Subgroups — C₄₁×#8386 #23784587

Order 10496.

802 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#559	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1801	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#1545	C₂	1	2	yes	maximal
4	C₄₁×#1487	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1461	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#559	—	2	4	no	—
7	C₄₁×#559	—	2	4	no	—
8	C₂×#127×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×#124×C₄₁	—	2	1	no	—
10	C₂×#124×C₄₁	C₄	1	2	yes	—
11	C₂×#123×C₄₁	—	2	2	no	—
12	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₄₁	—	2	1	no	—
15	C₂×C₄×D₄×C₄₁	C₄	1	2	yes	—
16	C₄₁×#431	C₂²	1	2	yes	—
17	C₄₁×#423	C₂²	1	2	yes	—
18	C₂×#101×C₄₁	—	2	1	no	—
19	C₄₁×#343	—	2	1	no	—
20	C₄₁×#331	—	2	4	no	—
21	C₄₁×Q₈⋊C₈	—	2	4	no	—
22	C₄₁×#324	—	2	2	no	—
23	C₂²×Q₈×C₄₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×Q₈×C₄₁	D₄	1	1	yes	characteristic
25	C₂²×D₄×C₄₁	—	2	1	no	—
26	C₂²×D₄×C₄₁	—	2	1	no	—
27	C₂²×D₄×C₄₁	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₄₁	—	2	2	no	—
29	C₂×M₁₆×C₄₁	—	2	2	no	—
30	#127×C₄₁	—	2	2	no	—
31	#127×C₄₁	D₄	1	4	yes	—
32	#124×C₄₁	—	2	8	no	—
33	#124×C₄₁	—	2	4	no	—
34	#124×C₄₁	—	2	4	no	—
35	#123×C₄₁	—	2	16	no	—
36	C₄×Q₈×C₄₁	—	2	2	no	—
37	C₄×Q₈×C₄₁	—	2	2	no	—
38	C₄×Q₈×C₄₁	—	2	2	no	—
39	C₄×Q₈×C₄₁	D₄	1	4	yes	—
40	C₄×D₄×C₄₁	—	2	8	no	—
41	C₄×D₄×C₄₁	—	2	4	no	—
42	C₄×D₄×C₄₁	—	2	4	no	—
43	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₄₁	D₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
50	C₂×C₄²×C₄₁	—	2	1	no	—
51	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
53	C₄⋊C₈×C₄₁	—	2	2	no	—
54	C₄⋊C₈×C₄₁	—	2	2	no	—
55	C₄⋊C₈×C₄₁	—	2	2	no	—
56	#101×C₄₁	—	4	2	no	—
57	C₂²⋊C₈×C₄₁	—	2	2	no	—
58	C₄×C₈×C₄₁	—	2	2	no	—
59	C₂⁴×C₄₁	—	2	2	no	—
60	C₂×Q₈×C₄₁	—	2	1	no	—
61	C₂×Q₈×C₄₁	—	2	2	no	—
62	C₂×Q₈×C₄₁	—	2	2	no	—
63	C₂×Q₈×C₄₁	—	2	1	no	—
64	C₂×Q₈×C₄₁	—	2	1	no	—
65	C₂×Q₈×C₄₁	—	4	2	no	—
66	C₂×Q₈×C₄₁	C₂²⋊C₄	1	2	yes	—
67	C₂×D₄×C₄₁	—	2	2	no	—
68	C₂×D₄×C₄₁	—	2	4	no	—
69	C₂×D₄×C₄₁	—	2	4	no	—
70	C₂×D₄×C₄₁	—	4	4	no	—
71	C₂×D₄×C₄₁	—	4	2	no	—
72	C₂×D₄×C₄₁	—	4	1	no	—
73	C₂×D₄×C₄₁	—	4	1	no	—
74	C₂×D₄×C₄₁	—	8	2	no	—
75	C₂×D₄×C₄₁	C₂²⋊C₄	1	4	yes	—
76	C₂²×C₄×C₄₁	—	2	1	no	—
77	C₂²×C₄×C₄₁	—	2	1	no	—
78	C₂²×C₄×C₄₁	—	2	8	no	—
79	C₂²×C₄×C₄₁	—	8	4	no	—
80	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
81	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
82	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
85	M₁₆×C₄₁	—	4	2	no	—
86	M₁₆×C₄₁	—	4	2	no	—
87	C₂×C₈×C₄₁	—	4	2	no	—
88	C₂×C₈×C₄₁	—	4	2	no	—
89	C₄⋊C₄×C₄₁	—	2	1	no	—
90	C₄⋊C₄×C₄₁	—	2	4	no	—
91	C₄⋊C₄×C₄₁	—	2	2	no	—
92	C₄⋊C₄×C₄₁	—	2	4	no	—
93	C₄⋊C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
94	C₂²⋊C₄×C₄₁	—	2	8	no	—
95	C₂²⋊C₄×C₄₁	—	2	8	no	—
96	C₂²⋊C₄×C₄₁	—	2	8	no	—
97	C₄²×C₄₁	—	2	2	no	—
98	C₄²×C₄₁	—	2	2	no	—
99	C₄²×C₄₁	—	2	4	no	—
100	C₄²×C₄₁	C₂×D₄	1	2	yes	—
101	C₄²×C₄₁	C₂×D₄	1	2	yes	—
102	C₂³×C₄₁	—	2	4	no	—
103	C₂³×C₄₁	—	4	4	no	—
104	C₂³×C₄₁	—	4	4	no	—
105	C₂³×C₄₁	—	4	2	no	—
106	C₂³×C₄₁	—	4	2	no	—
107	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	Q₈×C₄₁	—	2	4	no	—
