Subgroups — C₄₁×#61088 #23837289

Order 10496.

2714 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#577	C₂	1	2	yes	maximal, complemented
2	C₄₁×#3500	C₂	1	2	yes	maximal, complemented
3	C₄₁×#3282	C₂	1	8	yes	maximal, complemented
4	C₄₁×#3277	C₂	1	4	yes	maximal, complemented
5	C₄₁×#3273	C₂	1	4	yes	maximal, complemented
6	C₂×C₄₁×#495	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#3057	C₂	1	2	yes	maximal, complemented
8	C₄₁×#3040	C₂	1	2	yes	maximal, complemented
9	C₄₁×#3011	C₂	1	2	yes	maximal, complemented
10	C₄₁×#3008	C₂	1	2	yes	maximal, complemented
11	C₄×#135×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×2₋¹⁺⁴×C₄₁	C₂²	1	2	yes	—
14	C₂×H(5,𝔽₂)×C₄₁	C₂²	1	2	yes	complemented
15	C₄₁×#577	—	2	16	no	—
16	C₂×#137×C₄₁	C₂²	1	2	yes	complemented
17	C₂×#137×C₄₁	C₂²	1	4	yes	complemented
18	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	2	yes	—
19	C₂×AGL(1,ℤ/8)×C₄₁	C₂²	1	4	yes	complemented
20	C₂×#135×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×#135×C₄₁	C₂²	1	4	yes	complemented
22	C₂²×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic, complemented
23	C₂²×SD₁₆×C₄₁	C₂²	1	2	yes	complemented
24	C₂²×SD₁₆×C₄₁	C₂²	1	4	yes	complemented
25	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
26	C₂×#131×C₄₁	C₂²	1	2	yes	complemented
27	C₄₁×#548	C₂²	1	8	yes	complemented
28	C₄₁×#546	—	2	8	no	—
29	C₄₁×#546	C₂²	1	8	yes	complemented
30	C₄₁×#536	—	2	2	no	—
31	C₄₁×#536	C₂²	1	2	yes	complemented
32	C₄₁×#535	—	2	2	no	—
33	C₄₁×#535	C₂²	1	2	yes	—
34	C₄₁×#532	—	2	2	no	—
35	C₄₁×#532	C₂²	1	1	yes	characteristic
36	C₂×#124×C₄₁	—	2	2	no	—
37	C₂×#124×C₄₁	C₂²	1	2	yes	complemented
38	C₄×#41×C₄₁	C₂²	1	2	yes	complemented
39	C₄₁×#495	C₂²	1	4	yes	complemented
40	C₄₁×#495	C₂²	1	4	yes	complemented
41	C₄₁×#468	C₂²	1	8	yes	complemented
42	C₄₁×#465	C₂²	1	8	yes	complemented
43	C₄₁×#464	C₂²	1	8	yes	complemented
44	C₄₁×#463	C₂²	1	8	yes	complemented
45	C₄₁×#452	C₂²	1	8	yes	complemented
46	C₄₁×#451	C₂²	1	8	yes	complemented
47	C₄₁×#449	C₂²	1	8	yes	complemented
48	C₄₁×#447	C₂²	1	8	yes	complemented
49	C₄×Q₁₆×C₄₁	C₂²	1	2	yes	complemented
50	C₄×SD₁₆×C₄₁	C₂²	1	4	yes	complemented
51	C₄×D₈×C₄₁	C₂²	1	2	yes	complemented
52	C₄₁×#436	C₂²	1	4	yes	complemented
53	C₈×D₄×C₄₁	C₂²	1	4	yes	complemented
54	C₄₁×#427	C₂²	1	1	yes	characteristic, complemented
55	C₄₁×#423	C₂²	1	1	yes	characteristic, complemented
56	C₄₁×#419	C₂²	1	2	yes	complemented
57	C₄₁×#418	C₂²	1	2	yes	complemented
58	C₄₁×#416	C₂²	1	2	yes	complemented
59	C₄₁×#408	C₂²	1	2	yes	complemented
60	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic, complemented
61	2₋¹⁺⁴×C₄₁	—	2	8	no	—
62	2₋¹⁺⁴×C₄₁	—	4	4	no	—
63	H(5,𝔽₂)×C₄₁	—	2	8	no	—
64	H(5,𝔽₂)×C₄₁	—	4	4	no	—
65	C₂×#41×C₄₁	—	2	2	no	—
66	C₂×#41×C₄₁	—	2	2	no	—
67	C₂×#41×C₄₁	—	2	4	no	—
68	C₂×#41×C₄₁	C₂³	1	4	yes	—
69	C₂×#41×C₄₁	C₂³	1	4	yes	—
70	C₂×#41×C₄₁	C₂³	1	2	yes	—
