Subgroups — C₄₁×#21360 #23797561

Order 10496.

870 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2283	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1949	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1854	C₂	1	4	yes	maximal
6	C₄×#107×C₄₁	C₂	1	2	yes	maximal, complemented
7	C₄×#106×C₄₁	C₂	1	2	yes	maximal, complemented
8	C₂×#117×C₄₁	C₂²	1	2	yes	—
9	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
10	C₄₁×#431	C₄	1	8	yes	complemented
11	C₂×#107×C₄₁	C₂²	1	4	yes	—
12	C₂×#106×C₄₁	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#383	—	2	1	no	—
18	C₄₁×#382	C₂²	1	2	yes	—
19	C₄₁×#381	—	2	1	no	—
20	C₄₁×#380	C₂²	1	2	yes	—
21	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
22	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
24	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
25	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
26	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
27	C₄₁×#376	—	2	1	no	—
28	C₄₁×#376	—	2	1	no	—
29	C₄³×C₄₁	—	2	1	no	—
30	C₄₁×#340	C₂²	1	4	yes	—
31	C₄₁×#340	C₂²	1	4	yes	—
32	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
33	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
37	#117×C₄₁	—	2	8	no	—
38	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
40	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
41	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
42	C₂×C₄²×C₄₁	—	2	2	no	—
43	C₂×C₄²×C₄₁	—	2	2	no	—
44	C₂×C₄²×C₄₁	—	2	2	no	—
45	C₂×C₄²×C₄₁	—	2	2	no	—
46	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
47	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
48	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
50	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
52	#107×C₄₁	—	2	8	no	—
53	#106×C₄₁	—	2	8	no	—
54	C₄⋊C₈×C₄₁	C₂×C₄	1	8	yes	—
55	C₂²⋊C₈×C₄₁	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
58	#95×C₄₁	—	2	2	no	—
59	#95×C₄₁	—	2	2	no	—
60	#95×C₄₁	—	2	2	no	—
61	#95×C₄₁	—	2	2	no	—
62	#95×C₄₁	—	2	2	no	—
63	#95×C₄₁	—	2	2	no	—
64	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₁	—	2	4	no	—
66	C₂²×C₄×C₄₁	—	2	4	no	—
67	C₂²×C₄×C₄₁	—	2	4	no	—
68	C₂²×C₄×C₄₁	—	2	4	no	—
69	C₂²×C₄×C₄₁	—	4	2	no	—
70	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
71	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
72	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₄₁	—	2	4	no	—
81	C₂×C₈×C₄₁	—	2	4	no	—
82	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
83	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
84	C₄⋊C₄×C₄₁	—	2	4	no	—
85	C₄⋊C₄×C₄₁	—	2	4	no	—
86	C₄⋊C₄×C₄₁	—	2	4	no	—
87	C₄⋊C₄×C₄₁	#41	1	8	yes	—
88	C₂²⋊C₄×C₄₁	—	2	8	no	—
89	C₂²⋊C₄×C₄₁	—	4	4	no	—
90	C₄²×C₄₁	—	2	8	no	—
91	C₄²×C₄₁	—	4	4	no	—
92	C₄²×C₄₁	—	4	4	no	—
93	C₄²×C₄₁	—	4	4	no	—
94	C₄²×C₄₁	C₄⋊C₄	1	4	yes	—
95	C₄²×C₄₁	C₄⋊C₄	1	4	yes	—
96	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
97	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
98	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
99	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
100	C₂³×C₄₁	—	2	2	no	—
101	C₂³×C₄₁	—	2	2	no	—
102	C₂³×C₄₁	—	2	2	no	—
103	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
104	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₄₁	—	2	2	no	—
106	C₂×C₄×C₄₁	—	2	2	no	—
107	C₂×C₄×C₄₁	—	2	2	no	—
108	C₂×C₄×C₄₁	—	2	2	no	—
109	C₂×C₄×C₄₁	—	2	4	no	—
110	C₂×C₄×C₄₁	—	2	4	no	—
111	C₂×C₄×C₄₁	—	2	8	no	—
112	C₂×C₄×C₄₁	—	2	8	no	—
113	C₂×C₄×C₄₁	—	4	4	no	—
114	C₂×C₄×C₄₁	—	4	4	no	—
115	C₂×C₄×C₄₁	—	4	4	no	—
116	C₂×C₄×C₄₁	—	4	4	no	—
117	C₂×C₄×C₄₁	—	4	4	no	—
118	C₂×C₄×C₄₁	—	4	4	no	—
119	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₄₁	#135	1	4	yes	—
125	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
128	C₈×C₄₁	—	2	4	no	—
129	C₈×C₄₁	—	2	4	no	—
130	#21360	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
131	C₂²×C₄₁	—	2	2	no	—
132	C₂²×C₄₁	—	2	2	no	—
133	C₂²×C₄₁	—	2	2	no	—
134	C₂²×C₄₁	—	4	4	no	—
135	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₄₁	#525	1	2	yes	—
137	C₂²×C₄₁	C₂×#135	1	2	yes	—
138	C₂²×C₄₁	#575	1	2	yes	—
139	C₄×C₄₁	—	2	2	no	—
140	C₄×C₄₁	—	2	2	no	—
141	C₄×C₄₁	—	2	2	no	—
142	C₄×C₄₁	—	2	2	no	—
143	C₄×C₄₁	—	4	8	no	—
144	C₄×C₄₁	—	8	4	no	—
145	C₄×C₄₁	#427	1	4	yes	—
146	C₄×C₄₁	C₄×#41	1	2	yes	—
147	C₄×C₄₁	#533	1	2	yes	—
148	C₂×#431	C₂×C₄₁	1	1	yes	characteristic
149	#2283	C₂×C₄₁	1	1	yes	characteristic
150	C₄×#117	C₂×C₄₁	1	1	yes	characteristic
