Subgroups — C₄₁×#24151 #23800352

Order 10496.

1198 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	C₂×C₄₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#121×C₄₁	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#2035	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2000	C₂	1	2	yes	maximal
7	C₄₁×#1987	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1981	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1880	C₂	1	2	yes	maximal, complemented
10	C₄₁×#1865	C₂	1	2	yes	maximal, complemented
11	C₄₁×#1797	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	—
13	C₄₁×#533	—	2	4	no	—
14	C₄₁×#533	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×#125×C₄₁	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₄₁	—	2	1	no	—
19	C₂×C₂²⋊Q₈×C₄₁	C₂²	1	2	yes	—
20	C₂×#121×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₄₁	—	2	1	no	—
22	C₂×C₄×Q₈×C₄₁	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₄₁	—	2	1	no	—
24	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
25	C₂×#117×C₄₁	—	2	1	no	—
26	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#476	—	2	4	no	—
28	C₄₁×#461	—	2	4	no	—
29	C₂×#106×C₄₁	C₂²	1	2	yes	—
30	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
32	C₂×#103×C₄₁	C₂²	1	2	yes	—
33	C₂×#103×C₄₁	C₂²	1	2	yes	—
34	C₂×#103×C₄₁	C₂²	1	2	yes	—
35	C₂×#102×C₄₁	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
37	C₄₁×#390	C₂²	1	1	yes	characteristic
38	C₄₁×#389	C₂²	1	1	yes	characteristic
39	C₄₁×#388	—	2	1	no	—
40	C₄₁×#386	—	2	1	no	—
41	C₄₁×#380	C₂²	1	1	yes	characteristic
42	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
43	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
44	C₄₁×#340	C₂²	1	2	yes	—
45	C₄₁×#340	C₂²	1	2	yes	—
46	C₄₁×#336	C₂²	1	2	yes	—
47	C₂²×Q₈×C₄₁	—	2	1	no	—
48	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
49	C₂²×D₄×C₄₁	—	2	1	no	—
50	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₄₁	—	2	1	no	—
52	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
55	C₂×SD₁₆×C₄₁	—	2	4	no	—
56	C₂×SD₁₆×C₄₁	—	2	4	no	—
57	C₂×SD₁₆×C₄₁	—	4	2	no	—
58	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
59	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
60	C₄⋊Q₈×C₄₁	—	2	2	no	—
61	C₄⋊Q₈×C₄₁	—	2	2	no	—
62	C₄⋊Q₈×C₄₁	D₄	1	4	yes	complemented
63	C₄⋊Q₈×C₄₁	D₄	1	4	yes	complemented
64	#125×C₄₁	—	2	8	no	—
65	C₂²⋊Q₈×C₄₁	—	2	8	no	—
66	C₂²⋊Q₈×C₄₁	—	2	4	no	—
67	C₂²⋊Q₈×C₄₁	—	2	4	no	—
68	#121×C₄₁	—	2	2	no	—
69	#121×C₄₁	—	2	2	no	—
70	C₄×Q₈×C₄₁	—	2	8	no	—
71	C₄×Q₈×C₄₁	—	2	4	no	—
72	C₄×Q₈×C₄₁	—	2	4	no	—
73	C₄×D₄×C₄₁	—	2	2	no	—
74	C₄×D₄×C₄₁	—	2	2	no	—
75	C₄×D₄×C₄₁	—	4	4	no	—
76	#117×C₄₁	—	2	8	no	—
77	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
