Subgroups — C₄₁×#21933 #23798134

Order 10496.

1114 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×C₄₁×#407	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₂²⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#2280	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1851	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1850	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1798	C₂	1	4	yes	maximal, complemented
7	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
9	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
10	C₂×#117×C₄₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₄₁	C₂²	1	8	yes	complemented
12	C₄₁×#407	—	2	4	no	—
13	C₂×C₂²⋊C₈×C₄₁	—	2	4	no	—
14	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₈×C₄₁	C₂²	1	2	yes	—
18	C₄₁×#381	C₄	1	2	yes	—
19	C₄₁×#380	C₄	1	2	yes	—
20	C₄₁×#380	C₄	1	2	yes	—
21	C₄₁×#379	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
23	C₄×C₂²⋊C₄×C₄₁	C₂²	1	2	yes	—
24	C₄₁×#336	C₂²	1	8	yes	complemented
25	C₂⁵×C₄₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₄₁	—	4	4	no	—
27	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₁	D₄	1	4	yes	—
31	C₂³×C₄×C₄₁	Q₈	1	4	yes	—
32	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₄₁	—	2	4	no	—
36	C₂×M₁₆×C₄₁	—	4	4	no	—
37	C₂²×C₈×C₄₁	—	2	4	no	—
38	C₂²×C₈×C₄₁	—	4	4	no	—
39	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
40	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
41	#117×C₄₁	—	4	4	no	—
42	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
45	C₂×C₂²⋊C₄×C₄₁	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₄₁	C₂³	1	4	yes	—
47	C₄⋊C₈×C₄₁	—	4	8	no	—
48	C₂²⋊C₈×C₄₁	—	2	4	no	—
49	C₂²⋊C₈×C₄₁	—	2	4	no	—
50	C₂²⋊C₈×C₄₁	—	2	4	no	—
51	C₂²⋊C₈×C₄₁	—	2	4	no	—
52	#95×C₄₁	C₂×C₄	1	4	yes	—
53	#95×C₄₁	C₂×C₄	1	4	yes	—
54	C₂⁴×C₄₁	—	2	4	no	—
55	C₂⁴×C₄₁	—	4	2	no	—
56	C₂⁴×C₄₁	—	4	2	no	—
57	C₂⁴×C₄₁	C₄⋊C₄	1	4	yes	—
58	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
59	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
60	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₁	—	2	4	no	—
62	C₂²×C₄×C₄₁	—	2	4	no	—
63	C₂²×C₄×C₄₁	—	2	4	no	—
64	C₂²×C₄×C₄₁	—	4	8	no	—
65	C₂²×C₄×C₄₁	—	4	8	no	—
66	C₂²×C₄×C₄₁	—	4	2	no	—
67	C₂²×C₄×C₄₁	—	4	2	no	—
68	C₂²×C₄×C₄₁	—	4	2	no	—
69	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
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	8	yes	—
73	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₄₁	—	4	8	no	—
84	C₂×C₈×C₄₁	—	2	4	no	—
85	C₂×C₈×C₄₁	—	2	4	no	—
86	C₂×C₈×C₄₁	—	2	8	no	—
87	C₂×C₈×C₄₁	—	2	8	no	—
88	C₂×C₈×C₄₁	—	4	8	no	—
89	C₄⋊C₄×C₄₁	—	4	4	no	—
90	C₂²⋊C₄×C₄₁	—	2	8	no	—
91	C₂²⋊C₄×C₄₁	—	4	4	no	—
92	C₂²⋊C₄×C₄₁	—	4	4	no	—
93	C₄²×C₄₁	—	2	8	no	—
94	C₂³×C₄₁	—	2	2	no	—
95	C₂³×C₄₁	—	2	2	no	—
96	C₂³×C₄₁	—	2	2	no	—
97	C₂³×C₄₁	—	2	2	no	—
98	C₂³×C₄₁	—	2	2	no	—
99	C₂³×C₄₁	—	2	2	no	—
100	C₂³×C₄₁	—	2	2	no	—
101	C₂³×C₄₁	—	2	2	no	—
102	C₂³×C₄₁	—	2	8	no	—
103	C₂³×C₄₁	—	4	4	no	—
104	C₂³×C₄₁	—	4	4	no	—
105	C₂³×C₄₁	—	4	16	no	—
106	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
108	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
109	C₂³×C₄₁	#131	1	4	yes	—
110	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₁	—	2	2	no	—
112	C₂×C₄×C₄₁	—	2	2	no	—
113	C₂×C₄×C₄₁	—	2	2	no	—
114	C₂×C₄×C₄₁	—	2	4	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	—	2	4	no	—
120	C₂×C₄×C₄₁	—	2	8	no	—
