Subgroups — C₂×C₄₁×#1687 #23795332

Order 10496.

926 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄₁×#459	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₁×#1687	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄₁×#331	C₂	1	2	yes	maximal, complemented
7	C₂²×D₈×C₄₁	—	2	2	no	—
8	C₂²×D₈×C₄₁	—	2	1	no	—
9	C₂²×D₈×C₄₁	—	2	1	no	—
10	C₂²×M₁₆×C₄₁	—	2	1	no	—
11	C₂×#127×C₄₁	C₂²	1	2	yes	—
12	C₂×C₄×D₄×C₄₁	C₂²	1	1	yes	characteristic
13	C₄₁×#493	—	2	2	no	—
14	C₄₁×#493	C₂²	1	4	yes	complemented
15	C₄₁×#459	—	2	4	no	—
16	C₄₁×#459	C₂²	1	8	yes	complemented
17	C₄₁×#436	—	2	2	no	—
18	C₄₁×#436	C₂²	1	4	yes	complemented
19	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
21	C₂×#102×C₄₁	—	2	2	no	—
22	C₂×C₂²⋊C₈×C₄₁	—	2	1	no	—
23	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
24	C₄₁×#334	C₂²	1	4	yes	complemented
25	C₄₁×#331	C₂²	1	8	yes	complemented
26	C₂²×D₄×C₄₁	—	2	2	no	—
27	C₂²×D₄×C₄₁	—	2	2	no	—
28	C₂²×D₄×C₄₁	—	2	2	no	—
29	C₂²×D₄×C₄₁	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₁	—	2	1	no	—
31	C₂×D₈×C₄₁	—	2	8	no	—
32	C₂×D₈×C₄₁	—	2	4	no	—
33	C₂×D₈×C₄₁	—	2	4	no	—
34	C₂×D₈×C₄₁	—	4	4	no	—
35	C₂×D₈×C₄₁	—	4	8	no	—
36	C₂×D₈×C₄₁	—	4	4	no	—
37	C₂×M₁₆×C₄₁	—	2	4	no	—
38	C₂×M₁₆×C₄₁	—	4	4	no	—
39	C₂²×C₈×C₄₁	—	2	1	no	—
40	C₂²×C₈×C₄₁	—	2	2	no	—
41	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₄₁	D₄	1	1	yes	characteristic
43	#127×C₄₁	—	2	2	no	—
44	#127×C₄₁	—	2	2	no	—
45	#127×C₄₁	—	2	2	no	—
46	#127×C₄₁	C₂³	1	4	yes	—
47	C₄×D₄×C₄₁	—	2	1	no	—
48	C₄×D₄×C₄₁	—	2	1	no	—
49	C₄×D₄×C₄₁	D₄	1	2	yes	—
50	C₄×D₄×C₄₁	C₂³	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₁	D₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₄₁	—	2	1	no	—
53	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₁	—	2	1	no	—
55	C₄⋊C₈×C₄₁	—	2	2	no	—
56	C₄⋊C₈×C₄₁	C₂³	1	2	yes	—
57	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
58	#102×C₄₁	—	2	8	no	—
59	C₂²⋊C₈×C₄₁	—	2	4	no	—
60	C₄×C₈×C₄₁	D₄	1	2	yes	—
61	C₄×C₈×C₄₁	C₂³	1	2	yes	—
62	C₂⁴×C₄₁	—	2	1	no	—
63	C₂⁴×C₄₁	—	4	2	no	—
64	C₂×D₄×C₄₁	—	2	1	no	—
65	C₂×D₄×C₄₁	—	2	4	no	—
66	C₂×D₄×C₄₁	—	2	4	no	—
67	C₂×D₄×C₄₁	—	2	4	no	—
68	C₂×D₄×C₄₁	—	2	4	no	—
69	C₂×D₄×C₄₁	—	4	2	no	—
70	C₂×D₄×C₄₁	—	4	2	no	—
71	C₂×D₄×C₄₁	—	4	2	no	—
72	C₂×D₄×C₄₁	—	4	4	no	—
73	C₂×D₄×C₄₁	—	4	4	no	—
74	C₂×D₄×C₄₁	—	4	4	no	—
75	C₂×D₄×C₄₁	—	4	4	no	—
76	C₂×D₄×C₄₁	—	8	4	no	—
77	C₂×D₄×C₄₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₄₁	—	2	1	no	—
79	C₂²×C₄×C₄₁	—	2	2	no	—
80	C₂²×C₄×C₄₁	—	2	2	no	—
81	C₂²×C₄×C₄₁	—	4	2	no	—
82	C₂²×C₄×C₄₁	—	4	2	no	—
83	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
86	D₈×C₄₁	—	4	8	no	—
87	D₈×C₄₁	—	4	16	no	—
88	D₈×C₄₁	—	4	8	no	—
89	M₁₆×C₄₁	—	4	8	no	—
90	C₂×C₈×C₄₁	—	2	2	no	—
91	C₂×C₈×C₄₁	—	2	2	no	—
92	C₂×C₈×C₄₁	—	2	4	no	—
