Subgroups — C₄₁×#24291 #23800492

Order 10496.

1362 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#576	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#472	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#3061	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#3012	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₁×#443	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#2051	C₂	1	2	yes	maximal, complemented
7	C₄₁×#2006	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1974	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1970	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄₁×#1816	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₁×#1792	C₂	1	2	yes	maximal, complemented
12	C₂×H(5,𝔽₂)×C₄₁	C₂²	1	2	yes	—
13	C₄₁×#576	—	2	4	no	—
14	C₄₁×#576	—	2	4	no	—
15	C₂×AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
16	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
17	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	complemented
18	C₂×#135×C₄₁	—	2	1	no	—
19	C₂×#135×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×#135×C₄₁	C₂²	1	1	yes	characteristic
21	C₂²×D₈×C₄₁	—	2	1	no	—
22	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
23	C₂×#131×C₄₁	C₂²	1	1	yes	characteristic
24	C₂×#131×C₄₁	C₂²	1	1	yes	characteristic, complemented
25	C₄₁×#537	C₂²	1	2	yes	complemented
26	C₄₁×#472	—	2	4	no	—
27	C₄₁×#468	—	2	2	no	—
28	C₄₁×#467	—	2	1	no	—
29	C₄₁×#466	—	2	1	no	—
30	C₄₁×#452	—	2	2	no	—
31	C₄₁×#450	—	2	2	no	—
32	C₄₁×#449	—	2	2	no	—
33	C₄₁×#447	—	2	2	no	—
34	C₄₁×#443	—	2	4	no	—
35	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic, complemented
36	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic, complemented
37	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic, complemented
38	C₂×#104×C₄₁	C₂²	1	1	yes	characteristic
39	C₂×#104×C₄₁	C₂²	1	1	yes	characteristic, complemented
40	C₂×#104×C₄₁	C₂²	1	2	yes	complemented
41	C₄₁×#417	C₂²	1	2	yes	—
42	C₄₁×#417	C₂²	1	2	yes	complemented
43	C₂×#102×C₄₁	C₂²	1	2	yes	complemented
44	C₂×#100×C₄₁	C₂²	1	2	yes	complemented
45	C₄₁×#408	C₂²	1	2	yes	complemented
46	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
47	C₄₁×#390	C₂²	1	2	yes	complemented
48	C₄₁×#343	C₂²	1	2	yes	complemented
49	H(5,𝔽₂)×C₄₁	—	2	4	no	—
50	H(5,𝔽₂)×C₄₁	—	2	4	no	—
51	H(5,𝔽₂)×C₄₁	—	2	4	no	—
52	H(5,𝔽₂)×C₄₁	—	4	2	no	—
53	C₂×#41×C₄₁	—	2	2	no	—
54	C₂×#41×C₄₁	—	2	2	no	—
55	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
56	C₂×#41×C₄₁	C₂³	1	2	yes	—
57	C₂×#41×C₄₁	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₄₁	—	2	2	no	—
59	C₂²×D₄×C₄₁	—	2	2	no	—
60	C₂²×D₄×C₄₁	—	4	2	no	—
