Subgroups — C₄₁×#20765 #23796966

Order 10496.

926 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	C₂×C₄×M₁₆×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2285	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₁×#382	C₂	1	1	yes	maximal, characteristic
4	C₄₁×#1836	C₂	1	4	yes	maximal, complemented
5	C₄₁×#1835	C₂	1	4	yes	maximal, complemented
6	C₄₁×#1736	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₄₁	C₂²	1	2	yes	—
8	C₂×#117×C₄₁	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
10	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
11	C₄×M₁₆×C₄₁	C₄	1	8	yes	complemented
12	C₂×#97×C₄₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₄₁	C₂²	1	2	yes	—
14	C₄₁×#384	C₂²	1	2	yes	—
15	C₄₁×#382	—	2	2	no	—
16	C₄₁×#382	C₂²	1	2	yes	—
17	C₄₁×#382	C₂²	1	2	yes	—
18	C₄₁×#382	C₂²	1	2	yes	—
19	C₄₁×#380	—	2	2	no	—
20	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
21	C₂×#95×C₄₁	—	2	2	no	—
22	C₄₁×#340	C₂²	1	16	yes	—
23	C₂³×C₄×C₄₁	—	2	2	no	—
24	C₂³×C₄×C₄₁	D₄	1	2	yes	—
25	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
26	C₂×M₁₆×C₄₁	—	2	4	no	—
27	C₂×M₁₆×C₄₁	—	2	4	no	—
28	C₂×M₁₆×C₄₁	C₂×C₄	1	8	yes	—
29	C₂²×C₈×C₄₁	C₂³	1	4	yes	—
30	#117×C₄₁	—	2	8	no	—
31	C₂×C₄⋊C₄×C₄₁	—	2	8	no	—
32	C₂×C₄⋊C₄×C₄₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
35	C₂×C₄⋊C₄×C₄₁	C₂³	1	4	yes	—
36	C₂×C₂²⋊C₄×C₄₁	—	2	2	no	—
37	C₂×C₄²×C₄₁	—	2	2	no	—
38	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₄₁	D₄	1	2	yes	—
40	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
41	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
43	#97×C₄₁	C₂×C₄	1	8	yes	complemented
44	C₄×C₈×C₄₁	C₂×C₄	1	8	yes	complemented
45	#95×C₄₁	—	2	4	no	—
46	#95×C₄₁	—	2	4	no	—
47	#95×C₄₁	—	2	4	no	—
48	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₄₁	—	2	4	no	—
50	C₂²×C₄×C₄₁	—	2	4	no	—
51	C₂²×C₄×C₄₁	—	4	4	no	—
52	C₂²×C₄×C₄₁	—	4	8	no	—
53	C₂²×C₄×C₄₁	—	4	4	no	—
54	C₂²×C₄×C₄₁	C₄⋊C₄	1	8	yes	—
55	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₁	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
60	C₂²×C₄×C₄₁	#41	1	4	yes	—
61	C₂²×C₄×C₄₁	#41	1	4	yes	—
62	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
63	M₁₆×C₄₁	—	2	16	no	—
64	C₂×C₈×C₄₁	—	2	4	no	—
65	C₂×C₈×C₄₁	—	2	4	no	—
66	C₂×C₈×C₄₁	C₂²×C₄	1	8	yes	—
67	C₄⋊C₄×C₄₁	—	2	8	no	—
68	C₄⋊C₄×C₄₁	—	2	8	no	—
69	C₄⋊C₄×C₄₁	—	4	8	no	—
70	C₄⋊C₄×C₄₁	—	4	4	no	—
71	C₂²⋊C₄×C₄₁	—	4	4	no	—
72	C₄²×C₄₁	—	4	4	no	—
73	C₄²×C₄₁	C₄⋊C₄	1	8	yes	—
74	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
75	C₄²×C₄₁	C₂²×C₄	1	2	yes	—
76	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
77	C₄²×C₄₁	C₂×Q₈	1	2	yes	—
78	C₂³×C₄₁	—	2	4	no	—
79	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
80	C₂³×C₄₁	#124	1	2	yes	—
81	C₂³×C₄₁	#124	1	2	yes	—
82	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₄₁	—	2	4	no	—
84	C₂×C₄×C₄₁	—	2	4	no	—
85	C₂×C₄×C₄₁	—	4	8	no	—
86	C₂×C₄×C₄₁	—	4	4	no	—
87	C₂×C₄×C₄₁	—	4	4	no	—
88	C₂×C₄×C₄₁	—	4	8	no	—
89	C₂×C₄×C₄₁	—	4	8	no	—
90	C₂×C₄×C₄₁	—	8	8	no	—
91	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
92	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₄₁	#117	1	8	yes	—
95	C₂×C₄×C₄₁	#124	1	2	yes	—
96	C₂×C₄×C₄₁	#124	1	2	yes	—
97	C₂×C₄×C₄₁	#125	1	8	yes	—
98	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
101	C₂×C₄×C₄₁	C₂×#41	1	2	yes	—
102	C₈×C₄₁	—	2	8	no	—
103	#20765	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₄₁	—	2	4	no	—
105	C₂²×C₄₁	—	2	4	no	—
106	C₂²×C₄₁	—	2	4	no	—
107	C₂²×C₄₁	#382	1	4	yes	—
108	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
