Subgroups — C₄₁×#24177 #23800378

Order 10496.

1070 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3507	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
4	C₄×#124×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2038	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2003	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1994	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1962	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×#1801	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
13	C₄₁×#559	—	2	1	no	—
14	C₄₁×#557	—	2	1	no	—
15	C₄₁×#556	—	2	1	no	—
16	C₄₁×#556	—	2	2	no	—
17	C₄₁×#554	—	2	1	no	—
18	C₄₁×#552	—	2	2	no	—
19	C₄₁×#539	—	2	2	no	—
20	C₄₁×#536	C₂²	1	2	yes	—
21	C₄₁×#533	C₂²	1	1	yes	characteristic
22	C₄₁×#532	C₂²	1	1	yes	characteristic
23	C₄₁×#529	—	2	2	no	—
24	C₂×#124×C₄₁	C₂²	1	1	yes	characteristic
25	C₄₁×#520	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×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×#101×C₄₁	C₂²	1	2	yes	—
42	C₄₁×#391	C₂²	1	1	yes	characteristic
43	C₄₁×#390	C₂²	1	1	yes	characteristic
44	C₄₁×#388	—	2	2	no	—
45	C₄₁×#382	C₂²	1	1	yes	characteristic
46	C₄×C₂²⋊C₄×C₄₁	—	2	2	no	—
47	C₄³×C₄₁	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	C₂³	1	2	yes	—
52	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₄₁	—	2	1	no	—
55	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
56	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
57	C₂×SD₁₆×C₄₁	—	2	2	no	—
58	C₂×D₈×C₄₁	—	2	2	no	—
59	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
60	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
61	C₄⋊Q₈×C₄₁	—	2	2	no	—
62	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
64	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
65	#127×C₄₁	D₄	1	1	yes	characteristic
66	#126×C₄₁	—	2	4	no	—
67	#126×C₄₁	—	2	4	no	—
68	#125×C₄₁	—	2	2	no	—
69	#125×C₄₁	—	2	4	no	—
70	#125×C₄₁	D₄	1	2	yes	—
71	#124×C₄₁	—	2	2	no	—
72	#124×C₄₁	—	2	2	no	—
73	#124×C₄₁	—	2	2	no	—
74	#124×C₄₁	—	2	2	no	—
75	#124×C₄₁	D₄	1	2	yes	—
76	#123×C₄₁	—	2	4	no	—
77	#123×C₄₁	—	2	4	no	—
78	C₂²⋊Q₈×C₄₁	—	2	4	no	—
79	C₂²⋊Q₈×C₄₁	—	2	1	no	—
80	C₂²⋊Q₈×C₄₁	—	2	1	no	—
81	C₂²⋊Q₈×C₄₁	—	2	2	no	—
82	C₂²⋊Q₈×C₄₁	—	2	2	no	—
83	#121×C₄₁	—	2	1	no	—
84	#121×C₄₁	—	2	1	no	—
85	C₄×Q₈×C₄₁	—	2	2	no	—
86	C₄×Q₈×C₄₁	—	2	2	no	—
87	C₄×Q₈×C₄₁	—	2	1	no	—
88	C₄×Q₈×C₄₁	—	2	1	no	—
89	C₄×Q₈×C₄₁	—	2	1	no	—
90	C₄×Q₈×C₄₁	—	2	1	no	—
91	C₄×Q₈×C₄₁	—	2	1	no	—
92	C₄×Q₈×C₄₁	—	2	1	no	—
93	C₄×D₄×C₄₁	—	2	2	no	—
