Subgroups — C₄₁×#20857 #23797058

Order 10496.

1330 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×#519	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2911	C₂	1	2	yes	maximal
3	C₂²×#101×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#2310	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#1857	C₂	1	8	yes	maximal, complemented
6	C₄₁×#1733	C₂	1	2	yes	maximal
7	C₂³×Q₈×C₄₁	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
9	C₄₁×#519	—	2	2	no	—
10	C₄₁×#519	—	2	4	no	—
11	C₄₁×#519	C₂²	1	4	yes	—
12	C₂×C₄×Q₈×C₄₁	—	2	4	no	—
13	C₂×#117×C₄₁	—	2	2	no	—
14	C₂×#117×C₄₁	C₂²	1	2	yes	—
15	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×#117×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#428	—	2	4	no	—
18	C₄₁×#428	C₂²	1	8	yes	—
19	C₄₁×#427	—	2	4	no	—
20	C₂×#101×C₄₁	C₄	1	8	yes	complemented
21	C₂×#101×C₄₁	C₂²	1	4	yes	—
22	C₄₁×#391	—	2	2	no	—
23	C₄₁×#391	C₂²	1	4	yes	—
24	C₄₁×#381	—	2	2	no	—
25	C₄₁×#339	C₂²	1	8	yes	—
26	C₂²×Q₈×C₄₁	—	2	2	no	—
27	C₂²×Q₈×C₄₁	—	2	2	no	—
28	C₂²×Q₈×C₄₁	—	2	4	no	—
29	C₂²×Q₈×C₄₁	C₂×C₄	1	4	yes	—
30	C₂²×Q₈×C₄₁	D₄	1	2	yes	—
31	C₂²×Q₈×C₄₁	Q₈	1	4	yes	—
32	C₂²×Q₈×C₄₁	C₂³	1	2	yes	—
33	C₂³×C₄×C₄₁	D₄	1	2	yes	—
34	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₄₁	—	2	2	no	—
36	C₂×M₁₆×C₄₁	—	2	4	no	—
37	C₂×M₁₆×C₄₁	C₂×C₄	1	8	yes	—
38	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
39	C₂²×C₈×C₄₁	—	2	2	no	—
40	C₄×Q₈×C₄₁	—	2	16	no	—
41	C₄×Q₈×C₄₁	—	2	16	no	—
42	#117×C₄₁	—	2	8	no	—
43	#117×C₄₁	—	2	8	no	—
44	#117×C₄₁	—	2	2	no	—
45	#117×C₄₁	—	2	2	no	—
46	#117×C₄₁	—	2	4	no	—
47	#117×C₄₁	—	2	2	no	—
48	#117×C₄₁	—	2	2	no	—
49	#117×C₄₁	D₄	1	4	yes	—
50	#117×C₄₁	C₂³	1	4	yes	—
51	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
52	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₄₁	D₄	1	4	yes	—
54	C₂×C₄⋊C₄×C₄₁	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
56	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
58	C₂×C₄²×C₄₁	—	2	4	no	—
59	C₂×C₄²×C₄₁	—	2	2	no	—
60	C₂×C₄²×C₄₁	—	2	1	no	—
61	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
62	#107×C₄₁	—	2	8	no	—
63	#106×C₄₁	—	2	8	no	—
64	#101×C₄₁	C₂×C₄	1	16	yes	complemented
65	#95×C₄₁	—	2	8	no	—
66	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
67	C₂×Q₈×C₄₁	—	2	8	no	—
68	C₂×Q₈×C₄₁	—	2	8	no	—
69	C₂×Q₈×C₄₁	—	2	4	no	—
70	C₂×Q₈×C₄₁	—	2	8	no	—
71	C₂×Q₈×C₄₁	—	2	8	no	—
72	C₂×Q₈×C₄₁	—	2	4	no	—
73	C₂×Q₈×C₄₁	—	2	4	no	—
74	C₂×Q₈×C₄₁	—	2	4	no	—
75	C₂×Q₈×C₄₁	C₄⋊C₄	1	8	yes	—
76	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
77	C₂×Q₈×C₄₁	C₂×Q₈	1	4	yes	—
78	C₂²×C₄×C₄₁	—	2	2	no	—
79	C₂²×C₄×C₄₁	—	2	2	no	—
80	C₂²×C₄×C₄₁	—	2	4	no	—
81	C₂²×C₄×C₄₁	—	2	4	no	—
82	C₂²×C₄×C₄₁	—	2	2	no	—