71	C₂×#41×C₄₁	C₂³	1	2	yes	—
72	C₂²×Q₈×C₄₁	—	2	2	no	—
73	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
74	C₂²×Q₈×C₄₁	C₂³	1	4	yes	—
75	C₂²×D₄×C₄₁	—	2	2	no	—
76	C₂²×D₄×C₄₁	—	2	4	no	—
77	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
78	C₂²×D₄×C₄₁	C₂³	1	4	yes	—
79	#137×C₄₁	—	2	8	no	—
80	#137×C₄₁	—	2	16	no	—
81	AGL(1,ℤ/8)×C₄₁	—	2	8	no	—
82	AGL(1,ℤ/8)×C₄₁	—	2	16	no	—
83	#135×C₄₁	—	2	16	no	—
84	#135×C₄₁	D₄	1	8	yes	complemented
85	C₂×Q₁₆×C₄₁	—	4	2	no	—
86	C₂×Q₁₆×C₄₁	C₂³	1	4	yes	—
87	C₂×Q₁₆×C₄₁	C₂³	1	4	yes	—
88	C₂×Q₁₆×C₄₁	C₂³	1	1	yes	characteristic
89	C₂×SD₁₆×C₄₁	—	2	16	no	—
90	C₂×SD₁₆×C₄₁	—	2	8	no	—
91	C₂×SD₁₆×C₄₁	C₂³	1	4	yes	—
92	C₂×SD₁₆×C₄₁	C₂³	1	4	yes	—
93	C₂×SD₁₆×C₄₁	C₂³	1	4	yes	—
94	C₂×SD₁₆×C₄₁	C₂³	1	4	yes	—
95	C₂×SD₁₆×C₄₁	C₂³	1	2	yes	—
96	C₂×D₈×C₄₁	—	4	2	no	—
97	C₂×D₈×C₄₁	C₂³	1	4	yes	—
98	C₂×D₈×C₄₁	C₂³	1	4	yes	—
99	C₂×D₈×C₄₁	C₂³	1	1	yes	characteristic
100	#131×C₄₁	—	2	8	no	—
101	C₂×M₁₆×C₄₁	C₂³	1	2	yes	—
102	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
103	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
104	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
105	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
106	C₄⋊Q₈×C₄₁	—	2	2	no	—
107	C₄⋊Q₈×C₄₁	D₄	1	1	yes	characteristic
108	#127×C₄₁	—	2	2	no	—
109	#127×C₄₁	D₄	1	1	yes	characteristic
110	#124×C₄₁	—	2	2	no	—
111	#124×C₄₁	—	2	2	no	—
112	#124×C₄₁	—	2	8	no	—
113	#124×C₄₁	—	2	2	no	—
114	#124×C₄₁	—	2	2	no	—
115	#124×C₄₁	D₄	1	2	yes	—
116	#124×C₄₁	C₂³	1	8	yes	—
117	#123×C₄₁	—	2	4	no	—
118	#123×C₄₁	—	2	8	no	—
119	C₂²⋊Q₈×C₄₁	—	2	8	no	—
120	C₂²⋊Q₈×C₄₁	C₂³	1	8	yes	—
121	#121×C₄₁	—	2	8	no	—
122	#121×C₄₁	—	2	8	no	—
123	#121×C₄₁	D₄	1	4	yes	—
124	#121×C₄₁	C₂³	1	8	yes	—
125	#120×C₄₁	—	2	8	no	—
126	#120×C₄₁	—	2	4	no	—
127	C₄×Q₈×C₄₁	C₂³	1	4	yes	—
128	C₄×D₄×C₄₁	—	2	4	no	—
129	C₄×D₄×C₄₁	—	2	8	no	—
130	C₄×D₄×C₄₁	C₂³	1	4	yes	—
131	C₄×D₄×C₄₁	C₂³	1	4	yes	—
132	#117×C₄₁	—	2	2	no	—
133	#117×C₄₁	C₂³	1	2	yes	—
134	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
135	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
136	C₂×C₄²×C₄₁	—	2	2	no	—
137	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
138	#107×C₄₁	C₂³	1	2	yes	—
139	#106×C₄₁	C₂³	1	2	yes	—
140	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
141	#103×C₄₁	C₂³	1	4	yes	—
142	#103×C₄₁	C₂³	1	8	yes	—
143	#102×C₄₁	C₂³	1	4	yes	—
144	#102×C₄₁	C₂³	1	8	yes	—
145	C₂²⋊C₈×C₄₁	C₂³	1	8	yes	—
146	C₄×C₈×C₄₁	C₂³	1	2	yes	—
147	C₄×C₈×C₄₁	C₂³	1	2	yes	—
148	C₂⁴×C₄₁	—	2	4	no	—
149	C₂⁴×C₄₁	—	2	2	no	—
150	#41×C₄₁	—	2	4	no	—
151	#41×C₄₁	—	2	4	no	—
152	#41×C₄₁	—	2	4	no	—
153	#41×C₄₁	—	2	8	no	—
154	#41×C₄₁	—	2	8	no	—
155	#41×C₄₁	—	4	4	no	—
156	#41×C₄₁	—	4	4	no	—
157	#41×C₄₁	—	4	8	no	—
158	#41×C₄₁	—	4	8	no	—
159	#41×C₄₁	—	4	16	no	—
160	#41×C₄₁	C₂×D₄	1	8	yes	—
161	C₂×Q₈×C₄₁	—	2	4	no	—
162	C₂×Q₈×C₄₁	—	2	2	no	—
163	C₂×Q₈×C₄₁	—	2	8	no	—
164	C₂×Q₈×C₄₁	—	2	4	no	—
