Subgroups — C₄₁×#24217 #23800418

Order 10496.

1558 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#124×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#464	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₁×#451	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₁×#450	C₂	1	1	yes	maximal, characteristic
5	C₄×C₂²⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2042	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2015	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1983	C₂	1	2	yes	maximal, complemented
9	C₄₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1787	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1786	C₂	1	1	yes	maximal, characteristic
12	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₄₁	C₂²	1	2	yes	complemented
15	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
16	C₂×#124×C₄₁	—	2	8	no	—
17	C₂×#124×C₄₁	—	2	1	no	—
18	C₂×#124×C₄₁	—	2	1	no	—
19	C₂×#124×C₄₁	C₂²	1	2	yes	—
20	C₂×#124×C₄₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
23	C₂²×C₂²⋊C₄×C₄₁	—	2	1	no	—
24	C₂²×C₂²⋊C₄×C₄₁	—	2	1	no	—
25	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
26	C₄₁×#464	—	2	8	no	—
27	C₄₁×#451	—	2	4	no	—
28	C₄₁×#450	—	2	4	no	—
29	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	complemented
30	C₂×#103×C₄₁	C₂²	1	2	yes	complemented
31	C₂×#103×C₄₁	C₂²	1	2	yes	complemented
32	C₂×#103×C₄₁	C₂²	1	2	yes	complemented
33	C₂×#102×C₄₁	C₂²	1	2	yes	—
34	C₂×#102×C₄₁	C₂²	1	2	yes	—
35	C₂×#102×C₄₁	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
37	C₂×C₂²⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
38	C₄₁×#391	C₂²	1	1	yes	characteristic, complemented
39	C₄₁×#389	C₂²	1	1	yes	characteristic, complemented
40	C₄₁×#387	—	2	1	no	—
41	C₄₁×#385	—	2	1	no	—
42	C₄₁×#380	C₂²	1	1	yes	characteristic
43	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
44	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic, complemented
45	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
46	C₄₁×#336	C₂²	1	2	yes	complemented
47	C₂⁵×C₄₁	—	2	1	no	—
48	C₂²×Q₈×C₄₁	—	2	4	no	—
49	C₂²×Q₈×C₄₁	—	2	1	no	—
50	C₂²×Q₈×C₄₁	—	2	2	no	—
51	C₂²×Q₈×C₄₁	—	2	2	no	—
52	C₂²×Q₈×C₄₁	—	4	2	no	—
53	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
54	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
55	C₂²×D₄×C₄₁	—	2	4	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	—	4	2	no	—
60	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₄₁	—	2	1	no	—
62	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
