Subgroups — C₄₁×#24147 #23800348

Order 10496.

1006 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3509	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#3078	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#137×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2546	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2038	C₂	1	2	yes	maximal
6	C₄₁×#2005	C₂	1	2	yes	maximal
7	C₄₁×#1986	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1966	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1886	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1857	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1806	C₂	1	1	yes	maximal, characteristic
12	C₂×#137×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#557	—	2	1	no	—
14	C₄₁×#557	—	2	1	no	—
15	C₄₁×#556	—	2	1	no	—
16	C₄₁×#556	—	2	1	no	—
17	C₄₁×#552	—	2	2	no	—
18	C₄₁×#552	—	2	2	no	—
19	C₄₁×#541	—	2	2	no	—
20	C₄₁×#536	C₂²	1	2	yes	—
21	C₄₁×#533	—	2	1	no	—
22	C₄₁×#533	—	2	1	no	—
23	C₄₁×#533	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
25	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₄₁×#519	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×#479	—	2	2	no	—
30	C₄₁×#477	—	2	1	no	—
31	C₄₁×#475	—	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	2	yes	—
37	C₄₁×#428	C₂²	1	1	yes	characteristic
38	C₄₁×#423	C₂²	1	1	yes	characteristic
39	C₂×#104×C₄₁	C₂²	1	2	yes	—
40	C₄₁×#419	C₂²	1	2	yes	—
41	C₄₁×#419	C₂²	1	1	yes	characteristic
42	C₄₁×#417	C₂²	1	1	yes	characteristic
43	C₂×#101×C₄₁	C₂²	1	2	yes	—
44	C₂×#101×C₄₁	C₂²	1	2	yes	—
45	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
46	C₄₁×#398	—	2	2	no	—
47	C₄₁×#391	C₂²	1	2	yes	—
48	C₄₁×#389	—	2	2	no	—
49	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
50	C₄₁×#339	C₂²	1	2	yes	—
51	C₄₁×#339	C₂²	1	2	yes	—
52	C₄₁×#339	C₂²	1	2	yes	—
53	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
54	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
56	#137×C₄₁	—	2	2	no	—
57	#137×C₄₁	—	2	2	no	—
58	C₂×Q₁₆×C₄₁	—	2	1	no	—
59	C₂×SD₁₆×C₄₁	—	2	1	no	—
60	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
62	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
63	C₄⋊Q₈×C₄₁	—	2	2	no	—
64	C₄⋊Q₈×C₄₁	—	2	4	no	—
65	C₄⋊Q₈×C₄₁	—	2	1	no	—
66	C₄⋊Q₈×C₄₁	—	2	1	no	—
67	C₄⋊Q₈×C₄₁	—	2	1	no	—
68	C₄⋊Q₈×C₄₁	—	2	1	no	—
69	C₄⋊Q₈×C₄₁	—	4	2	no	—
70	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
71	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
72	#126×C₄₁	—	2	4	no	—
73	#126×C₄₁	—	2	4	no	—
74	#125×C₄₁	—	2	2	no	—
75	#125×C₄₁	—	2	2	no	—
76	#125×C₄₁	—	2	4	no	—
77	#125×C₄₁	—	2	4	no	—
78	#125×C₄₁	D₄	1	2	yes	—
