Subgroups — C₄₁×#8279 #23784480

Order 10496.

394 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#1767	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
3	C₄₁×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₁×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₁×#334	C₂	1	1	yes	maximal, characteristic
6	C₄₁×#1277	C₂	1	1	yes	maximal, characteristic
7	C₄₁×#1277	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×#97×C₄₁	—	2	1	no	—
13	C₂×C₄×C₈×C₄₁	—	2	1	no	—
14	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₄³×C₄₁	C₂²	1	1	yes	characteristic
17	C₄₁×#336	—	2	1	no	—
18	C₄₁×#336	—	2	1	no	—
19	C₄₁×#336	C₄	1	4	yes	—
20	C₄₁×#335	C₄	1	4	yes	complemented
21	C₄₁×#334	C₄	1	4	yes	complemented
22	C₂²×C₈×C₄₁	—	2	1	no	—
23	C₂²×C₈×C₄₁	—	2	1	no	—
24	C₂²×C₈×C₄₁	—	2	1	no	—
25	C₂²×C₈×C₄₁	—	2	1	no	—
26	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₄₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₄₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
31	C₂×C₄²×C₄₁	Q₈	1	2	yes	—
32	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₄₁	—	2	1	no	—
34	C₄⋊C₈×C₄₁	—	2	1	no	—
35	C₄⋊C₈×C₄₁	—	2	1	no	—
36	C₄⋊C₈×C₄₁	—	2	1	no	—
37	C₄⋊C₈×C₄₁	C₂×C₄	1	2	yes	—
38	C₄⋊C₈×C₄₁	C₂×C₄	1	2	yes	—
39	C₄⋊C₈×C₄₁	C₂×C₄	1	2	yes	—
40	C₄⋊C₈×C₄₁	C₂×C₄	1	2	yes	—
41	#97×C₄₁	—	4	2	no	—
42	C₄×C₈×C₄₁	—	2	1	no	—
43	C₄×C₈×C₄₁	—	2	1	no	—
44	C₄×C₈×C₄₁	—	4	2	no	—
45	C₄×C₈×C₄₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₄₁	C₂×C₄	1	2	yes	—
47	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₄₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₄₁	—	2	4	no	—
53	C₂×C₈×C₄₁	—	2	2	no	—
54	C₂×C₈×C₄₁	—	2	2	no	—
55	C₂×C₈×C₄₁	—	2	2	no	—
56	C₂×C₈×C₄₁	—	2	2	no	—
57	C₂×C₈×C₄₁	—	2	2	no	—
58	C₂×C₈×C₄₁	—	2	4	no	—
59	C₂×C₈×C₄₁	—	4	1	no	—
60	C₂×C₈×C₄₁	—	4	1	no	—
61	C₂×C₈×C₄₁	—	4	1	no	—
62	C₂×C₈×C₄₁	—	4	1	no	—
63	C₂×C₈×C₄₁	—	4	1	no	—
64	C₂×C₈×C₄₁	—	4	1	no	—
65	C₂×C₈×C₄₁	—	4	1	no	—
66	C₂×C₈×C₄₁	—	4	1	no	—
67	C₂×C₈×C₄₁	C₄²	1	4	yes	—
68	C₄²×C₄₁	—	2	4	no	—
69	C₄²×C₄₁	—	2	4	no	—
70	C₄²×C₄₁	—	2	1	no	—
71	C₄²×C₄₁	—	2	1	no	—
72	C₄²×C₄₁	—	2	1	no	—
73	C₄²×C₄₁	—	2	1	no	—
74	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₄₁	#41	1	1	yes	characteristic
77	C₄²×C₄₁	#41	1	1	yes	characteristic
78	C₂³×C₄₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₄₁	—	2	2	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	2	no	—
84	C₂×C₄×C₄₁	—	2	2	no	—
85	C₂×C₄×C₄₁	#108	1	4	yes	—
86	C₂×C₄×C₄₁	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₄₁	#117	1	1	yes	characteristic
88	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₄₁	C₄×D₄	1	1	yes	characteristic
90	C₂×C₄×C₄₁	C₄×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₄₁	C₄×Q₈	1	1	yes	characteristic
92	C₂×C₄×C₄₁	#131	1	4	yes	—
