Subgroups — C₂×C₄₁×#1571 #23794002

Order 10496.

770 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1571	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄₁×#335	C₂	1	4	yes	maximal, complemented
6	C₂²×M₁₆×C₄₁	—	2	1	no	—
7	C₂³×C₈×C₄₁	—	2	1	no	—
8	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
9	C₄₁×#431	C₄	1	4	yes	—
10	C₄₁×#431	C₂²	1	4	yes	—
11	C₄₁×#423	—	2	2	no	—
12	C₄₁×#423	C₂²	1	4	yes	—
13	C₂×C₄⋊C₈×C₄₁	—	2	2	no	—
14	C₂×C₄⋊C₈×C₄₁	—	2	1	no	—
15	C₂×C₄⋊C₈×C₄₁	—	2	1	no	—
16	C₂×C₄⋊C₈×C₄₁	C₄	1	2	yes	—
17	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₄₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₄₁	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₈×C₄₁	C₄	1	2	yes	—
21	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
22	C₄₁×#335	C₂²	1	16	yes	complemented
23	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₄₁	Q₈	1	1	yes	characteristic
26	C₂×M₁₆×C₄₁	—	2	4	no	—
27	C₂×M₁₆×C₄₁	—	4	4	no	—
28	C₂²×C₈×C₄₁	—	2	4	no	—
29	C₂²×C₈×C₄₁	—	2	2	no	—
30	C₂²×C₈×C₄₁	—	2	2	no	—
31	C₂²×C₈×C₄₁	—	4	4	no	—
32	C₂²×C₈×C₄₁	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₄₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₄₁	D₄	1	2	yes	—
36	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₁	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₄₁	Q₈	1	2	yes	—
39	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
40	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₄₁	—	2	2	no	—
42	C₄⋊C₈×C₄₁	—	2	4	no	—
43	C₄⋊C₈×C₄₁	—	2	4	no	—
44	C₄⋊C₈×C₄₁	—	2	2	no	—
45	C₄⋊C₈×C₄₁	C₂×C₄	1	8	yes	—
46	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
47	C₄⋊C₈×C₄₁	C₂³	1	4	yes	—
48	C₂²⋊C₈×C₄₁	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
50	C₄×C₈×C₄₁	C₂³	1	4	yes	—
51	C₂⁴×C₄₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₁	—	2	4	no	—
53	C₂²×C₄×C₄₁	—	4	2	no	—
54	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₄₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
60	C₂²×C₄×C₄₁	D₈	1	4	yes	—
61	C₂²×C₄×C₄₁	Q₁₆	1	4	yes	—
62	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
65	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
68	M₁₆×C₄₁	—	4	8	no	—
69	C₂×C₈×C₄₁	—	2	8	no	—
70	C₂×C₈×C₄₁	—	2	4	no	—
71	C₂×C₈×C₄₁	—	2	4	no	—
72	C₂×C₈×C₄₁	—	4	4	no	—
73	C₂×C₈×C₄₁	—	4	8	no	—
74	C₂×C₈×C₄₁	—	4	4	no	—
75	C₂×C₈×C₄₁	C₂²×C₄	1	8	yes	—
76	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
77	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
78	C₄²×C₄₁	C₄⋊C₄	1	2	yes	—
79	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
81	C₄²×C₄₁	C₂×D₄	1	2	yes	—
82	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
84	C₄²×C₄₁	C₂×Q₈	1	1	yes	characteristic
85	C₄²×C₄₁	C₂⁴	1	1	yes	characteristic
86	C₂³×C₄₁	—	2	2	no	—
87	C₂³×C₄₁	—	2	2	no	—
88	C₂³×C₄₁	#107	1	4	yes	—
89	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
90	C₂³×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₄₁	—	2	2	no	—
92	C₂×C₄×C₄₁	—	2	2	no	—
93	C₂×C₄×C₄₁	—	2	2	no	—
94	C₂×C₄×C₄₁	—	2	4	no	—
95	C₂×C₄×C₄₁	—	2	4	no	—
96	C₂×C₄×C₄₁	—	4	2	no	—
97	C₂×C₄×C₄₁	—	4	2	no	—
98	C₂×C₄×C₄₁	#107	1	8	yes	—
99	C₂×C₄×C₄₁	#107	1	4	yes	—
100	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
105	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₁	#131	1	4	yes	—