61	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
62	AGL(1,ℤ/8)×C₄₁	—	2	16	no	—
63	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
64	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
65	AGL(1,ℤ/8)×C₄₁	—	2	4	no	—
66	AGL(1,ℤ/8)×C₄₁	—	4	2	no	—
67	#135×C₄₁	—	2	8	no	—
68	#135×C₄₁	—	2	2	no	—
69	#135×C₄₁	—	2	2	no	—
70	#135×C₄₁	—	2	2	no	—
71	#135×C₄₁	—	2	2	no	—
72	C₂×Q₁₆×C₄₁	D₄	1	1	yes	characteristic
73	C₂×Q₁₆×C₄₁	D₄	1	1	yes	characteristic
74	C₂×SD₁₆×C₄₁	—	2	2	no	—
75	C₂×SD₁₆×C₄₁	—	2	2	no	—
76	C₂×SD₁₆×C₄₁	D₄	1	2	yes	—
77	C₂×SD₁₆×C₄₁	D₄	1	1	yes	characteristic
78	C₂×SD₁₆×C₄₁	D₄	1	1	yes	characteristic
79	C₂×D₈×C₄₁	—	2	8	no	—
80	C₂×D₈×C₄₁	—	2	2	no	—
81	C₂×D₈×C₄₁	—	2	2	no	—
82	C₂×D₈×C₄₁	—	2	1	no	—
83	C₂×D₈×C₄₁	—	2	1	no	—
84	C₂×D₈×C₄₁	—	2	2	no	—
85	C₂×D₈×C₄₁	D₄	1	1	yes	characteristic
86	C₂×D₈×C₄₁	D₄	1	1	yes	characteristic
87	C₂×D₈×C₄₁	D₄	1	4	yes	complemented
88	#131×C₄₁	—	2	2	no	—
89	#131×C₄₁	—	2	2	no	—
90	#131×C₄₁	—	2	2	no	—
91	#131×C₄₁	D₄	1	4	yes	complemented
92	C₂×M₁₆×C₄₁	—	2	1	no	—
93	C₂×M₁₆×C₄₁	—	2	1	no	—
94	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
95	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
96	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
97	C₂²×C₈×C₄₁	—	2	1	no	—
98	C₂²×C₈×C₄₁	—	2	1	no	—
99	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
100	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
101	C₂²⋊Q₈×C₄₁	—	2	2	no	—
102	C₂²⋊Q₈×C₄₁	—	2	2	no	—
103	#121×C₄₁	—	2	2	no	—
104	#121×C₄₁	—	2	2	no	—
105	#121×C₄₁	—	4	2	no	—
106	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₄₁	—	4	2	no	—
108	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
109	#108×C₄₁	—	2	2	no	—
110	#108×C₄₁	—	2	2	no	—
111	#107×C₄₁	—	2	1	no	—
112	#107×C₄₁	—	2	1	no	—
113	#104×C₄₁	—	2	2	no	—
114	#104×C₄₁	—	2	2	no	—
115	#104×C₄₁	—	4	2	no	—
116	#103×C₄₁	—	2	2	no	—
117	#103×C₄₁	—	2	2	no	—
118	#102×C₄₁	—	2	2	no	—
119	#102×C₄₁	—	2	2	no	—
120	#102×C₄₁	—	2	2	no	—
121	#102×C₄₁	—	2	2	no	—
122	#100×C₄₁	—	2	4	no	—
123	C₂²⋊C₈×C₄₁	—	2	2	no	—
124	C₂²⋊C₈×C₄₁	—	2	2	no	—
125	C₄×C₈×C₄₁	—	2	2	no	—
126	C₂⁴×C₄₁	—	2	2	no	—
127	C₂⁴×C₄₁	—	4	2	no	—
128	#41×C₄₁	—	2	1	no	—
129	#41×C₄₁	—	2	1	no	—
130	#41×C₄₁	—	2	8	no	—
131	#41×C₄₁	—	2	1	no	—
132	#41×C₄₁	—	2	1	no	—
133	#41×C₄₁	—	2	2	no	—
134	#41×C₄₁	—	2	2	no	—
135	#41×C₄₁	—	2	1	no	—
136	#41×C₄₁	—	2	2	no	—
137	#41×C₄₁	—	2	2	no	—
138	#41×C₄₁	—	2	1	no	—
139	#41×C₄₁	—	4	4	no	—
140	#41×C₄₁	—	4	4	no	—
141	#41×C₄₁	—	4	4	no	—
142	#41×C₄₁	C₂×D₄	1	2	yes	—
143	#41×C₄₁	#41	1	2	yes	—
144	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
145	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
146	C₂×Q₈×C₄₁	C₂×D₄	1	1	yes	characteristic
147	C₂×D₄×C₄₁	—	2	1	no	—
