Subgroups — C₄₁×#24133 #23800334

Order 10496.

1140 conjugacy classes of subgroups.

The subgroup lattice has 540 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×#475	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#440	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#2536	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2034	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2000	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1955	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×#1804	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×#475	—	2	4	no	—
26	C₄₁×#440	—	2	4	no	—
27	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
28	C₂×#106×C₄₁	C₂²	1	2	yes	—
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×#97×C₄₁	C₂²	1	1	yes	characteristic
37	C₄₁×#398	—	2	1	no	—
38	C₄₁×#394	—	2	1	no	—
39	C₄₁×#388	—	2	2	no	—
40	C₄₁×#382	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	D₄	1	1	yes	characteristic
79	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
83	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
85	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
86	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
89	C₂×C₄²×C₄₁	—	2	2	no	—
90	C₂×C₄²×C₄₁	—	2	1	no	—
91	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
92	#107×C₄₁	—	2	2	no	—
93	#106×C₄₁	—	2	4	no	—
94	C₄⋊C₈×C₄₁	—	2	2	no	—
95	#103×C₄₁	—	2	2	no	—
96	#103×C₄₁	—	2	4	no	—
97	#102×C₄₁	—	2	2	no	—
98	#102×C₄₁	—	4	2	no	—
99	C₂²⋊C₈×C₄₁	—	2	4	no	—
100	C₂²⋊C₈×C₄₁	—	4	2	no	—
101	#97×C₄₁	—	2	2	no	—
102	#95×C₄₁	—	2	2	no	—
103	#95×C₄₁	—	2	1	no	—
104	#95×C₄₁	—	2	1	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	2	no	—
114	C₂×Q₈×C₄₁	#41	1	2	yes	—
115	C₂×D₄×C₄₁	—	2	8	no	—
116	C₂×D₄×C₄₁	—	2	2	no	—
117	C₂×D₄×C₄₁	—	2	1	no	—
118	C₂×D₄×C₄₁	—	2	1	no	—
119	C₂×D₄×C₄₁	—	2	1	no	—
120	C₂×D₄×C₄₁	—	2	1	no	—
121	C₂×D₄×C₄₁	—	2	1	no	—
122	C₂×D₄×C₄₁	—	4	2	no	—
123	C₂×D₄×C₄₁	—	8	2	no	—
124	C₂×D₄×C₄₁	—	8	2	no	—
125	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₄₁	—	2	1	no	—
127	C₂²×C₄×C₄₁	—	2	1	no	—
128	C₂²×C₄×C₄₁	—	2	2	no	—
129	C₂²×C₄×C₄₁	—	2	4	no	—
130	C₂²×C₄×C₄₁	—	2	2	no	—
131	C₂²×C₄×C₄₁	—	2	2	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	1	no	—
138	C₂²×C₄×C₄₁	—	4	4	no	—
139	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
143	SD₁₆×C₄₁	—	2	8	no	—
144	C₂×C₈×C₄₁	—	2	1	no	—
145	C₂×C₈×C₄₁	—	2	1	no	—
146	C₂×C₈×C₄₁	—	2	1	no	—
147	C₂×C₈×C₄₁	—	2	2	no	—
148	C₂×C₈×C₄₁	—	2	2	no	—
149	C₂×C₈×C₄₁	—	2	1	no	—
150	C₂×C₈×C₄₁	—	2	2	no	—
151	C₂×C₈×C₄₁	—	2	1	no	—
152	C₂×C₈×C₄₁	#41	1	2	yes	—
153	C₄⋊C₄×C₄₁	—	2	4	no	—
154	C₄⋊C₄×C₄₁	—	2	2	no	—
155	C₄⋊C₄×C₄₁	—	2	1	no	—
156	C₄⋊C₄×C₄₁	—	2	2	no	—
157	C₄⋊C₄×C₄₁	—	2	1	no	—
158	C₄⋊C₄×C₄₁	—	2	1	no	—
159	C₄⋊C₄×C₄₁	—	2	2	no	—
160	C₄⋊C₄×C₄₁	—	2	2	no	—
161	C₄⋊C₄×C₄₁	—	4	4	no	—
162	C₄⋊C₄×C₄₁	—	4	2	no	—
163	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
