Subgroups — C₄₁×#24178 #23800379

Order 10496.

1102 conjugacy classes of subgroups.

The subgroup lattice has 638 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
2	C₄₁×#3078	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3037	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#124×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2037	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2004	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×#1882	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×#137×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×#124×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#525	—	2	2	no	—
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×#479	—	2	2	no	—
29	C₄₁×#477	—	2	1	no	—
30	C₄₁×#475	—	2	1	no	—
31	C₄₁×#464	—	2	2	no	—
32	C₄₁×#462	—	2	1	no	—
33	C₄₁×#460	—	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×#419	C₂²	1	2	yes	—
40	C₄₁×#417	C₂²	1	2	yes	—
41	C₂×#100×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	—	2	2	no	—
52	C₂×#41×C₄₁	C₂³	1	2	yes	—
53	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₄₁	D₄	1	2	yes	—
56	#137×C₄₁	—	2	4	no	—
57	#137×C₄₁	—	4	2	no	—
58	C₂×Q₁₆×C₄₁	—	2	2	no	—
59	C₂×SD₁₆×C₄₁	—	2	2	no	—
60	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
61	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
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	—	2	2	no	—
66	#127×C₄₁	D₄	1	1	yes	characteristic
67	#126×C₄₁	—	2	4	no	—
68	#125×C₄₁	D₄	1	2	yes	—
69	#124×C₄₁	—	2	4	no	—
70	#124×C₄₁	—	2	2	no	—
71	#124×C₄₁	—	2	2	no	—
72	#124×C₄₁	—	2	2	no	—
73	#124×C₄₁	D₄	1	2	yes	—
74	#123×C₄₁	—	2	4	no	—
75	#123×C₄₁	—	2	4	no	—
76	C₂²⋊Q₈×C₄₁	—	2	2	no	—
77	C₂²⋊Q₈×C₄₁	—	2	1	no	—
78	C₂²⋊Q₈×C₄₁	—	2	1	no	—
79	#121×C₄₁	—	2	2	no	—
80	#121×C₄₁	—	2	4	no	—
81	#121×C₄₁	—	2	1	no	—
82	#121×C₄₁	—	2	1	no	—
83	#121×C₄₁	—	2	2	no	—
84	#121×C₄₁	—	2	2	no	—
85	#120×C₄₁	—	2	4	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	4	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	2	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	—	2	1	no	—
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	2	no	—
111	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
112	C₂×C₄²×C₄₁	—	2	1	no	—
113	C₂×C₄²×C₄₁	—	2	1	no	—
114	C₂×C₄²×C₄₁	—	2	1	no	—
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	#103×C₄₁	—	2	2	no	—
128	#103×C₄₁	—	2	2	no	—
129	#102×C₄₁	—	2	2	no	—
130	#100×C₄₁	—	4	2	no	—
131	#95×C₄₁	—	2	2	no	—
132	#95×C₄₁	—	2	2	no	—
133	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
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	#41×C₄₁	—	8	4	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×Q₈×C₄₁	#41	1	2	yes	complemented
148	C₂×D₄×C₄₁	—	2	2	no	—
149	C₂×D₄×C₄₁	—	2	4	no	—
150	C₂×D₄×C₄₁	—	2	4	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	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₄₁	—	2	1	no	—
161	C₂²×C₄×C₄₁	—	2	1	no	—
162	C₂²×C₄×C₄₁	—	2	1	no	—
163	C₂²×C₄×C₄₁	—	2	4	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	1	no	—
168	C₂²×C₄×C₄₁	—	4	4	no	—
169	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
173	Q₁₆×C₄₁	—	4	4	no	—
174	SD₁₆×C₄₁	—	4	4	no	—
175	M₁₆×C₄₁	—	2	2	no	—
176	M₁₆×C₄₁	—	2	2	no	—
177	M₁₆×C₄₁	—	2	2	no	—
178	M₁₆×C₄₁	—	2	2	no	—
179	C₂×C₈×C₄₁	—	2	2	no	—
