Subgroups — C₄₁×#24180 #23800381

Order 10496.

1146 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3506	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3077	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3036	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2328	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2037	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2002	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1995	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1963	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1883	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1860	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1800	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
13	C₄₁×#559	—	2	2	no	—
14	C₄₁×#556	—	2	1	no	—
15	C₄₁×#555	—	2	1	no	—
16	C₄₁×#553	—	2	2	no	—
17	C₄₁×#550	—	2	1	no	—
18	C₄₁×#547	—	2	1	no	—
19	C₄₁×#538	—	2	2	no	—
20	C₄₁×#535	C₂²	1	2	yes	—
21	C₄₁×#533	C₂²	1	1	yes	characteristic
22	C₄₁×#532	C₂²	1	1	yes	characteristic
23	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#525	—	2	2	no	—
25	C₄₁×#520	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
28	C₄₁×#478	—	2	2	no	—
29	C₄₁×#476	—	2	1	no	—
30	C₄₁×#474	—	2	1	no	—
31	C₄₁×#463	—	2	2	no	—
32	C₄₁×#461	—	2	1	no	—
33	C₄₁×#459	—	2	1	no	—
34	C₄₁×#428	C₂²	1	2	yes	—
35	C₄₁×#423	C₂²	1	1	yes	characteristic
36	C₄₁×#423	C₂²	1	1	yes	characteristic
37	C₂×#104×C₄₁	C₂²	1	2	yes	—
38	C₂×#104×C₄₁	C₂²	1	2	yes	—
39	C₄₁×#418	C₂²	1	2	yes	—
40	C₄₁×#417	C₂²	1	2	yes	—
41	C₂×#100×C₄₁	C₂²	1	2	yes	—
42	C₄₁×#390	C₂²	1	1	yes	characteristic
43	C₄₁×#390	C₂²	1	1	yes	characteristic
44	C₄₁×#386	—	2	2	no	—
45	C₄₁×#386	—	2	2	no	—
46	C₄₁×#382	C₂²	1	1	yes	characteristic
47	C₄₁×#376	C₂²	1	1	yes	characteristic
48	C₄₁×#343	C₂²	1	2	yes	—
49	C₄₁×#339	C₂²	1	2	yes	—
50	C₄₁×#339	C₂²	1	2	yes	—
51	C₂×#41×C₄₁	—	2	2	no	—
52	C₂×#41×C₄₁	C₂³	1	2	yes	—
53	C₂²×D₄×C₄₁	—	2	2	no	—
54	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
55	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
56	C₂³×C₄×C₄₁	—	2	1	no	—
57	C₂³×C₄×C₄₁	D₄	1	2	yes	—
58	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
59	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
60	C₂×SD₁₆×C₄₁	—	2	2	no	—
61	C₂×D₈×C₄₁	—	2	2	no	—
62	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
63	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic, complemented
65	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic, complemented
66	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic, complemented
67	#127×C₄₁	—	2	2	no	—
68	#127×C₄₁	—	2	2	no	—