165	C₂×Q₈×C₄₁	—	2	4	no	—
166	C₂×Q₈×C₄₁	—	4	4	no	—
167	C₂×Q₈×C₄₁	—	4	8	no	—
168	C₂×Q₈×C₄₁	C₂×D₄	1	2	yes	—
169	C₂×Q₈×C₄₁	C₂⁴	1	4	yes	—
170	C₂×Q₈×C₄₁	C₂⁴	1	2	yes	—
171	C₂×D₄×C₄₁	—	2	2	no	—
172	C₂×D₄×C₄₁	—	2	8	no	—
173	C₂×D₄×C₄₁	—	2	4	no	—
174	C₂×D₄×C₄₁	—	2	4	no	—
175	C₂×D₄×C₄₁	—	2	8	no	—
176	C₂×D₄×C₄₁	—	2	4	no	—
177	C₂×D₄×C₄₁	—	2	4	no	—
178	C₂×D₄×C₄₁	—	2	2	no	—
179	C₂×D₄×C₄₁	—	2	4	no	—
180	C₂×D₄×C₄₁	—	4	4	no	—
181	C₂×D₄×C₄₁	—	4	8	no	—
182	C₂×D₄×C₄₁	—	4	16	no	—
183	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
184	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
185	C₂×D₄×C₄₁	C₂×D₄	1	4	yes	—
186	C₂×D₄×C₄₁	C₂⁴	1	4	yes	—
187	C₂×D₄×C₄₁	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₄₁	—	2	2	no	—
189	C₂²×C₄×C₄₁	—	2	2	no	—
190	C₂²×C₄×C₄₁	—	2	4	no	—
191	C₂²×C₄×C₄₁	—	2	4	no	—
192	C₂²×C₄×C₄₁	—	2	4	no	—
193	C₂²×C₄×C₄₁	—	2	4	no	—
194	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
195	C₂²×C₄×C₄₁	C₂⁴	1	4	yes	—
196	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
197	Q₁₆×C₄₁	—	2	8	no	—
198	Q₁₆×C₄₁	—	4	4	no	—
199	Q₁₆×C₄₁	C₂×D₄	1	4	yes	—
200	SD₁₆×C₄₁	—	2	8	no	—
201	SD₁₆×C₄₁	—	2	8	no	—
202	SD₁₆×C₄₁	—	2	8	no	—
203	SD₁₆×C₄₁	—	2	8	no	—
204	SD₁₆×C₄₁	C₂×D₄	1	8	yes	—
205	D₈×C₄₁	—	2	8	no	—
206	D₈×C₄₁	—	4	4	no	—
207	D₈×C₄₁	C₂×D₄	1	4	yes	—
208	M₁₆×C₄₁	—	2	4	no	—
209	M₁₆×C₄₁	—	2	8	no	—
210	C₂×C₈×C₄₁	—	2	2	no	—
211	C₂×C₈×C₄₁	—	2	8	no	—
212	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
213	C₂×C₈×C₄₁	C₂⁴	1	2	yes	—
214	C₂×C₈×C₄₁	C₂⁴	1	4	yes	—
215	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
216	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
217	C₄⋊C₄×C₄₁	—	2	4	no	—
218	C₄⋊C₄×C₄₁	—	2	8	no	—
219	C₄⋊C₄×C₄₁	C₂×D₄	1	4	yes	—
220	C₄⋊C₄×C₄₁	C₂⁴	1	4	yes	—
221	C₂²⋊C₄×C₄₁	—	2	8	no	—
222	C₂²⋊C₄×C₄₁	—	2	2	no	—
223	C₂²⋊C₄×C₄₁	—	2	8	no	—
224	C₂²⋊C₄×C₄₁	—	2	8	no	—
225	C₂²⋊C₄×C₄₁	—	2	8	no	—
226	C₂²⋊C₄×C₄₁	—	2	4	no	—
227	C₂²⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
228	C₄²×C₄₁	—	2	2	no	—
229	C₄²×C₄₁	—	2	4	no	—
230	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
231	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
232	C₄²×C₄₁	C₂⁴	1	2	yes	—
233	C₂³×C₄₁	—	2	2	no	—
234	C₂³×C₄₁	—	2	4	no	—
235	C₂³×C₄₁	—	4	8	no	—
236	C₂³×C₄₁	—	4	4	no	—
237	C₂³×C₄₁	—	8	4	no	—
238	C₂³×C₄₁	—	8	8	no	—
239	C₂³×C₄₁	C₂²×D₄	1	4	yes	—
240	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
241	Q₈×C₄₁	—	2	4	no	—
242	Q₈×C₄₁	—	2	2	no	—
243	Q₈×C₄₁	—	2	4	no	—
244	Q₈×C₄₁	—	2	4	no	—
245	Q₈×C₄₁	—	4	8	no	—
246	Q₈×C₄₁	—	4	4	no	—
247	Q₈×C₄₁	C₂²×D₄	1	4	yes	—
248	D₄×C₄₁	—	2	8	no	—
249	D₄×C₄₁	—	2	8	no	—
250	D₄×C₄₁	—	2	4	no	—
251	D₄×C₄₁	—	2	4	no	—
252	D₄×C₄₁	—	2	2	no	—
253	D₄×C₄₁	—	2	4	no	—
254	D₄×C₄₁	—	4	4	no	—
255	D₄×C₄₁	—	4	8	no	—
256	D₄×C₄₁	—	4	8	no	—
257	D₄×C₄₁	—	4	8	no	—
258	D₄×C₄₁	—	4	4	no	—
259	D₄×C₄₁	—	8	8	no	—
260	D₄×C₄₁	C₂²×D₄	1	4	yes	—
261	C₂×C₄×C₄₁	—	2	2	no	—
