Subgroups — C₄₁×#24356 #23800557

Order 10496.

1656 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×SD₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#460	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₁×#448	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2433	C₂	1	1	yes	maximal, characteristic
5	Q₈×C₂²⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2055	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2014	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1893	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1880	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1850	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1786	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₄₁	—	2	4	no	—
15	C₂²×SD₁₆×C₄₁	—	2	8	no	—
16	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
17	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
18	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic, complemented
19	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic, complemented
20	C₂³×C₈×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×#127×C₄₁	C₂²	1	2	yes	—
22	C₂×#121×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×#120×C₄₁	—	2	1	no	—
24	C₂×#120×C₄₁	—	2	1	no	—
25	C₂×C₄×Q₈×C₄₁	C₂²	1	2	yes	complemented
26	C₄₁×#460	—	2	8	no	—
27	C₄₁×#448	—	2	4	no	—
28	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	complemented
29	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic, complemented
30	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic, complemented
31	C₂×#102×C₄₁	C₂²	1	2	yes	—
32	C₂×#102×C₄₁	C₂²	1	2	yes	—
33	C₂×#102×C₄₁	C₂²	1	2	yes	complemented
34	C₂×C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
35	C₄₁×#392	—	2	2	no	—
36	C₄₁×#391	—	2	2	no	—
37	C₄₁×#391	C₂²	1	2	yes	complemented
38	C₄₁×#390	C₂²	1	2	yes	—
39	C₄₁×#380	—	2	1	no	—
40	C₄₁×#380	C₂²	1	1	yes	characteristic
41	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
42	C₄×C₂²⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
43	C₄₁×#340	C₂²	1	2	yes	complemented
44	C₄₁×#336	C₂²	1	2	yes	complemented
45	C₂⁵×C₄₁	—	2	1	no	—
46	C₂²×Q₈×C₄₁	—	2	1	no	—
47	C₂²×Q₈×C₄₁	—	2	2	no	—
48	C₂²×Q₈×C₄₁	—	2	4	no	—
49	C₂²×Q₈×C₄₁	—	4	2	no	—
50	C₂²×Q₈×C₄₁	D₄	1	4	yes	complemented
51	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
52	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₄₁	—	2	4	no	—
54	C₂²×D₄×C₄₁	—	2	1	no	—
55	C₂²×D₄×C₄₁	—	2	2	no	—
56	C₂²×D₄×C₄₁	—	2	1	no	—
57	C₂²×D₄×C₄₁	—	2	2	no	—
58	C₂²×D₄×C₄₁	—	2	2	no	—
59	C₂²×D₄×C₄₁	—	2	2	no	—