79	#124×C₄₁	—	2	2	no	—
80	#123×C₄₁	—	2	4	no	—
81	C₂²⋊Q₈×C₄₁	—	2	2	no	—
82	C₂²⋊Q₈×C₄₁	—	2	2	no	—
83	C₂²⋊Q₈×C₄₁	—	2	1	no	—
84	C₂²⋊Q₈×C₄₁	—	2	1	no	—
85	C₂²⋊Q₈×C₄₁	—	2	2	no	—
86	C₂²⋊Q₈×C₄₁	—	2	2	no	—
87	C₄×Q₈×C₄₁	—	2	2	no	—
88	C₄×Q₈×C₄₁	—	2	2	no	—
89	C₄×Q₈×C₄₁	—	2	2	no	—
90	C₄×Q₈×C₄₁	—	2	2	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	—	2	1	no	—
98	C₄×Q₈×C₄₁	—	2	1	no	—
99	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
100	C₄×Q₈×C₄₁	D₄	1	1	yes	characteristic
101	C₄×D₄×C₄₁	—	2	1	no	—
102	C₄×D₄×C₄₁	—	2	1	no	—
103	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
104	C₄×D₄×C₄₁	D₄	1	1	yes	characteristic
105	#117×C₄₁	—	2	2	no	—
106	#117×C₄₁	—	2	2	no	—
107	#117×C₄₁	—	2	4	no	—
108	#117×C₄₁	—	2	1	no	—
109	#117×C₄₁	C₂³	1	1	yes	characteristic
110	#117×C₄₁	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
112	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
113	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
114	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
115	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
116	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₄₁	—	2	1	no	—
118	C₂×C₄²×C₄₁	—	2	1	no	—
119	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
120	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
121	#107×C₄₁	—	2	1	no	—
122	#107×C₄₁	—	2	2	no	—
123	#106×C₄₁	—	2	1	no	—
124	#106×C₄₁	—	2	2	no	—
125	C₄⋊C₈×C₄₁	—	2	1	no	—
126	C₄⋊C₈×C₄₁	—	2	1	no	—
127	#104×C₄₁	—	4	2	no	—
128	#103×C₄₁	—	2	1	no	—
129	#103×C₄₁	—	2	1	no	—
130	#103×C₄₁	—	2	1	no	—
131	#103×C₄₁	—	2	2	no	—
132	#103×C₄₁	—	2	2	no	—
133	#102×C₄₁	—	2	1	no	—
134	#101×C₄₁	—	2	4	no	—
135	#101×C₄₁	—	4	2	no	—
136	#97×C₄₁	—	2	1	no	—
137	C₄×C₈×C₄₁	—	2	1	no	—
138	#95×C₄₁	—	2	1	no	—
139	#95×C₄₁	—	2	4	no	—
140	#41×C₄₁	—	2	1	no	—
141	#41×C₄₁	—	2	1	no	—
142	#41×C₄₁	—	2	1	no	—
143	#41×C₄₁	—	2	1	no	—
144	C₂×Q₈×C₄₁	—	2	1	no	—
145	C₂×Q₈×C₄₁	—	2	1	no	—
146	C₂×Q₈×C₄₁	—	2	2	no	—
147	C₂×Q₈×C₄₁	—	2	2	no	—
148	C₂×Q₈×C₄₁	—	2	2	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	2	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	—	2	1	no	—
157	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
158	C₂×Q₈×C₄₁	#41	1	2	yes	—
159	C₂×D₄×C₄₁	—	2	1	no	—
160	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₄₁	—	2	2	no	—
162	C₂²×C₄×C₄₁	—	2	1	no	—
163	C₂²×C₄×C₄₁	—	2	1	no	—
164	C₂²×C₄×C₄₁	—	2	1	no	—
165	C₂²×C₄×C₄₁	—	2	1	no	—
166	C₂²×C₄×C₄₁	—	2	1	no	—
167	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
168	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
171	Q₁₆×C₄₁	—	2	4	no	—
172	SD₁₆×C₄₁	—	2	4	no	—
173	M₁₆×C₄₁	—	2	1	no	—
174	M₁₆×C₄₁	—	2	2	no	—
