Subgroups — C₄₁×#24222 #23800423

Order 10496.

1586 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3496	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#3036	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₁×#453	C₂	1	2	yes	maximal, complemented
4	C₄×#127×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#2039	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2006	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1962	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1800	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1792	C₂	1	2	yes	maximal, complemented
10	C₂×H(5,𝔽₂)×C₄₁	C₂²	1	2	yes	—
11	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	4	yes	complemented
12	C₄₁×#547	—	2	2	no	—
13	C₄₁×#546	—	2	4	no	—
14	D₄²×C₄₁	—	2	2	no	—
15	C₄₁×#537	C₂²	1	2	yes	—
16	C₄₁×#534	—	2	2	no	—
17	C₄₁×#532	C₂²	1	2	yes	—
18	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×#121×C₄₁	—	2	2	no	—
20	C₂×C₄×D₄×C₄₁	—	2	2	no	—
21	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic, complemented
23	C₄₁×#463	—	2	4	no	—
24	C₄₁×#461	—	2	2	no	—
25	C₄₁×#459	—	2	2	no	—
26	C₄₁×#453	—	2	8	no	—
27	C₄₁×#423	C₂²	1	2	yes	complemented
28	C₂×#104×C₄₁	C₂²	1	4	yes	complemented
29	C₂×#104×C₄₁	C₂²	1	4	yes	complemented
30	C₄₁×#417	C₂²	1	4	yes	complemented
31	C₂×#100×C₄₁	C₂²	1	2	yes	—
32	C₄₁×#390	—	2	2	no	—
33	C₄₁×#390	C₂²	1	1	yes	characteristic
34	C₄₁×#390	C₂²	1	1	yes	characteristic, complemented
35	C₄₁×#384	C₂²	1	1	yes	characteristic, complemented
36	C₄³×C₄₁	C₂²	1	1	yes	characteristic, complemented
37	C₄₁×#343	C₂²	1	2	yes	complemented
38	H(5,𝔽₂)×C₄₁	—	2	4	no	—
39	H(5,𝔽₂)×C₄₁	—	2	4	no	—
40	H(5,𝔽₂)×C₄₁	—	4	4	no	—
41	C₂×#41×C₄₁	—	2	4	no	—
42	C₂×#41×C₄₁	C₂³	1	4	yes	—
43	C₂²×D₄×C₄₁	—	2	2	no	—
44	C₂²×D₄×C₄₁	—	2	4	no	—
45	C₂²×D₄×C₄₁	—	2	2	no	—
46	C₂²×D₄×C₄₁	—	2	4	no	—
47	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₄₁	C₂³	1	1	yes	characteristic
49	C₂³×C₄×C₄₁	—	2	1	no	—
50	C₂³×C₄×C₄₁	D₄	1	2	yes	—
51	AGL(1,ℤ/8)×C₄₁	—	2	8	no	—
52	AGL(1,ℤ/8)×C₄₁	—	4	4	no	—
53	C₂×SD₁₆×C₄₁	—	2	4	no	—
54	C₂×D₈×C₄₁	—	2	4	no	—
55	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
56	C₄⋊Q₈×C₄₁	D₄	1	2	yes	complemented
57	#127×C₄₁	—	2	2	no	—
58	#127×C₄₁	D₄	1	2	yes	complemented
59	#127×C₄₁	D₄	1	4	yes	complemented
60	#124×C₄₁	D₄	1	4	yes	complemented
61	#123×C₄₁	—	2	8	no	—
62	C₂²⋊Q₈×C₄₁	—	2	4	no	—
63	#121×C₄₁	—	2	4	no	—
64	#121×C₄₁	—	2	4	no	—
65	#121×C₄₁	—	2	4	no	—
66	#121×C₄₁	—	2	2	no	—