180	C₂×C₈×C₄₁	—	2	2	no	—
181	C₄⋊C₄×C₄₁	—	2	4	no	—
182	C₄⋊C₄×C₄₁	—	2	1	no	—
183	C₄⋊C₄×C₄₁	—	2	1	no	—
184	C₄⋊C₄×C₄₁	—	2	2	no	—
185	C₄⋊C₄×C₄₁	—	2	1	no	—
186	C₄⋊C₄×C₄₁	—	2	2	no	—
187	C₄⋊C₄×C₄₁	—	2	1	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	—	4	1	no	—
193	C₄⋊C₄×C₄₁	—	4	1	no	—
194	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
195	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
196	C₂²⋊C₄×C₄₁	—	2	8	no	—
197	C₂²⋊C₄×C₄₁	—	2	4	no	—
198	C₂²⋊C₄×C₄₁	—	2	4	no	—
199	C₂²⋊C₄×C₄₁	—	2	4	no	—
200	C₂²⋊C₄×C₄₁	—	2	2	no	—
201	C₂²⋊C₄×C₄₁	—	2	2	no	—
202	C₂²⋊C₄×C₄₁	—	2	2	no	—
203	C₂²⋊C₄×C₄₁	—	2	2	no	—
204	C₂²⋊C₄×C₄₁	—	2	2	no	—
205	C₂²⋊C₄×C₄₁	—	4	4	no	—
206	C₄²×C₄₁	—	2	4	no	—
207	C₄²×C₄₁	—	2	2	no	—
208	C₄²×C₄₁	—	2	1	no	—
209	C₄²×C₄₁	—	2	1	no	—
210	C₄²×C₄₁	—	2	1	no	—
211	C₄²×C₄₁	—	4	2	no	—
212	C₄²×C₄₁	—	4	2	no	—
213	C₄²×C₄₁	—	4	1	no	—
214	C₄²×C₄₁	—	4	1	no	—
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	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₄₁	#41	1	2	yes	complemented
220	C₂³×C₄₁	—	2	2	no	—
221	C₂³×C₄₁	—	2	2	no	—
222	C₂³×C₄₁	—	2	2	no	—
223	C₂³×C₄₁	—	2	2	no	—
224	C₂³×C₄₁	—	4	2	no	—
225	C₂³×C₄₁	—	4	2	no	—
226	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
227	Q₈×C₄₁	—	2	2	no	—
228	Q₈×C₄₁	—	2	2	no	—
229	Q₈×C₄₁	—	2	2	no	—
230	Q₈×C₄₁	—	2	2	no	—
231	Q₈×C₄₁	—	4	1	no	—
232	Q₈×C₄₁	—	4	1	no	—
233	Q₈×C₄₁	—	4	2	no	—
234	Q₈×C₄₁	—	8	1	no	—
235	D₄×C₄₁	—	2	2	no	—
236	D₄×C₄₁	—	2	2	no	—
237	D₄×C₄₁	—	4	2	no	—
238	D₄×C₄₁	—	4	2	no	—
239	D₄×C₄₁	—	4	1	no	—
240	D₄×C₄₁	—	4	1	no	—
241	D₄×C₄₁	—	4	1	no	—
242	D₄×C₄₁	—	4	1	no	—
243	D₄×C₄₁	—	4	2	no	—
244	D₄×C₄₁	—	8	1	no	—
245	D₄×C₄₁	—	8	4	no	—
246	D₄×C₄₁	—	8	4	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	—	2	2	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	1	no	—
260	C₂×C₄×C₄₁	—	4	1	no	—
261	C₂×C₄×C₄₁	—	4	1	no	—
262	C₂×C₄×C₄₁	—	4	2	no	—
263	C₂×C₄×C₄₁	—	4	2	no	—
264	C₂×C₄×C₄₁	—	4	4	no	—
265	C₂×C₄×C₄₁	—	4	2	no	—
266	C₂×C₄×C₄₁	—	4	2	no	—
267	C₂×C₄×C₄₁	—	4	1	no	—
268	C₂×C₄×C₄₁	—	8	2	no	—
269	C₂×C₄×C₄₁	—	8	4	no	—
270	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
271	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
272	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
273	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
274	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
277	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
278	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
279	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
280	C₈×C₄₁	—	4	2	no	—
281	C₈×C₄₁	—	4	2	no	—
282	#24178	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
283	C₂²×C₄₁	—	2	1	no	—
284	C₂²×C₄₁	—	2	1	no	—
285	C₂²×C₄₁	—	4	4	no	—
286	C₂²×C₄₁	—	4	2	no	—
287	C₂²×C₄₁	—	4	2	no	—
288	C₂²×C₄₁	—	4	2	no	—
289	C₂²×C₄₁	—	4	2	no	—
290	C₂²×C₄₁	—	4	2	no	—
291	C₂²×C₄₁	—	4	2	no	—
292	C₂²×C₄₁	—	8	2	no	—
293	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
294	C₂²×C₄₁	#525	1	1	yes	characteristic
295	C₂²×C₄₁	#537	1	1	yes	characteristic
296	C₄×C₄₁	—	2	1	no	—
297	C₄×C₄₁	—	2	1	no	—
298	C₄×C₄₁	—	2	1	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	—	4	2	no	—
304	C₄×C₄₁	—	4	2	no	—
305	C₄×C₄₁	—	4	2	no	—
306	C₄×C₄₁	—	8	1	no	—
307	C₄×C₄₁	#455	1	2	yes	—
308	C₄×C₄₁	#546	1	1	yes	characteristic