107	C₂×C₄×C₄₁	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₄₁	C₂×D₈	1	4	yes	—
109	C₂×C₄×C₄₁	C₂×Q₁₆	1	4	yes	—
110	C₂×C₄×C₄₁	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
114	C₈×C₄₁	—	2	8	no	—
115	C₈×C₄₁	—	4	4	no	—
116	C₈×C₄₁	—	4	4	no	—
117	C₂×#1571	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂²×C₄₁	—	2	8	no	—
119	C₂²×C₄₁	—	2	2	no	—
120	C₂²×C₄₁	—	2	2	no	—
121	C₂²×C₄₁	#424	1	2	yes	—
122	C₂²×C₄₁	C₂×#107	1	4	yes	—
123	C₂²×C₄₁	C₂×#107	1	2	yes	—
124	C₂²×C₄₁	#430	1	2	yes	—
125	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₄×C₄₁	—	2	2	no	—
127	C₄×C₄₁	—	2	2	no	—
128	C₄×C₄₁	—	2	1	no	—
129	C₄×C₄₁	—	2	1	no	—
130	C₄×C₄₁	#424	1	2	yes	—
131	C₄×C₄₁	C₂×#107	1	4	yes	—
132	C₄×C₄₁	C₂×#107	1	2	yes	—
133	C₄×C₄₁	C₂×#131	1	2	yes	—
134	C₄×C₄₁	C₂²×D₈	1	1	yes	characteristic
135	C₄×C₄₁	C₂²×Q₁₆	1	1	yes	characteristic
136	C₂×#431	C₂×C₄₁	1	1	yes	characteristic
137	C₂×#423	C₂×C₄₁	1	1	yes	characteristic
138	C₂²×C₄⋊C₈	C₂×C₄₁	1	1	yes	characteristic
139	#1571	C₂×C₄₁	1	8	yes	direct product factor
140	C₂×#335	C₂×C₄₁	1	4	yes	complemented
141	C₂×C₄₁	—	2	4	no	—
142	C₂×C₄₁	#1571	1	4	yes	direct product factor
143	C₂×C₄₁	C₂×#424	1	1	yes	characteristic
144	C₂×C₄₁	C₂²×#107	1	1	yes	characteristic
145	C₂×C₄₁	C₂×#430	1	1	yes	characteristic
146	C₂²×M₁₆	—	2	1	no	—
147	C₂³×C₈	—	2	1	no	—
148	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
149	#431	C₄×C₄₁	1	4	yes	—
150	#431	C₂²×C₄₁	1	4	yes	—
151	#423	—	2	2	no	—
152	#423	C₂²×C₄₁	1	4	yes	—
153	C₂×C₄⋊C₈	—	2	1	no	—
154	C₂×C₄⋊C₈	—	2	2	no	—
155	C₂×C₄⋊C₈	—	2	1	no	—
156	C₂×C₄⋊C₈	C₄×C₄₁	1	2	yes	—
157	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
158	C₂×C₄⋊C₈	C₂²×C₄₁	1	2	yes	—
159	C₂×C₄⋊C₈	C₂²×C₄₁	1	4	yes	—
160	C₂×C₂²⋊C₈	C₄×C₄₁	1	2	yes	—
161	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
162	#335	C₂²×C₄₁	1	16	yes	complemented
163	C₄₁	C₂×#1571	1	1	yes	characteristic, direct product factor, Sylow
164	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
165	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
166	C₂³×C₄	Q₈×C₄₁	1	1	yes	characteristic
167	C₂×M₁₆	—	2	4	no	—
168	C₂×M₁₆	—	4	4	no	—
169	C₂²×C₈	—	2	2	no	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	4	4	no	—
173	C₂²×C₈	C₂×C₄×C₄₁	1	4	yes	—
174	C₂×C₄²	C₂×C₄×C₄₁	1	1	yes	characteristic
175	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
176	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
177	C₂×C₄²	D₄×C₄₁	1	2	yes	—
178	C₂×C₄²	Q₈×C₄₁	1	1	yes	characteristic
179	C₂×C₄²	Q₈×C₄₁	1	2	yes	—
180	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
181	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
182	C₄⋊C₈	—	2	2	no	—
183	C₄⋊C₈	—	2	4	no	—
184	C₄⋊C₈	—	2	4	no	—
185	C₄⋊C₈	—	2	2	no	—
186	C₄⋊C₈	C₂×C₄×C₄₁	1	8	yes	—
187	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
188	C₄⋊C₈	C₂³×C₄₁	1	4	yes	—
189	C₂²⋊C₈	C₂×C₄×C₄₁	1	8	yes	—
190	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
191	C₄×C₈	C₂³×C₄₁	1	4	yes	—
192	C₂⁴	C₄⋊C₄×C₄₁	1	1	yes	characteristic
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
196	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
197	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
198	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
199	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
