Subgroups — C₄₁×#15153 #23791354

Order 10496.

1334 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#400	C₂	1	4	yes	maximal
2	C₂×C₄₁×#389	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#384	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1445	C₂	1	4	yes	maximal
5	C₄₁×C₄⋊#95	C₂	1	2	yes	maximal
6	C₄₁×C₄⋊#95	C₂	1	2	yes	maximal
7	C₄×#95×C₄₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
10	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
11	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄⋊C₄×C₄₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₄₁	C₂²	1	2	yes	—
15	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#400	—	2	16	no	—
17	C₄₁×#389	—	2	4	no	—
18	C₄₁×#384	—	2	4	no	—
19	C₂×#95×C₄₁	C₂²	1	2	yes	—
20	C₂×#95×C₄₁	C₂²	1	8	yes	—
21	C₂×#95×C₄₁	C₂²	1	8	yes	—
22	C₂×#95×C₄₁	C₂²	1	2	yes	—
23	C₂×#95×C₄₁	C₂²	1	2	yes	—
24	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
25	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
26	C₂³×C₄×C₄₁	C₂³	1	4	yes	—
27	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
28	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
39	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
40	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
42	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
43	C₂×C₄⋊C₄×C₄₁	Q₈	1	4	yes	—
44	C₂×C₄²×C₄₁	—	2	4	no	—
45	C₂×C₄²×C₄₁	—	2	4	no	—
46	C₂×C₄²×C₄₁	—	2	4	no	—
47	C₂×C₄²×C₄₁	—	2	4	no	—
48	C₂×C₄²×C₄₁	D₄	1	4	yes	—
49	#95×C₄₁	—	2	4	no	—
50	#95×C₄₁	—	2	16	no	—
51	#95×C₄₁	—	2	16	no	—
52	#95×C₄₁	—	2	4	no	—
53	#95×C₄₁	D₄	1	8	yes	—
54	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
55	C₂²×C₄×C₄₁	—	2	2	no	—
56	C₂²×C₄×C₄₁	—	2	2	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	2	no	—
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	—	2	4	no	—
65	C₂²×C₄×C₄₁	—	2	2	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	—	2	8	no	—
70	C₂²×C₄×C₄₁	—	2	4	no	—
71	C₂²×C₄×C₄₁	—	2	2	no	—
72	C₂²×C₄×C₄₁	—	2	4	no	—
73	C₂²×C₄×C₄₁	—	2	4	no	—
74	C₂²×C₄×C₄₁	—	2	8	no	—
75	C₂²×C₄×C₄₁	—	2	8	no	—
76	C₂²×C₄×C₄₁	—	2	4	no	—
77	C₂²×C₄×C₄₁	C₂×D₄	1	8	yes	—
78	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
83	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
84	C₂²×C₄×C₄₁	#41	1	4	yes	—
85	C₂²×C₄×C₄₁	#41	1	2	yes	—
86	C₂²×C₄×C₄₁	#41	1	4	yes	—
87	C₂²×C₄×C₄₁	#41	1	4	yes	—
88	C₄⋊C₄×C₄₁	—	2	8	no	—
89	C₄⋊C₄×C₄₁	—	4	4	no	—
90	C₄⋊C₄×C₄₁	—	4	8	no	—
91	C₄⋊C₄×C₄₁	—	4	8	no	—
92	C₄⋊C₄×C₄₁	—	4	8	no	—
93	C₄⋊C₄×C₄₁	—	4	8	no	—
94	C₄²×C₄₁	—	2	8	no	—
95	C₄²×C₄₁	—	4	8	no	—
96	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
97	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
98	C₂³×C₄₁	C₂²×Q₈	1	1	yes	characteristic
99	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
100	C₂³×C₄₁	C₂×#41	1	2	yes	—
101	C₂³×C₄₁	C₂×#41	1	2	yes	—
102	C₂³×C₄₁	C₂×#41	1	2	yes	—
103	C₂³×C₄₁	2₋¹⁺⁴	1	2	yes	—
104	C₂³×C₄₁	2₋¹⁺⁴	1	2	yes	—
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	8	no	—
110	C₂×C₄×C₄₁	—	2	8	no	—
111	C₂×C₄×C₄₁	—	2	4	no	—
112	C₂×C₄×C₄₁	—	2	8	no	—
113	C₂×C₄×C₄₁	—	2	8	no	—
114	C₂×C₄×C₄₁	—	2	8	no	—
115	C₂×C₄×C₄₁	—	2	8	no	—
116	C₂×C₄×C₄₁	—	4	4	no	—
117	C₂×C₄×C₄₁	—	4	2	no	—
118	C₂×C₄×C₄₁	—	4	2	no	—
119	C₂×C₄×C₄₁	—	4	4	no	—
120	C₂×C₄×C₄₁	—	4	4	no	—
