Subgroups — C₄₁×#24172 #23800373

Order 10496.

1058 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3511	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#3037	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#137×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2568	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2040	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2007	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1980	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1882	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1860	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1806	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×#137×C₄₁	C₂²	1	2	yes	—
14	C₄₁×#556	—	2	1	no	—
15	C₄₁×#555	—	2	1	no	—
16	C₄₁×#554	—	2	1	no	—
17	C₄₁×#553	—	2	2	no	—
18	C₄₁×#548	—	2	2	no	—
19	C₄₁×#547	—	2	1	no	—
20	C₄₁×#542	—	2	2	no	—
21	C₄₁×#537	C₂²	1	2	yes	—
22	C₄₁×#533	—	2	1	no	—
23	C₄₁×#532	—	2	1	no	—
24	C₄₁×#532	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₄₁×#520	C₂²	1	2	yes	—
27	C₄×#41×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
29	C₄₁×#464	—	2	2	no	—
30	C₄₁×#462	—	2	1	no	—
31	C₄₁×#460	—	2	1	no	—
32	C₄₁×#441	—	2	1	no	—
33	C₄₁×#440	—	2	1	no	—
34	C₄×Q₁₆×C₄₁	—	2	1	no	—
35	C₄×SD₁₆×C₄₁	—	2	1	no	—
36	C₄₁×#428	C₂²	1	1	yes	characteristic
37	C₄₁×#423	C₂²	1	1	yes	characteristic
38	C₂×#104×C₄₁	C₂²	1	2	yes	—
39	C₂×#104×C₄₁	C₂²	1	2	yes	—
40	C₂×#104×C₄₁	C₂²	1	2	yes	—
41	C₄₁×#419	C₂²	1	1	yes	characteristic
42	C₄₁×#417	C₂²	1	2	yes	—
43	C₄₁×#417	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
45	C₄₁×#400	—	2	2	no	—
46	C₄₁×#391	—	2	2	no	—
47	C₄₁×#391	C₂²	1	2	yes	—
48	C₄₁×#382	C₂²	1	2	yes	—
49	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
50	C₄₁×#343	C₂²	1	2	yes	—
51	C₄₁×#343	C₂²	1	2	yes	—
52	C₄₁×#339	C₂²	1	2	yes	—
53	C₂×#41×C₄₁	—	2	2	no	—
54	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
55	C₂×#41×C₄₁	C₂³	1	2	yes	—
56	C₂²×Q₈×C₄₁	D₄	1	2	yes	—
57	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
58	#137×C₄₁	—	2	2	no	—
59	#137×C₄₁	—	2	4	no	—
60	#137×C₄₁	—	2	2	no	—
61	#137×C₄₁	—	4	2	no	—
62	C₂×Q₁₆×C₄₁	—	2	1	no	—
63	C₂×Q₁₆×C₄₁	—	2	2	no	—
64	C₂×SD₁₆×C₄₁	—	2	1	no	—
65	C₂×SD₁₆×C₄₁	—	2	2	no	—
66	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
67	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₄₁	—	2	1	no	—
70	C₄⋊Q₈×C₄₁	—	2	1	no	—
71	C₄⋊Q₈×C₄₁	—	2	1	no	—
72	C₄⋊Q₈×C₄₁	—	2	1	no	—
73	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