67	#121×C₄₁	—	2	4	no	—
68	#121×C₄₁	—	2	2	no	—
69	#120×C₄₁	—	2	8	no	—
70	C₄×D₄×C₄₁	—	2	2	no	—
71	C₄×D₄×C₄₁	—	2	2	no	—
72	C₄×D₄×C₄₁	—	2	2	no	—
73	C₄×D₄×C₄₁	—	2	2	no	—
74	C₄×D₄×C₄₁	—	2	2	no	—
75	C₄×D₄×C₄₁	—	2	2	no	—
76	C₄×D₄×C₄₁	—	4	8	no	—
77	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
78	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
79	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₄₁	C₂³	1	1	yes	characteristic
81	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
82	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
84	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₄₁	D₄	1	2	yes	—
86	C₂×C₄²×C₄₁	—	2	1	no	—
87	C₂×C₄²×C₄₁	—	2	1	no	—
88	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
91	C₄⋊C₈×C₄₁	—	2	2	no	—
92	C₄⋊C₈×C₄₁	—	2	2	no	—
93	#104×C₄₁	—	2	8	no	—
94	#104×C₄₁	—	4	4	no	—
95	#103×C₄₁	—	2	4	no	—
96	#102×C₄₁	—	2	4	no	—
97	#100×C₄₁	—	2	4	no	—
98	C₂⁴×C₄₁	—	2	4	no	—
99	C₂⁴×C₄₁	—	2	1	no	—
100	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
101	#41×C₄₁	—	2	4	no	—
102	#41×C₄₁	—	2	4	no	—
103	#41×C₄₁	—	2	4	no	—
104	#41×C₄₁	—	2	4	no	—
105	#41×C₄₁	—	4	8	no	—
106	#41×C₄₁	—	4	8	no	—
107	C₂×Q₈×C₄₁	C₂×D₄	1	4	yes	—
108	C₂×D₄×C₄₁	—	2	8	no	—
109	C₂×D₄×C₄₁	—	2	8	no	—
110	C₂×D₄×C₄₁	—	2	4	no	—
111	C₂×D₄×C₄₁	—	2	4	no	—
112	C₂×D₄×C₄₁	—	2	2	no	—
113	C₂×D₄×C₄₁	—	2	2	no	—
114	C₂×D₄×C₄₁	—	2	2	no	—
115	C₂×D₄×C₄₁	—	2	2	no	—
116	C₂×D₄×C₄₁	—	2	1	no	—
117	C₂×D₄×C₄₁	—	2	2	no	—
118	C₂×D₄×C₄₁	—	2	2	no	—
119	C₂×D₄×C₄₁	—	4	4	no	—
120	C₂×D₄×C₄₁	—	4	4	no	—
121	C₂×D₄×C₄₁	—	4	8	no	—
122	C₂×D₄×C₄₁	—	4	8	no	—
123	C₂×D₄×C₄₁	—	4	8	no	—
124	C₂×D₄×C₄₁	—	4	8	no	—
125	C₂×D₄×C₄₁	—	8	4	no	—
126	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
127	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
128	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	complemented
129	C₂×D₄×C₄₁	#41	1	2	yes	—
130	C₂²×C₄×C₄₁	—	2	1	no	—
131	C₂²×C₄×C₄₁	—	2	4	no	—
132	C₂²×C₄×C₄₁	—	2	1	no	—
133	C₂²×C₄×C₄₁	—	2	2	no	—
134	C₂²×C₄×C₄₁	—	2	4	no	—
135	C₂²×C₄×C₄₁	—	2	2	no	—
136	C₂²×C₄×C₄₁	—	2	1	no	—
137	C₂²×C₄×C₄₁	—	4	4	no	—
138	C₂²×C₄×C₄₁	—	4	4	no	—
139	C₂²×C₄×C₄₁	—	4	1	no	—
140	C₂²×C₄×C₄₁	—	4	1	no	—
141	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
142	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
145	SD₁₆×C₄₁	—	4	8	no	—
146	D₈×C₄₁	—	4	8	no	—
147	M₁₆×C₄₁	—	2	4	no	—
148	M₁₆×C₄₁	—	2	4	no	—
149	C₂×C₈×C₄₁	—	2	4	no	—
150	C₄⋊C₄×C₄₁	—	2	4	no	—
151	C₄⋊C₄×C₄₁	—	2	2	no	—
