Subgroups — C₄₁×#13038 #23789239

Order 10496.

474 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	C₂×C₄₁×#510	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#441	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄₁×#362	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄₁×#360	C₂	1	1	yes	maximal, characteristic
5	C₄₁×#1975	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#1383	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₁×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#135×C₄₁	C₂²	1	1	yes	characteristic
9	C₂²×Q₁₆×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₄₁	—	2	1	no	—
11	C₄₁×#510	—	2	2	no	—
12	C₄₁×#510	—	2	2	no	—
13	C₂×Q₃₂×C₄₁	C₄	1	2	yes	complemented
14	C₂×SD₃₂×C₄₁	C₄	1	2	yes	complemented
15	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#441	—	2	2	no	—
18	C₄₁×#441	—	2	2	no	—
19	C₂×#108×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×#106×C₄₁	C₂²	1	1	yes	characteristic
21	C₂×#104×C₄₁	—	2	1	no	—
22	C₂×#103×C₄₁	—	2	1	no	—
23	C₄₁×#408	—	2	1	no	—
24	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
25	C₄₁×#362	—	2	2	no	—
26	C₄₁×#360	—	2	2	no	—
27	C₂×#41×C₄₁	—	2	1	no	—
28	C₂²×Q₈×C₄₁	—	2	1	no	—
29	#135×C₄₁	—	2	1	no	—
30	#135×C₄₁	—	2	1	no	—
31	#135×C₄₁	—	2	1	no	—
32	#135×C₄₁	—	2	1	no	—
33	C₂×Q₁₆×C₄₁	—	2	1	no	—
34	C₂×Q₁₆×C₄₁	—	2	1	no	—
35	C₂×Q₁₆×C₄₁	—	2	1	no	—
36	C₂×Q₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
37	C₂×Q₁₆×C₄₁	C₂×C₄	1	2	yes	—
38	C₂×Q₁₆×C₄₁	D₄	1	2	yes	—
39	C₂×Q₁₆×C₄₁	D₄	1	2	yes	—
40	C₂×SD₁₆×C₄₁	—	2	1	no	—
41	C₂×D₈×C₄₁	C₂×C₄	1	1	yes	characteristic
42	C₂×M₁₆×C₄₁	—	2	1	no	—
43	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
45	C₄×Q₈×C₄₁	—	2	4	no	—
46	C₄×Q₈×C₄₁	—	4	2	no	—
47	C₂×C₄⋊C₄×C₄₁	—	2	1	no	—
48	C₂×C₄⋊C₄×C₄₁	—	4	1	no	—
49	C₂×C₄²×C₄₁	—	4	1	no	—
50	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
51	Q₃₂×C₄₁	—	2	4	no	—
52	SD₃₂×C₄₁	—	2	4	no	—
53	M₃₂×C₄₁	—	2	1	no	—
54	M₃₂×C₄₁	—	2	1	no	—
55	M₃₂×C₄₁	—	2	1	no	—
56	M₃₂×C₄₁	—	2	1	no	—
57	C₂×C₁₆×C₄₁	—	2	1	no	—
58	C₂×C₁₆×C₄₁	C₂×C₄	1	2	yes	—
59	#108×C₄₁	—	2	1	no	—
60	#108×C₄₁	—	2	1	no	—
61	#106×C₄₁	—	2	1	no	—
62	#106×C₄₁	D₄	1	2	yes	—
63	#104×C₄₁	—	4	2	no	—
64	#103×C₄₁	—	2	4	no	—
65	C₂²⋊C₈×C₄₁	—	4	2	no	—
66	#97×C₄₁	—	2	1	no	—
67	#97×C₄₁	—	2	1	no	—
68	#41×C₄₁	—	4	1	no	—
69	#41×C₄₁	—	4	1	no	—
70	#41×C₄₁	—	4	1	no	—
71	#41×C₄₁	—	4	1	no	—
72	C₂×Q₈×C₄₁	—	2	2	no	—
73	C₂×Q₈×C₄₁	—	2	1	no	—
74	C₂×Q₈×C₄₁	—	2	2	no	—
75	C₂×Q₈×C₄₁	—	2	2	no	—
76	C₂×Q₈×C₄₁	—	4	1	no	—
77	C₂×Q₈×C₄₁	—	4	1	no	—
78	C₂×Q₈×C₄₁	—	4	1	no	—
79	C₂×D₄×C₄₁	—	2	1	no	—
80	C₂×D₄×C₄₁	—	4	1	no	—
81	C₂²×C₄×C₄₁	—	2	1	no	—
82	C₂²×C₄×C₄₁	—	4	1	no	—
83	C₂²×C₄×C₄₁	—	4	1	no	—
