Subgroups — C₄₁×#21363 #23797564

Order 10496.

938 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#432	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2283	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1949	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1853	C₂	1	2	yes	maximal
6	C₄₁×#1852	C₂	1	2	yes	maximal
7	C₄₁×#1776	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
9	C₂²×C₄²×C₄₁	—	2	1	no	—
10	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
11	C₄₁×#432	C₄	1	8	yes	complemented
12	C₂×#107×C₄₁	C₂²	1	4	yes	—
13	C₂×#106×C₄₁	C₂²	1	4	yes	—
14	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
16	C₂×#97×C₄₁	C₂²	1	2	yes	—
17	C₄₁×#383	—	2	1	no	—
18	C₄₁×#382	C₂²	1	2	yes	—
19	C₄₁×#381	—	2	1	no	—
20	C₄₁×#380	C₂²	1	1	yes	characteristic
21	C₄₁×#380	C₂²	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
23	C₄×C₄⋊C₄×C₄₁	—	2	1	no	—
24	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
25	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
27	C₄×C₂²⋊C₄×C₄₁	—	2	1	no	—
28	C₄₁×#376	—	2	1	no	—
29	C₂×#95×C₄₁	—	2	1	no	—
30	C₄₁×#340	C₂²	1	4	yes	—
31	C₄₁×#340	C₂²	1	4	yes	—
32	C₂³×C₄×C₄₁	—	2	2	no	—
33	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
34	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
39	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
42	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
43	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
44	C₂×C₄²×C₄₁	—	2	2	no	—
45	C₂×C₄²×C₄₁	—	2	2	no	—
46	C₂×C₄²×C₄₁	—	2	2	no	—
47	C₂×C₄²×C₄₁	—	4	4	no	—
48	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
52	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
54	#107×C₄₁	—	2	8	no	—
55	#106×C₄₁	—	2	8	no	—
56	C₄⋊C₈×C₄₁	C₂×C₄	1	8	yes	complemented
57	C₂²⋊C₈×C₄₁	C₂×C₄	1	8	yes	—
58	#97×C₄₁	C₂×C₄	1	8	yes	complemented
59	#95×C₄₁	—	2	2	no	—
60	#95×C₄₁	—	2	2	no	—
61	#95×C₄₁	—	2	1	no	—
62	#95×C₄₁	—	2	1	no	—
63	#95×C₄₁	—	2	2	no	—
64	#95×C₄₁	—	2	2	no	—
65	#95×C₄₁	—	2	2	no	—
66	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₁	—	2	2	no	—
68	C₂²×C₄×C₄₁	—	2	2	no	—
69	C₂²×C₄×C₄₁	—	2	4	no	—
70	C₂²×C₄×C₄₁	—	2	8	no	—
71	C₂²×C₄×C₄₁	—	2	4	no	—
72	C₂²×C₄×C₄₁	—	2	2	no	—
73	C₂²×C₄×C₄₁	—	2	2	no	—
74	C₂²×C₄×C₄₁	—	4	2	no	—
75	C₂²×C₄×C₄₁	—	4	4	no	—
76	C₂²×C₄×C₄₁	—	4	4	no	—
77	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
78	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
79	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
87	C₂×C₈×C₄₁	—	2	4	no	—
88	C₂×C₈×C₄₁	—	2	4	no	—
89	C₂×C₈×C₄₁	C₂²×C₄	1	8	yes	—
90	C₄⋊C₄×C₄₁	—	2	8	no	—
91	C₄⋊C₄×C₄₁	—	2	4	no	—
92	C₄⋊C₄×C₄₁	—	2	4	no	—
93	C₄⋊C₄×C₄₁	—	2	4	no	—
94	C₄⋊C₄×C₄₁	#41	1	8	yes	—
95	C₂²⋊C₄×C₄₁	—	4	2	no	—
96	C₂²⋊C₄×C₄₁	—	4	2	no	—
97	C₄²×C₄₁	—	4	4	no	—
98	C₄²×C₄₁	—	4	4	no	—
99	C₄²×C₄₁	—	4	4	no	—
100	C₄²×C₄₁	C₄⋊C₄	1	4	yes	—
101	C₄²×C₄₁	C₄⋊C₄	1	4	yes	—
102	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
103	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
104	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
105	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
106	C₂³×C₄₁	—	2	2	no	—
107	C₂³×C₄₁	—	2	2	no	—
108	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
109	C₂³×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
110	C₂³×C₄₁	#137	1	2	yes	—
111	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₁	—	2	2	no	—
113	C₂×C₄×C₄₁	—	2	1	no	—
114	C₂×C₄×C₄₁	—	2	1	no	—
115	C₂×C₄×C₄₁	—	2	1	no	—
116	C₂×C₄×C₄₁	—	2	1	no	—