164	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
165	C₂²⋊C₄×C₄₁	—	2	8	no	—
166	C₂²⋊C₄×C₄₁	—	2	8	no	—
167	C₂²⋊C₄×C₄₁	—	2	4	no	—
168	C₂²⋊C₄×C₄₁	—	2	4	no	—
169	C₂²⋊C₄×C₄₁	—	4	4	no	—
170	C₂²⋊C₄×C₄₁	—	4	4	no	—
171	C₄²×C₄₁	—	2	8	no	—
172	C₄²×C₄₁	—	4	2	no	—
173	C₄²×C₄₁	C₂×D₄	1	2	yes	—
174	C₄²×C₄₁	C₂×D₄	1	2	yes	—
175	C₂³×C₄₁	—	2	2	no	—
176	C₂³×C₄₁	—	2	2	no	—
177	C₂³×C₄₁	—	2	2	no	—
178	C₂³×C₄₁	—	2	2	no	—
179	C₂³×C₄₁	—	2	2	no	—
180	C₂³×C₄₁	—	2	2	no	—
181	C₂³×C₄₁	—	2	2	no	—
182	C₂³×C₄₁	—	2	2	no	—
183	C₂³×C₄₁	—	4	1	no	—
184	C₂³×C₄₁	—	4	1	no	—
185	C₂³×C₄₁	—	4	2	no	—
186	C₂³×C₄₁	—	4	2	no	—
187	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
188	Q₈×C₄₁	—	2	2	no	—
189	Q₈×C₄₁	—	2	2	no	—
190	Q₈×C₄₁	—	4	1	no	—
191	Q₈×C₄₁	—	4	1	no	—
192	Q₈×C₄₁	—	8	4	no	—
193	D₄×C₄₁	—	2	2	no	—
194	D₄×C₄₁	—	2	2	no	—
195	D₄×C₄₁	—	4	4	no	—
196	D₄×C₄₁	—	4	1	no	—
197	D₄×C₄₁	—	4	1	no	—
198	D₄×C₄₁	—	8	8	no	—
199	C₂×C₄×C₄₁	—	2	2	no	—
200	C₂×C₄×C₄₁	—	2	2	no	—
201	C₂×C₄×C₄₁	—	2	2	no	—
202	C₂×C₄×C₄₁	—	2	2	no	—
203	C₂×C₄×C₄₁	—	2	4	no	—
204	C₂×C₄×C₄₁	—	2	2	no	—
205	C₂×C₄×C₄₁	—	2	2	no	—
206	C₂×C₄×C₄₁	—	2	2	no	—
207	C₂×C₄×C₄₁	—	4	8	no	—
208	C₂×C₄×C₄₁	—	4	1	no	—
209	C₂×C₄×C₄₁	—	4	1	no	—
210	C₂×C₄×C₄₁	—	4	2	no	—
211	C₂×C₄×C₄₁	—	4	2	no	—
212	C₂×C₄×C₄₁	—	4	2	no	—
213	C₂×C₄×C₄₁	—	4	2	no	—
214	C₂×C₄×C₄₁	—	4	2	no	—
215	C₂×C₄×C₄₁	—	4	2	no	—
216	C₂×C₄×C₄₁	—	4	2	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	#120	1	2	yes	—
223	C₂×C₄×C₄₁	#120	1	2	yes	—
224	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
227	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
228	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
230	C₈×C₄₁	—	2	2	no	—
231	C₈×C₄₁	—	4	1	no	—
232	C₈×C₄₁	—	4	2	no	—
233	#24133	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
234	C₂²×C₄₁	—	4	2	no	—
235	C₂²×C₄₁	—	4	2	no	—
236	C₂²×C₄₁	—	4	2	no	—
237	C₂²×C₄₁	—	4	2	no	—
238	C₂²×C₄₁	—	4	2	no	—
239	C₂²×C₄₁	—	4	2	no	—
240	C₂²×C₄₁	—	4	2	no	—
241	C₂²×C₄₁	—	4	2	no	—
242	C₂²×C₄₁	—	4	2	no	—
243	C₂²×C₄₁	—	4	2	no	—
244	C₂²×C₄₁	—	8	2	no	—
245	C₂²×C₄₁	—	8	2	no	—
246	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
247	C₂²×C₄₁	#546	1	1	yes	characteristic
248	C₂²×C₄₁	#547	1	1	yes	characteristic
249	C₂²×C₄₁	#575	1	2	yes	—
250	C₂²×C₄₁	#577	1	2	yes	—
251	C₄×C₄₁	—	2	2	no	—
252	C₄×C₄₁	—	2	2	no	—
253	C₄×C₄₁	—	4	2	no	—
254	C₄×C₄₁	—	4	2	no	—
255	C₄×C₄₁	—	4	2	no	—
256	C₄×C₄₁	—	4	4	no	—
257	C₄×C₄₁	—	8	1	no	—
258	C₄×C₄₁	—	8	2	no	—
259	C₄×C₄₁	#525	1	1	yes	characteristic
260	C₄×C₄₁	#537	1	1	yes	characteristic
261	C₄×C₄₁	#548	1	1	yes	characteristic
262	C₄×C₄₁	#550	1	1	yes	characteristic
263	C₂×#555	C₂×C₄₁	1	1	yes	characteristic
264	C₂×#475	C₂×C₄₁	1	1	yes	characteristic, complemented
265	C₂×#440	C₂×C₄₁	1	1	yes	characteristic, complemented
266	#2536	C₂×C₄₁	1	1	yes	characteristic, complemented
267	#2034	C₂×C₄₁	1	2	yes	complemented
268	#2000	C₂×C₄₁	1	2	yes	complemented