148	C₂×D₄×C₄₁	—	2	8	no	—
149	C₂×D₄×C₄₁	—	2	1	no	—
150	C₂×D₄×C₄₁	—	2	2	no	—
151	C₂×D₄×C₄₁	—	2	2	no	—
152	C₂×D₄×C₄₁	—	2	2	no	—
153	C₂×D₄×C₄₁	—	2	2	no	—
154	C₂×D₄×C₄₁	—	2	2	no	—
155	C₂×D₄×C₄₁	—	2	2	no	—
156	C₂×D₄×C₄₁	—	4	4	no	—
157	C₂×D₄×C₄₁	—	4	2	no	—
158	C₂×D₄×C₄₁	—	4	2	no	—
159	C₂×D₄×C₄₁	—	4	4	no	—
160	C₂×D₄×C₄₁	—	4	8	no	—
161	C₂×D₄×C₄₁	—	4	8	no	—
162	C₂×D₄×C₄₁	—	4	4	no	—
163	C₂×D₄×C₄₁	—	4	2	no	—
164	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
165	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
166	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
167	C₂×D₄×C₄₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₄₁	—	2	1	no	—
169	C₂²×C₄×C₄₁	—	2	1	no	—
170	C₂²×C₄×C₄₁	—	2	2	no	—
171	C₂²×C₄×C₄₁	—	2	1	no	—
172	C₂²×C₄×C₄₁	—	2	2	no	—
173	C₂²×C₄×C₄₁	—	4	2	no	—
174	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
175	Q₁₆×C₄₁	—	2	2	no	—
176	Q₁₆×C₄₁	—	2	2	no	—
177	SD₁₆×C₄₁	—	2	8	no	—
178	SD₁₆×C₄₁	—	2	2	no	—
179	SD₁₆×C₄₁	—	2	2	no	—
180	SD₁₆×C₄₁	—	2	4	no	—
181	SD₁₆×C₄₁	—	4	4	no	—
182	D₈×C₄₁	—	2	8	no	—
183	D₈×C₄₁	—	2	2	no	—
184	D₈×C₄₁	—	2	2	no	—
185	D₈×C₄₁	—	2	4	no	—
186	D₈×C₄₁	—	2	4	no	—
187	D₈×C₄₁	—	4	4	no	—
188	M₁₆×C₄₁	—	2	1	no	—
189	M₁₆×C₄₁	—	2	4	no	—
190	M₁₆×C₄₁	—	2	1	no	—
191	M₁₆×C₄₁	—	2	2	no	—
192	M₁₆×C₄₁	—	2	1	no	—
193	M₁₆×C₄₁	—	2	2	no	—
194	M₁₆×C₄₁	—	4	2	no	—
195	M₁₆×C₄₁	C₂×D₄	1	2	yes	—
196	C₂×C₈×C₄₁	—	2	1	no	—
197	C₂×C₈×C₄₁	—	2	1	no	—
198	C₂×C₈×C₄₁	—	2	1	no	—
199	C₂×C₈×C₄₁	—	2	4	no	—
200	C₂×C₈×C₄₁	—	2	2	no	—
201	C₂×C₈×C₄₁	—	2	1	no	—
202	C₂×C₈×C₄₁	—	4	2	no	—
203	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
204	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
205	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
206	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
207	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
208	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
209	C₄⋊C₄×C₄₁	—	2	2	no	—
210	C₄⋊C₄×C₄₁	—	2	2	no	—
211	C₄⋊C₄×C₄₁	—	4	2	no	—
212	C₂²⋊C₄×C₄₁	—	4	2	no	—
213	C₂²⋊C₄×C₄₁	—	4	2	no	—
214	C₂²⋊C₄×C₄₁	—	8	4	no	—
215	C₄²×C₄₁	—	2	1	no	—
216	C₄²×C₄₁	—	2	1	no	—
217	C₂³×C₄₁	—	2	1	no	—
218	C₂³×C₄₁	—	2	1	no	—
219	C₂³×C₄₁	—	2	2	no	—
220	C₂³×C₄₁	—	2	4	no	—
221	C₂³×C₄₁	—	4	8	no	—
222	C₂³×C₄₁	—	4	2	no	—
223	C₂³×C₄₁	—	4	2	no	—
224	C₂³×C₄₁	—	8	2	no	—
225	C₂³×C₄₁	—	8	2	no	—
226	C₂³×C₄₁	—	16	2	no	—
227	C₂³×C₄₁	—	16	2	no	—
228	C₂³×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
229	Q₈×C₄₁	—	2	1	no	—
230	Q₈×C₄₁	—	2	2	no	—
231	Q₈×C₄₁	—	2	1	no	—
232	Q₈×C₄₁	—	2	1	no	—
233	Q₈×C₄₁	—	2	2	no	—
234	Q₈×C₄₁	#121	1	2	yes	—
235	D₄×C₄₁	—	2	2	no	—
236	D₄×C₄₁	—	2	2	no	—
237	D₄×C₄₁	—	2	2	no	—
238	D₄×C₄₁	—	2	4	no	—
