Subgroups — C₄₁×#24121 #23800322

Order 10496.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#555	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×SD₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2538	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2033	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2000	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1954	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1873	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1854	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1802	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#555	—	2	4	no	—
14	C₄₁×#555	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×#124×C₄₁	—	2	2	no	—
17	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
18	C₂×#121×C₄₁	—	2	1	no	—
19	C₂×#121×C₄₁	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×#117×C₄₁	—	2	1	no	—
24	C₂×#117×C₄₁	C₂²	1	2	yes	—
25	C₄₁×#477	—	2	4	no	—
26	C₄×SD₁₆×C₄₁	—	2	4	no	—
27	C₂×#107×C₄₁	C₂²	1	2	yes	—
28	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
30	C₂×#103×C₄₁	C₂²	1	2	yes	—
31	C₂×#103×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×#102×C₄₁	C₂²	1	1	yes	characteristic
33	C₂×#102×C₄₁	C₂²	1	2	yes	—
34	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
35	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
37	C₄₁×#396	—	2	2	no	—
38	C₄₁×#390	—	2	1	no	—
39	C₄₁×#389	—	2	1	no	—
40	C₄₁×#384	C₂²	1	1	yes	characteristic
41	C₄₁×#380	C₂²	1	2	yes	—
42	C₄₁×#340	C₂²	1	2	yes	—
43	C₄₁×#340	C₂²	1	2	yes	—
44	C₄₁×#340	C₂²	1	2	yes	—
45	C₂²×Q₈×C₄₁	—	2	1	no	—
46	C₂²×Q₈×C₄₁	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₄₁	—	2	2	no	—
48	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
49	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₄₁	—	2	1	no	—
51	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
52	C₂×SD₁₆×C₄₁	—	2	2	no	—
53	C₂×SD₁₆×C₄₁	—	2	2	no	—
54	C₂×SD₁₆×C₄₁	—	2	2	no	—
55	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
57	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
58	C₄⋊Q₈×C₄₁	—	2	2	no	—
59	C₄⋊Q₈×C₄₁	D₄	1	4	yes	—
60	#124×C₄₁	—	2	16	no	—
61	C₂²⋊Q₈×C₄₁	—	2	4	no	—
62	C₂²⋊Q₈×C₄₁	—	4	2	no	—
63	#121×C₄₁	—	2	8	no	—
64	#121×C₄₁	—	2	4	no	—
65	#121×C₄₁	—	2	4	no	—
66	C₄×Q₈×C₄₁	—	2	2	no	—
67	C₄×Q₈×C₄₁	—	2	2	no	—
68	C₄×D₄×C₄₁	—	2	2	no	—
69	C₄×D₄×C₄₁	—	2	2	no	—
70	C₄×D₄×C₄₁	—	2	2	no	—
71	C₄×D₄×C₄₁	D₄	1	4	yes	—
72	#117×C₄₁	—	2	8	no	—
73	#117×C₄₁	—	2	4	no	—
74	#117×C₄₁	—	2	4	no	—
75	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
76	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
78	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
85	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
88	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
90	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
91	C₂×C₄²×C₄₁	—	2	2	no	—
92	C₂×C₄²×C₄₁	—	2	1	no	—
93	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
