Subgroups — C₄₁×#22209 #23798410

Order 10496.

850 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₄×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2398	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1955	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#1923	C₂	1	4	yes	maximal, complemented
7	C₄₁×#1850	C₂	1	4	yes	maximal, complemented
8	C₄₁×#1843	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₈×Q₈×C₄₁	—	2	4	no	—
14	C₈×D₄×C₄₁	—	2	4	no	—
15	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
19	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
21	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#400	C₄	1	2	yes	—
25	C₄₁×#397	C₄	1	2	yes	—
26	C₄₁×#394	C₄	1	2	yes	—
27	C₄₁×#393	C₄	1	2	yes	—
28	C₄×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
29	C₄₁×#376	C₂²	1	1	yes	characteristic
30	C₄₁×#336	C₂²	1	2	yes	—
31	C₄₁×#336	C₂²	1	4	yes	—
32	C₄₁×#336	C₂²	1	2	yes	—
33	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₄×C₄₁	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₄₁	—	2	4	no	—
37	C₂²×C₈×C₄₁	—	4	4	no	—
38	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₄₁	—	2	2	no	—
44	C₄×Q₈×C₄₁	—	2	2	no	—
45	C₄×D₄×C₄₁	—	2	2	no	—
46	C₄×D₄×C₄₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
52	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
54	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
55	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₁	—	2	4	no	—
57	C₄⋊C₈×C₄₁	—	2	2	no	—
58	C₄⋊C₈×C₄₁	—	2	2	no	—
59	C₄⋊C₈×C₄₁	—	4	4	no	—
60	C₂²⋊C₈×C₄₁	—	2	4	no	—
61	C₂²⋊C₈×C₄₁	—	4	4	no	—
62	#97×C₄₁	—	2	2	no	—
63	C₄×C₈×C₄₁	—	2	4	no	—
64	C₄×C₈×C₄₁	D₄	1	4	yes	—
65	#95×C₄₁	C₂×C₄	1	2	yes	—
66	#95×C₄₁	C₂×C₄	1	2	yes	—
67	#95×C₄₁	C₂×C₄	1	4	yes	—
68	#95×C₄₁	C₂×C₄	1	4	yes	—
69	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
70	C₂×Q₈×C₄₁	—	2	2	no	—
71	C₂×Q₈×C₄₁	—	2	4	no	—
72	C₂×D₄×C₄₁	—	2	4	no	—
73	C₂×D₄×C₄₁	—	2	2	no	—
74	C₂²×C₄×C₄₁	—	2	2	no	—
75	C₂²×C₄×C₄₁	—	2	2	no	—
76	C₂²×C₄×C₄₁	—	2	4	no	—
77	C₂²×C₄×C₄₁	—	2	4	no	—
78	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
85	C₂×C₈×C₄₁	—	2	2	no	—
86	C₂×C₈×C₄₁	—	2	2	no	—
87	C₂×C₈×C₄₁	—	2	2	no	—
88	C₂×C₈×C₄₁	—	2	2	no	—
89	C₂×C₈×C₄₁	—	2	2	no	—
90	C₂×C₈×C₄₁	—	2	2	no	—
91	C₂×C₈×C₄₁	—	2	4	no	—
92	C₂×C₈×C₄₁	—	2	4	no	—
93	C₂×C₈×C₄₁	—	2	2	no	—
94	C₂×C₈×C₄₁	—	4	8	no	—
95	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
97	C₄⋊C₄×C₄₁	—	2	2	no	—
98	C₄⋊C₄×C₄₁	—	2	4	no	—
99	C₄⋊C₄×C₄₁	—	2	2	no	—
100	C₂²⋊C₄×C₄₁	—	2	8	no	—
101	C₂²⋊C₄×C₄₁	—	2	4	no	—
102	C₂²⋊C₄×C₄₁	—	2	8	no	—
103	C₄²×C₄₁	—	2	2	no	—
104	C₄²×C₄₁	—	2	2	no	—
105	C₄²×C₄₁	—	2	2	no	—
106	C₄²×C₄₁	—	2	2	no	—
107	C₄²×C₄₁	C₂×D₄	1	2	yes	—
108	C₄²×C₄₁	#41	1	2	yes	—
109	C₂³×C₄₁	—	2	2	no	—
110	C₂³×C₄₁	—	2	4	no	—
111	C₂³×C₄₁	—	2	4	no	—
112	C₂³×C₄₁	—	2	2	no	—
113	C₂³×C₄₁	#131	1	4	yes	—
114	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
115	Q₈×C₄₁	—	2	8	no	—
116	D₄×C₄₁	—	2	8	no	—
117	C₂×C₄×C₄₁	—	2	2	no	—
118	C₂×C₄×C₄₁	—	2	2	no	—
119	C₂×C₄×C₄₁	—	2	2	no	—
120	C₂×C₄×C₄₁	—	2	2	no	—
121	C₂×C₄×C₄₁	—	2	2	no	—
122	C₂×C₄×C₄₁	—	2	2	no	—
123	C₂×C₄×C₄₁	—	2	4	no	—
124	C₂×C₄×C₄₁	—	2	4	no	—
125	C₂×C₄×C₄₁	—	2	4	no	—
126	C₂×C₄×C₄₁	—	2	4	no	—
