Subgroups — C₄₁×#21005 #23797206

Order 10496.

1782 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/8)×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#568	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2896	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₁×#418	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#100×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1907	C₂	1	8	yes	maximal, complemented
7	C₄₁×#1742	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1738	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₄₁	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
12	C₂×AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
13	C₂×AGL(1,ℤ/8)×C₄₁	C₄	1	8	yes	complemented
14	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	4	yes	—
15	C₂²×SD₁₆×C₄₁	—	2	1	no	—
16	C₂²×D₈×C₄₁	—	2	1	no	—
17	C₄₁×#568	—	2	4	no	—
18	C₄₁×#568	—	2	4	no	—
19	C₂×#131×C₄₁	—	2	2	no	—
20	C₂×#131×C₄₁	C₂²	1	4	yes	complemented
21	C₂²×M₁₆×C₄₁	—	2	2	no	—
22	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
23	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
24	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
25	C₂²×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#419	—	2	1	no	—
28	C₄₁×#418	—	2	1	no	—
29	C₄₁×#418	—	2	2	no	—
30	C₄₁×#418	—	2	2	no	—
31	C₄₁×#417	C₂²	1	4	yes	complemented
32	C₄₁×#416	—	2	2	no	—
33	C₂×#102×C₄₁	C₂²	1	4	yes	complemented
34	C₂×#100×C₄₁	—	2	4	no	—
35	C₂×#100×C₄₁	C₂²	1	4	yes	complemented
36	C₄₁×#343	C₂²	1	4	yes	complemented
37	C₄₁×#341	C₂²	1	4	yes	complemented
38	C₂⁵×C₄₁	—	2	1	no	—
39	C₂×#41×C₄₁	—	2	1	no	—
40	C₂×#41×C₄₁	—	2	2	no	—
41	C₂×#41×C₄₁	—	2	2	no	—
42	C₂×#41×C₄₁	—	2	1	no	—
43	C₂×#41×C₄₁	C₂×C₄	1	4	yes	—
44	C₂×#41×C₄₁	D₄	1	4	yes	—
45	C₂×#41×C₄₁	D₄	1	2	yes	—
46	C₂×#41×C₄₁	C₂³	1	2	yes	—
47	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
48	C₂²×D₄×C₄₁	—	2	1	no	—
49	C₂²×D₄×C₄₁	—	2	1	no	—
50	C₂²×D₄×C₄₁	—	2	2	no	—
51	C₂²×D₄×C₄₁	—	2	1	no	—
52	C₂²×D₄×C₄₁	—	2	1	no	—
53	C₂²×D₄×C₄₁	—	2	1	no	—
54	C₂²×D₄×C₄₁	—	2	2	no	—
55	C₂²×D₄×C₄₁	—	2	2	no	—
56	C₂²×D₄×C₄₁	—	2	1	no	—
57	C₂²×D₄×C₄₁	C₂×C₄	1	4	yes	—
58	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
59	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
60	C₂³×C₄×C₄₁	—	2	1	no	—
61	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
62	AGL(1,ℤ/8)×C₄₁	—	2	16	no	—
63	AGL(1,ℤ/8)×C₄₁	—	2	8	no	—
64	AGL(1,ℤ/8)×C₄₁	C₂×C₄	1	16	yes	—
65	C₂×SD₁₆×C₄₁	—	2	8	no	—
66	C₂×SD₁₆×C₄₁	—	2	4	no	—
67	C₂×D₈×C₄₁	—	2	8	no	—
68	C₂×D₈×C₄₁	—	2	4	no	—
69	#131×C₄₁	—	2	16	no	—
70	#131×C₄₁	—	2	8	no	—
71	#131×C₄₁	—	2	8	no	—
72	C₂×M₁₆×C₄₁	—	2	16	no	—
73	C₂×M₁₆×C₄₁	—	2	1	no	—
74	C₂×M₁₆×C₄₁	—	2	2	no	—
75	C₂×M₁₆×C₄₁	—	2	2	no	—
76	C₂×M₁₆×C₄₁	—	2	2	no	—
