Subgroups — C₄₁×#9860 #23786061

Order 10496.

498 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₁×#3147	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₁×#1492	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1492	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#135×C₄₁	—	2	2	no	—
7	C₄₁×#532	C₂²	1	2	yes	—
8	C₄₁×#495	—	2	2	no	—
9	C₄₁×#494	C₄	1	1	yes	characteristic
10	C₄₁×#493	C₄	1	1	yes	characteristic
11	C₄₁×#492	C₄	1	2	yes	—
12	C₄₁×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
16	C₄₁×#341	—	2	2	no	—
17	C₄₁×D₄⋊C₈	—	2	4	no	—
18	C₄₁×D₄⋊C₈	—	2	4	no	—
19	C₄₁×#322	—	2	2	no	—
20	C₂×#41×C₄₁	—	2	2	no	—
21	#135×C₄₁	—	4	8	no	—
22	C₂×Q₁₆×C₄₁	—	2	2	no	—
23	C₂×SD₁₆×C₄₁	—	2	4	no	—
24	C₂×D₈×C₄₁	—	2	2	no	—
25	C₂×M₁₆×C₄₁	—	2	2	no	—
26	C₂²×C₈×C₄₁	—	2	1	no	—
27	C₂²×C₈×C₄₁	—	2	1	no	—
28	C₂²×C₈×C₄₁	D₄	1	2	yes	—
29	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
30	#127×C₄₁	C₂×C₄	1	2	yes	—
31	#124×C₄₁	—	2	2	no	—
32	#121×C₄₁	—	2	2	no	—
33	#121×C₄₁	—	2	2	no	—
34	C₄×D₄×C₄₁	—	2	2	no	—
35	C₄×D₄×C₄₁	—	2	2	no	—
36	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₄₁	—	2	2	no	—
38	C₄⋊C₈×C₄₁	—	2	2	no	—
39	C₄×C₈×C₄₁	—	2	1	no	—
40	C₄×C₈×C₄₁	—	2	1	no	—
41	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₄₁	D₄	1	2	yes	—
44	C₄×C₈×C₄₁	D₄	1	2	yes	—
45	C₄×C₈×C₄₁	D₄	1	2	yes	—
46	#41×C₄₁	—	4	8	no	—
47	C₂×Q₈×C₄₁	—	2	2	no	—
48	C₂×D₄×C₄₁	—	2	2	no	—
49	C₂×D₄×C₄₁	—	2	2	no	—
50	C₂×D₄×C₄₁	—	2	2	no	—
51	C₂×D₄×C₄₁	—	4	2	no	—
52	C₂²×C₄×C₄₁	—	4	2	no	—
53	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
55	C₂²×C₄×C₄₁	#41	1	1	yes	characteristic
56	Q₁₆×C₄₁	—	4	4	no	—
57	SD₁₆×C₄₁	—	4	8	no	—
58	D₈×C₄₁	—	4	4	no	—
59	M₁₆×C₄₁	—	4	4	no	—
60	C₂×C₈×C₄₁	—	2	4	no	—
61	C₂×C₈×C₄₁	—	2	2	no	—
62	C₂×C₈×C₄₁	—	2	4	no	—
63	C₂×C₈×C₄₁	—	2	2	no	—
64	C₂×C₈×C₄₁	—	4	2	no	—
65	C₂×C₈×C₄₁	—	4	2	no	—
66	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₄₁	C₂²⋊C₄	1	2	yes	—
68	C₄⋊C₄×C₄₁	—	2	2	no	—
69	C₄⋊C₄×C₄₁	—	2	2	no	—
70	C₂²⋊C₄×C₄₁	—	4	2	no	—
71	C₂²⋊C₄×C₄₁	—	4	2	no	—
72	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₄₁	C₂×D₄	1	1	yes	characteristic
76	C₂³×C₄₁	—	4	2	no	—
77	C₂³×C₄₁	#124	1	1	yes	characteristic
78	Q₈×C₄₁	—	4	4	no	—
79	D₄×C₄₁	—	4	4	no	—
80	D₄×C₄₁	—	4	4	no	—
81	D₄×C₄₁	—	4	4	no	—
82	D₄×C₄₁	—	8	4	no	—
83	C₂×C₄×C₄₁	—	2	2	no	—
84	C₂×C₄×C₄₁	—	2	2	no	—
85	C₂×C₄×C₄₁	—	4	2	no	—
86	C₂×C₄×C₄₁	—	8	4	no	—