60	C₂²×D₄×C₄₁	—	4	2	no	—
61	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
63	C₂³×C₄×C₄₁	—	2	1	no	—
64	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
65	C₂×SD₁₆×C₄₁	—	2	32	no	—
66	C₂×SD₁₆×C₄₁	—	2	4	no	—
67	C₂×SD₁₆×C₄₁	—	2	4	no	—
68	C₂×SD₁₆×C₄₁	—	2	4	no	—
69	C₂×SD₁₆×C₄₁	—	2	4	no	—
70	C₂×SD₁₆×C₄₁	—	2	2	no	—
71	C₂×SD₁₆×C₄₁	—	2	2	no	—
72	C₂×SD₁₆×C₄₁	—	2	4	no	—
73	C₂×SD₁₆×C₄₁	—	2	4	no	—
74	C₂×SD₁₆×C₄₁	—	4	2	no	—
75	C₂×SD₁₆×C₄₁	D₄	1	8	yes	complemented
76	C₂²×C₈×C₄₁	—	2	2	no	—
77	C₂²×C₈×C₄₁	—	2	4	no	—
78	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
81	#127×C₄₁	—	2	4	no	—
82	#127×C₄₁	—	4	2	no	—
83	#121×C₄₁	—	2	2	no	—
84	#121×C₄₁	—	2	2	no	—
85	#120×C₄₁	—	4	4	no	—
86	#120×C₄₁	—	4	4	no	—
87	C₄×Q₈×C₄₁	—	2	4	no	—
88	C₄×Q₈×C₄₁	—	4	2	no	—
89	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
90	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
92	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
93	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
96	C₂×C₂²⋊C₄×C₄₁	D₄	1	1	yes	characteristic
97	C₂×C₂²⋊C₄×C₄₁	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₄₁	—	2	2	no	—
99	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
100	C₄⋊C₈×C₄₁	—	2	4	no	—
101	#103×C₄₁	—	2	2	no	—
102	#103×C₄₁	—	2	2	no	—
103	#102×C₄₁	—	2	4	no	—
104	#102×C₄₁	—	2	4	no	—
105	#102×C₄₁	—	4	2	no	—
106	C₂²⋊C₈×C₄₁	—	2	2	no	—
107	#95×C₄₁	—	2	2	no	—
108	#95×C₄₁	—	2	2	no	—
109	#95×C₄₁	—	2	1	no	—
110	#95×C₄₁	D₄	1	2	yes	—
111	C₂⁴×C₄₁	—	2	2	no	—
112	C₂⁴×C₄₁	—	2	1	no	—
113	C₂⁴×C₄₁	—	4	4	no	—
114	C₂⁴×C₄₁	—	4	2	no	—
115	C₂⁴×C₄₁	—	4	2	no	—
116	C₂⁴×C₄₁	—	4	2	no	—
117	C₂⁴×C₄₁	—	8	2	no	—
118	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
119	C₂×Q₈×C₄₁	—	2	4	no	—
120	C₂×Q₈×C₄₁	—	2	1	no	—
121	C₂×Q₈×C₄₁	—	2	1	no	—
122	C₂×Q₈×C₄₁	—	2	2	no	—
123	C₂×Q₈×C₄₁	—	2	8	no	—
124	C₂×Q₈×C₄₁	—	2	2	no	—
125	C₂×Q₈×C₄₁	—	2	2	no	—
126	C₂×Q₈×C₄₁	—	4	4	no	—
127	C₂×Q₈×C₄₁	—	4	4	no	—
128	C₂×Q₈×C₄₁	—	4	8	no	—
129	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
130	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
131	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
132	C₂×D₄×C₄₁	—	2	8	no	—
133	C₂×D₄×C₄₁	—	2	2	no	—
134	C₂×D₄×C₄₁	—	2	2	no	—
135	C₂×D₄×C₄₁	—	2	2	no	—
136	C₂×D₄×C₄₁	—	2	1	no	—
137	C₂×D₄×C₄₁	—	2	2	no	—
138	C₂×D₄×C₄₁	—	2	1	no	—
139	C₂×D₄×C₄₁	—	2	2	no	—
140	C₂×D₄×C₄₁	—	2	2	no	—
141	C₂×D₄×C₄₁	—	2	2	no	—
142	C₂×D₄×C₄₁	—	4	8	no	—
143	C₂×D₄×C₄₁	—	4	4	no	—
144	C₂×D₄×C₄₁	—	4	4	no	—
145	C₂×D₄×C₄₁	—	4	4	no	—
146	C₂×D₄×C₄₁	—	4	4	no	—
147	C₂×D₄×C₄₁	—	4	4	no	—