239	D₄×C₄₁	—	2	1	no	—
240	D₄×C₄₁	—	2	1	no	—
241	D₄×C₄₁	—	2	1	no	—
242	D₄×C₄₁	—	2	4	no	—
243	D₄×C₄₁	—	2	2	no	—
244	D₄×C₄₁	—	4	8	no	—
245	D₄×C₄₁	—	4	8	no	—
246	D₄×C₄₁	—	4	2	no	—
247	D₄×C₄₁	—	4	1	no	—
248	D₄×C₄₁	—	4	2	no	—
249	D₄×C₄₁	—	4	1	no	—
250	D₄×C₄₁	—	4	2	no	—
251	D₄×C₄₁	—	4	2	no	—
252	D₄×C₄₁	—	16	2	no	—
253	D₄×C₄₁	#121	1	2	yes	—
254	C₂×C₄×C₄₁	—	2	1	no	—
255	C₂×C₄×C₄₁	—	2	2	no	—
256	C₂×C₄×C₄₁	—	2	2	no	—
257	C₂×C₄×C₄₁	—	2	2	no	—
258	C₂×C₄×C₄₁	—	2	1	no	—
259	C₂×C₄×C₄₁	—	2	2	no	—
260	C₂×C₄×C₄₁	—	4	8	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	2	no	—
265	C₂×C₄×C₄₁	—	4	2	no	—
266	C₂×C₄×C₄₁	—	4	1	no	—
267	C₂×C₄×C₄₁	—	4	2	no	—
268	C₂×C₄×C₄₁	—	8	2	no	—
269	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
274	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
275	C₈×C₄₁	—	2	2	no	—
276	C₈×C₄₁	—	2	1	no	—
277	C₈×C₄₁	—	2	1	no	—
278	C₈×C₄₁	—	4	1	no	—
279	C₈×C₄₁	—	4	2	no	—
280	C₈×C₄₁	#120	1	2	yes	—
281	C₈×C₄₁	#120	1	2	yes	—
282	#24291	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
283	C₂²×C₄₁	—	2	1	no	—
284	C₂²×C₄₁	—	2	1	no	—
285	C₂²×C₄₁	—	2	2	no	—
286	C₂²×C₄₁	—	4	1	no	—
287	C₂²×C₄₁	—	4	1	no	—
288	C₂²×C₄₁	—	4	2	no	—
289	C₂²×C₄₁	—	4	4	no	—
290	C₂²×C₄₁	—	4	1	no	—
291	C₂²×C₄₁	—	4	1	no	—
292	C₂²×C₄₁	—	4	2	no	—
293	C₂²×C₄₁	—	4	1	no	—
294	C₂²×C₄₁	—	4	2	no	—
295	C₂²×C₄₁	—	8	4	no	—
296	C₂²×C₄₁	—	8	2	no	—
297	C₂²×C₄₁	—	8	2	no	—
298	C₂²×C₄₁	—	16	2	no	—
299	C₂²×C₄₁	—	16	8	no	—
300	C₂²×C₄₁	#535	1	1	yes	characteristic
301	C₂²×C₄₁	D₄²	1	1	yes	characteristic
302	C₂²×C₄₁	#546	1	1	yes	characteristic
303	C₄×C₄₁	—	2	2	no	—
304	C₄×C₄₁	—	4	2	no	—
305	C₄×C₄₁	—	4	2	no	—
306	C₄×C₄₁	—	4	1	no	—
307	C₄×C₄₁	—	8	2	no	—
308	C₄×C₄₁	C₂×#120	1	1	yes	characteristic
309	C₄×C₄₁	C₂×#121	1	1	yes	characteristic
310	C₄×C₄₁	D₄²	1	1	yes	characteristic
311	C₄×C₄₁	#546	1	1	yes	characteristic
312	C₂×#576	C₂×C₄₁	1	1	yes	characteristic, complemented
313	C₂×#472	C₂×C₄₁	1	1	yes	characteristic, complemented
314	#3061	C₂×C₄₁	1	1	yes	characteristic, complemented
315	#3012	C₂×C₄₁	1	2	yes	complemented
316	C₂×#443	C₂×C₄₁	1	1	yes	characteristic, complemented
317	#2051	C₂×C₄₁	1	2	yes	complemented
318	#2006	C₂×C₄₁	1	2	yes	complemented
319	#1974	C₂×C₄₁	1	1	yes	characteristic, complemented
320	#1970	C₂×C₄₁	1	1	yes	characteristic, complemented
321	#1816	C₂×C₄₁	1	1	yes	characteristic, complemented
322	#1792	C₂×C₄₁	1	2	yes	complemented
323	C₂×C₄₁	—	2	1	no	—
324	C₂×C₄₁	—	2	1	no	—
325	C₂×C₄₁	—	4	2	no	—
326	C₂×C₄₁	—	8	1	no	—
327	C₂×C₄₁	—	8	2	no	—
328	C₂×C₄₁	—	8	4	no	—
329	C₂×C₄₁	#2408	1	1	yes	characteristic
330	C₂×C₄₁	#3290	1	1	yes	characteristic
331	C₂×C₄₁	#3292	1	1	yes	characteristic
332	C₂×H(5,𝔽₂)	C₂²×C₄₁	1	2	yes	—