78	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
79	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
80	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
81	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
82	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
84	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
86	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
87	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
88	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
89	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
91	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
93	C₂×C₄²×C₄₁	—	2	2	no	—
94	C₂×C₄²×C₄₁	—	2	2	no	—
95	C₂×C₄²×C₄₁	—	2	2	no	—
96	C₂×C₄²×C₄₁	—	2	1	no	—
97	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
101	#106×C₄₁	—	2	4	no	—
102	C₄⋊C₈×C₄₁	—	2	2	no	—
103	C₄⋊C₈×C₄₁	—	2	2	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	C₂²⋊C₈×C₄₁	—	4	2	no	—
109	#95×C₄₁	—	2	1	no	—
110	#95×C₄₁	—	2	1	no	—
111	#95×C₄₁	—	2	1	no	—
112	C₂⁴×C₄₁	—	2	1	no	—
113	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
114	C₂×Q₈×C₄₁	—	2	4	no	—
115	C₂×Q₈×C₄₁	—	2	2	no	—
116	C₂×Q₈×C₄₁	—	2	2	no	—
117	C₂×Q₈×C₄₁	—	2	2	no	—
118	C₂×Q₈×C₄₁	—	2	2	no	—
119	C₂×Q₈×C₄₁	—	2	2	no	—
120	C₂×Q₈×C₄₁	—	4	2	no	—
121	C₂×Q₈×C₄₁	—	4	2	no	—
122	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
123	C₂×D₄×C₄₁	—	2	4	no	—
124	C₂×D₄×C₄₁	—	2	1	no	—
125	C₂×D₄×C₄₁	—	2	1	no	—
126	C₂×D₄×C₄₁	—	2	1	no	—
127	C₂×D₄×C₄₁	—	2	1	no	—
128	C₂×D₄×C₄₁	—	2	1	no	—
129	C₂×D₄×C₄₁	—	4	2	no	—
130	C₂×D₄×C₄₁	—	4	2	no	—
131	C₂×D₄×C₄₁	#41	1	2	yes	—
132	C₂²×C₄×C₄₁	—	2	2	no	—
133	C₂²×C₄×C₄₁	—	2	2	no	—
134	C₂²×C₄×C₄₁	—	2	1	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	1	no	—
140	C₂²×C₄×C₄₁	—	2	1	no	—
141	C₂²×C₄×C₄₁	—	2	1	no	—
142	C₂²×C₄×C₄₁	—	2	1	no	—
143	C₂²×C₄×C₄₁	—	2	1	no	—
144	C₂²×C₄×C₄₁	—	2	1	no	—
145	C₂²×C₄×C₄₁	—	2	1	no	—
146	C₂²×C₄×C₄₁	—	4	1	no	—
147	C₂²×C₄×C₄₁	—	4	1	no	—
148	C₂²×C₄×C₄₁	—	4	2	no	—
149	C₂²×C₄×C₄₁	—	4	2	no	—
150	C₂²×C₄×C₄₁	—	4	2	no	—
151	C₂²×C₄×C₄₁	SD₁₆	1	4	yes	complemented
152	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
158	SD₁₆×C₄₁	—	4	8	no	—
159	C₂×C₈×C₄₁	—	2	2	no	—
160	C₂×C₈×C₄₁	—	2	2	no	—
161	C₂×C₈×C₄₁	—	2	2	no	—
162	C₂×C₈×C₄₁	—	2	2	no	—
163	C₂×C₈×C₄₁	—	2	2	no	—
164	C₂×C₈×C₄₁	—	2	2	no	—
165	C₄⋊C₄×C₄₁	—	2	8	no	—
166	C₄⋊C₄×C₄₁	—	2	8	no	—
167	C₄⋊C₄×C₄₁	—	2	4	no	—
168	C₄⋊C₄×C₄₁	—	2	4	no	—
169	C₄⋊C₄×C₄₁	—	2	4	no	—
170	C₄⋊C₄×C₄₁	—	2	2	no	—
171	C₄⋊C₄×C₄₁	—	2	1	no	—
172	C₄⋊C₄×C₄₁	—	2	1	no	—
173	C₄⋊C₄×C₄₁	—	2	1	no	—
174	C₄⋊C₄×C₄₁	—	2	1	no	—
175	C₄⋊C₄×C₄₁	—	2	1	no	—
176	C₄⋊C₄×C₄₁	—	4	2	no	—
177	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₄₁	#41	1	2	yes	—