148	C₂×D₄×C₄₁	—	4	2	no	—
149	C₂×D₄×C₄₁	—	4	2	no	—
150	C₂×D₄×C₄₁	—	4	2	no	—
151	C₂×D₄×C₄₁	—	4	2	no	—
152	C₂×D₄×C₄₁	—	4	4	no	—
153	C₂×D₄×C₄₁	—	8	2	no	—
154	C₂×D₄×C₄₁	—	8	2	no	—
155	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
156	C₂²×C₄×C₄₁	—	2	1	no	—
157	C₂²×C₄×C₄₁	—	2	2	no	—
158	C₂²×C₄×C₄₁	—	2	2	no	—
159	C₂²×C₄×C₄₁	—	2	4	no	—
160	C₂²×C₄×C₄₁	—	2	1	no	—
161	C₂²×C₄×C₄₁	—	2	2	no	—
162	C₂²×C₄×C₄₁	—	2	1	no	—
163	C₂²×C₄×C₄₁	—	4	4	no	—
164	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₄₁	#41	1	2	yes	—
169	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
170	SD₁₆×C₄₁	—	2	16	no	—
171	SD₁₆×C₄₁	—	2	16	no	—
172	SD₁₆×C₄₁	—	4	4	no	—
173	SD₁₆×C₄₁	—	4	4	no	—
174	C₂×C₈×C₄₁	—	2	8	no	—
175	C₂×C₈×C₄₁	—	2	2	no	—
176	C₂×C₈×C₄₁	—	2	2	no	—
177	C₂×C₈×C₄₁	—	2	1	no	—
178	C₂×C₈×C₄₁	—	2	1	no	—
179	C₂×C₈×C₄₁	—	2	2	no	—
180	C₂×C₈×C₄₁	—	2	2	no	—
181	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
182	C₄⋊C₄×C₄₁	—	2	2	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	—	4	4	no	—
187	C₂²⋊C₄×C₄₁	—	2	2	no	—
188	C₂²⋊C₄×C₄₁	—	4	4	no	—
189	C₂²⋊C₄×C₄₁	—	4	2	no	—
190	C₂²⋊C₄×C₄₁	—	4	2	no	—
191	C₂²⋊C₄×C₄₁	—	4	4	no	—
192	C₂²⋊C₄×C₄₁	SD₁₆	1	4	yes	complemented
193	C₄²×C₄₁	—	2	2	no	—
194	C₄²×C₄₁	—	2	2	no	—
195	C₄²×C₄₁	—	4	4	no	—
196	C₂³×C₄₁	—	2	2	no	—
197	C₂³×C₄₁	—	2	2	no	—
198	C₂³×C₄₁	—	2	4	no	—
199	C₂³×C₄₁	—	4	8	no	—
200	C₂³×C₄₁	—	4	2	no	—
201	C₂³×C₄₁	—	4	2	no	—
202	C₂³×C₄₁	—	4	2	no	—
203	C₂³×C₄₁	—	4	2	no	—
204	C₂³×C₄₁	—	4	2	no	—
205	C₂³×C₄₁	—	4	2	no	—
206	C₂³×C₄₁	—	4	1	no	—
207	C₂³×C₄₁	—	4	2	no	—
208	C₂³×C₄₁	—	4	1	no	—
209	C₂³×C₄₁	—	4	1	no	—
210	C₂³×C₄₁	—	4	2	no	—
211	C₂³×C₄₁	—	8	8	no	—
212	C₂³×C₄₁	—	8	8	no	—
213	C₂³×C₄₁	—	8	4	no	—
214	C₂³×C₄₁	—	8	4	no	—
215	C₂³×C₄₁	#121	1	2	yes	—
216	C₂³×C₄₁	C₂×SD₁₆	1	2	yes	—
217	C₂³×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
218	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
219	Q₈×C₄₁	—	2	2	no	—
220	Q₈×C₄₁	—	2	4	no	—
221	Q₈×C₄₁	—	2	2	no	—
222	Q₈×C₄₁	—	2	2	no	—
223	Q₈×C₄₁	—	4	8	no	—
224	Q₈×C₄₁	—	4	2	no	—
225	Q₈×C₄₁	#120	1	4	yes	complemented
226	D₄×C₄₁	—	2	4	no	—
227	D₄×C₄₁	—	2	4	no	—
228	D₄×C₄₁	—	4	8	no	—
229	D₄×C₄₁	—	4	1	no	—
230	D₄×C₄₁	—	4	2	no	—
231	D₄×C₄₁	—	4	1	no	—
232	D₄×C₄₁	—	4	2	no	—
233	D₄×C₄₁	—	4	2	no	—
234	D₄×C₄₁	—	8	8	no	—
235	D₄×C₄₁	—	8	8	no	—
236	D₄×C₄₁	—	8	4	no	—
237	C₂×C₄×C₄₁	—	2	2	no	—
238	C₂×C₄×C₄₁	—	2	2	no	—
239	C₂×C₄×C₄₁	—	2	4	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	4	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	—	4	8	no	—
250	C₂×C₄×C₄₁	—	4	2	no	—