94	C₄×D₄×C₄₁	—	2	2	no	—
95	C₄×D₄×C₄₁	—	2	4	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	#117×C₄₁	—	2	4	no	—
103	#117×C₄₁	—	2	2	no	—
104	#117×C₄₁	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
107	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
109	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
112	C₂×C₄²×C₄₁	—	2	1	no	—
113	C₂×C₄²×C₄₁	—	2	1	no	—
114	C₂×C₄²×C₄₁	D₄	1	2	yes	—
115	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
118	#107×C₄₁	—	2	2	no	—
119	#106×C₄₁	—	2	2	no	—
120	C₄⋊C₈×C₄₁	—	2	1	no	—
121	C₄⋊C₈×C₄₁	—	2	1	no	—
122	C₄⋊C₈×C₄₁	—	2	1	no	—
123	C₄⋊C₈×C₄₁	—	2	1	no	—
124	#104×C₄₁	—	2	4	no	—
125	#104×C₄₁	—	4	2	no	—
126	#103×C₄₁	—	2	2	no	—
127	#102×C₄₁	—	2	2	no	—
128	#102×C₄₁	—	2	2	no	—
129	#102×C₄₁	—	2	2	no	—
130	#101×C₄₁	—	4	2	no	—
131	#95×C₄₁	—	2	2	no	—
132	#95×C₄₁	—	2	2	no	—
133	C₂⁴×C₄₁	—	2	1	no	—
134	#41×C₄₁	—	2	2	no	—
135	#41×C₄₁	—	2	2	no	—
136	#41×C₄₁	—	2	2	no	—
137	#41×C₄₁	—	2	2	no	—
138	C₂×Q₈×C₄₁	—	2	1	no	—
139	C₂×Q₈×C₄₁	—	2	1	no	—
140	C₂×Q₈×C₄₁	—	2	1	no	—
141	C₂×Q₈×C₄₁	—	2	1	no	—
142	C₂×Q₈×C₄₁	—	2	1	no	—
143	C₂×Q₈×C₄₁	—	2	1	no	—
144	C₂×Q₈×C₄₁	—	2	1	no	—
145	C₂×Q₈×C₄₁	—	2	1	no	—
146	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
147	C₂×D₄×C₄₁	—	2	2	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	C₂×D₄	1	2	yes	—
156	C₂×D₄×C₄₁	#41	1	2	yes	—
157	C₂²×C₄×C₄₁	—	2	1	no	—
158	C₂²×C₄×C₄₁	—	2	1	no	—
159	C₂²×C₄×C₄₁	—	2	4	no	—
160	C₂²×C₄×C₄₁	—	2	2	no	—
161	C₂²×C₄×C₄₁	—	2	1	no	—
162	C₂²×C₄×C₄₁	—	4	4	no	—
163	C₂²×C₄×C₄₁	—	4	1	no	—
164	C₂²×C₄×C₄₁	—	4	1	no	—
165	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
170	SD₁₆×C₄₁	—	4	4	no	—
171	D₈×C₄₁	—	4	4	no	—
172	M₁₆×C₄₁	—	2	2	no	—
173	M₁₆×C₄₁	—	2	2	no	—
174	M₁₆×C₄₁	—	2	2	no	—
175	M₁₆×C₄₁	—	2	2	no	—
176	C₂×C₈×C₄₁	—	2	2	no	—
177	C₂×C₈×C₄₁	—	2	2	no	—
178	C₄⋊C₄×C₄₁	—	2	4	no	—
179	C₄⋊C₄×C₄₁	—	2	4	no	—
180	C₄⋊C₄×C₄₁	—	2	4	no	—
181	C₄⋊C₄×C₄₁	—	2	2	no	—
182	C₄⋊C₄×C₄₁	—	2	2	no	—
183	C₄⋊C₄×C₄₁	—	2	2	no	—
184	C₄⋊C₄×C₄₁	—	2	1	no	—
185	C₄⋊C₄×C₄₁	—	2	1	no	—
186	C₄⋊C₄×C₄₁	—	2	1	no	—
187	C₄⋊C₄×C₄₁	—	2	2	no	—
188	C₄⋊C₄×C₄₁	—	2	1	no	—
189	C₄⋊C₄×C₄₁	—	2	1	no	—
190	C₄⋊C₄×C₄₁	—	2	1	no	—
191	C₄⋊C₄×C₄₁	—	2	1	no	—
192	C₄⋊C₄×C₄₁	—	2	1	no	—
193	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
194	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
195	C₂²⋊C₄×C₄₁	—	2	8	no	—
196	C₂²⋊C₄×C₄₁	—	2	4	no	—
197	C₂²⋊C₄×C₄₁	—	2	4	no	—
198	C₂²⋊C₄×C₄₁	—	2	4	no	—
199	C₂²⋊C₄×C₄₁	—	2	2	no	—
200	C₂²⋊C₄×C₄₁	—	2	2	no	—
201	C₂²⋊C₄×C₄₁	—	2	2	no	—
