Subgroups — C₄₁×#22813 #23799014

Order 10496.

922 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2354	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1972	C₂	1	2	yes	maximal
4	C₄₁×#1926	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1844	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1802	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1776	C₂	1	2	yes	maximal, complemented
8	C₄₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₄₁	C₂²	1	2	yes	—
11	C₂×#117×C₄₁	—	2	2	no	—
12	C₄×D₈×C₄₁	C₄	1	8	yes	complemented
13	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×#106×C₄₁	C₂²	1	2	yes	—
15	C₂×#106×C₄₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₂×#102×C₄₁	C₂²	1	4	yes	—
18	C₂×#102×C₄₁	C₂²	1	2	yes	—
19	C₂×#97×C₄₁	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
21	C₄₁×#390	C₂²	1	4	yes	—
22	C₄₁×#388	—	2	4	no	—
23	C₄₁×#384	C₂²	1	2	yes	—
24	C₄₁×#380	—	2	2	no	—
25	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
26	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#340	C₂²	1	4	yes	—
28	C₄₁×#336	C₂²	1	2	yes	—
29	C₂²×D₄×C₄₁	C₂³	1	2	yes	—
30	C₂³×C₄×C₄₁	—	2	2	no	—
31	C₂×D₈×C₄₁	—	2	2	no	—
32	C₂×D₈×C₄₁	—	2	2	no	—
33	C₂×D₈×C₄₁	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
37	C₄×D₄×C₄₁	—	2	4	no	—
38	C₄×D₄×C₄₁	C₂×C₄	1	8	yes	—
39	#117×C₄₁	—	4	8	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
44	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
45	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
46	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
47	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
48	C₂×C₄²×C₄₁	—	2	4	no	—
49	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
51	#107×C₄₁	C₂×C₄	1	4	yes	—
52	#106×C₄₁	—	2	4	no	—
53	#106×C₄₁	—	4	2	no	—
54	C₄⋊C₈×C₄₁	—	2	2	no	—
55	#102×C₄₁	—	4	4	no	—
56	#102×C₄₁	C₂×C₄	1	8	yes	—
57	#97×C₄₁	—	2	4	no	—
58	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
59	#95×C₄₁	—	2	4	no	—
60	#95×C₄₁	D₄	1	2	yes	—
61	C₂⁴×C₄₁	—	2	2	no	—
62	C₂×D₄×C₄₁	—	2	2	no	—
63	C₂×D₄×C₄₁	—	2	2	no	—
64	C₂×D₄×C₄₁	—	2	2	no	—
65	C₂×D₄×C₄₁	—	2	2	no	—
66	C₂×D₄×C₄₁	—	2	2	no	—
67	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₄₁	—	2	2	no	—
69	C₂²×C₄×C₄₁	—	2	8	no	—
70	C₂²×C₄×C₄₁	—	2	4	no	—
71	C₂²×C₄×C₄₁	—	4	4	no	—
72	C₂²×C₄×C₄₁	—	4	4	no	—
73	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
77	D₈×C₄₁	—	2	8	no	—
78	C₂×C₈×C₄₁	—	2	1	no	—
79	C₂×C₈×C₄₁	—	2	1	no	—
80	C₂×C₈×C₄₁	—	2	2	no	—
81	C₂×C₈×C₄₁	—	2	2	no	—
82	C₂×C₈×C₄₁	—	2	2	no	—
83	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₄₁	#41	1	2	yes	—
86	C₂×C₈×C₄₁	#41	1	2	yes	—
87	C₄⋊C₄×C₄₁	—	2	4	no	—
88	C₄⋊C₄×C₄₁	—	2	2	no	—
89	C₄⋊C₄×C₄₁	—	2	4	no	—
90	C₄⋊C₄×C₄₁	—	2	4	no	—
91	C₄⋊C₄×C₄₁	—	4	4	no	—
92	C₄⋊C₄×C₄₁	—	8	2	no	—
93	C₄⋊C₄×C₄₁	C₂²×C₄	1	4	yes	—
94	C₂²⋊C₄×C₄₁	—	2	8	no	—
95	C₂²⋊C₄×C₄₁	—	4	8	no	—
96	C₂²⋊C₄×C₄₁	—	8	4	no	—
97	C₄²×C₄₁	—	2	2	no	—
98	C₄²×C₄₁	—	2	2	no	—
99	C₄²×C₄₁	—	4	8	no	—
100	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
101	C₄²×C₄₁	#41	1	2	yes	—
102	C₂³×C₄₁	—	2	4	no	—
103	C₂³×C₄₁	—	4	2	no	—
104	C₂³×C₄₁	—	4	2	no	—
105	C₂³×C₄₁	—	4	2	no	—
106	C₂³×C₄₁	—	4	2	no	—
107	C₂³×C₄₁	—	4	2	no	—
108	C₂³×C₄₁	—	4	2	no	—
109	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
110	D₄×C₄₁	—	2	4	no	—
111	D₄×C₄₁	—	2	4	no	—
112	D₄×C₄₁	—	4	2	no	—
113	D₄×C₄₁	—	4	2	no	—
114	C₂×C₄×C₄₁	—	2	2	no	—
115	C₂×C₄×C₄₁	—	2	4	no	—
116	C₂×C₄×C₄₁	—	2	4	no	—
117	C₂×C₄×C₄₁	—	2	4	no	—