94	#107×C₄₁	—	2	4	no	—
95	#106×C₄₁	—	2	2	no	—
96	C₄⋊C₈×C₄₁	—	2	2	no	—
97	#103×C₄₁	—	2	2	no	—
98	#103×C₄₁	—	2	4	no	—
99	#102×C₄₁	—	2	2	no	—
100	#102×C₄₁	—	4	2	no	—
101	C₂²⋊C₈×C₄₁	—	2	4	no	—
102	C₂²⋊C₈×C₄₁	—	4	2	no	—
103	C₄×C₈×C₄₁	—	2	2	no	—
104	#95×C₄₁	—	2	2	no	—
105	C₂⁴×C₄₁	—	2	1	no	—
106	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
107	C₂×Q₈×C₄₁	—	2	4	no	—
108	C₂×Q₈×C₄₁	—	2	1	no	—
109	C₂×Q₈×C₄₁	—	2	1	no	—
110	C₂×Q₈×C₄₁	—	2	1	no	—
111	C₂×Q₈×C₄₁	—	2	1	no	—
112	C₂×Q₈×C₄₁	—	2	1	no	—
113	C₂×Q₈×C₄₁	—	8	1	no	—
114	C₂×Q₈×C₄₁	—	8	1	no	—
115	C₂×Q₈×C₄₁	#41	1	2	yes	—
116	C₂×D₄×C₄₁	—	2	8	no	—
117	C₂×D₄×C₄₁	—	2	2	no	—
118	C₂×D₄×C₄₁	—	2	2	no	—
119	C₂×D₄×C₄₁	—	2	2	no	—
120	C₂×D₄×C₄₁	—	2	1	no	—
121	C₂×D₄×C₄₁	—	2	1	no	—
122	C₂×D₄×C₄₁	—	2	1	no	—
123	C₂×D₄×C₄₁	—	2	1	no	—
124	C₂×D₄×C₄₁	—	2	1	no	—
125	C₂×D₄×C₄₁	—	8	2	no	—
126	C₂×D₄×C₄₁	—	8	2	no	—
127	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₄₁	—	2	1	no	—
129	C₂²×C₄×C₄₁	—	2	1	no	—
130	C₂²×C₄×C₄₁	—	2	2	no	—
131	C₂²×C₄×C₄₁	—	2	4	no	—
132	C₂²×C₄×C₄₁	—	2	2	no	—
133	C₂²×C₄×C₄₁	—	2	2	no	—
134	C₂²×C₄×C₄₁	—	2	2	no	—
135	C₂²×C₄×C₄₁	—	2	2	no	—
136	C₂²×C₄×C₄₁	—	2	2	no	—
137	C₂²×C₄×C₄₁	—	2	2	no	—
138	C₂²×C₄×C₄₁	—	2	1	no	—
139	C₂²×C₄×C₄₁	—	2	2	no	—
140	C₂²×C₄×C₄₁	—	4	2	no	—
141	C₂²×C₄×C₄₁	—	4	2	no	—
142	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
146	SD₁₆×C₄₁	—	2	8	no	—
147	C₂×C₈×C₄₁	—	2	1	no	—
148	C₂×C₈×C₄₁	—	2	1	no	—
149	C₂×C₈×C₄₁	—	2	1	no	—
150	C₂×C₈×C₄₁	—	2	2	no	—
151	C₂×C₈×C₄₁	—	2	2	no	—
152	C₂×C₈×C₄₁	—	2	1	no	—
153	C₂×C₈×C₄₁	—	2	2	no	—
154	C₂×C₈×C₄₁	—	2	1	no	—
155	C₂×C₈×C₄₁	#41	1	2	yes	—
156	C₄⋊C₄×C₄₁	—	2	4	no	—
157	C₄⋊C₄×C₄₁	—	2	2	no	—
158	C₄⋊C₄×C₄₁	—	2	1	no	—
159	C₄⋊C₄×C₄₁	—	2	2	no	—
160	C₄⋊C₄×C₄₁	—	2	1	no	—
161	C₄⋊C₄×C₄₁	—	2	1	no	—
162	C₄⋊C₄×C₄₁	—	2	2	no	—
163	C₄⋊C₄×C₄₁	—	2	2	no	—
164	C₄⋊C₄×C₄₁	—	4	4	no	—
165	C₄⋊C₄×C₄₁	—	4	2	no	—
166	C₄⋊C₄×C₄₁	—	4	2	no	—
167	C₄⋊C₄×C₄₁	—	8	1	no	—
168	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
169	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
170	C₂²⋊C₄×C₄₁	—	2	8	no	—
171	C₂²⋊C₄×C₄₁	—	2	8	no	—
172	C₂²⋊C₄×C₄₁	—	2	4	no	—
173	C₂²⋊C₄×C₄₁	—	2	4	no	—
174	C₂²⋊C₄×C₄₁	—	4	4	no	—
175	C₂²⋊C₄×C₄₁	—	8	2	no	—
176	C₄²×C₄₁	—	2	8	no	—
177	C₄²×C₄₁	—	4	2	no	—
178	C₄²×C₄₁	C₂×D₄	1	2	yes	—
179	C₄²×C₄₁	C₂×D₄	1	2	yes	—
180	C₂³×C₄₁	—	2	2	no	—
181	C₂³×C₄₁	—	2	2	no	—
182	C₂³×C₄₁	—	2	2	no	—
183	C₂³×C₄₁	—	2	2	no	—
184	C₂³×C₄₁	—	2	2	no	—
185	C₂³×C₄₁	—	2	2	no	—
186	C₂³×C₄₁	—	2	2	no	—
187	C₂³×C₄₁	—	2	2	no	—
188	C₂³×C₄₁	—	4	1	no	—
189	C₂³×C₄₁	—	4	1	no	—
190	C₂³×C₄₁	—	4	1	no	—
191	C₂³×C₄₁	—	4	1	no	—
192	C₂³×C₄₁	—	4	1	no	—
193	C₂³×C₄₁	—	4	1	no	—
194	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
195	Q₈×C₄₁	—	2	2	no	—
196	Q₈×C₄₁	—	2	2	no	—
197	Q₈×C₄₁	—	4	1	no	—
198	Q₈×C₄₁	—	4	1	no	—
199	Q₈×C₄₁	—	8	4	no	—
200	D₄×C₄₁	—	2	2	no	—
201	D₄×C₄₁	—	2	2	no	—
202	D₄×C₄₁	—	4	1	no	—
203	D₄×C₄₁	—	4	1	no	—
204	D₄×C₄₁	—	4	4	no	—
205	D₄×C₄₁	—	8	8	no	—
206	C₂×C₄×C₄₁	—	2	1	no	—