202	C₂²⋊C₄×C₄₁	—	4	4	no	—
203	C₂²⋊C₄×C₄₁	—	4	1	no	—
204	C₂²⋊C₄×C₄₁	—	4	1	no	—
205	C₄²×C₄₁	—	2	4	no	—
206	C₄²×C₄₁	—	2	2	no	—
207	C₄²×C₄₁	—	2	2	no	—
208	C₄²×C₄₁	—	2	2	no	—
209	C₄²×C₄₁	—	2	1	no	—
210	C₄²×C₄₁	—	2	1	no	—
211	C₄²×C₄₁	—	2	1	no	—
212	C₄²×C₄₁	—	4	2	no	—
213	C₄²×C₄₁	—	4	2	no	—
214	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
215	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
218	C₄²×C₄₁	#41	1	2	yes	—
219	C₂³×C₄₁	—	2	2	no	—
220	C₂³×C₄₁	—	2	2	no	—
221	C₂³×C₄₁	—	4	2	no	—
222	C₂³×C₄₁	—	4	2	no	—
223	C₂³×C₄₁	—	4	1	no	—
224	C₂³×C₄₁	—	4	1	no	—
225	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
226	Q₈×C₄₁	—	2	2	no	—
227	Q₈×C₄₁	—	2	2	no	—
228	Q₈×C₄₁	—	4	1	no	—
229	Q₈×C₄₁	—	4	1	no	—
230	Q₈×C₄₁	—	4	1	no	—
231	Q₈×C₄₁	—	4	1	no	—
232	Q₈×C₄₁	—	4	2	no	—
233	Q₈×C₄₁	—	8	1	no	—
234	D₄×C₄₁	—	2	2	no	—
235	D₄×C₄₁	—	2	2	no	—
236	D₄×C₄₁	—	2	2	no	—
237	D₄×C₄₁	—	2	2	no	—
238	D₄×C₄₁	—	4	2	no	—
239	D₄×C₄₁	—	4	2	no	—
240	D₄×C₄₁	—	4	1	no	—
241	D₄×C₄₁	—	4	1	no	—
242	D₄×C₄₁	—	4	2	no	—
243	D₄×C₄₁	—	8	1	no	—
244	C₂×C₄×C₄₁	—	2	2	no	—
245	C₂×C₄×C₄₁	—	2	2	no	—
246	C₂×C₄×C₄₁	—	2	2	no	—
247	C₂×C₄×C₄₁	—	2	2	no	—
248	C₂×C₄×C₄₁	—	2	2	no	—
249	C₂×C₄×C₄₁	—	2	2	no	—
250	C₂×C₄×C₄₁	—	2	2	no	—
251	C₂×C₄×C₄₁	—	2	2	no	—
252	C₂×C₄×C₄₁	—	2	2	no	—
253	C₂×C₄×C₄₁	—	2	2	no	—
254	C₂×C₄×C₄₁	—	4	1	no	—
255	C₂×C₄×C₄₁	—	4	1	no	—
256	C₂×C₄×C₄₁	—	4	1	no	—
257	C₂×C₄×C₄₁	—	4	1	no	—
258	C₂×C₄×C₄₁	—	4	1	no	—
259	C₂×C₄×C₄₁	—	4	2	no	—
260	C₂×C₄×C₄₁	—	4	2	no	—
261	C₂×C₄×C₄₁	—	4	2	no	—
262	C₂×C₄×C₄₁	—	4	2	no	—
263	C₂×C₄×C₄₁	—	4	1	no	—
264	C₂×C₄×C₄₁	—	8	2	no	—
265	C₂×C₄×C₄₁	—	8	4	no	—
266	C₂×C₄×C₄₁	—	8	2	no	—
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	#120	1	1	yes	characteristic
271	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
275	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
276	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
277	C₈×C₄₁	—	4	2	no	—
278	C₈×C₄₁	—	4	2	no	—
279	#24177	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	4	no	—
283	C₂²×C₄₁	—	4	2	no	—
284	C₂²×C₄₁	—	4	2	no	—
285	C₂²×C₄₁	—	4	2	no	—
286	C₂²×C₄₁	—	4	2	no	—
287	C₂²×C₄₁	—	8	2	no	—
288	C₂²×C₄₁	—	8	1	no	—
289	C₂²×C₄₁	—	8	1	no	—
290	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
291	C₂²×C₄₁	#525	1	1	yes	characteristic
292	C₂²×C₄₁	#537	1	1	yes	characteristic
293	C₄×C₄₁	—	2	1	no	—
294	C₄×C₄₁	—	2	1	no	—
295	C₄×C₄₁	—	2	1	no	—
296	C₄×C₄₁	—	4	2	no	—
297	C₄×C₄₁	—	4	2	no	—
298	C₄×C₄₁	—	4	2	no	—
299	C₄×C₄₁	—	4	2	no	—
300	C₄×C₄₁	—	4	2	no	—