152	C₄⋊C₄×C₄₁	—	2	2	no	—
153	C₄⋊C₄×C₄₁	—	2	2	no	—
154	C₄⋊C₄×C₄₁	—	2	2	no	—
155	C₄⋊C₄×C₄₁	—	4	4	no	—
156	C₂²⋊C₄×C₄₁	—	2	8	no	—
157	C₂²⋊C₄×C₄₁	—	2	4	no	—
158	C₂²⋊C₄×C₄₁	—	2	4	no	—
159	C₂²⋊C₄×C₄₁	—	2	4	no	—
160	C₂²⋊C₄×C₄₁	—	4	4	no	—
161	C₂²⋊C₄×C₄₁	—	4	4	no	—
162	C₂²⋊C₄×C₄₁	—	4	2	no	—
163	C₄²×C₄₁	—	2	4	no	—
164	C₄²×C₄₁	—	2	1	no	—
165	C₄²×C₄₁	—	2	2	no	—
166	C₄²×C₄₁	—	4	2	no	—
167	C₄²×C₄₁	C₂×D₄	1	2	yes	—
168	C₄²×C₄₁	C₂×D₄	1	2	yes	—
169	C₄²×C₄₁	C₂×D₄	1	2	yes	complemented
170	C₂³×C₄₁	—	2	2	no	—
171	C₂³×C₄₁	—	2	2	no	—
172	C₂³×C₄₁	—	2	4	no	—
173	C₂³×C₄₁	—	2	2	no	—
174	C₂³×C₄₁	—	4	2	no	—
175	C₂³×C₄₁	—	4	2	no	—
176	C₂³×C₄₁	—	4	2	no	—
177	C₂³×C₄₁	—	4	2	no	—
178	C₂³×C₄₁	—	4	8	no	—
179	C₂³×C₄₁	—	4	1	no	—
180	C₂³×C₄₁	—	4	1	no	—
181	C₂³×C₄₁	—	8	4	no	—
182	C₂³×C₄₁	—	8	4	no	—
183	C₂³×C₄₁	#121	1	4	yes	—
184	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
185	Q₈×C₄₁	—	2	4	no	—
186	Q₈×C₄₁	—	2	4	no	—
187	D₄×C₄₁	—	2	4	no	—
188	D₄×C₄₁	—	2	4	no	—
189	D₄×C₄₁	—	2	4	no	—
190	D₄×C₄₁	—	2	4	no	—
191	D₄×C₄₁	—	2	4	no	—
192	D₄×C₄₁	—	4	16	no	—
193	D₄×C₄₁	—	4	16	no	—
194	D₄×C₄₁	—	4	4	no	—
195	D₄×C₄₁	—	4	4	no	—
196	D₄×C₄₁	—	4	2	no	—
197	D₄×C₄₁	—	4	4	no	—
198	D₄×C₄₁	—	8	2	no	—
199	D₄×C₄₁	—	8	4	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	2	no	—
204	C₂×C₄×C₄₁	—	2	2	no	—
205	C₂×C₄×C₄₁	—	2	2	no	—
206	C₂×C₄×C₄₁	—	2	4	no	—
207	C₂×C₄×C₄₁	—	4	1	no	—
208	C₂×C₄×C₄₁	—	4	1	no	—
209	C₂×C₄×C₄₁	—	4	8	no	—
210	C₂×C₄×C₄₁	—	4	1	no	—
211	C₂×C₄×C₄₁	—	4	4	no	—
212	C₂×C₄×C₄₁	—	4	4	no	—
213	C₂×C₄×C₄₁	—	4	4	no	—
214	C₂×C₄×C₄₁	—	4	2	no	—
215	C₂×C₄×C₄₁	—	4	2	no	—
216	C₂×C₄×C₄₁	—	4	1	no	—
217	C₂×C₄×C₄₁	—	8	2	no	—
218	C₂×C₄×C₄₁	—	8	2	no	—
219	C₂×C₄×C₄₁	—	8	2	no	—
220	C₂×C₄×C₄₁	#120	1	2	yes	—
221	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
222	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
223	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
224	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
226	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
227	C₈×C₄₁	—	4	4	no	—
228	#24222	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂²×C₄₁	—	2	1	no	—
230	C₂²×C₄₁	—	2	1	no	—
231	C₂²×C₄₁	—	2	4	no	—
232	C₂²×C₄₁	—	4	4	no	—
233	C₂²×C₄₁	—	4	4	no	—
234	C₂²×C₄₁	—	4	2	no	—
235	C₂²×C₄₁	—	4	2	no	—
236	C₂²×C₄₁	—	4	4	no	—
237	C₂²×C₄₁	—	4	2	no	—
238	C₂²×C₄₁	—	4	4	no	—
239	C₂²×C₄₁	—	4	4	no	—
240	C₂²×C₄₁	—	8	2	no	—
241	C₂²×C₄₁	—	8	2	no	—
242	C₂²×C₄₁	—	8	1	no	—
243	C₂²×C₄₁	—	8	1	no	—