121	C₂×C₄×C₄₁	—	4	4	no	—
122	C₂×C₄×C₄₁	—	4	4	no	—
123	C₂×C₄×C₄₁	—	4	2	no	—
124	C₂×C₄×C₄₁	—	4	4	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	16	no	—
129	C₂×C₄×C₄₁	—	4	2	no	—
130	C₂×C₄×C₄₁	—	4	8	no	—
131	C₂×C₄×C₄₁	C₂²⋊Q₈	1	4	yes	—
132	C₂×C₄×C₄₁	#123	1	4	yes	—
133	C₂×C₄×C₄₁	#124	1	4	yes	—
134	C₂×C₄×C₄₁	#124	1	4	yes	—
135	C₂×C₄×C₄₁	#125	1	8	yes	—
136	C₂×C₄×C₄₁	#126	1	8	yes	—
137	C₂×C₄×C₄₁	#127	1	4	yes	—
138	#15153	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₄₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
140	C₂²×C₄₁	C₂×#123	1	1	yes	characteristic
141	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
142	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
143	C₂²×C₄₁	C₂×#125	1	2	yes	—
144	C₂²×C₄₁	C₂×#126	1	2	yes	—
145	C₂²×C₄₁	C₂×#127	1	1	yes	characteristic
146	C₂²×C₄₁	#532	1	4	yes	—
147	C₂²×C₄₁	#536	1	1	yes	characteristic
148	C₂²×C₄₁	#536	1	1	yes	characteristic
149	C₂²×C₄₁	#541	1	4	yes	—
150	C₂²×C₄₁	#547	1	2	yes	—
151	C₂²×C₄₁	#547	1	4	yes	—
152	C₂²×C₄₁	D₄×Q₈	1	2	yes	—
153	C₂²×C₄₁	#552	1	4	yes	—
154	C₂²×C₄₁	#556	1	2	yes	—
155	C₂²×C₄₁	#557	1	2	yes	—
156	C₄×C₄₁	—	2	4	no	—
157	C₄×C₄₁	—	2	4	no	—
158	C₄×C₄₁	—	4	8	no	—
159	C₄×C₄₁	—	4	4	no	—
160	C₄×C₄₁	—	8	4	no	—
161	C₄×C₄₁	—	8	2	no	—
162	C₄×C₄₁	#400	1	8	yes	complemented
163	C₂×#400	C₂×C₄₁	1	4	yes	—
164	C₂×#389	C₂×C₄₁	1	1	yes	characteristic
165	C₂×#384	C₂×C₄₁	1	1	yes	characteristic
166	#1445	C₂×C₄₁	1	4	yes	—
167	C₄⋊#95	C₂×C₄₁	1	2	yes	—
168	C₄⋊#95	C₂×C₄₁	1	2	yes	—
169	C₄×#95	C₂×C₄₁	1	1	yes	characteristic
170	C₂×C₄₁	C₂×#400	1	1	yes	characteristic
171	C₂×C₄₁	#2496	1	4	yes	—
172	C₂×C₄₁	#2501	1	1	yes	characteristic
173	C₂×C₄₁	#2503	1	1	yes	characteristic
174	C₂×C₄₁	#2506	1	4	yes	—
175	C₂×C₄₁	#2542	1	1	yes	characteristic
176	C₂×C₄₁	#2568	1	1	yes	characteristic
177	C₂×C₄₁	#2642	1	1	yes	characteristic
178	C₂×C₄₁	#2858	1	1	yes	characteristic
179	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
180	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
181	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
182	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
183	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
184	C₂²×C₄⋊C₄	C₂²×C₄₁	1	1	yes	characteristic
185	C₂²×C₄²	C₂²×C₄₁	1	2	yes	—
186	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
187	#400	—	2	16	no	—
188	#389	—	2	4	no	—
189	#384	—	2	4	no	—
190	C₂×#95	C₂²×C₄₁	1	8	yes	—
191	C₂×#95	C₂²×C₄₁	1	8	yes	—
192	C₂×#95	C₂²×C₄₁	1	2	yes	—
193	C₂×#95	C₂²×C₄₁	1	2	yes	—
194	C₂×#95	C₂²×C₄₁	1	2	yes	—
195	C₄₁	#15153	1	1	yes	characteristic, direct product factor, Sylow
196	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
197	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
198	C₂³×C₄	C₂³×C₄₁	1	4	yes	—
199	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
200	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
201	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
202	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	4	no	—
210	C₂×C₄⋊C₄	—	2	8	no	—
211	C₂×C₄⋊C₄	—	2	8	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	8	no	—
214	C₂×C₄⋊C₄	—	2	8	no	—
215	C₂×C₄⋊C₄	Q₈×C₄₁	1	4	yes	—
216	C₂×C₄²	—	2	4	no	—
217	C₂×C₄²	—	2	4	no	—
218	C₂×C₄²	—	2	4	no	—
219	C₂×C₄²	—	2	4	no	—