118	C₂×C₄×C₄₁	—	2	4	no	—
119	C₂×C₄×C₄₁	—	4	2	no	—
120	C₂×C₄×C₄₁	—	4	2	no	—
121	C₂×C₄×C₄₁	—	4	16	no	—
122	C₂×C₄×C₄₁	—	4	4	no	—
123	C₂×C₄×C₄₁	—	4	4	no	—
124	C₂×C₄×C₄₁	—	4	4	no	—
125	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
128	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
129	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
130	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
133	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
134	C₈×C₄₁	—	2	2	no	—
135	C₈×C₄₁	—	4	2	no	—
136	C₈×C₄₁	#117	1	4	yes	complemented
137	#22813	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂²×C₄₁	—	4	4	no	—
139	C₂²×C₄₁	—	4	4	no	—
140	C₂²×C₄₁	—	4	4	no	—
141	C₂²×C₄₁	—	8	2	no	—
142	C₂²×C₄₁	—	8	2	no	—
143	C₂²×C₄₁	—	8	2	no	—
144	C₂²×C₄₁	—	8	2	no	—
145	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂²×C₄₁	#547	1	1	yes	characteristic
147	C₂²×C₄₁	#550	1	1	yes	characteristic
148	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
149	C₂²×C₄₁	#577	1	2	yes	—
150	C₄×C₄₁	—	2	2	no	—
151	C₄×C₄₁	—	4	4	no	—
152	C₄×C₄₁	—	4	4	no	—
153	C₄×C₄₁	—	8	4	no	—
154	C₄×C₄₁	#442	1	4	yes	—
155	C₄×C₄₁	C₂×#117	1	1	yes	characteristic
156	C₄×C₄₁	#520	1	1	yes	characteristic
157	C₄×C₄₁	#555	1	1	yes	characteristic
158	C₄×C₄₁	#556	1	1	yes	characteristic
159	C₂×C₄×D₈	C₂×C₄₁	1	1	yes	characteristic
160	#2354	C₂×C₄₁	1	2	yes	complemented
161	#1972	C₂×C₄₁	1	2	yes	—
162	#1926	C₂×C₄₁	1	4	yes	complemented
163	#1844	C₂×C₄₁	1	1	yes	characteristic, complemented
164	#1802	C₂×C₄₁	1	2	yes	complemented
165	#1776	C₂×C₄₁	1	2	yes	complemented
166	#1770	C₂×C₄₁	1	1	yes	characteristic, complemented
167	C₂×C₄₁	—	8	4	no	—
168	C₂×C₄₁	#2354	1	1	yes	characteristic
169	C₂×C₄₁	C₂×#442	1	1	yes	characteristic
170	C₂×C₄₁	#2952	1	1	yes	characteristic
171	C₂×C₄₁	#3354	1	2	yes	—
172	C₂×C₄₁	#3397	1	2	yes	—
173	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
174	C₂×C₄×D₄	C₂²×C₄₁	1	2	yes	—
175	C₂×#117	—	2	2	no	—
176	C₄×D₈	C₄×C₄₁	1	8	yes	complemented
177	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
178	C₂×#106	C₂²×C₄₁	1	2	yes	—
179	C₂×#106	C₂²×C₄₁	1	2	yes	—
180	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
181	C₂×#102	C₂²×C₄₁	1	4	yes	—
182	C₂×#102	C₂²×C₄₁	1	2	yes	—
183	C₂×#97	C₂²×C₄₁	1	2	yes	—
184	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
185	#390	C₂²×C₄₁	1	4	yes	—
186	#388	—	2	4	no	—
187	#384	C₂²×C₄₁	1	2	yes	—
188	#380	—	2	2	no	—
189	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
190	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
191	#340	C₂²×C₄₁	1	4	yes	—
192	#336	C₂²×C₄₁	1	2	yes	—
193	C₄₁	#22813	1	1	yes	characteristic, direct product factor, Sylow
194	C₂²×D₄	C₂³×C₄₁	1	2	yes	—
195	C₂³×C₄	—	2	2	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	—	2	2	no	—
198	C₂×D₈	C₂×C₄×C₄₁	1	4	yes	—
199	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
200	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
202	C₄×D₄	—	2	4	no	—
203	C₄×D₄	C₂×C₄×C₄₁	1	8	yes	—
204	#117	—	4	8	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
208	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
209	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
210	C₂×C₂²⋊C₄	—	2	2	no	—
211	C₂×C₂²⋊C₄	—	2	4	no	—
212	C₂×C₂²⋊C₄	—	2	4	no	—
213	C₂×C₄²	—	2	4	no	—
214	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
215	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
216	#107	C₂×C₄×C₄₁	1	4	yes	—
217	#106	—	2	4	no	—