74	#127×C₄₁	D₄	1	1	yes	characteristic
75	#126×C₄₁	—	2	4	no	—
76	#125×C₄₁	—	2	2	no	—
77	#124×C₄₁	—	2	2	no	—
78	#124×C₄₁	—	2	4	no	—
79	#124×C₄₁	D₄	1	2	yes	—
80	#123×C₄₁	—	2	4	no	—
81	C₂²⋊Q₈×C₄₁	—	2	2	no	—
82	C₂²⋊Q₈×C₄₁	—	2	4	no	—
83	C₂²⋊Q₈×C₄₁	—	2	2	no	—
84	C₂²⋊Q₈×C₄₁	—	2	2	no	—
85	#121×C₄₁	—	2	2	no	—
86	#121×C₄₁	—	2	2	no	—
87	#121×C₄₁	—	2	2	no	—
88	#121×C₄₁	—	2	1	no	—
89	#121×C₄₁	—	2	1	no	—
90	#121×C₄₁	—	2	4	no	—
91	C₄×Q₈×C₄₁	—	2	2	no	—
92	C₄×Q₈×C₄₁	—	2	2	no	—
93	C₄×Q₈×C₄₁	—	2	1	no	—
94	C₄×Q₈×C₄₁	—	2	1	no	—
95	C₄×Q₈×C₄₁	—	2	1	no	—
96	C₄×Q₈×C₄₁	—	2	1	no	—
97	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
98	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
99	C₄×D₄×C₄₁	—	2	2	no	—
100	C₄×D₄×C₄₁	—	2	2	no	—
101	C₄×D₄×C₄₁	—	2	1	no	—
102	C₄×D₄×C₄₁	—	2	1	no	—
103	C₄×D₄×C₄₁	—	2	4	no	—
104	C₄×D₄×C₄₁	—	2	1	no	—
105	C₄×D₄×C₄₁	—	2	1	no	—
106	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
107	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
108	#117×C₄₁	—	2	2	no	—
109	#117×C₄₁	—	2	2	no	—
110	#117×C₄₁	—	2	1	no	—
111	#117×C₄₁	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
113	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
114	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
115	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
116	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
118	C₂×C₄²×C₄₁	—	2	1	no	—
119	C₂×C₄²×C₄₁	—	2	1	no	—
120	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
121	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
122	#107×C₄₁	—	2	1	no	—
123	#106×C₄₁	—	2	1	no	—
124	C₄⋊C₈×C₄₁	—	2	1	no	—
125	C₄⋊C₈×C₄₁	—	2	1	no	—
126	#104×C₄₁	—	2	4	no	—
127	#104×C₄₁	—	4	2	no	—
128	#104×C₄₁	—	4	2	no	—
129	#103×C₄₁	—	2	1	no	—
130	#103×C₄₁	—	2	2	no	—
131	#103×C₄₁	—	2	1	no	—
132	#103×C₄₁	—	2	1	no	—
133	#102×C₄₁	—	2	1	no	—
134	#102×C₄₁	—	2	2	no	—
135	#97×C₄₁	—	2	1	no	—
136	C₄×C₈×C₄₁	—	2	1	no	—
137	#95×C₄₁	—	2	1	no	—
138	#95×C₄₁	—	2	4	no	—
139	#95×C₄₁	—	2	4	no	—
140	#41×C₄₁	—	2	2	no	—
141	#41×C₄₁	—	2	2	no	—
142	#41×C₄₁	—	2	1	no	—
143	#41×C₄₁	—	2	1	no	—
144	#41×C₄₁	—	2	1	no	—
145	#41×C₄₁	—	2	2	no	—
146	#41×C₄₁	—	2	2	no	—
147	#41×C₄₁	—	2	1	no	—
148	#41×C₄₁	—	8	4	no	—
149	C₂×Q₈×C₄₁	—	2	2	no	—
150	C₂×Q₈×C₄₁	—	2	2	no	—
151	C₂×Q₈×C₄₁	—	2	1	no	—
152	C₂×Q₈×C₄₁	—	2	1	no	—
153	C₂×Q₈×C₄₁	—	2	1	no	—
154	C₂×Q₈×C₄₁	—	2	1	no	—
155	C₂×Q₈×C₄₁	—	2	1	no	—
156	C₂×Q₈×C₄₁	—	4	4	no	—
157	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