301	C₄×C₄₁	—	4	2	no	—
302	C₄×C₄₁	—	4	2	no	—
303	C₄×C₄₁	—	8	1	no	—
304	C₄×C₄₁	#454	1	2	yes	—
305	C₄×C₄₁	#546	1	1	yes	characteristic
306	C₄×C₄₁	#547	1	1	yes	characteristic
307	C₄×C₄₁	#548	1	1	yes	characteristic
308	C₄×C₄₁	#550	1	1	yes	characteristic
309	#3507	C₂×C₄₁	1	1	yes	characteristic, complemented
310	#3077	C₂×C₄₁	1	1	yes	characteristic, complemented
311	#3036	C₂×C₄₁	1	1	yes	characteristic
312	C₄×#124	C₂×C₄₁	1	1	yes	characteristic, complemented
313	#2038	C₂×C₄₁	1	2	yes	complemented
314	#2003	C₂×C₄₁	1	2	yes	complemented
315	#1994	C₂×C₄₁	1	1	yes	characteristic, complemented
316	#1962	C₂×C₄₁	1	1	yes	characteristic, complemented
317	#1883	C₂×C₄₁	1	2	yes	complemented
318	#1860	C₂×C₄₁	1	2	yes	complemented
319	#1801	C₂×C₄₁	1	1	yes	characteristic, complemented
320	C₂×C₄₁	—	2	1	no	—
321	C₂×C₄₁	—	2	1	no	—
322	C₂×C₄₁	—	8	2	no	—
323	C₂×C₄₁	—	8	2	no	—
324	C₂×C₄₁	#2407	1	1	yes	characteristic
325	C₂×C₄₁	C₂×#454	1	1	yes	characteristic
326	C₂×C₄₁	#3018	1	1	yes	characteristic
327	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
328	#559	—	2	1	no	—
329	#557	—	2	1	no	—
330	#556	—	2	1	no	—
331	#556	—	2	2	no	—
332	#554	—	2	1	no	—
333	#552	—	2	2	no	—
334	#539	—	2	2	no	—
335	#536	C₂²×C₄₁	1	2	yes	—
336	#533	C₂²×C₄₁	1	1	yes	characteristic
337	#532	C₂²×C₄₁	1	1	yes	characteristic
338	#529	—	2	2	no	—
339	C₂×#124	C₂²×C₄₁	1	1	yes	characteristic
340	#520	C₂²×C₄₁	1	2	yes	—
341	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
342	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
343	#478	—	2	2	no	—
344	#476	—	2	1	no	—
345	#474	—	2	1	no	—
346	#463	—	2	2	no	—
347	#461	—	2	1	no	—
348	#459	—	2	1	no	—
349	#428	C₂²×C₄₁	1	2	yes	—
350	#423	C₂²×C₄₁	1	1	yes	characteristic
351	#423	C₂²×C₄₁	1	1	yes	characteristic
352	C₂×#104	C₂²×C₄₁	1	2	yes	—
353	C₂×#104	C₂²×C₄₁	1	2	yes	—
354	#418	C₂²×C₄₁	1	2	yes	—
355	#417	C₂²×C₄₁	1	2	yes	—
356	C₂×#101	C₂²×C₄₁	1	2	yes	—
357	#391	C₂²×C₄₁	1	1	yes	characteristic
358	#390	C₂²×C₄₁	1	1	yes	characteristic
359	#388	—	2	2	no	—
360	#382	C₂²×C₄₁	1	1	yes	characteristic
361	C₄×C₂²⋊C₄	—	2	2	no	—
362	C₄³	C₂²×C₄₁	1	1	yes	characteristic
363	#343	C₂²×C₄₁	1	2	yes	—
364	#339	C₂²×C₄₁	1	2	yes	—
365	#339	C₂²×C₄₁	1	2	yes	—
366	C₄₁	#24177	1	1	yes	characteristic, direct product factor, Sylow
367	C₂×#41	C₂³×C₄₁	1	2	yes	—
368	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
369	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
370	C₂³×C₄	—	2	1	no	—
371	AGL(1,ℤ/8)	—	2	4	no	—
372	AGL(1,ℤ/8)	—	4	2	no	—
373	C₂×SD₁₆	—	2	2	no	—