207	C₂×C₄×C₄₁	—	2	1	no	—
208	C₂×C₄×C₄₁	—	2	2	no	—
209	C₂×C₄×C₄₁	—	2	2	no	—
210	C₂×C₄×C₄₁	—	2	4	no	—
211	C₂×C₄×C₄₁	—	2	2	no	—
212	C₂×C₄×C₄₁	—	2	2	no	—
213	C₂×C₄×C₄₁	—	2	2	no	—
214	C₂×C₄×C₄₁	—	4	1	no	—
215	C₂×C₄×C₄₁	—	4	1	no	—
216	C₂×C₄×C₄₁	—	4	1	no	—
217	C₂×C₄×C₄₁	—	4	1	no	—
218	C₂×C₄×C₄₁	—	4	2	no	—
219	C₂×C₄×C₄₁	—	4	2	no	—
220	C₂×C₄×C₄₁	—	4	2	no	—
221	C₂×C₄×C₄₁	—	4	2	no	—
222	C₂×C₄×C₄₁	—	4	2	no	—
223	C₂×C₄×C₄₁	—	4	2	no	—
224	C₂×C₄×C₄₁	—	4	8	no	—
225	C₂×C₄×C₄₁	—	4	1	no	—
226	C₂×C₄×C₄₁	—	4	2	no	—
227	C₂×C₄×C₄₁	—	4	2	no	—
228	C₂×C₄×C₄₁	—	4	1	no	—
229	C₂×C₄×C₄₁	—	4	2	no	—
230	C₂×C₄×C₄₁	—	4	1	no	—
231	C₂×C₄×C₄₁	#120	1	2	yes	—
232	C₂×C₄×C₄₁	#120	1	2	yes	—
233	C₂×C₄×C₄₁	#135	1	2	yes	—
234	C₂×C₄×C₄₁	#137	1	2	yes	—
235	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
236	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
238	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
239	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
240	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
241	C₈×C₄₁	—	2	2	no	—
242	C₈×C₄₁	—	4	1	no	—
243	C₈×C₄₁	—	4	2	no	—
244	#24121	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
245	C₂²×C₄₁	—	4	2	no	—
246	C₂²×C₄₁	—	4	2	no	—
247	C₂²×C₄₁	—	4	2	no	—
248	C₂²×C₄₁	—	4	2	no	—
249	C₂²×C₄₁	—	4	2	no	—
250	C₂²×C₄₁	—	4	2	no	—
251	C₂²×C₄₁	—	4	2	no	—
252	C₂²×C₄₁	—	4	2	no	—
253	C₂²×C₄₁	—	4	2	no	—
254	C₂²×C₄₁	—	4	2	no	—
255	C₂²×C₄₁	—	8	1	no	—
256	C₂²×C₄₁	—	8	1	no	—
257	C₂²×C₄₁	—	8	1	no	—
258	C₂²×C₄₁	—	8	1	no	—
259	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
260	C₂²×C₄₁	#546	1	1	yes	characteristic
261	C₂²×C₄₁	#547	1	1	yes	characteristic
262	C₂²×C₄₁	C₂×#135	1	1	yes	characteristic
263	C₂²×C₄₁	C₂×#137	1	1	yes	characteristic
264	C₂²×C₄₁	#576	1	1	yes	characteristic
265	C₂²×C₄₁	#577	1	1	yes	characteristic
266	C₄×C₄₁	—	2	2	no	—
267	C₄×C₄₁	—	4	2	no	—
268	C₄×C₄₁	—	4	2	no	—
269	C₄×C₄₁	—	4	2	no	—
270	C₄×C₄₁	—	4	4	no	—
271	C₄×C₄₁	—	8	1	no	—
272	C₄×C₄₁	—	8	2	no	—
273	C₄×C₄₁	#452	1	4	yes	—
274	C₄×C₄₁	#525	1	1	yes	characteristic
275	C₄×C₄₁	#537	1	1	yes	characteristic
276	C₄×C₄₁	#548	1	1	yes	characteristic
277	C₄×C₄₁	#550	1	1	yes	characteristic
278	C₂×#555	C₂×C₄₁	1	1	yes	characteristic
279	C₂×#477	C₂×C₄₁	1	1	yes	characteristic, complemented
280	C₂×C₄×SD₁₆	C₂×C₄₁	1	1	yes	characteristic, complemented
281	#2538	C₂×C₄₁	1	1	yes	characteristic, complemented
282	#2033	C₂×C₄₁	1	2	yes	complemented
283	#2000	C₂×C₄₁	1	2	yes	complemented
284	#1990	C₂×C₄₁	1	1	yes	characteristic, complemented
285	#1954	C₂×C₄₁	1	1	yes	characteristic, complemented
286	#1873	C₂×C₄₁	1	2	yes	complemented
287	#1854	C₂×C₄₁	1	2	yes	complemented
288	#1802	C₂×C₄₁	1	1	yes	characteristic, complemented
289	C₂×C₄₁	—	8	2	no	—
290	C₂×C₄₁	—	8	2	no	—
291	C₂×C₄₁	#2407	1	1	yes	characteristic
292	C₂×C₄₁	C₂×#452	1	1	yes	characteristic
293	C₂×C₄₁	#3012	1	1	yes	characteristic
294	C₂×C₄₁	#3305	1	1	yes	characteristic
295	C₂×C₄₁	#3329	1	1	yes	characteristic