77	C₂×M₁₆×C₄₁	—	2	2	no	—
78	C₂×M₁₆×C₄₁	—	2	1	no	—
79	C₂×M₁₆×C₄₁	—	2	1	no	—
80	C₂×M₁₆×C₄₁	—	2	4	no	—
81	C₂×M₁₆×C₄₁	—	2	2	no	—
82	C₂×M₁₆×C₄₁	—	2	2	no	—
83	C₂×M₁₆×C₄₁	C₂×C₄	1	4	yes	—
84	C₂×M₁₆×C₄₁	D₄	1	4	yes	—
85	C₂×M₁₆×C₄₁	D₄	1	2	yes	—
86	C₂×M₁₆×C₄₁	D₄	1	2	yes	—
87	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
88	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
89	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
90	C₂²×C₈×C₄₁	—	2	1	no	—
91	C₂²×C₈×C₄₁	—	2	1	no	—
92	C₂²×C₈×C₄₁	—	2	4	no	—
93	C₂²×C₈×C₄₁	—	2	1	no	—
94	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
95	#117×C₄₁	—	2	1	no	—
96	#117×C₄₁	—	2	1	no	—
97	#117×C₄₁	—	2	1	no	—
98	#117×C₄₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
100	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
101	C₂×C₄²×C₄₁	—	2	1	no	—
102	#103×C₄₁	—	2	4	no	—
103	#102×C₄₁	—	2	4	no	—
104	#102×C₄₁	—	2	8	no	—
105	#100×C₄₁	—	2	8	no	—
106	C₂⁴×C₄₁	—	2	1	no	—
107	C₂⁴×C₄₁	—	2	4	no	—
108	C₂⁴×C₄₁	—	2	2	no	—
109	C₂⁴×C₄₁	—	4	1	no	—
110	C₂⁴×C₄₁	—	4	1	no	—
111	C₂⁴×C₄₁	—	4	2	no	—
112	C₂⁴×C₄₁	—	8	4	no	—
113	C₂⁴×C₄₁	#41	1	1	yes	characteristic
114	#41×C₄₁	—	2	4	no	—
115	#41×C₄₁	—	2	4	no	—
116	#41×C₄₁	—	2	4	no	—
117	#41×C₄₁	—	2	4	no	—
118	#41×C₄₁	—	2	4	no	—
119	#41×C₄₁	—	2	2	no	—
120	#41×C₄₁	—	2	2	no	—
121	#41×C₄₁	C₂²⋊C₄	1	8	yes	—
122	#41×C₄₁	C₂²×C₄	1	4	yes	—
123	#41×C₄₁	C₂×D₄	1	4	yes	—
124	C₂×Q₈×C₄₁	—	2	1	no	—
125	C₂×Q₈×C₄₁	—	2	2	no	—
126	C₂×Q₈×C₄₁	C₂²⋊C₄	1	4	yes	—
127	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
128	C₂×D₄×C₄₁	—	2	4	no	—
129	C₂×D₄×C₄₁	—	2	2	no	—
130	C₂×D₄×C₄₁	—	2	4	no	—
131	C₂×D₄×C₄₁	—	2	2	no	—
132	C₂×D₄×C₄₁	—	2	4	no	—
133	C₂×D₄×C₄₁	—	2	4	no	—
134	C₂×D₄×C₄₁	—	2	4	no	—
135	C₂×D₄×C₄₁	—	2	4	no	—
136	C₂×D₄×C₄₁	—	2	4	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	—	2	2	no	—
143	C₂×D₄×C₄₁	—	2	2	no	—
144	C₂×D₄×C₄₁	—	4	2	no	—
145	C₂×D₄×C₄₁	—	4	2	no	—
146	C₂×D₄×C₄₁	—	4	2	no	—
147	C₂×D₄×C₄₁	—	4	2	no	—
148	C₂×D₄×C₄₁	—	4	1	no	—
149	C₂×D₄×C₄₁	—	4	1	no	—
150	C₂×D₄×C₄₁	—	4	2	no	—
151	C₂×D₄×C₄₁	C₂²⋊C₄	1	4	yes	—
152	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
153	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₄₁	—	2	1	no	—
155	C₂²×C₄×C₄₁	—	2	4	no	—
156	C₂²×C₄×C₄₁	—	2	4	no	—
157	C₂²×C₄×C₄₁	—	2	2	no	—
158	C₂²×C₄×C₄₁	—	2	2	no	—
159	C₂²×C₄×C₄₁	—	2	2	no	—
160	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
161	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
168	C₂²×C₄×C₄₁	#41	1	2	yes	—
169	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
170	SD₁₆×C₄₁	—	2	16	no	—
171	D₈×C₄₁	—	2	16	no	—
172	M₁₆×C₄₁	—	2	16	no	—
173	M₁₆×C₄₁	—	2	4	no	—
174	M₁₆×C₄₁	—	2	4	no	—
175	M₁₆×C₄₁	—	2	4	no	—
176	M₁₆×C₄₁	—	2	4	no	—
177	M₁₆×C₄₁	—	2	2	no	—
178	M₁₆×C₄₁	—	2	2	no	—