65	C₂×Q₁₆×C₄₁	—	2	4	no	—
66	C₂×Q₁₆×C₄₁	—	2	4	no	—
67	C₂×Q₁₆×C₄₁	—	4	2	no	—
68	C₂×SD₁₆×C₄₁	—	2	4	no	—
69	C₂×SD₁₆×C₄₁	—	2	4	no	—
70	C₂×SD₁₆×C₄₁	—	4	2	no	—
71	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
72	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
73	#124×C₄₁	—	2	8	no	—
74	#124×C₄₁	—	2	8	no	—
75	#124×C₄₁	—	2	4	no	—
76	#124×C₄₁	—	2	4	no	—
77	#124×C₄₁	—	2	4	no	—
78	#124×C₄₁	D₄	1	8	yes	—
79	C₂²⋊Q₈×C₄₁	—	2	2	no	—
80	C₂²⋊Q₈×C₄₁	D₄	1	4	yes	—
81	C₄×Q₈×C₄₁	—	2	4	no	—
82	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
85	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
87	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
89	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
90	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
92	C₂×C₂²⋊C₄×C₄₁	—	2	2	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×C₄²×C₄₁	—	2	4	no	—
97	C₂×C₄²×C₄₁	—	2	1	no	—
98	C₂×C₄²×C₄₁	—	2	1	no	—
99	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₈×C₄₁	—	2	4	no	—
104	#103×C₄₁	—	2	4	no	—
105	#103×C₄₁	—	2	4	no	—
106	#103×C₄₁	—	4	2	no	—
107	#102×C₄₁	—	2	4	no	—
108	#102×C₄₁	—	2	4	no	—
109	#102×C₄₁	—	4	2	no	—
110	C₂²⋊C₈×C₄₁	—	2	2	no	—
111	C₂²⋊C₈×C₄₁	—	2	2	no	—
112	#95×C₄₁	—	2	1	no	—
113	#95×C₄₁	—	2	1	no	—
114	#95×C₄₁	—	2	1	no	—
115	#95×C₄₁	—	2	1	no	—
116	#95×C₄₁	—	2	1	no	—
117	C₂⁴×C₄₁	—	2	4	no	—
118	C₂⁴×C₄₁	—	2	2	no	—
119	C₂⁴×C₄₁	—	4	2	no	—
120	C₂⁴×C₄₁	—	4	2	no	—
121	C₂⁴×C₄₁	—	4	2	no	—
122	C₂⁴×C₄₁	—	4	2	no	—
123	C₂⁴×C₄₁	—	4	2	no	—
124	C₂⁴×C₄₁	—	4	2	no	—
125	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Q₈×C₄₁	—	2	4	no	—
127	C₂×Q₈×C₄₁	—	2	2	no	—
128	C₂×Q₈×C₄₁	—	2	1	no	—
129	C₂×Q₈×C₄₁	—	2	2	no	—
130	C₂×Q₈×C₄₁	—	2	2	no	—
131	C₂×Q₈×C₄₁	—	2	2	no	—
132	C₂×Q₈×C₄₁	—	2	2	no	—
133	C₂×Q₈×C₄₁	—	2	2	no	—
134	C₂×Q₈×C₄₁	—	2	2	no	—
135	C₂×Q₈×C₄₁	—	4	8	no	—
136	C₂×Q₈×C₄₁	—	4	8	no	—
137	C₂×Q₈×C₄₁	—	4	2	no	—
138	C₂×Q₈×C₄₁	—	4	2	no	—
139	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
140	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
141	C₂×D₄×C₄₁	—	2	4	no	—
142	C₂×D₄×C₄₁	—	2	2	no	—
143	C₂×D₄×C₄₁	—	2	2	no	—
144	C₂×D₄×C₄₁	—	2	2	no	—
145	C₂×D₄×C₄₁	—	2	2	no	—
146	C₂×D₄×C₄₁	—	2	2	no	—
147	C₂×D₄×C₄₁	—	4	8	no	—
148	C₂×D₄×C₄₁	—	4	8	no	—
149	C₂×D₄×C₄₁	—	4	2	no	—
150	C₂×D₄×C₄₁	—	4	2	no	—
151	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
152	C₂²×C₄×C₄₁	—	2	2	no	—
153	C₂²×C₄×C₄₁	—	2	2	no	—
154	C₂²×C₄×C₄₁	—	2	4	no	—
155	C₂²×C₄×C₄₁	—	2	2	no	—
156	C₂²×C₄×C₄₁	—	2	2	no	—