269	#1993	C₂×C₄₁	1	1	yes	characteristic, complemented
270	#1955	C₂×C₄₁	1	1	yes	characteristic, complemented
271	#1873	C₂×C₄₁	1	2	yes	complemented
272	#1854	C₂×C₄₁	1	2	yes	complemented
273	#1804	C₂×C₄₁	1	1	yes	characteristic, complemented
274	C₂×C₄₁	—	8	2	no	—
275	C₂×C₄₁	—	8	2	no	—
276	C₂×C₄₁	#2407	1	1	yes	characteristic
277	C₂×C₄₁	#3005	1	1	yes	characteristic
278	C₂×C₄₁	#3009	1	1	yes	characteristic
279	C₂×C₄₁	#3321	1	2	yes	—
280	C₂×C₄₁	#3348	1	2	yes	—
281	C₂²×SD₁₆	C₂²×C₄₁	1	1	yes	characteristic
282	#555	—	2	4	no	—
283	#555	—	2	4	no	—
284	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
285	C₂×#124	—	2	2	no	—
286	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
287	C₂×#121	—	2	1	no	—
288	C₂×#121	C₂²×C₄₁	1	2	yes	—
289	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
290	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
291	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
292	C₂×#117	—	2	1	no	—
293	C₂×#117	C₂²×C₄₁	1	2	yes	—
294	#475	—	2	4	no	—
295	#440	—	2	4	no	—
296	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
297	C₂×#106	C₂²×C₄₁	1	2	yes	—
298	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
299	C₂×#103	C₂²×C₄₁	1	2	yes	—
300	C₂×#103	C₂²×C₄₁	1	1	yes	characteristic
301	C₂×#102	C₂²×C₄₁	1	1	yes	characteristic
302	C₂×#102	C₂²×C₄₁	1	2	yes	—
303	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
304	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
305	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
306	#398	—	2	1	no	—
307	#394	—	2	1	no	—
308	#388	—	2	2	no	—
309	#382	C₂²×C₄₁	1	1	yes	characteristic
310	#380	C₂²×C₄₁	1	2	yes	—
311	#340	C₂²×C₄₁	1	2	yes	—
312	#340	C₂²×C₄₁	1	2	yes	—
313	#340	C₂²×C₄₁	1	2	yes	—
314	C₄₁	#24133	1	1	yes	characteristic, direct product factor, Sylow
315	C₂²×Q₈	—	2	1	no	—
316	C₂²×Q₈	C₂³×C₄₁	1	1	yes	characteristic
317	C₂²×D₄	—	2	2	no	—
318	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
319	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
320	C₂³×C₄	—	2	1	no	—
321	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
322	C₂×SD₁₆	—	2	2	no	—
323	C₂×SD₁₆	—	2	2	no	—
324	C₂×SD₁₆	—	2	2	no	—
325	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
326	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
327	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
328	C₄⋊Q₈	—	2	2	no	—
329	C₄⋊Q₈	D₄×C₄₁	1	4	yes	—
330	#124	—	2	16	no	—
331	C₂²⋊Q₈	—	2	4	no	—
332	C₂²⋊Q₈	—	4	2	no	—
333	#121	—	2	8	no	—
334	#121	—	2	4	no	—
335	#121	—	2	4	no	—
336	C₄×Q₈	—	2	2	no	—
337	C₄×Q₈	—	2	2	no	—
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	—	2	2	no	—
340	C₄×D₄	—	2	2	no	—
341	C₄×D₄	D₄×C₄₁	1	4	yes	—
342	#117	—	2	8	no	—
343	#117	—	2	4	no	—
344	#117	—	2	4	no	—
345	C₂×C₄⋊C₄	—	2	1	no	—
346	C₂×C₄⋊C₄	—	2	1	no	—
347	C₂×C₄⋊C₄	—	2	2	no	—
348	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
349	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