158	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
159	C₂×Q₈×C₄₁	#41	1	2	yes	complemented
160	C₂×D₄×C₄₁	—	2	1	no	—
161	C₂×D₄×C₄₁	—	2	2	no	—
162	C₂×D₄×C₄₁	—	2	4	no	—
163	C₂×D₄×C₄₁	—	2	1	no	—
164	C₂×D₄×C₄₁	—	2	1	no	—
165	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
166	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₄₁	—	2	1	no	—
168	C₂²×C₄×C₄₁	—	2	4	no	—
169	C₂²×C₄×C₄₁	—	2	1	no	—
170	C₂²×C₄×C₄₁	—	2	1	no	—
171	C₂²×C₄×C₄₁	—	2	1	no	—
172	C₂²×C₄×C₄₁	—	2	2	no	—
173	C₂²×C₄×C₄₁	—	2	2	no	—
174	C₂²×C₄×C₄₁	—	2	1	no	—
175	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
177	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
179	Q₁₆×C₄₁	—	2	4	no	—
180	Q₁₆×C₄₁	—	4	4	no	—
181	SD₁₆×C₄₁	—	2	4	no	—
182	SD₁₆×C₄₁	—	4	4	no	—
183	M₁₆×C₄₁	—	2	1	no	—
184	M₁₆×C₄₁	—	2	2	no	—
185	M₁₆×C₄₁	—	2	1	no	—
186	M₁₆×C₄₁	—	2	2	no	—
187	M₁₆×C₄₁	—	2	1	no	—
188	M₁₆×C₄₁	#41	1	2	yes	complemented
189	C₂×C₈×C₄₁	—	2	1	no	—
190	C₂×C₈×C₄₁	—	2	1	no	—
191	C₂×C₈×C₄₁	—	2	2	no	—
192	C₄⋊C₄×C₄₁	—	2	4	no	—
193	C₄⋊C₄×C₄₁	—	2	4	no	—
194	C₄⋊C₄×C₄₁	—	2	2	no	—
195	C₄⋊C₄×C₄₁	—	2	2	no	—
196	C₄⋊C₄×C₄₁	—	2	1	no	—
197	C₄⋊C₄×C₄₁	—	2	1	no	—
198	C₄⋊C₄×C₄₁	—	2	1	no	—
199	C₄⋊C₄×C₄₁	—	2	1	no	—
200	C₄⋊C₄×C₄₁	—	2	2	no	—
201	C₄⋊C₄×C₄₁	—	2	2	no	—
202	C₄⋊C₄×C₄₁	—	2	1	no	—
203	C₄⋊C₄×C₄₁	—	2	1	no	—
204	C₄⋊C₄×C₄₁	—	4	2	no	—
205	C₄⋊C₄×C₄₁	—	4	1	no	—
206	C₄⋊C₄×C₄₁	—	4	1	no	—
207	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
208	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
209	C₂²⋊C₄×C₄₁	—	2	4	no	—
210	C₂²⋊C₄×C₄₁	—	2	4	no	—
211	C₂²⋊C₄×C₄₁	—	2	4	no	—
212	C₂²⋊C₄×C₄₁	—	2	1	no	—
213	C₂²⋊C₄×C₄₁	—	2	1	no	—
214	C₂²⋊C₄×C₄₁	—	2	1	no	—
215	C₂²⋊C₄×C₄₁	—	2	2	no	—
216	C₂²⋊C₄×C₄₁	—	2	2	no	—
217	C₄²×C₄₁	—	2	4	no	—
218	C₄²×C₄₁	—	2	1	no	—
219	C₄²×C₄₁	—	2	1	no	—
220	C₄²×C₄₁	—	2	1	no	—
221	C₄²×C₄₁	—	2	2	no	—
222	C₄²×C₄₁	—	2	2	no	—
223	C₄²×C₄₁	—	4	1	no	—
224	C₄²×C₄₁	—	4	1	no	—
225	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
226	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
227	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
228	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
229	C₂³×C₄₁	—	2	1	no	—
230	C₂³×C₄₁	—	2	2	no	—
231	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
232	Q₈×C₄₁	—	2	2	no	—
233	Q₈×C₄₁	—	2	1	no	—
234	Q₈×C₄₁	—	2	1	no	—
235	Q₈×C₄₁	—	2	2	no	—
236	Q₈×C₄₁	—	2	2	no	—
237	Q₈×C₄₁	—	2	2	no	—
238	Q₈×C₄₁	—	4	1	no	—
239	Q₈×C₄₁	—	4	1	no	—
240	Q₈×C₄₁	—	4	4	no	—
241	Q₈×C₄₁	—	8	2	no	—
242	D₄×C₄₁	—	2	1	no	—
243	D₄×C₄₁	—	2	1	no	—
