Subgroups — C₄₁×#23157 #23799358

Order 10496.

806 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#500	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2365	C₂	1	2	yes	maximal
3	C₄₁×#1988	C₂	1	2	yes	maximal
4	C₄₁×#1942	C₂	1	2	yes	maximal
5	C₄₁×#1918	C₂	1	4	yes	maximal
6	C₄×#107×C₄₁	C₂	1	1	yes	maximal, characteristic
7	C₄×#107×C₄₁	C₂	1	2	yes	maximal
8	C₈×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₄₁	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₄₁	—	2	2	no	—
11	C₄₁×#500	C₄	1	8	yes	complemented
12	C₂×#107×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×#107×C₄₁	C₂²	1	2	yes	—
14	C₂×#107×C₄₁	C₂²	1	2	yes	—
15	C₂×#107×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	1	yes	characteristic
19	C₄₁×#391	—	2	2	no	—
20	C₄₁×#389	C₂²	1	4	yes	—
21	C₄₁×#387	—	2	4	no	—
22	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
23	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
24	C₄×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
25	C₄₁×#340	C₂²	1	2	yes	—
26	C₄₁×#340	C₂²	1	4	yes	—
27	C₄₁×#340	C₂²	1	4	yes	—
28	C₄₁×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₄₁	—	2	2	no	—
30	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
33	C₄⋊Q₈×C₄₁	—	2	4	no	—
34	C₄⋊Q₈×C₄₁	C₂×C₄	1	8	yes	—
35	C₄×Q₈×C₄₁	—	4	8	no	—
36	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
42	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
43	C₂×C₄²×C₄₁	—	2	4	no	—
44	C₂×C₄²×C₄₁	—	2	2	no	—
45	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
47	#107×C₄₁	—	2	2	no	—
48	#107×C₄₁	C₂×C₄	1	8	yes	—
49	#107×C₄₁	C₂×C₄	1	8	yes	—
50	#107×C₄₁	Q₈	1	4	yes	complemented
51	C₄⋊C₈×C₄₁	—	2	2	no	—
52	C₄×C₈×C₄₁	—	2	4	no	—
53	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
54	#95×C₄₁	—	2	2	no	—
55	#95×C₄₁	—	2	4	no	—
56	#95×C₄₁	—	2	4	no	—
57	#95×C₄₁	D₄	1	2	yes	—
58	C₂×Q₈×C₄₁	—	2	8	no	—
59	C₂×Q₈×C₄₁	—	4	4	no	—
60	C₂×Q₈×C₄₁	—	4	4	no	—
61	C₂²×C₄×C₄₁	—	2	2	no	—
62	C₂²×C₄×C₄₁	—	2	4	no	—
63	C₂²×C₄×C₄₁	—	2	2	no	—
64	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
69	C₂×C₈×C₄₁	—	2	1	no	—
70	C₂×C₈×C₄₁	—	2	1	no	—
71	C₂×C₈×C₄₁	—	2	2	no	—
72	C₂×C₈×C₄₁	—	2	2	no	—
73	C₂×C₈×C₄₁	—	2	2	no	—
74	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₁	C₂×Q₈	1	2	yes	—
77	C₂×C₈×C₄₁	#41	1	2	yes	—
78	C₄⋊C₄×C₄₁	—	2	8	no	—
79	C₄⋊C₄×C₄₁	—	2	2	no	—
80	C₄⋊C₄×C₄₁	—	2	4	no	—
81	C₄⋊C₄×C₄₁	—	2	2	no	—
82	C₄⋊C₄×C₄₁	—	2	2	no	—
83	C₄⋊C₄×C₄₁	—	2	2	no	—
84	C₄⋊C₄×C₄₁	—	4	8	no	—
85	C₄⋊C₄×C₄₁	D₈	1	4	yes	—
86	C₄⋊C₄×C₄₁	C₂²×C₄	1	8	yes	—
87	C₄⋊C₄×C₄₁	C₂×Q₈	1	4	yes	—
88	C₄²×C₄₁	—	2	2	no	—
89	C₄²×C₄₁	—	2	2	no	—
90	C₄²×C₄₁	—	4	8	no	—
91	C₄²×C₄₁	—	4	4	no	—
92	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
93	C₄²×C₄₁	#41	1	2	yes	—
94	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
95	Q₈×C₄₁	—	4	16	no	—
96	C₂×C₄×C₄₁	—	2	2	no	—
97	C₂×C₄×C₄₁	—	2	2	no	—
98	C₂×C₄×C₄₁	—	2	1	no	—
99	C₂×C₄×C₄₁	—	2	1	no	—
100	C₂×C₄×C₄₁	—	2	8	no	—
101	C₂×C₄×C₄₁	—	2	2	no	—
102	C₂×C₄×C₄₁	—	2	4	no	—
103	C₂×C₄×C₄₁	—	2	2	no	—
104	C₂×C₄×C₄₁	—	2	2	no	—
105	C₂×C₄×C₄₁	—	4	2	no	—
106	C₂×C₄×C₄₁	—	4	4	no	—