179	C₂²⋊C₄×C₄₁	—	2	8	no	—
180	C₂²⋊C₄×C₄₁	—	2	4	no	—
181	C₂²⋊C₄×C₄₁	—	4	2	no	—
182	C₂²⋊C₄×C₄₁	—	4	2	no	—
183	C₂²⋊C₄×C₄₁	—	4	2	no	—
184	C₂²⋊C₄×C₄₁	—	4	2	no	—
185	C₄²×C₄₁	—	2	8	no	—
186	C₄²×C₄₁	—	2	2	no	—
187	C₄²×C₄₁	—	2	2	no	—
188	C₄²×C₄₁	—	2	1	no	—
189	C₄²×C₄₁	—	2	1	no	—
190	C₄²×C₄₁	—	4	2	no	—
191	C₄²×C₄₁	C₂×D₄	1	2	yes	—
192	C₄²×C₄₁	C₂×D₄	1	2	yes	—
193	C₂³×C₄₁	—	2	1	no	—
194	C₂³×C₄₁	—	2	1	no	—
195	C₂³×C₄₁	—	2	1	no	—
196	C₂³×C₄₁	—	2	1	no	—
197	C₂³×C₄₁	—	2	1	no	—
198	C₂³×C₄₁	—	2	1	no	—
199	C₂³×C₄₁	—	2	2	no	—
200	C₂³×C₄₁	—	4	1	no	—
201	C₂³×C₄₁	—	4	1	no	—
202	C₂³×C₄₁	—	4	1	no	—
203	C₂³×C₄₁	—	4	1	no	—
204	C₂³×C₄₁	—	4	1	no	—
205	C₂³×C₄₁	—	4	1	no	—
206	C₂³×C₄₁	C₂×SD₁₆	1	2	yes	—
207	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
208	Q₈×C₄₁	—	2	4	no	—
209	Q₈×C₄₁	—	2	4	no	—
210	Q₈×C₄₁	—	4	8	no	—
211	Q₈×C₄₁	—	4	2	no	—
212	Q₈×C₄₁	—	4	2	no	—
213	D₄×C₄₁	—	2	2	no	—
214	D₄×C₄₁	—	2	2	no	—
215	D₄×C₄₁	—	4	8	no	—
216	D₄×C₄₁	—	4	1	no	—
217	D₄×C₄₁	—	4	1	no	—
218	C₂×C₄×C₄₁	—	2	1	no	—
219	C₂×C₄×C₄₁	—	2	1	no	—
220	C₂×C₄×C₄₁	—	2	1	no	—
221	C₂×C₄×C₄₁	—	2	1	no	—
222	C₂×C₄×C₄₁	—	2	8	no	—
223	C₂×C₄×C₄₁	—	2	1	no	—
224	C₂×C₄×C₄₁	—	2	1	no	—
225	C₂×C₄×C₄₁	—	2	4	no	—
226	C₂×C₄×C₄₁	—	2	4	no	—
227	C₂×C₄×C₄₁	—	2	1	no	—
228	C₂×C₄×C₄₁	—	2	2	no	—
229	C₂×C₄×C₄₁	—	2	1	no	—
230	C₂×C₄×C₄₁	—	2	1	no	—
231	C₂×C₄×C₄₁	—	2	1	no	—
232	C₂×C₄×C₄₁	—	4	4	no	—
233	C₂×C₄×C₄₁	—	4	2	no	—
234	C₂×C₄×C₄₁	—	4	2	no	—
235	C₂×C₄×C₄₁	—	4	1	no	—
236	C₂×C₄×C₄₁	—	4	1	no	—
237	C₂×C₄×C₄₁	—	4	2	no	—
238	C₂×C₄×C₄₁	—	4	2	no	—
239	C₂×C₄×C₄₁	—	4	1	no	—
240	C₂×C₄×C₄₁	—	4	1	no	—
241	C₂×C₄×C₄₁	—	4	1	no	—
242	C₂×C₄×C₄₁	—	4	1	no	—
243	C₂×C₄×C₄₁	—	4	1	no	—
244	C₂×C₄×C₄₁	—	4	1	no	—
245	C₂×C₄×C₄₁	—	4	1	no	—
246	C₂×C₄×C₄₁	—	4	1	no	—
247	C₂×C₄×C₄₁	—	4	1	no	—
248	C₂×C₄×C₄₁	—	4	1	no	—
249	C₂×C₄×C₄₁	—	8	4	no	—
250	C₂×C₄×C₄₁	#120	1	2	yes	—
251	C₂×C₄×C₄₁	#120	1	2	yes	—
252	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
253	C₂×C₄×C₄₁	C₂×SD₁₆	1	2	yes	—
254	C₂×C₄×C₄₁	#137	1	2	yes	—
255	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
258	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
259	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
260	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
261	C₈×C₄₁	—	4	2	no	—
262	C₈×C₄₁	—	4	2	no	—
263	#24151	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂²×C₄₁	—	2	2	no	—
265	C₂²×C₄₁	—	2	2	no	—
266	C₂²×C₄₁	—	2	2	no	—
267	C₂²×C₄₁	—	4	1	no	—
268	C₂²×C₄₁	—	4	1	no	—
269	C₂²×C₄₁	—	4	1	no	—
270	C₂²×C₄₁	—	4	1	no	—
271	C₂²×C₄₁	—	4	2	no	—
272	C₂²×C₄₁	—	4	2	no	—
273	C₂²×C₄₁	—	4	2	no	—
274	C₂²×C₄₁	—	8	1	no	—
275	C₂²×C₄₁	—	8	1	no	—
276	C₂²×C₄₁	—	8	1	no	—
277	C₂²×C₄₁	—	8	1	no	—
278	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