179	M₁₆×C₄₁	—	2	2	no	—
180	M₁₆×C₄₁	—	2	2	no	—
181	M₁₆×C₄₁	—	4	2	no	—
182	M₁₆×C₄₁	C₂²×C₄	1	4	yes	—
183	M₁₆×C₄₁	C₂×D₄	1	4	yes	—
184	C₂×C₈×C₄₁	—	2	16	no	—
185	C₂×C₈×C₄₁	—	2	4	no	—
186	C₂×C₈×C₄₁	—	2	4	no	—
187	C₂×C₈×C₄₁	—	2	2	no	—
188	C₂×C₈×C₄₁	—	2	2	no	—
189	C₂×C₈×C₄₁	—	2	2	no	—
190	C₂×C₈×C₄₁	—	2	2	no	—
191	C₂×C₈×C₄₁	—	2	2	no	—
192	C₂×C₈×C₄₁	—	4	2	no	—
193	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
194	C₄⋊C₄×C₄₁	—	2	2	no	—
195	C₄⋊C₄×C₄₁	—	2	2	no	—
196	C₂²⋊C₄×C₄₁	—	4	1	no	—
197	C₂²⋊C₄×C₄₁	—	4	1	no	—
198	C₄²×C₄₁	—	4	1	no	—
199	C₄²×C₄₁	—	4	1	no	—
200	C₂³×C₄₁	—	2	2	no	—
201	C₂³×C₄₁	—	2	4	no	—
202	C₂³×C₄₁	—	2	4	no	—
203	C₂³×C₄₁	—	4	2	no	—
204	C₂³×C₄₁	—	4	2	no	—
205	C₂³×C₄₁	—	4	2	no	—
206	C₂³×C₄₁	—	4	4	no	—
207	C₂³×C₄₁	—	4	4	no	—
208	C₂³×C₄₁	—	8	2	no	—
209	C₂³×C₄₁	—	8	4	no	—
210	C₂³×C₄₁	—	8	4	no	—
211	C₂³×C₄₁	—	8	4	no	—
212	C₂³×C₄₁	—	8	8	no	—
213	C₂³×C₄₁	—	8	8	no	—
214	C₂³×C₄₁	C₄×D₄	1	2	yes	—
215	C₂³×C₄₁	#121	1	1	yes	characteristic
216	C₂³×C₄₁	#121	1	1	yes	characteristic
217	C₂³×C₄₁	#123	1	2	yes	—
218	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
219	Q₈×C₄₁	—	2	2	no	—
220	Q₈×C₄₁	—	2	4	no	—
221	Q₈×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
222	D₄×C₄₁	—	2	4	no	—
223	D₄×C₄₁	—	2	4	no	—
224	D₄×C₄₁	—	2	8	no	—
225	D₄×C₄₁	—	2	4	no	—
226	D₄×C₄₁	—	2	4	no	—
227	D₄×C₄₁	—	2	2	no	—
228	D₄×C₄₁	—	2	4	no	—
229	D₄×C₄₁	—	4	2	no	—
230	D₄×C₄₁	—	4	2	no	—
231	D₄×C₄₁	—	4	4	no	—
232	D₄×C₄₁	—	4	4	no	—
233	D₄×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
234	C₂×C₄×C₄₁	—	2	2	no	—
235	C₂×C₄×C₄₁	—	2	4	no	—
236	C₂×C₄×C₄₁	—	2	4	no	—
237	C₂×C₄×C₄₁	—	2	4	no	—
238	C₂×C₄×C₄₁	—	2	8	no	—
239	C₂×C₄×C₄₁	—	4	2	no	—
240	C₂×C₄×C₄₁	—	8	2	no	—
241	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
242	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	2	yes	—
243	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
244	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
245	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
246	C₂×C₄×C₄₁	#120	1	2	yes	—
247	C₂×C₄×C₄₁	#120	1	2	yes	—
248	C₂×C₄×C₄₁	#121	1	2	yes	—
249	C₂×C₄×C₄₁	#121	1	2	yes	—
250	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
251	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
252	C₂×C₄×C₄₁	#123	1	2	yes	—
253	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
254	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
256	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
258	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
259	C₈×C₄₁	—	2	4	no	—
260	C₈×C₄₁	—	2	4	no	—
261	C₈×C₄₁	—	2	4	no	—
262	C₈×C₄₁	—	4	2	no	—
263	#21005	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	—	2	4	no	—
268	C₂²×C₄₁	—	4	2	no	—
269	C₂²×C₄₁	—	4	2	no	—