157	C₂²×C₄×C₄₁	—	2	1	no	—
158	C₂²×C₄×C₄₁	—	2	1	no	—
159	C₂²×C₄×C₄₁	—	2	2	no	—
160	C₂²×C₄×C₄₁	—	2	2	no	—
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	—	4	1	no	—
165	C₂²×C₄×C₄₁	—	4	1	no	—
166	C₂²×C₄×C₄₁	—	4	2	no	—
167	C₂²×C₄×C₄₁	—	4	2	no	—
168	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
169	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
170	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₄₁	#41	1	2	yes	—
175	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
176	Q₁₆×C₄₁	—	4	8	no	—
177	SD₁₆×C₄₁	—	4	8	no	—
178	C₂×C₈×C₄₁	—	2	2	no	—
179	C₂×C₈×C₄₁	—	2	2	no	—
180	C₂×C₈×C₄₁	—	2	2	no	—
181	C₂×C₈×C₄₁	—	2	2	no	—
182	C₂×C₈×C₄₁	—	2	2	no	—
183	C₂×C₈×C₄₁	—	2	2	no	—
184	C₄⋊C₄×C₄₁	—	2	4	no	—
185	C₄⋊C₄×C₄₁	—	2	1	no	—
186	C₄⋊C₄×C₄₁	—	2	2	no	—
187	C₄⋊C₄×C₄₁	—	2	2	no	—
188	C₄⋊C₄×C₄₁	—	4	2	no	—
189	C₄⋊C₄×C₄₁	—	4	2	no	—
190	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
191	C₂²⋊C₄×C₄₁	—	2	8	no	—
192	C₂²⋊C₄×C₄₁	—	2	8	no	—
193	C₂²⋊C₄×C₄₁	—	2	4	no	—
194	C₂²⋊C₄×C₄₁	—	2	8	no	—
195	C₂²⋊C₄×C₄₁	—	2	8	no	—
196	C₂²⋊C₄×C₄₁	—	4	2	no	—
197	C₄²×C₄₁	—	2	2	no	—
198	C₄²×C₄₁	—	2	2	no	—
199	C₄²×C₄₁	—	2	2	no	—
200	C₄²×C₄₁	—	4	2	no	—
201	C₄²×C₄₁	—	4	2	no	—
202	C₄²×C₄₁	C₂×D₄	1	4	yes	—
203	C₂³×C₄₁	—	2	1	no	—
204	C₂³×C₄₁	—	2	1	no	—
205	C₂³×C₄₁	—	2	1	no	—
206	C₂³×C₄₁	—	2	1	no	—
207	C₂³×C₄₁	—	2	4	no	—
208	C₂³×C₄₁	—	4	8	no	—
209	C₂³×C₄₁	—	4	8	no	—
210	C₂³×C₄₁	—	4	8	no	—
211	C₂³×C₄₁	—	4	2	no	—
212	C₂³×C₄₁	—	4	2	no	—
213	C₂³×C₄₁	—	4	2	no	—
214	C₂³×C₄₁	—	4	2	no	—
215	C₂³×C₄₁	—	4	2	no	—
216	C₂³×C₄₁	—	4	2	no	—
217	C₂³×C₄₁	—	8	4	no	—
218	C₂³×C₄₁	—	8	4	no	—
219	C₂³×C₄₁	—	8	4	no	—
220	C₂³×C₄₁	—	8	4	no	—
221	C₂³×C₄₁	#121	1	2	yes	—
222	C₂³×C₄₁	#135	1	2	yes	—
223	C₂³×C₄₁	#137	1	2	yes	—
224	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
225	Q₈×C₄₁	—	2	4	no	—
226	Q₈×C₄₁	—	2	4	no	—
227	Q₈×C₄₁	—	2	4	no	—
228	Q₈×C₄₁	—	4	8	no	—
229	Q₈×C₄₁	—	4	2	no	—
230	Q₈×C₄₁	—	4	2	no	—
231	Q₈×C₄₁	—	4	2	no	—
232	D₄×C₄₁	—	2	4	no	—
233	D₄×C₄₁	—	2	4	no	—
234	D₄×C₄₁	—	4	8	no	—
235	D₄×C₄₁	—	4	2	no	—
236	D₄×C₄₁	—	4	2	no	—
237	C₂×C₄×C₄₁	—	2	1	no	—
238	C₂×C₄×C₄₁	—	2	1	no	—
239	C₂×C₄×C₄₁	—	2	1	no	—
240	C₂×C₄×C₄₁	—	2	1	no	—
241	C₂×C₄×C₄₁	—	2	1	no	—
242	C₂×C₄×C₄₁	—	2	1	no	—
243	C₂×C₄×C₄₁	—	2	4	no	—
244	C₂×C₄×C₄₁	—	2	4	no	—
245	C₂×C₄×C₄₁	—	2	1	no	—
246	C₂×C₄×C₄₁	—	2	1	no	—
247	C₂×C₄×C₄₁	—	2	4	no	—
248	C₂×C₄×C₄₁	—	2	2	no	—
249	C₂×C₄×C₄₁	—	2	2	no	—
250	C₂×C₄×C₄₁	—	2	1	no	—
251	C₂×C₄×C₄₁	—	2	1	no	—
252	C₂×C₄×C₄₁	—	2	1	no	—
253	C₂×C₄×C₄₁	—	2	1	no	—
