Subgroups — C₄₁×#14707 #23790908

Order 10496.

1566 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#389	C₂	1	2	yes	maximal
2	C₂×C₄₁×#382	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#382	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄³×C₄₁	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#1436	C₂	1	8	yes	maximal
6	C₄₁×C₄⋊#95	C₂	1	2	yes	maximal
7	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
9	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
10	C₂²×C₄²×C₄₁	C₂²	1	4	yes	—
11	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
14	C₄₁×#389	—	2	8	no	—
15	C₄₁×#382	—	2	4	no	—
16	C₄₁×#382	C₄	1	8	yes	complemented
17	C₂×#95×C₄₁	C₂²	1	4	yes	—
18	C₂×#95×C₄₁	C₂²	1	4	yes	—
19	C₂×#95×C₄₁	C₂²	1	8	yes	—
20	C₂×#95×C₄₁	C₂²	1	4	yes	—
21	C₄³×C₄₁	—	2	4	no	—
22	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
23	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
24	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
25	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
26	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
27	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
29	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
33	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₄₁	—	2	16	no	—
35	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	8	yes	—
36	C₂×C₄²×C₄₁	—	2	8	no	—
37	C₂×C₄²×C₄₁	—	2	2	no	—
38	C₂×C₄²×C₄₁	—	2	2	no	—
39	C₂×C₄²×C₄₁	—	2	8	no	—
40	C₂×C₄²×C₄₁	—	2	8	no	—
41	C₂×C₄²×C₄₁	—	2	2	no	—
42	C₂×C₄²×C₄₁	—	2	2	no	—
43	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
44	C₂×C₄²×C₄₁	D₄	1	4	yes	—
45	C₂×C₄²×C₄₁	D₄	1	4	yes	—
46	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
47	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
48	#95×C₄₁	—	2	8	no	—
49	#95×C₄₁	—	2	8	no	—
50	#95×C₄₁	—	2	16	no	—
51	#95×C₄₁	C₂×C₄	1	16	yes	—
52	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
53	C₂²×C₄×C₄₁	—	2	2	no	—
54	C₂²×C₄×C₄₁	—	2	2	no	—
55	C₂²×C₄×C₄₁	—	2	4	no	—
56	C₂²×C₄×C₄₁	—	2	4	no	—
57	C₂²×C₄×C₄₁	—	2	2	no	—
58	C₂²×C₄×C₄₁	—	2	2	no	—
59	C₂²×C₄×C₄₁	—	2	4	no	—
60	C₂²×C₄×C₄₁	—	2	4	no	—
61	C₂²×C₄×C₄₁	—	2	4	no	—
62	C₂²×C₄×C₄₁	—	2	8	no	—
63	C₂²×C₄×C₄₁	—	2	4	no	—
64	C₂²×C₄×C₄₁	—	2	8	no	—
65	C₂²×C₄×C₄₁	—	2	4	no	—
66	C₂²×C₄×C₄₁	—	2	4	no	—
67	C₂²×C₄×C₄₁	—	2	8	no	—
68	C₂²×C₄×C₄₁	—	2	4	no	—
69	C₂²×C₄×C₄₁	—	2	8	no	—
70	C₂²×C₄×C₄₁	C₂²×C₄	1	8	yes	—
71	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
78	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₄₁	#41	1	8	yes	—
82	C₂²×C₄×C₄₁	#41	1	8	yes	—
83	C₂²×C₄×C₄₁	#41	1	2	yes	—
84	C₂²×C₄×C₄₁	#41	1	2	yes	—
85	C₄⋊C₄×C₄₁	—	2	16	no	—
86	C₄⋊C₄×C₄₁	—	4	8	no	—
87	C₄⋊C₄×C₄₁	—	4	16	no	—
88	C₄²×C₄₁	—	2	32	no	—
89	C₄²×C₄₁	—	2	8	no	—
90	C₄²×C₄₁	—	2	8	no	—
91	C₄²×C₄₁	C₄⋊C₄	1	16	yes	complemented
92	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
93	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
94	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
95	C₂³×C₄₁	C₂²×Q₈	1	1	yes	characteristic
96	C₂³×C₄₁	C₂²×Q₈	1	1	yes	characteristic
97	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
98	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
99	C₂³×C₄₁	C₂×#41	1	4	yes	—
100	C₂³×C₄₁	C₂×#41	1	4	yes	—
101	C₂×C₄×C₄₁	—	2	8	no	—
102	C₂×C₄×C₄₁	—	2	4	no	—
103	C₂×C₄×C₄₁	—	2	4	no	—
104	C₂×C₄×C₄₁	—	2	8	no	—
105	C₂×C₄×C₄₁	—	2	8	no	—
106	C₂×C₄×C₄₁	—	2	8	no	—
107	C₂×C₄×C₄₁	—	2	8	no	—
108	C₂×C₄×C₄₁	—	2	8	no	—
109	C₂×C₄×C₄₁	—	2	4	no	—
110	C₂×C₄×C₄₁	—	2	4	no	—
111	C₂×C₄×C₄₁	—	2	4	no	—
112	C₂×C₄×C₄₁	—	2	4	no	—
