Subgroups — C₄₁×#24179 #23800380

Order 10496.

1038 conjugacy classes of subgroups.

The subgroup lattice has 612 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
2	C₄₁×#3078	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3037	C₂	1	1	yes	maximal, characteristic
4	C₄×C₄⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2038	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2005	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
9	C₄₁×#1882	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1859	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1801	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×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	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#529	—	2	2	no	—
25	C₄₁×#520	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×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×#101×C₄₁	C₂²	1	2	yes	—
42	C₄₁×#391	C₂²	1	1	yes	characteristic
43	C₄₁×#391	C₂²	1	1	yes	characteristic
44	C₄₁×#389	—	2	2	no	—
45	C₄₁×#384	C₂²	1	1	yes	characteristic
46	C₄×C₄⋊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×Q₈×C₄₁	C₂³	1	1	yes	characteristic
54	#137×C₄₁	—	2	4	no	—
55	#137×C₄₁	—	4	2	no	—
56	C₂×Q₁₆×C₄₁	—	2	2	no	—
57	C₂×SD₁₆×C₄₁	—	2	2	no	—
58	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
59	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
60	C₄⋊Q₈×C₄₁	—	2	2	no	—
61	C₄⋊Q₈×C₄₁	—	2	2	no	—
62	C₄⋊Q₈×C₄₁	—	2	2	no	—
63	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
64	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
65	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
66	#127×C₄₁	D₄	1	1	yes	characteristic
67	#126×C₄₁	—	2	4	no	—
68	#126×C₄₁	—	2	4	no	—
69	#125×C₄₁	—	2	2	no	—
70	#125×C₄₁	—	2	4	no	—
71	#125×C₄₁	D₄	1	2	yes	—
72	#124×C₄₁	—	2	2	no	—
73	#124×C₄₁	—	2	2	no	—
74	#124×C₄₁	D₄	1	2	yes	—
75	#123×C₄₁	—	2	4	no	—
76	#123×C₄₁	—	2	4	no	—
77	C₂²⋊Q₈×C₄₁	—	2	4	no	—
78	C₂²⋊Q₈×C₄₁	—	2	1	no	—
79	C₂²⋊Q₈×C₄₁	—	2	1	no	—
80	C₂²⋊Q₈×C₄₁	—	2	2	no	—
81	C₂²⋊Q₈×C₄₁	—	2	2	no	—
82	#121×C₄₁	—	2	1	no	—
83	#121×C₄₁	—	2	1	no	—
84	C₄×Q₈×C₄₁	—	2	2	no	—
85	C₄×Q₈×C₄₁	—	2	2	no	—
86	C₄×Q₈×C₄₁	—	2	1	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×Q₈×C₄₁	—	2	1	no	—
94	C₄×Q₈×C₄₁	—	2	1	no	—
95	C₄×Q₈×C₄₁	—	2	1	no	—
96	C₄×D₄×C₄₁	—	2	2	no	—
97	C₄×D₄×C₄₁	—	2	2	no	—
98	C₄×D₄×C₄₁	—	2	4	no	—
99	C₄×D₄×C₄₁	—	2	1	no	—
100	C₄×D₄×C₄₁	—	2	1	no	—
101	#117×C₄₁	—	2	4	no	—
102	#117×C₄₁	—	2	2	no	—
103	#117×C₄₁	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
107	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
108	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
109	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₄₁	—	2	1	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	#103×C₄₁	—	2	2	no	—
128	#103×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	#41×C₄₁	—	2	2	no	—
134	#41×C₄₁	—	2	2	no	—
135	#41×C₄₁	—	2	2	no	—
136	#41×C₄₁	—	2	2	no	—
137	C₂×Q₈×C₄₁	—	2	1	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	—	2	1	no	—
147	C₂×Q₈×C₄₁	—	2	1	no	—
148	C₂×Q₈×C₄₁	—	2	1	no	—
149	C₂×Q₈×C₄₁	—	2	1	no	—
150	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
151	C₂×Q₈×C₄₁	#41	1	2	yes	—