262	C₂×C₄×C₄₁	—	2	8	no	—
263	C₂×C₄×C₄₁	—	2	4	no	—
264	C₂×C₄×C₄₁	—	2	4	no	—
265	C₂×C₄×C₄₁	—	2	4	no	—
266	C₂×C₄×C₄₁	—	2	4	no	—
267	C₂×C₄×C₄₁	—	4	2	no	—
268	C₂×C₄×C₄₁	—	4	8	no	—
269	C₂×C₄×C₄₁	—	4	8	no	—
270	C₂×C₄×C₄₁	—	4	8	no	—
271	C₂×C₄×C₄₁	—	4	4	no	—
272	C₂×C₄×C₄₁	—	8	4	no	—
273	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₄₁	C₂²×D₄	1	4	yes	—
276	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
277	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
278	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
279	C₂×C₄×C₄₁	C₂⁵	1	1	yes	characteristic
280	C₈×C₄₁	—	2	4	no	—
281	C₈×C₄₁	—	2	2	no	—
282	C₈×C₄₁	C₂²×D₄	1	2	yes	—
283	C₈×C₄₁	C₂²×D₄	1	2	yes	—
284	#61088	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
285	C₂²×C₄₁	—	2	2	no	—
286	C₂²×C₄₁	—	2	8	no	—
287	C₂²×C₄₁	—	2	4	no	—
288	C₂²×C₄₁	—	2	4	no	—
289	C₂²×C₄₁	—	4	4	no	—
290	C₂²×C₄₁	—	4	8	no	—
291	C₂²×C₄₁	—	4	4	no	—
292	C₂²×C₄₁	—	16	4	no	—
293	C₂²×C₄₁	—	16	8	no	—
294	C₂²×C₄₁	D₄²	1	2	yes	—
295	C₂²×C₄₁	C₂³×D₄	1	1	yes	characteristic
296	C₄×C₄₁	—	2	2	no	—
297	C₄×C₄₁	—	2	2	no	—
298	C₄×C₄₁	—	2	4	no	—
299	C₄×C₄₁	—	2	4	no	—
300	C₄×C₄₁	—	4	4	no	—
301	C₄×C₄₁	—	4	4	no	—
302	C₄×C₄₁	D₄²	1	2	yes	—
303	C₄×C₄₁	C₂³×D₄	1	1	yes	characteristic
304	C₄×C₄₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
305	C₂×#577	C₂×C₄₁	1	2	yes	complemented
306	#3500	C₂×C₄₁	1	2	yes	complemented
307	#3282	C₂×C₄₁	1	8	yes	complemented
308	#3277	C₂×C₄₁	1	4	yes	complemented
309	#3273	C₂×C₄₁	1	4	yes	complemented
310	C₂×#495	C₂×C₄₁	1	1	yes	characteristic, complemented
311	#3057	C₂×C₄₁	1	2	yes	complemented
312	#3040	C₂×C₄₁	1	2	yes	complemented
313	#3011	C₂×C₄₁	1	2	yes	complemented
314	#3008	C₂×C₄₁	1	2	yes	complemented
315	C₄×#135	C₂×C₄₁	1	1	yes	characteristic, complemented
316	#2931	C₂×C₄₁	1	1	yes	characteristic, complemented
317	C₂×C₄₁	—	2	2	no	—
318	C₂×C₄₁	—	4	4	no	—
319	C₂×C₄₁	—	4	4	no	—
320	C₂×C₄₁	—	8	4	no	—
321	C₂×C₄₁	C₂×D₄²	1	1	yes	characteristic
322	C₂×C₄₁	#3584	1	2	yes	—
323	C₂×2₋¹⁺⁴	C₂²×C₄₁	1	2	yes	—
324	C₂×H(5,𝔽₂)	C₂²×C₄₁	1	2	yes	complemented
325	#577	—	2	16	no	—
326	C₂×#137	C₂²×C₄₁	1	2	yes	complemented
327	C₂×#137	C₂²×C₄₁	1	4	yes	complemented
328	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	2	yes	—
329	C₂×AGL(1,ℤ/8)	C₂²×C₄₁	1	4	yes	complemented
330	C₂×#135	C₂²×C₄₁	1	1	yes	characteristic
331	C₂×#135	C₂²×C₄₁	1	4	yes	complemented
332	C₂²×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic, complemented
333	C₂²×SD₁₆	C₂²×C₄₁	1	2	yes	complemented
334	C₂²×SD₁₆	C₂²×C₄₁	1	4	yes	complemented
335	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
336	C₂×#131	C₂²×C₄₁	1	2	yes	complemented
337	#548	C₂²×C₄₁	1	8	yes	complemented
338	#546	—	2	8	no	—
339	#546	C₂²×C₄₁	1	8	yes	complemented
340	#536	—	2	2	no	—
341	#536	C₂²×C₄₁	1	2	yes	complemented
342	#535	—	2	2	no	—
343	#535	C₂²×C₄₁	1	2	yes	—
344	#532	—	2	2	no	—