220	C₂×C₄²	D₄×C₄₁	1	4	yes	—
221	#95	—	2	4	no	—
222	#95	—	2	16	no	—
223	#95	—	2	16	no	—
224	#95	—	2	4	no	—
225	#95	D₄×C₄₁	1	8	yes	—
226	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	2	8	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	8	no	—
247	C₂²×C₄	—	2	8	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	C₂×D₄×C₄₁	1	8	yes	—
250	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
252	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
254	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
255	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
256	C₂²×C₄	#41×C₄₁	1	4	yes	—
257	C₂²×C₄	#41×C₄₁	1	2	yes	—
258	C₂²×C₄	#41×C₄₁	1	4	yes	—
259	C₂²×C₄	#41×C₄₁	1	4	yes	—
260	C₄⋊C₄	—	2	8	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	—	4	8	no	—
263	C₄⋊C₄	—	4	8	no	—
264	C₄⋊C₄	—	4	8	no	—
265	C₄⋊C₄	—	4	8	no	—
266	C₄²	—	2	8	no	—
267	C₄²	—	4	8	no	—
268	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
269	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
270	C₂³	C₂²×Q₈×C₄₁	1	1	yes	characteristic
271	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
272	C₂³	C₂×#41×C₄₁	1	2	yes	—
273	C₂³	C₂×#41×C₄₁	1	2	yes	—
274	C₂³	C₂×#41×C₄₁	1	2	yes	—
275	C₂³	2₋¹⁺⁴×C₄₁	1	2	yes	—
276	C₂³	2₋¹⁺⁴×C₄₁	1	2	yes	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	8	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	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	4	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	16	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	8	no	—
303	C₂×C₄	C₂²⋊Q₈×C₄₁	1	4	yes	—
304	C₂×C₄	#123×C₄₁	1	4	yes	—
305	C₂×C₄	#124×C₄₁	1	4	yes	—
306	C₂×C₄	#124×C₄₁	1	4	yes	—
307	C₂×C₄	#125×C₄₁	1	8	yes	—
308	C₂×C₄	#126×C₄₁	1	8	yes	—
309	C₂×C₄	#127×C₄₁	1	4	yes	—
310	C₂²	C₂×C₂²⋊Q₈×C₄₁	1	1	yes	characteristic
311	C₂²	C₂×#123×C₄₁	1	1	yes	characteristic
312	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
313	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
314	C₂²	C₂×#125×C₄₁	1	2	yes	—
315	C₂²	C₂×#126×C₄₁	1	2	yes	—
316	C₂²	C₂×#127×C₄₁	1	1	yes	characteristic
317	C₂²	C₄₁×#532	1	4	yes	—
318	C₂²	C₄₁×#536	1	1	yes	characteristic
319	C₂²	C₄₁×#536	1	1	yes	characteristic
320	C₂²	C₄₁×#541	1	4	yes	—
321	C₂²	C₄₁×#547	1	2	yes	—
322	C₂²	C₄₁×#547	1	4	yes	—
323	C₂²	D₄×Q₈×C₄₁	1	2	yes	—
324	C₂²	C₄₁×#552	1	4	yes	—
325	C₂²	C₄₁×#556	1	2	yes	—
326	C₂²	C₄₁×#557	1	2	yes	—
327	C₄	—	2	4	no	—
328	C₄	—	2	4	no	—
329	C₄	—	4	4	no	—
330	C₄	—	4	8	no	—
331	C₄	—	8	4	no	—
332	C₄	—	8	2	no	—
333	C₄	C₄₁×#400	1	8	yes	complemented
334	C₂	C₂×C₄₁×#400	1	1	yes	characteristic
335	C₂	C₄₁×#2496	1	4	yes	—
336	C₂	C₄₁×#2501	1	1	yes	characteristic
337	C₂	C₄₁×#2503	1	1	yes	characteristic
338	C₂	C₄₁×#2506	1	4	yes	—
339	C₂	C₄₁×#2542	1	1	yes	characteristic
340	C₂	C₄₁×#2568	1	1	yes	characteristic
341	C₂	C₄₁×#2642	1	1	yes	characteristic
342	C₂	C₄₁×#2858	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
138. #15153	C₄₁	direct (N × H)
162. C₄×C₄₁	#400	semidirect
195. C₄₁	#15153	direct (N × H)
333. C₄	C₄₁×#400	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄₁×#400	2
C₂×C₄₁×#389	2
C₂×C₄₁×#384	2
C₄₁×#1445	2
C₄₁×C₄⋊#95	2
C₄×#95×C₄₁	2
#15153	41

Subgroups — C41×#15153 #23791354

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#15153 #23791354