Subgroups — C₄₁×#20314 #23796515

Order 10496.

716 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3354	C₂	1	2	yes	maximal, complemented
2	C₄₁×#3272	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1705	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1691	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1685	C₂	1	2	yes	maximal, complemented
6	C₄₁×#1654	C₂	1	2	yes	maximal, complemented
7	C₄₁×#1629	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄₁×#1624	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#1589	C₂	1	2	yes	maximal, complemented
10	C₂×AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
11	C₄₁×#555	C₂²	1	1	yes	characteristic, complemented
12	C₄₁×#547	C₂²	1	2	yes	—
13	D₄²×C₄₁	C₂²	1	1	yes	characteristic
14	C₄₁×#498	C₂²	1	2	yes	complemented
15	C₄₁×#498	C₂²	1	1	yes	characteristic, complemented
16	C₄₁×#492	C₂²	1	2	yes	—
17	C₄₁×#492	C₂²	1	1	yes	characteristic
18	C₄₁×#482	—	2	2	no	—
19	C₄₁×#481	—	2	2	no	—
20	C₄₁×#476	C₂²	1	2	yes	—
21	C₄₁×#474	C₂²	1	2	yes	complemented
22	C₄₁×#469	—	2	2	no	—
23	C₄₁×#468	—	2	2	no	—
24	C₄₁×#461	C₂²	1	2	yes	—
25	C₄₁×#459	C₂²	1	2	yes	—
26	C₄₁×#450	—	2	2	no	—
27	C₄₁×#449	—	2	2	no	—
28	C₄×D₈×C₄₁	C₂²	1	2	yes	—
29	C₄₁×#436	C₂²	1	2	yes	—
30	C₄₁×#436	C₂²	1	2	yes	complemented
31	C₄₁×#428	—	2	2	no	—
32	C₄₁×#335	C₂²	1	2	yes	complemented
33	C₄₁×#333	C₂²	1	1	yes	characteristic, complemented
34	C₄₁×#332	C₂²	1	2	yes	complemented
35	C₄₁×#331	C₂²	1	1	yes	characteristic
36	C₄₁×D₄⋊C₈	C₂²	1	2	yes	—
37	C₄₁×D₄⋊C₈	C₂²	1	2	yes	complemented
38	C₈²×C₄₁	C₂²	1	1	yes	characteristic, complemented
39	C₂×#41×C₄₁	—	2	2	no	—
40	C₂²×D₄×C₄₁	—	2	2	no	—
41	AGL(1,ℤ/8)×C₄₁	—	4	8	no	—
42	C₂×SD₁₆×C₄₁	—	2	2	no	—
43	C₂×SD₁₆×C₄₁	—	2	2	no	—
44	C₂×SD₁₆×C₄₁	—	2	2	no	—
45	C₂×SD₁₆×C₄₁	—	4	1	no	—
46	C₂×D₈×C₄₁	—	2	2	no	—
47	C₂×D₈×C₄₁	D₄	1	2	yes	—
48	C₂×M₁₆×C₄₁	—	2	2	no	—
49	C₂×M₁₆×C₄₁	—	2	2	no	—
50	C₄⋊Q₈×C₄₁	C₂³	1	2	yes	—
51	C₄⋊Q₈×C₄₁	C₂³	1	1	yes	characteristic
52	#127×C₄₁	C₂³	1	1	yes	characteristic
53	#124×C₄₁	—	4	1	no	—
54	#123×C₄₁	—	4	2	no	—
55	C₂²⋊Q₈×C₄₁	—	2	2	no	—
56	#121×C₄₁	—	2	2	no	—
57	#121×C₄₁	—	2	2	no	—
58	#121×C₄₁	—	2	2	no	—
59	#120×C₄₁	—	4	1	no	—
60	C₄×D₄×C₄₁	C₂³	1	2	yes	—
61	C₄×D₄×C₄₁	C₂³	1	2	yes	—
62	#117×C₄₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
64	#107×C₄₁	—	2	2	no	—
65	#107×C₄₁	D₄	1	2	yes	—
66	#106×C₄₁	—	2	2	no	—
67	#106×C₄₁	—	2	2	no	—