175	M₁₆×C₄₁	—	2	1	no	—
176	M₁₆×C₄₁	—	2	2	no	—
177	M₁₆×C₄₁	—	2	1	no	—
178	M₁₆×C₄₁	#41	1	2	yes	—
179	C₂×C₈×C₄₁	—	2	1	no	—
180	C₂×C₈×C₄₁	—	2	1	no	—
181	C₂×C₈×C₄₁	—	2	2	no	—
182	C₄⋊C₄×C₄₁	—	2	4	no	—
183	C₄⋊C₄×C₄₁	—	2	4	no	—
184	C₄⋊C₄×C₄₁	—	2	4	no	—
185	C₄⋊C₄×C₄₁	—	2	4	no	—
186	C₄⋊C₄×C₄₁	—	2	4	no	—
187	C₄⋊C₄×C₄₁	—	2	2	no	—
188	C₄⋊C₄×C₄₁	—	2	2	no	—
189	C₄⋊C₄×C₄₁	—	2	2	no	—
190	C₄⋊C₄×C₄₁	—	2	2	no	—
191	C₄⋊C₄×C₄₁	—	2	1	no	—
192	C₄⋊C₄×C₄₁	—	2	1	no	—
193	C₄⋊C₄×C₄₁	—	2	1	no	—
194	C₄⋊C₄×C₄₁	—	2	1	no	—
195	C₄⋊C₄×C₄₁	—	2	2	no	—
196	C₄⋊C₄×C₄₁	—	2	2	no	—
197	C₄⋊C₄×C₄₁	—	2	1	no	—
198	C₄⋊C₄×C₄₁	—	2	1	no	—
199	C₄⋊C₄×C₄₁	—	4	8	no	—
200	C₄⋊C₄×C₄₁	—	4	1	no	—
201	C₄⋊C₄×C₄₁	—	4	1	no	—
202	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
203	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
204	C₄⋊C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
205	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
206	C₂²⋊C₄×C₄₁	—	2	4	no	—
207	C₂²⋊C₄×C₄₁	—	2	4	no	—
208	C₂²⋊C₄×C₄₁	—	2	1	no	—
209	C₂²⋊C₄×C₄₁	—	2	2	no	—
210	C₂²⋊C₄×C₄₁	—	2	1	no	—
211	C₂²⋊C₄×C₄₁	—	2	1	no	—
212	C₄²×C₄₁	—	2	4	no	—
213	C₄²×C₄₁	—	2	4	no	—
214	C₄²×C₄₁	—	2	1	no	—
215	C₄²×C₄₁	—	2	1	no	—
216	C₄²×C₄₁	—	2	1	no	—
217	C₄²×C₄₁	—	2	2	no	—
218	C₄²×C₄₁	—	2	2	no	—
219	C₄²×C₄₁	—	2	2	no	—
220	C₄²×C₄₁	—	4	1	no	—
221	C₄²×C₄₁	—	4	1	no	—
222	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
223	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
224	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
225	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
226	C₂³×C₄₁	—	2	1	no	—
227	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
228	Q₈×C₄₁	—	2	2	no	—
229	Q₈×C₄₁	—	2	1	no	—
230	Q₈×C₄₁	—	2	1	no	—
231	Q₈×C₄₁	—	2	2	no	—
232	Q₈×C₄₁	—	4	2	no	—
233	Q₈×C₄₁	—	4	2	no	—
234	Q₈×C₄₁	—	4	2	no	—
235	Q₈×C₄₁	—	4	1	no	—
236	Q₈×C₄₁	—	4	2	no	—
237	Q₈×C₄₁	—	4	2	no	—
238	Q₈×C₄₁	—	4	1	no	—
239	Q₈×C₄₁	—	4	2	no	—
240	D₄×C₄₁	—	2	1	no	—
241	D₄×C₄₁	—	2	1	no	—
242	D₄×C₄₁	—	4	1	no	—
243	D₄×C₄₁	—	4	1	no	—
244	C₂×C₄×C₄₁	—	2	2	no	—
245	C₂×C₄×C₄₁	—	2	4	no	—
246	C₂×C₄×C₄₁	—	2	1	no	—
247	C₂×C₄×C₄₁	—	2	1	no	—
248	C₂×C₄×C₄₁	—	2	2	no	—
249	C₂×C₄×C₄₁	—	2	2	no	—
250	C₂×C₄×C₄₁	—	2	4	no	—
251	C₂×C₄×C₄₁	—	2	1	no	—
252	C₂×C₄×C₄₁	—	2	2	no	—
253	C₂×C₄×C₄₁	—	2	4	no	—
254	C₂×C₄×C₄₁	—	2	4	no	—
255	C₂×C₄×C₄₁	—	4	1	no	—
256	C₂×C₄×C₄₁	—	4	2	no	—
257	C₂×C₄×C₄₁	—	4	2	no	—
258	C₂×C₄×C₄₁	—	4	1	no	—
259	C₂×C₄×C₄₁	—	4	2	no	—
260	C₂×C₄×C₄₁	—	4	1	no	—
261	C₂×C₄×C₄₁	—	4	1	no	—
262	C₂×C₄×C₄₁	—	4	1	no	—
263	C₂×C₄×C₄₁	—	4	2	no	—
264	C₂×C₄×C₄₁	—	4	2	no	—
265	C₂×C₄×C₄₁	—	4	1	no	—
266	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
