Subgroups — C₄₁×#33688 #23809889

Order 10496.

564 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3184	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₁×#2956	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#2238	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2230	C₂	1	2	yes	maximal, complemented
5	C₄₁×#2213	C₂	1	2	yes	maximal, complemented
6	C₄₁×#2190	C₂	1	2	yes	maximal, complemented
7	C₄₁×AGL(1,ℤ/16)	C₂	1	4	yes	maximal, complemented
8	C₄₁×#2165	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₁×#575	C₂²	1	1	yes	characteristic
10	C₄₁×#568	—	2	1	no	—
11	C₄₁×#532	—	2	1	no	—
12	C₄₁×#509	C₂²	1	4	yes	—
13	C₄₁×#498	C₂²	1	1	yes	characteristic, complemented
14	C₄₁×#493	C₂²	1	1	yes	characteristic, complemented
15	C₄₁×#486	—	2	1	no	—
16	C₄₁×#469	—	2	1	no	—
17	C₄₁×#469	C₂²	1	2	yes	—
18	C₄₁×#444	—	2	1	no	—
19	C₄₁×#444	C₂²	1	2	yes	—
20	C₄₁×#443	—	2	1	no	—
21	C₄₁×#443	C₂²	1	2	yes	—
22	C₄₁×#442	—	2	1	no	—
23	C₄₁×#442	C₂²	1	2	yes	complemented
24	C₄₁×#430	C₂²	1	1	yes	characteristic
25	C₂×#104×C₄₁	—	2	1	no	—
26	C₄×M₁₆×C₄₁	C₂²	1	1	yes	characteristic
27	C₄₁×#365	C₂²	1	4	yes	complemented
28	C₄₁×#364	C₂²	1	2	yes	complemented
29	C₄₁×#361	C₂²	1	4	yes	complemented
30	C₄₁×#360	C₂²	1	4	yes	—
31	C₄₁×#349	C₂²	1	2	yes	—
32	C₄₁×#347	C₂²	1	2	yes	complemented
33	C₂×#41×C₄₁	—	2	1	no	—
34	C₂×#41×C₄₁	—	4	1	no	—
35	#137×C₄₁	—	2	2	no	—
36	AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
37	#135×C₄₁	—	2	1	no	—
38	#135×C₄₁	C₂³	1	2	yes	—
39	C₂×D₈×C₄₁	—	2	1	no	—
40	C₂×D₈×C₄₁	C₂³	1	2	yes	—
41	#131×C₄₁	—	2	2	no	—
42	#131×C₄₁	—	2	2	no	—
43	#131×C₄₁	—	4	2	no	—
44	C₂×M₁₆×C₄₁	—	2	1	no	—
45	C₂×M₁₆×C₄₁	—	4	2	no	—
46	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
47	C₂×M₁₆×C₄₁	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₄₁	—	2	1	no	—
49	#127×C₄₁	—	2	1	no	—
50	#124×C₄₁	—	4	1	no	—
51	#121×C₄₁	—	2	2	no	—
52	#121×C₄₁	—	4	1	no	—
53	C₄×D₄×C₄₁	—	2	2	no	—
54	C₄×D₄×C₄₁	—	4	1	no	—
55	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
56	SD₃₂×C₄₁	—	2	4	no	—
57	D₁₆×C₄₁	—	2	4	no	—
58	M₃₂×C₄₁	C₂³	1	4	yes	—
59	#108×C₄₁	—	2	1	no	—
60	#108×C₄₁	C₂³	1	2	yes	—
61	#106×C₄₁	—	2	1	no	—
62	#106×C₄₁	C₂³	1	2	yes	—
63	#104×C₄₁	—	2	2	no	—
64	#104×C₄₁	—	2	2	no	—
65	#102×C₄₁	—	2	2	no	—
66	#102×C₄₁	—	2	2	no	—
67	#97×C₄₁	D₄	1	1	yes	characteristic
68	#97×C₄₁	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₄₁	D₄	1	1	yes	characteristic
70	C₄×C₈×C₄₁	C₂³	1	1	yes	characteristic
71	#41×C₄₁	—	2	2	no	—
72	#41×C₄₁	—	2	2	no	—
73	#41×C₄₁	—	4	1	no	—
74	#41×C₄₁	—	4	1	no	—
75	#41×C₄₁	—	8	4	no	—
76	C₂×Q₈×C₄₁	—	2	1	no	—
77	C₂×Q₈×C₄₁	—	4	1	no	—
78	C₂×D₄×C₄₁	—	2	1	no	—