109	Q₈×C₄₁	—	4	2	no	—
110	Q₈×C₄₁	—	4	2	no	—
111	Q₈×C₄₁	—	4	2	no	—
112	D₄×C₄₁	—	4	4	no	—
113	D₄×C₄₁	—	4	4	no	—
114	D₄×C₄₁	—	4	4	no	—
115	D₄×C₄₁	—	4	4	no	—
116	D₄×C₄₁	—	8	4	no	—
117	C₂×C₄×C₄₁	—	2	2	no	—
118	C₂×C₄×C₄₁	—	2	2	no	—
119	C₂×C₄×C₄₁	—	2	2	no	—
120	C₂×C₄×C₄₁	—	2	2	no	—
121	C₂×C₄×C₄₁	—	2	2	no	—
122	C₂×C₄×C₄₁	—	2	2	no	—
123	C₂×C₄×C₄₁	—	2	2	no	—
124	C₂×C₄×C₄₁	—	4	1	no	—
125	C₂×C₄×C₄₁	—	4	2	no	—
126	C₂×C₄×C₄₁	—	4	2	no	—
127	C₂×C₄×C₄₁	—	4	2	no	—
128	C₂×C₄×C₄₁	—	8	8	no	—
129	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₁	#120	1	2	yes	—
132	C₂×C₄×C₄₁	#120	1	2	yes	—
133	C₂×C₄×C₄₁	#135	1	2	yes	—
134	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₄₁	#137	1	1	yes	characteristic
136	C₈×C₄₁	—	8	2	no	—
137	C₈×C₄₁	—	8	2	no	—
138	#8386	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂²×C₄₁	—	2	1	no	—
140	C₂²×C₄₁	—	2	1	no	—
141	C₂²×C₄₁	—	4	4	no	—
142	C₂²×C₄₁	—	4	4	no	—
143	C₂²×C₄₁	—	4	4	no	—
144	C₂²×C₄₁	—	8	4	no	—
145	C₂²×C₄₁	—	8	4	no	—
146	C₂²×C₄₁	#379	1	1	yes	characteristic
147	C₂²×C₄₁	#416	1	1	yes	characteristic
148	C₂²×C₄₁	#417	1	1	yes	characteristic
149	C₄×C₄₁	—	2	2	no	—
150	C₄×C₄₁	—	2	1	no	—
151	C₄×C₄₁	—	4	2	no	—
152	C₄×C₄₁	—	4	2	no	—
153	C₄×C₄₁	—	4	2	no	—
154	C₄×C₄₁	—	4	2	no	—
155	C₄×C₄₁	#413	1	2	yes	—
156	C₄×C₄₁	#449	1	2	yes	—
157	C₄×C₄₁	#452	1	2	yes	—
158	C₂×#559	C₂×C₄₁	1	1	yes	characteristic
159	#1801	C₂×C₄₁	1	1	yes	characteristic, complemented
160	#1545	C₂×C₄₁	1	2	yes	—
161	#1487	C₂×C₄₁	1	2	yes	complemented
162	#1461	C₂×C₄₁	1	1	yes	characteristic, complemented
163	C₂×C₄₁	—	2	1	no	—
164	C₂×C₄₁	—	2	1	no	—
165	C₂×C₄₁	—	8	4	no	—
166	C₂×C₄₁	#1785	1	1	yes	characteristic
167	C₂×C₄₁	#1788	1	1	yes	characteristic
168	C₂×C₄₁	#1795	1	1	yes	characteristic
169	#559	—	2	4	no	—
170	#559	—	2	4	no	—
171	C₂×#127	C₂²×C₄₁	1	1	yes	characteristic
172	C₂×#124	—	2	1	no	—
173	C₂×#124	C₄×C₄₁	1	2	yes	—
174	C₂×#123	—	2	2	no	—
175	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
176	C₂×C₄×Q₈	C₂²×C₄₁	1	1	yes	characteristic
177	C₂×C₄×D₄	—	2	1	no	—
178	C₂×C₄×D₄	C₄×C₄₁	1	2	yes	—
179	#431	C₂²×C₄₁	1	2	yes	—
180	#423	C₂²×C₄₁	1	2	yes	—
181	C₂×#101	—	2	1	no	—
182	#343	—	2	1	no	—
183	#331	—	2	4	no	—
184	Q₈⋊C₈	—	2	4	no	—
185	#324	—	2	2	no	—
186	C₄₁	#8386	1	1	yes	characteristic, direct product factor, Sylow
187	C₂²×Q₈	C₂×C₄×C₄₁	1	1	yes	characteristic
188	C₂²×Q₈	D₄×C₄₁	1	1	yes	characteristic
189	C₂²×D₄	—	2	1	no	—
190	C₂²×D₄	—	2	1	no	—
191	C₂²×D₄	C₂×C₄×C₄₁	1	2	yes	—
192	C₂³×C₄	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	#127	—	2	2	no	—
195	#127	D₄×C₄₁	1	4	yes	—
196	#124	—	2	8	no	—
197	#124	—	2	4	no	—
198	#124	—	2	4	no	—
199	#123	—	2	16	no	—
200	C₄×Q₈	—	2	2	no	—
201	C₄×Q₈	—	2	2	no	—
202	C₄×Q₈	—	2	2	no	—
203	C₄×Q₈	D₄×C₄₁	1	4	yes	—
204	C₄×D₄	—	2	8	no	—
205	C₄×D₄	—	2	4	no	—
206	C₄×D₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
209	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
210	C₂×C₄⋊C₄	D₄×C₄₁	1	2	yes	—
211	C₂×C₂²⋊C₄	—	2	2	no	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₄²	—	2	1	no	—
215	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
