Subgroups — C₄₁×#12743 #23788944

Order 10496.

926 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#576	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#361	C₂	1	2	yes	maximal, complemented
3	C₄₁×#2143	C₂	1	2	yes	maximal, complemented
4	C₄₁×#2115	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₁×#1813	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×H(5,𝔽₂)×C₄₁	—	2	1	no	—
7	C₄₁×#576	—	2	4	no	—
8	C₄₁×#576	—	2	4	no	—
9	C₂×AGL(1,ℤ/8)×C₄₁	—	2	1	no	—
10	C₂×AGL(1,ℤ/8)×C₄₁	—	2	2	no	—
11	C₂×#135×C₄₁	C₄	1	2	yes	—
12	C₂×#135×C₄₁	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₄₁	C₄	1	2	yes	—
14	C₂²×D₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×#131×C₄₁	C₂²	1	1	yes	characteristic
16	C₂×M₃₂×C₄₁	C₂²	1	2	yes	complemented
17	C₂×#108×C₄₁	C₂²	1	2	yes	complemented
18	C₄₁×#413	—	2	1	no	—
19	C₄₁×#361	—	2	4	no	—
20	C₄₁×#359	—	2	4	no	—
21	C₄₁×#349	—	2	2	no	—
22	C₄₁×#338	—	2	1	no	—
23	H(5,𝔽₂)×C₄₁	—	2	8	no	—
24	H(5,𝔽₂)×C₄₁	—	4	4	no	—
25	C₂×#41×C₄₁	—	2	2	no	—
26	C₂×#41×C₄₁	—	2	1	no	—
27	C₂×#41×C₄₁	—	4	1	no	—
28	C₂×#41×C₄₁	D₄	1	1	yes	characteristic
29	C₂²×D₄×C₄₁	—	2	2	no	—
30	C₂²×D₄×C₄₁	—	2	1	no	—
31	C₂²×D₄×C₄₁	—	4	1	no	—
32	C₂²×D₄×C₄₁	—	4	2	no	—
33	AGL(1,ℤ/8)×C₄₁	—	2	8	no	—
34	AGL(1,ℤ/8)×C₄₁	—	2	16	no	—
35	#135×C₄₁	—	2	4	no	—
36	#135×C₄₁	—	2	4	no	—
37	#135×C₄₁	—	2	2	no	—
38	#135×C₄₁	D₄	1	4	yes	—
39	C₂×Q₁₆×C₄₁	C₂×C₄	1	1	yes	characteristic
40	C₂×SD₁₆×C₄₁	—	2	2	no	—
41	C₂×SD₁₆×C₄₁	—	2	1	no	—
42	C₂×D₈×C₄₁	—	2	4	no	—
43	C₂×D₈×C₄₁	—	2	4	no	—
44	C₂×D₈×C₄₁	—	2	2	no	—
45	C₂×D₈×C₄₁	—	2	1	no	—
46	C₂×D₈×C₄₁	—	2	2	no	—
47	C₂×D₈×C₄₁	C₂×C₄	1	1	yes	characteristic
48	C₂×D₈×C₄₁	C₂×C₄	1	2	yes	—
49	C₂×D₈×C₄₁	D₄	1	4	yes	—
50	#131×C₄₁	—	2	2	no	—
51	#131×C₄₁	—	2	2	no	—
52	C₂×M₁₆×C₄₁	—	2	2	no	—
53	C₂×M₁₆×C₄₁	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₄₁	D₄	1	2	yes	—
55	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₄₁	C₂³	1	1	yes	characteristic
57	M₃₂×C₄₁	—	2	2	no	—
58	M₃₂×C₄₁	—	2	2	no	—
59	C₂×C₁₆×C₄₁	—	2	2	no	—
60	#108×C₄₁	—	2	2	no	—
61	#108×C₄₁	—	2	2	no	—
62	#101×C₄₁	—	4	1	no	—
63	#100×C₄₁	—	4	1	no	—
64	C₂⁴×C₄₁	—	4	2	no	—
65	C₂⁴×C₄₁	—	4	1	no	—
66	#41×C₄₁	—	2	2	no	—
67	#41×C₄₁	—	2	2	no	—
68	#41×C₄₁	—	2	8	no	—
69	#41×C₄₁	—	2	4	no	—
70	#41×C₄₁	—	4	4	no	—
71	#41×C₄₁	—	4	8	no	—
72	#41×C₄₁	—	4	2	no	—
73	C₂×Q₈×C₄₁	—	2	1	no	—
74	C₂×Q₈×C₄₁	D₈	1	1	yes	characteristic
75	C₂×D₄×C₄₁	—	2	8	no	—
76	C₂×D₄×C₄₁	—	2	1	no	—
77	C₂×D₄×C₄₁	—	2	2	no	—
78	C₂×D₄×C₄₁	—	2	4	no	—
79	C₂×D₄×C₄₁	—	4	4	no	—
80	C₂×D₄×C₄₁	—	4	16	no	—
81	C₂×D₄×C₄₁	—	4	8	no	—
82	C₂×D₄×C₄₁	—	4	2	no	—
83	C₂×D₄×C₄₁	—	4	1	no	—
84	C₂×D₄×C₄₁	—	4	2	no	—
85	C₂×D₄×C₄₁	—	4	1	no	—
86	C₂×D₄×C₄₁	—	4	2	no	—
87	C₂×D₄×C₄₁	D₈	1	1	yes	characteristic