84	C₂²×C₄×C₄₁	—	4	1	no	—
85	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
86	Q₁₆×C₄₁	—	2	1	no	—
87	Q₁₆×C₄₁	—	2	2	no	—
88	Q₁₆×C₄₁	—	2	1	no	—
89	Q₁₆×C₄₁	—	2	2	no	—
90	Q₁₆×C₄₁	—	2	2	no	—
91	Q₁₆×C₄₁	C₂²⋊C₄	1	4	yes	—
92	SD₁₆×C₄₁	—	4	1	no	—
93	SD₁₆×C₄₁	—	4	1	no	—
94	D₈×C₄₁	—	2	1	no	—
95	D₈×C₄₁	—	2	1	no	—
96	M₁₆×C₄₁	—	4	1	no	—
97	M₁₆×C₄₁	—	4	1	no	—
98	C₂×C₈×C₄₁	—	2	1	no	—
99	C₂×C₈×C₄₁	—	2	1	no	—
100	C₂×C₈×C₄₁	—	2	1	no	—
101	C₂×C₈×C₄₁	—	4	1	no	—
102	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₁	#41	1	1	yes	characteristic
107	C₂×C₈×C₄₁	#41	1	1	yes	characteristic
108	C₄⋊C₄×C₄₁	—	2	2	no	—
109	C₄⋊C₄×C₄₁	—	4	4	no	—
110	C₄⋊C₄×C₄₁	—	4	1	no	—
111	C₄²×C₄₁	—	2	1	no	—
112	C₄²×C₄₁	—	4	4	no	—
113	C₄²×C₄₁	D₈	1	2	yes	—
114	C₁₆×C₄₁	—	2	2	no	—
115	C₁₆×C₄₁	—	4	1	no	—
116	C₂³×C₄₁	—	4	1	no	—
117	C₂³×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
118	Q₈×C₄₁	—	2	4	no	—
119	Q₈×C₄₁	—	4	2	no	—
120	Q₈×C₄₁	—	4	2	no	—
121	Q₈×C₄₁	—	4	2	no	—
122	Q₈×C₄₁	—	4	1	no	—
123	Q₈×C₄₁	—	4	1	no	—
124	D₄×C₄₁	—	4	1	no	—
125	D₄×C₄₁	—	4	1	no	—
126	D₄×C₄₁	—	8	1	no	—
127	D₄×C₄₁	—	8	1	no	—
128	C₂×C₄×C₄₁	—	2	2	no	—
129	C₂×C₄×C₄₁	—	4	2	no	—
130	C₂×C₄×C₄₁	—	4	2	no	—
131	C₂×C₄×C₄₁	—	4	2	no	—
132	C₂×C₄×C₄₁	—	4	2	no	—
133	C₂×C₄×C₄₁	—	4	1	no	—
134	C₂×C₄×C₄₁	—	4	2	no	—
135	C₂×C₄×C₄₁	—	8	2	no	—
136	C₂×C₄×C₄₁	—	8	1	no	—
137	C₂×C₄×C₄₁	—	8	1	no	—
138	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₁	#120	1	1	yes	characteristic
140	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
141	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₄₁	#135	1	1	yes	characteristic
143	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₈×C₄₁	—	4	1	no	—
145	C₈×C₄₁	—	8	1	no	—
146	C₈×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₈×C₄₁	C₄×D₄	1	1	yes	characteristic
148	C₈×C₄₁	#121	1	1	yes	characteristic
149	C₈×C₄₁	#123	1	1	yes	characteristic
150	#13038	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂²×C₄₁	—	2	2	no	—
152	C₂²×C₄₁	—	8	1	no	—
153	C₂²×C₄₁	—	8	2	no	—
154	C₂²×C₄₁	#442	1	1	yes	characteristic
155	C₂²×C₄₁	#447	1	1	yes	characteristic
156	C₂²×C₄₁	#463	1	1	yes	characteristic
157	C₄×C₄₁	—	4	2	no	—
158	C₄×C₄₁	—	4	4	no	—
159	C₄×C₄₁	—	8	2	no	—
160	C₄×C₄₁	—	8	1	no	—
161	C₄×C₄₁	#386	1	1	yes	characteristic
162	C₄×C₄₁	C₄×D₈	1	1	yes	characteristic
163	C₄×C₄₁	#449	1	1	yes	characteristic
164	C₄×C₄₁	#459	1	1	yes	characteristic
165	C₂×#510	C₂×C₄₁	1	1	yes	characteristic
166	C₂×#441	C₂×C₄₁	1	1	yes	characteristic, complemented
167	C₂×#362	C₂×C₄₁	1	1	yes	characteristic, complemented
168	C₂×#360	C₂×C₄₁	1	1	yes	characteristic
169	#1975	C₂×C₄₁	1	1	yes	characteristic