254	C₂×C₄×C₄₁	—	4	4	no	—
255	C₂×C₄×C₄₁	—	4	4	no	—
256	C₂×C₄×C₄₁	—	4	8	no	—
257	C₂×C₄×C₄₁	—	4	1	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	—	4	1	no	—
262	C₂×C₄×C₄₁	—	4	1	no	—
263	C₂×C₄×C₄₁	—	4	1	no	—
264	C₂×C₄×C₄₁	—	4	1	no	—
265	C₂×C₄×C₄₁	—	4	1	no	—
266	C₂×C₄×C₄₁	—	4	1	no	—
267	C₂×C₄×C₄₁	—	4	1	no	—
268	C₂×C₄×C₄₁	—	4	1	no	—
269	C₂×C₄×C₄₁	#120	1	2	yes	—
270	C₂×C₄×C₄₁	#120	1	2	yes	—
271	C₂×C₄×C₄₁	#121	1	2	yes	—
272	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
275	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
276	C₈×C₄₁	—	4	2	no	—
277	C₈×C₄₁	—	4	2	no	—
278	#24217	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
279	C₂²×C₄₁	—	2	2	no	—
280	C₂²×C₄₁	—	2	2	no	—
281	C₂²×C₄₁	—	2	2	no	—
282	C₂²×C₄₁	—	2	2	no	—
283	C₂²×C₄₁	—	2	2	no	—
284	C₂²×C₄₁	—	2	2	no	—
285	C₂²×C₄₁	—	4	4	no	—
286	C₂²×C₄₁	—	4	4	no	—
287	C₂²×C₄₁	—	4	4	no	—
288	C₂²×C₄₁	—	8	16	no	—
289	C₂²×C₄₁	—	8	2	no	—
290	C₂²×C₄₁	—	8	2	no	—
291	C₂²×C₄₁	—	8	2	no	—
292	C₂²×C₄₁	—	8	2	no	—
293	C₂²×C₄₁	#464	1	4	yes	complemented
294	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
295	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
296	C₂²×C₄₁	#535	1	1	yes	characteristic
297	C₂²×C₄₁	C₂×#135	1	1	yes	characteristic
298	C₂²×C₄₁	C₂×#137	1	1	yes	characteristic
299	C₂²×C₄₁	#575	1	1	yes	characteristic
300	C₂²×C₄₁	#575	1	1	yes	characteristic
301	C₄×C₄₁	—	2	2	no	—
302	C₄×C₄₁	—	2	2	no	—
303	C₄×C₄₁	—	2	2	no	—
304	C₄×C₄₁	—	4	4	no	—
305	C₄×C₄₁	—	4	2	no	—
306	C₄×C₄₁	—	4	1	no	—
307	C₄×C₄₁	—	4	1	no	—
308	C₄×C₄₁	—	4	2	no	—
309	C₄×C₄₁	—	8	1	no	—
310	C₄×C₄₁	—	8	1	no	—
311	C₄×C₄₁	D₄²	1	2	yes	—
312	C₄×C₄₁	#546	1	2	yes	—
313	C₂²×#124	C₂×C₄₁	1	1	yes	characteristic
314	C₂×#464	C₂×C₄₁	1	2	yes	complemented
315	C₂×#451	C₂×C₄₁	1	1	yes	characteristic, complemented
316	C₂×#450	C₂×C₄₁	1	1	yes	characteristic
317	C₄×C₂²⋊Q₈	C₂×C₄₁	1	1	yes	characteristic, complemented
318	#2042	C₂×C₄₁	1	2	yes	complemented
319	#2015	C₂×C₄₁	1	2	yes	complemented
320	#1983	C₂×C₄₁	1	2	yes	complemented
321	#1797	C₂×C₄₁	1	1	yes	characteristic, complemented
322	#1787	C₂×C₄₁	1	1	yes	characteristic, complemented
323	#1786	C₂×C₄₁	1	1	yes	characteristic
324	C₂×C₄₁	—	2	4	no	—
325	C₂×C₄₁	—	8	4	no	—
326	C₂×C₄₁	#2408	1	1	yes	characteristic
327	C₂×C₄₁	#3002	1	1	yes	characteristic
328	C₂×C₄₁	#3007	1	1	yes	characteristic
329	C₂×C₄₁	C₂×#464	1	1	yes	characteristic
330	C₂×C₄₁	#3040	1	1	yes	characteristic
331	C₂×C₄₁	#3211	1	1	yes	characteristic
332	C₂×C₄₁	#3215	1	1	yes	characteristic