244	C₂²×C₄₁	—	16	8	no	—
245	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
246	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
247	C₂²×C₄₁	#535	1	1	yes	characteristic
248	C₄×C₄₁	—	2	4	no	—
249	C₄×C₄₁	—	2	1	no	—
250	C₄×C₄₁	—	2	2	no	—
251	C₄×C₄₁	—	4	2	no	—
252	C₄×C₄₁	—	4	4	no	—
253	C₄×C₄₁	—	4	2	no	—
254	C₄×C₄₁	—	8	1	no	—
255	C₄×C₄₁	#453	1	2	yes	complemented
256	C₄×C₄₁	D₄²	1	2	yes	—
257	C₄×C₄₁	#546	1	2	yes	—
258	#3496	C₂×C₄₁	1	1	yes	characteristic, complemented
259	#3036	C₂×C₄₁	1	2	yes	complemented
260	C₂×#453	C₂×C₄₁	1	2	yes	complemented
261	C₄×#127	C₂×C₄₁	1	1	yes	characteristic, complemented
262	#2039	C₂×C₄₁	1	2	yes	complemented
263	#2006	C₂×C₄₁	1	2	yes	complemented
264	#1962	C₂×C₄₁	1	2	yes	complemented
265	#1800	C₂×C₄₁	1	1	yes	characteristic, complemented
266	#1792	C₂×C₄₁	1	2	yes	complemented
267	C₂×C₄₁	—	2	1	no	—
268	C₂×C₄₁	—	2	1	no	—
269	C₂×C₄₁	—	4	4	no	—
270	C₂×C₄₁	—	8	2	no	—
271	C₂×C₄₁	—	8	4	no	—
272	C₂×C₄₁	#2408	1	1	yes	characteristic
273	C₂×C₄₁	C₂×#453	1	1	yes	characteristic
274	C₂×C₄₁	#3018	1	1	yes	characteristic
275	C₂×H(5,𝔽₂)	C₂²×C₄₁	1	2	yes	—
276	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	4	yes	complemented
277	#547	—	2	2	no	—
278	#546	—	2	4	no	—
279	D₄²	—	2	2	no	—
280	#537	C₂²×C₄₁	1	2	yes	—
281	#534	—	2	2	no	—
282	#532	C₂²×C₄₁	1	2	yes	—
283	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
284	C₂×#121	—	2	2	no	—
285	C₂×C₄×D₄	—	2	2	no	—
286	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
287	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic, complemented
288	#463	—	2	4	no	—
289	#461	—	2	2	no	—
290	#459	—	2	2	no	—
291	#453	—	2	8	no	—
292	#423	C₂²×C₄₁	1	2	yes	complemented
293	C₂×#104	C₂²×C₄₁	1	4	yes	complemented
294	C₂×#104	C₂²×C₄₁	1	4	yes	complemented
295	#417	C₂²×C₄₁	1	4	yes	complemented
296	C₂×#100	C₂²×C₄₁	1	2	yes	—
297	#390	—	2	2	no	—
298	#390	C₂²×C₄₁	1	1	yes	characteristic
299	#390	C₂²×C₄₁	1	1	yes	characteristic, complemented
300	#384	C₂²×C₄₁	1	1	yes	characteristic, complemented
301	C₄³	C₂²×C₄₁	1	1	yes	characteristic, complemented
302	#343	C₂²×C₄₁	1	2	yes	complemented
303	C₄₁	#24222	1	1	yes	characteristic, direct product factor, Sylow
304	H(5,𝔽₂)	—	2	4	no	—
305	H(5,𝔽₂)	—	2	4	no	—
306	H(5,𝔽₂)	—	4	4	no	—
307	C₂×#41	—	2	4	no	—
308	C₂×#41	C₂³×C₄₁	1	4	yes	—
309	C₂²×D₄	—	2	2	no	—
310	C₂²×D₄	—	2	4	no	—
311	C₂²×D₄	—	2	2	no	—
312	C₂²×D₄	—	2	4	no	—
313	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