333	#576	—	2	4	no	—
334	#576	—	2	4	no	—
335	C₂×AGL(1,ℤ/8)	—	2	2	no	—
336	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
337	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	complemented
338	C₂×#135	—	2	1	no	—
339	C₂×#135	C₂²×C₄₁	1	1	yes	characteristic
340	C₂×#135	C₂²×C₄₁	1	1	yes	characteristic
341	C₂²×D₈	—	2	1	no	—
342	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
343	C₂×#131	C₂²×C₄₁	1	1	yes	characteristic
344	C₂×#131	C₂²×C₄₁	1	1	yes	characteristic, complemented
345	#537	C₂²×C₄₁	1	2	yes	complemented
346	#472	—	2	4	no	—
347	#468	—	2	2	no	—
348	#467	—	2	1	no	—
349	#466	—	2	1	no	—
350	#452	—	2	2	no	—
351	#450	—	2	2	no	—
352	#449	—	2	2	no	—
353	#447	—	2	2	no	—
354	#443	—	2	4	no	—
355	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic, complemented
356	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic, complemented
357	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic, complemented
358	C₂×#104	C₂²×C₄₁	1	1	yes	characteristic
359	C₂×#104	C₂²×C₄₁	1	1	yes	characteristic, complemented
360	C₂×#104	C₂²×C₄₁	1	2	yes	complemented
361	#417	C₂²×C₄₁	1	2	yes	—
362	#417	C₂²×C₄₁	1	2	yes	complemented
363	C₂×#102	C₂²×C₄₁	1	2	yes	complemented
364	C₂×#100	C₂²×C₄₁	1	2	yes	complemented
365	#408	C₂²×C₄₁	1	2	yes	complemented
366	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
367	#390	C₂²×C₄₁	1	2	yes	complemented
368	#343	C₂²×C₄₁	1	2	yes	complemented
369	C₄₁	#24291	1	1	yes	characteristic, direct product factor, Sylow
370	H(5,𝔽₂)	—	2	4	no	—
371	H(5,𝔽₂)	—	2	4	no	—
372	H(5,𝔽₂)	—	2	4	no	—
373	H(5,𝔽₂)	—	4	2	no	—
374	C₂×#41	—	2	2	no	—
375	C₂×#41	—	2	2	no	—
376	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
377	C₂×#41	C₂³×C₄₁	1	2	yes	—
378	C₂×#41	C₂³×C₄₁	1	1	yes	characteristic
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	AGL(1,ℤ/8)	—	2	16	no	—
384	AGL(1,ℤ/8)	—	2	4	no	—
385	AGL(1,ℤ/8)	—	2	4	no	—
386	AGL(1,ℤ/8)	—	2	4	no	—
387	AGL(1,ℤ/8)	—	4	2	no	—
388	#135	—	2	8	no	—
389	#135	—	2	2	no	—
390	#135	—	2	2	no	—
391	#135	—	2	2	no	—
392	#135	—	2	2	no	—
393	C₂×Q₁₆	D₄×C₄₁	1	1	yes	characteristic
394	C₂×Q₁₆	D₄×C₄₁	1	1	yes	characteristic
395	C₂×SD₁₆	—	2	2	no	—
396	C₂×SD₁₆	—	2	2	no	—
397	C₂×SD₁₆	D₄×C₄₁	1	2	yes	—
398	C₂×SD₁₆	D₄×C₄₁	1	1	yes	characteristic
399	C₂×SD₁₆	D₄×C₄₁	1	1	yes	characteristic
400	C₂×D₈	—	2	8	no	—
401	C₂×D₈	—	2	2	no	—
402	C₂×D₈	—	2	2	no	—
403	C₂×D₈	—	2	1	no	—
404	C₂×D₈	—	2	1	no	—
405	C₂×D₈	—	2	2	no	—
406	C₂×D₈	D₄×C₄₁	1	1	yes	characteristic
407	C₂×D₈	D₄×C₄₁	1	1	yes	characteristic
408	C₂×D₈	D₄×C₄₁	1	4	yes	complemented
409	#131	—	2	2	no	—
410	#131	—	2	2	no	—
411	#131	—	2	2	no	—
412	#131	D₄×C₄₁	1	4	yes	complemented
413	C₂×M₁₆	—	2	1	no	—