270	C₂²×C₄₁	—	4	4	no	—
271	C₂²×C₄₁	—	4	4	no	—
272	C₂²×C₄₁	—	8	4	no	—
273	C₂²×C₄₁	—	8	4	no	—
274	C₂²×C₄₁	—	8	8	no	—
275	C₂²×C₄₁	—	8	8	no	—
276	C₂²×C₄₁	—	8	4	no	—
277	C₂²×C₄₁	—	8	4	no	—
278	C₂²×C₄₁	#386	1	2	yes	—
279	C₂²×C₄₁	#386	1	2	yes	—
280	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
281	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
282	C₂²×C₄₁	C₂×#123	1	1	yes	characteristic
283	C₄×C₄₁	—	2	4	no	—
284	C₄×C₄₁	—	8	2	no	—
285	C₄×C₄₁	#386	1	2	yes	—
286	C₄×C₄₁	#386	1	2	yes	—
287	C₄×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
288	C₄×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
289	C₄×C₄₁	C₂×#120	1	1	yes	characteristic
290	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
291	C₂²×AGL(1,ℤ/8)	C₂×C₄₁	1	1	yes	characteristic
292	C₂×#568	C₂×C₄₁	1	1	yes	characteristic, complemented
293	#2896	C₂×C₄₁	1	1	yes	characteristic, complemented
294	C₂×#418	C₂×C₄₁	1	1	yes	characteristic, complemented
295	C₂²×#100	C₂×C₄₁	1	1	yes	characteristic, complemented
296	#1907	C₂×C₄₁	1	8	yes	complemented
297	#1742	C₂×C₄₁	1	1	yes	characteristic, complemented
298	#1738	C₂×C₄₁	1	1	yes	characteristic, complemented
299	C₂×C₄₁	—	2	2	no	—
300	C₂×C₄₁	—	2	1	no	—
301	C₂×C₄₁	—	2	1	no	—
302	C₂×C₄₁	—	2	2	no	—
303	C₂×C₄₁	—	4	4	no	—
304	C₂×C₄₁	—	8	4	no	—
305	C₂×C₄₁	C₂×#386	1	1	yes	characteristic
306	C₂×C₄₁	#3068	1	2	yes	—
307	C₂²×#41	C₂²×C₄₁	1	1	yes	characteristic
308	C₂³×D₄	C₂²×C₄₁	1	1	yes	characteristic
309	C₂×AGL(1,ℤ/8)	—	2	4	no	—
310	C₂×AGL(1,ℤ/8)	—	2	2	no	—
311	C₂×AGL(1,ℤ/8)	C₄×C₄₁	1	8	yes	complemented
312	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	4	yes	—
313	C₂²×SD₁₆	—	2	1	no	—
314	C₂²×D₈	—	2	1	no	—
315	#568	—	2	4	no	—
316	#568	—	2	4	no	—
317	C₂×#131	—	2	2	no	—
318	C₂×#131	C₂²×C₄₁	1	4	yes	complemented
319	C₂²×M₁₆	—	2	2	no	—
320	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
321	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
322	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
323	C₂²×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
324	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
325	#419	—	2	1	no	—
326	#418	—	2	1	no	—
327	#418	—	2	2	no	—
328	#418	—	2	2	no	—
329	#417	C₂²×C₄₁	1	4	yes	complemented
330	#416	—	2	2	no	—
331	C₂×#102	C₂²×C₄₁	1	4	yes	complemented
332	C₂×#100	—	2	4	no	—
333	C₂×#100	C₂²×C₄₁	1	4	yes	complemented
334	#343	C₂²×C₄₁	1	4	yes	complemented
335	#341	C₂²×C₄₁	1	4	yes	complemented
336	C₄₁	#21005	1	1	yes	characteristic, direct product factor, Sylow
337	C₂⁵	—	2	1	no	—
338	C₂×#41	—	2	1	no	—
339	C₂×#41	—	2	2	no	—
340	C₂×#41	—	2	2	no	—
341	C₂×#41	—	2	1	no	—
342	C₂×#41	C₂×C₄×C₄₁	1	4	yes	—
343	C₂×#41	D₄×C₄₁	1	4	yes	—
344	C₂×#41	D₄×C₄₁	1	2	yes	—
345	C₂×#41	C₂³×C₄₁	1	2	yes	—