333	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
334	C₂³×D₄	C₂²×C₄₁	1	1	yes	characteristic
335	C₂²×Q₁₆	C₂²×C₄₁	1	2	yes	complemented
336	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
337	C₂×#124	—	2	8	no	—
338	C₂×#124	—	2	1	no	—
339	C₂×#124	—	2	1	no	—
340	C₂×#124	C₂²×C₄₁	1	2	yes	—
341	C₂×#124	C₂²×C₄₁	1	2	yes	—
342	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
343	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
344	C₂²×C₂²⋊C₄	—	2	1	no	—
345	C₂²×C₂²⋊C₄	—	2	1	no	—
346	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
347	#464	—	2	8	no	—
348	#451	—	2	4	no	—
349	#450	—	2	4	no	—
350	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	complemented
351	C₂×#103	C₂²×C₄₁	1	2	yes	complemented
352	C₂×#103	C₂²×C₄₁	1	2	yes	complemented
353	C₂×#103	C₂²×C₄₁	1	2	yes	complemented
354	C₂×#102	C₂²×C₄₁	1	2	yes	—
355	C₂×#102	C₂²×C₄₁	1	2	yes	—
356	C₂×#102	C₂²×C₄₁	1	2	yes	—
357	C₂×C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
358	C₂×C₂²⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
359	#391	C₂²×C₄₁	1	1	yes	characteristic, complemented
360	#389	C₂²×C₄₁	1	1	yes	characteristic, complemented
361	#387	—	2	1	no	—
362	#385	—	2	1	no	—
363	#380	C₂²×C₄₁	1	1	yes	characteristic
364	C₄×C₄⋊C₄	—	2	1	no	—
365	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic, complemented
366	C₄×C₂²⋊C₄	—	2	1	no	—
367	#336	C₂²×C₄₁	1	2	yes	complemented
368	C₄₁	#24217	1	1	yes	characteristic, direct product factor, Sylow
369	C₂⁵	—	2	1	no	—
370	C₂²×Q₈	—	2	4	no	—
371	C₂²×Q₈	—	2	1	no	—
372	C₂²×Q₈	—	2	2	no	—
373	C₂²×Q₈	—	2	2	no	—
374	C₂²×Q₈	—	4	2	no	—
375	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
376	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
377	C₂²×D₄	—	2	4	no	—
378	C₂²×D₄	—	2	1	no	—
379	C₂²×D₄	—	2	2	no	—
380	C₂²×D₄	—	2	2	no	—
381	C₂²×D₄	—	4	2	no	—
382	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
383	C₂³×C₄	—	2	1	no	—
384	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
385	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
386	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
387	C₂×Q₁₆	—	2	4	no	—
388	C₂×Q₁₆	—	2	4	no	—
389	C₂×Q₁₆	—	4	2	no	—
390	C₂×SD₁₆	—	2	4	no	—
391	C₂×SD₁₆	—	2	4	no	—
392	C₂×SD₁₆	—	4	2	no	—
393	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
394	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
395	#124	—	2	8	no	—
396	#124	—	2	8	no	—
397	#124	—	2	4	no	—
398	#124	—	2	4	no	—
399	#124	—	2	4	no	—
400	#124	D₄×C₄₁	1	8	yes	—
401	C₂²⋊Q₈	—	2	2	no	—