113	C₂×C₄×C₄₁	—	2	8	no	—
114	C₂×C₄×C₄₁	—	2	16	no	—
115	C₂×C₄×C₄₁	—	4	4	no	—
116	C₂×C₄×C₄₁	—	4	8	no	—
117	C₂×C₄×C₄₁	—	4	4	no	—
118	C₂×C₄×C₄₁	—	4	4	no	—
119	C₂×C₄×C₄₁	—	4	8	no	—
120	C₂×C₄×C₄₁	—	4	4	no	—
121	C₂×C₄×C₄₁	—	4	8	no	—
122	C₂×C₄×C₄₁	—	4	4	no	—
123	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
124	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×C₄×C₄₁	C₄×D₄	1	4	yes	—
126	C₂×C₄×C₄₁	C₄×Q₈	1	4	yes	—
127	C₂×C₄×C₄₁	#121	1	4	yes	—
128	C₂×C₄×C₄₁	C₂²⋊Q₈	1	4	yes	—
129	C₂×C₄×C₄₁	#124	1	8	yes	—
130	C₂×C₄×C₄₁	#125	1	8	yes	—
131	C₂×C₄×C₄₁	#125	1	4	yes	—
132	C₂×C₄×C₄₁	C₄⋊Q₈	1	4	yes	—
133	#14707	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₄₁	C₂×C₄×Q₈	1	1	yes	characteristic
137	C₂²×C₄₁	C₄×#41	1	4	yes	—
138	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
139	C₂²×C₄₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
140	C₂²×C₄₁	#525	1	4	yes	—
141	C₂²×C₄₁	C₂×#124	1	2	yes	—
142	C₂²×C₄₁	C₂×#125	1	2	yes	—
143	C₂²×C₄₁	C₂×#125	1	1	yes	characteristic
144	C₂²×C₄₁	#529	1	4	yes	—
145	C₂²×C₄₁	#529	1	4	yes	—
146	C₂²×C₄₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	C₂²×C₄₁	#532	1	2	yes	—
148	C₂²×C₄₁	#533	1	2	yes	—
149	C₂²×C₄₁	#533	1	2	yes	—
150	C₂²×C₄₁	#533	1	2	yes	—
151	C₄×C₄₁	—	2	8	no	—
152	C₄×C₄₁	—	2	8	no	—
153	C₄×C₄₁	—	2	8	no	—
154	C₄×C₄₁	—	4	8	no	—
155	C₄×C₄₁	—	8	4	no	—
156	C₄×C₄₁	#389	1	8	yes	—
157	C₂×#389	C₂×C₄₁	1	2	yes	—
158	C₂×#382	C₂×C₄₁	1	1	yes	characteristic
159	C₂×#382	C₂×C₄₁	1	1	yes	characteristic
160	C₂×C₄³	C₂×C₄₁	1	1	yes	characteristic
161	#1436	C₂×C₄₁	1	8	yes	—
162	C₄⋊#95	C₂×C₄₁	1	2	yes	—
163	C₂×C₄₁	C₂×#389	1	1	yes	characteristic
164	C₂×C₄₁	#2296	1	2	yes	—
165	C₂×C₄₁	C₄×#124	1	2	yes	—
166	C₂×C₄₁	C₄×#125	1	2	yes	—
167	C₂×C₄₁	#2850	1	2	yes	—
168	C₂×C₄₁	#2851	1	2	yes	—
169	C₂×C₄₁	#2859	1	1	yes	characteristic
170	C₂×C₄₁	#2862	1	1	yes	characteristic
171	C₂×C₄₁	#2864	1	2	yes	—
172	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
173	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
174	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
175	C₂²×C₄²	C₂²×C₄₁	1	4	yes	—
176	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
177	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
178	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
179	#389	—	2	8	no	—
180	#382	—	2	4	no	—
181	#382	C₄×C₄₁	1	8	yes	complemented
182	C₂×#95	C₂²×C₄₁	1	8	yes	—
183	C₂×#95	C₂²×C₄₁	1	4	yes	—
184	C₂×#95	C₂²×C₄₁	1	4	yes	—
185	C₂×#95	C₂²×C₄₁	1	4	yes	—
186	C₄³	—	2	4	no	—
187	C₄₁	#14707	1	1	yes	characteristic, direct product factor, Sylow
188	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
189	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
190	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
191	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
192	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
193	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	4	no	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	8	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	—	2	16	no	—
201	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	8	yes	—
202	C₂×C₄²	—	2	8	no	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	8	no	—
206	C₂×C₄²	—	2	8	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
210	C₂×C₄²	D₄×C₄₁	1	4	yes	—
211	C₂×C₄²	D₄×C₄₁	1	4	yes	—
212	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
213	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
214	#95	—	2	8	no	—
215	#95	—	2	8	no	—
216	#95	—	2	16	no	—
217	#95	C₂×C₄×C₄₁	1	16	yes	—