69	#127×C₄₁	—	2	2	no	—
70	#127×C₄₁	D₄	1	1	yes	characteristic, complemented
71	#126×C₄₁	—	2	4	no	—
72	#125×C₄₁	D₄	1	2	yes	complemented
73	#124×C₄₁	—	2	4	no	—
74	#124×C₄₁	—	2	2	no	—
75	#124×C₄₁	D₄	1	2	yes	complemented
76	#123×C₄₁	—	2	4	no	—
77	#123×C₄₁	—	2	4	no	—
78	C₂²⋊Q₈×C₄₁	—	2	2	no	—
79	C₂²⋊Q₈×C₄₁	—	2	1	no	—
80	C₂²⋊Q₈×C₄₁	—	2	1	no	—
81	#121×C₄₁	—	2	2	no	—
82	#121×C₄₁	—	2	4	no	—
83	#121×C₄₁	—	2	1	no	—
84	#121×C₄₁	—	2	1	no	—
85	#121×C₄₁	—	2	2	no	—
86	#121×C₄₁	—	2	2	no	—
87	#120×C₄₁	—	2	4	no	—
88	C₄×Q₈×C₄₁	—	2	2	no	—
89	C₄×Q₈×C₄₁	—	2	1	no	—
90	C₄×Q₈×C₄₁	—	2	1	no	—
91	C₄×D₄×C₄₁	—	2	4	no	—
92	C₄×D₄×C₄₁	—	2	2	no	—
93	C₄×D₄×C₄₁	—	2	4	no	—
94	C₄×D₄×C₄₁	—	2	1	no	—
95	C₄×D₄×C₄₁	—	2	1	no	—
96	C₄×D₄×C₄₁	—	2	1	no	—
97	C₄×D₄×C₄₁	—	2	1	no	—
98	C₄×D₄×C₄₁	—	2	1	no	—
99	C₄×D₄×C₄₁	—	2	1	no	—
100	C₄×D₄×C₄₁	—	2	1	no	—
101	C₄×D₄×C₄₁	—	2	1	no	—
102	C₄×D₄×C₄₁	—	2	1	no	—
103	C₄×D₄×C₄₁	—	2	1	no	—
104	#117×C₄₁	—	2	2	no	—
105	#117×C₄₁	—	2	2	no	—
106	#117×C₄₁	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
108	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
110	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
114	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
117	#107×C₄₁	—	2	2	no	—
118	#106×C₄₁	—	2	2	no	—
119	C₄⋊C₈×C₄₁	—	2	1	no	—
120	C₄⋊C₈×C₄₁	—	2	1	no	—
121	C₄⋊C₈×C₄₁	—	2	1	no	—
122	C₄⋊C₈×C₄₁	—	2	1	no	—
123	#104×C₄₁	—	2	4	no	—
124	#104×C₄₁	—	4	2	no	—
125	#103×C₄₁	—	2	2	no	—
126	#102×C₄₁	—	2	2	no	—
127	#102×C₄₁	—	2	2	no	—
128	#102×C₄₁	—	2	2	no	—
129	#100×C₄₁	—	4	2	no	—
130	#95×C₄₁	—	2	1	no	—
131	#95×C₄₁	—	2	1	no	—
132	#95×C₄₁	—	2	2	no	—
133	C₂⁴×C₄₁	—	2	1	no	—
134	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
135	#41×C₄₁	—	2	2	no	—
136	#41×C₄₁	—	2	2	no	—
137	#41×C₄₁	—	2	2	no	—
138	#41×C₄₁	—	2	2	no	—
139	#41×C₄₁	—	8	4	no	—
140	C₂×Q₈×C₄₁	—	2	1	no	—
141	C₂×Q₈×C₄₁	—	2	1	no	—
142	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
143	C₂×D₄×C₄₁	—	2	2	no	—
144	C₂×D₄×C₄₁	—	2	4	no	—
145	C₂×D₄×C₄₁	—	2	4	no	—
146	C₂×D₄×C₄₁	—	2	8	no	—
147	C₂×D₄×C₄₁	—	2	1	no	—
148	C₂×D₄×C₄₁	—	2	1	no	—
149	C₂×D₄×C₄₁	—	2	1	no	—
150	C₂×D₄×C₄₁	—	2	1	no	—
151	C₂×D₄×C₄₁	—	2	1	no	—
152	C₂×D₄×C₄₁	—	2	1	no	—
153	C₂×D₄×C₄₁	—	2	1	no	—
154	C₂×D₄×C₄₁	—	2	1	no	—
155	C₂×D₄×C₄₁	—	2	1	no	—
156	C₂×D₄×C₄₁	—	2	1	no	—
157	C₂×D₄×C₄₁	—	2	1	no	—
158	C₂×D₄×C₄₁	—	2	1	no	—
159	C₂×D₄×C₄₁	—	2	1	no	—
160	C₂×D₄×C₄₁	—	8	2	no	—
161	C₂×D₄×C₄₁	—	8	2	no	—
162	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
163	C₂×D₄×C₄₁	#41	1	2	yes	complemented
164	C₂²×C₄×C₄₁	—	2	1	no	—
165	C₂²×C₄×C₄₁	—	2	4	no	—
166	C₂²×C₄×C₄₁	—	2	2	no	—
167	C₂²×C₄×C₄₁	—	2	2	no	—