414	C₂×M₁₆	—	2	1	no	—
415	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
416	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
417	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
418	C₂²×C₈	—	2	1	no	—
419	C₂²×C₈	—	2	1	no	—
420	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
421	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
422	C₂²⋊Q₈	—	2	2	no	—
423	C₂²⋊Q₈	—	2	2	no	—
424	#121	—	2	2	no	—
425	#121	—	2	2	no	—
426	#121	—	4	2	no	—
427	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
428	C₂×C₂²⋊C₄	—	4	2	no	—
429	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
430	#108	—	2	2	no	—
431	#108	—	2	2	no	—
432	#107	—	2	1	no	—
433	#107	—	2	1	no	—
434	#104	—	2	2	no	—
435	#104	—	2	2	no	—
436	#104	—	4	2	no	—
437	#103	—	2	2	no	—
438	#103	—	2	2	no	—
439	#102	—	2	2	no	—
440	#102	—	2	2	no	—
441	#102	—	2	2	no	—
442	#102	—	2	2	no	—
443	#100	—	2	4	no	—
444	C₂²⋊C₈	—	2	2	no	—
445	C₂²⋊C₈	—	2	2	no	—
446	C₄×C₈	—	2	2	no	—
447	C₂⁴	—	2	2	no	—
448	C₂⁴	—	4	2	no	—
449	#41	—	2	1	no	—
450	#41	—	2	1	no	—
451	#41	—	2	8	no	—
452	#41	—	2	1	no	—
453	#41	—	2	1	no	—
454	#41	—	2	2	no	—
455	#41	—	2	2	no	—
456	#41	—	2	1	no	—
457	#41	—	2	2	no	—
458	#41	—	2	2	no	—
459	#41	—	2	1	no	—
460	#41	—	4	4	no	—
461	#41	—	4	4	no	—
462	#41	—	4	4	no	—
463	#41	C₂×D₄×C₄₁	1	2	yes	—
464	#41	#41×C₄₁	1	2	yes	—
465	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
466	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
467	C₂×Q₈	C₂×D₄×C₄₁	1	1	yes	characteristic
468	C₂×D₄	—	2	1	no	—
469	C₂×D₄	—	2	8	no	—
470	C₂×D₄	—	2	1	no	—
471	C₂×D₄	—	2	2	no	—
472	C₂×D₄	—	2	2	no	—
473	C₂×D₄	—	2	2	no	—
474	C₂×D₄	—	2	2	no	—
475	C₂×D₄	—	2	2	no	—
476	C₂×D₄	—	2	2	no	—
477	C₂×D₄	—	4	4	no	—
478	C₂×D₄	—	4	2	no	—
479	C₂×D₄	—	4	2	no	—
480	C₂×D₄	—	4	4	no	—
481	C₂×D₄	—	4	8	no	—
482	C₂×D₄	—	4	8	no	—
483	C₂×D₄	—	4	4	no	—
484	C₂×D₄	—	4	2	no	—
485	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
486	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
487	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
488	C₂×D₄	C₂×D₄×C₄₁	1	1	yes	characteristic
489	C₂²×C₄	—	2	1	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	2	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	2	no	—
494	C₂²×C₄	—	4	2	no	—
495	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
496	Q₁₆	—	2	2	no	—
497	Q₁₆	—	2	2	no	—
498	SD₁₆	—	2	8	no	—
499	SD₁₆	—	2	2	no	—
500	SD₁₆	—	2	2	no	—
501	SD₁₆	—	2	4	no	—
502	SD₁₆	—	4	4	no	—
503	D₈	—	2	8	no	—
504	D₈	—	2	2	no	—
505	D₈	—	2	2	no	—
506	D₈	—	2	4	no	—
507	D₈	—	2	4	no	—
508	D₈	—	4	4	no	—
509	M₁₆	—	2	1	no	—
510	M₁₆	—	2	4	no	—
511	M₁₆	—	2	1	no	—
512	M₁₆	—	2	2	no	—
513	M₁₆	—	2	1	no	—
514	M₁₆	—	2	2	no	—