402	C₂²⋊Q₈	D₄×C₄₁	1	4	yes	—
403	C₄×Q₈	—	2	4	no	—
404	C₂×C₄⋊C₄	—	2	1	no	—
405	C₂×C₄⋊C₄	—	2	1	no	—
406	C₂×C₄⋊C₄	—	2	1	no	—
407	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
408	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
410	C₂×C₂²⋊C₄	—	2	8	no	—
411	C₂×C₂²⋊C₄	—	2	8	no	—
412	C₂×C₂²⋊C₄	—	2	1	no	—
413	C₂×C₂²⋊C₄	—	2	2	no	—
414	C₂×C₂²⋊C₄	—	2	2	no	—
415	C₂×C₂²⋊C₄	—	2	2	no	—
416	C₂×C₂²⋊C₄	—	2	2	no	—
417	C₂×C₂²⋊C₄	—	2	1	no	—
418	C₂×C₄²	—	2	4	no	—
419	C₂×C₄²	—	2	1	no	—
420	C₂×C₄²	—	2	1	no	—
421	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
422	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
423	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
424	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
425	C₄⋊C₈	—	2	4	no	—
426	#103	—	2	4	no	—
427	#103	—	2	4	no	—
428	#103	—	4	2	no	—
429	#102	—	2	4	no	—
430	#102	—	2	4	no	—
431	#102	—	4	2	no	—
432	C₂²⋊C₈	—	2	2	no	—
433	C₂²⋊C₈	—	2	2	no	—
434	#95	—	2	1	no	—
435	#95	—	2	1	no	—
436	#95	—	2	1	no	—
437	#95	—	2	1	no	—
438	#95	—	2	1	no	—
439	C₂⁴	—	2	4	no	—
440	C₂⁴	—	2	2	no	—
441	C₂⁴	—	4	2	no	—
442	C₂⁴	—	4	2	no	—
443	C₂⁴	—	4	2	no	—
444	C₂⁴	—	4	2	no	—
445	C₂⁴	—	4	2	no	—
446	C₂⁴	—	4	2	no	—
447	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
448	C₂×Q₈	—	2	4	no	—
449	C₂×Q₈	—	2	2	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	2	no	—
452	C₂×Q₈	—	2	2	no	—
453	C₂×Q₈	—	2	2	no	—
454	C₂×Q₈	—	2	2	no	—
455	C₂×Q₈	—	2	2	no	—
456	C₂×Q₈	—	2	2	no	—
457	C₂×Q₈	—	4	8	no	—
458	C₂×Q₈	—	4	8	no	—
459	C₂×Q₈	—	4	2	no	—
460	C₂×Q₈	—	4	2	no	—
461	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
462	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
463	C₂×D₄	—	2	4	no	—
464	C₂×D₄	—	2	2	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	4	8	no	—
470	C₂×D₄	—	4	8	no	—
471	C₂×D₄	—	4	2	no	—
472	C₂×D₄	—	4	2	no	—
473	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
474	C₂²×C₄	—	2	2	no	—
475	C₂²×C₄	—	2	2	no	—
476	C₂²×C₄	—	2	4	no	—
477	C₂²×C₄	—	2	2	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	1	no	—
480	C₂²×C₄	—	2	1	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	2	no	—
483	C₂²×C₄	—	2	2	no	—
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	4	1	no	—
487	C₂²×C₄	—	4	1	no	—
488	C₂²×C₄	—	4	2	no	—
489	C₂²×C₄	—	4	2	no	—
490	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
491	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
492	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
493	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