152	C₂×D₄×C₄₁	—	2	2	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	C₂²×C₄×C₄₁	—	2	1	no	—
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	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
167	Q₁₆×C₄₁	—	4	4	no	—
168	SD₁₆×C₄₁	—	4	4	no	—
169	M₁₆×C₄₁	—	2	2	no	—
170	M₁₆×C₄₁	—	2	2	no	—
171	M₁₆×C₄₁	—	2	2	no	—
172	M₁₆×C₄₁	—	2	2	no	—
173	C₂×C₈×C₄₁	—	2	2	no	—
174	C₂×C₈×C₄₁	—	2	2	no	—
175	C₄⋊C₄×C₄₁	—	2	4	no	—
176	C₄⋊C₄×C₄₁	—	2	8	no	—
177	C₄⋊C₄×C₄₁	—	2	4	no	—
178	C₄⋊C₄×C₄₁	—	2	4	no	—
179	C₄⋊C₄×C₄₁	—	2	2	no	—
180	C₄⋊C₄×C₄₁	—	2	2	no	—
181	C₄⋊C₄×C₄₁	—	2	2	no	—
182	C₄⋊C₄×C₄₁	—	2	1	no	—
183	C₄⋊C₄×C₄₁	—	2	1	no	—
184	C₄⋊C₄×C₄₁	—	2	1	no	—
185	C₄⋊C₄×C₄₁	—	2	2	no	—
186	C₄⋊C₄×C₄₁	—	2	1	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	—	4	4	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	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	C₄²×C₄₁	—	2	4	no	—
203	C₄²×C₄₁	—	2	4	no	—
204	C₄²×C₄₁	—	2	2	no	—
205	C₄²×C₄₁	—	2	2	no	—
206	C₄²×C₄₁	—	2	2	no	—
207	C₄²×C₄₁	—	2	1	no	—
208	C₄²×C₄₁	—	2	1	no	—
209	C₄²×C₄₁	—	2	1	no	—
210	C₄²×C₄₁	—	4	2	no	—
211	C₄²×C₄₁	—	4	1	no	—
212	C₄²×C₄₁	—	4	1	no	—
213	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
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	#41	1	2	yes	—
218	C₂³×C₄₁	—	2	2	no	—
219	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
220	Q₈×C₄₁	—	2	2	no	—
221	Q₈×C₄₁	—	2	2	no	—
222	Q₈×C₄₁	—	2	2	no	—
223	Q₈×C₄₁	—	2	2	no	—
224	Q₈×C₄₁	—	4	1	no	—
225	Q₈×C₄₁	—	4	1	no	—
226	Q₈×C₄₁	—	4	1	no	—
227	Q₈×C₄₁	—	4	1	no	—
228	Q₈×C₄₁	—	4	1	no	—
229	Q₈×C₄₁	—	4	1	no	—
230	Q₈×C₄₁	—	4	2	no	—
231	Q₈×C₄₁	—	8	1	no	—
232	D₄×C₄₁	—	2	2	no	—
233	D₄×C₄₁	—	2	2	no	—
234	D₄×C₄₁	—	4	2	no	—
235	D₄×C₄₁	—	4	2	no	—
236	D₄×C₄₁	—	4	2	no	—
237	D₄×C₄₁	—	8	1	no	—
238	C₂×C₄×C₄₁	—	2	2	no	—
239	C₂×C₄×C₄₁	—	2	2	no	—
240	C₂×C₄×C₄₁	—	2	2	no	—
241	C₂×C₄×C₄₁	—	2	2	no	—
242	C₂×C₄×C₄₁	—	2	2	no	—
243	C₂×C₄×C₄₁	—	2	2	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	—	4	1	no	—
252	C₂×C₄×C₄₁	—	4	1	no	—
253	C₂×C₄×C₄₁	—	4	1	no	—
254	C₂×C₄×C₄₁	—	4	1	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	2	no	—
259	C₂×C₄×C₄₁	—	4	2	no	—
260	C₂×C₄×C₄₁	—	4	1	no	—
261	C₂×C₄×C₄₁	—	8	4	no	—
262	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
263	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
264	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
265	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
266	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
267	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
269	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
270	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
271	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
272	C₈×C₄₁	—	4	2	no	—
273	C₈×C₄₁	—	4	2	no	—
274	#24179	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
275	C₂²×C₄₁	—	2	1	no	—
276	C₂²×C₄₁	—	2	1	no	—
277	C₂²×C₄₁	—	4	2	no	—
278	C₂²×C₄₁	—	4	2	no	—
279	C₂²×C₄₁	—	4	2	no	—
280	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
281	C₂²×C₄₁	#525	1	1	yes	characteristic
282	C₂²×C₄₁	#537	1	1	yes	characteristic
283	C₄×C₄₁	—	2	4	no	—
284	C₄×C₄₁	—	2	1	no	—
285	C₄×C₄₁	—	2	1	no	—
286	C₄×C₄₁	—	2	1	no	—
287	C₄×C₄₁	—	4	2	no	—