346	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
347	C₂²×D₄	—	2	1	no	—
348	C₂²×D₄	—	2	1	no	—
349	C₂²×D₄	—	2	2	no	—
350	C₂²×D₄	—	2	1	no	—
351	C₂²×D₄	—	2	1	no	—
352	C₂²×D₄	—	2	1	no	—
353	C₂²×D₄	—	2	2	no	—
354	C₂²×D₄	—	2	2	no	—
355	C₂²×D₄	—	2	1	no	—
356	C₂²×D₄	C₂×C₄×C₄₁	1	4	yes	—
357	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
358	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
359	C₂³×C₄	—	2	1	no	—
360	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
361	AGL(1,ℤ/8)	—	2	16	no	—
362	AGL(1,ℤ/8)	—	2	8	no	—
363	AGL(1,ℤ/8)	C₂×C₄×C₄₁	1	16	yes	—
364	C₂×SD₁₆	—	2	8	no	—
365	C₂×SD₁₆	—	2	4	no	—
366	C₂×D₈	—	2	8	no	—
367	C₂×D₈	—	2	4	no	—
368	#131	—	2	16	no	—
369	#131	—	2	8	no	—
370	#131	—	2	8	no	—
371	C₂×M₁₆	—	2	16	no	—
372	C₂×M₁₆	—	2	1	no	—
373	C₂×M₁₆	—	2	2	no	—
374	C₂×M₁₆	—	2	2	no	—
375	C₂×M₁₆	—	2	2	no	—
376	C₂×M₁₆	—	2	2	no	—
377	C₂×M₁₆	—	2	1	no	—
378	C₂×M₁₆	—	2	1	no	—
379	C₂×M₁₆	—	2	4	no	—
380	C₂×M₁₆	—	2	2	no	—
381	C₂×M₁₆	—	2	2	no	—
382	C₂×M₁₆	C₂×C₄×C₄₁	1	4	yes	—
383	C₂×M₁₆	D₄×C₄₁	1	4	yes	—
384	C₂×M₁₆	D₄×C₄₁	1	2	yes	—
385	C₂×M₁₆	D₄×C₄₁	1	2	yes	—
386	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
387	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
388	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
389	C₂²×C₈	—	2	1	no	—
390	C₂²×C₈	—	2	1	no	—
391	C₂²×C₈	—	2	4	no	—
392	C₂²×C₈	—	2	1	no	—
393	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
394	#117	—	2	1	no	—
395	#117	—	2	1	no	—
396	#117	—	2	1	no	—
397	#117	—	2	1	no	—
398	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
399	C₂×C₂²⋊C₄	—	2	1	no	—
400	C₂×C₄²	—	2	1	no	—
401	#103	—	2	4	no	—
402	#102	—	2	4	no	—
403	#102	—	2	8	no	—
404	#100	—	2	8	no	—
405	C₂⁴	—	2	1	no	—
406	C₂⁴	—	2	4	no	—
407	C₂⁴	—	2	2	no	—
408	C₂⁴	—	4	1	no	—
409	C₂⁴	—	4	1	no	—
410	C₂⁴	—	4	2	no	—
411	C₂⁴	—	8	4	no	—
412	C₂⁴	#41×C₄₁	1	1	yes	characteristic
413	#41	—	2	4	no	—
414	#41	—	2	4	no	—
415	#41	—	2	4	no	—
416	#41	—	2	4	no	—
417	#41	—	2	4	no	—
418	#41	—	2	2	no	—
419	#41	—	2	2	no	—
420	#41	C₂²⋊C₄×C₄₁	1	8	yes	—
421	#41	C₂²×C₄×C₄₁	1	4	yes	—
422	#41	C₂×D₄×C₄₁	1	4	yes	—
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	—	2	2	no	—
425	C₂×Q₈	C₂²⋊C₄×C₄₁	1	4	yes	—
426	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	—	2	4	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	2	4	no	—
433	C₂×D₄	—	2	4	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	4	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	2	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	2	2	no	—
441	C₂×D₄	—	2	2	no	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	4	2	no	—
444	C₂×D₄	—	4	2	no	—
445	C₂×D₄	—	4	2	no	—
446	C₂×D₄	—	4	2	no	—
447	C₂×D₄	—	4	1	no	—
448	C₂×D₄	—	4	1	no	—
449	C₂×D₄	—	4	2	no	—