345	#532	C₂²×C₄₁	1	1	yes	characteristic
346	C₂×#124	—	2	2	no	—
347	C₂×#124	C₂²×C₄₁	1	2	yes	complemented
348	C₄×#41	C₂²×C₄₁	1	2	yes	complemented
349	#495	C₂²×C₄₁	1	4	yes	complemented
350	#495	C₂²×C₄₁	1	4	yes	complemented
351	#468	C₂²×C₄₁	1	8	yes	complemented
352	#465	C₂²×C₄₁	1	8	yes	complemented
353	#464	C₂²×C₄₁	1	8	yes	complemented
354	#463	C₂²×C₄₁	1	8	yes	complemented
355	#452	C₂²×C₄₁	1	8	yes	complemented
356	#451	C₂²×C₄₁	1	8	yes	complemented
357	#449	C₂²×C₄₁	1	8	yes	complemented
358	#447	C₂²×C₄₁	1	8	yes	complemented
359	C₄×Q₁₆	C₂²×C₄₁	1	2	yes	complemented
360	C₄×SD₁₆	C₂²×C₄₁	1	4	yes	complemented
361	C₄×D₈	C₂²×C₄₁	1	2	yes	complemented
362	#436	C₂²×C₄₁	1	4	yes	complemented
363	C₈×D₄	C₂²×C₄₁	1	4	yes	complemented
364	#427	C₂²×C₄₁	1	1	yes	characteristic, complemented
365	#423	C₂²×C₄₁	1	1	yes	characteristic, complemented
366	#419	C₂²×C₄₁	1	2	yes	complemented
367	#418	C₂²×C₄₁	1	2	yes	complemented
368	#416	C₂²×C₄₁	1	2	yes	complemented
369	#408	C₂²×C₄₁	1	2	yes	complemented
370	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic, complemented
371	C₄₁	#61088	1	1	yes	characteristic, direct product factor, Sylow
372	2₋¹⁺⁴	—	2	8	no	—
373	2₋¹⁺⁴	—	4	4	no	—
374	H(5,𝔽₂)	—	2	8	no	—
375	H(5,𝔽₂)	—	4	4	no	—
376	C₂×#41	—	2	2	no	—
377	C₂×#41	—	2	2	no	—
378	C₂×#41	—	2	4	no	—
379	C₂×#41	C₂³×C₄₁	1	4	yes	—
380	C₂×#41	C₂³×C₄₁	1	4	yes	—
381	C₂×#41	C₂³×C₄₁	1	2	yes	—
382	C₂×#41	C₂³×C₄₁	1	2	yes	—
383	C₂²×Q₈	—	2	2	no	—
384	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
385	C₂²×Q₈	C₂³×C₄₁	1	4	yes	—
386	C₂²×D₄	—	2	2	no	—
387	C₂²×D₄	—	2	4	no	—
388	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
389	C₂²×D₄	C₂³×C₄₁	1	4	yes	—
390	#137	—	2	8	no	—
391	#137	—	2	16	no	—
392	AGL(1,ℤ/8)	—	2	8	no	—
393	AGL(1,ℤ/8)	—	2	16	no	—
394	#135	—	2	16	no	—
395	#135	D₄×C₄₁	1	8	yes	complemented
396	C₂×Q₁₆	—	4	2	no	—
397	C₂×Q₁₆	C₂³×C₄₁	1	4	yes	—
398	C₂×Q₁₆	C₂³×C₄₁	1	4	yes	—
399	C₂×Q₁₆	C₂³×C₄₁	1	1	yes	characteristic
400	C₂×SD₁₆	—	2	16	no	—
401	C₂×SD₁₆	—	2	8	no	—
402	C₂×SD₁₆	C₂³×C₄₁	1	4	yes	—
403	C₂×SD₁₆	C₂³×C₄₁	1	4	yes	—
404	C₂×SD₁₆	C₂³×C₄₁	1	4	yes	—
405	C₂×SD₁₆	C₂³×C₄₁	1	4	yes	—
406	C₂×SD₁₆	C₂³×C₄₁	1	2	yes	—
407	C₂×D₈	—	4	2	no	—
408	C₂×D₈	C₂³×C₄₁	1	4	yes	—
409	C₂×D₈	C₂³×C₄₁	1	4	yes	—
410	C₂×D₈	C₂³×C₄₁	1	1	yes	characteristic
411	#131	—	2	8	no	—
412	C₂×M₁₆	C₂³×C₄₁	1	2	yes	—
413	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
414	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
415	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
416	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
417	C₄⋊Q₈	—	2	2	no	—
418	C₄⋊Q₈	D₄×C₄₁	1	1	yes	characteristic
419	#127	—	2	2	no	—
420	#127	D₄×C₄₁	1	1	yes	characteristic
421	#124	—	2	2	no	—
422	#124	—	2	2	no	—
423	#124	—	2	8	no	—
424	#124	—	2	2	no	—
425	#124	—	2	2	no	—
426	#124	D₄×C₄₁	1	2	yes	—
427	#124	C₂³×C₄₁	1	8	yes	—
428	#123	—	2	4	no	—
429	#123	—	2	8	no	—