168	C₂²×C₄×C₄₁	—	2	2	no	—
169	C₂²×C₄×C₄₁	—	2	1	no	—
170	C₂²×C₄×C₄₁	—	2	1	no	—
171	C₂²×C₄×C₄₁	—	4	1	no	—
172	C₂²×C₄×C₄₁	—	4	1	no	—
173	C₂²×C₄×C₄₁	—	8	2	no	—
174	C₂²×C₄×C₄₁	—	8	2	no	—
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	C₂⁴	1	1	yes	characteristic
179	SD₁₆×C₄₁	—	4	4	no	—
180	D₈×C₄₁	—	4	4	no	—
181	M₁₆×C₄₁	—	2	2	no	—
182	M₁₆×C₄₁	—	2	2	no	—
183	M₁₆×C₄₁	—	2	2	no	—
184	M₁₆×C₄₁	—	2	2	no	—
185	C₂×C₈×C₄₁	—	2	2	no	—
186	C₂×C₈×C₄₁	—	2	2	no	—
187	C₄⋊C₄×C₄₁	—	2	4	no	—
188	C₄⋊C₄×C₄₁	—	2	1	no	—
189	C₄⋊C₄×C₄₁	—	2	1	no	—
190	C₄⋊C₄×C₄₁	—	2	2	no	—
191	C₄⋊C₄×C₄₁	—	2	1	no	—
192	C₄⋊C₄×C₄₁	—	2	2	no	—
193	C₄⋊C₄×C₄₁	—	2	1	no	—
194	C₄⋊C₄×C₄₁	—	2	1	no	—
195	C₄⋊C₄×C₄₁	—	2	1	no	—
196	C₄⋊C₄×C₄₁	—	2	1	no	—
197	C₄⋊C₄×C₄₁	—	2	1	no	—
198	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
199	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
200	C₂²⋊C₄×C₄₁	—	2	4	no	—
201	C₂²⋊C₄×C₄₁	—	2	4	no	—
202	C₂²⋊C₄×C₄₁	—	2	4	no	—
203	C₂²⋊C₄×C₄₁	—	2	2	no	—
204	C₂²⋊C₄×C₄₁	—	2	2	no	—
205	C₂²⋊C₄×C₄₁	—	2	2	no	—
206	C₂²⋊C₄×C₄₁	—	2	2	no	—
207	C₂²⋊C₄×C₄₁	—	2	2	no	—
208	C₂²⋊C₄×C₄₁	—	4	1	no	—
209	C₂²⋊C₄×C₄₁	—	4	1	no	—
210	C₂²⋊C₄×C₄₁	—	8	2	no	—
211	C₂²⋊C₄×C₄₁	—	8	2	no	—
212	C₄²×C₄₁	—	2	4	no	—
213	C₄²×C₄₁	—	2	2	no	—
214	C₄²×C₄₁	—	2	1	no	—
215	C₄²×C₄₁	—	2	1	no	—
216	C₄²×C₄₁	—	2	1	no	—
217	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
218	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
220	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
221	C₄²×C₄₁	#41	1	2	yes	complemented
222	C₂³×C₄₁	—	2	2	no	—
223	C₂³×C₄₁	—	2	2	no	—
224	C₂³×C₄₁	—	2	2	no	—
225	C₂³×C₄₁	—	2	2	no	—
226	C₂³×C₄₁	—	2	2	no	—
227	C₂³×C₄₁	—	4	2	no	—
228	C₂³×C₄₁	—	4	2	no	—
229	C₂³×C₄₁	—	4	2	no	—
230	C₂³×C₄₁	—	4	2	no	—
231	C₂³×C₄₁	—	4	1	no	—
232	C₂³×C₄₁	—	4	1	no	—
233	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
234	Q₈×C₄₁	—	2	2	no	—
235	Q₈×C₄₁	—	2	2	no	—
236	Q₈×C₄₁	—	4	2	no	—
237	Q₈×C₄₁	—	8	1	no	—
238	D₄×C₄₁	—	2	2	no	—
239	D₄×C₄₁	—	2	2	no	—
240	D₄×C₄₁	—	2	2	no	—
241	D₄×C₄₁	—	2	2	no	—
242	D₄×C₄₁	—	4	2	no	—
243	D₄×C₄₁	—	4	2	no	—
244	D₄×C₄₁	—	4	2	no	—
245	D₄×C₄₁	—	4	1	no	—
246	D₄×C₄₁	—	4	1	no	—
247	D₄×C₄₁	—	4	1	no	—
248	D₄×C₄₁	—	4	1	no	—
249	D₄×C₄₁	—	4	1	no	—
250	D₄×C₄₁	—	4	1	no	—
251	D₄×C₄₁	—	8	1	no	—
252	D₄×C₄₁	—	8	4	no	—
253	D₄×C₄₁	—	8	8	no	—
254	D₄×C₄₁	—	8	4	no	—
255	C₂×C₄×C₄₁	—	2	2	no	—
256	C₂×C₄×C₄₁	—	2	2	no	—
257	C₂×C₄×C₄₁	—	2	2	no	—
258	C₂×C₄×C₄₁	—	2	2	no	—
259	C₂×C₄×C₄₁	—	2	2	no	—
260	C₂×C₄×C₄₁	—	4	1	no	—
261	C₂×C₄×C₄₁	—	4	1	no	—
262	C₂×C₄×C₄₁	—	4	4	no	—
263	C₂×C₄×C₄₁	—	4	1	no	—
264	C₂×C₄×C₄₁	—	4	1	no	—
265	C₂×C₄×C₄₁	—	4	2	no	—
266	C₂×C₄×C₄₁	—	4	2	no	—