93	C₂×C₈×C₄₁	—	4	2	no	—
94	C₂×C₈×C₄₁	—	4	4	no	—
95	C₂×C₈×C₄₁	D₈	1	4	yes	complemented
96	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₄₁	C₂×D₄	1	2	yes	—
98	C₄⋊C₄×C₄₁	—	2	1	no	—
99	C₄⋊C₄×C₄₁	C₂×D₄	1	2	yes	—
100	C₂²⋊C₄×C₄₁	—	2	2	no	—
101	C₂²⋊C₄×C₄₁	—	2	2	no	—
102	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₄₁	#41	1	1	yes	characteristic
105	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
106	C₂³×C₄₁	—	2	2	no	—
107	C₂³×C₄₁	—	4	2	no	—
108	C₂³×C₄₁	—	4	2	no	—
109	C₂³×C₄₁	—	4	2	no	—
110	C₂³×C₄₁	—	4	4	no	—
111	C₂³×C₄₁	—	8	4	no	—
112	C₂³×C₄₁	—	8	4	no	—
113	C₂³×C₄₁	—	8	4	no	—
114	C₂³×C₄₁	#121	1	1	yes	characteristic
115	D₄×C₄₁	—	2	4	no	—
116	D₄×C₄₁	—	4	4	no	—
117	D₄×C₄₁	—	4	4	no	—
118	D₄×C₄₁	—	4	4	no	—
119	D₄×C₄₁	—	4	4	no	—
120	D₄×C₄₁	—	4	2	no	—
121	D₄×C₄₁	—	8	4	no	—
122	D₄×C₄₁	—	8	4	no	—
123	D₄×C₄₁	—	8	4	no	—
124	D₄×C₄₁	—	8	4	no	—
125	C₂×C₄×C₄₁	—	2	2	no	—
126	C₂×C₄×C₄₁	—	2	2	no	—
127	C₂×C₄×C₄₁	—	4	2	no	—
128	C₂×C₄×C₄₁	—	4	2	no	—
129	C₂×C₄×C₄₁	—	4	2	no	—
130	C₂×C₄×C₄₁	—	4	4	no	—
131	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
132	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	2	yes	—
136	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
137	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₁	C₂×#41	1	1	yes	characteristic
140	C₈×C₄₁	—	2	2	no	—
141	C₈×C₄₁	—	4	2	no	—
142	C₈×C₄₁	—	4	4	no	—
143	C₈×C₄₁	C₂×D₈	1	4	yes	complemented
144	C₂×#1687	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂²×C₄₁	—	4	2	no	—
146	C₂²×C₄₁	—	4	2	no	—
147	C₂²×C₄₁	—	4	2	no	—
148	C₂²×C₄₁	—	8	4	no	—
149	C₂²×C₄₁	—	8	4	no	—
150	C₂²×C₄₁	—	8	4	no	—
151	C₂²×C₄₁	—	8	4	no	—
152	C₂²×C₄₁	—	16	8	no	—
153	C₂²×C₄₁	#459	1	2	yes	—
154	C₂²×C₄₁	#469	1	2	yes	—
155	C₂²×C₄₁	#472	1	2	yes	—
156	C₂²×C₄₁	C₂×#121	1	1	yes	characteristic
157	C₄×C₄₁	—	2	1	no	—
158	C₄×C₄₁	—	2	1	no	—
159	C₄×C₄₁	—	4	2	no	—
160	C₄×C₄₁	#459	1	2	yes	—
161	C₄×C₄₁	#469	1	2	yes	—
162	C₄×C₄₁	C₂²×D₈	1	1	yes	characteristic
163	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₄×C₄₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
165	C₂×#493	C₂×C₄₁	1	1	yes	characteristic, complemented
166	C₂×#459	C₂×C₄₁	1	2	yes	complemented
167	C₂×#436	C₂×C₄₁	1	1	yes	characteristic, complemented
168	#1687	C₂×C₄₁	1	8	yes	direct product factor
169	C₂×#334	C₂×C₄₁	1	1	yes	characteristic, complemented
170	C₂×#331	C₂×C₄₁	1	2	yes	complemented
171	C₂×C₄₁	—	8	2	no	—
172	C₂×C₄₁	—	16	4	no	—
173	C₂×C₄₁	#1687	1	4	yes	direct product factor
174	C₂×C₄₁	C₂×#459	1	1	yes	characteristic
175	C₂×C₄₁	C₂×#469	1	1	yes	characteristic
176	C₂×C₄₁	C₂×#472	1	1	yes	characteristic
177	C₂²×D₈	—	2	2	no	—
178	C₂²×D₈	—	2	1	no	—
179	C₂²×D₈	—	2	1	no	—
180	C₂²×M₁₆	—	2	1	no	—
181	C₂×#127	C₂²×C₄₁	1	2	yes	—