350	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
351	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
352	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
353	C₂×C₂²⋊C₄	—	2	2	no	—
354	C₂×C₂²⋊C₄	—	2	2	no	—
355	C₂×C₂²⋊C₄	—	2	1	no	—
356	C₂×C₂²⋊C₄	—	2	2	no	—
357	C₂×C₂²⋊C₄	—	2	2	no	—
358	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
359	C₂×C₄²	—	2	2	no	—
360	C₂×C₄²	—	2	1	no	—
361	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
362	#107	—	2	2	no	—
363	#106	—	2	4	no	—
364	C₄⋊C₈	—	2	2	no	—
365	#103	—	2	2	no	—
366	#103	—	2	4	no	—
367	#102	—	2	2	no	—
368	#102	—	4	2	no	—
369	C₂²⋊C₈	—	2	4	no	—
370	C₂²⋊C₈	—	4	2	no	—
371	#97	—	2	2	no	—
372	#95	—	2	2	no	—
373	#95	—	2	1	no	—
374	#95	—	2	1	no	—
375	C₂⁴	—	2	1	no	—
376	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
377	C₂×Q₈	—	2	4	no	—
378	C₂×Q₈	—	2	1	no	—
379	C₂×Q₈	—	2	1	no	—
380	C₂×Q₈	—	2	1	no	—
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	—	2	1	no	—
383	C₂×Q₈	—	8	2	no	—
384	C₂×Q₈	#41×C₄₁	1	2	yes	—
385	C₂×D₄	—	2	8	no	—
386	C₂×D₄	—	2	2	no	—
387	C₂×D₄	—	2	1	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	2	1	no	—
391	C₂×D₄	—	2	1	no	—
392	C₂×D₄	—	4	2	no	—
393	C₂×D₄	—	8	2	no	—
394	C₂×D₄	—	8	2	no	—
395	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	2	4	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	2	no	—
403	C₂²×C₄	—	2	2	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	2	no	—
407	C₂²×C₄	—	2	1	no	—
408	C₂²×C₄	—	4	4	no	—
409	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
410	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
411	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
412	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
413	SD₁₆	—	2	8	no	—
414	C₂×C₈	—	2	1	no	—
415	C₂×C₈	—	2	1	no	—
416	C₂×C₈	—	2	1	no	—
417	C₂×C₈	—	2	2	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	—	2	1	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	1	no	—
422	C₂×C₈	#41×C₄₁	1	2	yes	—
423	C₄⋊C₄	—	2	4	no	—
424	C₄⋊C₄	—	2	2	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	2	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	2	no	—
430	C₄⋊C₄	—	2	2	no	—
431	C₄⋊C₄	—	4	4	no	—
432	C₄⋊C₄	—	4	2	no	—
433	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
434	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
435	C₂²⋊C₄	—	2	8	no	—
436	C₂²⋊C₄	—	2	8	no	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₂²⋊C₄	—	2	4	no	—
439	C₂²⋊C₄	—	4	4	no	—
440	C₂²⋊C₄	—	4	4	no	—
441	C₄²	—	2	8	no	—
442	C₄²	—	4	2	no	—
443	C₄²	C₂×D₄×C₄₁	1	2	yes	—
444	C₄²	C₂×D₄×C₄₁	1	2	yes	—
445	C₂³	—	2	2	no	—
446	C₂³	—	2	2	no	—
447	C₂³	—	2	2	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	2	2	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	2	2	no	—
453	C₂³	—	4	1	no	—
454	C₂³	—	4	1	no	—
455	C₂³	—	4	2	no	—
456	C₂³	—	4	2	no	—
457	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