267	C₂×C₄×C₄₁	—	4	2	no	—
268	C₂×C₄×C₄₁	—	4	1	no	—
269	C₂×C₄×C₄₁	—	4	1	no	—
270	C₂×C₄×C₄₁	—	4	2	no	—
271	C₂×C₄×C₄₁	—	4	1	no	—
272	C₂×C₄×C₄₁	—	4	1	no	—
273	C₂×C₄×C₄₁	—	8	2	no	—
274	C₂×C₄×C₄₁	—	8	2	no	—
275	C₂×C₄×C₄₁	—	8	2	no	—
276	C₂×C₄×C₄₁	—	8	4	no	—
277	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
278	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
279	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
280	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
281	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
283	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
284	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
285	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
286	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
287	C₈×C₄₁	—	4	2	no	—
288	C₈×C₄₁	—	4	2	no	—
289	#24180	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
290	C₂²×C₄₁	—	2	1	no	—
291	C₂²×C₄₁	—	2	1	no	—
292	C₂²×C₄₁	—	4	4	no	—
293	C₂²×C₄₁	—	4	4	no	—
294	C₂²×C₄₁	—	4	2	no	—
295	C₂²×C₄₁	—	4	2	no	—
296	C₂²×C₄₁	—	4	2	no	—
297	C₂²×C₄₁	—	4	2	no	—
298	C₂²×C₄₁	—	4	2	no	—
299	C₂²×C₄₁	—	4	2	no	—
300	C₂²×C₄₁	—	4	2	no	—
301	C₂²×C₄₁	—	8	2	no	—
302	C₂²×C₄₁	—	8	2	no	—
303	C₂²×C₄₁	—	8	1	no	—
304	C₂²×C₄₁	—	8	1	no	—
305	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
306	C₂²×C₄₁	#525	1	1	yes	characteristic
307	C₂²×C₄₁	#537	1	1	yes	characteristic
308	C₄×C₄₁	—	2	1	no	—
309	C₄×C₄₁	—	2	1	no	—
310	C₄×C₄₁	—	2	1	no	—
311	C₄×C₄₁	—	2	1	no	—
312	C₄×C₄₁	—	4	2	no	—
313	C₄×C₄₁	—	4	2	no	—
314	C₄×C₄₁	—	4	2	no	—
315	C₄×C₄₁	—	4	2	no	—
316	C₄×C₄₁	—	4	2	no	—
317	C₄×C₄₁	—	8	1	no	—
318	C₄×C₄₁	—	8	1	no	—
319	C₄×C₄₁	#546	1	1	yes	characteristic
320	C₄×C₄₁	#547	1	1	yes	characteristic
321	C₄×C₄₁	#548	1	1	yes	characteristic
322	C₄×C₄₁	#550	1	1	yes	characteristic
323	#3506	C₂×C₄₁	1	1	yes	characteristic, complemented
324	#3077	C₂×C₄₁	1	1	yes	characteristic, complemented
325	#3036	C₂×C₄₁	1	1	yes	characteristic, complemented
326	#2328	C₂×C₄₁	1	1	yes	characteristic, complemented
327	#2037	C₂×C₄₁	1	2	yes	complemented
328	#2002	C₂×C₄₁	1	2	yes	complemented
329	#1995	C₂×C₄₁	1	1	yes	characteristic, complemented
330	#1963	C₂×C₄₁	1	1	yes	characteristic, complemented
331	#1883	C₂×C₄₁	1	2	yes	complemented
332	#1860	C₂×C₄₁	1	2	yes	complemented
333	#1800	C₂×C₄₁	1	1	yes	characteristic, complemented
334	C₂×C₄₁	—	2	1	no	—
335	C₂×C₄₁	—	2	1	no	—
336	C₂×C₄₁	—	8	2	no	—
337	C₂×C₄₁	—	8	2	no	—
338	C₂×C₄₁	—	8	2	no	—
339	C₂×C₄₁	#2407	1	1	yes	characteristic
340	C₂×C₄₁	#3019	1	1	yes	characteristic
341	C₂×C₄₁	#3021	1	1	yes	characteristic