288	C₄×C₄₁	—	4	2	no	—
289	C₄×C₄₁	—	4	2	no	—
290	C₄×C₄₁	—	4	2	no	—
291	C₄×C₄₁	—	4	2	no	—
292	C₄×C₄₁	—	4	2	no	—
293	C₄×C₄₁	—	4	2	no	—
294	C₄×C₄₁	—	8	1	no	—
295	C₄×C₄₁	#456	1	2	yes	—
296	C₄×C₄₁	#546	1	1	yes	characteristic
297	C₄×C₄₁	#547	1	1	yes	characteristic
298	C₄×C₄₁	#548	1	1	yes	characteristic
299	C₄×C₄₁	#550	1	1	yes	characteristic
300	#3507	C₂×C₄₁	1	1	yes	characteristic
301	#3078	C₂×C₄₁	1	1	yes	characteristic, complemented
302	#3037	C₂×C₄₁	1	1	yes	characteristic
303	C₄×C₄⋊Q₈	C₂×C₄₁	1	1	yes	characteristic, complemented
304	#2038	C₂×C₄₁	1	2	yes	complemented
305	#2005	C₂×C₄₁	1	2	yes	complemented
306	#1994	C₂×C₄₁	1	1	yes	characteristic, complemented
307	#1962	C₂×C₄₁	1	1	yes	characteristic
308	#1882	C₂×C₄₁	1	2	yes	complemented
309	#1859	C₂×C₄₁	1	2	yes	complemented
310	#1801	C₂×C₄₁	1	1	yes	characteristic, complemented
311	C₂×C₄₁	—	2	1	no	—
312	C₂×C₄₁	—	2	1	no	—
313	C₂×C₄₁	—	8	2	no	—
314	C₂×C₄₁	#2407	1	1	yes	characteristic
315	C₂×C₄₁	C₂×#456	1	1	yes	characteristic
316	C₂×C₄₁	#3018	1	1	yes	characteristic
317	C₂×#137	C₂²×C₄₁	1	2	yes	—
318	#559	—	2	1	no	—
319	#557	—	2	1	no	—
320	#556	—	2	1	no	—
321	#556	—	2	2	no	—
322	#554	—	2	1	no	—
323	#552	—	2	2	no	—
324	#539	—	2	2	no	—
325	#536	C₂²×C₄₁	1	2	yes	—
326	#533	C₂²×C₄₁	1	1	yes	characteristic
327	#532	C₂²×C₄₁	1	1	yes	characteristic
328	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
329	#529	—	2	2	no	—
330	#520	C₂²×C₄₁	1	2	yes	—
331	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
332	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
333	#479	—	2	2	no	—
334	#477	—	2	1	no	—
335	#475	—	2	1	no	—
336	#464	—	2	2	no	—
337	#462	—	2	1	no	—
338	#460	—	2	1	no	—
339	#428	C₂²×C₄₁	1	2	yes	—
340	#423	C₂²×C₄₁	1	1	yes	characteristic
341	#423	C₂²×C₄₁	1	1	yes	characteristic
342	C₂×#104	C₂²×C₄₁	1	2	yes	—
343	C₂×#104	C₂²×C₄₁	1	2	yes	—
344	#419	C₂²×C₄₁	1	2	yes	—
345	#417	C₂²×C₄₁	1	2	yes	—
346	C₂×#101	C₂²×C₄₁	1	2	yes	—
347	#391	C₂²×C₄₁	1	1	yes	characteristic
348	#391	C₂²×C₄₁	1	1	yes	characteristic
349	#389	—	2	2	no	—
350	#384	C₂²×C₄₁	1	1	yes	characteristic
351	C₄×C₄⋊C₄	—	2	2	no	—
352	C₄³	C₂²×C₄₁	1	1	yes	characteristic
353	#343	C₂²×C₄₁	1	2	yes	—
354	#339	C₂²×C₄₁	1	2	yes	—
355	#339	C₂²×C₄₁	1	2	yes	—
356	C₄₁	#24179	1	1	yes	characteristic, direct product factor, Sylow
357	C₂×#41	C₂³×C₄₁	1	2	yes	—
358	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
359	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
360	#137	—	2	4	no	—
361	#137	—	4	2	no	—
362	C₂×Q₁₆	—	2	2	no	—
363	C₂×SD₁₆	—	2	2	no	—
364	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
365	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
366	C₄⋊Q₈	—	2	2	no	—
367	C₄⋊Q₈	—	2	2	no	—
368	C₄⋊Q₈	—	2	2	no	—
369	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
370	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
371	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
372	#127	D₄×C₄₁	1	1	yes	characteristic
373	#126	—	2	4	no	—
374	#126	—	2	4	no	—
375	#125	—	2	2	no	—
376	#125	—	2	4	no	—
377	#125	D₄×C₄₁	1	2	yes	—
378	#124	—	2	2	no	—
379	#124	—	2	2	no	—
380	#124	D₄×C₄₁	1	2	yes	—
381	#123	—	2	4	no	—
382	#123	—	2	4	no	—
383	C₂²⋊Q₈	—	2	4	no	—
384	C₂²⋊Q₈	—	2	1	no	—