450	C₂×D₄	C₂²⋊C₄×C₄₁	1	4	yes	—
451	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
452	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	4	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	2	no	—
458	C₂²×C₄	—	2	2	no	—
459	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
460	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
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	C₂×D₄×C₄₁	1	2	yes	—
466	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
467	C₂²×C₄	#41×C₄₁	1	2	yes	—
468	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
469	SD₁₆	—	2	16	no	—
470	D₈	—	2	16	no	—
471	M₁₆	—	2	16	no	—
472	M₁₆	—	2	4	no	—
473	M₁₆	—	2	4	no	—
474	M₁₆	—	2	4	no	—
475	M₁₆	—	2	4	no	—
476	M₁₆	—	2	2	no	—
477	M₁₆	—	2	2	no	—
478	M₁₆	—	2	2	no	—
479	M₁₆	—	2	2	no	—
480	M₁₆	—	4	2	no	—
481	M₁₆	C₂²×C₄×C₄₁	1	4	yes	—
482	M₁₆	C₂×D₄×C₄₁	1	4	yes	—
483	C₂×C₈	—	2	16	no	—
484	C₂×C₈	—	2	4	no	—
485	C₂×C₈	—	2	4	no	—
486	C₂×C₈	—	2	2	no	—
487	C₂×C₈	—	2	2	no	—
488	C₂×C₈	—	2	2	no	—
489	C₂×C₈	—	2	2	no	—
490	C₂×C₈	—	2	2	no	—
491	C₂×C₈	—	4	2	no	—
492	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₂²⋊C₄	—	4	1	no	—
496	C₂²⋊C₄	—	4	1	no	—
497	C₄²	—	4	1	no	—
498	C₄²	—	4	1	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	2	4	no	—
501	C₂³	—	2	4	no	—
502	C₂³	—	4	2	no	—
503	C₂³	—	4	2	no	—
504	C₂³	—	4	2	no	—
505	C₂³	—	4	4	no	—
506	C₂³	—	4	4	no	—
507	C₂³	—	8	2	no	—
508	C₂³	—	8	4	no	—
509	C₂³	—	8	4	no	—
510	C₂³	—	8	4	no	—
511	C₂³	—	8	8	no	—
512	C₂³	—	8	8	no	—
513	C₂³	C₄×D₄×C₄₁	1	2	yes	—
514	C₂³	#121×C₄₁	1	1	yes	characteristic
515	C₂³	#121×C₄₁	1	1	yes	characteristic
516	C₂³	#123×C₄₁	1	2	yes	—
517	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
518	Q₈	—	2	2	no	—
519	Q₈	—	2	4	no	—
520	Q₈	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
521	D₄	—	2	4	no	—
522	D₄	—	2	4	no	—
523	D₄	—	2	8	no	—
524	D₄	—	2	4	no	—
525	D₄	—	2	4	no	—
526	D₄	—	2	2	no	—
527	D₄	—	2	4	no	—
528	D₄	—	4	2	no	—
529	D₄	—	4	2	no	—
530	D₄	—	4	4	no	—
531	D₄	—	4	4	no	—
532	D₄	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	2	8	no	—
538	C₂×C₄	—	4	2	no	—
539	C₂×C₄	—	8	2	no	—
540	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
541	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	2	yes	—
542	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
543	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
544	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
545	C₂×C₄	#120×C₄₁	1	2	yes	—
546	C₂×C₄	#120×C₄₁	1	2	yes	—
547	C₂×C₄	#121×C₄₁	1	2	yes	—
548	C₂×C₄	#121×C₄₁	1	2	yes	—
549	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
550	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
551	C₂×C₄	#123×C₄₁	1	2	yes	—
552	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