342	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
343	#559	—	2	2	no	—
344	#556	—	2	1	no	—
345	#555	—	2	1	no	—
346	#553	—	2	2	no	—
347	#550	—	2	1	no	—
348	#547	—	2	1	no	—
349	#538	—	2	2	no	—
350	#535	C₂²×C₄₁	1	2	yes	—
351	#533	C₂²×C₄₁	1	1	yes	characteristic
352	#532	C₂²×C₄₁	1	1	yes	characteristic
353	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
354	#525	—	2	2	no	—
355	#520	C₂²×C₄₁	1	2	yes	—
356	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
357	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
358	#478	—	2	2	no	—
359	#476	—	2	1	no	—
360	#474	—	2	1	no	—
361	#463	—	2	2	no	—
362	#461	—	2	1	no	—
363	#459	—	2	1	no	—
364	#428	C₂²×C₄₁	1	2	yes	—
365	#423	C₂²×C₄₁	1	1	yes	characteristic
366	#423	C₂²×C₄₁	1	1	yes	characteristic
367	C₂×#104	C₂²×C₄₁	1	2	yes	—
368	C₂×#104	C₂²×C₄₁	1	2	yes	—
369	#418	C₂²×C₄₁	1	2	yes	—
370	#417	C₂²×C₄₁	1	2	yes	—
371	C₂×#100	C₂²×C₄₁	1	2	yes	—
372	#390	C₂²×C₄₁	1	1	yes	characteristic
373	#390	C₂²×C₄₁	1	1	yes	characteristic
374	#386	—	2	2	no	—
375	#386	—	2	2	no	—
376	#382	C₂²×C₄₁	1	1	yes	characteristic
377	#376	C₂²×C₄₁	1	1	yes	characteristic
378	#343	C₂²×C₄₁	1	2	yes	—
379	#339	C₂²×C₄₁	1	2	yes	—
380	#339	C₂²×C₄₁	1	2	yes	—
381	C₄₁	#24180	1	1	yes	characteristic, direct product factor, Sylow
382	C₂×#41	—	2	2	no	—
383	C₂×#41	C₂³×C₄₁	1	2	yes	—
384	C₂²×D₄	—	2	2	no	—
385	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
386	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
387	C₂³×C₄	—	2	1	no	—
388	C₂³×C₄	D₄×C₄₁	1	2	yes	—
389	AGL(1,ℤ/8)	—	2	4	no	—
390	AGL(1,ℤ/8)	—	4	2	no	—
391	C₂×SD₁₆	—	2	2	no	—
392	C₂×D₈	—	2	2	no	—
393	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
394	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
395	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic, complemented
396	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic, complemented
397	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic, complemented
398	#127	—	2	2	no	—
399	#127	—	2	2	no	—
400	#127	—	2	2	no	—
401	#127	D₄×C₄₁	1	1	yes	characteristic, complemented
402	#126	—	2	4	no	—
403	#125	D₄×C₄₁	1	2	yes	complemented
404	#124	—	2	4	no	—
405	#124	—	2	2	no	—
406	#124	D₄×C₄₁	1	2	yes	complemented
407	#123	—	2	4	no	—
408	#123	—	2	4	no	—
409	C₂²⋊Q₈	—	2	2	no	—
410	C₂²⋊Q₈	—	2	1	no	—
411	C₂²⋊Q₈	—	2	1	no	—
412	#121	—	2	2	no	—
413	#121	—	2	4	no	—
414	#121	—	2	1	no	—
415	#121	—	2	1	no	—
416	#121	—	2	2	no	—
417	#121	—	2	2	no	—
418	#120	—	2	4	no	—
419	C₄×Q₈	—	2	2	no	—
420	C₄×Q₈	—	2	1	no	—
421	C₄×Q₈	—	2	1	no	—
422	C₄×D₄	—	2	4	no	—
423	C₄×D₄	—	2	2	no	—
424	C₄×D₄	—	2	4	no	—
425	C₄×D₄	—	2	1	no	—
