Subgroups — C₄₁×#8701 #23784902

Order 10496.

642 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊C₄×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1817	C₂	1	2	yes	maximal
3	C₄₁×#1280	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₄₁	C₄	1	1	yes	characteristic
5	C₂²×C₄⋊C₄×C₄₁	C₄	1	1	yes	characteristic
6	C₂²×C₄²×C₄₁	C₄	1	2	yes	—
7	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
8	C₂×C₂²⋊C₈×C₄₁	C₂²	1	4	yes	—
9	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
10	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
11	C₂×#95×C₄₁	C₂²	1	2	yes	—
12	C₄₁×#336	—	2	2	no	—
13	C₄₁×#336	—	2	2	no	—
14	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
17	C₂³×C₄×C₄₁	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
19	C₂²×C₈×C₄₁	—	2	4	no	—
20	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
21	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
22	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
23	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
24	C₂×C₄⋊C₄×C₄₁	C₈	1	4	yes	complemented
25	C₂×C₄²×C₄₁	—	2	4	no	—
26	C₂×C₄²×C₄₁	—	2	2	no	—
27	C₂×C₄²×C₄₁	—	2	2	no	—
28	C₂×C₄²×C₄₁	—	4	4	no	—
29	C₂×C₄²×C₄₁	D₄	1	2	yes	—
30	C₂×C₄²×C₄₁	D₄	1	2	yes	—
31	C₂²⋊C₈×C₄₁	—	2	8	no	—
32	#95×C₄₁	—	2	4	no	—
33	C₂⁴×C₄₁	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₁	—	2	2	no	—
35	C₂²×C₄×C₄₁	—	2	2	no	—
36	C₂²×C₄×C₄₁	—	2	2	no	—
37	C₂²×C₄×C₄₁	—	2	2	no	—
38	C₂²×C₄×C₄₁	—	2	2	no	—
39	C₂²×C₄×C₄₁	—	2	2	no	—
40	C₂²×C₄×C₄₁	—	2	2	no	—
41	C₂²×C₄×C₄₁	—	2	2	no	—
42	C₂²×C₄×C₄₁	—	2	2	no	—
43	C₂²×C₄×C₄₁	—	4	1	no	—
44	C₂²×C₄×C₄₁	—	4	1	no	—
45	C₂²×C₄×C₄₁	—	4	2	no	—
46	C₂²×C₄×C₄₁	—	4	2	no	—
47	C₂²×C₄×C₄₁	—	4	2	no	—
48	C₂²×C₄×C₄₁	—	4	2	no	—
49	C₂²×C₄×C₄₁	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₁	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₄₁	C₂×C₈	1	4	yes	—
54	C₂²×C₄×C₄₁	M₁₆	1	2	yes	—
55	C₂²×C₄×C₄₁	M₁₆	1	4	yes	—
56	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₁	#41	1	4	yes	—
58	C₂×C₈×C₄₁	—	4	4	no	—
59	C₂×C₈×C₄₁	—	4	4	no	—
60	C₂×C₈×C₄₁	—	4	4	no	—
61	C₄⋊C₄×C₄₁	—	2	8	no	—
62	C₄⋊C₄×C₄₁	—	2	8	no	—
63	C₄²×C₄₁	—	2	4	no	—
64	C₄²×C₄₁	—	2	4	no	—
65	C₄²×C₄₁	—	4	4	no	—
66	C₄²×C₄₁	—	4	4	no	—
67	C₂³×C₄₁	—	2	1	no	—
68	C₂³×C₄₁	—	2	1	no	—
69	C₂³×C₄₁	—	2	1	no	—
70	C₂³×C₄₁	—	2	1	no	—
71	C₂³×C₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₂³×C₄₁	#117	1	2	yes	—
73	C₂³×C₄₁	C₂²×C₈	1	1	yes	characteristic
74	C₂³×C₄₁	C₂×M₁₆	1	1	yes	characteristic
75	C₂³×C₄₁	C₂×M₁₆	1	2	yes	—
76	C₂×C₄×C₄₁	—	2	8	no	—
77	C₂×C₄×C₄₁	—	2	2	no	—
78	C₂×C₄×C₄₁	—	2	1	no	—
79	C₂×C₄×C₄₁	—	2	1	no	—
80	C₂×C₄×C₄₁	—	2	2	no	—