267	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
268	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
269	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
270	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
275	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
276	C₈×C₄₁	—	2	2	no	—
277	C₈×C₄₁	—	4	1	no	—
278	C₈×C₄₁	—	4	2	no	—
279	#24147	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂²×C₄₁	—	2	1	no	—
281	C₂²×C₄₁	—	2	1	no	—
282	C₂²×C₄₁	—	4	1	no	—
283	C₂²×C₄₁	—	4	1	no	—
284	C₂²×C₄₁	—	4	1	no	—
285	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
286	C₂²×C₄₁	#546	1	1	yes	characteristic
287	C₂²×C₄₁	#547	1	1	yes	characteristic
288	C₄×C₄₁	—	2	1	no	—
289	C₄×C₄₁	—	2	1	no	—
290	C₄×C₄₁	—	2	1	no	—
291	C₄×C₄₁	—	4	4	no	—
292	C₄×C₄₁	—	4	4	no	—
293	C₄×C₄₁	—	4	1	no	—
294	C₄×C₄₁	—	4	2	no	—
295	C₄×C₄₁	—	4	1	no	—
296	C₄×C₄₁	—	4	2	no	—
297	C₄×C₄₁	—	4	1	no	—
298	C₄×C₄₁	—	4	2	no	—
299	C₄×C₄₁	—	8	1	no	—
300	C₄×C₄₁	#456	1	2	yes	—
301	C₄×C₄₁	#525	1	1	yes	characteristic
302	C₄×C₄₁	#537	1	1	yes	characteristic
303	C₄×C₄₁	#548	1	1	yes	characteristic
304	C₄×C₄₁	#550	1	1	yes	characteristic
305	#3509	C₂×C₄₁	1	1	yes	characteristic
306	#3078	C₂×C₄₁	1	1	yes	characteristic, complemented
307	C₄×#137	C₂×C₄₁	1	1	yes	characteristic
308	#2546	C₂×C₄₁	1	1	yes	characteristic, complemented
309	#2038	C₂×C₄₁	1	2	yes	—
310	#2005	C₂×C₄₁	1	2	yes	—
311	#1986	C₂×C₄₁	1	1	yes	characteristic, complemented
312	#1966	C₂×C₄₁	1	1	yes	characteristic, complemented
313	#1886	C₂×C₄₁	1	2	yes	complemented
314	#1857	C₂×C₄₁	1	2	yes	complemented
315	#1806	C₂×C₄₁	1	1	yes	characteristic
316	C₂×C₄₁	—	2	1	no	—
317	C₂×C₄₁	—	2	1	no	—
318	C₂×C₄₁	—	8	1	no	—
319	C₂×C₄₁	#2407	1	1	yes	characteristic
320	C₂×C₄₁	C₂×#456	1	1	yes	characteristic
321	C₂×C₄₁	#3020	1	1	yes	characteristic
322	C₂×#137	C₂²×C₄₁	1	1	yes	characteristic
323	#557	—	2	1	no	—
324	#557	—	2	1	no	—
325	#556	—	2	1	no	—
326	#556	—	2	1	no	—
327	#552	—	2	2	no	—
328	#552	—	2	2	no	—
329	#541	—	2	2	no	—
330	#536	C₂²×C₄₁	1	2	yes	—
331	#533	—	2	1	no	—
332	#533	—	2	1	no	—
333	#533	C₂²×C₄₁	1	1	yes	characteristic
334	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
335	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
336	#519	C₂²×C₄₁	1	2	yes	—
337	C₄×#41	C₂²×C₄₁	1	1	yes	characteristic
338	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
339	#479	—	2	2	no	—
340	#477	—	2	1	no	—
341	#475	—	2	1	no	—
342	#441	—	2	1	no	—
343	#440	—	2	1	no	—
344	C₄×Q₁₆	—	2	1	no	—
345	C₄×SD₁₆	—	2	1	no	—
346	#428	C₂²×C₄₁	1	2	yes	—
347	#428	C₂²×C₄₁	1	1	yes	characteristic
348	#423	C₂²×C₄₁	1	1	yes	characteristic
349	C₂×#104	C₂²×C₄₁	1	2	yes	—
