Subgroups — C₄₁×#14339 #23790540

Order 10496.

2142 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#381	C₂	1	2	yes	maximal
3	C₂×C₄₁×#380	C₂	1	2	yes	maximal
4	C₄₁×#1441	C₂	1	8	yes	maximal, complemented
5	C₄₁×C₂²⋊#95	C₂	1	2	yes	maximal
6	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
7	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
8	C₂⁴×C₄×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄⋊C₄×C₄₁	—	2	8	no	—
10	C₂²×C₄⋊C₄×C₄₁	C₄	1	8	yes	complemented
11	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
12	C₂²×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
13	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	4	yes	—
14	C₄₁×#381	—	2	8	no	—
15	C₄₁×#380	—	2	8	no	—
16	C₂×#95×C₄₁	C₂²	1	4	yes	—
17	C₂×#95×C₄₁	C₂²	1	8	yes	—
18	C₂×#95×C₄₁	C₂²	1	8	yes	—
19	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
20	C₂³×C₄×C₄₁	—	2	4	no	—
21	C₂³×C₄×C₄₁	—	2	4	no	—
22	C₂³×C₄×C₄₁	—	2	4	no	—
23	C₂³×C₄×C₄₁	—	4	4	no	—
24	C₂³×C₄×C₄₁	—	4	4	no	—
25	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
27	C₂³×C₄×C₄₁	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₄₁	D₄	1	4	yes	—
29	C₂³×C₄×C₄₁	D₄	1	8	yes	—
30	C₂³×C₄×C₄₁	Q₈	1	4	yes	—
31	C₂³×C₄×C₄₁	C₂³	1	8	yes	—
32	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
33	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
34	C₂³×C₄×C₄₁	C₂³	1	2	yes	—
35	C₂×C₄⋊C₄×C₄₁	—	2	32	no	—
36	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
37	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
38	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
39	C₂×C₄⋊C₄×C₄₁	—	4	8	no	—
40	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	16	yes	—
41	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
42	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
43	C₂×C₂²⋊C₄×C₄₁	—	2	8	no	—
44	#95×C₄₁	—	2	8	no	—
45	#95×C₄₁	—	2	16	no	—
46	#95×C₄₁	—	4	8	no	—
47	C₂⁴×C₄₁	—	2	4	no	—
48	C₂⁴×C₄₁	—	2	4	no	—
49	C₂⁴×C₄₁	C₂²×C₄	1	2	yes	—
50	C₂⁴×C₄₁	C₂×D₄	1	4	yes	—
51	C₂⁴×C₄₁	C₂×D₄	1	2	yes	—
52	C₂⁴×C₄₁	C₂×Q₈	1	2	yes	—
53	C₂⁴×C₄₁	#41	1	4	yes	—
54	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
55	C₂²×C₄×C₄₁	—	2	8	no	—
56	C₂²×C₄×C₄₁	—	2	8	no	—
57	C₂²×C₄×C₄₁	—	2	16	no	—
58	C₂²×C₄×C₄₁	—	2	16	no	—
59	C₂²×C₄×C₄₁	—	2	4	no	—
60	C₂²×C₄×C₄₁	—	2	4	no	—
61	C₂²×C₄×C₄₁	—	2	8	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	8	no	—
66	C₂²×C₄×C₄₁	—	2	8	no	—
67	C₂²×C₄×C₄₁	—	2	8	no	—
68	C₂²×C₄×C₄₁	—	2	16	no	—
69	C₂²×C₄×C₄₁	—	2	16	no	—
70	C₂²×C₄×C₄₁	—	4	8	no	—
71	C₂²×C₄×C₄₁	—	4	8	no	—
72	C₂²×C₄×C₄₁	—	4	16	no	—
73	C₂²×C₄×C₄₁	—	4	16	no	—
74	C₂²×C₄×C₄₁	C₂²⋊C₄	1	16	yes	—