218	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	8	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	8	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	8	no	—
236	C₂²×C₄	C₂²×C₄×C₄₁	1	8	yes	—
237	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
238	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
240	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
241	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
243	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
244	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
245	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
246	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
247	C₂²×C₄	#41×C₄₁	1	8	yes	—
248	C₂²×C₄	#41×C₄₁	1	8	yes	—
249	C₂²×C₄	#41×C₄₁	1	2	yes	—
250	C₂²×C₄	#41×C₄₁	1	2	yes	—
251	C₄⋊C₄	—	2	16	no	—
252	C₄⋊C₄	—	4	8	no	—
253	C₄⋊C₄	—	4	16	no	—
254	C₄²	—	2	32	no	—
255	C₄²	—	2	8	no	—
256	C₄²	—	2	8	no	—
257	C₄²	C₄⋊C₄×C₄₁	1	16	yes	complemented
258	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
259	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
260	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
261	C₂³	C₂²×Q₈×C₄₁	1	1	yes	characteristic
262	C₂³	C₂²×Q₈×C₄₁	1	1	yes	characteristic
263	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
264	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
265	C₂³	C₂×#41×C₄₁	1	4	yes	—
266	C₂³	C₂×#41×C₄₁	1	4	yes	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	16	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	8	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	8	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	8	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
290	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
291	C₂×C₄	C₄×D₄×C₄₁	1	4	yes	—
292	C₂×C₄	C₄×Q₈×C₄₁	1	4	yes	—
293	C₂×C₄	#121×C₄₁	1	4	yes	—
294	C₂×C₄	C₂²⋊Q₈×C₄₁	1	4	yes	—
295	C₂×C₄	#124×C₄₁	1	8	yes	—
296	C₂×C₄	#125×C₄₁	1	8	yes	—
297	C₂×C₄	#125×C₄₁	1	4	yes	—
298	C₂×C₄	C₄⋊Q₈×C₄₁	1	4	yes	—
299	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
300	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
301	C₂²	C₂×C₄×Q₈×C₄₁	1	1	yes	characteristic
302	C₂²	C₄×#41×C₄₁	1	4	yes	—
303	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
304	C₂²	C₂×C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
305	C₂²	C₄₁×#525	1	4	yes	—
306	C₂²	C₂×#124×C₄₁	1	2	yes	—
307	C₂²	C₂×#125×C₄₁	1	1	yes	characteristic
308	C₂²	C₂×#125×C₄₁	1	2	yes	—
309	C₂²	C₄₁×#529	1	4	yes	—
310	C₂²	C₄₁×#529	1	4	yes	—
311	C₂²	C₂×C₄⋊Q₈×C₄₁	1	1	yes	characteristic
312	C₂²	C₄₁×#532	1	2	yes	—
313	C₂²	C₄₁×#533	1	2	yes	—
314	C₂²	C₄₁×#533	1	2	yes	—
315	C₂²	C₄₁×#533	1	2	yes	—
316	C₄	—	2	8	no	—
317	C₄	—	2	8	no	—
318	C₄	—	2	8	no	—
319	C₄	—	4	8	no	—
320	C₄	—	8	4	no	—
321	C₄	C₄₁×#389	1	8	yes	—
322	C₂	C₂×C₄₁×#389	1	1	yes	characteristic
323	C₂	C₄₁×#2296	1	2	yes	—
324	C₂	C₄×#124×C₄₁	1	2	yes	—
325	C₂	C₄×#125×C₄₁	1	2	yes	—
326	C₂	C₄₁×#2850	1	2	yes	—
327	C₂	C₄₁×#2851	1	2	yes	—
328	C₂	C₄₁×#2859	1	1	yes	characteristic
329	C₂	C₄₁×#2862	1	1	yes	characteristic
330	C₂	C₄₁×#2864	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
16. C₄₁×#382	C₄	semidirect
91. C₄²×C₄₁	C₄⋊C₄	semidirect
133. #14707	C₄₁	direct (N × H)
181. #382	C₄×C₄₁	semidirect
187. C₄₁	#14707	direct (N × H)
257. C₄²	C₄⋊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×#389	2
C₂×C₄₁×#382	2
C₂×C₄³×C₄₁	2
C₄₁×#1436	2
C₄₁×C₄⋊#95	2
#14707	41

Subgroups — C41×#14707 #23790908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#14707 #23790908