426	C₄×D₄	—	2	1	no	—
427	C₄×D₄	—	2	1	no	—
428	C₄×D₄	—	2	1	no	—
429	C₄×D₄	—	2	1	no	—
430	C₄×D₄	—	2	1	no	—
431	C₄×D₄	—	2	1	no	—
432	C₄×D₄	—	2	1	no	—
433	C₄×D₄	—	2	1	no	—
434	C₄×D₄	—	2	1	no	—
435	#117	—	2	2	no	—
436	#117	—	2	2	no	—
437	#117	C₂³×C₄₁	1	2	yes	—
438	C₂×C₄⋊C₄	—	2	2	no	—
439	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
440	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
441	C₂×C₂²⋊C₄	—	2	1	no	—
442	C₂×C₂²⋊C₄	—	2	2	no	—
443	C₂×C₂²⋊C₄	—	2	2	no	—
444	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
445	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
447	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
448	#107	—	2	2	no	—
449	#106	—	2	2	no	—
450	C₄⋊C₈	—	2	1	no	—
451	C₄⋊C₈	—	2	1	no	—
452	C₄⋊C₈	—	2	1	no	—
453	C₄⋊C₈	—	2	1	no	—
454	#104	—	2	4	no	—
455	#104	—	4	2	no	—
456	#103	—	2	2	no	—
457	#102	—	2	2	no	—
458	#102	—	2	2	no	—
459	#102	—	2	2	no	—
460	#100	—	4	2	no	—
461	#95	—	2	1	no	—
462	#95	—	2	1	no	—
463	#95	—	2	2	no	—
464	C₂⁴	—	2	1	no	—
465	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
466	#41	—	2	2	no	—
467	#41	—	2	2	no	—
468	#41	—	2	2	no	—
469	#41	—	2	2	no	—
470	#41	—	8	4	no	—
471	C₂×Q₈	—	2	1	no	—
472	C₂×Q₈	—	2	1	no	—
473	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
474	C₂×D₄	—	2	2	no	—
475	C₂×D₄	—	2	4	no	—
476	C₂×D₄	—	2	4	no	—
477	C₂×D₄	—	2	8	no	—
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	2	1	no	—
480	C₂×D₄	—	2	1	no	—
481	C₂×D₄	—	2	1	no	—
482	C₂×D₄	—	2	1	no	—
483	C₂×D₄	—	2	1	no	—
484	C₂×D₄	—	2	1	no	—
485	C₂×D₄	—	2	1	no	—
486	C₂×D₄	—	2	1	no	—
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	—	2	1	no	—
489	C₂×D₄	—	2	1	no	—
490	C₂×D₄	—	2	1	no	—
491	C₂×D₄	—	8	2	no	—
492	C₂×D₄	—	8	2	no	—
493	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
494	C₂×D₄	#41×C₄₁	1	2	yes	complemented
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	4	no	—
497	C₂²×C₄	—	2	2	no	—
498	C₂²×C₄	—	2	2	no	—
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	2	1	no	—
501	C₂²×C₄	—	2	1	no	—
502	C₂²×C₄	—	4	1	no	—
503	C₂²×C₄	—	4	1	no	—
504	C₂²×C₄	—	8	2	no	—
505	C₂²×C₄	—	8	2	no	—
506	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
507	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
508	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
509	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
510	SD₁₆	—	4	4	no	—
511	D₈	—	4	4	no	—
512	M₁₆	—	2	2	no	—
513	M₁₆	—	2	2	no	—
514	M₁₆	—	2	2	no	—
515	M₁₆	—	2	2	no	—
516	C₂×C₈	—	2	2	no	—
517	C₂×C₈	—	2	2	no	—