121	C₂×C₄×C₄₁	—	2	8	no	—
122	C₂×C₄×C₄₁	—	2	8	no	—
123	C₂×C₄×C₄₁	—	4	4	no	—
124	C₂×C₄×C₄₁	—	4	4	no	—
125	C₂×C₄×C₄₁	—	4	4	no	—
126	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₄₁	H(5,𝔽₂)	1	4	yes	—
132	C₈×C₄₁	—	4	4	no	—
133	C₈×C₄₁	—	4	4	no	—
134	#21933	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂²×C₄₁	—	2	2	no	—
136	C₂²×C₄₁	—	2	2	no	—
137	C₂²×C₄₁	—	2	8	no	—
138	C₂²×C₄₁	—	2	2	no	—
139	C₂²×C₄₁	—	2	2	no	—
140	C₂²×C₄₁	—	2	2	no	—
141	C₂²×C₄₁	—	2	2	no	—
142	C₂²×C₄₁	—	2	2	no	—
143	C₂²×C₄₁	—	2	2	no	—
144	C₂²×C₄₁	—	4	16	no	—
145	C₂²×C₄₁	—	4	4	no	—
146	C₂²×C₄₁	—	4	4	no	—
147	C₂²×C₄₁	#424	1	4	yes	—
148	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₄₁	#518	1	2	yes	—
150	C₂²×C₄₁	C₂×#131	1	2	yes	—
151	C₂²×C₄₁	#568	1	2	yes	—
152	C₄×C₄₁	—	2	2	no	—
153	C₄×C₄₁	—	2	2	no	—
154	C₄×C₄₁	—	2	2	no	—
155	C₄×C₄₁	—	4	8	no	—
156	C₄×C₄₁	#534	1	2	yes	—
157	C₄×C₄₁	#543	1	2	yes	—
158	C₂×#407	C₂×C₄₁	1	1	yes	characteristic
159	C₂²×C₂²⋊C₈	C₂×C₄₁	1	1	yes	characteristic
160	#2280	C₂×C₄₁	1	1	yes	characteristic
161	#1851	C₂×C₄₁	1	4	yes	complemented
162	#1850	C₂×C₄₁	1	4	yes	complemented
163	#1798	C₂×C₄₁	1	4	yes	complemented
164	C₂×C₄₁	—	2	4	no	—
165	C₂×C₄₁	—	4	2	no	—
166	C₂×C₄₁	—	4	2	no	—
167	C₂×C₄₁	#2315	1	1	yes	characteristic
168	C₂×C₄₁	C₂×#424	1	1	yes	characteristic
169	C₂×C₄₁	#2905	1	1	yes	characteristic
170	C₂×C₄₁	#2976	1	4	yes	—
171	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
172	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
173	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
174	C₂×#117	C₂²×C₄₁	1	2	yes	—
175	C₂×C₄⋊C₈	C₂²×C₄₁	1	8	yes	complemented
176	#407	—	2	4	no	—
177	C₂×C₂²⋊C₈	—	2	4	no	—
178	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
179	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
180	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
181	C₂×C₂²⋊C₈	C₂²×C₄₁	1	2	yes	—
182	#381	C₄×C₄₁	1	2	yes	—
183	#380	C₄×C₄₁	1	2	yes	—
184	#380	C₄×C₄₁	1	2	yes	—
185	#379	C₄×C₄₁	1	2	yes	—
186	C₄×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
187	C₄×C₂²⋊C₄	C₂²×C₄₁	1	2	yes	—
188	#336	C₂²×C₄₁	1	8	yes	complemented
189	C₄₁	#21933	1	1	yes	characteristic, direct product factor, Sylow
190	C₂⁵	C₂×C₄×C₄₁	1	1	yes	characteristic
191	C₂³×C₄	—	4	4	no	—
192	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
193	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
194	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
195	C₂³×C₄	D₄×C₄₁	1	4	yes	—
196	C₂³×C₄	Q₈×C₄₁	1	4	yes	—
197	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
198	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
199	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
200	C₂×M₁₆	—	2	4	no	—
201	C₂×M₁₆	—	4	4	no	—
202	C₂²×C₈	—	2	4	no	—
203	C₂²×C₈	—	4	4	no	—
204	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
205	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
206	#117	—	4	4	no	—
207	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
208	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
209	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
210	C₂×C₂²⋊C₄	C₂×C₄×C₄₁	1	4	yes	—
211	C₂×C₄²	C₂³×C₄₁	1	4	yes	—
212	C₄⋊C₈	—	4	8	no	—
213	C₂²⋊C₈	—	2	4	no	—
214	C₂²⋊C₈	—	2	4	no	—
215	C₂²⋊C₈	—	2	4	no	—
216	C₂²⋊C₈	—	2	4	no	—
217	#95	C₂×C₄×C₄₁	1	4	yes	—
218	#95	C₂×C₄×C₄₁	1	4	yes	—
219	C₂⁴	—	2	4	no	—
220	C₂⁴	—	4	2	no	—