494	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
496	C₂²×C₄	#41×C₄₁	1	2	yes	—
497	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
498	Q₁₆	—	4	8	no	—
499	SD₁₆	—	4	8	no	—
500	C₂×C₈	—	2	2	no	—
501	C₂×C₈	—	2	2	no	—
502	C₂×C₈	—	2	2	no	—
503	C₂×C₈	—	2	2	no	—
504	C₂×C₈	—	2	2	no	—
505	C₂×C₈	—	2	2	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	4	2	no	—
511	C₄⋊C₄	—	4	2	no	—
512	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
513	C₂²⋊C₄	—	2	8	no	—
514	C₂²⋊C₄	—	2	8	no	—
515	C₂²⋊C₄	—	2	4	no	—
516	C₂²⋊C₄	—	2	8	no	—
517	C₂²⋊C₄	—	2	8	no	—
518	C₂²⋊C₄	—	4	2	no	—
519	C₄²	—	2	2	no	—
520	C₄²	—	2	2	no	—
521	C₄²	—	2	2	no	—
522	C₄²	—	4	2	no	—
523	C₄²	—	4	2	no	—
524	C₄²	C₂×D₄×C₄₁	1	4	yes	—
525	C₂³	—	2	1	no	—
526	C₂³	—	2	1	no	—
527	C₂³	—	2	1	no	—
528	C₂³	—	2	1	no	—
529	C₂³	—	2	4	no	—
530	C₂³	—	4	8	no	—
531	C₂³	—	4	8	no	—
532	C₂³	—	4	8	no	—
533	C₂³	—	4	2	no	—
534	C₂³	—	4	2	no	—
535	C₂³	—	4	2	no	—
536	C₂³	—	4	2	no	—
537	C₂³	—	4	2	no	—
538	C₂³	—	4	2	no	—
539	C₂³	—	8	4	no	—
540	C₂³	—	8	4	no	—
541	C₂³	—	8	4	no	—
542	C₂³	—	8	4	no	—
543	C₂³	#121×C₄₁	1	2	yes	—
544	C₂³	#135×C₄₁	1	2	yes	—
545	C₂³	#137×C₄₁	1	2	yes	—
546	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
547	Q₈	—	2	4	no	—
548	Q₈	—	2	4	no	—
549	Q₈	—	2	4	no	—
550	Q₈	—	4	8	no	—
551	Q₈	—	4	2	no	—
552	Q₈	—	4	2	no	—
553	Q₈	—	4	2	no	—
554	D₄	—	2	4	no	—
555	D₄	—	2	4	no	—
556	D₄	—	4	8	no	—
557	D₄	—	4	2	no	—
558	D₄	—	4	2	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	1	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	4	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	4	4	no	—
577	C₂×C₄	—	4	4	no	—
578	C₂×C₄	—	4	8	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	4	2	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	#120×C₄₁	1	2	yes	—
592	C₂×C₄	#120×C₄₁	1	2	yes	—
593	C₂×C₄	#121×C₄₁	1	2	yes	—
594	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
595	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
596	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
597	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
598	C₈	—	4	2	no	—
599	C₈	—	4	2	no	—
600	C₂²	—	2	2	no	—
601	C₂²	—	2	2	no	—
602	C₂²	—	2	2	no	—
603	C₂²	—	2	2	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	2	2	no	—
606	C₂²	—	4	4	no	—
607	C₂²	—	4	4	no	—
608	C₂²	—	4	4	no	—
609	C₂²	—	8	16	no	—
610	C₂²	—	8	2	no	—
611	C₂²	—	8	2	no	—
612	C₂²	—	8	2	no	—
613	C₂²	—	8	2	no	—
614	C₂²	C₄₁×#464	1	4	yes	complemented
615	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
