄₁	—	2	2	no	—
46	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
47	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₄₁	—	2	1	no	—
49	C₂³×C₄×C₄₁	—	2	1	no	—
50	AGL(1,ℤ/8)×C₄₁	—	4	4	no	—
51	C₂×SD₁₆×C₄₁	—	2	2	no	—
52	C₂×SD₁₆×C₄₁	—	2	4	no	—
53	C₂×D₈×C₄₁	—	2	2	no	—
54	C₂×D₈×C₄₁	—	2	2	no	—
55	C₂×M₁₆×C₄₁	—	2	1	no	—
56	C₄⋊Q₈×C₄₁	—	2	1	no	—
57	C₄⋊Q₈×C₄₁	—	2	1	no	—
58	C₄⋊Q₈×C₄₁	D₄	1	2	yes	—
59	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
61	#127×C₄₁	C₂³	1	1	yes	characteristic
62	#124×C₄₁	—	2	8	no	—
63	#124×C₄₁	—	2	8	no	—
64	#124×C₄₁	—	2	1	no	—
65	#121×C₄₁	—	2	4	no	—
66	#121×C₄₁	—	2	4	no	—
67	#121×C₄₁	—	2	4	no	—
68	#121×C₄₁	—	2	4	no	—
69	#121×C₄₁	—	2	1	no	—
70	#121×C₄₁	—	2	1	no	—
71	C₄×D₄×C₄₁	—	2	1	no	—
72	C₄×D₄×C₄₁	—	2	1	no	—
73	C₄×D₄×C₄₁	—	2	1	no	—
74	C₄×D₄×C₄₁	—	2	1	no	—
75	C₄×D₄×C₄₁	—	2	1	no	—
76	C₄×D₄×C₄₁	—	2	1	no	—
77	C₄×D₄×C₄₁	D₄	1	2	yes	—
78	C₄×D₄×C₄₁	D₄	1	2	yes	—
79	C₄×D₄×C₄₁	C₂³	1	2	yes	—
80	C₄×D₄×C₄₁	C₂³	1	2	yes	—
81	#117×C₄₁	—	2	4	no	—
82	#117×C₄₁	—	2	4	no	—
83	#117×C₄₁	—	2	4	no	—
84	#117×C₄₁	—	2	4	no	—
85	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
88	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
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×C₂²⋊C₄×C₄₁	—	2	1	no	—
93	C₂×C₄²×C₄₁	—	2	2	no	—
94	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
95	#107×C₄₁	—	2	2	no	—
96	C₄⋊C₈×C₄₁	—	2	1	no	—
97	C₄⋊C₈×C₄₁	—	2	1	no	—
98	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
99	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
100	#103×C₄₁	—	2	4	no	—
101	#103×C₄₁	—	2	2	no	—
102	#102×C₄₁	—	2	2	no	—
103	#102×C₄₁	—	2	2	no	—
104	#102×C₄₁	—	2	2	no	—
105	C₂²⋊C₈×C₄₁	—	2	1	no	—
106	C₂²⋊C₈×C₄₁	—	2	1	no	—
107	C₄×C₈×C₄₁	—	2	1	no	—
108	C₄×C₈×C₄₁	C₂³	1	2	yes	—
109	C₂⁴×C₄₁	—	2	1	no	—
110	C₂⁴×C₄₁	—	2	1	no	—
111	#41×C₄₁	—	4	2	no	—
112	#41×C₄₁	—	4	2	no	—
113	C₂×Q₈×C₄₁	—	2	2	no	—
114	C₂×Q₈×C₄₁	—	2	1	no	—
115	C₂×Q₈×C₄₁	—	8	2	no	—
116	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
117	C₂×D₄×C₄₁	—	2	8	no	—
118	C₂×D₄×C₄₁	—	2	1	no	—
119	C₂×D₄×C₄₁	—	2	1	no	—
120	C₂×D₄×C₄₁	—	2	2	no	—
121	C₂×D₄×C₄₁	—	2	2	no	—
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	2	no	—
126	C₂×D₄×C₄₁	—	4	1	no	—
127	C₂×D₄×C₄₁	—	8	4	no	—
128	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
129	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
130	C₂²×C₄×C₄₁	—	2	1	no	—
131	C₂²×C₄×C₄₁	—	2	1	no	—
132	C₂²×C₄×C₄₁	—	2	2	no	—
133	C₂²×C₄×C₄₁	—	2	2	no	—
134	C₂²×C₄×C₄₁	—	2	2	no	—
135	C₂²×C₄×C₄₁	—	2	2	no	—
136	C₂²×C₄×C₄₁	—	2	2	no	—
137	C₂²×C₄×C₄₁	—	4	4	no	—
138	C₂²×C₄×C₄₁	—	4	2	no	—
139	C₂²×C₄×C₄₁	—	4	2	no	—
140	C₂²×C₄×C₄₁	—	4	1	no	—
141	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
144	SD₁₆×C₄₁	—	4	4	no	—
145	SD₁₆×C₄₁	—	8	4	no	—
146	D₈×C₄₁	—	4	4	no	—
147	D₈×C₄₁	—	4	4	no	—
148	M₁₆×C₄₁	—	4	2	no	—
149	C₂×C₈×C₄₁	—	2	2	no	—
150	C₂×C₈×C₄₁	—	2	2	no	—
151	C₂×C₈×C₄₁	—	2	2	no	—