251	C₂×C₄×C₄₁	—	4	1	no	—
252	C₂×C₄×C₄₁	—	4	2	no	—
253	C₂×C₄×C₄₁	—	4	1	no	—
254	C₂×C₄×C₄₁	—	4	1	no	—
255	C₂×C₄×C₄₁	—	4	2	no	—
256	C₂×C₄×C₄₁	—	4	2	no	—
257	C₂×C₄×C₄₁	#121	1	2	yes	—
258	C₂×C₄×C₄₁	C₂×SD₁₆	1	4	yes	complemented
259	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
262	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
263	C₈×C₄₁	—	2	4	no	—
264	C₈×C₄₁	—	4	1	no	—
265	C₈×C₄₁	—	4	1	no	—
266	#24356	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
267	C₂²×C₄₁	—	2	2	no	—
268	C₂²×C₄₁	—	2	4	no	—
269	C₂²×C₄₁	—	2	4	no	—
270	C₂²×C₄₁	—	2	2	no	—
271	C₂²×C₄₁	—	4	4	no	—
272	C₂²×C₄₁	—	4	4	no	—
273	C₂²×C₄₁	—	4	4	no	—
274	C₂²×C₄₁	—	8	16	no	—
275	C₂²×C₄₁	—	8	2	no	—
276	C₂²×C₄₁	—	8	2	no	—
277	C₂²×C₄₁	—	8	2	no	—
278	C₂²×C₄₁	—	8	2	no	—
279	C₂²×C₄₁	—	8	1	no	—
280	C₂²×C₄₁	—	8	2	no	—
281	C₂²×C₄₁	—	8	2	no	—
282	C₂²×C₄₁	—	8	1	no	—
283	C₂²×C₄₁	—	8	1	no	—
284	C₂²×C₄₁	#460	1	4	yes	complemented
285	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
286	C₂²×C₄₁	D₄²	1	2	yes	—
287	C₂²×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
288	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
289	C₂²×C₄₁	#577	1	2	yes	—
290	C₄×C₄₁	—	2	2	no	—
291	C₄×C₄₁	—	2	4	no	—
292	C₄×C₄₁	—	2	4	no	—
293	C₄×C₄₁	—	4	2	no	—
294	C₄×C₄₁	—	4	2	no	—
295	C₄×C₄₁	—	8	2	no	—
296	C₄×C₄₁	—	8	1	no	—
297	C₄×C₄₁	C₂×#120	1	1	yes	characteristic
298	C₄×C₄₁	#534	1	1	yes	characteristic
299	C₄×C₄₁	#546	1	2	yes	—
300	C₂³×SD₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
301	C₂×#460	C₂×C₄₁	1	2	yes	complemented
302	C₂×#448	C₂×C₄₁	1	1	yes	characteristic, complemented
303	#2433	C₂×C₄₁	1	1	yes	characteristic
304	Q₈×C₂²⋊C₄	C₂×C₄₁	1	1	yes	characteristic, complemented
305	#2055	C₂×C₄₁	1	2	yes	complemented
306	#2014	C₂×C₄₁	1	2	yes	complemented
307	#1893	C₂×C₄₁	1	1	yes	characteristic, complemented
308	#1880	C₂×C₄₁	1	2	yes	complemented
309	#1850	C₂×C₄₁	1	1	yes	characteristic, complemented
310	#1786	C₂×C₄₁	1	1	yes	characteristic, complemented
311	C₂×C₄₁	—	2	4	no	—
312	C₂×C₄₁	—	8	4	no	—
313	C₂×C₄₁	—	16	1	no	—
314	C₂×C₄₁	#2410	1	1	yes	characteristic
315	C₂×C₄₁	C₂×#460	1	1	yes	characteristic
316	C₂×C₄₁	#3043	1	1	yes	characteristic
317	C₂×C₄₁	#3265	1	2	yes	—
318	C₂×C₄₁	D₄×SD₁₆	1	2	yes	—
319	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
320	C₂³×D₄	C₂²×C₄₁	1	1	yes	characteristic
321	C₂²×SD₁₆	—	2	4	no	—
322	C₂²×SD₁₆	—	2	8	no	—
323	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
324	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
325	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic, complemented