88	C₂×D₄×C₄₁	SD₁₆	1	2	yes	—
89	C₂²×C₄×C₄₁	—	4	2	no	—
90	C₂²×C₄×C₄₁	—	4	1	no	—
91	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
92	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
93	Q₁₆×C₄₁	—	2	1	no	—
94	Q₁₆×C₄₁	C₂²⋊C₄	1	2	yes	—
95	SD₁₆×C₄₁	—	2	8	no	—
96	SD₁₆×C₄₁	—	2	4	no	—
97	D₈×C₄₁	—	2	8	no	—
98	D₈×C₄₁	—	2	2	no	—
99	D₈×C₄₁	—	2	1	no	—
100	D₈×C₄₁	—	2	4	no	—
101	D₈×C₄₁	C₂²⋊C₄	1	2	yes	—
102	D₈×C₄₁	C₂²⋊C₄	1	4	yes	—
103	M₁₆×C₄₁	—	2	2	no	—
104	M₁₆×C₄₁	—	2	4	no	—
105	M₁₆×C₄₁	—	4	2	no	—
106	M₁₆×C₄₁	—	4	2	no	—
107	C₂×C₈×C₄₁	—	2	2	no	—
108	C₂×C₈×C₄₁	—	2	2	no	—
109	C₂×C₈×C₄₁	—	4	2	no	—
110	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
111	C₂×C₈×C₄₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₄₁	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₄₁	—	4	2	no	—
116	C₂³×C₄₁	—	4	8	no	—
117	C₂³×C₄₁	—	4	2	no	—
118	C₂³×C₄₁	—	4	1	no	—
119	C₂³×C₄₁	—	4	4	no	—
120	C₂³×C₄₁	—	16	2	no	—
121	C₂³×C₄₁	—	16	4	no	—
122	C₂³×C₄₁	#102	1	2	yes	—
123	C₂³×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
124	Q₈×C₄₁	—	2	2	no	—
125	Q₈×C₄₁	—	2	2	no	—
126	Q₈×C₄₁	—	4	1	no	—
127	D₄×C₄₁	—	2	4	no	—
128	D₄×C₄₁	—	2	2	no	—
129	D₄×C₄₁	—	2	4	no	—
130	D₄×C₄₁	—	2	2	no	—
131	D₄×C₄₁	—	4	8	no	—
132	D₄×C₄₁	—	4	8	no	—
133	D₄×C₄₁	—	4	4	no	—
134	D₄×C₄₁	—	4	1	no	—
135	D₄×C₄₁	—	4	2	no	—
136	D₄×C₄₁	—	4	4	no	—
137	C₂×C₄×C₄₁	—	2	2	no	—
138	C₂×C₄×C₄₁	—	2	2	no	—
139	C₂×C₄×C₄₁	—	4	8	no	—
140	C₂×C₄×C₄₁	—	4	1	no	—
141	C₂×C₄×C₄₁	—	4	4	no	—
142	C₂×C₄×C₄₁	#102	1	2	yes	—
143	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₄₁	#120	1	2	yes	—
145	C₂×C₄×C₄₁	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
147	C₈×C₄₁	—	2	2	no	—
148	C₈×C₄₁	—	8	2	no	—
149	C₈×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
150	C₈×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
151	C₈×C₄₁	#120	1	2	yes	—
152	#12743	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂²×C₄₁	—	2	2	no	—
154	C₂²×C₄₁	—	4	1	no	—
155	C₂²×C₄₁	—	4	2	no	—
156	C₂²×C₄₁	—	4	4	no	—
157	C₂²×C₄₁	—	4	2	no	—
158	C₂²×C₄₁	—	8	4	no	—
159	C₂²×C₄₁	—	8	2	no	—
160	C₂²×C₄₁	—	16	4	no	—
161	C₂²×C₄₁	—	16	8	no	—
162	C₂²×C₄₁	C₂×#102	1	1	yes	characteristic
163	C₂²×C₄₁	#450	1	2	yes	—
164	C₄×C₄₁	—	2	2	no	—
165	C₄×C₄₁	—	4	2	no	—
166	C₄×C₄₁	#379	1	1	yes	characteristic
167	C₄×C₄₁	#418	1	1	yes	characteristic
168	C₄×C₄₁	#447	1	2	yes	—
169	C₂×#576	C₂×C₄₁	1	1	yes	characteristic
170	C₂×#361	C₂×C₄₁	1	2	yes	complemented
171	#2143	C₂×C₄₁	1	2	yes	complemented
172	#2115	C₂×C₄₁	1	1	yes	characteristic, complemented
173	#1813	C₂×C₄₁	1	1	yes	characteristic, complemented
174	C₂×C₄₁	—	2	1	no	—
175	C₂×C₄₁	—	2	2	no	—
176	C₂×C₄₁	—	4	2	no	—
177	C₂×C₄₁	—	8	4	no	—
178	C₂×C₄₁	—	8	2	no	—
179	C₂×C₄₁	#1786	1	1	yes	characteristic