374	C₂×D₈	—	2	2	no	—
375	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
376	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
377	C₄⋊Q₈	—	2	2	no	—
378	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
379	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
380	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
381	#127	D₄×C₄₁	1	1	yes	characteristic
382	#126	—	2	4	no	—
383	#126	—	2	4	no	—
384	#125	—	2	2	no	—
385	#125	—	2	4	no	—
386	#125	D₄×C₄₁	1	2	yes	—
387	#124	—	2	2	no	—
388	#124	—	2	2	no	—
389	#124	—	2	2	no	—
390	#124	—	2	2	no	—
391	#124	D₄×C₄₁	1	2	yes	—
392	#123	—	2	4	no	—
393	#123	—	2	4	no	—
394	C₂²⋊Q₈	—	2	4	no	—
395	C₂²⋊Q₈	—	2	1	no	—
396	C₂²⋊Q₈	—	2	1	no	—
397	C₂²⋊Q₈	—	2	2	no	—
398	C₂²⋊Q₈	—	2	2	no	—
399	#121	—	2	1	no	—
400	#121	—	2	1	no	—
401	C₄×Q₈	—	2	2	no	—
402	C₄×Q₈	—	2	2	no	—
403	C₄×Q₈	—	2	1	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×D₄	—	2	2	no	—
410	C₄×D₄	—	2	2	no	—
411	C₄×D₄	—	2	4	no	—
412	C₄×D₄	—	2	1	no	—
413	C₄×D₄	—	2	1	no	—
414	C₄×D₄	—	2	1	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	—	2	1	no	—
418	#117	—	2	4	no	—
419	#117	—	2	2	no	—
420	#117	C₂³×C₄₁	1	2	yes	—
421	C₂×C₄⋊C₄	—	2	2	no	—
422	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
423	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
424	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
425	C₂×C₂²⋊C₄	—	2	2	no	—
426	C₂×C₂²⋊C₄	—	2	1	no	—
427	C₂×C₂²⋊C₄	—	2	2	no	—
428	C₂×C₄²	—	2	1	no	—
429	C₂×C₄²	—	2	1	no	—
430	C₂×C₄²	D₄×C₄₁	1	2	yes	—
431	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
433	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
434	#107	—	2	2	no	—
435	#106	—	2	2	no	—
436	C₄⋊C₈	—	2	1	no	—
437	C₄⋊C₈	—	2	1	no	—
438	C₄⋊C₈	—	2	1	no	—
439	C₄⋊C₈	—	2	1	no	—
440	#104	—	2	4	no	—
441	#104	—	4	2	no	—
442	#103	—	2	2	no	—
443	#102	—	2	2	no	—
444	#102	—	2	2	no	—
445	#102	—	2	2	no	—
446	#101	—	4	2	no	—
447	#95	—	2	2	no	—
448	#95	—	2	2	no	—
449	C₂⁴	—	2	1	no	—
450	#41	—	2	2	no	—
451	#41	—	2	2	no	—
452	#41	—	2	2	no	—
453	#41	—	2	2	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	—	2	1	no	—
457	C₂×Q₈	—	2	1	no	—
458	C₂×Q₈	—	2	1	no	—
459	C₂×Q₈	—	2	1	no	—
460	C₂×Q₈	—	2	1	no	—
461	C₂×Q₈	—	2	1	no	—
462	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	2	1	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	2	1	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
472	C₂×D₄	#41×C₄₁	1	2	yes	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	4	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	1	no	—