515	M₁₆	—	4	2	no	—
516	M₁₆	C₂×D₄×C₄₁	1	2	yes	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	2	1	no	—
519	C₂×C₈	—	2	1	no	—
520	C₂×C₈	—	2	4	no	—
521	C₂×C₈	—	2	2	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	4	2	no	—
524	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
525	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
526	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
527	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
528	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
529	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
530	C₄⋊C₄	—	2	2	no	—
531	C₄⋊C₄	—	2	2	no	—
532	C₄⋊C₄	—	4	2	no	—
533	C₂²⋊C₄	—	4	2	no	—
534	C₂²⋊C₄	—	4	2	no	—
535	C₂²⋊C₄	—	8	4	no	—
536	C₄²	—	2	1	no	—
537	C₄²	—	2	1	no	—
538	C₂³	—	2	1	no	—
539	C₂³	—	2	1	no	—
540	C₂³	—	2	2	no	—
541	C₂³	—	2	4	no	—
542	C₂³	—	4	8	no	—
543	C₂³	—	4	2	no	—
544	C₂³	—	4	2	no	—
545	C₂³	—	8	2	no	—
546	C₂³	—	8	2	no	—
547	C₂³	—	16	2	no	—
548	C₂³	—	16	2	no	—
549	C₂³	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
550	Q₈	—	2	1	no	—
551	Q₈	—	2	2	no	—
552	Q₈	—	2	1	no	—
553	Q₈	—	2	1	no	—
554	Q₈	—	2	2	no	—
555	Q₈	#121×C₄₁	1	2	yes	—
556	D₄	—	2	2	no	—
557	D₄	—	2	2	no	—
558	D₄	—	2	2	no	—
559	D₄	—	2	4	no	—
560	D₄	—	2	1	no	—
561	D₄	—	2	1	no	—
562	D₄	—	2	1	no	—
563	D₄	—	2	4	no	—
564	D₄	—	2	2	no	—
565	D₄	—	4	8	no	—
566	D₄	—	4	8	no	—
567	D₄	—	4	2	no	—
568	D₄	—	4	1	no	—
569	D₄	—	4	2	no	—
570	D₄	—	4	1	no	—
571	D₄	—	4	2	no	—
572	D₄	—	4	2	no	—
573	D₄	—	16	2	no	—
574	D₄	#121×C₄₁	1	2	yes	—
575	C₂×C₄	—	2	1	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	1	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	4	8	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	2	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	8	2	no	—
590	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
591	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
592	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
593	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
594	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
595	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
596	C₈	—	2	2	no	—
597	C₈	—	2	1	no	—
598	C₈	—	2	1	no	—
599	C₈	—	4	1	no	—
600	C₈	—	4	2	no	—
601	C₈	#120×C₄₁	1	2	yes	—
602	C₈	#120×C₄₁	1	2	yes	—
603	C₂²	—	2	1	no	—
604	C₂²	—	2	1	no	—
605	C₂²	—	2	2	no	—
606	C₂²	—	4	1	no	—
607	C₂²	—	4	1	no	—
608	C₂²	—	4	1	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	2	no	—
611	C₂²	—	4	4	no	—
612	C₂²	—	4	1	no	—
613	C₂²	—	4	2	no	—
614	C₂²	—	4	1	no	—
615	C₂²	—	8	4	no	—
616	C₂²	—	8	2	no	—
617	C₂²	—	8	2	no	—
618	C₂²	—	16	2	no	—
619	C₂²	—	16	8	no	—
620	C₂²	C₄₁×#535	1	1	yes	characteristic