152	C₄⋊C₄×C₄₁	—	2	2	no	—
153	C₄⋊C₄×C₄₁	—	2	2	no	—
154	C₄⋊C₄×C₄₁	—	2	1	no	—
155	C₄⋊C₄×C₄₁	—	2	1	no	—
156	C₄⋊C₄×C₄₁	—	2	2	no	—
157	C₄⋊C₄×C₄₁	—	2	2	no	—
158	C₄⋊C₄×C₄₁	—	2	1	no	—
159	C₄⋊C₄×C₄₁	—	2	1	no	—
160	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
161	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
162	C₂²⋊C₄×C₄₁	—	2	8	no	—
163	C₂²⋊C₄×C₄₁	—	2	8	no	—
164	C₂²⋊C₄×C₄₁	—	2	2	no	—
165	C₂²⋊C₄×C₄₁	—	2	2	no	—
166	C₂²⋊C₄×C₄₁	—	2	2	no	—
167	C₂²⋊C₄×C₄₁	—	2	2	no	—
168	C₂²⋊C₄×C₄₁	—	4	1	no	—
169	C₂²⋊C₄×C₄₁	—	4	1	no	—
170	C₄²×C₄₁	—	2	8	no	—
171	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
174	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
175	C₂³×C₄₁	—	2	2	no	—
176	C₂³×C₄₁	—	2	2	no	—
177	C₂³×C₄₁	—	4	2	no	—
178	C₂³×C₄₁	—	4	2	no	—
179	C₂³×C₄₁	—	4	2	no	—
180	C₂³×C₄₁	—	4	2	no	—
181	C₂³×C₄₁	—	4	2	no	—
182	C₂³×C₄₁	—	4	2	no	—
183	C₂³×C₄₁	—	4	1	no	—
184	C₂³×C₄₁	#120	1	1	yes	characteristic
185	Q₈×C₄₁	—	4	4	no	—
186	Q₈×C₄₁	—	4	1	no	—
187	Q₈×C₄₁	—	4	1	no	—
188	Q₈×C₄₁	—	8	2	no	—
189	D₄×C₄₁	—	2	2	no	—
190	D₄×C₄₁	—	2	2	no	—
191	D₄×C₄₁	—	2	2	no	—
192	D₄×C₄₁	—	2	2	no	—
193	D₄×C₄₁	—	4	2	no	—
194	D₄×C₄₁	—	4	2	no	—
195	D₄×C₄₁	—	4	1	no	—
196	D₄×C₄₁	—	4	1	no	—
197	D₄×C₄₁	—	4	2	no	—
198	D₄×C₄₁	—	4	2	no	—
199	D₄×C₄₁	—	8	8	no	—
200	D₄×C₄₁	—	8	2	no	—
201	C₂×C₄×C₄₁	—	2	2	no	—
202	C₂×C₄×C₄₁	—	2	2	no	—
203	C₂×C₄×C₄₁	—	2	2	no	—
204	C₂×C₄×C₄₁	—	2	4	no	—
205	C₂×C₄×C₄₁	—	4	1	no	—
206	C₂×C₄×C₄₁	—	4	4	no	—
207	C₂×C₄×C₄₁	—	4	2	no	—
208	C₂×C₄×C₄₁	—	4	4	no	—
209	C₂×C₄×C₄₁	—	4	2	no	—
210	C₂×C₄×C₄₁	—	4	2	no	—
211	C₂×C₄×C₄₁	—	4	2	no	—
212	C₂×C₄×C₄₁	—	4	4	no	—
213	C₂×C₄×C₄₁	—	4	4	no	—
214	C₂×C₄×C₄₁	—	4	1	no	—
215	C₂×C₄×C₄₁	—	8	2	no	—
216	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
217	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
218	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
219	C₂×C₄×C₄₁	#135	1	2	yes	—
220	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
221	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
224	C₈×C₄₁	—	4	2	no	—
225	C₈×C₄₁	—	4	2	no	—
226	C₈×C₄₁	—	4	2	no	—
227	#18772	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂²×C₄₁	—	2	1	no	—
229	C₂²×C₄₁	—	2	1	no	—
230	C₂²×C₄₁	—	4	2	no	—
231	C₂²×C₄₁	—	4	2	no	—
232	C₂²×C₄₁	—	4	2	no	—
233	C₂²×C₄₁	—	4	2	no	—
234	C₂²×C₄₁	—	4	2	no	—
235	C₂²×C₄₁	—	4	2	no	—
236	C₂²×C₄₁	—	8	4	no	—
237	C₂²×C₄₁	—	8	4	no	—
238	C₂²×C₄₁	—	8	1	no	—
239	C₂²×C₄₁	—	8	1	no	—
240	C₂²×C₄₁	—	8	1	no	—
241	C₂²×C₄₁	#449	1	2	yes	—
242	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
243	C₄×C₄₁	—	2	1	no	—
244	C₄×C₄₁	—	2	1	no	—
245	C₄×C₄₁	—	2	1	no	—
246	C₄×C₄₁	—	4	2	no	—
247	C₄×C₄₁	—	4	2	no	—
248	C₄×C₄₁	—	4	4	no	—
249	C₄×C₄₁	—	8	1	no	—
250	C₄×C₄₁	#449	1	2	yes	—
251	C₄×C₄₁	C₂×#135	1	1	yes	characteristic
252	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
253	C₄×C₄₁	#575	1	2	yes	—
254	C₂×#555	C₂×C₄₁	1	1	yes	characteristic, complemented