75	C₂²×C₄×C₄₁	C₄⋊C₄	1	16	yes	complemented
76	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₄₁	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₄₁	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₄₁	C₂×Q₈	1	4	yes	—
83	C₄⋊C₄×C₄₁	—	2	32	no	—
84	C₄⋊C₄×C₄₁	—	4	16	no	—
85	C₂²⋊C₄×C₄₁	—	4	16	no	—
86	C₂³×C₄₁	—	2	4	no	—
87	C₂³×C₄₁	—	2	4	no	—
88	C₂³×C₄₁	—	2	8	no	—
89	C₂³×C₄₁	—	2	8	no	—
90	C₂³×C₄₁	—	2	8	no	—
91	C₂³×C₄₁	—	2	16	no	—
92	C₂³×C₄₁	—	2	8	no	—
93	C₂³×C₄₁	C₂×C₂²⋊C₄	1	4	yes	—
94	C₂³×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
95	C₂³×C₄₁	#117	1	4	yes	—
96	C₂³×C₄₁	#121	1	8	yes	—
97	C₂³×C₄₁	C₂²⋊Q₈	1	8	yes	—
98	C₂³×C₄₁	C₂³×C₄	1	1	yes	characteristic
99	C₂³×C₄₁	C₂²×D₄	1	2	yes	—
100	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
101	C₂³×C₄₁	C₂²×Q₈	1	1	yes	characteristic
102	C₂³×C₄₁	C₂×#41	1	2	yes	—
103	C₂³×C₄₁	H(5,𝔽₂)	1	4	yes	—
104	C₂³×C₄₁	2₋¹⁺⁴	1	4	yes	—
105	C₂×C₄×C₄₁	—	2	4	no	—
106	C₂×C₄×C₄₁	—	2	4	no	—
107	C₂×C₄×C₄₁	—	2	8	no	—
108	C₂×C₄×C₄₁	—	2	8	no	—
109	C₂×C₄×C₄₁	—	2	8	no	—
110	C₂×C₄×C₄₁	—	2	8	no	—
111	C₂×C₄×C₄₁	—	2	8	no	—
112	C₂×C₄×C₄₁	—	2	32	no	—
113	C₂×C₄×C₄₁	—	2	4	no	—
114	C₂×C₄×C₄₁	—	2	8	no	—
115	C₂×C₄×C₄₁	—	4	16	no	—
116	C₂×C₄×C₄₁	—	4	16	no	—
117	C₂×C₄×C₄₁	—	4	8	no	—
118	C₂×C₄×C₄₁	—	4	8	no	—
119	C₂×C₄×C₄₁	—	4	8	no	—
120	C₂×C₄×C₄₁	—	4	8	no	—
121	C₂×C₄×C₄₁	—	4	16	no	—
122	C₂×C₄×C₄₁	—	4	16	no	—
123	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	8	yes	—
124	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	8	yes	—
125	#14339	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
126	C₂²×C₄₁	—	2	4	no	—
127	C₂²×C₄₁	—	2	4	no	—
128	C₂²×C₄₁	—	2	8	no	—
129	C₂²×C₄₁	—	2	8	no	—
130	C₂²×C₄₁	—	2	16	no	—
131	C₂²×C₄₁	—	2	16	no	—
132	C₂²×C₄₁	#380	1	8	yes	—
133	C₂²×C₄₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₄₁	C₂×#117	1	1	yes	characteristic
136	C₂²×C₄₁	#518	1	2	yes	—
137	C₂²×C₄₁	#519	1	2	yes	—
138	C₂²×C₄₁	C₂×#121	1	2	yes	—
139	C₂²×C₄₁	C₂×C₂²⋊Q₈	1	2	yes	—
140	C₂²×C₄₁	#534	1	4	yes	—
141	C₂²×C₄₁	#536	1	4	yes	—
142	C₂²×C₄₁	#537	1	4	yes	—
143	C₂²×C₄₁	#539	1	8	yes	—
144	C₂²×C₄₁	#544	1	4	yes	—
145	C₄×C₄₁	—	2	8	no	—
146	C₄×C₄₁	—	2	8	no	—
147	C₄×C₄₁	—	4	4	no	—
148	C₄×C₄₁	—	8	8	no	—
149	C₂³×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
150	C₂×#381	C₂×C₄₁	1	2	yes	—
151	C₂×#380	C₂×C₄₁	1	2	yes	—
152	#1441	C₂×C₄₁	1	8	yes	complemented
153	C₂²⋊#95	C₂×C₄₁	1	2	yes	—
154	C₂×C₄₁	—	2	8	no	—
155	C₂×C₄₁	C₂×#380	1	1	yes	characteristic
156	C₂×C₄₁	#2308	1	1	yes	characteristic
157	C₂×C₄₁	#2310	1	1	yes	characteristic
158	C₂×C₄₁	#2313	1	2	yes	—
159	C₂×C₄₁	#2316	1	2	yes	—