87	C₂×C₄×C₄₁	C₂×C₂²⋊C₄	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	#121	1	1	yes	characteristic
91	C₂×C₄×C₄₁	#121	1	1	yes	characteristic
92	C₂×C₄×C₄₁	#127	1	1	yes	characteristic
93	C₂×C₄×C₄₁	AGL(1,ℤ/8)	1	4	yes	—
94	C₈×C₄₁	—	2	4	no	—
95	C₈×C₄₁	—	2	2	no	—
96	C₈×C₄₁	—	2	2	no	—
97	C₈×C₄₁	—	4	2	no	—
98	C₈×C₄₁	—	8	2	no	—
99	#9860	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂²×C₄₁	—	2	1	no	—
101	C₂²×C₄₁	—	2	1	no	—
102	C₂²×C₄₁	—	8	2	no	—
103	C₂²×C₄₁	—	8	2	no	—
104	C₂²×C₄₁	—	8	2	no	—
105	C₂²×C₄₁	#390	1	1	yes	characteristic
106	C₂²×C₄₁	#490	1	2	yes	—
107	C₄×C₄₁	—	2	1	no	—
108	C₄×C₄₁	—	2	1	no	—
109	C₄×C₄₁	—	8	2	no	—
110	C₄×C₄₁	#418	1	2	yes	—
111	C₄×C₄₁	#443	1	2	yes	—
112	C₄×C₄₁	#444	1	2	yes	—
113	C₄×C₄₁	#496	1	2	yes	—
114	#3147	C₂×C₄₁	1	1	yes	characteristic
115	#1837	C₂×C₄₁	1	1	yes	characteristic, complemented
116	#1492	C₂×C₄₁	1	2	yes	complemented
117	#1492	C₂×C₄₁	1	2	yes	complemented
118	C₂×#322	C₂×C₄₁	1	1	yes	characteristic, complemented
119	C₂×C₄₁	—	2	2	no	—
120	C₂×C₄₁	—	16	2	no	—
121	C₂×C₄₁	#1961	1	1	yes	characteristic
122	C₂×C₄₁	#1974	1	1	yes	characteristic
123	C₂×C₄₁	#1975	1	1	yes	characteristic
124	C₂×#135	—	2	2	no	—
125	#532	C₂²×C₄₁	1	2	yes	—
126	#495	—	2	2	no	—
127	#494	C₄×C₄₁	1	1	yes	characteristic
128	#493	C₄×C₄₁	1	1	yes	characteristic
129	#492	C₄×C₄₁	1	2	yes	—
130	#423	C₂²×C₄₁	1	2	yes	—
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	#341	—	2	2	no	—
135	D₄⋊C₈	—	2	4	no	—
136	D₄⋊C₈	—	2	4	no	—
137	#322	—	2	2	no	—
138	C₄₁	#9860	1	1	yes	characteristic, direct product factor, Sylow
139	C₂×#41	—	2	2	no	—
140	#135	—	4	8	no	—
141	C₂×Q₁₆	—	2	2	no	—
142	C₂×SD₁₆	—	2	4	no	—
143	C₂×D₈	—	2	2	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂²×C₈	—	2	1	no	—
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	D₄×C₄₁	1	2	yes	—
148	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
149	#127	C₂×C₄×C₄₁	1	2	yes	—
150	#124	—	2	2	no	—
151	#121	—	2	2	no	—
152	#121	—	2	2	no	—
153	C₄×D₄	—	2	2	no	—
154	C₄×D₄	—	2	2	no	—
155	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄×C₈	—	2	1	no	—
159	C₄×C₈	—	2	1	no	—
160	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
161	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
162	C₄×C₈	D₄×C₄₁	1	2	yes	—
163	C₄×C₈	D₄×C₄₁	1	2	yes	—
164	C₄×C₈	D₄×C₄₁	1	2	yes	—
165	#41	—	4	8	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	—	2	2	no	—
169	C₂×D₄	—	2	2	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