81	C₂×C₄×C₄₁	—	2	2	no	—
82	C₂×C₄×C₄₁	—	2	2	no	—
83	C₂×C₄×C₄₁	—	2	1	no	—
84	C₂×C₄×C₄₁	—	2	1	no	—
85	C₂×C₄×C₄₁	—	2	4	no	—
86	C₂×C₄×C₄₁	—	2	4	no	—
87	C₂×C₄×C₄₁	—	2	4	no	—
88	C₂×C₄×C₄₁	—	2	4	no	—
89	C₂×C₄×C₄₁	—	4	4	no	—
90	C₂×C₄×C₄₁	—	4	4	no	—
91	C₂×C₄×C₄₁	—	4	4	no	—
92	C₂×C₄×C₄₁	—	4	2	no	—
93	C₂×C₄×C₄₁	—	4	2	no	—
94	C₂×C₄×C₄₁	—	4	2	no	—
95	C₂×C₄×C₄₁	—	4	2	no	—
96	C₂×C₄×C₄₁	—	4	4	no	—
97	C₂×C₄×C₄₁	—	4	4	no	—
98	C₂×C₄×C₄₁	C₂²⋊C₈	1	4	yes	—
99	C₂×C₄×C₄₁	#100	1	2	yes	—
100	C₂×C₄×C₄₁	#101	1	2	yes	—
101	C₂×C₄×C₄₁	#123	1	4	yes	—
102	C₈×C₄₁	—	8	4	no	—
103	#8701	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₄₁	—	2	2	no	—
105	C₂²×C₄₁	—	2	2	no	—
106	C₂²×C₄₁	—	2	1	no	—
107	C₂²×C₄₁	—	2	1	no	—
108	C₂²×C₄₁	—	2	1	no	—
109	C₂²×C₄₁	—	2	1	no	—
110	C₂²×C₄₁	—	2	2	no	—
111	C₂²×C₄₁	—	2	2	no	—
112	C₂²×C₄₁	#381	1	1	yes	characteristic
113	C₂²×C₄₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²×C₄₁	#407	1	1	yes	characteristic
115	C₂²×C₄₁	#410	1	1	yes	characteristic
116	C₂²×C₄₁	#410	1	1	yes	characteristic
117	C₂²×C₄₁	C₂×#100	1	1	yes	characteristic
118	C₂²×C₄₁	C₂×#101	1	1	yes	characteristic
119	C₂²×C₄₁	#431	1	2	yes	—
120	C₂²×C₄₁	#432	1	2	yes	—
121	C₄×C₄₁	—	2	4	no	—
122	C₄×C₄₁	—	2	4	no	—
123	C₄×C₄₁	—	2	2	no	—
124	C₄×C₄₁	—	2	2	no	—
125	C₄×C₄₁	—	4	4	no	—
126	C₄×C₄₁	—	4	4	no	—
127	C₂×C₄×C₄⋊C₄	C₂×C₄₁	1	1	yes	characteristic
128	#1817	C₂×C₄₁	1	2	yes	—
129	#1280	C₂×C₄₁	1	4	yes	—
130	C₂×C₄₁	—	2	2	no	—
131	C₂×C₄₁	—	2	2	no	—
132	C₂×C₄₁	#1460	1	1	yes	characteristic
133	C₂×C₄₁	#1461	1	1	yes	characteristic
134	C₂×C₄₁	#1466	1	1	yes	characteristic
135	C₂×C₄₁	#1467	1	1	yes	characteristic
136	C₂×C₄₁	#1817	1	1	yes	characteristic
137	C₂×C₄₁	#1820	1	1	yes	characteristic
138	C₂×C₄₁	#1821	1	1	yes	characteristic
139	C₂²×C₄⋊C₄	C₄×C₄₁	1	1	yes	characteristic
140	C₂²×C₄⋊C₄	C₄×C₄₁	1	1	yes	characteristic
141	C₂²×C₄²	C₄×C₄₁	1	2	yes	—
142	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
143	C₂×C₂²⋊C₈	C₂²×C₄₁	1	4	yes	—
144	C₄×C₄⋊C₄	—	2	2	no	—
145	C₄×C₄⋊C₄	—	2	2	no	—
146	C₂×#95	C₂²×C₄₁	1	2	yes	—
147	#336	—	2	2	no	—
148	#336	—	2	2	no	—
149	C₄₁	#8701	1	1	yes	characteristic, direct product factor, Sylow
150	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
151	C₂³×C₄	C₂×C₄×C₄₁	1	1	yes	characteristic
152	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
153	C₂³×C₄	C₂×C₄×C₄₁	1	2	yes	—
154	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
155	C₂²×C₈	—	2	4	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	C₈×C₄₁	1	4	yes	complemented
161	C₂×C₄²	—	2	4	no	—
162	C₂×C₄²	—	2	2	no	—
163	C₂×C₄²	—	2	2	no	—
164	C₂×C₄²	—	4	4	no	—
165	C₂×C₄²	D₄×C₄₁	1	2	yes	—