160	C₂×C₄₁	#2618	1	4	yes	—
161	C₂×C₄₁	#2627	1	4	yes	—
162	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
163	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
164	C₂⁴×C₄	C₂²×C₄₁	1	1	yes	characteristic
165	C₂²×C₄⋊C₄	—	2	8	no	—
166	C₂²×C₄⋊C₄	C₄×C₄₁	1	8	yes	complemented
167	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
168	C₂²×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
169	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	4	yes	—
170	#381	—	2	8	no	—
171	#380	—	2	8	no	—
172	C₂×#95	C₂²×C₄₁	1	4	yes	—
173	C₂×#95	C₂²×C₄₁	1	8	yes	—
174	C₂×#95	C₂²×C₄₁	1	8	yes	—
175	C₄₁	#14339	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
177	C₂³×C₄	—	2	4	no	—
178	C₂³×C₄	—	2	4	no	—
179	C₂³×C₄	—	2	4	no	—
180	C₂³×C₄	—	4	4	no	—
181	C₂³×C₄	—	4	4	no	—
182	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
183	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
184	C₂³×C₄	C₂×C₄×C₄₁	1	4	yes	—
185	C₂³×C₄	D₄×C₄₁	1	4	yes	—
186	C₂³×C₄	D₄×C₄₁	1	8	yes	—
187	C₂³×C₄	Q₈×C₄₁	1	4	yes	—
188	C₂³×C₄	C₂³×C₄₁	1	8	yes	—
189	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
190	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
191	C₂³×C₄	C₂³×C₄₁	1	2	yes	—
192	C₂×C₄⋊C₄	—	2	32	no	—
193	C₂×C₄⋊C₄	—	2	8	no	—
194	C₂×C₄⋊C₄	—	2	8	no	—
195	C₂×C₄⋊C₄	—	2	8	no	—
196	C₂×C₄⋊C₄	—	4	8	no	—
197	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	16	yes	—
198	C₂×C₂²⋊C₄	—	2	8	no	—
199	C₂×C₂²⋊C₄	—	2	8	no	—
200	C₂×C₂²⋊C₄	—	2	8	no	—
201	#95	—	2	8	no	—
202	#95	—	2	16	no	—
203	#95	—	4	8	no	—
204	C₂⁴	—	2	4	no	—
205	C₂⁴	—	2	4	no	—
206	C₂⁴	C₂²×C₄×C₄₁	1	2	yes	—
207	C₂⁴	C₂×D₄×C₄₁	1	4	yes	—
208	C₂⁴	C₂×D₄×C₄₁	1	2	yes	—
209	C₂⁴	C₂×Q₈×C₄₁	1	2	yes	—
210	C₂⁴	#41×C₄₁	1	4	yes	—
211	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
212	C₂²×C₄	—	2	8	no	—
213	C₂²×C₄	—	2	8	no	—
214	C₂²×C₄	—	2	16	no	—
215	C₂²×C₄	—	2	16	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	2	8	no	—
223	C₂²×C₄	—	2	8	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	16	no	—
226	C₂²×C₄	—	2	16	no	—
227	C₂²×C₄	—	4	8	no	—
228	C₂²×C₄	—	4	8	no	—
229	C₂²×C₄	—	4	16	no	—
230	C₂²×C₄	—	4	16	no	—
231	C₂²×C₄	C₂²⋊C₄×C₄₁	1	16	yes	—
232	C₂²×C₄	C₄⋊C₄×C₄₁	1	16	yes	complemented
233	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
234	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
235	C₂²×C₄	C₂²×C₄×C₄₁	1	4	yes	—
236	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
237	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
238	C₂²×C₄	C₂×D₄×C₄₁	1	4	yes	—
239	C₂²×C₄	C₂×Q₈×C₄₁	1	4	yes	—
240	C₄⋊C₄	—	2	32	no	—
241	C₄⋊C₄	—	4	16	no	—
242	C₂²⋊C₄	—	4	16	no	—