79	C₂×D₄×C₄₁	—	2	2	no	—
80	C₂×D₄×C₄₁	—	4	1	no	—
81	C₂×D₄×C₄₁	—	4	1	no	—
82	C₂×D₄×C₄₁	—	4	1	no	—
83	C₂×D₄×C₄₁	—	4	2	no	—
84	C₂²×C₄×C₄₁	—	2	1	no	—
85	C₂²×C₄×C₄₁	—	4	1	no	—
86	C₂²×C₄×C₄₁	—	4	2	no	—
87	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
88	Q₁₆×C₄₁	—	2	1	no	—
89	Q₁₆×C₄₁	#41	1	2	yes	complemented
90	SD₁₆×C₄₁	—	2	2	no	—
91	SD₁₆×C₄₁	—	2	2	no	—
92	D₈×C₄₁	—	2	1	no	—
93	D₈×C₄₁	—	2	4	no	—
94	D₈×C₄₁	—	4	2	no	—
95	D₈×C₄₁	#41	1	2	yes	complemented
96	M₁₆×C₄₁	—	2	2	no	—
97	M₁₆×C₄₁	—	2	2	no	—
98	M₁₆×C₄₁	—	2	2	no	—
99	M₁₆×C₄₁	—	4	2	no	—
100	M₁₆×C₄₁	—	4	1	no	—
101	M₁₆×C₄₁	—	4	2	no	—
102	C₂×C₈×C₄₁	—	4	2	no	—
103	C₂×C₈×C₄₁	—	4	1	no	—
104	C₂×C₈×C₄₁	—	4	2	no	—
105	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₄₁	C₂⁴	1	1	yes	characteristic
109	C₄⋊C₄×C₄₁	—	2	2	no	—
110	C₄⋊C₄×C₄₁	—	4	1	no	—
111	C₂²⋊C₄×C₄₁	—	4	2	no	—
112	C₂²⋊C₄×C₄₁	—	4	2	no	—
113	C₄²×C₄₁	—	2	1	no	—
114	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₄₁	—	2	4	no	—
117	C₂³×C₄₁	—	4	1	no	—
118	C₂³×C₄₁	—	4	2	no	—
119	C₂³×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
120	Q₈×C₄₁	—	2	2	no	—
121	Q₈×C₄₁	—	4	1	no	—
122	Q₈×C₄₁	—	8	2	no	—
123	D₄×C₄₁	—	2	2	no	—
124	D₄×C₄₁	—	4	1	no	—
125	D₄×C₄₁	—	4	2	no	—
126	D₄×C₄₁	—	4	2	no	—
127	D₄×C₄₁	—	4	2	no	—
128	D₄×C₄₁	—	4	2	no	—
129	D₄×C₄₁	—	8	2	no	—
130	D₄×C₄₁	—	8	2	no	—
131	D₄×C₄₁	—	8	4	no	—
132	C₂×C₄×C₄₁	—	2	1	no	—
133	C₂×C₄×C₄₁	—	2	1	no	—
134	C₂×C₄×C₄₁	—	2	1	no	—
135	C₂×C₄×C₄₁	—	2	1	no	—
136	C₂×C₄×C₄₁	—	4	2	no	—
137	C₂×C₄×C₄₁	—	4	2	no	—
138	C₂×C₄×C₄₁	—	4	1	no	—
139	C₂×C₄×C₄₁	—	4	2	no	—
140	C₂×C₄×C₄₁	—	4	2	no	—
141	C₂×C₄×C₄₁	—	8	4	no	—
142	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
144	C₈×C₄₁	—	2	1	no	—
145	C₈×C₄₁	—	2	1	no	—
146	C₈×C₄₁	—	4	2	no	—
147	C₈×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
148	C₈×C₄₁	C₂×#41	1	2	yes	—
149	#33688	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂²×C₄₁	—	2	1	no	—
151	C₂²×C₄₁	—	4	1	no	—
152	C₂²×C₄₁	—	4	2	no	—
153	C₂²×C₄₁	—	8	2	no	—
154	C₂²×C₄₁	—	8	2	no	—
155	C₂²×C₄₁	—	8	2	no	—
156	C₂²×C₄₁	—	16	1	no	—
157	C₂²×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
158	C₄×C₄₁	—	2	2	no	—
159	C₄×C₄₁	—	2	1	no	—
160	C₄×C₄₁	—	4	1	no	—
161	C₄×C₄₁	—	4	2	no	—
162	C₄×C₄₁	—	8	2	no	—
163	C₄×C₄₁	#525	1	1	yes	characteristic
164	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	#3184	C₂×C₄₁	1	1	yes	characteristic, complemented
166	#2956	C₂×C₄₁	1	1	yes	characteristic, complemented
167	#2238	C₂×C₄₁	1	2	yes	complemented
168	#2230	C₂×C₄₁	1	2	yes	complemented
169	#2213	C₂×C₄₁	1	2	yes	complemented
170	#2190	C₂×C₄₁	1	2	yes	complemented
171	AGL(1,ℤ/16)	C₂×C₄₁	1	4	yes	complemented