68	#106×C₄₁	—	2	2	no	—
69	#106×C₄₁	—	4	1	no	—
70	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
71	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
72	#103×C₄₁	—	2	2	no	—
73	#102×C₄₁	—	2	2	no	—
74	#102×C₄₁	—	2	2	no	—
75	#102×C₄₁	—	2	2	no	—
76	#102×C₄₁	D₄	1	2	yes	—
77	#102×C₄₁	D₄	1	2	yes	—
78	C₂²⋊C₈×C₄₁	—	2	2	no	—
79	C₂²⋊C₈×C₄₁	—	2	2	no	—
80	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₄₁	C₂³	1	2	yes	—
82	C₂⁴×C₄₁	—	4	1	no	—
83	#41×C₄₁	—	4	4	no	—
84	#41×C₄₁	—	8	2	no	—
85	C₂×Q₈×C₄₁	—	2	2	no	—
86	C₂×Q₈×C₄₁	—	2	1	no	—
87	C₂×D₄×C₄₁	—	2	2	no	—
88	C₂×D₄×C₄₁	—	2	1	no	—
89	C₂×D₄×C₄₁	—	4	2	no	—
90	C₂×D₄×C₄₁	—	4	4	no	—
91	C₂×D₄×C₄₁	—	4	2	no	—
92	C₂×D₄×C₄₁	—	4	1	no	—
93	C₂×D₄×C₄₁	—	8	2	no	—
94	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
95	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₄₁	—	2	2	no	—
97	C₂²×C₄×C₄₁	—	2	2	no	—
98	C₂²×C₄×C₄₁	—	4	2	no	—
99	SD₁₆×C₄₁	—	4	4	no	—
100	SD₁₆×C₄₁	—	4	4	no	—
101	SD₁₆×C₄₁	—	8	2	no	—
102	SD₁₆×C₄₁	—	8	2	no	—
103	D₈×C₄₁	—	2	4	no	—
104	D₈×C₄₁	—	4	4	no	—
105	M₁₆×C₄₁	—	4	4	no	—
106	M₁₆×C₄₁	—	4	4	no	—
107	C₂×C₈×C₄₁	—	2	1	no	—
108	C₂×C₈×C₄₁	—	2	2	no	—
109	C₂×C₈×C₄₁	—	2	2	no	—
110	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₄₁	—	2	2	no	—
113	C₄⋊C₄×C₄₁	—	2	2	no	—
114	C₄⋊C₄×C₄₁	—	2	2	no	—
115	C₄⋊C₄×C₄₁	—	4	2	no	—
116	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
117	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
118	C₂²⋊C₄×C₄₁	—	2	2	no	—
119	C₂²⋊C₄×C₄₁	—	2	2	no	—
120	C₂²⋊C₄×C₄₁	—	4	1	no	—
121	C₂²⋊C₄×C₄₁	—	4	2	no	—
122	C₂²⋊C₄×C₄₁	—	4	2	no	—
123	C₄²×C₄₁	—	4	1	no	—
124	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
125	C₂³×C₄₁	—	2	2	no	—
126	C₂³×C₄₁	—	2	2	no	—
127	C₂³×C₄₁	—	4	1	no	—
128	C₂³×C₄₁	—	8	4	no	—
129	C₂³×C₄₁	—	16	1	no	—
130	Q₈×C₄₁	—	4	2	no	—
131	Q₈×C₄₁	—	4	2	no	—
132	Q₈×C₄₁	—	8	1	no	—
133	D₄×C₄₁	—	2	2	no	—
134	D₄×C₄₁	—	2	2	no	—
135	D₄×C₄₁	—	2	2	no	—
136	D₄×C₄₁	—	2	2	no	—
137	D₄×C₄₁	—	4	2	no	—
138	D₄×C₄₁	—	4	2	no	—
139	D₄×C₄₁	—	8	1	no	—
140	D₄×C₄₁	—	8	4	no	—
141	D₄×C₄₁	—	8	4	no	—
142	D₄×C₄₁	—	16	1	no	—
143	C₂×C₄×C₄₁	—	2	2	no	—
144	C₂×C₄×C₄₁	—	2	2	no	—
145	C₂×C₄×C₄₁	—	4	2	no	—
146	C₂×C₄×C₄₁	—	4	2	no	—
147	C₂×C₄×C₄₁	—	4	2	no	—
148	C₂×C₄×C₄₁	—	4	2	no	—
149	C₂×C₄×C₄₁	—	4	1	no	—
150	C₂×C₄×C₄₁	—	4	2	no	—
151	C₂×C₄×C₄₁	—	8	4	no	—
152	C₂×C₄×C₄₁	C₂²×D₄	1	2	yes	—
153	C₂×C₄×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