350	#419	C₂²×C₄₁	1	2	yes	—
351	#419	C₂²×C₄₁	1	1	yes	characteristic
352	#417	C₂²×C₄₁	1	1	yes	characteristic
353	C₂×#101	C₂²×C₄₁	1	2	yes	—
354	C₂×#101	C₂²×C₄₁	1	2	yes	—
355	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
356	#398	—	2	2	no	—
357	#391	C₂²×C₄₁	1	2	yes	—
358	#389	—	2	2	no	—
359	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
360	#339	C₂²×C₄₁	1	2	yes	—
361	#339	C₂²×C₄₁	1	2	yes	—
362	#339	C₂²×C₄₁	1	2	yes	—
363	C₄₁	#24147	1	1	yes	characteristic, direct product factor, Sylow
364	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
365	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
366	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
367	#137	—	2	2	no	—
368	#137	—	2	2	no	—
369	C₂×Q₁₆	—	2	1	no	—
370	C₂×SD₁₆	—	2	1	no	—
371	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
372	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
373	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
374	C₄⋊Q₈	—	2	2	no	—
375	C₄⋊Q₈	—	2	4	no	—
376	C₄⋊Q₈	—	2	1	no	—
377	C₄⋊Q₈	—	2	1	no	—
378	C₄⋊Q₈	—	2	1	no	—
379	C₄⋊Q₈	—	2	1	no	—
380	C₄⋊Q₈	—	4	2	no	—
381	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
382	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
383	#126	—	2	4	no	—
384	#126	—	2	4	no	—
385	#125	—	2	2	no	—
386	#125	—	2	2	no	—
387	#125	—	2	4	no	—
388	#125	—	2	4	no	—
389	#125	D₄×C₄₁	1	2	yes	—
390	#124	—	2	2	no	—
391	#123	—	2	4	no	—
392	C₂²⋊Q₈	—	2	2	no	—
393	C₂²⋊Q₈	—	2	2	no	—
394	C₂²⋊Q₈	—	2	1	no	—
395	C₂²⋊Q₈	—	2	1	no	—
396	C₂²⋊Q₈	—	2	2	no	—
397	C₂²⋊Q₈	—	2	2	no	—
398	C₄×Q₈	—	2	2	no	—
399	C₄×Q₈	—	2	2	no	—
400	C₄×Q₈	—	2	2	no	—
401	C₄×Q₈	—	2	2	no	—
402	C₄×Q₈	—	2	2	no	—
403	C₄×Q₈	—	2	2	no	—
404	C₄×Q₈	—	2	1	no	—
405	C₄×Q₈	—	2	1	no	—
406	C₄×Q₈	—	2	1	no	—
407	C₄×Q₈	—	2	1	no	—
408	C₄×Q₈	—	2	1	no	—
409	C₄×Q₈	—	2	1	no	—
410	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
411	C₄×Q₈	D₄×C₄₁	1	1	yes	characteristic
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
415	C₄×D₄	D₄×C₄₁	1	1	yes	characteristic
416	#117	—	2	2	no	—
417	#117	—	2	2	no	—
418	#117	—	2	4	no	—
419	#117	—	2	1	no	—
420	#117	C₂³×C₄₁	1	1	yes	characteristic
421	#117	C₂³×C₄₁	1	2	yes	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄⋊C₄	—	2	1	no	—
424	C₂×C₄⋊C₄	—	2	4	no	—
425	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
426	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
427	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
428	C₂×C₄²	—	2	1	no	—
429	C₂×C₄²	—	2	1	no	—
430	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
431	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
432	#107	—	2	1	no	—
433	#107	—	2	2	no	—
434	#106	—	2	1	no	—