244	D₄×C₄₁	—	2	2	no	—
245	D₄×C₄₁	—	2	2	no	—
246	D₄×C₄₁	—	4	2	no	—
247	D₄×C₄₁	—	4	2	no	—
248	D₄×C₄₁	—	4	1	no	—
249	D₄×C₄₁	—	4	2	no	—
250	D₄×C₄₁	—	4	1	no	—
251	D₄×C₄₁	—	4	2	no	—
252	D₄×C₄₁	—	8	4	no	—
253	C₂×C₄×C₄₁	—	2	2	no	—
254	C₂×C₄×C₄₁	—	2	1	no	—
255	C₂×C₄×C₄₁	—	2	2	no	—
256	C₂×C₄×C₄₁	—	2	1	no	—
257	C₂×C₄×C₄₁	—	2	1	no	—
258	C₂×C₄×C₄₁	—	2	4	no	—
259	C₂×C₄×C₄₁	—	2	2	no	—
260	C₂×C₄×C₄₁	—	4	1	no	—
261	C₂×C₄×C₄₁	—	4	2	no	—
262	C₂×C₄×C₄₁	—	4	1	no	—
263	C₂×C₄×C₄₁	—	4	1	no	—
264	C₂×C₄×C₄₁	—	4	2	no	—
265	C₂×C₄×C₄₁	—	4	2	no	—
266	C₂×C₄×C₄₁	—	4	1	no	—
267	C₂×C₄×C₄₁	—	4	1	no	—
268	C₂×C₄×C₄₁	—	4	2	no	—
269	C₂×C₄×C₄₁	—	4	4	no	—
270	C₂×C₄×C₄₁	—	4	2	no	—
271	C₂×C₄×C₄₁	—	4	2	no	—
272	C₂×C₄×C₄₁	—	4	2	no	—
273	C₂×C₄×C₄₁	—	4	1	no	—
274	C₂×C₄×C₄₁	—	4	2	no	—
275	C₂×C₄×C₄₁	—	8	4	no	—
276	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
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	C₂²×D₄	1	1	yes	characteristic
281	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
282	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
283	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
284	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
285	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
286	C₈×C₄₁	—	2	2	no	—
287	C₈×C₄₁	—	4	1	no	—
288	C₈×C₄₁	—	4	2	no	—
289	#24172	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	2	no	—
293	C₂²×C₄₁	—	4	1	no	—
294	C₂²×C₄₁	—	4	1	no	—
295	C₂²×C₄₁	—	4	2	no	—
296	C₂²×C₄₁	—	4	1	no	—
297	C₂²×C₄₁	—	4	2	no	—
298	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
299	C₂²×C₄₁	#548	1	1	yes	characteristic
300	C₂²×C₄₁	#550	1	1	yes	characteristic
301	C₄×C₄₁	—	2	1	no	—
302	C₄×C₄₁	—	2	1	no	—
303	C₄×C₄₁	—	2	1	no	—
304	C₄×C₄₁	—	4	2	no	—
305	C₄×C₄₁	—	4	1	no	—
306	C₄×C₄₁	—	4	2	no	—
307	C₄×C₄₁	—	4	1	no	—
308	C₄×C₄₁	—	4	4	no	—
309	C₄×C₄₁	—	4	1	no	—
310	C₄×C₄₁	—	8	2	no	—
311	C₄×C₄₁	—	8	1	no	—
312	C₄×C₄₁	#455	1	2	yes	—
313	C₄×C₄₁	#525	1	1	yes	characteristic
314	C₄×C₄₁	#537	1	1	yes	characteristic
315	C₄×C₄₁	#546	1	1	yes	characteristic
316	C₄×C₄₁	#547	1	1	yes	characteristic
317	#3511	C₂×C₄₁	1	1	yes	characteristic
318	#3037	C₂×C₄₁	1	1	yes	characteristic, complemented
319	C₄×#137	C₂×C₄₁	1	1	yes	characteristic, complemented
320	#2568	C₂×C₄₁	1	1	yes	characteristic, complemented
321	#2040	C₂×C₄₁	1	2	yes	complemented
322	#2007	C₂×C₄₁	1	2	yes	complemented
323	#1997	C₂×C₄₁	1	1	yes	characteristic, complemented
324	#1980	C₂×C₄₁	1	1	yes	characteristic, complemented
325	#1882	C₂×C₄₁	1	2	yes	complemented