430	C₂²⋊Q₈	—	2	8	no	—
431	C₂²⋊Q₈	C₂³×C₄₁	1	8	yes	—
432	#121	—	2	8	no	—
433	#121	—	2	8	no	—
434	#121	D₄×C₄₁	1	4	yes	—
435	#121	C₂³×C₄₁	1	8	yes	—
436	#120	—	2	8	no	—
437	#120	—	2	4	no	—
438	C₄×Q₈	C₂³×C₄₁	1	4	yes	—
439	C₄×D₄	—	2	4	no	—
440	C₄×D₄	—	2	8	no	—
441	C₄×D₄	C₂³×C₄₁	1	4	yes	—
442	C₄×D₄	C₂³×C₄₁	1	4	yes	—
443	#117	—	2	2	no	—
444	#117	C₂³×C₄₁	1	2	yes	—
445	C₂×C₂²⋊C₄	—	2	4	no	—
446	C₂×C₂²⋊C₄	—	2	8	no	—
447	C₂×C₄²	—	2	2	no	—
448	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
449	#107	C₂³×C₄₁	1	2	yes	—
450	#106	C₂³×C₄₁	1	2	yes	—
451	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
452	#103	C₂³×C₄₁	1	4	yes	—
453	#103	C₂³×C₄₁	1	8	yes	—
454	#102	C₂³×C₄₁	1	4	yes	—
455	#102	C₂³×C₄₁	1	8	yes	—
456	C₂²⋊C₈	C₂³×C₄₁	1	8	yes	—
457	C₄×C₈	C₂³×C₄₁	1	2	yes	—
458	C₄×C₈	C₂³×C₄₁	1	2	yes	—
459	C₂⁴	—	2	4	no	—
460	C₂⁴	—	2	2	no	—
461	#41	—	2	4	no	—
462	#41	—	2	4	no	—
463	#41	—	2	4	no	—
464	#41	—	2	8	no	—
465	#41	—	2	8	no	—
466	#41	—	4	4	no	—
467	#41	—	4	4	no	—
468	#41	—	4	8	no	—
469	#41	—	4	8	no	—
470	#41	—	4	16	no	—
471	#41	C₂×D₄×C₄₁	1	8	yes	—
472	C₂×Q₈	—	2	4	no	—
473	C₂×Q₈	—	2	2	no	—
474	C₂×Q₈	—	2	8	no	—
475	C₂×Q₈	—	2	4	no	—
476	C₂×Q₈	—	2	4	no	—
477	C₂×Q₈	—	4	4	no	—
478	C₂×Q₈	—	4	8	no	—
479	C₂×Q₈	C₂×D₄×C₄₁	1	2	yes	—
480	C₂×Q₈	C₂⁴×C₄₁	1	4	yes	—
481	C₂×Q₈	C₂⁴×C₄₁	1	2	yes	—
482	C₂×D₄	—	2	2	no	—
483	C₂×D₄	—	2	8	no	—
484	C₂×D₄	—	2	4	no	—
485	C₂×D₄	—	2	4	no	—
486	C₂×D₄	—	2	8	no	—
487	C₂×D₄	—	2	4	no	—
488	C₂×D₄	—	2	4	no	—
489	C₂×D₄	—	2	2	no	—
490	C₂×D₄	—	2	4	no	—
491	C₂×D₄	—	4	4	no	—
492	C₂×D₄	—	4	8	no	—
493	C₂×D₄	—	4	16	no	—
494	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
495	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
496	C₂×D₄	C₂×D₄×C₄₁	1	4	yes	—
497	C₂×D₄	C₂⁴×C₄₁	1	4	yes	—
498	C₂×D₄	C₂⁴×C₄₁	1	2	yes	—
499	C₂²×C₄	—	2	2	no	—
500	C₂²×C₄	—	2	2	no	—
501	C₂²×C₄	—	2	4	no	—
502	C₂²×C₄	—	2	4	no	—
503	C₂²×C₄	—	2	4	no	—
504	C₂²×C₄	—	2	4	no	—
505	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
506	C₂²×C₄	C₂⁴×C₄₁	1	4	yes	—
507	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
508	Q₁₆	—	2	8	no	—
509	Q₁₆	—	4	4	no	—
510	Q₁₆	C₂×D₄×C₄₁	1	4	yes	—
511	SD₁₆	—	2	8	no	—
512	SD₁₆	—	2	8	no	—
513	SD₁₆	—	2	8	no	—
514	SD₁₆	—	2	8	no	—
515	SD₁₆	C₂×D₄×C₄₁	1	8	yes	—
516	D₈	—	2	8	no	—
517	D₈	—	4	4	no	—
518	D₈	C₂×D₄×C₄₁	1	4	yes	—
519	M₁₆	—	2	4	no	—
520	M₁₆	—	2	8	no	—
521	C₂×C₈	—	2	2	no	—
522	C₂×C₈	—	2	8	no	—
523	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
524	C₂×C₈	C₂⁴×C₄₁	1	2	yes	—
525	C₂×C₈	C₂⁴×C₄₁	1	4	yes	—
526	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
527	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
528	C₄⋊C₄	—	2	4	no	—
529	C₄⋊C₄	—	2	8	no	—
530	C₄⋊C₄	C₂×D₄×C₄₁	1	4	yes	—
531	C₄⋊C₄	C₂⁴×C₄₁	1	4	yes	—
532	C₂²⋊C₄	—	2	8	no	—
533	C₂²⋊C₄	—	2	2	no	—
534	C₂²⋊C₄	—	2	8	no	—
535	C₂²⋊C₄	—	2	8	no	—
536	C₂²⋊C₄	—	2	8	no	—
537	C₂²⋊C₄	—	2	4	no	—