154	C₈×C₄₁	—	2	2	no	—
155	C₈×C₄₁	—	2	2	no	—
156	C₈×C₄₁	—	4	2	no	—
157	C₈×C₄₁	—	4	2	no	—
158	C₈×C₄₁	AGL(1,ℤ/8)	1	4	yes	complemented
159	#20314	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂²×C₄₁	—	4	2	no	—
161	C₂²×C₄₁	—	4	2	no	—
162	C₂²×C₄₁	—	4	2	no	—
163	C₂²×C₄₁	—	4	2	no	—
164	C₂²×C₄₁	—	4	2	no	—
165	C₂²×C₄₁	—	4	2	no	—
166	C₂²×C₄₁	—	8	2	no	—
167	C₂²×C₄₁	—	8	1	no	—
168	C₂²×C₄₁	—	16	4	no	—
169	C₂²×C₄₁	D₄²	1	1	yes	characteristic
170	C₄×C₄₁	—	2	1	no	—
171	C₄×C₄₁	—	4	2	no	—
172	C₄×C₄₁	—	4	2	no	—
173	C₄×C₄₁	—	8	2	no	—
174	C₄×C₄₁	—	8	1	no	—
175	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
176	C₄×C₄₁	#577	1	2	yes	—
177	#3354	C₂×C₄₁	1	2	yes	complemented
178	#3272	C₂×C₄₁	1	2	yes	complemented
179	#1705	C₂×C₄₁	1	1	yes	characteristic, complemented
180	#1691	C₂×C₄₁	1	2	yes	complemented
181	#1685	C₂×C₄₁	1	2	yes	complemented
182	#1654	C₂×C₄₁	1	2	yes	complemented
183	#1629	C₂×C₄₁	1	1	yes	characteristic, complemented
184	#1624	C₂×C₄₁	1	1	yes	characteristic, complemented
185	#1589	C₂×C₄₁	1	2	yes	complemented
186	C₂×C₄₁	—	8	2	no	—
187	C₂×C₄₁	—	8	2	no	—
188	C₂×C₄₁	—	16	1	no	—
189	C₂×C₄₁	#3272	1	2	yes	—
190	C₂×C₄₁	#3291	1	1	yes	characteristic
191	C₂×AGL(1,ℤ/8)	—	2	2	no	—
192	#555	C₂²×C₄₁	1	1	yes	characteristic, complemented
193	#547	C₂²×C₄₁	1	2	yes	—
194	D₄²	C₂²×C₄₁	1	1	yes	characteristic
195	#498	C₂²×C₄₁	1	2	yes	complemented
196	#498	C₂²×C₄₁	1	1	yes	characteristic, complemented
197	#492	C₂²×C₄₁	1	2	yes	—
198	#492	C₂²×C₄₁	1	1	yes	characteristic
199	#482	—	2	2	no	—
200	#481	—	2	2	no	—
201	#476	C₂²×C₄₁	1	2	yes	—
202	#474	C₂²×C₄₁	1	2	yes	complemented
203	#469	—	2	2	no	—
204	#468	—	2	2	no	—
205	#461	C₂²×C₄₁	1	2	yes	—
206	#459	C₂²×C₄₁	1	2	yes	—
207	#450	—	2	2	no	—
208	#449	—	2	2	no	—
209	C₄×D₈	C₂²×C₄₁	1	2	yes	—
210	#436	C₂²×C₄₁	1	2	yes	—
211	#436	C₂²×C₄₁	1	2	yes	complemented
212	#428	—	2	2	no	—
213	#335	C₂²×C₄₁	1	2	yes	complemented
214	#333	C₂²×C₄₁	1	1	yes	characteristic, complemented
215	#332	C₂²×C₄₁	1	2	yes	complemented
216	#331	C₂²×C₄₁	1	1	yes	characteristic
217	D₄⋊C₈	C₂²×C₄₁	1	2	yes	—
218	D₄⋊C₈	C₂²×C₄₁	1	2	yes	complemented
219	C₈²	C₂²×C₄₁	1	1	yes	characteristic, complemented
220	C₄₁	#20314	1	1	yes	characteristic, direct product factor, Sylow
221	C₂×#41	—	2	2	no	—
222	C₂²×D₄	—	2	2	no	—
223	AGL(1,ℤ/8)	—	4	8	no	—
224	C₂×SD₁₆	—	2	2	no	—
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×SD₁₆	—	4	1	no	—