255	#3206	C₂×C₄₁	1	1	yes	characteristic, complemented
256	#3100	C₂×C₄₁	1	1	yes	characteristic, complemented
257	#3036	C₂×C₄₁	1	1	yes	characteristic, complemented
258	#1646	C₂×C₄₁	1	4	yes	complemented
259	#1624	C₂×C₄₁	1	4	yes	complemented
260	#1548	C₂×C₄₁	1	1	yes	characteristic, complemented
261	#1486	C₂×C₄₁	1	1	yes	characteristic, complemented
262	#1478	C₂×C₄₁	1	1	yes	characteristic, complemented
263	C₂×C₄₁	—	2	2	no	—
264	C₂×C₄₁	—	8	2	no	—
265	C₂×C₄₁	—	8	2	no	—
266	C₂×C₄₁	—	16	1	no	—
267	C₂×C₄₁	C₂×#449	1	1	yes	characteristic
268	C₂×C₄₁	#3005	1	1	yes	characteristic
269	C₂×C₄₁	#3019	1	1	yes	characteristic
270	C₂×AGL(1,ℤ/8)	—	2	1	no	—
271	#555	—	2	2	no	—
272	#555	—	2	2	no	—
273	#555	—	2	2	no	—
274	#555	C₂²×C₄₁	1	4	yes	—
275	#532	C₂²×C₄₁	1	1	yes	characteristic
276	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
277	C₂×#124	—	2	2	no	—
278	C₂×#121	—	2	1	no	—
279	C₂×#121	—	2	1	no	—
280	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
281	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
282	C₂×#117	—	2	1	no	—
283	C₂×#117	—	2	1	no	—
284	#492	C₂²×C₄₁	1	2	yes	complemented
285	#487	C₂²×C₄₁	1	2	yes	complemented
286	#480	—	2	1	no	—
287	#474	C₂²×C₄₁	1	2	yes	complemented
288	#463	—	2	2	no	—
289	#461	—	2	2	no	—
290	#461	C₂²×C₄₁	1	2	yes	complemented
291	#459	C₂²×C₄₁	1	2	yes	complemented
292	#459	C₂²×C₄₁	1	2	yes	complemented
293	#448	—	2	2	no	—
294	#447	—	2	1	no	—
295	C₄×D₈	—	2	2	no	—
296	#431	C₂²×C₄₁	1	1	yes	characteristic
297	#431	C₂²×C₄₁	1	1	yes	characteristic
298	#423	C₂²×C₄₁	1	1	yes	characteristic
299	#417	—	2	1	no	—
300	#333	C₂²×C₄₁	1	4	yes	complemented
301	D₄⋊C₈	C₂²×C₄₁	1	4	yes	complemented
302	D₄⋊C₈	C₂²×C₄₁	1	4	yes	complemented
303	C₄₁	#18772	1	1	yes	characteristic, direct product factor, Sylow
304	C₂×#41	—	2	1	no	—
305	C₂²×Q₈	D₄×C₄₁	1	2	yes	—
306	C₂²×D₄	—	2	2	no	—
307	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
308	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
309	C₂³×C₄	—	2	1	no	—
310	C₂³×C₄	—	2	1	no	—
311	AGL(1,ℤ/8)	—	4	4	no	—
312	C₂×SD₁₆	—	2	2	no	—
313	C₂×SD₁₆	—	2	4	no	—
314	C₂×D₈	—	2	2	no	—
315	C₂×D₈	—	2	2	no	—
316	C₂×M₁₆	—	2	1	no	—
317	C₄⋊Q₈	—	2	1	no	—
318	C₄⋊Q₈	—	2	1	no	—
319	C₄⋊Q₈	D₄×C₄₁	1	2	yes	—
320	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
321	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
322	#127	C₂³×C₄₁	1	1	yes	characteristic
323	#124	—	2	8	no	—
324	#124	—	2	8	no	—
325	#124	—	2	1	no	—
326	#121	—	2	4	no	—
327	#121	—	2	4	no	—
328	#121	—	2	4	no	—
329	#121	—	2	4	no	—
330	#121	—	2	1	no	—
331	#121	—	2	1	no	—
332	C₄×D₄	—	2	1	no	—
333	C₄×D₄	—	2	1	no	—
334	C₄×D₄	—	2	1	no	—
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	—	2	1	no	—
338	C₄×D₄	D₄×C₄₁	1	2	yes	—
339	C₄×D₄	D₄×C₄₁	1	2	yes	—
340	C₄×D₄	C₂³×C₄₁	1	2	yes	—
341	C₄×D₄	C₂³×C₄₁	1	2	yes	—
342	#117	—	2	4	no	—
343	#117	—	2	4	no	—
344	#117	—	2	4	no	—