127	C₂×C₄×C₄₁	—	2	4	no	—
128	C₂×C₄×C₄₁	—	4	2	no	—
129	C₂×C₄×C₄₁	—	4	2	no	—
130	C₂×C₄×C₄₁	—	4	4	no	—
131	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₄₁	#131	1	4	yes	—
134	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₄₁	H(5,𝔽₂)	1	2	yes	—
137	C₂×C₄×C₄₁	2₋¹⁺⁴	1	2	yes	—
138	C₈×C₄₁	—	2	2	no	—
139	C₈×C₄₁	—	2	2	no	—
140	C₈×C₄₁	—	4	2	no	—
141	C₈×C₄₁	—	4	2	no	—
142	C₈×C₄₁	—	4	2	no	—
143	#22209	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂²×C₄₁	—	2	4	no	—
145	C₂²×C₄₁	—	2	8	no	—
146	C₂²×C₄₁	—	4	2	no	—
147	C₂²×C₄₁	—	4	2	no	—
148	C₂²×C₄₁	—	4	4	no	—
149	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
150	C₂²×C₄₁	#520	1	2	yes	—
151	C₂²×C₄₁	C₂×#131	1	2	yes	—
152	C₂²×C₄₁	C₂×#131	1	2	yes	—
153	C₄×C₄₁	—	2	2	no	—
154	C₄×C₄₁	—	2	2	no	—
155	C₄×C₄₁	—	2	4	no	—
156	C₄×C₄₁	—	4	2	no	—
157	C₄×C₄₁	—	4	2	no	—
158	C₄×C₄₁	—	4	4	no	—
159	C₄×C₄₁	#535	1	1	yes	characteristic
160	C₄×C₄₁	#536	1	1	yes	characteristic
161	C₄×C₄₁	#542	1	2	yes	—
162	C₂×C₈×Q₈	C₂×C₄₁	1	1	yes	characteristic, complemented
163	C₂×C₈×D₄	C₂×C₄₁	1	1	yes	characteristic
164	#2398	C₂×C₄₁	1	1	yes	characteristic
165	#1985	C₂×C₄₁	1	1	yes	characteristic, complemented
166	#1955	C₂×C₄₁	1	1	yes	characteristic
167	#1923	C₂×C₄₁	1	4	yes	complemented
168	#1850	C₂×C₄₁	1	4	yes	complemented
169	#1843	C₂×C₄₁	1	1	yes	characteristic, complemented
170	#1746	C₂×C₄₁	1	1	yes	characteristic, complemented
171	C₂×C₄₁	—	4	4	no	—
172	C₂×C₄₁	#2326	1	1	yes	characteristic
173	C₂×C₄₁	#2929	1	1	yes	characteristic
174	C₂×C₄₁	#2929	1	1	yes	characteristic
175	C₂×C₄₁	#2994	1	4	yes	—
176	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
177	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
178	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
179	C₈×Q₈	—	2	4	no	—
180	C₈×D₄	—	2	4	no	—
181	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
182	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
183	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
184	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
185	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
186	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
187	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
189	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
190	#400	C₄×C₄₁	1	2	yes	—
191	#397	C₄×C₄₁	1	2	yes	—
192	#394	C₄×C₄₁	1	2	yes	—
193	#393	C₄×C₄₁	1	2	yes	—
194	C₄×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
195	#376	C₂²×C₄₁	1	1	yes	characteristic
196	#336	C₂²×C₄₁	1	2	yes	—
197	#336	C₂²×C₄₁	1	4	yes	—
198	#336	C₂²×C₄₁	1	2	yes	—
199	C₄₁	#22209	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
201	C₂²×D₄	C₂×C₄×C₄₁	1	1	yes	characteristic
202	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
203	C₂²×C₈	—	2	4	no	—
204	C₂²×C₈	—	4	4	no	—
205	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
206	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
207	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
208	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	2	2	no	—
214	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
216	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	2	yes	—