279	C₂²×C₄₁	#525	1	1	yes	characteristic
280	C₂²×C₄₁	#537	1	1	yes	characteristic
281	C₂²×C₄₁	C₂²×SD₁₆	1	1	yes	characteristic
282	C₂²×C₄₁	C₂×#137	1	1	yes	characteristic
283	C₂²×C₄₁	#575	1	1	yes	characteristic
284	C₂²×C₄₁	#575	1	1	yes	characteristic
285	C₄×C₄₁	—	2	2	no	—
286	C₄×C₄₁	—	2	2	no	—
287	C₄×C₄₁	—	4	2	no	—
288	C₄×C₄₁	—	4	1	no	—
289	C₄×C₄₁	—	4	4	no	—
290	C₄×C₄₁	—	4	1	no	—
291	C₄×C₄₁	—	4	4	no	—
292	C₄×C₄₁	—	8	1	no	—
293	C₄×C₄₁	—	8	1	no	—
294	C₄×C₄₁	#448	1	4	yes	—
295	C₄×C₄₁	#546	1	1	yes	characteristic
296	C₄×C₄₁	#547	1	1	yes	characteristic
297	C₄×C₄₁	#548	1	1	yes	characteristic
298	C₄×C₄₁	#550	1	1	yes	characteristic
299	C₂×#533	C₂×C₄₁	1	1	yes	characteristic, complemented
300	C₂×#476	C₂×C₄₁	1	1	yes	characteristic, complemented
301	C₂×#461	C₂×C₄₁	1	1	yes	characteristic, complemented
302	C₄×#121	C₂×C₄₁	1	1	yes	characteristic
303	#2035	C₂×C₄₁	1	2	yes	complemented
304	#2000	C₂×C₄₁	1	2	yes	—
305	#1987	C₂×C₄₁	1	1	yes	characteristic, complemented
306	#1981	C₂×C₄₁	1	1	yes	characteristic, complemented
307	#1880	C₂×C₄₁	1	2	yes	complemented
308	#1865	C₂×C₄₁	1	2	yes	complemented
309	#1797	C₂×C₄₁	1	1	yes	characteristic, complemented
310	C₂×C₄₁	—	4	2	no	—
311	C₂×C₄₁	—	8	2	no	—
312	C₂×C₄₁	#2407	1	1	yes	characteristic
313	C₂×C₄₁	C₂×#448	1	1	yes	characteristic
314	C₂×C₄₁	#3005	1	1	yes	characteristic
315	C₂×C₄₁	#3101	1	1	yes	characteristic
316	C₂×C₄₁	#3116	1	1	yes	characteristic
317	C₂×C₄₁	#3246	1	1	yes	characteristic
318	C₂×C₄₁	#3252	1	1	yes	characteristic
319	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	—
320	#533	—	2	4	no	—
321	#533	—	2	4	no	—
322	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
323	C₂×C₄⋊Q₈	C₂²×C₄₁	1	1	yes	characteristic
324	C₂×#125	—	2	1	no	—
325	C₂×C₂²⋊Q₈	—	2	1	no	—
326	C₂×C₂²⋊Q₈	C₂²×C₄₁	1	2	yes	—
327	C₂×#121	C₂²×C₄₁	1	1	yes	characteristic
328	C₂×C₄×Q₈	—	2	1	no	—
329	C₂×C₄×Q₈	C₂²×C₄₁	1	2	yes	—
330	C₂×C₄×D₄	—	2	1	no	—
331	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
332	C₂×#117	—	2	1	no	—
333	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
334	#476	—	2	4	no	—
335	#461	—	2	4	no	—
336	C₂×#106	C₂²×C₄₁	1	2	yes	—
337	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
338	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
339	C₂×#103	C₂²×C₄₁	1	2	yes	—
340	C₂×#103	C₂²×C₄₁	1	2	yes	—
341	C₂×#103	C₂²×C₄₁	1	2	yes	—
342	C₂×#102	C₂²×C₄₁	1	2	yes	—
343	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
344	#390	C₂²×C₄₁	1	1	yes	characteristic
345	#389	C₂²×C₄₁	1	1	yes	characteristic
346	#388	—	2	1	no	—
347	#386	—	2	1	no	—
348	#380	C₂²×C₄₁	1	1	yes	characteristic
349	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
350	C₄×C₂²⋊C₄	—	2	1	no	—