314	C₂²×D₄	C₂³×C₄₁	1	1	yes	characteristic
315	C₂³×C₄	—	2	1	no	—
316	C₂³×C₄	D₄×C₄₁	1	2	yes	—
317	AGL(1,ℤ/8)	—	2	8	no	—
318	AGL(1,ℤ/8)	—	4	4	no	—
319	C₂×SD₁₆	—	2	4	no	—
320	C₂×D₈	—	2	4	no	—
321	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
322	C₄⋊Q₈	D₄×C₄₁	1	2	yes	complemented
323	#127	—	2	2	no	—
324	#127	D₄×C₄₁	1	2	yes	complemented
325	#127	D₄×C₄₁	1	4	yes	complemented
326	#124	D₄×C₄₁	1	4	yes	complemented
327	#123	—	2	8	no	—
328	C₂²⋊Q₈	—	2	4	no	—
329	#121	—	2	4	no	—
330	#121	—	2	4	no	—
331	#121	—	2	4	no	—
332	#121	—	2	2	no	—
333	#121	—	2	4	no	—
334	#121	—	2	2	no	—
335	#120	—	2	8	no	—
336	C₄×D₄	—	2	2	no	—
337	C₄×D₄	—	2	2	no	—
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	—	2	2	no	—
340	C₄×D₄	—	2	2	no	—
341	C₄×D₄	—	2	2	no	—
342	C₄×D₄	—	4	8	no	—
343	C₂×C₄⋊C₄	—	2	2	no	—
344	C₂×C₄⋊C₄	—	2	2	no	—
345	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
346	C₂×C₄⋊C₄	C₂³×C₄₁	1	1	yes	characteristic
347	C₂×C₂²⋊C₄	—	2	4	no	—
348	C₂×C₂²⋊C₄	—	2	2	no	—
349	C₂×C₂²⋊C₄	—	2	2	no	—
350	C₂×C₂²⋊C₄	—	2	1	no	—
351	C₂×C₂²⋊C₄	D₄×C₄₁	1	2	yes	—
352	C₂×C₄²	—	2	1	no	—
353	C₂×C₄²	—	2	1	no	—
354	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
355	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
356	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
357	C₄⋊C₈	—	2	2	no	—
358	C₄⋊C₈	—	2	2	no	—
359	#104	—	2	8	no	—
360	#104	—	4	4	no	—
361	#103	—	2	4	no	—
362	#102	—	2	4	no	—
363	#100	—	2	4	no	—
364	C₂⁴	—	2	4	no	—
365	C₂⁴	—	2	1	no	—
366	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
367	#41	—	2	4	no	—
368	#41	—	2	4	no	—
369	#41	—	2	4	no	—
370	#41	—	2	4	no	—
371	#41	—	4	8	no	—
372	#41	—	4	8	no	—
373	C₂×Q₈	C₂×D₄×C₄₁	1	4	yes	—
374	C₂×D₄	—	2	8	no	—
375	C₂×D₄	—	2	8	no	—
376	C₂×D₄	—	2	4	no	—
377	C₂×D₄	—	2	4	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	2	2	no	—
384	C₂×D₄	—	2	2	no	—
385	C₂×D₄	—	4	4	no	—
386	C₂×D₄	—	4	4	no	—
387	C₂×D₄	—	4	8	no	—
388	C₂×D₄	—	4	8	no	—
389	C₂×D₄	—	4	8	no	—
390	C₂×D₄	—	4	8	no	—
391	C₂×D₄	—	8	4	no	—
392	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
393	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
394	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	complemented
395	C₂×D₄	#41×C₄₁	1	2	yes	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	4	no	—
401	C₂²×C₄	—	2	2	no	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	4	4	no	—
404	C₂²×C₄	—	4	4	no	—
405	C₂²×C₄	—	4	1	no	—
406	C₂²×C₄	—	4	1	no	—
407	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