216	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
217	C₄⋊C₈	—	2	2	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	#101	—	4	2	no	—
221	C₂²⋊C₈	—	2	2	no	—
222	C₄×C₈	—	2	2	no	—
223	C₂⁴	—	2	2	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	2	1	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×Q₈	C₂²⋊C₄×C₄₁	1	2	yes	—
231	C₂×D₄	—	2	2	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	4	4	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂×D₄	—	4	1	no	—
238	C₂×D₄	—	8	2	no	—
239	C₂×D₄	C₂²⋊C₄×C₄₁	1	4	yes	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	8	4	no	—
244	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
245	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
246	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
249	M₁₆	—	4	2	no	—
250	M₁₆	—	4	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
258	C₂²⋊C₄	—	2	8	no	—
259	C₂²⋊C₄	—	2	8	no	—
260	C₂²⋊C₄	—	2	8	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	4	no	—
264	C₄²	C₂×D₄×C₄₁	1	2	yes	—
265	C₄²	C₂×D₄×C₄₁	1	2	yes	—
266	C₂³	—	2	4	no	—
267	C₂³	—	4	4	no	—
268	C₂³	—	4	4	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	4	2	no	—
271	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
272	Q₈	—	2	4	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	D₄	—	4	4	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	4	no	—
279	D₄	—	4	4	no	—
280	D₄	—	8	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	8	8	no	—
293	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
295	C₂×C₄	#120×C₄₁	1	2	yes	—
296	C₂×C₄	#120×C₄₁	1	2	yes	—
297	C₂×C₄	#135×C₄₁	1	2	yes	—
298	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
299	C₂×C₄	#137×C₄₁	1	1	yes	characteristic
300	C₈	—	8	2	no	—
301	C₈	—	8	2	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	4	4	no	—
305	C₂²	—	4	4	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	8	4	no	—
308	C₂²	—	8	4	no	—
309	C₂²	C₄₁×#379	1	1	yes	characteristic
310	C₂²	C₄₁×#416	1	1	yes	characteristic
311	C₂²	C₄₁×#417	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	2	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	2	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	C₄₁×#413	1	2	yes	—
319	C₄	C₄₁×#449	1	2	yes	—
320	C₄	C₄₁×#452	1	2	yes	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	8	4	no	—
324	C₂	C₄₁×#1785	1	1	yes	characteristic
325	C₂	C₄₁×#1788	1	1	yes	characteristic
326	C₂	C₄₁×#1795	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	#379
commutator	C₂²×C₄	C₂²×C₄×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
prime radical	C₂×#127×C₄₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#1801	C₂	semidirect
4. C₄₁×#1487	C₂	semidirect
5. C₄₁×#1461	C₂	semidirect
138. #8386	C₄₁	direct (N × H)
159. #1801	C₂×C₄₁	semidirect
161. #1487	C₂×C₄₁	semidirect
162. #1461	C₂×C₄₁	semidirect
186. C₄₁	#8386	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×#559	2
C₄₁×#1801	2
C₄₁×#1545	2
C₄₁×#1487	2
C₄₁×#1461	2
#8386	41

Subgroups — C41×#8386 #23784587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#8386 #23784587