93	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
95	C₂×C₄×C₄₁	#137	1	1	yes	characteristic
96	C₂×C₄×C₄₁	#137	1	1	yes	characteristic
97	C₈×C₄₁	—	2	4	no	—
98	C₈×C₄₁	—	4	2	no	—
99	C₈×C₄₁	—	4	2	no	—
100	C₈×C₄₁	—	4	2	no	—
101	C₈×C₄₁	—	4	2	no	—
102	C₈×C₄₁	—	4	2	no	—
103	#8279	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₄₁	C₄×C₄⋊C₄	1	1	yes	characteristic
105	C₂²×C₄₁	#424	1	2	yes	—
106	C₂²×C₄₁	#428	1	1	yes	characteristic
107	C₂²×C₄₁	#428	1	1	yes	characteristic
108	C₂²×C₄₁	C₂×#108	1	2	yes	—
109	C₄×C₄₁	—	2	4	no	—
110	C₄×C₄₁	—	2	1	no	—
111	C₄×C₄₁	—	2	1	no	—
112	C₄×C₄₁	—	4	2	no	—
113	C₄×C₄₁	#405	1	2	yes	—
114	C₄×C₄₁	#433	1	2	yes	—
115	C₄×C₄₁	#440	1	1	yes	characteristic
116	C₄×C₄₁	#440	1	1	yes	characteristic
117	C₄×C₄₁	#441	1	1	yes	characteristic
118	C₄×C₄₁	#442	1	1	yes	characteristic
119	#1767	C₂×C₄₁	1	1	yes	characteristic
120	C₄×C₄⋊C₈	C₂×C₄₁	1	1	yes	characteristic
121	#1744	C₂×C₄₁	1	1	yes	characteristic
122	C₂×#335	C₂×C₄₁	1	1	yes	characteristic
123	C₂×#334	C₂×C₄₁	1	1	yes	characteristic
124	#1277	C₂×C₄₁	1	1	yes	characteristic
125	#1277	C₂×C₄₁	1	1	yes	characteristic
126	C₂×C₄₁	#1573	1	2	yes	—
127	C₂×C₄₁	#1574	1	2	yes	—
128	C₂×C₄₁	#1768	1	1	yes	characteristic
129	C₂×C₄₁	#1776	1	1	yes	characteristic
130	C₂×C₄₁	C₄×#108	1	1	yes	characteristic
131	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
132	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
133	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
134	C₂×C₄⋊C₈	C₂²×C₄₁	1	1	yes	characteristic
135	C₂×#97	—	2	1	no	—
136	C₂×C₄×C₈	—	2	1	no	—
137	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
139	C₄³	C₂²×C₄₁	1	1	yes	characteristic
140	#336	—	2	1	no	—
141	#336	—	2	1	no	—
142	#336	C₄×C₄₁	1	4	yes	—
143	#335	C₄×C₄₁	1	4	yes	complemented
144	#334	C₄×C₄₁	1	4	yes	complemented
145	C₄₁	#8279	1	1	yes	characteristic, direct product factor, Sylow
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	2	1	no	—
149	C₂²×C₈	—	2	1	no	—
150	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
151	C₂²×C₈	C₂×C₄×C₄₁	1	2	yes	—
152	C₂×C₄²	C₂×C₄×C₄₁	1	2	yes	—
153	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
154	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
155	C₂×C₄²	Q₈×C₄₁	1	2	yes	—
156	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
157	C₄⋊C₈	—	2	1	no	—
158	C₄⋊C₈	—	2	1	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	2	1	no	—
161	C₄⋊C₈	C₂×C₄×C₄₁	1	2	yes	—
162	C₄⋊C₈	C₂×C₄×C₄₁	1	2	yes	—
163	C₄⋊C₈	C₂×C₄×C₄₁	1	2	yes	—
164	C₄⋊C₈	C₂×C₄×C₄₁	1	2	yes	—
165	#97	—	4	2	no	—
166	C₄×C₈	—	2	1	no	—
167	C₄×C₈	—	2	1	no	—
168	C₄×C₈	—	4	2	no	—
169	C₄×C₈	C₂×C₄×C₄₁	1	2	yes	—
170	C₄×C₈	C₂×C₄×C₄₁	1	2	yes	—
171	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
172	C₂²×C₄	C₄⋊C₄×C₄₁	1	2	yes	—