218	#106	—	4	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	#102	—	4	4	no	—
221	#102	C₂×C₄×C₄₁	1	8	yes	—
222	#97	—	2	4	no	—
223	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
224	#95	—	2	4	no	—
225	#95	D₄×C₄₁	1	2	yes	—
226	C₂⁴	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	2	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	8	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
239	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
241	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
242	D₈	—	2	8	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
249	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
250	C₂×C₈	#41×C₄₁	1	2	yes	—
251	C₂×C₈	#41×C₄₁	1	2	yes	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	8	2	no	—
258	C₄⋊C₄	C₂²×C₄×C₄₁	1	4	yes	—
259	C₂²⋊C₄	—	2	8	no	—
260	C₂²⋊C₄	—	4	8	no	—
261	C₂²⋊C₄	—	8	4	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	8	no	—
265	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
266	C₄²	#41×C₄₁	1	2	yes	—
267	C₂³	—	2	4	no	—
268	C₂³	—	4	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	—	4	2	no	—
272	C₂³	—	4	2	no	—
273	C₂³	—	4	2	no	—
274	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
275	D₄	—	2	4	no	—
276	D₄	—	2	4	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	16	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
291	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
292	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
293	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
294	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
295	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
297	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
298	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
299	C₈	—	2	2	no	—
300	C₈	—	4	2	no	—
301	C₈	#117×C₄₁	1	4	yes	complemented
302	C₂²	—	4	4	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
310	C₂²	C₄₁×#547	1	1	yes	characteristic
311	C₂²	C₄₁×#550	1	1	yes	characteristic
312	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	2	yes	—
313	C₂²	C₄₁×#577	1	2	yes	—
314	C₄	—	2	2	no	—
315	C₄	—	4	4	no	—
316	C₄	—	4	4	no	—
317	C₄	—	8	4	no	—
318	C₄	C₄₁×#442	1	4	yes	—
319	C₄	C₂×#117×C₄₁	1	1	yes	characteristic
320	C₄	C₄₁×#520	1	1	yes	characteristic
321	C₄	C₄₁×#555	1	1	yes	characteristic
322	C₄	C₄₁×#556	1	1	yes	characteristic
323	C₂	—	8	4	no	—
324	C₂	C₄₁×#2354	1	1	yes	characteristic
325	C₂	C₂×C₄₁×#442	1	1	yes	characteristic
326	C₂	C₄₁×#2952	1	1	yes	characteristic
327	C₂	C₄₁×#3354	1	2	yes	—
328	C₂	C₄₁×#3397	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂²×D₈×C₄₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₄×C₄⋊C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2354	C₂	semidirect
4. C₄₁×#1926	C₂	semidirect
5. C₄₁×#1844	C₂	semidirect
6. C₄₁×#1802	C₂	semidirect
7. C₄₁×#1776	C₂	semidirect
8. C₄₁×#1770	C₂	semidirect
12. C₄×D₈×C₄₁	C₄	semidirect
136. C₈×C₄₁	#117	semidirect
137. #22813	C₄₁	direct (N × H)
160. #2354	C₂×C₄₁	semidirect
162. #1926	C₂×C₄₁	semidirect
163. #1844	C₂×C₄₁	semidirect
164. #1802	C₂×C₄₁	semidirect
165. #1776	C₂×C₄₁	semidirect
166. #1770	C₂×C₄₁	semidirect
176. C₄×D₈	C₄×C₄₁	semidirect
193. C₄₁	#22813	direct (N × H)
301. C₈	#117×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₈×C₄₁	2
C₄₁×#2354	2
C₄₁×#1972	2
C₄₁×#1926	2
C₄₁×#1844	2
C₄₁×#1802	2
C₄₁×#1776	2
C₄₁×#1770	2
#22813	41

Subgroups — C41×#22813 #23799014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22813 #23799014