83	C₂²×C₄×C₄₁	—	2	2	no	—
84	C₂²×C₄×C₄₁	C₄⋊C₄	1	8	yes	—
85	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
87	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
91	C₂²×C₄×C₄₁	#41	1	4	yes	—
92	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
93	M₁₆×C₄₁	—	2	8	no	—
94	M₁₆×C₄₁	—	2	4	no	—
95	M₁₆×C₄₁	C₂²×C₄	1	8	yes	—
96	C₂×C₈×C₄₁	—	2	8	no	—
97	C₂×C₈×C₄₁	—	2	4	no	—
98	C₄⋊C₄×C₄₁	—	2	16	no	—
99	C₄⋊C₄×C₄₁	—	2	4	no	—
100	C₄⋊C₄×C₄₁	—	2	4	no	—
101	C₄⋊C₄×C₄₁	—	4	2	no	—
102	C₄⋊C₄×C₄₁	C₂×D₄	1	8	yes	—
103	C₂²⋊C₄×C₄₁	—	2	4	no	—
104	C₂²⋊C₄×C₄₁	—	2	4	no	—
105	C₂²⋊C₄×C₄₁	—	4	1	no	—
106	C₂²⋊C₄×C₄₁	—	4	1	no	—
107	C₂²⋊C₄×C₄₁	—	4	2	no	—
108	C₄²×C₄₁	—	2	16	no	—
109	C₄²×C₄₁	—	2	4	no	—
110	C₄²×C₄₁	—	2	4	no	—
111	C₄²×C₄₁	—	2	4	no	—
112	C₄²×C₄₁	—	4	2	no	—
113	C₂³×C₄₁	—	4	2	no	—
114	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
115	C₂³×C₄₁	#120	1	1	yes	characteristic
116	C₂³×C₄₁	#120	1	1	yes	characteristic
117	C₂³×C₄₁	#123	1	2	yes	—
118	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
119	Q₈×C₄₁	—	2	8	no	—
120	Q₈×C₄₁	—	2	8	no	—
121	Q₈×C₄₁	—	2	16	no	—
122	C₂×C₄×C₄₁	—	2	4	no	—
123	C₂×C₄×C₄₁	—	2	4	no	—
124	C₂×C₄×C₄₁	—	2	8	no	—
125	C₂×C₄×C₄₁	—	2	8	no	—
126	C₂×C₄×C₄₁	—	2	8	no	—
127	C₂×C₄×C₄₁	—	4	8	no	—
128	C₂×C₄×C₄₁	—	4	4	no	—
129	C₂×C₄×C₄₁	—	4	4	no	—
130	C₂×C₄×C₄₁	—	4	2	no	—
131	C₂×C₄×C₄₁	—	4	2	no	—
132	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
133	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
134	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
135	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
136	C₂×C₄×C₄₁	#120	1	2	yes	—
137	C₂×C₄×C₄₁	C₂²⋊Q₈	1	4	yes	—
138	C₂×C₄×C₄₁	C₂²⋊Q₈	1	4	yes	—
139	C₂×C₄×C₄₁	#123	1	2	yes	—
140	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
142	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
143	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
144	C₈×C₄₁	—	2	8	no	—
145	#20857	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂²×C₄₁	—	2	2	no	—
147	C₂²×C₄₁	—	2	2	no	—
148	C₂²×C₄₁	—	2	2	no	—
149	C₂²×C₄₁	—	4	2	no	—
150	C₂²×C₄₁	—	4	2	no	—
151	C₂²×C₄₁	#385	1	2	yes	—
152	C₂²×C₄₁	#385	1	2	yes	—
153	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₂²×C₄₁	C₂×#120	1	1	yes	characteristic
155	C₂²×C₄₁	C₂×#123	1	1	yes	characteristic
156	C₄×C₄₁	—	2	8	no	—
157	C₄×C₄₁	—	4	8	no	—
158	C₄×C₄₁	—	8	2	no	—
159	C₄×C₄₁	—	8	2	no	—
160	C₄×C₄₁	#385	1	4	yes	—
161	C₄×C₄₁	C₂×C₄×D₄	1	2	yes	—
162	C₄×C₄₁	C₂×C₂²⋊Q₈	1	2	yes	—
163	C₂×#519	C₂×C₄₁	1	1	yes	characteristic
164	#2911	C₂×C₄₁	1	2	yes	—
165	C₂²×#101	C₂×C₄₁	1	1	yes	characteristic
166	#2310	C₂×C₄₁	1	1	yes	characteristic
167	#1857	C₂×C₄₁	1	8	yes	complemented
168	#1733	C₂×C₄₁	1	2	yes	—