107	C₂×C₄×C₄₁	—	4	4	no	—
108	C₂×C₄×C₄₁	—	4	2	no	—
109	C₂×C₄×C₄₁	—	4	2	no	—
110	C₂×C₄×C₄₁	—	4	2	no	—
111	C₂×C₄×C₄₁	—	4	2	no	—
112	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₁	C₂×D₈	1	4	yes	—
115	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₄₁	#135	1	2	yes	—
117	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
121	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×C₄×C₄₁	2₋¹⁺⁴	1	1	yes	characteristic
123	C₈×C₄₁	—	2	2	no	—
124	C₈×C₄₁	—	4	2	no	—
125	C₈×C₄₁	C₄×Q₈	1	4	yes	complemented
126	#23157	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
128	C₂²×C₄₁	#547	1	1	yes	characteristic
129	C₂²×C₄₁	D₄×Q₈	1	1	yes	characteristic
130	C₂²×C₄₁	C₂²×D₈	1	1	yes	characteristic
131	C₂²×C₄₁	C₂×#135	1	1	yes	characteristic
132	C₂²×C₄₁	#578	1	1	yes	characteristic
133	C₂²×C₄₁	#578	1	1	yes	characteristic
134	C₄×C₄₁	—	2	4	no	—
135	C₄×C₄₁	—	2	2	no	—
136	C₄×C₄₁	—	4	2	no	—
137	C₄×C₄₁	—	4	4	no	—
138	C₄×C₄₁	—	4	8	no	—
139	C₄×C₄₁	—	8	2	no	—
140	C₄×C₄₁	C₄×D₈	1	4	yes	—
141	C₄×C₄₁	C₂×C₄×Q₈	1	1	yes	characteristic
142	C₄×C₄₁	#519	1	1	yes	characteristic
143	C₄×C₄₁	#556	1	1	yes	characteristic
144	C₄×C₄₁	#557	1	1	yes	characteristic
145	C₂×#500	C₂×C₄₁	1	1	yes	characteristic
146	#2365	C₂×C₄₁	1	2	yes	—
147	#1988	C₂×C₄₁	1	2	yes	—
148	#1942	C₂×C₄₁	1	2	yes	—
149	#1918	C₂×C₄₁	1	4	yes	—
150	C₄×#107	C₂×C₄₁	1	1	yes	characteristic
151	C₄×#107	C₂×C₄₁	1	2	yes	—
152	C₈×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
153	C₂×C₄₁	#2365	1	1	yes	characteristic
154	C₂×C₄₁	C₂×C₄×D₈	1	1	yes	characteristic
155	C₂×C₄₁	#2950	1	1	yes	characteristic
156	C₂×C₄₁	#3334	1	1	yes	characteristic
157	C₂×C₄₁	#3342	1	1	yes	characteristic
158	C₂×C₄₁	Q₈×D₈	1	1	yes	characteristic
159	C₂×C₄₁	#3383	1	1	yes	characteristic
160	C₂×C₄⋊Q₈	C₂²×C₄₁	1	2	yes	—
161	C₂×C₄×Q₈	—	2	2	no	—
162	#500	C₄×C₄₁	1	8	yes	complemented
163	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
164	C₂×#107	C₂²×C₄₁	1	2	yes	—
165	C₂×#107	C₂²×C₄₁	1	2	yes	—
166	C₂×#107	C₂²×C₄₁	1	1	yes	characteristic
167	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
170	#391	—	2	2	no	—
171	#389	C₂²×C₄₁	1	4	yes	—
172	#387	—	2	4	no	—
173	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
174	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
175	C₄×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
176	#340	C₂²×C₄₁	1	2	yes	—
177	#340	C₂²×C₄₁	1	4	yes	—
178	#340	C₂²×C₄₁	1	4	yes	—
179	#336	C₂²×C₄₁	1	2	yes	—
180	C₄₁	#23157	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²×Q₈	—	2	2	no	—
182	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
183	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
185	C₄⋊Q₈	—	2	4	no	—
186	C₄⋊Q₈	C₂×C₄×C₄₁	1	8	yes	—
187	C₄×Q₈	—	4	8	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
191	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
193	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
194	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
195	C₂×C₄²	—	2	4	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
198	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
199	#107	—	2	2	no	—