478	C₂²×C₄	—	4	4	no	—
479	C₂²×C₄	—	4	1	no	—
480	C₂²×C₄	—	4	1	no	—
481	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
482	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
484	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
486	SD₁₆	—	4	4	no	—
487	D₈	—	4	4	no	—
488	M₁₆	—	2	2	no	—
489	M₁₆	—	2	2	no	—
490	M₁₆	—	2	2	no	—
491	M₁₆	—	2	2	no	—
492	C₂×C₈	—	2	2	no	—
493	C₂×C₈	—	2	2	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	4	no	—
497	C₄⋊C₄	—	2	2	no	—
498	C₄⋊C₄	—	2	2	no	—
499	C₄⋊C₄	—	2	2	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	2	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
510	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
511	C₂²⋊C₄	—	2	8	no	—
512	C₂²⋊C₄	—	2	4	no	—
513	C₂²⋊C₄	—	2	4	no	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	2	2	no	—
517	C₂²⋊C₄	—	2	2	no	—
518	C₂²⋊C₄	—	4	4	no	—
519	C₂²⋊C₄	—	4	1	no	—
520	C₂²⋊C₄	—	4	1	no	—
521	C₄²	—	2	4	no	—
522	C₄²	—	2	2	no	—
523	C₄²	—	2	2	no	—
524	C₄²	—	2	2	no	—
525	C₄²	—	2	1	no	—
526	C₄²	—	2	1	no	—
527	C₄²	—	2	1	no	—
528	C₄²	—	4	2	no	—
529	C₄²	—	4	2	no	—
530	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
531	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
532	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
533	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
534	C₄²	#41×C₄₁	1	2	yes	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	2	no	—
537	C₂³	—	4	2	no	—
538	C₂³	—	4	2	no	—
539	C₂³	—	4	1	no	—
540	C₂³	—	4	1	no	—
541	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
542	Q₈	—	2	2	no	—
543	Q₈	—	2	2	no	—
544	Q₈	—	4	1	no	—
545	Q₈	—	4	1	no	—
546	Q₈	—	4	1	no	—
547	Q₈	—	4	1	no	—
548	Q₈	—	4	2	no	—
549	Q₈	—	8	1	no	—
550	D₄	—	2	2	no	—
551	D₄	—	2	2	no	—
552	D₄	—	2	2	no	—
553	D₄	—	2	2	no	—
554	D₄	—	4	2	no	—
555	D₄	—	4	2	no	—
556	D₄	—	4	1	no	—
557	D₄	—	4	1	no	—
558	D₄	—	4	2	no	—
559	D₄	—	8	1	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	2	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	2	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	2	no	—
570	C₂×C₄	—	4	1	no	—
571	C₂×C₄	—	4	1	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	1	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	2	no	—
576	C₂×C₄	—	4	2	no	—
577	C₂×C₄	—	4	2	no	—
578	C₂×C₄	—	4	2	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	8	2	no	—
581	C₂×C₄	—	8	4	no	—
582	C₂×C₄	—	8	2	no	—
583	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
584	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