170	#1383	C₂×C₄₁	1	1	yes	characteristic, complemented
171	#1375	C₂×C₄₁	1	1	yes	characteristic, complemented
172	C₂×C₄₁	—	2	1	no	—
173	C₂×C₄₁	—	2	1	no	—
174	C₂×C₄₁	—	16	1	no	—
175	C₂×C₄₁	#1879	1	1	yes	characteristic
176	C₂×C₄₁	#2180	1	2	yes	—
177	C₂×#135	C₂²×C₄₁	1	1	yes	characteristic
178	C₂²×Q₁₆	C₂²×C₄₁	1	1	yes	characteristic
179	C₂×C₄×Q₈	—	2	1	no	—
180	#510	—	2	2	no	—
181	#510	—	2	2	no	—
182	C₂×Q₃₂	C₄×C₄₁	1	2	yes	complemented
183	C₂×SD₃₂	C₄×C₄₁	1	2	yes	complemented
184	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
185	C₂×M₃₂	C₂²×C₄₁	1	1	yes	characteristic
186	#441	—	2	2	no	—
187	#441	—	2	2	no	—
188	C₂×#108	C₂²×C₄₁	1	1	yes	characteristic
189	C₂×#106	C₂²×C₄₁	1	1	yes	characteristic
190	C₂×#104	—	2	1	no	—
191	C₂×#103	—	2	1	no	—
192	#408	—	2	1	no	—
193	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
194	#362	—	2	2	no	—
195	#360	—	2	2	no	—
196	C₄₁	#13038	1	1	yes	characteristic, direct product factor, Sylow
197	C₂×#41	—	2	1	no	—
198	C₂²×Q₈	—	2	1	no	—
199	#135	—	2	1	no	—
200	#135	—	2	1	no	—
201	#135	—	2	1	no	—
202	#135	—	2	1	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	—	2	1	no	—
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
207	C₂×Q₁₆	C₂×C₄×C₄₁	1	2	yes	—
208	C₂×Q₁₆	D₄×C₄₁	1	2	yes	—
209	C₂×Q₁₆	D₄×C₄₁	1	2	yes	—
210	C₂×SD₁₆	—	2	1	no	—
211	C₂×D₈	C₂×C₄×C₄₁	1	1	yes	characteristic
212	C₂×M₁₆	—	2	1	no	—
213	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
214	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
215	C₄×Q₈	—	2	4	no	—
216	C₄×Q₈	—	4	2	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	4	1	no	—
219	C₂×C₄²	—	4	1	no	—
220	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
221	Q₃₂	—	2	4	no	—
222	SD₃₂	—	2	4	no	—
223	M₃₂	—	2	1	no	—
224	M₃₂	—	2	1	no	—
225	M₃₂	—	2	1	no	—
226	M₃₂	—	2	1	no	—
227	C₂×C₁₆	—	2	1	no	—
228	C₂×C₁₆	C₂×C₄×C₄₁	1	2	yes	—
229	#108	—	2	1	no	—
230	#108	—	2	1	no	—
231	#106	—	2	1	no	—
232	#106	D₄×C₄₁	1	2	yes	—
233	#104	—	4	2	no	—
234	#103	—	2	4	no	—
235	C₂²⋊C₈	—	4	2	no	—
236	#97	—	2	1	no	—
237	#97	—	2	1	no	—
238	#41	—	4	1	no	—
239	#41	—	4	1	no	—
240	#41	—	4	1	no	—
241	#41	—	4	1	no	—
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	2	1	no	—
259	Q₁₆	—	2	2	no	—
260	Q₁₆	—	2	2	no	—
261	Q₁₆	C₂²⋊C₄×C₄₁	1	4	yes	—
262	SD₁₆	—	4	1	no	—
263	SD₁₆	—	4	1	no	—
264	D₈	—	2	1	no	—
265	D₈	—	2	1	no	—
266	M₁₆	—	4	1	no	—
267	M₁₆	—	4	1	no	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	4	1	no	—
272	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
273	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
274	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
276	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