351	#340	C₂²×C₄₁	1	2	yes	—
352	#340	C₂²×C₄₁	1	2	yes	—
353	#336	C₂²×C₄₁	1	2	yes	—
354	C₄₁	#24151	1	1	yes	characteristic, direct product factor, Sylow
355	C₂²×Q₈	—	2	1	no	—
356	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
357	C₂²×D₄	—	2	1	no	—
358	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
359	C₂³×C₄	—	2	1	no	—
360	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
361	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
362	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
363	C₂×SD₁₆	—	2	4	no	—
364	C₂×SD₁₆	—	2	4	no	—
365	C₂×SD₁₆	—	4	2	no	—
366	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
367	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
368	C₄⋊Q₈	—	2	2	no	—
369	C₄⋊Q₈	—	2	2	no	—
370	C₄⋊Q₈	D₄×C₄₁	1	4	yes	complemented
371	C₄⋊Q₈	D₄×C₄₁	1	4	yes	complemented
372	#125	—	2	8	no	—
373	C₂²⋊Q₈	—	2	8	no	—
374	C₂²⋊Q₈	—	2	4	no	—
375	C₂²⋊Q₈	—	2	4	no	—
376	#121	—	2	2	no	—
377	#121	—	2	2	no	—
378	C₄×Q₈	—	2	8	no	—
379	C₄×Q₈	—	2	4	no	—
380	C₄×Q₈	—	2	4	no	—
381	C₄×D₄	—	2	2	no	—
382	C₄×D₄	—	2	2	no	—
383	C₄×D₄	—	4	4	no	—
384	#117	—	2	8	no	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	C₂×C₄⋊C₄	—	2	1	no	—
387	C₂×C₄⋊C₄	—	2	1	no	—
388	C₂×C₄⋊C₄	—	2	2	no	—
389	C₂×C₄⋊C₄	—	2	2	no	—
390	C₂×C₄⋊C₄	—	2	1	no	—
391	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
395	C₂×C₂²⋊C₄	—	2	2	no	—
396	C₂×C₂²⋊C₄	—	2	1	no	—
397	C₂×C₂²⋊C₄	—	2	1	no	—
398	C₂×C₂²⋊C₄	—	2	1	no	—
399	C₂×C₂²⋊C₄	—	2	1	no	—
400	C₂×C₂²⋊C₄	—	2	1	no	—
401	C₂×C₄²	—	2	2	no	—
402	C₂×C₄²	—	2	2	no	—
403	C₂×C₄²	—	2	2	no	—
404	C₂×C₄²	—	2	1	no	—
405	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
406	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
407	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
408	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
409	#106	—	2	4	no	—
410	C₄⋊C₈	—	2	2	no	—
411	C₄⋊C₈	—	2	2	no	—
412	#103	—	2	4	no	—
413	#103	—	2	4	no	—
414	#103	—	4	2	no	—
415	#102	—	2	4	no	—
416	C₂²⋊C₈	—	4	2	no	—
417	#95	—	2	1	no	—
418	#95	—	2	1	no	—
419	#95	—	2	1	no	—
420	C₂⁴	—	2	1	no	—
421	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
422	C₂×Q₈	—	2	4	no	—
423	C₂×Q₈	—	2	2	no	—
424	C₂×Q₈	—	2	2	no	—
425	C₂×Q₈	—	2	2	no	—
426	C₂×Q₈	—	2	2	no	—
427	C₂×Q₈	—	2	2	no	—
428	C₂×Q₈	—	4	2	no	—
429	C₂×Q₈	—	4	2	no	—
430	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	2	1	no	—
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	4	2	no	—
438	C₂×D₄	—	4	2	no	—
439	C₂×D₄	#41×C₄₁	1	2	yes	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	2	no	—
445	C₂²×C₄	—	2	2	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	1	no	—
448	C₂²×C₄	—	2	1	no	—
449	C₂²×C₄	—	2	1	no	—
450	C₂²×C₄	—	2	1	no	—