345	#117	—	2	4	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	1	no	—
348	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
349	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
350	C₂×C₂²⋊C₄	—	2	2	no	—
351	C₂×C₂²⋊C₄	—	2	2	no	—
352	C₂×C₂²⋊C₄	—	2	1	no	—
353	C₂×C₂²⋊C₄	—	2	1	no	—
354	C₂×C₄²	—	2	2	no	—
355	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
356	#107	—	2	2	no	—
357	C₄⋊C₈	—	2	1	no	—
358	C₄⋊C₈	—	2	1	no	—
359	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
360	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
361	#103	—	2	4	no	—
362	#103	—	2	2	no	—
363	#102	—	2	2	no	—
364	#102	—	2	2	no	—
365	#102	—	2	2	no	—
366	C₂²⋊C₈	—	2	1	no	—
367	C₂²⋊C₈	—	2	1	no	—
368	C₄×C₈	—	2	1	no	—
369	C₄×C₈	C₂³×C₄₁	1	2	yes	—
370	C₂⁴	—	2	1	no	—
371	C₂⁴	—	2	1	no	—
372	#41	—	4	2	no	—
373	#41	—	4	2	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	2	1	no	—
376	C₂×Q₈	—	8	2	no	—
377	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
378	C₂×D₄	—	2	8	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	—	2	1	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	2	no	—
383	C₂×D₄	—	2	1	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	4	2	no	—
387	C₂×D₄	—	4	1	no	—
388	C₂×D₄	—	8	4	no	—
389	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
390	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	2	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	4	4	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	—	4	2	no	—
401	C₂²×C₄	—	4	1	no	—
402	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
403	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
405	SD₁₆	—	4	4	no	—
406	SD₁₆	—	8	4	no	—
407	D₈	—	4	4	no	—
408	D₈	—	4	4	no	—
409	M₁₆	—	4	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	1	no	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
422	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
423	C₂²⋊C₄	—	2	8	no	—
424	C₂²⋊C₄	—	2	8	no	—
425	C₂²⋊C₄	—	2	2	no	—
426	C₂²⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	2	2	no	—
428	C₂²⋊C₄	—	2	2	no	—
429	C₂²⋊C₄	—	4	1	no	—
430	C₂²⋊C₄	—	4	1	no	—
431	C₄²	—	2	8	no	—
432	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
433	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
434	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
435	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
436	C₂³	—	2	2	no	—
437	C₂³	—	2	2	no	—
438	C₂³	—	4	2	no	—
439	C₂³	—	4	2	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	4	2	no	—
442	C₂³	—	4	2	no	—
443	C₂³	—	4	2	no	—
444	C₂³	—	4	1	no	—
445	C₂³	#120×C₄₁	1	1	yes	characteristic
446	Q₈	—	4	4	no	—
447	Q₈	—	4	1	no	—
448	Q₈	—	4	1	no	—
449	Q₈	—	8	2	no	—
450	D₄	—	2	2	no	—
451	D₄	—	2	2	no	—
452	D₄	—	2	2	no	—
453	D₄	—	2	2	no	—
454	D₄	—	4	2	no	—
455	D₄	—	4	2	no	—
456	D₄	—	4	1	no	—
457	D₄	—	4	1	no	—
458	D₄	—	4	2	no	—
459	D₄	—	4	2	no	—
460	D₄	—	8	8	no	—