385	C₂²⋊Q₈	—	2	1	no	—
386	C₂²⋊Q₈	—	2	2	no	—
387	C₂²⋊Q₈	—	2	2	no	—
388	#121	—	2	1	no	—
389	#121	—	2	1	no	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×Q₈	—	2	2	no	—
392	C₄×Q₈	—	2	1	no	—
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	—	2	1	no	—
398	C₄×Q₈	—	2	1	no	—
399	C₄×Q₈	—	2	1	no	—
400	C₄×Q₈	—	2	1	no	—
401	C₄×Q₈	—	2	1	no	—
402	C₄×D₄	—	2	2	no	—
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	—	2	4	no	—
405	C₄×D₄	—	2	1	no	—
406	C₄×D₄	—	2	1	no	—
407	#117	—	2	4	no	—
408	#117	—	2	2	no	—
409	#117	C₂³×C₄₁	1	2	yes	—
410	C₂×C₄⋊C₄	—	2	2	no	—
411	C₂×C₄⋊C₄	—	2	2	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	2	no	—
414	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
415	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
416	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
417	C₂×C₄²	—	2	1	no	—
418	C₂×C₄²	—	2	1	no	—
419	C₂×C₄²	—	2	1	no	—
420	C₂×C₄²	D₄×C₄₁	1	2	yes	—
421	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
422	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
423	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
424	#107	—	2	2	no	—
425	#106	—	2	2	no	—
426	C₄⋊C₈	—	2	1	no	—
427	C₄⋊C₈	—	2	1	no	—
428	C₄⋊C₈	—	2	1	no	—
429	C₄⋊C₈	—	2	1	no	—
430	#104	—	2	4	no	—
431	#104	—	4	2	no	—
432	#103	—	2	2	no	—
433	#103	—	2	2	no	—
434	#103	—	2	2	no	—
435	#102	—	2	2	no	—
436	#101	—	4	2	no	—
437	#95	—	2	2	no	—
438	#95	—	2	2	no	—
439	#41	—	2	2	no	—
440	#41	—	2	2	no	—
441	#41	—	2	2	no	—
442	#41	—	2	2	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	—	2	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	—	2	1	no	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
457	C₂×Q₈	#41×C₄₁	1	2	yes	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	4	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
469	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
470	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
471	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
472	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
473	Q₁₆	—	4	4	no	—
474	SD₁₆	—	4	4	no	—
475	M₁₆	—	2	2	no	—
476	M₁₆	—	2	2	no	—
477	M₁₆	—	2	2	no	—
478	M₁₆	—	2	2	no	—
479	C₂×C₈	—	2	2	no	—
480	C₂×C₈	—	2	2	no	—
481	C₄⋊C₄	—	2	4	no	—
482	C₄⋊C₄	—	2	8	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	2	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	2	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	2	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	4	4	no	—
498	C₄⋊C₄	—	4	1	no	—
499	C₄⋊C₄	—	4	1	no	—
500	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
501	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
502	C₂²⋊C₄	—	2	4	no	—
503	C₂²⋊C₄	—	2	4	no	—
504	C₂²⋊C₄	—	2	4	no	—
505	C₂²⋊C₄	—	2	2	no	—
506	C₂²⋊C₄	—	2	2	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₄²	—	2	4	no	—
509	C₄²	—	2	4	no	—
510	C₄²	—	2	2	no	—
511	C₄²	—	2	2	no	—
512	C₄²	—	2	2	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	—	2	1	no	—
516	C₄²	—	4	2	no	—
517	C₄²	—	4	1	no	—
518	C₄²	—	4	1	no	—
519	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
520	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
521	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