228	C₂×D₈	—	2	2	no	—
229	C₂×D₈	D₄×C₄₁	1	2	yes	—
230	C₂×M₁₆	—	2	2	no	—
231	C₂×M₁₆	—	2	2	no	—
232	C₄⋊Q₈	C₂³×C₄₁	1	2	yes	—
233	C₄⋊Q₈	C₂³×C₄₁	1	1	yes	characteristic
234	#127	C₂³×C₄₁	1	1	yes	characteristic
235	#124	—	4	1	no	—
236	#123	—	4	2	no	—
237	C₂²⋊Q₈	—	2	2	no	—
238	#121	—	2	2	no	—
239	#121	—	2	2	no	—
240	#121	—	2	2	no	—
241	#120	—	4	1	no	—
242	C₄×D₄	C₂³×C₄₁	1	2	yes	—
243	C₄×D₄	C₂³×C₄₁	1	2	yes	—
244	#117	—	2	2	no	—
245	C₂×C₄⋊C₄	—	2	2	no	—
246	#107	—	2	2	no	—
247	#107	D₄×C₄₁	1	2	yes	—
248	#106	—	2	2	no	—
249	#106	—	2	2	no	—
250	#106	—	2	2	no	—
251	#106	—	4	1	no	—
252	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
253	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
254	#103	—	2	2	no	—
255	#102	—	2	2	no	—
256	#102	—	2	2	no	—
257	#102	—	2	2	no	—
258	#102	D₄×C₄₁	1	2	yes	—
259	#102	D₄×C₄₁	1	2	yes	—
260	C₂²⋊C₈	—	2	2	no	—
261	C₂²⋊C₈	—	2	2	no	—
262	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
263	C₄×C₈	C₂³×C₄₁	1	2	yes	—
264	C₂⁴	—	4	1	no	—
265	#41	—	4	4	no	—
266	#41	—	8	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	1	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	1	no	—
271	C₂×D₄	—	4	2	no	—
272	C₂×D₄	—	4	4	no	—
273	C₂×D₄	—	4	2	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂×D₄	—	8	2	no	—
276	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
277	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
278	C₂²×C₄	—	2	2	no	—
279	C₂²×C₄	—	2	2	no	—
280	C₂²×C₄	—	4	2	no	—
281	SD₁₆	—	4	4	no	—
282	SD₁₆	—	4	4	no	—
283	SD₁₆	—	8	2	no	—
284	SD₁₆	—	8	2	no	—
285	D₈	—	2	4	no	—
286	D₈	—	4	4	no	—
287	M₁₆	—	4	4	no	—
288	M₁₆	—	4	4	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
293	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
299	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
300	C₂²⋊C₄	—	2	2	no	—
301	C₂²⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	4	1	no	—
303	C₂²⋊C₄	—	4	2	no	—
304	C₂²⋊C₄	—	4	2	no	—
305	C₄²	—	4	1	no	—
306	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
307	C₂³	—	2	2	no	—
308	C₂³	—	2	2	no	—
309	C₂³	—	4	1	no	—
310	C₂³	—	8	4	no	—
311	C₂³	—	16	1	no	—
312	Q₈	—	4	2	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	8	1	no	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	2	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	4	2	no	—
321	D₄	—	8	1	no	—
322	D₄	—	8	4	no	—
323	D₄	—	8	4	no	—