182	C₂×C₄×D₄	C₂²×C₄₁	1	1	yes	characteristic
183	#493	—	2	2	no	—
184	#493	C₂²×C₄₁	1	4	yes	complemented
185	#459	—	2	4	no	—
186	#459	C₂²×C₄₁	1	8	yes	complemented
187	#436	—	2	2	no	—
188	#436	C₂²×C₄₁	1	4	yes	complemented
189	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
190	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
191	C₂×#102	—	2	2	no	—
192	C₂×C₂²⋊C₈	—	2	1	no	—
193	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
194	#334	C₂²×C₄₁	1	4	yes	complemented
195	#331	C₂²×C₄₁	1	8	yes	complemented
196	C₄₁	C₂×#1687	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²×D₄	—	2	2	no	—
198	C₂²×D₄	—	2	2	no	—
199	C₂²×D₄	—	2	2	no	—
200	C₂²×D₄	D₄×C₄₁	1	1	yes	characteristic
201	C₂³×C₄	—	2	1	no	—
202	C₂×D₈	—	2	8	no	—
203	C₂×D₈	—	2	4	no	—
204	C₂×D₈	—	2	4	no	—
205	C₂×D₈	—	4	4	no	—
206	C₂×D₈	—	4	8	no	—
207	C₂×D₈	—	4	4	no	—
208	C₂×M₁₆	—	2	4	no	—
209	C₂×M₁₆	—	4	4	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	2	2	no	—
212	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
213	C₂²×C₈	D₄×C₄₁	1	1	yes	characteristic
214	#127	—	2	2	no	—
215	#127	—	2	2	no	—
216	#127	—	2	2	no	—
217	#127	C₂³×C₄₁	1	4	yes	—
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	—	2	1	no	—
220	C₄×D₄	D₄×C₄₁	1	2	yes	—
221	C₄×D₄	C₂³×C₄₁	1	2	yes	—
222	C₂×C₄⋊C₄	D₄×C₄₁	1	1	yes	characteristic
223	C₂×C₂²⋊C₄	—	2	1	no	—
224	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	2	no	—
227	C₄⋊C₈	C₂³×C₄₁	1	2	yes	—
228	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
229	#102	—	2	8	no	—
230	C₂²⋊C₈	—	2	4	no	—
231	C₄×C₈	D₄×C₄₁	1	2	yes	—
232	C₄×C₈	C₂³×C₄₁	1	2	yes	—
233	C₂⁴	—	2	1	no	—
234	C₂⁴	—	4	2	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	4	no	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	4	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	4	no	—
244	C₂×D₄	—	4	4	no	—
245	C₂×D₄	—	4	4	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂×D₄	—	8	4	no	—
248	C₂×D₄	C₂×D₄×C₄₁	1	2	yes	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
256	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
257	D₈	—	4	8	no	—
258	D₈	—	4	16	no	—
259	D₈	—	4	8	no	—
260	M₁₆	—	4	8	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	4	4	no	—
266	C₂×C₈	D₈×C₄₁	1	4	yes	complemented
267	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₄₁	1	2	yes	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	C₂×D₄×C₄₁	1	2	yes	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
275	C₄²	#41×C₄₁	1	1	yes	characteristic
276	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
277	C₂³	—	2	2	no	—
278	C₂³	—	4	2	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	4	no	—
282	C₂³	—	8	4	no	—
283	C₂³	—	8	4	no	—
284	C₂³	—	8	4	no	—
285	C₂³	#121×C₄₁	1	1	yes	characteristic
286	D₄	—	2	4	no	—