326	#1860	C₂×C₄₁	1	2	yes	complemented
327	#1806	C₂×C₄₁	1	1	yes	characteristic, complemented
328	C₂×C₄₁	—	2	1	no	—
329	C₂×C₄₁	—	2	1	no	—
330	C₂×C₄₁	—	8	1	no	—
331	C₂×C₄₁	—	8	2	no	—
332	C₂×C₄₁	#2407	1	1	yes	characteristic
333	C₂×C₄₁	C₂×#455	1	1	yes	characteristic
334	C₂×C₄₁	#3020	1	1	yes	characteristic
335	C₂×#137	C₂²×C₄₁	1	1	yes	characteristic
336	C₂×#137	C₂²×C₄₁	1	2	yes	—
337	#556	—	2	1	no	—
338	#555	—	2	1	no	—
339	#554	—	2	1	no	—
340	#553	—	2	2	no	—
341	#548	—	2	2	no	—
342	#547	—	2	1	no	—
343	#542	—	2	2	no	—
344	#537	C₂²×C₄₁	1	2	yes	—
345	#533	—	2	1	no	—
346	#532	—	2	1	no	—
347	#532	C₂²×C₄₁	1	1	yes	characteristic
348	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
349	#520	C₂²×C₄₁	1	2	yes	—
350	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
351	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
352	#464	—	2	2	no	—
353	#462	—	2	1	no	—
354	#460	—	2	1	no	—
355	#441	—	2	1	no	—
356	#440	—	2	1	no	—
357	C₄×Q₁₆	—	2	1	no	—
358	C₄×SD₁₆	—	2	1	no	—
359	#428	C₂²×C₄₁	1	1	yes	characteristic
360	#423	C₂²×C₄₁	1	1	yes	characteristic
361	C₂×#104	C₂²×C₄₁	1	2	yes	—
362	C₂×#104	C₂²×C₄₁	1	2	yes	—
363	C₂×#104	C₂²×C₄₁	1	2	yes	—
364	#419	C₂²×C₄₁	1	1	yes	characteristic
365	#417	C₂²×C₄₁	1	2	yes	—
366	#417	C₂²×C₄₁	1	1	yes	characteristic
367	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
368	#400	—	2	2	no	—
369	#391	—	2	2	no	—
370	#391	C₂²×C₄₁	1	2	yes	—
371	#382	C₂²×C₄₁	1	2	yes	—
372	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
373	#343	C₂²×C₄₁	1	2	yes	—
374	#343	C₂²×C₄₁	1	2	yes	—
375	#339	C₂²×C₄₁	1	2	yes	—
376	C₄₁	#24172	1	1	yes	characteristic, direct product factor, Sylow
377	C₂×#41	—	2	2	no	—
378	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
379	C₂×#41	C₂³×C₄₁	1	2	yes	—
380	C₂²×Q₈	D₄×C₄₁	1	2	yes	—
381	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
382	#137	—	2	2	no	—
383	#137	—	2	4	no	—
384	#137	—	2	2	no	—
385	#137	—	4	2	no	—
386	C₂×Q₁₆	—	2	1	no	—
387	C₂×Q₁₆	—	2	2	no	—
388	C₂×SD₁₆	—	2	1	no	—
389	C₂×SD₁₆	—	2	2	no	—
390	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
391	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
392	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
393	C₄⋊Q₈	—	2	1	no	—
394	C₄⋊Q₈	—	2	1	no	—
395	C₄⋊Q₈	—	2	1	no	—
396	C₄⋊Q₈	—	2	1	no	—
397	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
398	#127	D₄×C₄₁	1	1	yes	characteristic
399	#126	—	2	4	no	—
400	#125	—	2	2	no	—
401	#124	—	2	2	no	—
402	#124	—	2	4	no	—
403	#124	D₄×C₄₁	1	2	yes	—