200	C₂²×C₄	C₄⋊C₄×C₄₁	1	1	yes	characteristic
201	C₂²×C₄	D₈×C₄₁	1	4	yes	—
202	C₂²×C₄	Q₁₆×C₄₁	1	4	yes	—
203	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
205	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
207	C₂²×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
208	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
209	M₁₆	—	4	8	no	—
210	C₂×C₈	—	2	8	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	4	4	no	—
214	C₂×C₈	—	4	8	no	—
215	C₂×C₈	—	4	4	no	—
216	C₂×C₈	C₂²×C₄×C₄₁	1	8	yes	—
217	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
218	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
219	C₄²	C₄⋊C₄×C₄₁	1	2	yes	—
220	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
221	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
222	C₄²	C₂×D₄×C₄₁	1	2	yes	—
223	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
224	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
225	C₄²	C₂×Q₈×C₄₁	1	1	yes	characteristic
226	C₄²	C₂⁴×C₄₁	1	1	yes	characteristic
227	C₂³	—	2	2	no	—
228	C₂³	—	2	2	no	—
229	C₂³	#107×C₄₁	1	4	yes	—
230	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
231	C₂³	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	#107×C₄₁	1	8	yes	—
240	C₂×C₄	#107×C₄₁	1	4	yes	—
241	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
242	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
244	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
245	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
246	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
247	C₂×C₄	#131×C₄₁	1	4	yes	—
248	C₂×C₄	C₂×D₈×C₄₁	1	2	yes	—
249	C₂×C₄	C₂×D₈×C₄₁	1	4	yes	—
250	C₂×C₄	C₂×Q₁₆×C₄₁	1	4	yes	—
251	C₂×C₄	C₂×Q₁₆×C₄₁	1	2	yes	—
252	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
253	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
254	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
255	C₈	—	2	8	no	—
256	C₈	—	4	4	no	—
257	C₈	—	4	4	no	—
258	C₂²	—	2	8	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	C₄₁×#424	1	2	yes	—
262	C₂²	C₂×#107×C₄₁	1	4	yes	—
263	C₂²	C₂×#107×C₄₁	1	2	yes	—
264	C₂²	C₄₁×#430	1	2	yes	—
265	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	C₄₁×#424	1	2	yes	—
271	C₄	C₂×#107×C₄₁	1	2	yes	—
272	C₄	C₂×#107×C₄₁	1	4	yes	—
273	C₄	C₂×#131×C₄₁	1	2	yes	—
274	C₄	C₂²×D₈×C₄₁	1	1	yes	characteristic
275	C₄	C₂²×Q₁₆×C₄₁	1	1	yes	characteristic
276	C₂	—	2	4	no	—
277	C₂	C₄₁×#1571	1	4	yes	direct product factor
278	C₂	C₂×C₄₁×#424	1	1	yes	characteristic
279	C₂	C₂²×#107×C₄₁	1	1	yes	characteristic
280	C₂	C₂×C₄₁×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₄²	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₁	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄₁×#1571	C₂	direct (N × H)
5. C₂×C₄₁×#335	C₂	semidirect
22. C₄₁×#335	C₂²	semidirect
117. C₂×#1571	C₄₁	direct (N × H)
139. #1571	C₂×C₄₁	direct (N × H)
140. C₂×#335	C₂×C₄₁	semidirect
142. C₂×C₄₁	#1571	direct (N × H)
162. #335	C₂²×C₄₁	semidirect
163. C₄₁	C₂×#1571	direct (N × H)
277. C₂	C₄₁×#1571	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×#431	2
C₂×C₄₁×#423	2
C₂²×C₄⋊C₈×C₄₁	2
C₄₁×#1571	2
C₂×C₄₁×#335	2
C₂×#1571	41

Subgroups — C2×C41×#1571 #23794002

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄₁×#1571 #23794002