169	C₂×C₄₁	—	2	1	no	—
170	C₂×C₄₁	—	2	1	no	—
171	C₂×C₄₁	—	2	2	no	—
172	C₂×C₄₁	—	2	2	no	—
173	C₂×C₄₁	C₂×#385	1	1	yes	characteristic
174	C₂×C₄₁	#3020	1	2	yes	—
175	C₂³×Q₈	C₂²×C₄₁	1	1	yes	characteristic
176	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
177	#519	—	2	2	no	—
178	#519	—	2	4	no	—
179	#519	C₂²×C₄₁	1	4	yes	—
180	C₂×C₄×Q₈	—	2	4	no	—
181	C₂×#117	—	2	2	no	—
182	C₂×#117	C₂²×C₄₁	1	2	yes	—
183	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
184	C₂×#117	C₂²×C₄₁	1	1	yes	characteristic
185	#428	—	2	4	no	—
186	#428	C₂²×C₄₁	1	8	yes	—
187	#427	—	2	4	no	—
188	C₂×#101	C₄×C₄₁	1	8	yes	complemented
189	C₂×#101	C₂²×C₄₁	1	4	yes	—
190	#391	—	2	2	no	—
191	#391	C₂²×C₄₁	1	4	yes	—
192	#381	—	2	2	no	—
193	#339	C₂²×C₄₁	1	8	yes	—
194	C₄₁	#20857	1	1	yes	characteristic, direct product factor, Sylow
195	C₂²×Q₈	—	2	2	no	—
196	C₂²×Q₈	—	2	2	no	—
197	C₂²×Q₈	—	2	4	no	—
198	C₂²×Q₈	C₂×C₄×C₄₁	1	4	yes	—
199	C₂²×Q₈	D₄×C₄₁	1	2	yes	—
200	C₂²×Q₈	Q₈×C₄₁	1	4	yes	—
201	C₂²×Q₈	C₂³×C₄₁	1	2	yes	—
202	C₂³×C₄	D₄×C₄₁	1	2	yes	—
203	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
204	C₂×M₁₆	—	2	2	no	—
205	C₂×M₁₆	—	2	4	no	—
206	C₂×M₁₆	C₂×C₄×C₄₁	1	8	yes	—
207	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
208	C₂²×C₈	—	2	2	no	—
209	C₄×Q₈	—	2	16	no	—
210	C₄×Q₈	—	2	16	no	—
211	#117	—	2	8	no	—
212	#117	—	2	8	no	—
213	#117	—	2	2	no	—
214	#117	—	2	2	no	—
215	#117	—	2	4	no	—
216	#117	—	2	2	no	—
217	#117	—	2	2	no	—
218	#117	D₄×C₄₁	1	4	yes	—
219	#117	C₂³×C₄₁	1	4	yes	—
220	C₂×C₄⋊C₄	—	2	4	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	D₄×C₄₁	1	4	yes	—
223	C₂×C₄⋊C₄	C₂³×C₄₁	1	2	yes	—
224	C₂×C₂²⋊C₄	—	2	2	no	—
225	C₂×C₂²⋊C₄	—	2	1	no	—
226	C₂×C₂²⋊C₄	—	2	1	no	—
227	C₂×C₄²	—	2	4	no	—
228	C₂×C₄²	—	2	2	no	—
229	C₂×C₄²	—	2	1	no	—
230	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
231	#107	—	2	8	no	—
232	#106	—	2	8	no	—
233	#101	C₂×C₄×C₄₁	1	16	yes	complemented
234	#95	—	2	8	no	—
235	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
236	C₂×Q₈	—	2	8	no	—
237	C₂×Q₈	—	2	8	no	—
238	C₂×Q₈	—	2	4	no	—
239	C₂×Q₈	—	2	8	no	—
240	C₂×Q₈	—	2	8	no	—
241	C₂×Q₈	—	2	4	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	2	4	no	—
244	C₂×Q₈	C₄⋊C₄×C₄₁	1	8	yes	—
245	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
246	C₂×Q₈	C₂×Q₈×C₄₁	1	4	yes	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	C₄⋊C₄×C₄₁	1	8	yes	—
254	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
255	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
256	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
258	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
259	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