404	#123	—	2	4	no	—
405	C₂²⋊Q₈	—	2	2	no	—
406	C₂²⋊Q₈	—	2	4	no	—
407	C₂²⋊Q₈	—	2	2	no	—
408	C₂²⋊Q₈	—	2	2	no	—
409	#121	—	2	2	no	—
410	#121	—	2	2	no	—
411	#121	—	2	2	no	—
412	#121	—	2	1	no	—
413	#121	—	2	1	no	—
414	#121	—	2	4	no	—
415	C₄×Q₈	—	2	2	no	—
416	C₄×Q₈	—	2	2	no	—
417	C₄×Q₈	—	2	1	no	—
418	C₄×Q₈	—	2	1	no	—
419	C₄×Q₈	—	2	1	no	—
420	C₄×Q₈	—	2	1	no	—
421	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
422	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
423	C₄×D₄	—	2	2	no	—
424	C₄×D₄	—	2	2	no	—
425	C₄×D₄	—	2	1	no	—
426	C₄×D₄	—	2	1	no	—
427	C₄×D₄	—	2	4	no	—
428	C₄×D₄	—	2	1	no	—
429	C₄×D₄	—	2	1	no	—
430	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
431	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
432	#117	—	2	2	no	—
433	#117	—	2	2	no	—
434	#117	—	2	1	no	—
435	#117	C₂³×C₄₁	1	1	yes	characteristic
436	C₂×C₄⋊C₄	—	2	2	no	—
437	C₂×C₄⋊C₄	—	2	1	no	—
438	C₂×C₄⋊C₄	—	2	1	no	—
439	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
440	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
441	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
442	C₂×C₄²	—	2	1	no	—
443	C₂×C₄²	—	2	1	no	—
444	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
445	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
446	#107	—	2	1	no	—
447	#106	—	2	1	no	—
448	C₄⋊C₈	—	2	1	no	—
449	C₄⋊C₈	—	2	1	no	—
450	#104	—	2	4	no	—
451	#104	—	4	2	no	—
452	#104	—	4	2	no	—
453	#103	—	2	1	no	—
454	#103	—	2	2	no	—
455	#103	—	2	1	no	—
456	#103	—	2	1	no	—
457	#102	—	2	1	no	—
458	#102	—	2	2	no	—
459	#97	—	2	1	no	—
460	C₄×C₈	—	2	1	no	—
461	#95	—	2	1	no	—
462	#95	—	2	4	no	—
463	#95	—	2	4	no	—
464	#41	—	2	2	no	—
465	#41	—	2	2	no	—
466	#41	—	2	1	no	—
467	#41	—	2	1	no	—
468	#41	—	2	1	no	—
469	#41	—	2	2	no	—
470	#41	—	2	2	no	—
471	#41	—	2	1	no	—
472	#41	—	8	4	no	—
473	C₂×Q₈	—	2	2	no	—
474	C₂×Q₈	—	2	2	no	—
475	C₂×Q₈	—	2	1	no	—
476	C₂×Q₈	—	2	1	no	—
477	C₂×Q₈	—	2	1	no	—
478	C₂×Q₈	—	2	1	no	—
479	C₂×Q₈	—	2	1	no	—
480	C₂×Q₈	—	4	4	no	—
481	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
482	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
483	C₂×Q₈	#41×C₄₁	1	2	yes	complemented
484	C₂×D₄	—	2	1	no	—
485	C₂×D₄	—	2	2	no	—
486	C₂×D₄	—	2	4	no	—
487	C₂×D₄	—	2	1	no	—
488	C₂×D₄	—	2	1	no	—
489	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
490	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
491	C₂²×C₄	—	2	1	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	2	no	—