326	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic, complemented
327	C₂³×C₈	C₂²×C₄₁	1	1	yes	characteristic
328	C₂×#127	C₂²×C₄₁	1	2	yes	—
329	C₂×#121	C₂²×C₄₁	1	1	yes	characteristic
330	C₂×#120	—	2	1	no	—
331	C₂×#120	—	2	1	no	—
332	C₂×C₄×Q₈	C₂²×C₄₁	1	2	yes	complemented
333	#460	—	2	8	no	—
334	#448	—	2	4	no	—
335	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	complemented
336	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic, complemented
337	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic, complemented
338	C₂×#102	C₂²×C₄₁	1	2	yes	—
339	C₂×#102	C₂²×C₄₁	1	2	yes	—
340	C₂×#102	C₂²×C₄₁	1	2	yes	complemented
341	C₂×C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
342	#392	—	2	2	no	—
343	#391	—	2	2	no	—
344	#391	C₂²×C₄₁	1	2	yes	complemented
345	#390	C₂²×C₄₁	1	2	yes	—
346	#380	—	2	1	no	—
347	#380	C₂²×C₄₁	1	1	yes	characteristic
348	C₄×C₂²⋊C₄	—	2	1	no	—
349	C₄×C₂²⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
350	#340	C₂²×C₄₁	1	2	yes	complemented
351	#336	C₂²×C₄₁	1	2	yes	complemented
352	C₄₁	#24356	1	1	yes	characteristic, direct product factor, Sylow
353	C₂⁵	—	2	1	no	—
354	C₂²×Q₈	—	2	1	no	—
355	C₂²×Q₈	—	2	2	no	—
356	C₂²×Q₈	—	2	4	no	—
357	C₂²×Q₈	—	4	2	no	—
358	C₂²×Q₈	D₄×C₄₁	1	4	yes	complemented
359	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
360	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
361	C₂²×D₄	—	2	4	no	—
362	C₂²×D₄	—	2	1	no	—
363	C₂²×D₄	—	2	2	no	—
364	C₂²×D₄	—	2	1	no	—
365	C₂²×D₄	—	2	2	no	—
366	C₂²×D₄	—	2	2	no	—
367	C₂²×D₄	—	2	2	no	—
368	C₂²×D₄	—	4	2	no	—
369	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
370	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
371	C₂³×C₄	—	2	1	no	—
372	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
373	C₂×SD₁₆	—	2	32	no	—
374	C₂×SD₁₆	—	2	4	no	—
375	C₂×SD₁₆	—	2	4	no	—
376	C₂×SD₁₆	—	2	4	no	—
377	C₂×SD₁₆	—	2	4	no	—
378	C₂×SD₁₆	—	2	2	no	—
379	C₂×SD₁₆	—	2	2	no	—
380	C₂×SD₁₆	—	2	4	no	—
381	C₂×SD₁₆	—	2	4	no	—
382	C₂×SD₁₆	—	4	2	no	—
383	C₂×SD₁₆	D₄×C₄₁	1	8	yes	complemented
384	C₂²×C₈	—	2	2	no	—
385	C₂²×C₈	—	2	4	no	—
386	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
387	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
388	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
389	#127	—	2	4	no	—
390	#127	—	4	2	no	—
391	#121	—	2	2	no	—
392	#121	—	2	2	no	—
393	#120	—	4	4	no	—
394	#120	—	4	4	no	—
395	C₄×Q₈	—	2	4	no	—
396	C₄×Q₈	—	4	2	no	—
397	C₂×C₄⋊C₄	—	2	2	no	—
398	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