435	#106	—	2	2	no	—
436	C₄⋊C₈	—	2	1	no	—
437	C₄⋊C₈	—	2	1	no	—
438	#104	—	4	2	no	—
439	#103	—	2	1	no	—
440	#103	—	2	1	no	—
441	#103	—	2	1	no	—
442	#103	—	2	2	no	—
443	#103	—	2	2	no	—
444	#102	—	2	1	no	—
445	#101	—	2	4	no	—
446	#101	—	4	2	no	—
447	#97	—	2	1	no	—
448	C₄×C₈	—	2	1	no	—
449	#95	—	2	1	no	—
450	#95	—	2	4	no	—
451	#41	—	2	1	no	—
452	#41	—	2	1	no	—
453	#41	—	2	1	no	—
454	#41	—	2	1	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	—	2	1	no	—
457	C₂×Q₈	—	2	2	no	—
458	C₂×Q₈	—	2	2	no	—
459	C₂×Q₈	—	2	2	no	—
460	C₂×Q₈	—	2	2	no	—
461	C₂×Q₈	—	2	2	no	—
462	C₂×Q₈	—	2	1	no	—
463	C₂×Q₈	—	2	2	no	—
464	C₂×Q₈	—	2	1	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	—	2	1	no	—
468	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
469	C₂×Q₈	#41×C₄₁	1	2	yes	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
472	C₂²×C₄	—	2	2	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	1	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
479	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
480	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
481	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
482	Q₁₆	—	2	4	no	—
483	SD₁₆	—	2	4	no	—
484	M₁₆	—	2	1	no	—
485	M₁₆	—	2	2	no	—
486	M₁₆	—	2	1	no	—
487	M₁₆	—	2	2	no	—
488	M₁₆	—	2	1	no	—
489	M₁₆	#41×C₄₁	1	2	yes	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	2	no	—
493	C₄⋊C₄	—	2	4	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	4	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	2	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	4	8	no	—
511	C₄⋊C₄	—	4	1	no	—
512	C₄⋊C₄	—	4	1	no	—
513	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
514	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
515	C₄⋊C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
516	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	1	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	1	no	—
522	C₂²⋊C₄	—	2	1	no	—
523	C₄²	—	2	4	no	—
524	C₄²	—	2	4	no	—
525	C₄²	—	2	1	no	—
526	C₄²	—	2	1	no	—
527	C₄²	—	2	1	no	—
528	C₄²	—	2	2	no	—
529	C₄²	—	2	2	no	—
530	C₄²	—	2	2	no	—
531	C₄²	—	4	1	no	—
532	C₄²	—	4	1	no	—
533	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
534	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
535	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
536	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
537	C₂³	—	2	1	no	—
538	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
539	Q₈	—	2	2	no	—
540	Q₈	—	2	1	no	—
541	Q₈	—	2	1	no	—
542	Q₈	—	2	2	no	—
543	Q₈	—	4	2	no	—
544	Q₈	—	4	2	no	—
545	Q₈	—	4	2	no	—
546	Q₈	—	4	1	no	—
547	Q₈	—	4	2	no	—
548	Q₈	—	4	2	no	—