497	C₂²×C₄	—	2	2	no	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
500	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
501	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
502	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
503	Q₁₆	—	2	4	no	—
504	Q₁₆	—	4	4	no	—
505	SD₁₆	—	2	4	no	—
506	SD₁₆	—	4	4	no	—
507	M₁₆	—	2	1	no	—
508	M₁₆	—	2	2	no	—
509	M₁₆	—	2	1	no	—
510	M₁₆	—	2	2	no	—
511	M₁₆	—	2	1	no	—
512	M₁₆	#41×C₄₁	1	2	yes	complemented
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	2	no	—
516	C₄⋊C₄	—	2	4	no	—
517	C₄⋊C₄	—	2	4	no	—
518	C₄⋊C₄	—	2	2	no	—
519	C₄⋊C₄	—	2	2	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	1	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	2	no	—
525	C₄⋊C₄	—	2	2	no	—
526	C₄⋊C₄	—	2	1	no	—
527	C₄⋊C₄	—	2	1	no	—
528	C₄⋊C₄	—	4	2	no	—
529	C₄⋊C₄	—	4	1	no	—
530	C₄⋊C₄	—	4	1	no	—
531	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
532	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
533	C₂²⋊C₄	—	2	4	no	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₂²⋊C₄	—	2	1	no	—
537	C₂²⋊C₄	—	2	1	no	—
538	C₂²⋊C₄	—	2	1	no	—
539	C₂²⋊C₄	—	2	2	no	—
540	C₂²⋊C₄	—	2	2	no	—
541	C₄²	—	2	4	no	—
542	C₄²	—	2	1	no	—
543	C₄²	—	2	1	no	—
544	C₄²	—	2	1	no	—
545	C₄²	—	2	2	no	—
546	C₄²	—	2	2	no	—
547	C₄²	—	4	1	no	—
548	C₄²	—	4	1	no	—
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	C₂×D₄×C₄₁	1	1	yes	characteristic
553	C₂³	—	2	1	no	—
554	C₂³	—	2	2	no	—
555	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
556	Q₈	—	2	2	no	—
557	Q₈	—	2	1	no	—
558	Q₈	—	2	1	no	—
559	Q₈	—	2	2	no	—
560	Q₈	—	2	2	no	—
561	Q₈	—	2	2	no	—
562	Q₈	—	4	1	no	—
563	Q₈	—	4	1	no	—
564	Q₈	—	4	4	no	—
565	Q₈	—	8	2	no	—
566	D₄	—	2	1	no	—
567	D₄	—	2	1	no	—
568	D₄	—	2	2	no	—
569	D₄	—	2	2	no	—
570	D₄	—	4	2	no	—
571	D₄	—	4	2	no	—
572	D₄	—	4	1	no	—
573	D₄	—	4	2	no	—
574	D₄	—	4	1	no	—
575	D₄	—	4	2	no	—
576	D₄	—	8	4	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	1	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	1	no	—
581	C₂×C₄	—	2	1	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	2	2	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	4	1	no	—
592	C₂×C₄	—	4	2	no	—
593	C₂×C₄	—	4	4	no	—
594	C₂×C₄	—	4	2	no	—
595	C₂×C₄	—	4	2	no	—
596	C₂×C₄	—	4	2	no	—
597	C₂×C₄	—	4	1	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	8	4	no	—
600	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
601	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
602	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