180	C₂×H(5,𝔽₂)	—	2	1	no	—
181	#576	—	2	4	no	—
182	#576	—	2	4	no	—
183	C₂×AGL(1,ℤ/8)	—	2	1	no	—
184	C₂×AGL(1,ℤ/8)	—	2	2	no	—
185	C₂×#135	C₄×C₄₁	1	2	yes	—
186	C₂×#135	C₂²×C₄₁	1	1	yes	characteristic
187	C₂²×D₈	C₄×C₄₁	1	2	yes	—
188	C₂²×D₈	C₂²×C₄₁	1	1	yes	characteristic
189	C₂×#131	C₂²×C₄₁	1	1	yes	characteristic
190	C₂×M₃₂	C₂²×C₄₁	1	2	yes	complemented
191	C₂×#108	C₂²×C₄₁	1	2	yes	complemented
192	#413	—	2	1	no	—
193	#361	—	2	4	no	—
194	#359	—	2	4	no	—
195	#349	—	2	2	no	—
196	#338	—	2	1	no	—
197	C₄₁	#12743	1	1	yes	characteristic, direct product factor, Sylow
198	H(5,𝔽₂)	—	2	8	no	—
199	H(5,𝔽₂)	—	4	4	no	—
200	C₂×#41	—	2	2	no	—
201	C₂×#41	—	2	1	no	—
202	C₂×#41	—	4	1	no	—
203	C₂×#41	D₄×C₄₁	1	1	yes	characteristic
204	C₂²×D₄	—	2	2	no	—
205	C₂²×D₄	—	2	1	no	—
206	C₂²×D₄	—	4	1	no	—
207	C₂²×D₄	—	4	2	no	—
208	AGL(1,ℤ/8)	—	2	8	no	—
209	AGL(1,ℤ/8)	—	2	16	no	—
210	#135	—	2	4	no	—
211	#135	—	2	4	no	—
212	#135	—	2	2	no	—
213	#135	D₄×C₄₁	1	4	yes	—
214	C₂×Q₁₆	C₂×C₄×C₄₁	1	1	yes	characteristic
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×SD₁₆	—	2	1	no	—
217	C₂×D₈	—	2	4	no	—
218	C₂×D₈	—	2	4	no	—
219	C₂×D₈	—	2	2	no	—
220	C₂×D₈	—	2	1	no	—
221	C₂×D₈	—	2	2	no	—
222	C₂×D₈	C₂×C₄×C₄₁	1	1	yes	characteristic
223	C₂×D₈	C₂×C₄×C₄₁	1	2	yes	—
224	C₂×D₈	D₄×C₄₁	1	4	yes	—
225	#131	—	2	2	no	—
226	#131	—	2	2	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	D₄×C₄₁	1	1	yes	characteristic
229	C₂×M₁₆	D₄×C₄₁	1	2	yes	—
230	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
231	C₂²×C₈	C₂³×C₄₁	1	1	yes	characteristic
232	M₃₂	—	2	2	no	—
233	M₃₂	—	2	2	no	—
234	C₂×C₁₆	—	2	2	no	—
235	#108	—	2	2	no	—
236	#108	—	2	2	no	—
237	#101	—	4	1	no	—
238	#100	—	4	1	no	—
239	C₂⁴	—	4	2	no	—
240	C₂⁴	—	4	1	no	—
241	#41	—	2	2	no	—
242	#41	—	2	2	no	—
243	#41	—	2	8	no	—
244	#41	—	2	4	no	—
245	#41	—	4	4	no	—
246	#41	—	4	8	no	—
247	#41	—	4	2	no	—
248	C₂×Q₈	—	2	1	no	—
249	C₂×Q₈	D₈×C₄₁	1	1	yes	characteristic
250	C₂×D₄	—	2	8	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	4	no	—
254	C₂×D₄	—	4	4	no	—
255	C₂×D₄	—	4	16	no	—
256	C₂×D₄	—	4	8	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	1	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂×D₄	D₈×C₄₁	1	1	yes	characteristic
263	C₂×D₄	SD₁₆×C₄₁	1	2	yes	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	—	4	1	no	—
266	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
268	Q₁₆	—	2	1	no	—
269	Q₁₆	C₂²⋊C₄×C₄₁	1	2	yes	—
270	SD₁₆	—	2	8	no	—
271	SD₁₆	—	2	4	no	—
272	D₈	—	2	8	no	—
273	D₈	—	2	2	no	—
274	D₈	—	2	1	no	—
275	D₈	—	2	4	no	—
276	D₈	C₂²⋊C₄×C₄₁	1	2	yes	—
277	D₈	C₂²⋊C₄×C₄₁	1	4	yes	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	4	no	—
280	M₁₆	—	4	2	no	—
281	M₁₆	—	4	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	2	no	—