585	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
586	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
587	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
588	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
589	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
590	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
591	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
592	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
593	C₈	—	4	2	no	—
594	C₈	—	4	2	no	—
595	C₂²	—	2	1	no	—
596	C₂²	—	2	1	no	—
597	C₂²	—	4	4	no	—
598	C₂²	—	4	2	no	—
599	C₂²	—	4	2	no	—
600	C₂²	—	4	2	no	—
601	C₂²	—	4	2	no	—
602	C₂²	—	8	2	no	—
603	C₂²	—	8	1	no	—
604	C₂²	—	8	1	no	—
605	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
606	C₂²	C₄₁×#525	1	1	yes	characteristic
607	C₂²	C₄₁×#537	1	1	yes	characteristic
608	C₄	—	2	1	no	—
609	C₄	—	2	1	no	—
610	C₄	—	2	1	no	—
611	C₄	—	4	2	no	—
612	C₄	—	4	2	no	—
613	C₄	—	4	2	no	—
614	C₄	—	4	2	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	2	no	—
617	C₄	—	4	2	no	—
618	C₄	—	8	1	no	—
619	C₄	C₄₁×#454	1	2	yes	—
620	C₄	C₄₁×#546	1	1	yes	characteristic
621	C₄	C₄₁×#547	1	1	yes	characteristic
622	C₄	C₄₁×#548	1	1	yes	characteristic
623	C₄	C₄₁×#550	1	1	yes	characteristic
624	C₂	—	2	1	no	—
625	C₂	—	2	1	no	—
626	C₂	—	8	2	no	—
627	C₂	—	8	2	no	—
628	C₂	C₄₁×#2407	1	1	yes	characteristic
629	C₂	C₂×C₄₁×#454	1	1	yes	characteristic
630	C₂	C₄₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#3036	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
1. C₄₁×#3507	C₂	semidirect
2. C₄₁×#3077	C₂	semidirect
4. C₄×#124×C₄₁	C₂	semidirect
5. C₄₁×#2038	C₂	semidirect
6. C₄₁×#2003	C₂	semidirect
7. C₄₁×#1994	C₂	semidirect
8. C₄₁×#1962	C₂	semidirect
9. C₄₁×#1883	C₂	semidirect
10. C₄₁×#1860	C₂	semidirect
11. C₄₁×#1801	C₂	semidirect
279. #24177	C₄₁	direct (N × H)
309. #3507	C₂×C₄₁	semidirect
310. #3077	C₂×C₄₁	semidirect
312. C₄×#124	C₂×C₄₁	semidirect
313. #2038	C₂×C₄₁	semidirect
314. #2003	C₂×C₄₁	semidirect
315. #1994	C₂×C₄₁	semidirect
316. #1962	C₂×C₄₁	semidirect
317. #1883	C₂×C₄₁	semidirect
318. #1860	C₂×C₄₁	semidirect
319. #1801	C₂×C₄₁	semidirect
366. C₄₁	#24177	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×#3507	2
C₄₁×#3077	2
C₄₁×#3036	2
C₄×#124×C₄₁	2
C₄₁×#2038	2
C₄₁×#2003	2
C₄₁×#1994	2
C₄₁×#1962	2
C₄₁×#1883	2
C₄₁×#1860	2
C₄₁×#1801	2
#24177	41

Subgroups — C41×#24177 #23800378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24177 #23800378