151	#1949	C₂×C₄₁	1	4	yes	complemented
152	#1854	C₂×C₄₁	1	4	yes	—
153	C₄×#107	C₂×C₄₁	1	2	yes	complemented
154	C₄×#106	C₂×C₄₁	1	2	yes	complemented
155	C₂×C₄₁	—	4	2	no	—
156	C₂×C₄₁	#2296	1	1	yes	characteristic
157	C₂×C₄₁	C₂×#427	1	1	yes	characteristic
158	C₂×C₄₁	#2911	1	1	yes	characteristic
159	C₂×C₄₁	#3102	1	4	yes	—
160	C₂×#117	C₂²×C₄₁	1	2	yes	—
161	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
162	#431	C₄×C₄₁	1	8	yes	complemented
163	C₂×#107	C₂²×C₄₁	1	4	yes	—
164	C₂×#106	C₂²×C₄₁	1	4	yes	—
165	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
166	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
169	#383	—	2	1	no	—
170	#382	C₂²×C₄₁	1	2	yes	—
171	#381	—	2	1	no	—
172	#380	C₂²×C₄₁	1	2	yes	—
173	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
174	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
175	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
176	C₄×C₂²⋊C₄	—	2	1	no	—
177	C₄×C₂²⋊C₄	—	2	1	no	—
178	C₄×C₂²⋊C₄	—	2	1	no	—
179	#376	—	2	1	no	—
180	#376	—	2	1	no	—
181	C₄³	—	2	1	no	—
182	#340	C₂²×C₄₁	1	4	yes	—
183	#340	C₂²×C₄₁	1	4	yes	—
184	C₄₁	#21360	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
186	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
187	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
189	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
190	#117	—	2	8	no	—
191	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
192	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
200	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
201	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
202	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
203	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
204	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
205	#107	—	2	8	no	—
206	#106	—	2	8	no	—
207	C₄⋊C₈	C₂×C₄×C₄₁	1	8	yes	—
208	C₂²⋊C₈	C₂×C₄×C₄₁	1	8	yes	—
209	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
210	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	2	2	no	—
214	#95	—	2	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
224	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
225	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
231	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
232	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
236	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	#41×C₄₁	1	8	yes	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₄²	—	2	8	no	—
244	C₄²	—	4	4	no	—
245	C₄²	—	4	4	no	—
246	C₄²	—	4	4	no	—
247	C₄²	C₄⋊C₄×C₄₁	1	4	yes	—
248	C₄²	C₄⋊C₄×C₄₁	1	4	yes	—
249	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
250	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
251	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
252	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
257	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
275	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
276	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
277	C₂×C₄	#135×C₄₁	1	4	yes	—
278	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
279	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
280	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	2	4	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	4	4	no	—
287	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
288	C₂²	C₄₁×#525	1	2	yes	—
289	C₂²	C₂×#135×C₄₁	1	2	yes	—
290	C₂²	C₄₁×#575	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	4	8	no	—
296	C₄	—	8	4	no	—
297	C₄	C₄₁×#427	1	4	yes	—
298	C₄	C₄×#41×C₄₁	1	2	yes	—
299	C₄	C₄₁×#533	1	2	yes	—
300	C₂	—	4	2	no	—
301	C₂	C₄₁×#2296	1	1	yes	characteristic
302	C₂	C₂×C₄₁×#427	1	1	yes	characteristic
303	C₂	C₄₁×#2911	1	1	yes	characteristic
304	C₂	C₄₁×#3102	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂²×D₄
prime radical	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1949	C₂	semidirect
6. C₄×#107×C₄₁	C₂	semidirect
7. C₄×#106×C₄₁	C₂	semidirect
10. C₄₁×#431	C₄	semidirect
130. #21360	C₄₁	direct (N × H)
151. #1949	C₂×C₄₁	semidirect
153. C₄×#107	C₂×C₄₁	semidirect
154. C₄×#106	C₂×C₄₁	semidirect
162. #431	C₄×C₄₁	semidirect
184. C₄₁	#21360	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×C₄₁×#431	2
C₄₁×#2283	2
C₄×#117×C₄₁	2
C₄₁×#1949	2
C₄₁×#1854	2
C₄×#107×C₄₁	2
C₄×#106×C₄₁	2
#21360	41

Subgroups — C41×#21360 #23797561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21360 #23797561