309	C₄×C₄₁	#547	1	1	yes	characteristic
310	C₄×C₄₁	#548	1	1	yes	characteristic
311	C₄×C₄₁	#550	1	1	yes	characteristic
312	#3506	C₂×C₄₁	1	1	yes	characteristic
313	#3078	C₂×C₄₁	1	1	yes	characteristic, complemented
314	#3037	C₂×C₄₁	1	1	yes	characteristic, complemented
315	C₄×#124	C₂×C₄₁	1	1	yes	characteristic, complemented
316	#2037	C₂×C₄₁	1	2	yes	complemented
317	#2004	C₂×C₄₁	1	2	yes	complemented
318	#1994	C₂×C₄₁	1	1	yes	characteristic, complemented
319	#1962	C₂×C₄₁	1	1	yes	characteristic, complemented
320	#1882	C₂×C₄₁	1	2	yes	complemented
321	#1860	C₂×C₄₁	1	2	yes	complemented
322	#1800	C₂×C₄₁	1	1	yes	characteristic, complemented
323	C₂×C₄₁	—	2	1	no	—
324	C₂×C₄₁	—	2	1	no	—
325	C₂×C₄₁	—	8	2	no	—
326	C₂×C₄₁	—	8	2	no	—
327	C₂×C₄₁	#2407	1	1	yes	characteristic
328	C₂×C₄₁	C₂×#455	1	1	yes	characteristic
329	C₂×C₄₁	#3018	1	1	yes	characteristic
330	C₂×#137	C₂²×C₄₁	1	2	yes	—
331	#559	—	2	2	no	—
332	#556	—	2	1	no	—
333	#555	—	2	1	no	—
334	#553	—	2	2	no	—
335	#550	—	2	1	no	—
336	#547	—	2	1	no	—
337	#538	—	2	2	no	—
338	#535	C₂²×C₄₁	1	2	yes	—
339	#533	C₂²×C₄₁	1	1	yes	characteristic
340	#532	C₂²×C₄₁	1	1	yes	characteristic
341	C₂×#124	C₂²×C₄₁	1	1	yes	characteristic
342	#525	—	2	2	no	—
343	#520	C₂²×C₄₁	1	2	yes	—
344	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
345	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
346	#479	—	2	2	no	—
347	#477	—	2	1	no	—
348	#475	—	2	1	no	—
349	#464	—	2	2	no	—
350	#462	—	2	1	no	—
351	#460	—	2	1	no	—
352	#428	C₂²×C₄₁	1	2	yes	—
353	#423	C₂²×C₄₁	1	1	yes	characteristic
354	#423	C₂²×C₄₁	1	1	yes	characteristic
355	C₂×#104	C₂²×C₄₁	1	2	yes	—
356	C₂×#104	C₂²×C₄₁	1	2	yes	—
357	#419	C₂²×C₄₁	1	2	yes	—
358	#417	C₂²×C₄₁	1	2	yes	—
359	C₂×#100	C₂²×C₄₁	1	2	yes	—
360	#391	C₂²×C₄₁	1	1	yes	characteristic
361	#390	C₂²×C₄₁	1	1	yes	characteristic
362	#388	—	2	2	no	—
363	#382	C₂²×C₄₁	1	1	yes	characteristic
364	C₄×C₂²⋊C₄	—	2	2	no	—
365	C₄³	C₂²×C₄₁	1	1	yes	characteristic
366	#343	C₂²×C₄₁	1	2	yes	—
367	#339	C₂²×C₄₁	1	2	yes	—
368	#339	C₂²×C₄₁	1	2	yes	—
369	C₄₁	#24178	1	1	yes	characteristic, direct product factor, Sylow
370	C₂×#41	—	2	2	no	—
371	C₂×#41	C₂³×C₄₁	1	2	yes	—
372	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
373	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
374	C₂³×C₄	D₄×C₄₁	1	2	yes	—
375	#137	—	2	4	no	—
376	#137	—	4	2	no	—
377	C₂×Q₁₆	—	2	2	no	—
378	C₂×SD₁₆	—	2	2	no	—
379	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
380	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
381	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
382	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
383	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
384	#127	—	2	2	no	—
385	#127	D₄×C₄₁	1	1	yes	characteristic
386	#126	—	2	4	no	—
387	#125	D₄×C₄₁	1	2	yes	—
388	#124	—	2	4	no	—
389	#124	—	2	2	no	—
390	#124	—	2	2	no	—
391	#124	—	2	2	no	—
392	#124	D₄×C₄₁	1	2	yes	—
393	#123	—	2	4	no	—
394	#123	—	2	4	no	—
395	C₂²⋊Q₈	—	2	2	no	—
396	C₂²⋊Q₈	—	2	1	no	—
397	C₂²⋊Q₈	—	2	1	no	—
398	#121	—	2	2	no	—
399	#121	—	2	4	no	—
400	#121	—	2	1	no	—
401	#121	—	2	1	no	—
402	#121	—	2	2	no	—
403	#121	—	2	2	no	—
404	#120	—	2	4	no	—
405	C₄×Q₈	—	2	2	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	—	2	1	no	—
411	C₄×Q₈	—	2	1	no	—
412	C₄×D₄	—	2	4	no	—
413	C₄×D₄	—	2	2	no	—