408	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
409	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
410	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
411	SD₁₆	—	4	8	no	—
412	D₈	—	4	8	no	—
413	M₁₆	—	2	4	no	—
414	M₁₆	—	2	4	no	—
415	C₂×C₈	—	2	4	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	2	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	4	4	no	—
422	C₂²⋊C₄	—	2	8	no	—
423	C₂²⋊C₄	—	2	4	no	—
424	C₂²⋊C₄	—	2	4	no	—
425	C₂²⋊C₄	—	2	4	no	—
426	C₂²⋊C₄	—	4	4	no	—
427	C₂²⋊C₄	—	4	4	no	—
428	C₂²⋊C₄	—	4	2	no	—
429	C₄²	—	2	4	no	—
430	C₄²	—	2	1	no	—
431	C₄²	—	2	2	no	—
432	C₄²	—	4	2	no	—
433	C₄²	C₂×D₄×C₄₁	1	2	yes	—
434	C₄²	C₂×D₄×C₄₁	1	2	yes	—
435	C₄²	C₂×D₄×C₄₁	1	2	yes	complemented
436	C₂³	—	2	2	no	—
437	C₂³	—	2	2	no	—
438	C₂³	—	2	4	no	—
439	C₂³	—	2	2	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	4	2	no	—
442	C₂³	—	4	2	no	—
443	C₂³	—	4	2	no	—
444	C₂³	—	4	8	no	—
445	C₂³	—	4	1	no	—
446	C₂³	—	4	1	no	—
447	C₂³	—	8	4	no	—
448	C₂³	—	8	4	no	—
449	C₂³	#121×C₄₁	1	4	yes	—
450	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
451	Q₈	—	2	4	no	—
452	Q₈	—	2	4	no	—
453	D₄	—	2	4	no	—
454	D₄	—	2	4	no	—
455	D₄	—	2	4	no	—
456	D₄	—	2	4	no	—
457	D₄	—	2	4	no	—
458	D₄	—	4	16	no	—
459	D₄	—	4	16	no	—
460	D₄	—	4	4	no	—
461	D₄	—	4	4	no	—
462	D₄	—	4	2	no	—
463	D₄	—	4	4	no	—
464	D₄	—	8	2	no	—
465	D₄	—	8	4	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	8	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	4	4	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	2	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	8	2	no	—
484	C₂×C₄	—	8	2	no	—
485	C₂×C₄	—	8	2	no	—
486	C₂×C₄	#120×C₄₁	1	2	yes	—
487	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
488	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
489	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
490	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
491	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
492	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
493	C₈	—	4	4	no	—
494	C₂²	—	2	1	no	—
495	C₂²	—	2	1	no	—
496	C₂²	—	2	4	no	—
497	C₂²	—	4	4	no	—
498	C₂²	—	4	4	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	4	2	no	—
501	C₂²	—	4	4	no	—
502	C₂²	—	4	2	no	—
503	C₂²	—	4	4	no	—
504	C₂²	—	4	4	no	—
505	C₂²	—	8	2	no	—
506	C₂²	—	8	2	no	—
507	C₂²	—	8	1	no	—
508	C₂²	—	8	1	no	—
509	C₂²	—	16	8	no	—
510	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