296	C₂×C₄₁	#3340	1	1	yes	characteristic
297	C₂×C₄₁	#3357	1	1	yes	characteristic
298	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
299	#555	—	2	4	no	—
300	#555	—	2	4	no	—
301	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
302	C₂×#124	—	2	2	no	—
303	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
304	C₂×#121	—	2	1	no	—
305	C₂×#121	C₂²×C₄₁	1	2	yes	—
306	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
307	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
308	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
309	C₂×#117	—	2	1	no	—
310	C₂×#117	C₂²×C₄₁	1	2	yes	—
311	#477	—	2	4	no	—
312	C₄×SD₁₆	—	2	4	no	—
313	C₂×#107	C₂²×C₄₁	1	2	yes	—
314	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
315	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
316	C₂×#103	C₂²×C₄₁	1	2	yes	—
317	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
318	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
319	C₂×#102	C₂²×C₄₁	1	2	yes	—
320	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
321	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
322	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
323	#396	—	2	2	no	—
324	#390	—	2	1	no	—
325	#389	—	2	1	no	—
326	#384	C₂²×C₄₁	1	1	yes	characteristic
327	#380	C₂²×C₄₁	1	2	yes	—
328	#340	C₂²×C₄₁	1	2	yes	—
329	#340	C₂²×C₄₁	1	2	yes	—
330	#340	C₂²×C₄₁	1	2	yes	—
331	C₄₁	#24121	1	1	yes	characteristic, direct product factor, Sylow
332	C₂²×Q₈	—	2	1	no	—
333	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
334	C₂²×D₄	—	2	2	no	—
335	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
336	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
337	C₂³×C₄	—	2	1	no	—
338	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
339	C₂×SD₁₆	—	2	2	no	—
340	C₂×SD₁₆	—	2	2	no	—
341	C₂×SD₁₆	—	2	2	no	—
342	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
343	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
344	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
345	C₄⋊Q₈	—	2	2	no	—
346	C₄⋊Q₈	D₄×C₄₁	1	4	yes	—
347	#124	—	2	16	no	—
348	C₂²⋊Q₈	—	2	4	no	—
349	C₂²⋊Q₈	—	4	2	no	—
350	#121	—	2	8	no	—
351	#121	—	2	4	no	—
352	#121	—	2	4	no	—
353	C₄×Q₈	—	2	2	no	—
354	C₄×Q₈	—	2	2	no	—
355	C₄×D₄	—	2	2	no	—
356	C₄×D₄	—	2	2	no	—
357	C₄×D₄	—	2	2	no	—
358	C₄×D₄	D₄×C₄₁	1	4	yes	—
359	#117	—	2	8	no	—
360	#117	—	2	4	no	—
361	#117	—	2	4	no	—
362	C₂×C₄⋊C₄	—	2	1	no	—
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄⋊C₄	—	2	2	no	—
365	C₂×C₄⋊C₄	—	2	1	no	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
369	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
371	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
372	C₂×C₂²⋊C₄	—	2	2	no	—
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	—	2	1	no	—
375	C₂×C₂²⋊C₄	—	2	2	no	—
376	C₂×C₂²⋊C₄	—	2	2	no	—
377	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
378	C₂×C₄²	—	2	2	no	—
379	C₂×C₄²	—	2	1	no	—
380	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