518	C₄⋊C₄	—	2	4	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	2	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	—	2	1	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	2	1	no	—
529	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
530	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
531	C₂²⋊C₄	—	2	4	no	—
532	C₂²⋊C₄	—	2	4	no	—
533	C₂²⋊C₄	—	2	4	no	—
534	C₂²⋊C₄	—	2	2	no	—
535	C₂²⋊C₄	—	2	2	no	—
536	C₂²⋊C₄	—	2	2	no	—
537	C₂²⋊C₄	—	2	2	no	—
538	C₂²⋊C₄	—	2	2	no	—
539	C₂²⋊C₄	—	4	1	no	—
540	C₂²⋊C₄	—	4	1	no	—
541	C₂²⋊C₄	—	8	2	no	—
542	C₂²⋊C₄	—	8	2	no	—
543	C₄²	—	2	4	no	—
544	C₄²	—	2	2	no	—
545	C₄²	—	2	1	no	—
546	C₄²	—	2	1	no	—
547	C₄²	—	2	1	no	—
548	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
549	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
550	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
551	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
552	C₄²	#41×C₄₁	1	2	yes	complemented
553	C₂³	—	2	2	no	—
554	C₂³	—	2	2	no	—
555	C₂³	—	2	2	no	—
556	C₂³	—	2	2	no	—
557	C₂³	—	2	2	no	—
558	C₂³	—	4	2	no	—
559	C₂³	—	4	2	no	—
560	C₂³	—	4	2	no	—
561	C₂³	—	4	2	no	—
562	C₂³	—	4	1	no	—
563	C₂³	—	4	1	no	—
564	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
565	Q₈	—	2	2	no	—
566	Q₈	—	2	2	no	—
567	Q₈	—	4	2	no	—
568	Q₈	—	8	1	no	—
569	D₄	—	2	2	no	—
570	D₄	—	2	2	no	—
571	D₄	—	2	2	no	—
572	D₄	—	2	2	no	—
573	D₄	—	4	2	no	—
574	D₄	—	4	2	no	—
575	D₄	—	4	2	no	—
576	D₄	—	4	1	no	—
577	D₄	—	4	1	no	—
578	D₄	—	4	1	no	—
579	D₄	—	4	1	no	—
580	D₄	—	4	1	no	—
581	D₄	—	4	1	no	—
582	D₄	—	8	1	no	—
583	D₄	—	8	8	no	—
584	D₄	—	8	4	no	—
585	D₄	—	8	4	no	—
586	C₂×C₄	—	2	2	no	—
587	C₂×C₄	—	2	2	no	—
588	C₂×C₄	—	2	2	no	—
589	C₂×C₄	—	2	2	no	—
590	C₂×C₄	—	2	2	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	1	no	—
593	C₂×C₄	—	4	4	no	—
594	C₂×C₄	—	4	1	no	—
595	C₂×C₄	—	4	1	no	—
596	C₂×C₄	—	4	2	no	—
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	4	1	no	—
600	C₂×C₄	—	4	1	no	—
601	C₂×C₄	—	4	2	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	8	2	no	—
605	C₂×C₄	—	8	2	no	—
606	C₂×C₄	—	8	2	no	—
607	C₂×C₄	—	8	4	no	—
608	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
609	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
610	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
611	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
612	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
613	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
614	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
615	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
616	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
617	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