277	C₂×C₈	#41×C₄₁	1	1	yes	characteristic
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	4	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	4	4	no	—
283	C₄²	D₈×C₄₁	1	2	yes	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	4	1	no	—
286	C₂³	—	4	1	no	—
287	C₂³	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
288	Q₈	—	2	4	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	4	1	no	—
293	Q₈	—	4	1	no	—
294	D₄	—	4	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	8	1	no	—
297	D₄	—	8	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	8	2	no	—
306	C₂×C₄	—	8	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
309	C₂×C₄	#120×C₄₁	1	1	yes	characteristic
310	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
311	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
312	C₂×C₄	#135×C₄₁	1	1	yes	characteristic
313	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
314	C₈	—	4	1	no	—
315	C₈	—	8	1	no	—
316	C₈	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
317	C₈	C₄×D₄×C₄₁	1	1	yes	characteristic
318	C₈	#121×C₄₁	1	1	yes	characteristic
319	C₈	#123×C₄₁	1	1	yes	characteristic
320	C₂²	—	2	2	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	8	2	no	—
323	C₂²	C₄₁×#442	1	1	yes	characteristic
324	C₂²	C₄₁×#447	1	1	yes	characteristic
325	C₂²	C₄₁×#463	1	1	yes	characteristic
326	C₄	—	4	2	no	—
327	C₄	—	4	4	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	1	no	—
330	C₄	C₄₁×#386	1	1	yes	characteristic
331	C₄	C₄×D₈×C₄₁	1	1	yes	characteristic
332	C₄	C₄₁×#449	1	1	yes	characteristic
333	C₄	C₄₁×#459	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	2	1	no	—
336	C₂	—	16	1	no	—
337	C₂	C₄₁×#1879	1	1	yes	characteristic
338	C₂	C₄₁×#2180	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₄₁	#442
commutator	C₂×C₈	C₂²×C₄×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
prime radical	C₂×#135×C₄₁	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ C₂×#97×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#441	C₂	semidirect
3. C₂×C₄₁×#362	C₂	semidirect
6. C₄₁×#1383	C₂	semidirect
7. C₄₁×#1375	C₂	semidirect
13. C₂×Q₃₂×C₄₁	C₄	semidirect
14. C₂×SD₃₂×C₄₁	C₄	semidirect
150. #13038	C₄₁	direct (N × H)
166. C₂×#441	C₂×C₄₁	semidirect
167. C₂×#362	C₂×C₄₁	semidirect
170. #1383	C₂×C₄₁	semidirect
171. #1375	C₂×C₄₁	semidirect
182. C₂×Q₃₂	C₄×C₄₁	semidirect
183. C₂×SD₃₂	C₄×C₄₁	semidirect
196. C₄₁	#13038	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×#510	2
C₂×C₄₁×#441	2
C₂×C₄₁×#362	2
C₂×C₄₁×#360	2
C₄₁×#1975	2
C₄₁×#1383	2
C₄₁×#1375	2
#13038	41

Subgroups — C41×#13038 #23789239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#13038 #23789239