381	#107	—	2	4	no	—
382	#106	—	2	2	no	—
383	C₄⋊C₈	—	2	2	no	—
384	#103	—	2	2	no	—
385	#103	—	2	4	no	—
386	#102	—	2	2	no	—
387	#102	—	4	2	no	—
388	C₂²⋊C₈	—	2	4	no	—
389	C₂²⋊C₈	—	4	2	no	—
390	C₄×C₈	—	2	2	no	—
391	#95	—	2	2	no	—
392	C₂⁴	—	2	1	no	—
393	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
394	C₂×Q₈	—	2	4	no	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	1	no	—
398	C₂×Q₈	—	2	1	no	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×Q₈	—	8	1	no	—
401	C₂×Q₈	—	8	1	no	—
402	C₂×Q₈	#41×C₄₁	1	2	yes	—
403	C₂×D₄	—	2	8	no	—
404	C₂×D₄	—	2	2	no	—
405	C₂×D₄	—	2	2	no	—
406	C₂×D₄	—	2	2	no	—
407	C₂×D₄	—	2	1	no	—
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	—	2	1	no	—
410	C₂×D₄	—	2	1	no	—
411	C₂×D₄	—	2	1	no	—
412	C₂×D₄	—	8	2	no	—
413	C₂×D₄	—	8	2	no	—
414	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	1	no	—
417	C₂²×C₄	—	2	2	no	—
418	C₂²×C₄	—	2	4	no	—
419	C₂²×C₄	—	2	2	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	2	no	—
423	C₂²×C₄	—	2	2	no	—
424	C₂²×C₄	—	2	2	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	4	2	no	—
428	C₂²×C₄	—	4	2	no	—
429	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
430	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
431	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
432	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
433	SD₁₆	—	2	8	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	1	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	2	no	—
438	C₂×C₈	—	2	2	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	#41×C₄₁	1	2	yes	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	4	4	no	—
452	C₄⋊C₄	—	4	2	no	—
453	C₄⋊C₄	—	4	2	no	—
454	C₄⋊C₄	—	8	1	no	—
455	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
456	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
457	C₂²⋊C₄	—	2	8	no	—
458	C₂²⋊C₄	—	2	8	no	—
459	C₂²⋊C₄	—	2	4	no	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	4	4	no	—
462	C₂²⋊C₄	—	8	2	no	—
463	C₄²	—	2	8	no	—
464	C₄²	—	4	2	no	—
465	C₄²	C₂×D₄×C₄₁	1	2	yes	—
466	C₄²	C₂×D₄×C₄₁	1	2	yes	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	2	no	—
469	C₂³	—	2	2	no	—
470	C₂³	—	2	2	no	—
471	C₂³	—	2	2	no	—
472	C₂³	—	2	2	no	—
473	C₂³	—	2	2	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	4	1	no	—
476	C₂³	—	4	1	no	—
477	C₂³	—	4	1	no	—
478	C₂³	—	4	1	no	—
479	C₂³	—	4	1	no	—
480	C₂³	—	4	1	no	—
481	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
482	Q₈	—	2	2	no	—
483	Q₈	—	2	2	no	—
484	Q₈	—	4	1	no	—
485	Q₈	—	4	1	no	—
486	Q₈	—	8	4	no	—
487	D₄	—	2	2	no	—
488	D₄	—	2	2	no	—
489	D₄	—	4	1	no	—
490	D₄	—	4	1	no	—
491	D₄	—	4	4	no	—
492	D₄	—	8	8	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	1	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	2	no	—
506	C₂×C₄	—	4	2	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	2	no	—
509	C₂×C₄	—	4	2	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	—	4	8	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	4	1	no	—