414	C₄×D₄	—	2	4	no	—
415	C₄×D₄	—	2	1	no	—
416	C₄×D₄	—	2	1	no	—
417	C₄×D₄	—	2	1	no	—
418	C₄×D₄	—	2	1	no	—
419	C₄×D₄	—	2	1	no	—
420	C₄×D₄	—	2	1	no	—
421	#117	—	2	2	no	—
422	#117	—	2	2	no	—
423	#117	C₂³×C₄₁	1	2	yes	—
424	C₂×C₄⋊C₄	—	2	2	no	—
425	C₂×C₄⋊C₄	—	2	1	no	—
426	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
427	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
428	C₂×C₂²⋊C₄	—	2	2	no	—
429	C₂×C₂²⋊C₄	—	2	2	no	—
430	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
431	C₂×C₄²	—	2	1	no	—
432	C₂×C₄²	—	2	1	no	—
433	C₂×C₄²	—	2	1	no	—
434	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
435	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
436	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
437	#107	—	2	2	no	—
438	#106	—	2	2	no	—
439	C₄⋊C₈	—	2	1	no	—
440	C₄⋊C₈	—	2	1	no	—
441	C₄⋊C₈	—	2	1	no	—
442	C₄⋊C₈	—	2	1	no	—
443	#104	—	2	4	no	—
444	#104	—	4	2	no	—
445	#103	—	2	2	no	—
446	#103	—	2	2	no	—
447	#103	—	2	2	no	—
448	#102	—	2	2	no	—
449	#100	—	4	2	no	—
450	#95	—	2	2	no	—
451	#95	—	2	2	no	—
452	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
453	#41	—	2	2	no	—
454	#41	—	2	2	no	—
455	#41	—	2	2	no	—
456	#41	—	2	2	no	—
457	#41	—	8	4	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	—	2	1	no	—
463	C₂×Q₈	—	2	1	no	—
464	C₂×Q₈	—	2	1	no	—
465	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
466	C₂×Q₈	#41×C₄₁	1	2	yes	complemented
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	4	no	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	1	no	—
472	C₂×D₄	—	2	1	no	—
473	C₂×D₄	—	2	1	no	—
474	C₂×D₄	—	2	1	no	—
475	C₂×D₄	—	2	1	no	—
476	C₂×D₄	—	2	1	no	—
477	C₂×D₄	—	2	1	no	—
478	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	1	no	—
482	C₂²×C₄	—	2	4	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	2	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	4	4	no	—
488	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
489	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
490	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
491	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
492	Q₁₆	—	4	4	no	—
493	SD₁₆	—	4	4	no	—
494	M₁₆	—	2	2	no	—
495	M₁₆	—	2	2	no	—
496	M₁₆	—	2	2	no	—
497	M₁₆	—	2	2	no	—
498	C₂×C₈	—	2	2	no	—
499	C₂×C₈	—	2	2	no	—
500	C₄⋊C₄	—	2	4	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	2	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	4	1	no	—
512	C₄⋊C₄	—	4	1	no	—
513	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
514	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
515	C₂²⋊C₄	—	2	8	no	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	2	no	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	2	no	—
523	C₂²⋊C₄	—	2	2	no	—
524	C₂²⋊C₄	—	4	4	no	—
525	C₄²	—	2	4	no	—
526	C₄²	—	2	2	no	—
527	C₄²	—	2	1	no	—
528	C₄²	—	2	1	no	—
529	C₄²	—	2	1	no	—
530	C₄²	—	4	2	no	—
531	C₄²	—	4	2	no	—
532	C₄²	—	4	1	no	—
533	C₄²	—	4	1	no	—
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	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
538	C₄²	#41×C₄₁	1	2	yes	complemented
539	C₂³	—	2	2	no	—
540	C₂³	—	2	2	no	—
541	C₂³	—	2	2	no	—
542	C₂³	—	2	2	no	—
543	C₂³	—	4	2	no	—
544	C₂³	—	4	2	no	—
545	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
546	Q₈	—	2	2	no	—
547	Q₈	—	2	2	no	—
548	Q₈	—	2	2	no	—