549	Q₈	—	4	1	no	—
550	Q₈	—	4	2	no	—
551	D₄	—	2	1	no	—
552	D₄	—	2	1	no	—
553	D₄	—	4	1	no	—
554	D₄	—	4	1	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	2	1	no	—
558	C₂×C₄	—	2	1	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	4	2	no	—
568	C₂×C₄	—	4	2	no	—
569	C₂×C₄	—	4	1	no	—
570	C₂×C₄	—	4	2	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	2	no	—
575	C₂×C₄	—	4	2	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
578	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
579	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
580	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
581	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
582	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
583	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
584	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
585	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
586	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
587	C₈	—	2	2	no	—
588	C₈	—	4	1	no	—
589	C₈	—	4	2	no	—
590	C₂²	—	2	1	no	—
591	C₂²	—	2	1	no	—
592	C₂²	—	4	1	no	—
593	C₂²	—	4	1	no	—
594	C₂²	—	4	1	no	—
595	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
596	C₂²	C₄₁×#546	1	1	yes	characteristic
597	C₂²	C₄₁×#547	1	1	yes	characteristic
598	C₄	—	2	1	no	—
599	C₄	—	2	1	no	—
600	C₄	—	2	1	no	—
601	C₄	—	4	4	no	—
602	C₄	—	4	4	no	—
603	C₄	—	4	1	no	—
604	C₄	—	4	2	no	—
605	C₄	—	4	1	no	—
606	C₄	—	4	2	no	—
607	C₄	—	4	1	no	—
608	C₄	—	4	2	no	—
609	C₄	—	8	1	no	—
610	C₄	C₄₁×#456	1	2	yes	—
611	C₄	C₄₁×#525	1	1	yes	characteristic
612	C₄	C₄₁×#537	1	1	yes	characteristic
613	C₄	C₄₁×#548	1	1	yes	characteristic
614	C₄	C₄₁×#550	1	1	yes	characteristic
615	C₂	—	2	1	no	—
616	C₂	—	2	1	no	—
617	C₂	—	8	1	no	—
618	C₂	C₄₁×#2407	1	1	yes	characteristic
619	C₂	C₂×C₄₁×#456	1	1	yes	characteristic
620	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	C₂×#41×C₄₁	C₂³
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×#3078	C₂	semidirect
4. C₄₁×#2546	C₂	semidirect
7. C₄₁×#1986	C₂	semidirect
8. C₄₁×#1966	C₂	semidirect
9. C₄₁×#1886	C₂	semidirect
10. C₄₁×#1857	C₂	semidirect
279. #24147	C₄₁	direct (N × H)
306. #3078	C₂×C₄₁	semidirect
308. #2546	C₂×C₄₁	semidirect
311. #1986	C₂×C₄₁	semidirect
312. #1966	C₂×C₄₁	semidirect
313. #1886	C₂×C₄₁	semidirect
314. #1857	C₂×C₄₁	semidirect
363. C₄₁	#24147	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×#3509	2
C₄₁×#3078	2
C₄×#137×C₄₁	2
C₄₁×#2546	2
C₄₁×#2038	2
C₄₁×#2005	2
C₄₁×#1986	2
C₄₁×#1966	2
C₄₁×#1886	2
C₄₁×#1857	2
C₄₁×#1806	2
#24147	41

Subgroups — C41×#24147 #23800348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24147 #23800348