451	C₂²×C₄	—	2	1	no	—
452	C₂²×C₄	—	2	1	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	4	1	no	—
455	C₂²×C₄	—	4	1	no	—
456	C₂²×C₄	—	4	2	no	—
457	C₂²×C₄	—	4	2	no	—
458	C₂²×C₄	—	4	2	no	—
459	C₂²×C₄	SD₁₆×C₄₁	1	4	yes	complemented
460	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
461	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
462	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
463	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
464	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
465	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
466	SD₁₆	—	4	8	no	—
467	C₂×C₈	—	2	2	no	—
468	C₂×C₈	—	2	2	no	—
469	C₂×C₈	—	2	2	no	—
470	C₂×C₈	—	2	2	no	—
471	C₂×C₈	—	2	2	no	—
472	C₂×C₈	—	2	2	no	—
473	C₄⋊C₄	—	2	8	no	—
474	C₄⋊C₄	—	2	8	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	2	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	2	1	no	—
482	C₄⋊C₄	—	2	1	no	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	4	2	no	—
485	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
486	C₄⋊C₄	#41×C₄₁	1	2	yes	—
487	C₂²⋊C₄	—	2	8	no	—
488	C₂²⋊C₄	—	2	4	no	—
489	C₂²⋊C₄	—	4	2	no	—
490	C₂²⋊C₄	—	4	2	no	—
491	C₂²⋊C₄	—	4	2	no	—
492	C₂²⋊C₄	—	4	2	no	—
493	C₄²	—	2	8	no	—
494	C₄²	—	2	2	no	—
495	C₄²	—	2	2	no	—
496	C₄²	—	2	1	no	—
497	C₄²	—	2	1	no	—
498	C₄²	—	4	2	no	—
499	C₄²	C₂×D₄×C₄₁	1	2	yes	—
500	C₄²	C₂×D₄×C₄₁	1	2	yes	—
501	C₂³	—	2	1	no	—
502	C₂³	—	2	1	no	—
503	C₂³	—	2	1	no	—
504	C₂³	—	2	1	no	—
505	C₂³	—	2	1	no	—
506	C₂³	—	2	1	no	—
507	C₂³	—	2	2	no	—
508	C₂³	—	4	1	no	—
509	C₂³	—	4	1	no	—
510	C₂³	—	4	1	no	—
511	C₂³	—	4	1	no	—
512	C₂³	—	4	1	no	—
513	C₂³	—	4	1	no	—
514	C₂³	C₂×SD₁₆×C₄₁	1	2	yes	—
515	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
516	Q₈	—	2	4	no	—
517	Q₈	—	2	4	no	—
518	Q₈	—	4	8	no	—
519	Q₈	—	4	2	no	—
520	Q₈	—	4	2	no	—
521	D₄	—	2	2	no	—
522	D₄	—	2	2	no	—
523	D₄	—	4	8	no	—
524	D₄	—	4	1	no	—
525	D₄	—	4	1	no	—
526	C₂×C₄	—	2	1	no	—
527	C₂×C₄	—	2	1	no	—
528	C₂×C₄	—	2	1	no	—
529	C₂×C₄	—	2	1	no	—
530	C₂×C₄	—	2	8	no	—
531	C₂×C₄	—	2	1	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	4	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	1	no	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	4	4	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	—	4	2	no	—
543	C₂×C₄	—	4	1	no	—
544	C₂×C₄	—	4	1	no	—
545	C₂×C₄	—	4	2	no	—
546	C₂×C₄	—	4	2	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	—	8	4	no	—
558	C₂×C₄	#120×C₄₁	1	2	yes	—
559	C₂×C₄	#120×C₄₁	1	2	yes	—
560	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
561	C₂×C₄	C₂×SD₁₆×C₄₁	1	2	yes	—
562	C₂×C₄	#137×C₄₁	1	2	yes	—
563	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