603	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
604	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
605	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
606	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
607	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
608	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
609	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
610	C₈	—	2	2	no	—
611	C₈	—	4	1	no	—
612	C₈	—	4	2	no	—
613	C₂²	—	2	1	no	—
614	C₂²	—	2	1	no	—
615	C₂²	—	4	2	no	—
616	C₂²	—	4	1	no	—
617	C₂²	—	4	1	no	—
618	C₂²	—	4	2	no	—
619	C₂²	—	4	1	no	—
620	C₂²	—	4	2	no	—
621	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
622	C₂²	C₄₁×#548	1	1	yes	characteristic
623	C₂²	C₄₁×#550	1	1	yes	characteristic
624	C₄	—	2	1	no	—
625	C₄	—	2	1	no	—
626	C₄	—	2	1	no	—
627	C₄	—	4	2	no	—
628	C₄	—	4	1	no	—
629	C₄	—	4	2	no	—
630	C₄	—	4	1	no	—
631	C₄	—	4	4	no	—
632	C₄	—	4	1	no	—
633	C₄	—	8	2	no	—
634	C₄	—	8	1	no	—
635	C₄	C₄₁×#455	1	2	yes	—
636	C₄	C₄₁×#525	1	1	yes	characteristic
637	C₄	C₄₁×#537	1	1	yes	characteristic
638	C₄	C₄₁×#546	1	1	yes	characteristic
639	C₄	C₄₁×#547	1	1	yes	characteristic
640	C₂	—	2	1	no	—
641	C₂	—	2	1	no	—
642	C₂	—	8	1	no	—
643	C₂	—	8	2	no	—
644	C₂	C₄₁×#2407	1	1	yes	characteristic
645	C₂	C₂×C₄₁×#455	1	1	yes	characteristic
646	C₂	C₄₁×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#3511	C₂
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
2. C₄₁×#3037	C₂	semidirect
3. C₄×#137×C₄₁	C₂	semidirect
4. C₄₁×#2568	C₂	semidirect
5. C₄₁×#2040	C₂	semidirect
6. C₄₁×#2007	C₂	semidirect
7. C₄₁×#1997	C₂	semidirect
8. C₄₁×#1980	C₂	semidirect
9. C₄₁×#1882	C₂	semidirect
10. C₄₁×#1860	C₂	semidirect
11. C₄₁×#1806	C₂	semidirect
159. C₂×Q₈×C₄₁	#41	semidirect
188. M₁₆×C₄₁	#41	semidirect
289. #24172	C₄₁	direct (N × H)
318. #3037	C₂×C₄₁	semidirect
319. C₄×#137	C₂×C₄₁	semidirect
320. #2568	C₂×C₄₁	semidirect
321. #2040	C₂×C₄₁	semidirect
322. #2007	C₂×C₄₁	semidirect
323. #1997	C₂×C₄₁	semidirect
324. #1980	C₂×C₄₁	semidirect
325. #1882	C₂×C₄₁	semidirect
326. #1860	C₂×C₄₁	semidirect
327. #1806	C₂×C₄₁	semidirect
376. C₄₁	#24172	direct (N × H)
483. C₂×Q₈	#41×C₄₁	semidirect
512. M₁₆	#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×#3511	2
C₄₁×#3037	2
C₄×#137×C₄₁	2
C₄₁×#2568	2
C₄₁×#2040	2
C₄₁×#2007	2
C₄₁×#1997	2
C₄₁×#1980	2
C₄₁×#1882	2
C₄₁×#1860	2
C₄₁×#1806	2
#24172	41

Subgroups — C41×#24172 #23800373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24172 #23800373