200	#107	C₂×C₄×C₄₁	1	8	yes	—
201	#107	C₂×C₄×C₄₁	1	8	yes	—
202	#107	Q₈×C₄₁	1	4	yes	complemented
203	C₄⋊C₈	—	2	2	no	—
204	C₄×C₈	—	2	4	no	—
205	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
206	#95	—	2	2	no	—
207	#95	—	2	4	no	—
208	#95	—	2	4	no	—
209	#95	D₄×C₄₁	1	2	yes	—
210	C₂×Q₈	—	2	8	no	—
211	C₂×Q₈	—	4	4	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
219	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
220	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
227	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
228	C₂×C₈	C₂×Q₈×C₄₁	1	2	yes	—
229	C₂×C₈	#41×C₄₁	1	2	yes	—
230	C₄⋊C₄	—	2	8	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	8	no	—
237	C₄⋊C₄	D₈×C₄₁	1	4	yes	—
238	C₄⋊C₄	C₂²×C₄×C₄₁	1	8	yes	—
239	C₄⋊C₄	C₂×Q₈×C₄₁	1	4	yes	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	4	8	no	—
243	C₄²	—	4	4	no	—
244	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
245	C₄²	#41×C₄₁	1	2	yes	—
246	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
247	Q₈	—	4	16	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
265	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
266	C₂×C₄	C₂×D₈×C₄₁	1	4	yes	—
267	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
268	C₂×C₄	#135×C₄₁	1	2	yes	—
269	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
270	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
272	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
273	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
274	C₂×C₄	2₋¹⁺⁴×C₄₁	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	4	2	no	—
277	C₈	C₄×Q₈×C₄₁	1	4	yes	complemented
278	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
279	C₂²	C₄₁×#547	1	1	yes	characteristic
280	C₂²	D₄×Q₈×C₄₁	1	1	yes	characteristic
281	C₂²	C₂²×D₈×C₄₁	1	1	yes	characteristic
282	C₂²	C₂×#135×C₄₁	1	1	yes	characteristic
283	C₂²	C₄₁×#578	1	1	yes	characteristic
284	C₂²	C₄₁×#578	1	1	yes	characteristic
285	C₄	—	2	4	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	4	8	no	—
290	C₄	—	8	2	no	—
291	C₄	C₄×D₈×C₄₁	1	4	yes	—
292	C₄	C₂×C₄×Q₈×C₄₁	1	1	yes	characteristic
293	C₄	C₄₁×#519	1	1	yes	characteristic
294	C₄	C₄₁×#556	1	1	yes	characteristic
295	C₄	C₄₁×#557	1	1	yes	characteristic
296	C₂	C₄₁×#2365	1	1	yes	characteristic
297	C₂	C₂×C₄×D₈×C₄₁	1	1	yes	characteristic
298	C₂	C₄₁×#2950	1	1	yes	characteristic
299	C₂	C₄₁×#3334	1	1	yes	characteristic
300	C₂	C₄₁×#3342	1	1	yes	characteristic
301	C₂	Q₈×D₈×C₄₁	1	1	yes	characteristic
302	C₂	C₄₁×#3383	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
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
11. C₄₁×#500	C₄	semidirect
50. #107×C₄₁	Q₈	semidirect
125. C₈×C₄₁	C₄×Q₈	semidirect
126. #23157	C₄₁	direct (N × H)
162. #500	C₄×C₄₁	semidirect
180. C₄₁	#23157	direct (N × H)
202. #107	Q₈×C₄₁	semidirect
277. C₈	C₄×Q₈×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×#500	2
C₄₁×#2365	2
C₄₁×#1988	2
C₄₁×#1942	2
C₄₁×#1918	2
C₄×#107×C₄₁	2
C₈×C₄⋊C₄×C₄₁	2
#23157	41

Subgroups — C41×#23157 #23799358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#23157 #23799358