324	D₄	—	16	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	8	4	no	—
334	C₂×C₄	C₂²×D₄×C₄₁	1	2	yes	—
335	C₂×C₄	H(5,𝔽₂)×C₄₁	1	1	yes	characteristic
336	C₈	—	2	2	no	—
337	C₈	—	2	2	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	AGL(1,ℤ/8)×C₄₁	1	4	yes	complemented
341	C₂²	—	4	2	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	4	2	no	—
346	C₂²	—	4	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	1	no	—
349	C₂²	—	16	4	no	—
350	C₂²	D₄²×C₄₁	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	4	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	1	no	—
356	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	2	yes	—
357	C₄	C₄₁×#577	1	2	yes	—
358	C₂	—	8	2	no	—
359	C₂	—	8	2	no	—
360	C₂	—	16	1	no	—
361	C₂	C₄₁×#3272	1	2	yes	—
362	C₂	C₄₁×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₄₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3354	C₂	semidirect
2. C₄₁×#3272	C₂	semidirect
3. C₄₁×#1705	C₂	semidirect
4. C₄₁×#1691	C₂	semidirect
5. C₄₁×#1685	C₂	semidirect
6. C₄₁×#1654	C₂	semidirect
7. C₄₁×#1629	C₂	semidirect
8. C₄₁×#1624	C₂	semidirect
9. C₄₁×#1589	C₂	semidirect
11. C₄₁×#555	C₂²	semidirect
14. C₄₁×#498	C₂²	semidirect
15. C₄₁×#498	C₂²	semidirect
21. C₄₁×#474	C₂²	semidirect
30. C₄₁×#436	C₂²	semidirect
32. C₄₁×#335	C₂²	semidirect
33. C₄₁×#333	C₂²	semidirect
34. C₄₁×#332	C₂²	semidirect
37. C₄₁×D₄⋊C₈	C₂²	semidirect
38. C₈²×C₄₁	C₂²	semidirect
158. C₈×C₄₁	AGL(1,ℤ/8)	semidirect
159. #20314	C₄₁	direct (N × H)
177. #3354	C₂×C₄₁	semidirect
178. #3272	C₂×C₄₁	semidirect
179. #1705	C₂×C₄₁	semidirect
180. #1691	C₂×C₄₁	semidirect
181. #1685	C₂×C₄₁	semidirect
182. #1654	C₂×C₄₁	semidirect
183. #1629	C₂×C₄₁	semidirect
184. #1624	C₂×C₄₁	semidirect
185. #1589	C₂×C₄₁	semidirect
192. #555	C₂²×C₄₁	semidirect
195. #498	C₂²×C₄₁	semidirect
196. #498	C₂²×C₄₁	semidirect
202. #474	C₂²×C₄₁	semidirect
211. #436	C₂²×C₄₁	semidirect
213. #335	C₂²×C₄₁	semidirect
214. #333	C₂²×C₄₁	semidirect
215. #332	C₂²×C₄₁	semidirect
218. D₄⋊C₈	C₂²×C₄₁	semidirect
219. C₈²	C₂²×C₄₁	semidirect
220. C₄₁	#20314	direct (N × H)
340. C₈	AGL(1,ℤ/8)×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3354	2
C₄₁×#3272	2
C₄₁×#1705	2
C₄₁×#1691	2
C₄₁×#1685	2
C₄₁×#1654	2
C₄₁×#1629	2
C₄₁×#1624	2
C₄₁×#1589	2
#20314	41

Subgroups — C41×#20314 #23796515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20314 #23796515