458	Q₈	—	2	2	no	—
459	Q₈	—	2	2	no	—
460	Q₈	—	4	1	no	—
461	Q₈	—	4	1	no	—
462	Q₈	—	8	4	no	—
463	D₄	—	2	2	no	—
464	D₄	—	2	2	no	—
465	D₄	—	4	4	no	—
466	D₄	—	4	1	no	—
467	D₄	—	4	1	no	—
468	D₄	—	8	8	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	4	8	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	2	no	—
483	C₂×C₄	—	4	2	no	—
484	C₂×C₄	—	4	2	no	—
485	C₂×C₄	—	4	2	no	—
486	C₂×C₄	—	4	2	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	2	no	—
491	C₂×C₄	—	4	2	no	—
492	C₂×C₄	#120×C₄₁	1	2	yes	—
493	C₂×C₄	#120×C₄₁	1	2	yes	—
494	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
495	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
496	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
497	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
498	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
499	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
500	C₈	—	2	2	no	—
501	C₈	—	4	1	no	—
502	C₈	—	4	2	no	—
503	C₂²	—	4	2	no	—
504	C₂²	—	4	2	no	—
505	C₂²	—	4	2	no	—
506	C₂²	—	4	2	no	—
507	C₂²	—	4	2	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	4	2	no	—
510	C₂²	—	4	2	no	—
511	C₂²	—	4	2	no	—
512	C₂²	—	4	2	no	—
513	C₂²	—	8	2	no	—
514	C₂²	—	8	2	no	—
515	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
516	C₂²	C₄₁×#546	1	1	yes	characteristic
517	C₂²	C₄₁×#547	1	1	yes	characteristic
518	C₂²	C₄₁×#575	1	2	yes	—
519	C₂²	C₄₁×#577	1	2	yes	—
520	C₄	—	2	2	no	—
521	C₄	—	2	2	no	—
522	C₄	—	4	2	no	—
523	C₄	—	4	2	no	—
524	C₄	—	4	2	no	—
525	C₄	—	4	4	no	—
526	C₄	—	8	2	no	—
527	C₄	—	8	1	no	—
528	C₄	C₄₁×#525	1	1	yes	characteristic
529	C₄	C₄₁×#537	1	1	yes	characteristic
530	C₄	C₄₁×#548	1	1	yes	characteristic
531	C₄	C₄₁×#550	1	1	yes	characteristic
532	C₂	—	8	2	no	—
533	C₂	—	8	2	no	—
534	C₂	C₄₁×#2407	1	1	yes	characteristic
535	C₂	C₄₁×#3005	1	1	yes	characteristic
536	C₂	C₄₁×#3009	1	1	yes	characteristic
537	C₂	C₄₁×#3321	1	2	yes	—
538	C₂	C₄₁×#3348	1	2	yes	—

	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×#475	C₂	semidirect
3. C₂×C₄₁×#440	C₂	semidirect
4. C₄₁×#2536	C₂	semidirect
5. C₄₁×#2034	C₂	semidirect
6. C₄₁×#2000	C₂	semidirect
7. C₄₁×#1993	C₂	semidirect
8. C₄₁×#1955	C₂	semidirect
9. C₄₁×#1873	C₂	semidirect
10. C₄₁×#1854	C₂	semidirect
11. C₄₁×#1804	C₂	semidirect
233. #24133	C₄₁	direct (N × H)
264. C₂×#475	C₂×C₄₁	semidirect
265. C₂×#440	C₂×C₄₁	semidirect
266. #2536	C₂×C₄₁	semidirect
267. #2034	C₂×C₄₁	semidirect
268. #2000	C₂×C₄₁	semidirect
269. #1993	C₂×C₄₁	semidirect
270. #1955	C₂×C₄₁	semidirect
271. #1873	C₂×C₄₁	semidirect
272. #1854	C₂×C₄₁	semidirect
273. #1804	C₂×C₄₁	semidirect
314. C₄₁	#24133	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×#475	2
C₂×C₄₁×#440	2
C₄₁×#2536	2
C₄₁×#2034	2
C₄₁×#2000	2
C₄₁×#1993	2
C₄₁×#1955	2
C₄₁×#1873	2
C₄₁×#1854	2
C₄₁×#1804	2
#24133	41

Subgroups — C41×#24133 #23800334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24133 #23800334