399	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
400	C₂×C₂²⋊C₄	—	2	1	no	—
401	C₂×C₂²⋊C₄	—	2	2	no	—
402	C₂×C₂²⋊C₄	—	2	2	no	—
403	C₂×C₂²⋊C₄	—	2	1	no	—
404	C₂×C₂²⋊C₄	D₄×C₄₁	1	1	yes	characteristic
405	C₂×C₂²⋊C₄	D₄×C₄₁	1	1	yes	characteristic
406	C₂×C₄²	—	2	2	no	—
407	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
408	C₄⋊C₈	—	2	4	no	—
409	#103	—	2	2	no	—
410	#103	—	2	2	no	—
411	#102	—	2	4	no	—
412	#102	—	2	4	no	—
413	#102	—	4	2	no	—
414	C₂²⋊C₈	—	2	2	no	—
415	#95	—	2	2	no	—
416	#95	—	2	2	no	—
417	#95	—	2	1	no	—
418	#95	D₄×C₄₁	1	2	yes	—
419	C₂⁴	—	2	2	no	—
420	C₂⁴	—	2	1	no	—
421	C₂⁴	—	4	4	no	—
422	C₂⁴	—	4	2	no	—
423	C₂⁴	—	4	2	no	—
424	C₂⁴	—	4	2	no	—
425	C₂⁴	—	8	2	no	—
426	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
427	C₂×Q₈	—	2	4	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×Q₈	—	2	1	no	—
430	C₂×Q₈	—	2	2	no	—
431	C₂×Q₈	—	2	8	no	—
432	C₂×Q₈	—	2	2	no	—
433	C₂×Q₈	—	2	2	no	—
434	C₂×Q₈	—	4	4	no	—
435	C₂×Q₈	—	4	4	no	—
436	C₂×Q₈	—	4	8	no	—
437	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
438	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
439	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
440	C₂×D₄	—	2	8	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	—	2	1	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	1	no	—
447	C₂×D₄	—	2	2	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	4	8	no	—
451	C₂×D₄	—	4	4	no	—
452	C₂×D₄	—	4	4	no	—
453	C₂×D₄	—	4	4	no	—
454	C₂×D₄	—	4	4	no	—
455	C₂×D₄	—	4	4	no	—
456	C₂×D₄	—	4	2	no	—
457	C₂×D₄	—	4	2	no	—
458	C₂×D₄	—	4	2	no	—
459	C₂×D₄	—	4	2	no	—
460	C₂×D₄	—	4	4	no	—
461	C₂×D₄	—	8	2	no	—
462	C₂×D₄	—	8	2	no	—
463	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	2	no	—
467	C₂²×C₄	—	2	4	no	—
468	C₂²×C₄	—	2	1	no	—
469	C₂²×C₄	—	2	2	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	4	4	no	—
472	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
473	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
474	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
475	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
476	C₂²×C₄	#41×C₄₁	1	2	yes	—
477	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
478	SD₁₆	—	2	16	no	—
479	SD₁₆	—	2	16	no	—
480	SD₁₆	—	4	4	no	—
481	SD₁₆	—	4	4	no	—
482	C₂×C₈	—	2	8	no	—
483	C₂×C₈	—	2	2	no	—
484	C₂×C₈	—	2	2	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	2	no	—
488	C₂×C₈	—	2	2	no	—
489	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
490	C₄⋊C₄	—	2	2	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	4	4	no	—