461	D₄	—	8	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	4	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	4	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	4	no	—
474	C₂×C₄	—	4	4	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	8	2	no	—
477	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
478	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
479	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
480	C₂×C₄	#135×C₄₁	1	2	yes	—
481	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
482	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
483	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
485	C₈	—	4	2	no	—
486	C₈	—	4	2	no	—
487	C₈	—	4	2	no	—
488	C₂²	—	2	1	no	—
489	C₂²	—	2	1	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	8	4	no	—
497	C₂²	—	8	4	no	—
498	C₂²	—	8	1	no	—
499	C₂²	—	8	1	no	—
500	C₂²	—	8	1	no	—
501	C₂²	C₄₁×#449	1	2	yes	—
502	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
503	C₄	—	2	1	no	—
504	C₄	—	2	1	no	—
505	C₄	—	2	1	no	—
506	C₄	—	4	2	no	—
507	C₄	—	4	2	no	—
508	C₄	—	4	4	no	—
509	C₄	—	8	1	no	—
510	C₄	C₄₁×#449	1	2	yes	—
511	C₄	C₂×#135×C₄₁	1	1	yes	characteristic
512	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
513	C₄	C₄₁×#575	1	2	yes	—
514	C₂	—	2	2	no	—
515	C₂	—	8	2	no	—
516	C₂	—	8	2	no	—
517	C₂	—	16	1	no	—
518	C₂	C₂×C₄₁×#449	1	1	yes	characteristic
519	C₂	C₄₁×#3005	1	1	yes	characteristic
520	C₂	C₄₁×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
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
1. C₂×C₄₁×#555	C₂	semidirect
2. C₄₁×#3206	C₂	semidirect
3. C₄₁×#3100	C₂	semidirect
4. C₄₁×#3036	C₂	semidirect
5. C₄₁×#1646	C₂	semidirect
6. C₄₁×#1624	C₂	semidirect
7. C₄₁×#1548	C₂	semidirect
8. C₄₁×#1486	C₂	semidirect
9. C₄₁×#1478	C₂	semidirect
24. C₄₁×#492	C₂²	semidirect
25. C₄₁×#487	C₂²	semidirect
27. C₄₁×#474	C₂²	semidirect
30. C₄₁×#461	C₂²	semidirect
31. C₄₁×#459	C₂²	semidirect
32. C₄₁×#459	C₂²	semidirect
40. C₄₁×#333	C₂²	semidirect
41. C₄₁×D₄⋊C₈	C₂²	semidirect
42. C₄₁×D₄⋊C₈	C₂²	semidirect
227. #18772	C₄₁	direct (N × H)
254. C₂×#555	C₂×C₄₁	semidirect
255. #3206	C₂×C₄₁	semidirect
256. #3100	C₂×C₄₁	semidirect
257. #3036	C₂×C₄₁	semidirect
258. #1646	C₂×C₄₁	semidirect
259. #1624	C₂×C₄₁	semidirect
260. #1548	C₂×C₄₁	semidirect
261. #1486	C₂×C₄₁	semidirect
262. #1478	C₂×C₄₁	semidirect
284. #492	C₂²×C₄₁	semidirect
285. #487	C₂²×C₄₁	semidirect
287. #474	C₂²×C₄₁	semidirect
290. #461	C₂²×C₄₁	semidirect
291. #459	C₂²×C₄₁	semidirect
292. #459	C₂²×C₄₁	semidirect
300. #333	C₂²×C₄₁	semidirect
301. D₄⋊C₈	C₂²×C₄₁	semidirect
302. D₄⋊C₈	C₂²×C₄₁	semidirect
303. C₄₁	#18772	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#555	2
C₄₁×#3206	2
C₄₁×#3100	2
C₄₁×#3036	2
C₄₁×#1646	2
C₄₁×#1624	2
C₄₁×#1548	2
C₄₁×#1486	2
C₄₁×#1478	2
#18772	41

Subgroups — C41×#18772 #23794973

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#18772 #23794973