285	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
286	C₂×C₈	C₂²×C₄×C₄₁	1	1	yes	characteristic
287	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
288	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
289	C₂×C₈	C₂×D₄×C₄₁	1	1	yes	characteristic
290	C₁₆	—	4	2	no	—
291	C₂³	—	4	8	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	4	1	no	—
294	C₂³	—	4	4	no	—
295	C₂³	—	16	2	no	—
296	C₂³	—	16	4	no	—
297	C₂³	#102×C₄₁	1	2	yes	—
298	C₂³	C₂×SD₁₆×C₄₁	1	1	yes	characteristic
299	Q₈	—	2	2	no	—
300	Q₈	—	2	2	no	—
301	Q₈	—	4	1	no	—
302	D₄	—	2	4	no	—
303	D₄	—	2	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	8	no	—
307	D₄	—	4	8	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	1	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	8	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	#102×C₄₁	1	2	yes	—
318	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
319	C₂×C₄	#120×C₄₁	1	2	yes	—
320	C₂×C₄	C₂×D₈×C₄₁	1	1	yes	characteristic
321	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
322	C₈	—	2	2	no	—
323	C₈	—	8	2	no	—
324	C₈	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
325	C₈	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
326	C₈	#120×C₄₁	1	2	yes	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	4	4	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	8	4	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	16	4	no	—
335	C₂²	—	16	8	no	—
336	C₂²	C₂×#102×C₄₁	1	1	yes	characteristic
337	C₂²	C₄₁×#450	1	2	yes	—
338	C₄	—	2	2	no	—
339	C₄	—	4	2	no	—
340	C₄	C₄₁×#379	1	1	yes	characteristic
341	C₄	C₄₁×#418	1	1	yes	characteristic
342	C₄	C₄₁×#447	1	2	yes	—
343	C₂	—	2	1	no	—
344	C₂	—	2	2	no	—
345	C₂	—	4	2	no	—
346	C₂	—	8	4	no	—
347	C₂	—	8	2	no	—
348	C₂	C₄₁×#1786	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₄₁	#1786
commutator	C₂×C₈	C₂²×C₄×C₄₁
Frattini, cosocle	C₂²×C₈	C₂³×C₄₁
prime radical	C₂×C₄₁×#576	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄₁ ◁ C₂²×C₄×C₄₁ ◁ C₂×#131×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄₁×#361	C₂	semidirect
3. C₄₁×#2143	C₂	semidirect
4. C₄₁×#2115	C₂	semidirect
5. C₄₁×#1813	C₂	semidirect
16. C₂×M₃₂×C₄₁	C₂²	semidirect
17. C₂×#108×C₄₁	C₂²	semidirect
152. #12743	C₄₁	direct (N × H)
170. C₂×#361	C₂×C₄₁	semidirect
171. #2143	C₂×C₄₁	semidirect
172. #2115	C₂×C₄₁	semidirect
173. #1813	C₂×C₄₁	semidirect
190. C₂×M₃₂	C₂²×C₄₁	semidirect
191. C₂×#108	C₂²×C₄₁	semidirect
197. C₄₁	#12743	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×#576	2
C₂×C₄₁×#361	2
C₄₁×#2143	2
C₄₁×#2115	2
C₄₁×#1813	2
#12743	41

Subgroups — C41×#12743 #23788944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#12743 #23788944