564	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
565	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
566	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
567	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
568	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
569	C₈	—	4	2	no	—
570	C₈	—	4	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	2	2	no	—
573	C₂²	—	2	2	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	1	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	2	no	—
579	C₂²	—	4	2	no	—
580	C₂²	—	4	2	no	—
581	C₂²	—	8	1	no	—
582	C₂²	—	8	1	no	—
583	C₂²	—	8	1	no	—
584	C₂²	—	8	1	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	C₂²	C₂²×SD₁₆×C₄₁	1	1	yes	characteristic
589	C₂²	C₂×#137×C₄₁	1	1	yes	characteristic
590	C₂²	C₄₁×#575	1	1	yes	characteristic
591	C₂²	C₄₁×#575	1	1	yes	characteristic
592	C₄	—	2	2	no	—
593	C₄	—	2	2	no	—
594	C₄	—	4	2	no	—
595	C₄	—	4	1	no	—
596	C₄	—	4	4	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	4	no	—
599	C₄	—	8	1	no	—
600	C₄	—	8	1	no	—
601	C₄	C₄₁×#448	1	4	yes	—
602	C₄	C₄₁×#546	1	1	yes	characteristic
603	C₄	C₄₁×#547	1	1	yes	characteristic
604	C₄	C₄₁×#548	1	1	yes	characteristic
605	C₄	C₄₁×#550	1	1	yes	characteristic
606	C₂	—	4	2	no	—
607	C₂	—	8	2	no	—
608	C₂	C₄₁×#2407	1	1	yes	characteristic
609	C₂	C₂×C₄₁×#448	1	1	yes	characteristic
610	C₂	C₄₁×#3005	1	1	yes	characteristic
611	C₂	C₄₁×#3101	1	1	yes	characteristic
612	C₂	C₄₁×#3116	1	1	yes	characteristic
613	C₂	C₄₁×#3246	1	1	yes	characteristic
614	C₂	C₄₁×#3252	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂×#121×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
1. C₂×C₄₁×#533	C₂	semidirect
2. C₂×C₄₁×#476	C₂	semidirect
3. C₂×C₄₁×#461	C₂	semidirect
5. C₄₁×#2035	C₂	semidirect
7. C₄₁×#1987	C₂	semidirect
8. C₄₁×#1981	C₂	semidirect
9. C₄₁×#1880	C₂	semidirect
10. C₄₁×#1865	C₂	semidirect
11. C₄₁×#1797	C₂	semidirect
62. C₄⋊Q₈×C₄₁	D₄	semidirect
63. C₄⋊Q₈×C₄₁	D₄	semidirect
151. C₂²×C₄×C₄₁	SD₁₆	semidirect
263. #24151	C₄₁	direct (N × H)
299. C₂×#533	C₂×C₄₁	semidirect
300. C₂×#476	C₂×C₄₁	semidirect
301. C₂×#461	C₂×C₄₁	semidirect
303. #2035	C₂×C₄₁	semidirect
305. #1987	C₂×C₄₁	semidirect
306. #1981	C₂×C₄₁	semidirect
307. #1880	C₂×C₄₁	semidirect
308. #1865	C₂×C₄₁	semidirect
309. #1797	C₂×C₄₁	semidirect
354. C₄₁	#24151	direct (N × H)
370. C₄⋊Q₈	D₄×C₄₁	semidirect
371. C₄⋊Q₈	D₄×C₄₁	semidirect
459. C₂²×C₄	SD₁₆×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#533	2
C₂×C₄₁×#476	2
C₂×C₄₁×#461	2
C₄×#121×C₄₁	2
C₄₁×#2035	2
C₄₁×#2000	2
C₄₁×#1987	2
C₄₁×#1981	2
C₄₁×#1880	2
C₄₁×#1865	2
C₄₁×#1797	2
#24151	41

Subgroups — C41×#24151 #23800352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24151 #23800352