538	C₂²⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
539	C₄²	—	2	2	no	—
540	C₄²	—	2	4	no	—
541	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
542	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
543	C₄²	C₂⁴×C₄₁	1	2	yes	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	4	no	—
546	C₂³	—	4	8	no	—
547	C₂³	—	4	4	no	—
548	C₂³	—	8	4	no	—
549	C₂³	—	8	8	no	—
550	C₂³	C₂²×D₄×C₄₁	1	4	yes	—
551	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
552	Q₈	—	2	4	no	—
553	Q₈	—	2	2	no	—
554	Q₈	—	2	4	no	—
555	Q₈	—	2	4	no	—
556	Q₈	—	4	8	no	—
557	Q₈	—	4	4	no	—
558	Q₈	C₂²×D₄×C₄₁	1	4	yes	—
559	D₄	—	2	8	no	—
560	D₄	—	2	8	no	—
561	D₄	—	2	4	no	—
562	D₄	—	2	4	no	—
563	D₄	—	2	2	no	—
564	D₄	—	2	4	no	—
565	D₄	—	4	4	no	—
566	D₄	—	4	8	no	—
567	D₄	—	4	8	no	—
568	D₄	—	4	8	no	—
569	D₄	—	4	4	no	—
570	D₄	—	8	8	no	—
571	D₄	C₂²×D₄×C₄₁	1	4	yes	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	4	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	4	no	—
577	C₂×C₄	—	2	4	no	—
578	C₂×C₄	—	4	2	no	—
579	C₂×C₄	—	4	8	no	—
580	C₂×C₄	—	4	8	no	—
581	C₂×C₄	—	4	8	no	—
582	C₂×C₄	—	4	4	no	—
583	C₂×C₄	—	8	4	no	—
584	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
585	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
586	C₂×C₄	C₂²×D₄×C₄₁	1	4	yes	—
587	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
588	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
589	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
590	C₂×C₄	C₂⁵×C₄₁	1	1	yes	characteristic
591	C₈	—	2	4	no	—
592	C₈	—	2	2	no	—
593	C₈	C₂²×D₄×C₄₁	1	2	yes	—
594	C₈	C₂²×D₄×C₄₁	1	2	yes	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	8	no	—
597	C₂²	—	2	4	no	—
598	C₂²	—	2	4	no	—
599	C₂²	—	4	4	no	—
600	C₂²	—	4	8	no	—
601	C₂²	—	4	4	no	—
602	C₂²	—	16	4	no	—
603	C₂²	—	16	8	no	—
604	C₂²	D₄²×C₄₁	1	2	yes	—
605	C₂²	C₂³×D₄×C₄₁	1	1	yes	characteristic
606	C₄	—	2	2	no	—
607	C₄	—	2	2	no	—
608	C₄	—	2	4	no	—
609	C₄	—	2	4	no	—
610	C₄	—	4	4	no	—
611	C₄	—	4	4	no	—
612	C₄	D₄²×C₄₁	1	2	yes	—
613	C₄	C₂³×D₄×C₄₁	1	1	yes	characteristic
614	C₄	C₂×H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
615	C₂	—	2	2	no	—
616	C₂	—	4	4	no	—
617	C₂	—	4	4	no	—
618	C₂	—	8	4	no	—
619	C₂	C₂×D₄²×C₄₁	1	1	yes	characteristic
620	C₂	C₄₁×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₄₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄₁×#532 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#577	C₂	semidirect
2. C₄₁×#3500	C₂	semidirect
3. C₄₁×#3282	C₂	semidirect
4. C₄₁×#3277	C₂	semidirect
5. C₄₁×#3273	C₂	semidirect
6. C₂×C₄₁×#495	C₂	semidirect
7. C₄₁×#3057	C₂	semidirect
8. C₄₁×#3040	C₂	semidirect
9. C₄₁×#3011	C₂	semidirect
10. C₄₁×#3008	C₂	semidirect
11. C₄×#135×C₄₁	C₂	semidirect
12. C₄₁×#2931	C₂	semidirect
14. C₂×H(5,𝔽₂)×C₄₁	C₂²	semidirect
16. C₂×#137×C₄₁	C₂²	semidirect
17. C₂×#137×C₄₁	C₂²	semidirect
19. C₂×AGL(1,ℤ/8)×C₄₁	C₂²	semidirect