221	C₂⁴	—	4	2	no	—
222	C₂⁴	C₄⋊C₄×C₄₁	1	4	yes	—
223	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
224	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
225	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	4	8	no	—
230	C₂²×C₄	—	4	8	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
235	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
236	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
237	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
238	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
239	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
243	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
244	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
245	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
246	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
247	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
248	M₁₆	—	4	8	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	8	no	—
252	C₂×C₈	—	2	8	no	—
253	C₂×C₈	—	4	8	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₂²⋊C₄	—	2	8	no	—
256	C₂²⋊C₄	—	4	4	no	—
257	C₂²⋊C₄	—	4	4	no	—
258	C₄²	—	2	8	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	8	no	—
268	C₂³	—	4	4	no	—
269	C₂³	—	4	4	no	—
270	C₂³	—	4	16	no	—
271	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
272	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
273	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
274	C₂³	#131×C₄₁	1	4	yes	—
275	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	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	—	2	4	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	2	8	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
292	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
293	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
294	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
295	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
296	C₂×C₄	H(5,𝔽₂)×C₄₁	1	4	yes	—
297	C₈	—	4	4	no	—
298	C₈	—	4	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	8	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	4	16	no	—
309	C₂²	—	4	4	no	—
310	C₂²	—	4	4	no	—
311	C₂²	C₄₁×#424	1	4	yes	—
312	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
313	C₂²	C₄₁×#518	1	2	yes	—
314	C₂²	C₂×#131×C₄₁	1	2	yes	—
315	C₂²	C₄₁×#568	1	2	yes	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	4	8	no	—
320	C₄	C₄₁×#534	1	2	yes	—
321	C₄	C₄₁×#543	1	2	yes	—
322	C₂	—	2	4	no	—
323	C₂	—	4	2	no	—
324	C₂	—	4	2	no	—
325	C₂	C₄₁×#2315	1	1	yes	characteristic
326	C₂	C₂×C₄₁×#424	1	1	yes	characteristic
327	C₂	C₄₁×#2905	1	1	yes	characteristic
328	C₂	C₄₁×#2976	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	C₂⁵×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
4. C₄₁×#1851	C₂	semidirect
5. C₄₁×#1850	C₂	semidirect
6. C₄₁×#1798	C₂	semidirect
11. C₂×C₄⋊C₈×C₄₁	C₂²	semidirect
24. C₄₁×#336	C₂²	semidirect
134. #21933	C₄₁	direct (N × H)
161. #1851	C₂×C₄₁	semidirect
162. #1850	C₂×C₄₁	semidirect
163. #1798	C₂×C₄₁	semidirect
175. C₂×C₄⋊C₈	C₂²×C₄₁	semidirect
188. #336	C₂²×C₄₁	semidirect
189. C₄₁	#21933	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×#407	2
C₂²×C₂²⋊C₈×C₄₁	2
C₄₁×#2280	2
C₄₁×#1851	2
C₄₁×#1850	2
C₄₁×#1798	2
#21933	41

Subgroups — C41×#21933 #23798134

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#21933 #23798134