516	C₂×C₄	—	4	2	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	#120×C₄₁	1	2	yes	—
519	C₂×C₄	#120×C₄₁	1	2	yes	—
520	C₂×C₄	#135×C₄₁	1	2	yes	—
521	C₂×C₄	#137×C₄₁	1	2	yes	—
522	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
523	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
524	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
525	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
526	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
527	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
528	C₈	—	2	2	no	—
529	C₈	—	4	1	no	—
530	C₈	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	2	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	2	no	—
538	C₂²	—	4	2	no	—
539	C₂²	—	4	2	no	—
540	C₂²	—	4	2	no	—
541	C₂²	—	8	1	no	—
542	C₂²	—	8	1	no	—
543	C₂²	—	8	1	no	—
544	C₂²	—	8	1	no	—
545	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
546	C₂²	C₄₁×#546	1	1	yes	characteristic
547	C₂²	C₄₁×#547	1	1	yes	characteristic
548	C₂²	C₂×#135×C₄₁	1	1	yes	characteristic
549	C₂²	C₂×#137×C₄₁	1	1	yes	characteristic
550	C₂²	C₄₁×#576	1	1	yes	characteristic
551	C₂²	C₄₁×#577	1	1	yes	characteristic
552	C₄	—	2	2	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	2	no	—
555	C₄	—	4	2	no	—
556	C₄	—	4	4	no	—
557	C₄	—	8	2	no	—
558	C₄	—	8	1	no	—
559	C₄	C₄₁×#452	1	4	yes	—
560	C₄	C₄₁×#525	1	1	yes	characteristic
561	C₄	C₄₁×#537	1	1	yes	characteristic
562	C₄	C₄₁×#548	1	1	yes	characteristic
563	C₄	C₄₁×#550	1	1	yes	characteristic
564	C₂	—	8	2	no	—
565	C₂	—	8	2	no	—
566	C₂	C₄₁×#2407	1	1	yes	characteristic
567	C₂	C₂×C₄₁×#452	1	1	yes	characteristic
568	C₂	C₄₁×#3012	1	1	yes	characteristic
569	C₂	C₄₁×#3305	1	1	yes	characteristic
570	C₂	C₄₁×#3329	1	1	yes	characteristic
571	C₂	C₄₁×#3340	1	1	yes	characteristic
572	C₂	C₄₁×#3357	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×C₄₁×#555	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄×D₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#477	C₂	semidirect
3. C₂×C₄×SD₁₆×C₄₁	C₂	semidirect
4. C₄₁×#2538	C₂	semidirect
5. C₄₁×#2033	C₂	semidirect
6. C₄₁×#2000	C₂	semidirect
7. C₄₁×#1990	C₂	semidirect
8. C₄₁×#1954	C₂	semidirect
9. C₄₁×#1873	C₂	semidirect
10. C₄₁×#1854	C₂	semidirect
11. C₄₁×#1802	C₂	semidirect
244. #24121	C₄₁	direct (N × H)
279. C₂×#477	C₂×C₄₁	semidirect
280. C₂×C₄×SD₁₆	C₂×C₄₁	semidirect
281. #2538	C₂×C₄₁	semidirect
282. #2033	C₂×C₄₁	semidirect
283. #2000	C₂×C₄₁	semidirect
284. #1990	C₂×C₄₁	semidirect
285. #1954	C₂×C₄₁	semidirect
286. #1873	C₂×C₄₁	semidirect
287. #1854	C₂×C₄₁	semidirect
288. #1802	C₂×C₄₁	semidirect
331. C₄₁	#24121	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#555	2
C₂×C₄₁×#477	2
C₂×C₄×SD₁₆×C₄₁	2
C₄₁×#2538	2
C₄₁×#2033	2
C₄₁×#2000	2
C₄₁×#1990	2
C₄₁×#1954	2
C₄₁×#1873	2
C₄₁×#1854	2
C₄₁×#1802	2
#24121	41

Subgroups — C41×#24121 #23800322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24121 #23800322