621	C₂²	D₄²×C₄₁	1	1	yes	characteristic
622	C₂²	C₄₁×#546	1	1	yes	characteristic
623	C₄	—	2	2	no	—
624	C₄	—	4	2	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	1	no	—
627	C₄	—	8	2	no	—
628	C₄	C₂×#120×C₄₁	1	1	yes	characteristic
629	C₄	C₂×#121×C₄₁	1	1	yes	characteristic
630	C₄	D₄²×C₄₁	1	1	yes	characteristic
631	C₄	C₄₁×#546	1	1	yes	characteristic
632	C₂	—	2	1	no	—
633	C₂	—	2	1	no	—
634	C₂	—	4	2	no	—
635	C₂	—	8	2	no	—
636	C₂	—	8	1	no	—
637	C₂	—	8	4	no	—
638	C₂	C₄₁×#2408	1	1	yes	characteristic
639	C₂	C₄₁×#3290	1	1	yes	characteristic
640	C₂	C₄₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#576	C₂	semidirect
2. C₂×C₄₁×#472	C₂	semidirect
3. C₄₁×#3061	C₂	semidirect
4. C₄₁×#3012	C₂	semidirect
5. C₂×C₄₁×#443	C₂	semidirect
6. C₄₁×#2051	C₂	semidirect
7. C₄₁×#2006	C₂	semidirect
8. C₄₁×#1974	C₂	semidirect
9. C₄₁×#1970	C₂	semidirect
10. C₄₁×#1816	C₂	semidirect
11. C₄₁×#1792	C₂	semidirect
17. C₂×AGL(1,ℤ/8)×C₄₁	C₂²	semidirect
24. C₂×#131×C₄₁	C₂²	semidirect
25. C₄₁×#537	C₂²	semidirect
35. C₂×#108×C₄₁	C₂²	semidirect
36. C₂×#108×C₄₁	C₂²	semidirect
37. C₂×#107×C₄₁	C₂²	semidirect
39. C₂×#104×C₄₁	C₂²	semidirect
40. C₂×#104×C₄₁	C₂²	semidirect
42. C₄₁×#417	C₂²	semidirect
43. C₂×#102×C₄₁	C₂²	semidirect
44. C₂×#100×C₄₁	C₂²	semidirect
45. C₄₁×#408	C₂²	semidirect
46. C₂×C₄×C₈×C₄₁	C₂²	semidirect
47. C₄₁×#390	C₂²	semidirect
48. C₄₁×#343	C₂²	semidirect
87. C₂×D₈×C₄₁	D₄	semidirect
91. #131×C₄₁	D₄	semidirect
282. #24291	C₄₁	direct (N × H)
312. C₂×#576	C₂×C₄₁	semidirect
313. C₂×#472	C₂×C₄₁	semidirect
314. #3061	C₂×C₄₁	semidirect
315. #3012	C₂×C₄₁	semidirect
316. C₂×#443	C₂×C₄₁	semidirect
317. #2051	C₂×C₄₁	semidirect
318. #2006	C₂×C₄₁	semidirect
319. #1974	C₂×C₄₁	semidirect
320. #1970	C₂×C₄₁	semidirect
321. #1816	C₂×C₄₁	semidirect
322. #1792	C₂×C₄₁	semidirect
337. C₂×AGL(1,ℤ/8)	C₂²×C₄₁	semidirect
344. C₂×#131	C₂²×C₄₁	semidirect
345. #537	C₂²×C₄₁	semidirect
355. C₂×#108	C₂²×C₄₁	semidirect
356. C₂×#108	C₂²×C₄₁	semidirect
357. C₂×#107	C₂²×C₄₁	semidirect
359. C₂×#104	C₂²×C₄₁	semidirect
360. C₂×#104	C₂²×C₄₁	semidirect
362. #417	C₂²×C₄₁	semidirect
363. C₂×#102	C₂²×C₄₁	semidirect
364. C₂×#100	C₂²×C₄₁	semidirect
365. #408	C₂²×C₄₁	semidirect
366. C₂×C₄×C₈	C₂²×C₄₁	semidirect
367. #390	C₂²×C₄₁	semidirect
368. #343	C₂²×C₄₁	semidirect
369. C₄₁	#24291	direct (N × H)
408. C₂×D₈	D₄×C₄₁	semidirect
412. #131	D₄×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×#576	2
C₂×C₄₁×#472	2
C₄₁×#3061	2
C₄₁×#3012	2
C₂×C₄₁×#443	2
C₄₁×#2051	2
C₄₁×#2006	2
C₄₁×#1974	2
C₄₁×#1970	2
C₄₁×#1816	2
C₄₁×#1792	2
#24291	41

Subgroups — C41×#24291 #23800492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24291 #23800492