243	C₂³	—	2	4	no	—
244	C₂³	—	2	4	no	—
245	C₂³	—	2	8	no	—
246	C₂³	—	2	8	no	—
247	C₂³	—	2	8	no	—
248	C₂³	—	2	16	no	—
249	C₂³	—	2	8	no	—
250	C₂³	C₂×C₂²⋊C₄×C₄₁	1	4	yes	—
251	C₂³	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
252	C₂³	#117×C₄₁	1	4	yes	—
253	C₂³	#121×C₄₁	1	8	yes	—
254	C₂³	C₂²⋊Q₈×C₄₁	1	8	yes	—
255	C₂³	C₂³×C₄×C₄₁	1	1	yes	characteristic
256	C₂³	C₂²×D₄×C₄₁	1	2	yes	—
257	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
258	C₂³	C₂²×Q₈×C₄₁	1	1	yes	characteristic
259	C₂³	C₂×#41×C₄₁	1	2	yes	—
260	C₂³	H(5,𝔽₂)×C₄₁	1	4	yes	—
261	C₂³	2₋¹⁺⁴×C₄₁	1	4	yes	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	32	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	8	no	—
272	C₂×C₄	—	4	16	no	—
273	C₂×C₄	—	4	16	no	—
274	C₂×C₄	—	4	8	no	—
275	C₂×C₄	—	4	8	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	—	4	16	no	—
279	C₂×C₄	—	4	16	no	—
280	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	8	yes	—
281	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	8	yes	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	8	no	—
285	C₂²	—	2	8	no	—
286	C₂²	—	2	16	no	—
287	C₂²	—	2	16	no	—
288	C₂²	C₄₁×#380	1	8	yes	—
289	C₂²	C₂²×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
290	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
291	C₂²	C₂×#117×C₄₁	1	1	yes	characteristic
292	C₂²	C₄₁×#518	1	2	yes	—
293	C₂²	C₄₁×#519	1	2	yes	—
294	C₂²	C₂×#121×C₄₁	1	2	yes	—
295	C₂²	C₂×C₂²⋊Q₈×C₄₁	1	2	yes	—
296	C₂²	C₄₁×#534	1	4	yes	—
297	C₂²	C₄₁×#536	1	4	yes	—
298	C₂²	C₄₁×#537	1	4	yes	—
299	C₂²	C₄₁×#539	1	8	yes	—
300	C₂²	C₄₁×#544	1	4	yes	—
301	C₄	—	2	8	no	—
302	C₄	—	2	8	no	—
303	C₄	—	4	4	no	—
304	C₄	—	8	8	no	—
305	C₂	—	2	8	no	—
306	C₂	C₂×C₄₁×#380	1	1	yes	characteristic
307	C₂	C₄₁×#2308	1	1	yes	characteristic
308	C₂	C₄₁×#2310	1	1	yes	characteristic
309	C₂	C₄₁×#2313	1	2	yes	—
310	C₂	C₄₁×#2316	1	2	yes	—
311	C₂	C₄₁×#2618	1	4	yes	—
312	C₂	C₄₁×#2627	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₄₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₄₁
Frattini, cosocle	C₂⁴	C₂⁴×C₄₁
prime radical	C₂⁵×C₄₁	C₂³
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
4. C₄₁×#1441	C₂	semidirect
10. C₂²×C₄⋊C₄×C₄₁	C₄	semidirect
75. C₂²×C₄×C₄₁	C₄⋊C₄	semidirect
125. #14339	C₄₁	direct (N × H)
152. #1441	C₂×C₄₁	semidirect
166. C₂²×C₄⋊C₄	C₄×C₄₁	semidirect
175. C₄₁	#14339	direct (N × H)
232. C₂²×C₄	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
C₂³×C₄⋊C₄×C₄₁	2
C₂×C₄₁×#381	2
C₂×C₄₁×#380	2
C₄₁×#1441	2
C₄₁×C₂²⋊#95	2
#14339	41

Subgroups — C41×#14339 #23790540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#14339 #23790540