495	C₂²⋊C₄	—	2	2	no	—
496	C₂²⋊C₄	—	4	4	no	—
497	C₂²⋊C₄	—	4	2	no	—
498	C₂²⋊C₄	—	4	2	no	—
499	C₂²⋊C₄	—	4	4	no	—
500	C₂²⋊C₄	SD₁₆×C₄₁	1	4	yes	complemented
501	C₄²	—	2	2	no	—
502	C₄²	—	2	2	no	—
503	C₄²	—	4	4	no	—
504	C₂³	—	2	2	no	—
505	C₂³	—	2	2	no	—
506	C₂³	—	2	4	no	—
507	C₂³	—	4	8	no	—
508	C₂³	—	4	2	no	—
509	C₂³	—	4	2	no	—
510	C₂³	—	4	2	no	—
511	C₂³	—	4	2	no	—
512	C₂³	—	4	2	no	—
513	C₂³	—	4	2	no	—
514	C₂³	—	4	1	no	—
515	C₂³	—	4	2	no	—
516	C₂³	—	4	1	no	—
517	C₂³	—	4	1	no	—
518	C₂³	—	4	2	no	—
519	C₂³	—	8	8	no	—
520	C₂³	—	8	8	no	—
521	C₂³	—	8	4	no	—
522	C₂³	—	8	4	no	—
523	C₂³	#121×C₄₁	1	2	yes	—
524	C₂³	C₂×SD₁₆×C₄₁	1	2	yes	—
525	C₂³	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
526	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
527	Q₈	—	2	2	no	—
528	Q₈	—	2	4	no	—
529	Q₈	—	2	2	no	—
530	Q₈	—	2	2	no	—
531	Q₈	—	4	8	no	—
532	Q₈	—	4	2	no	—
533	Q₈	#120×C₄₁	1	4	yes	complemented
534	D₄	—	2	4	no	—
535	D₄	—	2	4	no	—
536	D₄	—	4	8	no	—
537	D₄	—	4	1	no	—
538	D₄	—	4	2	no	—
539	D₄	—	4	1	no	—
540	D₄	—	4	2	no	—
541	D₄	—	4	2	no	—
542	D₄	—	8	8	no	—
543	D₄	—	8	8	no	—
544	D₄	—	8	4	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	4	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	4	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	8	no	—
558	C₂×C₄	—	4	2	no	—
559	C₂×C₄	—	4	1	no	—
560	C₂×C₄	—	4	2	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	1	no	—
563	C₂×C₄	—	4	2	no	—
564	C₂×C₄	—	4	2	no	—
565	C₂×C₄	#121×C₄₁	1	2	yes	—
566	C₂×C₄	C₂×SD₁₆×C₄₁	1	4	yes	complemented
567	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
568	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
569	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
570	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
571	C₈	—	2	4	no	—
572	C₈	—	4	1	no	—
573	C₈	—	4	1	no	—
574	C₂²	—	2	2	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	2	4	no	—
577	C₂²	—	2	4	no	—
578	C₂²	—	4	4	no	—
579	C₂²	—	4	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	8	16	no	—
582	C₂²	—	8	2	no	—
583	C₂²	—	8	2	no	—
584	C₂²	—	8	2	no	—
585	C₂²	—	8	2	no	—
586	C₂²	—	8	1	no	—
587	C₂²	—	8	1	no	—
588	C₂²	—	8	2	no	—
589	C₂²	—	8	2	no	—
590	C₂²	—	8	1	no	—
591	C₂²	C₄₁×#460	1	4	yes	complemented
592	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
593	C₂²	D₄²×C₄₁	1	2	yes	—
594	C₂²	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
595	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
596	C₂²	C₄₁×#577	1	2	yes	—
597	C₄	—	2	2	no	—
598	C₄	—	2	4	no	—
599	C₄	—	2	4	no	—
600	C₄	—	4	2	no	—