117	C₂×C₄×C₄₁	—	2	2	no	—
118	C₂×C₄×C₄₁	—	2	2	no	—
119	C₂×C₄×C₄₁	—	2	8	no	—
120	C₂×C₄×C₄₁	—	2	4	no	—
121	C₂×C₄×C₄₁	—	2	4	no	—
122	C₂×C₄×C₄₁	—	2	8	no	—
123	C₂×C₄×C₄₁	—	4	4	no	—
124	C₂×C₄×C₄₁	—	4	16	no	—
125	C₂×C₄×C₄₁	—	4	4	no	—
126	C₂×C₄×C₄₁	—	4	4	no	—
127	C₂×C₄×C₄₁	—	4	2	no	—
128	C₂×C₄×C₄₁	—	4	2	no	—
129	C₂×C₄×C₄₁	—	4	2	no	—
130	C₂×C₄×C₄₁	—	4	2	no	—
131	C₂×C₄×C₄₁	—	4	2	no	—
132	C₂×C₄×C₄₁	—	4	2	no	—
133	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
135	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
137	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
138	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
141	C₈×C₄₁	—	2	8	no	—
142	#21363	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂²×C₄₁	—	2	4	no	—
144	C₂²×C₄₁	—	2	4	no	—
145	C₂²×C₄₁	—	2	2	no	—
146	C₂²×C₄₁	—	2	2	no	—
147	C₂²×C₄₁	#428	1	4	yes	—
148	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄₁	#525	1	2	yes	—
150	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
151	C₂²×C₄₁	C₂×#137	1	1	yes	characteristic
152	C₂²×C₄₁	#575	1	2	yes	—
153	C₄×C₄₁	—	2	2	no	—
154	C₄×C₄₁	—	2	2	no	—
155	C₄×C₄₁	—	2	2	no	—
156	C₄×C₄₁	—	2	2	no	—
157	C₄×C₄₁	—	2	2	no	—
158	C₄×C₄₁	—	4	8	no	—
159	C₄×C₄₁	—	8	2	no	—
160	C₄×C₄₁	—	8	2	no	—
161	C₄×C₄₁	C₄×#41	1	2	yes	—
162	C₄×C₄₁	#533	1	2	yes	—
163	C₂×#432	C₂×C₄₁	1	1	yes	characteristic
164	#2283	C₂×C₄₁	1	1	yes	characteristic
165	C₂×C₄×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
166	#1949	C₂×C₄₁	1	4	yes	complemented
167	#1853	C₂×C₄₁	1	2	yes	—
168	#1852	C₂×C₄₁	1	2	yes	—
169	#1776	C₂×C₄₁	1	4	yes	complemented
170	C₂×C₄₁	—	2	4	no	—
171	C₂×C₄₁	#2296	1	1	yes	characteristic
172	C₂×C₄₁	C₂×#428	1	1	yes	characteristic
173	C₂×C₄₁	#2911	1	1	yes	characteristic
174	C₂×C₄₁	#3114	1	2	yes	—
175	C₂×C₄₁	#3117	1	2	yes	—
176	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
177	C₂²×C₄²	—	2	1	no	—
178	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
179	#432	C₄×C₄₁	1	8	yes	complemented
180	C₂×#107	C₂²×C₄₁	1	4	yes	—
181	C₂×#106	C₂²×C₄₁	1	4	yes	—
182	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
183	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
184	C₂×#97	C₂²×C₄₁	1	2	yes	—
185	#383	—	2	1	no	—
186	#382	C₂²×C₄₁	1	2	yes	—
187	#381	—	2	1	no	—
188	#380	C₂²×C₄₁	1	1	yes	characteristic
189	#380	C₂²×C₄₁	1	1	yes	characteristic
190	C₄×C₄⋊C₄	—	2	1	no	—
191	C₄×C₄⋊C₄	—	2	1	no	—
192	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
193	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
194	C₄×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
195	C₄×C₂²⋊C₄	—	2	1	no	—
196	#376	—	2	1	no	—
197	C₂×#95	—	2	1	no	—
198	#340	C₂²×C₄₁	1	4	yes	—
199	#340	C₂²×C₄₁	1	4	yes	—
200	C₄₁	#21363	1	1	yes	characteristic, direct product factor, Sylow
201	C₂³×C₄	—	2	2	no	—
202	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
203	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
204	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	8	no	—
208	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
209	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
210	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
211	C₂×C₂²⋊C₄	—	2	1	no	—
212	C₂×C₂²⋊C₄	—	2	1	no	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	2	2	no	—
216	C₂×C₄²	—	4	4	no	—
217	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
218	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
219	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