549	Q₈	—	2	2	no	—
550	Q₈	—	4	1	no	—
551	Q₈	—	4	1	no	—
552	Q₈	—	4	2	no	—
553	Q₈	—	8	1	no	—
554	D₄	—	2	2	no	—
555	D₄	—	2	2	no	—
556	D₄	—	4	2	no	—
557	D₄	—	4	2	no	—
558	D₄	—	4	1	no	—
559	D₄	—	4	1	no	—
560	D₄	—	4	1	no	—
561	D₄	—	4	1	no	—
562	D₄	—	4	2	no	—
563	D₄	—	8	1	no	—
564	D₄	—	8	4	no	—
565	D₄	—	8	4	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	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	4	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	2	no	—
582	C₂×C₄	—	4	2	no	—
583	C₂×C₄	—	4	4	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	8	2	no	—
588	C₂×C₄	—	8	4	no	—
589	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
590	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
591	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
592	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
593	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
594	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
595	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
596	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
597	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
598	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
599	C₈	—	4	2	no	—
600	C₈	—	4	2	no	—
601	C₂²	—	2	1	no	—
602	C₂²	—	2	1	no	—
603	C₂²	—	4	4	no	—
604	C₂²	—	4	2	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	2	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	8	2	no	—
611	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
612	C₂²	C₄₁×#525	1	1	yes	characteristic
613	C₂²	C₄₁×#537	1	1	yes	characteristic
614	C₄	—	2	1	no	—
615	C₄	—	2	1	no	—
616	C₄	—	2	1	no	—
617	C₄	—	4	2	no	—
618	C₄	—	4	2	no	—
619	C₄	—	4	2	no	—
620	C₄	—	4	2	no	—
621	C₄	—	4	2	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	2	no	—
624	C₄	—	8	1	no	—
625	C₄	C₄₁×#455	1	2	yes	—
626	C₄	C₄₁×#546	1	1	yes	characteristic
627	C₄	C₄₁×#547	1	1	yes	characteristic
628	C₄	C₄₁×#548	1	1	yes	characteristic
629	C₄	C₄₁×#550	1	1	yes	characteristic
630	C₂	—	2	1	no	—
631	C₂	—	2	1	no	—
632	C₂	—	8	2	no	—
633	C₂	—	8	2	no	—
634	C₂	C₄₁×#2407	1	1	yes	characteristic
635	C₂	C₂×C₄₁×#455	1	1	yes	characteristic
636	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×#3506	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×#3078	C₂	semidirect
3. C₄₁×#3037	C₂	semidirect
4. C₄×#124×C₄₁	C₂	semidirect
5. C₄₁×#2037	C₂	semidirect
6. C₄₁×#2004	C₂	semidirect
7. C₄₁×#1994	C₂	semidirect
8. C₄₁×#1962	C₂	semidirect
9. C₄₁×#1882	C₂	semidirect
10. C₄₁×#1860	C₂	semidirect
11. C₄₁×#1800	C₂	semidirect
147. C₂×Q₈×C₄₁	#41	semidirect
219. C₄²×C₄₁	#41	semidirect
282. #24178	C₄₁	direct (N × H)
313. #3078	C₂×C₄₁	semidirect
314. #3037	C₂×C₄₁	semidirect
315. C₄×#124	C₂×C₄₁	semidirect
316. #2037	C₂×C₄₁	semidirect
317. #2004	C₂×C₄₁	semidirect
318. #1994	C₂×C₄₁	semidirect
319. #1962	C₂×C₄₁	semidirect
320. #1882	C₂×C₄₁	semidirect
321. #1860	C₂×C₄₁	semidirect
322. #1800	C₂×C₄₁	semidirect
369. C₄₁	#24178	direct (N × H)
466. C₂×Q₈	#41×C₄₁	semidirect
538. 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×#3078	2
C₄₁×#3037	2
C₄×#124×C₄₁	2
C₄₁×#2037	2
C₄₁×#2004	2
C₄₁×#1994	2
C₄₁×#1962	2
C₄₁×#1882	2
C₄₁×#1860	2
C₄₁×#1800	2
#24178	41

Subgroups — C41×#24178 #23800379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24178 #23800379