618	C₈	—	4	2	no	—
619	C₈	—	4	2	no	—
620	C₂²	—	2	1	no	—
621	C₂²	—	2	1	no	—
622	C₂²	—	4	4	no	—
623	C₂²	—	4	4	no	—
624	C₂²	—	4	2	no	—
625	C₂²	—	4	2	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	4	2	no	—
628	C₂²	—	4	2	no	—
629	C₂²	—	4	2	no	—
630	C₂²	—	4	2	no	—
631	C₂²	—	8	2	no	—
632	C₂²	—	8	2	no	—
633	C₂²	—	8	1	no	—
634	C₂²	—	8	1	no	—
635	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
636	C₂²	C₄₁×#525	1	1	yes	characteristic
637	C₂²	C₄₁×#537	1	1	yes	characteristic
638	C₄	—	2	1	no	—
639	C₄	—	2	1	no	—
640	C₄	—	2	1	no	—
641	C₄	—	2	1	no	—
642	C₄	—	4	2	no	—
643	C₄	—	4	2	no	—
644	C₄	—	4	2	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	2	no	—
647	C₄	—	8	1	no	—
648	C₄	—	8	1	no	—
649	C₄	C₄₁×#546	1	1	yes	characteristic
650	C₄	C₄₁×#547	1	1	yes	characteristic
651	C₄	C₄₁×#548	1	1	yes	characteristic
652	C₄	C₄₁×#550	1	1	yes	characteristic
653	C₂	—	2	1	no	—
654	C₂	—	2	1	no	—
655	C₂	—	8	2	no	—
656	C₂	—	8	2	no	—
657	C₂	—	8	2	no	—
658	C₂	C₄₁×#2407	1	1	yes	characteristic
659	C₂	C₄₁×#3019	1	1	yes	characteristic
660	C₂	C₄₁×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
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×#3506	C₂	semidirect
2. C₄₁×#3077	C₂	semidirect
3. C₄₁×#3036	C₂	semidirect
4. C₄₁×#2328	C₂	semidirect
5. C₄₁×#2037	C₂	semidirect
6. C₄₁×#2002	C₂	semidirect
7. C₄₁×#1995	C₂	semidirect
8. C₄₁×#1963	C₂	semidirect
9. C₄₁×#1883	C₂	semidirect
10. C₄₁×#1860	C₂	semidirect
11. C₄₁×#1800	C₂	semidirect
64. C₄⋊Q₈×C₄₁	D₄	semidirect
65. C₄⋊Q₈×C₄₁	D₄	semidirect
66. C₄⋊Q₈×C₄₁	D₄	semidirect
70. #127×C₄₁	D₄	semidirect
72. #125×C₄₁	D₄	semidirect
75. #124×C₄₁	D₄	semidirect
163. C₂×D₄×C₄₁	#41	semidirect
221. C₄²×C₄₁	#41	semidirect
289. #24180	C₄₁	direct (N × H)
323. #3506	C₂×C₄₁	semidirect
324. #3077	C₂×C₄₁	semidirect
325. #3036	C₂×C₄₁	semidirect
326. #2328	C₂×C₄₁	semidirect
327. #2037	C₂×C₄₁	semidirect
328. #2002	C₂×C₄₁	semidirect
329. #1995	C₂×C₄₁	semidirect
330. #1963	C₂×C₄₁	semidirect
331. #1883	C₂×C₄₁	semidirect
332. #1860	C₂×C₄₁	semidirect
333. #1800	C₂×C₄₁	semidirect
381. C₄₁	#24180	direct (N × H)
395. C₄⋊Q₈	D₄×C₄₁	semidirect
396. C₄⋊Q₈	D₄×C₄₁	semidirect
397. C₄⋊Q₈	D₄×C₄₁	semidirect
401. #127	D₄×C₄₁	semidirect
403. #125	D₄×C₄₁	semidirect
406. #124	D₄×C₄₁	semidirect
494. C₂×D₄	#41×C₄₁	semidirect
552. C₄²	#41×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3506	2
C₄₁×#3077	2
C₄₁×#3036	2
C₄₁×#2328	2
C₄₁×#2037	2
C₄₁×#2002	2
C₄₁×#1995	2
C₄₁×#1963	2
C₄₁×#1883	2
C₄₁×#1860	2
C₄₁×#1800	2
#24180	41

Subgroups — C41×#24180 #23800381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24180 #23800381