601	C₄	—	4	2	no	—
602	C₄	—	8	2	no	—
603	C₄	—	8	1	no	—
604	C₄	C₂×#120×C₄₁	1	1	yes	characteristic
605	C₄	C₄₁×#534	1	1	yes	characteristic
606	C₄	C₄₁×#546	1	2	yes	—
607	C₂	—	2	4	no	—
608	C₂	—	8	4	no	—
609	C₂	—	16	1	no	—
610	C₂	C₄₁×#2410	1	1	yes	characteristic
611	C₂	C₂×C₄₁×#460	1	1	yes	characteristic
612	C₂	C₄₁×#3043	1	1	yes	characteristic
613	C₂	C₄₁×#3265	1	2	yes	—
614	C₂	D₄×SD₁₆×C₄₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₄₁×#2433	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄×C₂²⋊C₄×C₄₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂³×SD₁₆×C₄₁	C₂	semidirect
2. C₂×C₄₁×#460	C₂	semidirect
3. C₂×C₄₁×#448	C₂	semidirect
5. Q₈×C₂²⋊C₄×C₄₁	C₂	semidirect
6. C₄₁×#2055	C₂	semidirect
7. C₄₁×#2014	C₂	semidirect
8. C₄₁×#1893	C₂	semidirect
9. C₄₁×#1880	C₂	semidirect
10. C₄₁×#1850	C₂	semidirect
11. C₄₁×#1786	C₂	semidirect
18. C₂²×SD₁₆×C₄₁	C₂²	semidirect
19. C₂²×SD₁₆×C₄₁	C₂²	semidirect
25. C₂×C₄×Q₈×C₄₁	C₂²	semidirect
28. C₂×C₄⋊C₈×C₄₁	C₂²	semidirect
29. C₂×#103×C₄₁	C₂²	semidirect
30. C₂×#103×C₄₁	C₂²	semidirect
33. C₂×#102×C₄₁	C₂²	semidirect
37. C₄₁×#391	C₂²	semidirect
43. C₄₁×#340	C₂²	semidirect
44. C₄₁×#336	C₂²	semidirect
50. C₂²×Q₈×C₄₁	D₄	semidirect
75. C₂×SD₁₆×C₄₁	D₄	semidirect
192. C₂²⋊C₄×C₄₁	SD₁₆	semidirect
225. Q₈×C₄₁	#120	semidirect
258. C₂×C₄×C₄₁	C₂×SD₁₆	semidirect
266. #24356	C₄₁	direct (N × H)
284. C₂²×C₄₁	#460	semidirect
300. C₂³×SD₁₆	C₂×C₄₁	semidirect
301. C₂×#460	C₂×C₄₁	semidirect
302. C₂×#448	C₂×C₄₁	semidirect
304. Q₈×C₂²⋊C₄	C₂×C₄₁	semidirect
305. #2055	C₂×C₄₁	semidirect
306. #2014	C₂×C₄₁	semidirect
307. #1893	C₂×C₄₁	semidirect
308. #1880	C₂×C₄₁	semidirect
309. #1850	C₂×C₄₁	semidirect
310. #1786	C₂×C₄₁	semidirect
325. C₂²×SD₁₆	C₂²×C₄₁	semidirect
326. C₂²×SD₁₆	C₂²×C₄₁	semidirect
332. C₂×C₄×Q₈	C₂²×C₄₁	semidirect
335. C₂×C₄⋊C₈	C₂²×C₄₁	semidirect
336. C₂×#103	C₂²×C₄₁	semidirect
337. C₂×#103	C₂²×C₄₁	semidirect
340. C₂×#102	C₂²×C₄₁	semidirect
344. #391	C₂²×C₄₁	semidirect
350. #340	C₂²×C₄₁	semidirect
351. #336	C₂²×C₄₁	semidirect
352. C₄₁	#24356	direct (N × H)
358. C₂²×Q₈	D₄×C₄₁	semidirect
383. C₂×SD₁₆	D₄×C₄₁	semidirect
500. C₂²⋊C₄	SD₁₆×C₄₁	semidirect
533. Q₈	#120×C₄₁	semidirect
566. C₂×C₄	C₂×SD₁₆×C₄₁	semidirect
591. C₂²	C₄₁×#460	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×SD₁₆×C₄₁	2
C₂×C₄₁×#460	2
C₂×C₄₁×#448	2
C₄₁×#2433	2
Q₈×C₂²⋊C₄×C₄₁	2
C₄₁×#2055	2
C₄₁×#2014	2
C₄₁×#1893	2
C₄₁×#1880	2
C₄₁×#1850	2
C₄₁×#1786	2
#24356	41

Subgroups — C41×#24356 #23800557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24356 #23800557