616	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
617	C₂²	C₄₁×#535	1	1	yes	characteristic
618	C₂²	C₂×#135×C₄₁	1	1	yes	characteristic
619	C₂²	C₂×#137×C₄₁	1	1	yes	characteristic
620	C₂²	C₄₁×#575	1	1	yes	characteristic
621	C₂²	C₄₁×#575	1	1	yes	characteristic
622	C₄	—	2	2	no	—
623	C₄	—	2	2	no	—
624	C₄	—	2	2	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	4	no	—
627	C₄	—	4	2	no	—
628	C₄	—	4	1	no	—
629	C₄	—	4	1	no	—
630	C₄	—	8	1	no	—
631	C₄	—	8	1	no	—
632	C₄	D₄²×C₄₁	1	2	yes	—
633	C₄	C₄₁×#546	1	2	yes	—
634	C₂	—	2	4	no	—
635	C₂	—	8	4	no	—
636	C₂	C₄₁×#2408	1	1	yes	characteristic
637	C₂	C₄₁×#3002	1	1	yes	characteristic
638	C₂	C₄₁×#3007	1	1	yes	characteristic
639	C₂	C₂×C₄₁×#464	1	1	yes	characteristic
640	C₂	C₄₁×#3040	1	1	yes	characteristic
641	C₂	C₄₁×#3211	1	1	yes	characteristic
642	C₂	C₄₁×#3215	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂³×D₄×C₄₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#464	C₂	semidirect
3. C₂×C₄₁×#451	C₂	semidirect
5. C₄×C₂²⋊Q₈×C₄₁	C₂	semidirect
6. C₄₁×#2042	C₂	semidirect
7. C₄₁×#2015	C₂	semidirect
8. C₄₁×#1983	C₂	semidirect
9. C₄₁×#1797	C₂	semidirect
10. C₄₁×#1787	C₂	semidirect
14. C₂²×Q₁₆×C₄₁	C₂²	semidirect
22. C₂×C₄×Q₈×C₄₁	C₂²	semidirect
29. C₂×C₄⋊C₈×C₄₁	C₂²	semidirect
30. C₂×#103×C₄₁	C₂²	semidirect
31. C₂×#103×C₄₁	C₂²	semidirect
32. C₂×#103×C₄₁	C₂²	semidirect
38. C₄₁×#391	C₂²	semidirect
39. C₄₁×#389	C₂²	semidirect
44. C₄×C₄⋊C₄×C₄₁	C₂²	semidirect
46. C₄₁×#336	C₂²	semidirect
278. #24217	C₄₁	direct (N × H)
293. C₂²×C₄₁	#464	semidirect
314. C₂×#464	C₂×C₄₁	semidirect
315. C₂×#451	C₂×C₄₁	semidirect
317. C₄×C₂²⋊Q₈	C₂×C₄₁	semidirect
318. #2042	C₂×C₄₁	semidirect
319. #2015	C₂×C₄₁	semidirect
320. #1983	C₂×C₄₁	semidirect
321. #1797	C₂×C₄₁	semidirect
322. #1787	C₂×C₄₁	semidirect
335. C₂²×Q₁₆	C₂²×C₄₁	semidirect
343. C₂×C₄×Q₈	C₂²×C₄₁	semidirect
350. C₂×C₄⋊C₈	C₂²×C₄₁	semidirect
351. C₂×#103	C₂²×C₄₁	semidirect
352. C₂×#103	C₂²×C₄₁	semidirect
353. C₂×#103	C₂²×C₄₁	semidirect
359. #391	C₂²×C₄₁	semidirect
360. #389	C₂²×C₄₁	semidirect
365. C₄×C₄⋊C₄	C₂²×C₄₁	semidirect
367. #336	C₂²×C₄₁	semidirect
368. C₄₁	#24217	direct (N × H)
614. C₂²	C₄₁×#464	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#124×C₄₁	2
C₂×C₄₁×#464	2
C₂×C₄₁×#451	2
C₂×C₄₁×#450	2
C₄×C₂²⋊Q₈×C₄₁	2
C₄₁×#2042	2
C₄₁×#2015	2
C₄₁×#1983	2
C₄₁×#1797	2
C₄₁×#1787	2
C₄₁×#1786	2
#24217	41

Subgroups — C41×#24217 #23800418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24217 #23800418