511	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
512	C₂²	C₄₁×#535	1	1	yes	characteristic
513	C₄	—	2	4	no	—
514	C₄	—	2	1	no	—
515	C₄	—	2	2	no	—
516	C₄	—	4	2	no	—
517	C₄	—	4	4	no	—
518	C₄	—	4	2	no	—
519	C₄	—	8	1	no	—
520	C₄	C₄₁×#453	1	2	yes	complemented
521	C₄	D₄²×C₄₁	1	2	yes	—
522	C₄	C₄₁×#546	1	2	yes	—
523	C₂	—	2	1	no	—
524	C₂	—	2	1	no	—
525	C₂	—	4	4	no	—
526	C₂	—	8	2	no	—
527	C₂	—	8	4	no	—
528	C₂	C₄₁×#2408	1	1	yes	characteristic
529	C₂	C₂×C₄₁×#453	1	1	yes	characteristic
530	C₂	C₄₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3496	C₂	semidirect
2. C₄₁×#3036	C₂	semidirect
3. C₂×C₄₁×#453	C₂	semidirect
4. C₄×#127×C₄₁	C₂	semidirect
5. C₄₁×#2039	C₂	semidirect
6. C₄₁×#2006	C₂	semidirect
7. C₄₁×#1962	C₂	semidirect
8. C₄₁×#1800	C₂	semidirect
9. C₄₁×#1792	C₂	semidirect
11. C₂×AGL(1,ℤ/8)×C₄₁	C₂²	semidirect
22. C₂×C₄×D₄×C₄₁	C₂²	semidirect
27. C₄₁×#423	C₂²	semidirect
28. C₂×#104×C₄₁	C₂²	semidirect
29. C₂×#104×C₄₁	C₂²	semidirect
30. C₄₁×#417	C₂²	semidirect
34. C₄₁×#390	C₂²	semidirect
35. C₄₁×#384	C₂²	semidirect
36. C₄³×C₄₁	C₂²	semidirect
37. C₄₁×#343	C₂²	semidirect
56. C₄⋊Q₈×C₄₁	D₄	semidirect
58. #127×C₄₁	D₄	semidirect
59. #127×C₄₁	D₄	semidirect
60. #124×C₄₁	D₄	semidirect
128. C₂×D₄×C₄₁	C₂×D₄	semidirect
169. C₄²×C₄₁	C₂×D₄	semidirect
228. #24222	C₄₁	direct (N × H)
255. C₄×C₄₁	#453	semidirect
258. #3496	C₂×C₄₁	semidirect
259. #3036	C₂×C₄₁	semidirect
260. C₂×#453	C₂×C₄₁	semidirect
261. C₄×#127	C₂×C₄₁	semidirect
262. #2039	C₂×C₄₁	semidirect
263. #2006	C₂×C₄₁	semidirect
264. #1962	C₂×C₄₁	semidirect
265. #1800	C₂×C₄₁	semidirect
266. #1792	C₂×C₄₁	semidirect
276. C₂×AGL(1,ℤ/8)	C₂²×C₄₁	semidirect
287. C₂×C₄×D₄	C₂²×C₄₁	semidirect
292. #423	C₂²×C₄₁	semidirect
293. C₂×#104	C₂²×C₄₁	semidirect
294. C₂×#104	C₂²×C₄₁	semidirect
295. #417	C₂²×C₄₁	semidirect
299. #390	C₂²×C₄₁	semidirect
300. #384	C₂²×C₄₁	semidirect
301. C₄³	C₂²×C₄₁	semidirect
302. #343	C₂²×C₄₁	semidirect
303. C₄₁	#24222	direct (N × H)
322. C₄⋊Q₈	D₄×C₄₁	semidirect
324. #127	D₄×C₄₁	semidirect
325. #127	D₄×C₄₁	semidirect
326. #124	D₄×C₄₁	semidirect
394. C₂×D₄	C₂×D₄×C₄₁	semidirect
435. C₄²	C₂×D₄×C₄₁	semidirect
520. C₄	C₄₁×#453	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3496	2
C₄₁×#3036	2
C₂×C₄₁×#453	2
C₄×#127×C₄₁	2
C₄₁×#2039	2
C₄₁×#2006	2
C₄₁×#1962	2
C₄₁×#1800	2
C₄₁×#1792	2
#24222	41

Subgroups — C41×#24222 #23800423

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#24222 #23800423