173	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
174	C₂²×C₄	#41×C₄₁	1	1	yes	characteristic
175	Q₁₆	—	4	4	no	—
176	SD₁₆	—	4	8	no	—
177	D₈	—	4	4	no	—
178	M₁₆	—	4	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
186	C₂×C₈	C₂²⋊C₄×C₄₁	1	2	yes	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	—	4	2	no	—
191	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
192	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
193	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
194	C₄²	C₂×D₄×C₄₁	1	1	yes	characteristic
195	C₂³	—	4	2	no	—
196	C₂³	#124×C₄₁	1	1	yes	characteristic
197	Q₈	—	4	4	no	—
198	D₄	—	4	4	no	—
199	D₄	—	4	4	no	—
200	D₄	—	4	4	no	—
201	D₄	—	8	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	8	4	no	—
206	C₂×C₄	C₂×C₂²⋊C₄×C₄₁	1	1	yes	characteristic
207	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
208	C₂×C₄	C₄×D₄×C₄₁	1	1	yes	characteristic
209	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
210	C₂×C₄	#121×C₄₁	1	1	yes	characteristic
211	C₂×C₄	#127×C₄₁	1	1	yes	characteristic
212	C₂×C₄	AGL(1,ℤ/8)×C₄₁	1	4	yes	—
213	C₈	—	2	4	no	—
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	—	4	2	no	—
217	C₈	—	8	2	no	—
218	C₂²	—	2	1	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	8	2	no	—
221	C₂²	—	8	2	no	—
222	C₂²	—	8	2	no	—
223	C₂²	C₄₁×#390	1	1	yes	characteristic
224	C₂²	C₄₁×#490	1	2	yes	—
225	C₄	—	2	1	no	—
226	C₄	—	2	1	no	—
227	C₄	—	8	2	no	—
228	C₄	C₄₁×#418	1	2	yes	—
229	C₄	C₄₁×#443	1	2	yes	—
230	C₄	C₄₁×#444	1	2	yes	—
231	C₄	C₄₁×#496	1	2	yes	—
232	C₂	—	2	2	no	—
233	C₂	—	16	2	no	—
234	C₂	C₄₁×#1961	1	1	yes	characteristic
235	C₂	C₄₁×#1974	1	1	yes	characteristic
236	C₂	C₄₁×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₄₁	#127
commutator	C₄²	C₂²×C₄×C₄₁
Frattini, cosocle	C₂×C₄²	C₂³×C₄₁
socle	C₂²×C₄₁	#390
prime radical	C₄₁×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂×C₄²×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#1837	C₂	semidirect
3. C₄₁×#1492	C₂	semidirect
4. C₄₁×#1492	C₂	semidirect
5. C₂×C₄₁×#322	C₂	semidirect
99. #9860	C₄₁	direct (N × H)
115. #1837	C₂×C₄₁	semidirect
116. #1492	C₂×C₄₁	semidirect
117. #1492	C₂×C₄₁	semidirect
118. C₂×#322	C₂×C₄₁	semidirect
138. C₄₁	#9860	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×#3147	2
C₄₁×#1837	2
C₄₁×#1492	2
C₂×C₄₁×#322	2
#9860	41

Subgroups — C41×#9860 #23786061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#9860 #23786061