218	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
219	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
220	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
221	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
222	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
223	C₄⋊C₈	—	2	4	no	—
224	C₄⋊C₈	—	2	2	no	—
225	C₄⋊C₈	—	2	2	no	—
226	C₄⋊C₈	—	4	4	no	—
227	C₂²⋊C₈	—	2	4	no	—
228	C₂²⋊C₈	—	4	4	no	—
229	#97	—	2	2	no	—
230	C₄×C₈	—	2	4	no	—
231	C₄×C₈	D₄×C₄₁	1	4	yes	—
232	#95	C₂×C₄×C₄₁	1	2	yes	—
233	#95	C₂×C₄×C₄₁	1	2	yes	—
234	#95	C₂×C₄×C₄₁	1	4	yes	—
235	#95	C₂×C₄×C₄₁	1	4	yes	—
236	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
246	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
249	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
250	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
251	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	4	8	no	—
262	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
263	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	2	8	no	—
268	C₂²⋊C₄	—	2	4	no	—
269	C₂²⋊C₄	—	2	8	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₄²	—	2	2	no	—
274	C₄²	C₂×D₄×C₄₁	1	2	yes	—
275	C₄²	#41×C₄₁	1	2	yes	—
276	C₂³	—	2	2	no	—
277	C₂³	—	2	4	no	—
278	C₂³	—	2	4	no	—
279	C₂³	—	2	2	no	—
280	C₂³	#131×C₄₁	1	4	yes	—
281	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
282	Q₈	—	2	8	no	—
283	D₄	—	2	8	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
299	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
300	C₂×C₄	#131×C₄₁	1	4	yes	—
301	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
302	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
303	C₂×C₄	H(5,𝔽₂)×C₄₁	1	2	yes	—
304	C₂×C₄	2₋¹⁺⁴×C₄₁	1	2	yes	—
305	C₈	—	2	2	no	—
306	C₈	—	2	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	2	8	no	—
312	C₂²	—	4	2	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	4	no	—
315	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
316	C₂²	C₄₁×#520	1	2	yes	—
317	C₂²	C₂×#131×C₄₁	1	2	yes	—
318	C₂²	C₂×#131×C₄₁	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	4	no	—
325	C₄	C₄₁×#535	1	1	yes	characteristic
326	C₄	C₄₁×#536	1	1	yes	characteristic
327	C₄	C₄₁×#542	1	2	yes	—
328	C₂	—	4	4	no	—
329	C₂	C₄₁×#2326	1	1	yes	characteristic
330	C₂	C₄₁×#2929	1	1	yes	characteristic
331	C₂	C₄₁×#2929	1	1	yes	characteristic
332	C₂	C₄₁×#2994	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂³×C₄
prime radical	C₂²×D₄×C₄₁	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₈×Q₈×C₄₁	C₂	semidirect
4. C₄₁×#1985	C₂	semidirect
6. C₄₁×#1923	C₂	semidirect
7. C₄₁×#1850	C₂	semidirect
8. C₄₁×#1843	C₂	semidirect
9. C₄₁×#1746	C₂	semidirect
143. #22209	C₄₁	direct (N × H)
162. C₂×C₈×Q₈	C₂×C₄₁	semidirect
165. #1985	C₂×C₄₁	semidirect
167. #1923	C₂×C₄₁	semidirect
168. #1850	C₂×C₄₁	semidirect
169. #1843	C₂×C₄₁	semidirect
170. #1746	C₂×C₄₁	semidirect
199. C₄₁	#22209	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₄₁	2
C₂×C₈×D₄×C₄₁	2
C₄₁×#2398	2
C₄₁×#1985	2
C₄₁×#1955	2
C₄₁×#1923	2
C₄₁×#1850	2
C₄₁×#1843	2
C₄₁×#1746	2
#22209	41

Subgroups — C41×#22209 #23798410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22209 #23798410