553	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
554	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
555	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
556	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
557	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
558	C₈	—	2	4	no	—
559	C₈	—	2	4	no	—
560	C₈	—	2	4	no	—
561	C₈	—	4	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	2	2	no	—
565	C₂²	—	2	4	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	4	no	—
569	C₂²	—	4	4	no	—
570	C₂²	—	8	4	no	—
571	C₂²	—	8	4	no	—
572	C₂²	—	8	8	no	—
573	C₂²	—	8	4	no	—
574	C₂²	—	8	8	no	—
575	C₂²	—	8	4	no	—
576	C₂²	C₄₁×#386	1	2	yes	—
577	C₂²	C₄₁×#386	1	2	yes	—
578	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
579	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
580	C₂²	C₂×#123×C₄₁	1	1	yes	characteristic
581	C₄	—	2	4	no	—
582	C₄	—	8	2	no	—
583	C₄	C₄₁×#386	1	2	yes	—
584	C₄	C₄₁×#386	1	2	yes	—
585	C₄	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
586	C₄	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
587	C₄	C₂×#120×C₄₁	1	1	yes	characteristic
588	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
589	C₂	—	2	2	no	—
590	C₂	—	2	1	no	—
591	C₂	—	2	1	no	—
592	C₂	—	2	2	no	—
593	C₂	—	4	4	no	—
594	C₂	—	8	4	no	—
595	C₂	C₂×C₄₁×#386	1	1	yes	characteristic
596	C₂	C₄₁×#3068	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#121
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂²×AGL(1,ℤ/8)×C₄₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#568	C₂	semidirect
3. C₄₁×#2896	C₂	semidirect
4. C₂×C₄₁×#418	C₂	semidirect
5. C₂²×#100×C₄₁	C₂	semidirect
6. C₄₁×#1907	C₂	semidirect
7. C₄₁×#1742	C₂	semidirect
8. C₄₁×#1738	C₂	semidirect
13. C₂×AGL(1,ℤ/8)×C₄₁	C₄	semidirect
20. C₂×#131×C₄₁	C₂²	semidirect
31. C₄₁×#417	C₂²	semidirect
33. C₂×#102×C₄₁	C₂²	semidirect
35. C₂×#100×C₄₁	C₂²	semidirect
36. C₄₁×#343	C₂²	semidirect
37. C₄₁×#341	C₂²	semidirect
263. #21005	C₄₁	direct (N × H)
292. C₂×#568	C₂×C₄₁	semidirect
293. #2896	C₂×C₄₁	semidirect
294. C₂×#418	C₂×C₄₁	semidirect
295. C₂²×#100	C₂×C₄₁	semidirect
296. #1907	C₂×C₄₁	semidirect
297. #1742	C₂×C₄₁	semidirect
298. #1738	C₂×C₄₁	semidirect
311. C₂×AGL(1,ℤ/8)	C₄×C₄₁	semidirect
318. C₂×#131	C₂²×C₄₁	semidirect
329. #417	C₂²×C₄₁	semidirect
331. C₂×#102	C₂²×C₄₁	semidirect
333. C₂×#100	C₂²×C₄₁	semidirect
334. #343	C₂²×C₄₁	semidirect
335. #341	C₂²×C₄₁	semidirect
336. C₄₁	#21005	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×AGL(1,ℤ/8)×C₄₁	2
C₂×C₄₁×#568	2
C₄₁×#2896	2
C₂×C₄₁×#418	2
C₂²×#100×C₄₁	2
C₄₁×#1907	2
C₄₁×#1742	2
C₄₁×#1738	2
#21005	41

Subgroups — C41×#21005 #23797206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21005 #23797206