287	D₄	—	4	4	no	—
288	D₄	—	4	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	2	no	—
292	D₄	—	8	4	no	—
293	D₄	—	8	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	8	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
303	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
304	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
305	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
306	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	2	yes	—
307	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
308	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
310	C₂×C₄	C₂×#41×C₄₁	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	4	2	no	—
313	C₈	—	4	4	no	—
314	C₈	C₂×D₈×C₄₁	1	4	yes	complemented
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	8	4	no	—
319	C₂²	—	8	4	no	—
320	C₂²	—	8	4	no	—
321	C₂²	—	8	4	no	—
322	C₂²	—	16	8	no	—
323	C₂²	C₄₁×#459	1	2	yes	—
324	C₂²	C₄₁×#469	1	2	yes	—
325	C₂²	C₄₁×#472	1	2	yes	—
326	C₂²	C₂×#121×C₄₁	1	1	yes	characteristic
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	4	2	no	—
330	C₄	C₄₁×#459	1	2	yes	—
331	C₄	C₄₁×#469	1	2	yes	—
332	C₄	C₂²×D₈×C₄₁	1	1	yes	characteristic
333	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	2	yes	—
334	C₄	C₂×AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
335	C₂	—	8	2	no	—
336	C₂	—	16	4	no	—
337	C₂	C₄₁×#1687	1	4	yes	direct product factor
338	C₂	C₂×C₄₁×#459	1	1	yes	characteristic
339	C₂	C₂×C₄₁×#469	1	1	yes	characteristic
340	C₂	C₂×C₄₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₄₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#493	C₂	semidirect
2. C₂×C₄₁×#459	C₂	semidirect
3. C₂×C₄₁×#436	C₂	semidirect
4. C₄₁×#1687	C₂	direct (N × H)
5. C₂×C₄₁×#334	C₂	semidirect
6. C₂×C₄₁×#331	C₂	semidirect
14. C₄₁×#493	C₂²	semidirect
16. C₄₁×#459	C₂²	semidirect
18. C₄₁×#436	C₂²	semidirect
24. C₄₁×#334	C₂²	semidirect
25. C₄₁×#331	C₂²	semidirect
95. C₂×C₈×C₄₁	D₈	semidirect
143. C₈×C₄₁	C₂×D₈	semidirect
144. C₂×#1687	C₄₁	direct (N × H)
165. C₂×#493	C₂×C₄₁	semidirect
166. C₂×#459	C₂×C₄₁	semidirect
167. C₂×#436	C₂×C₄₁	semidirect
168. #1687	C₂×C₄₁	direct (N × H)
169. C₂×#334	C₂×C₄₁	semidirect
170. C₂×#331	C₂×C₄₁	semidirect
173. C₂×C₄₁	#1687	direct (N × H)
184. #493	C₂²×C₄₁	semidirect
186. #459	C₂²×C₄₁	semidirect
188. #436	C₂²×C₄₁	semidirect
194. #334	C₂²×C₄₁	semidirect
195. #331	C₂²×C₄₁	semidirect
196. C₄₁	C₂×#1687	direct (N × H)
266. C₂×C₈	D₈×C₄₁	semidirect
314. C₈	C₂×D₈×C₄₁	semidirect
337. C₂	C₄₁×#1687	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×#493	2
C₂×C₄₁×#459	2
C₂×C₄₁×#436	2
C₄₁×#1687	2
C₂×C₄₁×#334	2
C₂×C₄₁×#331	2
C₂×#1687	41

Subgroups — C2×C41×#1687 #23795332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1687 #23795332