172	#2165	C₂×C₄₁	1	1	yes	characteristic, complemented
173	C₂×C₄₁	—	2	1	no	—
174	C₂×C₄₁	—	4	1	no	—
175	C₂×C₄₁	—	8	2	no	—
176	C₂×C₄₁	—	16	2	no	—
177	C₂×C₄₁	#3109	1	1	yes	characteristic
178	#575	C₂²×C₄₁	1	1	yes	characteristic
179	#568	—	2	1	no	—
180	#532	—	2	1	no	—
181	#509	C₂²×C₄₁	1	4	yes	—
182	#498	C₂²×C₄₁	1	1	yes	characteristic, complemented
183	#493	C₂²×C₄₁	1	1	yes	characteristic, complemented
184	#486	—	2	1	no	—
185	#469	—	2	1	no	—
186	#469	C₂²×C₄₁	1	2	yes	—
187	#444	—	2	1	no	—
188	#444	C₂²×C₄₁	1	2	yes	—
189	#443	—	2	1	no	—
190	#443	C₂²×C₄₁	1	2	yes	—
191	#442	—	2	1	no	—
192	#442	C₂²×C₄₁	1	2	yes	complemented
193	#430	C₂²×C₄₁	1	1	yes	characteristic
194	C₂×#104	—	2	1	no	—
195	C₄×M₁₆	C₂²×C₄₁	1	1	yes	characteristic
196	#365	C₂²×C₄₁	1	4	yes	complemented
197	#364	C₂²×C₄₁	1	2	yes	complemented
198	#361	C₂²×C₄₁	1	4	yes	complemented
199	#360	C₂²×C₄₁	1	4	yes	—
200	#349	C₂²×C₄₁	1	2	yes	—
201	#347	C₂²×C₄₁	1	2	yes	complemented
202	C₄₁	#33688	1	1	yes	characteristic, direct product factor, Sylow
203	C₂×#41	—	2	1	no	—
204	C₂×#41	—	4	1	no	—
205	#137	—	2	2	no	—
206	AGL(1,ℤ/8)	—	2	2	no	—
207	#135	—	2	1	no	—
208	#135	C₂³×C₄₁	1	2	yes	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	C₂³×C₄₁	1	2	yes	—
211	#131	—	2	2	no	—
212	#131	—	2	2	no	—
213	#131	—	4	2	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	4	2	no	—
216	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
217	C₂×M₁₆	C₂³×C₄₁	1	1	yes	characteristic
218	C₄⋊Q₈	—	2	1	no	—
219	#127	—	2	1	no	—
220	#124	—	4	1	no	—
221	#121	—	2	2	no	—
222	#121	—	4	1	no	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	4	1	no	—
225	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
226	SD₃₂	—	2	4	no	—
227	D₁₆	—	2	4	no	—
228	M₃₂	C₂³×C₄₁	1	4	yes	—
229	#108	—	2	1	no	—
230	#108	C₂³×C₄₁	1	2	yes	—
231	#106	—	2	1	no	—
232	#106	C₂³×C₄₁	1	2	yes	—
233	#104	—	2	2	no	—
234	#104	—	2	2	no	—
235	#102	—	2	2	no	—
236	#102	—	2	2	no	—
237	#97	D₄×C₄₁	1	1	yes	characteristic
238	#97	C₂³×C₄₁	1	1	yes	characteristic
239	C₄×C₈	D₄×C₄₁	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₄₁	1	1	yes	characteristic
241	#41	—	2	2	no	—
242	#41	—	2	2	no	—
243	#41	—	4	1	no	—
244	#41	—	4	1	no	—
245	#41	—	8	4	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
258	Q₁₆	—	2	1	no	—
259	Q₁₆	#41×C₄₁	1	2	yes	complemented
260	SD₁₆	—	2	2	no	—
261	SD₁₆	—	2	2	no	—
262	D₈	—	2	1	no	—
263	D₈	—	2	4	no	—
264	D₈	—	4	2	no	—
265	D₈	#41×C₄₁	1	2	yes	complemented
266	M₁₆	—	2	2	no	—
267	M₁₆	—	2	2	no	—
268	M₁₆	—	2	2	no	—
269	M₁₆	—	4	2	no	—
270	M₁₆	—	4	1	no	—
271	M₁₆	—	4	2	no	—
272	C₂×C₈	—	4	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
278	C₂×C₈	C₂⁴×C₄₁	1	1	yes	characteristic