21. C₂×#135×C₄₁	C₂²	semidirect
22. C₂²×Q₁₆×C₄₁	C₂²	semidirect
23. C₂²×SD₁₆×C₄₁	C₂²	semidirect
24. C₂²×SD₁₆×C₄₁	C₂²	semidirect
26. C₂×#131×C₄₁	C₂²	semidirect
27. C₄₁×#548	C₂²	semidirect
29. C₄₁×#546	C₂²	semidirect
31. C₄₁×#536	C₂²	semidirect
37. C₂×#124×C₄₁	C₂²	semidirect
38. C₄×#41×C₄₁	C₂²	semidirect
39. C₄₁×#495	C₂²	semidirect
40. C₄₁×#495	C₂²	semidirect
41. C₄₁×#468	C₂²	semidirect
42. C₄₁×#465	C₂²	semidirect
43. C₄₁×#464	C₂²	semidirect
44. C₄₁×#463	C₂²	semidirect
45. C₄₁×#452	C₂²	semidirect
46. C₄₁×#451	C₂²	semidirect
47. C₄₁×#449	C₂²	semidirect
48. C₄₁×#447	C₂²	semidirect
49. C₄×Q₁₆×C₄₁	C₂²	semidirect
50. C₄×SD₁₆×C₄₁	C₂²	semidirect
51. C₄×D₈×C₄₁	C₂²	semidirect
52. C₄₁×#436	C₂²	semidirect
53. C₈×D₄×C₄₁	C₂²	semidirect
54. C₄₁×#427	C₂²	semidirect
55. C₄₁×#423	C₂²	semidirect
56. C₄₁×#419	C₂²	semidirect
57. C₄₁×#418	C₂²	semidirect
58. C₄₁×#416	C₂²	semidirect
59. C₄₁×#408	C₂²	semidirect
60. C₂×C₄×C₈×C₄₁	C₂²	semidirect
84. #135×C₄₁	D₄	semidirect
284. #61088	C₄₁	direct (N × H)
305. C₂×#577	C₂×C₄₁	semidirect
306. #3500	C₂×C₄₁	semidirect
307. #3282	C₂×C₄₁	semidirect
308. #3277	C₂×C₄₁	semidirect
309. #3273	C₂×C₄₁	semidirect
310. C₂×#495	C₂×C₄₁	semidirect
311. #3057	C₂×C₄₁	semidirect
312. #3040	C₂×C₄₁	semidirect
313. #3011	C₂×C₄₁	semidirect
314. #3008	C₂×C₄₁	semidirect
315. C₄×#135	C₂×C₄₁	semidirect
316. #2931	C₂×C₄₁	semidirect
324. C₂×H(5,𝔽₂)	C₂²×C₄₁	semidirect
326. C₂×#137	C₂²×C₄₁	semidirect
327. C₂×#137	C₂²×C₄₁	semidirect
329. C₂×AGL(1,ℤ/8)	C₂²×C₄₁	semidirect
331. C₂×#135	C₂²×C₄₁	semidirect
332. C₂²×Q₁₆	C₂²×C₄₁	semidirect
333. C₂²×SD₁₆	C₂²×C₄₁	semidirect
334. C₂²×SD₁₆	C₂²×C₄₁	semidirect
336. C₂×#131	C₂²×C₄₁	semidirect
337. #548	C₂²×C₄₁	semidirect
339. #546	C₂²×C₄₁	semidirect
341. #536	C₂²×C₄₁	semidirect
347. C₂×#124	C₂²×C₄₁	semidirect
348. C₄×#41	C₂²×C₄₁	semidirect
349. #495	C₂²×C₄₁	semidirect
350. #495	C₂²×C₄₁	semidirect
351. #468	C₂²×C₄₁	semidirect
352. #465	C₂²×C₄₁	semidirect
353. #464	C₂²×C₄₁	semidirect
354. #463	C₂²×C₄₁	semidirect
355. #452	C₂²×C₄₁	semidirect
356. #451	C₂²×C₄₁	semidirect
357. #449	C₂²×C₄₁	semidirect
358. #447	C₂²×C₄₁	semidirect
359. C₄×Q₁₆	C₂²×C₄₁	semidirect
360. C₄×SD₁₆	C₂²×C₄₁	semidirect
361. C₄×D₈	C₂²×C₄₁	semidirect
362. #436	C₂²×C₄₁	semidirect
363. C₈×D₄	C₂²×C₄₁	semidirect
364. #427	C₂²×C₄₁	semidirect
365. #423	C₂²×C₄₁	semidirect
366. #419	C₂²×C₄₁	semidirect
367. #418	C₂²×C₄₁	semidirect
368. #416	C₂²×C₄₁	semidirect
369. #408	C₂²×C₄₁	semidirect
370. C₂×C₄×C₈	C₂²×C₄₁	semidirect
371. C₄₁	#61088	direct (N × H)
395. #135	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×#577	2
C₄₁×#3500	2
C₄₁×#3282	2
C₄₁×#3277	2
C₄₁×#3273	2
C₂×C₄₁×#495	2
C₄₁×#3057	2
C₄₁×#3040	2
C₄₁×#3011	2
C₄₁×#3008	2
C₄×#135×C₄₁	2
C₄₁×#2931	2
#61088	41

Subgroups — C41×#61088 #23837289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#61088 #23837289