522	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
523	C₄²	#41×C₄₁	1	2	yes	—
524	C₂³	—	2	2	no	—
525	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
526	Q₈	—	2	2	no	—
527	Q₈	—	2	2	no	—
528	Q₈	—	2	2	no	—
529	Q₈	—	2	2	no	—
530	Q₈	—	4	1	no	—
531	Q₈	—	4	1	no	—
532	Q₈	—	4	1	no	—
533	Q₈	—	4	1	no	—
534	Q₈	—	4	1	no	—
535	Q₈	—	4	1	no	—
536	Q₈	—	4	2	no	—
537	Q₈	—	8	1	no	—
538	D₄	—	2	2	no	—
539	D₄	—	2	2	no	—
540	D₄	—	4	2	no	—
541	D₄	—	4	2	no	—
542	D₄	—	4	2	no	—
543	D₄	—	8	1	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	2	no	—
554	C₂×C₄	—	2	2	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	2	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	1	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	—	4	2	no	—
564	C₂×C₄	—	4	2	no	—
565	C₂×C₄	—	4	2	no	—
566	C₂×C₄	—	4	1	no	—
567	C₂×C₄	—	8	4	no	—
568	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
569	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
570	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
571	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
572	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
573	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
574	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
575	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
576	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
577	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
578	C₈	—	4	2	no	—
579	C₈	—	4	2	no	—
580	C₂²	—	2	1	no	—
581	C₂²	—	2	1	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	4	2	no	—
584	C₂²	—	4	2	no	—
585	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
586	C₂²	C₄₁×#525	1	1	yes	characteristic
587	C₂²	C₄₁×#537	1	1	yes	characteristic
588	C₄	—	2	4	no	—
589	C₄	—	2	1	no	—
590	C₄	—	2	1	no	—
591	C₄	—	2	1	no	—
592	C₄	—	4	2	no	—
593	C₄	—	4	2	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	2	no	—
597	C₄	—	4	2	no	—
598	C₄	—	4	2	no	—
599	C₄	—	8	1	no	—
600	C₄	C₄₁×#456	1	2	yes	—
601	C₄	C₄₁×#546	1	1	yes	characteristic
602	C₄	C₄₁×#547	1	1	yes	characteristic
603	C₄	C₄₁×#548	1	1	yes	characteristic
604	C₄	C₄₁×#550	1	1	yes	characteristic
605	C₂	—	2	1	no	—
606	C₂	—	2	1	no	—
607	C₂	—	8	2	no	—
608	C₂	C₄₁×#2407	1	1	yes	characteristic
609	C₂	C₂×C₄₁×#456	1	1	yes	characteristic
610	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×#532	C₂²
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. C₄×C₄⋊Q₈×C₄₁	C₂	semidirect
5. C₄₁×#2038	C₂	semidirect
6. C₄₁×#2005	C₂	semidirect
7. C₄₁×#1994	C₂	semidirect
9. C₄₁×#1882	C₂	semidirect
10. C₄₁×#1859	C₂	semidirect
11. C₄₁×#1801	C₂	semidirect
274. #24179	C₄₁	direct (N × H)
301. #3078	C₂×C₄₁	semidirect
303. C₄×C₄⋊Q₈	C₂×C₄₁	semidirect
304. #2038	C₂×C₄₁	semidirect
305. #2005	C₂×C₄₁	semidirect
306. #1994	C₂×C₄₁	semidirect
308. #1882	C₂×C₄₁	semidirect
309. #1859	C₂×C₄₁	semidirect
310. #1801	C₂×C₄₁	semidirect
356. C₄₁	#24179	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×#3078	2
C₄₁×#3037	2
C₄×C₄⋊Q₈×C₄₁	2
C₄₁×#2038	2
C₄₁×#2005	2
C₄₁×#1994	2
C₄₁×#1962	2
C₄₁×#1882	2
C₄₁×#1859	2
C₄₁×#1801	2
#24179	41

Subgroups — C41×#24179 #23800380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24179 #23800380