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	2	1	no	—
284	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
285	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
286	C₁₆	—	2	4	no	—
287	C₂³	—	4	1	no	—
288	C₂³	—	4	2	no	—
289	C₂³	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
290	Q₈	—	2	2	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	8	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	4	2	no	—
299	D₄	—	8	2	no	—
300	D₄	—	8	2	no	—
301	D₄	—	8	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
313	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	4	2	no	—
317	C₈	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
318	C₈	C₂×#41×C₄₁	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	16	1	no	—
326	C₂²	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	2	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	2	no	—
331	C₄	—	8	2	no	—
332	C₄	C₄₁×#525	1	1	yes	characteristic
333	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	4	1	no	—
336	C₂	—	8	2	no	—
337	C₂	—	16	2	no	—
338	C₂	C₄₁×#3109	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁	#3109
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₄₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁ ◁ C₂×C₄×C₄₁ ◁ C₄×M₁₆×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄₁×#3184	C₂	semidirect
2. C₄₁×#2956	C₂	semidirect
3. C₄₁×#2238	C₂	semidirect
4. C₄₁×#2230	C₂	semidirect
5. C₄₁×#2213	C₂	semidirect
6. C₄₁×#2190	C₂	semidirect
7. C₄₁×AGL(1,ℤ/16)	C₂	semidirect
8. C₄₁×#2165	C₂	semidirect
13. C₄₁×#498	C₂²	semidirect
14. C₄₁×#493	C₂²	semidirect
23. C₄₁×#442	C₂²	semidirect
27. C₄₁×#365	C₂²	semidirect
28. C₄₁×#364	C₂²	semidirect
29. C₄₁×#361	C₂²	semidirect
32. C₄₁×#347	C₂²	semidirect
89. Q₁₆×C₄₁	#41	semidirect
95. D₈×C₄₁	#41	semidirect
149. #33688	C₄₁	direct (N × H)
165. #3184	C₂×C₄₁	semidirect
166. #2956	C₂×C₄₁	semidirect
167. #2238	C₂×C₄₁	semidirect
168. #2230	C₂×C₄₁	semidirect
169. #2213	C₂×C₄₁	semidirect
170. #2190	C₂×C₄₁	semidirect
171. AGL(1,ℤ/16)	C₂×C₄₁	semidirect
172. #2165	C₂×C₄₁	semidirect
182. #498	C₂²×C₄₁	semidirect
183. #493	C₂²×C₄₁	semidirect
192. #442	C₂²×C₄₁	semidirect
196. #365	C₂²×C₄₁	semidirect
197. #364	C₂²×C₄₁	semidirect
198. #361	C₂²×C₄₁	semidirect
201. #347	C₂²×C₄₁	semidirect
202. C₄₁	#33688	direct (N × H)
259. Q₁₆	#41×C₄₁	semidirect
265. D₈	#41×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₄₁×#3184	2
C₄₁×#2956	2
C₄₁×#2238	2
C₄₁×#2230	2
C₄₁×#2213	2
C₄₁×#2190	2
C₄₁×AGL(1,ℤ/16)	2
C₄₁×#2165	2
#33688	41

Subgroups — C41×#33688 #23809889

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#33688 #23809889