260	C₂²×C₄	#41×C₄₁	1	4	yes	—
261	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
262	M₁₆	—	2	8	no	—
263	M₁₆	—	2	4	no	—
264	M₁₆	C₂²×C₄×C₄₁	1	8	yes	—
265	C₂×C₈	—	2	8	no	—
266	C₂×C₈	—	2	4	no	—
267	C₄⋊C₄	—	2	16	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	C₂×D₄×C₄₁	1	8	yes	—
272	C₂²⋊C₄	—	2	4	no	—
273	C₂²⋊C₄	—	2	4	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	2	16	no	—
278	C₄²	—	2	4	no	—
279	C₄²	—	2	4	no	—
280	C₄²	—	2	4	no	—
281	C₄²	—	4	2	no	—
282	C₂³	—	4	2	no	—
283	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
284	C₂³	#120×C₄₁	1	1	yes	characteristic
285	C₂³	#120×C₄₁	1	1	yes	characteristic
286	C₂³	#123×C₄₁	1	2	yes	—
287	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
288	Q₈	—	2	8	no	—
289	Q₈	—	2	8	no	—
290	Q₈	—	2	16	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	2	8	no	—
295	C₂×C₄	—	2	8	no	—
296	C₂×C₄	—	4	8	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
302	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
303	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
304	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
305	C₂×C₄	#120×C₄₁	1	2	yes	—
306	C₂×C₄	C₂²⋊Q₈×C₄₁	1	4	yes	—
307	C₂×C₄	C₂²⋊Q₈×C₄₁	1	4	yes	—
308	C₂×C₄	#123×C₄₁	1	2	yes	—
309	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
311	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
312	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
313	C₈	—	2	8	no	—
314	C₂²	—	2	2	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	2	no	—
319	C₂²	C₄₁×#385	1	2	yes	—
320	C₂²	C₄₁×#385	1	2	yes	—
321	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
322	C₂²	C₂×#120×C₄₁	1	1	yes	characteristic
323	C₂²	C₂×#123×C₄₁	1	1	yes	characteristic
324	C₄	—	2	8	no	—
325	C₄	—	4	8	no	—
326	C₄	—	8	2	no	—
327	C₄	—	8	2	no	—
328	C₄	C₄₁×#385	1	4	yes	—
329	C₄	C₂×C₄×D₄×C₄₁	1	2	yes	—
330	C₄	C₂×C₂²⋊Q₈×C₄₁	1	2	yes	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	2	2	no	—
334	C₂	—	2	2	no	—
335	C₂	C₂×C₄₁×#385	1	1	yes	characteristic
336	C₂	C₄₁×#3020	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	C₂×#120
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 ◁ C₂²×C₄₁ ◁ C₂³×C₄×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₄₁×#1857	C₂	semidirect
20. C₂×#101×C₄₁	C₄	semidirect
64. #101×C₄₁	C₂×C₄	semidirect
145. #20857	C₄₁	direct (N × H)
167. #1857	C₂×C₄₁	semidirect
188. C₂×#101	C₄×C₄₁	semidirect
194. C₄₁	#20857	direct (N × H)
233. #101	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×#519	2
C₄₁×#2911	2
C₂²×#101×C₄₁	2
C₄₁×#2310	2
C₄₁×#1857	2
C₄₁×#1733	2
#20857	41

Subgroups — C41×#20857 #23797058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20857 #23797058