173	C₂²×C₄	C₂²×C₄×C₄₁	1	1	yes	characteristic
174	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
175	C₂²×C₄	C₂×Q₈×C₄₁	1	1	yes	characteristic
176	C₂×C₈	—	2	4	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	1	no	—
184	C₂×C₈	—	4	1	no	—
185	C₂×C₈	—	4	1	no	—
186	C₂×C₈	—	4	1	no	—
187	C₂×C₈	—	4	1	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	4	1	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	C₄²×C₄₁	1	4	yes	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	4	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	1	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
199	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
200	C₄²	#41×C₄₁	1	1	yes	characteristic
201	C₄²	#41×C₄₁	1	1	yes	characteristic
202	C₂³	C₂×C₄⋊C₄×C₄₁	1	1	yes	characteristic
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	#108×C₄₁	1	4	yes	—
210	C₂×C₄	C₂×C₄²×C₄₁	1	1	yes	characteristic
211	C₂×C₄	#117×C₄₁	1	1	yes	characteristic
212	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
213	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
214	C₂×C₄	C₄×Q₈×C₄₁	1	1	yes	characteristic
215	C₂×C₄	C₄×Q₈×C₄₁	1	1	yes	characteristic
216	C₂×C₄	#131×C₄₁	1	4	yes	—
217	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
218	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	1	yes	characteristic
219	C₂×C₄	#137×C₄₁	1	1	yes	characteristic
220	C₂×C₄	#137×C₄₁	1	1	yes	characteristic
221	C₈	—	2	4	no	—
222	C₈	—	4	2	no	—
223	C₈	—	4	2	no	—
224	C₈	—	4	2	no	—
225	C₈	—	4	2	no	—
226	C₈	—	4	2	no	—
227	C₂²	C₄×C₄⋊C₄×C₄₁	1	1	yes	characteristic
228	C₂²	C₄₁×#424	1	2	yes	—
229	C₂²	C₄₁×#428	1	1	yes	characteristic
230	C₂²	C₄₁×#428	1	1	yes	characteristic
231	C₂²	C₂×#108×C₄₁	1	2	yes	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	4	no	—
235	C₄	—	4	2	no	—
236	C₄	C₄₁×#405	1	2	yes	—
237	C₄	C₄₁×#433	1	2	yes	—
238	C₄	C₄₁×#440	1	1	yes	characteristic
239	C₄	C₄₁×#440	1	1	yes	characteristic
240	C₄	C₄₁×#441	1	1	yes	characteristic
241	C₄	C₄₁×#442	1	1	yes	characteristic
242	C₂	C₄₁×#1573	1	2	yes	—
243	C₂	C₄₁×#1574	1	2	yes	—
244	C₂	C₄₁×#1768	1	1	yes	characteristic
245	C₂	C₄₁×#1776	1	1	yes	characteristic
246	C₂	C₄×#108×C₄₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
socle, prime radical	C₂³×C₄₁	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₄₁ ◁ C₄³×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
20. C₄₁×#335	C₄	semidirect
21. C₄₁×#334	C₄	semidirect
103. #8279	C₄₁	direct (N × H)
143. #335	C₄×C₄₁	semidirect
144. #334	C₄×C₄₁	semidirect
145. C₄₁	#8279	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄₁×#1767	2
C₄×C₄⋊C₈×C₄₁	2
C₄₁×#1744	2
C₂×C₄₁×#335	2
C₂×C₄₁×#334	2
C₄₁×#1277	2
#8279	41

Subgroups — C41×#8279 #23784480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#8279 #23784480