220	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
221	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
223	#107	—	2	8	no	—
224	#106	—	2	8	no	—
225	C₄⋊C₈	C₂×C₄×C₄₁	1	8	yes	complemented
226	C₂²⋊C₈	C₂×C₄×C₄₁	1	8	yes	—
227	#97	C₂×C₄×C₄₁	1	8	yes	complemented
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	1	no	—
231	#95	—	2	1	no	—
232	#95	—	2	2	no	—
233	#95	—	2	2	no	—
234	#95	—	2	2	no	—
235	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
247	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
248	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
250	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
251	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
252	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
254	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
255	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	C₂²×C₄×C₄₁	1	8	yes	—
259	C₄⋊C₄	—	2	8	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	#41×C₄₁	1	8	yes	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	4	4	no	—
267	C₄²	—	4	4	no	—
268	C₄²	—	4	4	no	—
269	C₄²	C₄⋊C₄×C₄₁	1	4	yes	—
270	C₄²	C₄⋊C₄×C₄₁	1	4	yes	—
271	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
272	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
273	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
274	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
278	C₂³	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
279	C₂³	#137×C₄₁	1	2	yes	—
280	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	8	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	8	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	16	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	4	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
303	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
304	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
305	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
306	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
307	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
308	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
310	C₈	—	2	8	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	2	no	—
315	C₂²	C₄₁×#428	1	4	yes	—
316	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
317	C₂²	C₄₁×#525	1	2	yes	—
318	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
319	C₂²	C₂×#137×C₄₁	1	1	yes	characteristic
320	C₂²	C₄₁×#575	1	2	yes	—
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	4	8	no	—
327	C₄	—	8	2	no	—
328	C₄	—	8	2	no	—
329	C₄	C₄×#41×C₄₁	1	2	yes	—
330	C₄	C₄₁×#533	1	2	yes	—
331	C₂	—	2	4	no	—
332	C₂	C₄₁×#2296	1	1	yes	characteristic
333	C₂	C₂×C₄₁×#428	1	1	yes	characteristic
334	C₂	C₄₁×#2911	1	1	yes	characteristic
335	C₂	C₄₁×#3114	1	2	yes	—
336	C₂	C₄₁×#3117	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	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
7. C₄₁×#1776	C₂	semidirect
11. C₄₁×#432	C₄	semidirect
56. C₄⋊C₈×C₄₁	C₂×C₄	semidirect
58. #97×C₄₁	C₂×C₄	semidirect
142. #21363	C₄₁	direct (N × H)
166. #1949	C₂×C₄₁	semidirect
169. #1776	C₂×C₄₁	semidirect
179. #432	C₄×C₄₁	semidirect
200. C₄₁	#21363	direct (N × H)
225. C₄⋊C₈	C₂×C₄×C₄₁	semidirect
227. #97	C₂×C₄×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×#432	2
C₄₁×#2283	2
C₂×C₄×C₄⋊C₄×C₄₁	2
C₄₁×#1949	2
C₄₁×#1853	2
C₄₁×#1852	2
C₄₁×#1776	2
#21363	41

Subgroups — C41×#21363 #23797564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21363 #23797564