166	C₂×C₄²	D₄×C₄₁	1	2	yes	—
167	C₂²⋊C₈	—	2	8	no	—
168	#95	—	2	4	no	—
169	C₂⁴	C₂²×C₄×C₄₁	1	1	yes	characteristic
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	4	1	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₂²⋊C₄×C₄₁	1	2	yes	—
186	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
187	C₂²×C₄	C₂²⋊C₄×C₄₁	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₈×C₄₁	1	2	yes	—
189	C₂²×C₄	C₂×C₈×C₄₁	1	4	yes	—
190	C₂²×C₄	M₁₆×C₄₁	1	2	yes	—
191	C₂²×C₄	M₁₆×C₄₁	1	4	yes	—
192	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
193	C₂²×C₄	#41×C₄₁	1	4	yes	—
194	C₂×C₈	—	4	4	no	—
195	C₂×C₈	—	4	4	no	—
196	C₂×C₈	—	4	4	no	—
197	C₄⋊C₄	—	2	8	no	—
198	C₄⋊C₄	—	2	8	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	—	4	4	no	—
202	C₄²	—	4	4	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	1	no	—
206	C₂³	—	2	1	no	—
207	C₂³	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
208	C₂³	#117×C₄₁	1	2	yes	—
209	C₂³	C₂²×C₈×C₄₁	1	1	yes	characteristic
210	C₂³	C₂×M₁₆×C₄₁	1	1	yes	characteristic
211	C₂³	C₂×M₁₆×C₄₁	1	2	yes	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	C₂²⋊C₈×C₄₁	1	4	yes	—
235	C₂×C₄	#100×C₄₁	1	2	yes	—
236	C₂×C₄	#101×C₄₁	1	2	yes	—
237	C₂×C₄	#123×C₄₁	1	4	yes	—
238	C₈	—	8	4	no	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	C₄₁×#381	1	1	yes	characteristic
248	C₂²	C₂×C₂²⋊C₈×C₄₁	1	1	yes	characteristic
249	C₂²	C₄₁×#407	1	1	yes	characteristic
250	C₂²	C₄₁×#410	1	1	yes	characteristic
251	C₂²	C₄₁×#410	1	1	yes	characteristic
252	C₂²	C₂×#100×C₄₁	1	1	yes	characteristic
253	C₂²	C₂×#101×C₄₁	1	1	yes	characteristic
254	C₂²	C₄₁×#431	1	2	yes	—
255	C₂²	C₄₁×#432	1	2	yes	—
256	C₄	—	2	4	no	—
257	C₄	—	2	4	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	4	no	—
261	C₄	—	4	4	no	—
262	C₂	—	2	2	no	—
263	C₂	—	2	2	no	—
264	C₂	C₄₁×#1460	1	1	yes	characteristic
265	C₂	C₄₁×#1461	1	1	yes	characteristic
266	C₂	C₄₁×#1466	1	1	yes	characteristic
267	C₂	C₄₁×#1467	1	1	yes	characteristic
268	C₂	C₄₁×#1817	1	1	yes	characteristic
269	C₂	C₄₁×#1820	1	1	yes	characteristic
270	C₂	C₄₁×#1821	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₈×C₄₁
Frattini, cosocle	C₂³×C₄	C₂³×C₄₁
prime radical	C₂⁴×C₄₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₁ ◁ C₂²×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
24. C₂×C₄⋊C₄×C₄₁	C₈	semidirect
103. #8701	C₄₁	direct (N × H)
149. C₄₁	#8701	direct (N × H)
160. C₂×C₄⋊C₄	C₈×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₄⋊C₄×C₄₁	2
C₄₁×#1817	2
C₄₁×#1280	2
#8701	41

Subgroups — C41×#8701 #23784902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#8701 #23784902