109	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
110	C₂²×C₄₁	C₂×#124	1	1	yes	characteristic
111	C₂²×C₄₁	#575	1	4	yes	—
112	C₄×C₄₁	—	2	4	no	—
113	C₄×C₄₁	—	2	4	no	—
114	C₄×C₄₁	—	8	4	no	—
115	C₄×C₄₁	—	8	4	no	—
116	C₄×C₄₁	#382	1	4	yes	—
117	C₄×C₄₁	C₂×#117	1	2	yes	—
118	C₄×C₄₁	C₂×#125	1	2	yes	—
119	C₂×C₄×M₁₆	C₂×C₄₁	1	1	yes	characteristic
120	#2285	C₂×C₄₁	1	1	yes	characteristic
121	C₂×#382	C₂×C₄₁	1	1	yes	characteristic
122	#1836	C₂×C₄₁	1	4	yes	complemented
123	#1835	C₂×C₄₁	1	4	yes	complemented
124	#1736	C₂×C₄₁	1	4	yes	—
125	C₂×C₄₁	—	2	4	no	—
126	C₂×C₄₁	C₂×#382	1	1	yes	characteristic
127	C₂×C₄₁	#2911	1	2	yes	—
128	C₂×C₄₁	#3140	1	2	yes	—
129	C₂×C₄₁	#3142	1	2	yes	—
130	C₂²×M₁₆	C₂²×C₄₁	1	2	yes	—
131	C₂×#117	C₂²×C₄₁	1	2	yes	—
132	C₂²×C₄⋊C₄	C₂²×C₄₁	1	2	yes	—
133	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
134	C₄×M₁₆	C₄×C₄₁	1	8	yes	complemented
135	C₂×#97	C₂²×C₄₁	1	2	yes	—
136	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
137	#384	C₂²×C₄₁	1	2	yes	—
138	#382	—	2	2	no	—
139	#382	C₂²×C₄₁	1	2	yes	—
140	#382	C₂²×C₄₁	1	2	yes	—
141	#382	C₂²×C₄₁	1	2	yes	—
142	#380	—	2	2	no	—
143	C₄×C₄⋊C₄	—	2	2	no	—
144	C₂×#95	—	2	2	no	—
145	#340	C₂²×C₄₁	1	16	yes	—
146	C₄₁	#20765	1	1	yes	characteristic, direct product factor, Sylow
147	C₂³×C₄	—	2	2	no	—
148	C₂³×C₄	D₄×C₄₁	1	2	yes	—
149	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
150	C₂×M₁₆	—	2	4	no	—
151	C₂×M₁₆	—	2	4	no	—
152	C₂×M₁₆	C₂×C₄×C₄₁	1	8	yes	—
153	C₂²×C₈	C₂³×C₄₁	1	4	yes	—
154	#117	—	2	8	no	—
155	C₂×C₄⋊C₄	—	2	8	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
159	C₂×C₄⋊C₄	C₂³×C₄₁	1	4	yes	—
160	C₂×C₂²⋊C₄	—	2	2	no	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
163	C₂×C₄²	D₄×C₄₁	1	2	yes	—
164	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
165	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
166	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
167	#97	C₂×C₄×C₄₁	1	8	yes	complemented
168	C₄×C₈	C₂×C₄×C₄₁	1	8	yes	complemented
169	#95	—	2	4	no	—
170	#95	—	2	4	no	—
171	#95	—	2	4	no	—
172	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	—	4	4	no	—
176	C₂²×C₄	—	4	8	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	C₄⋊C₄×C₄₁	1	8	yes	—
179	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
180	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₄₁	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
183	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
184	C₂²×C₄	#41×C₄₁	1	4	yes	—
185	C₂²×C₄	#41×C₄₁	1	4	yes	—
186	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
187	M₁₆	—	2	16	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	C₂²×C₄×C₄₁	1	8	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	8	no	—
193	C₄⋊C₄	—	4	8	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₂²⋊C₄	—	4	4	no	—
196	C₄²	—	4	4	no	—
197	C₄²	C₄⋊C₄×C₄₁	1	8	yes	—
198	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
199	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
200	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
201	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
202	C₂³	—	2	4	no	—
203	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
204	C₂³	#124×C₄₁	1	2	yes	—
205	C₂³	#124×C₄₁	1	2	yes	—
206	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	8	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	8	no	—
213	C₂×C₄	—	4	8	no	—
214	C₂×C₄	—	8	8	no	—
215	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
216	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	4	yes	—
217	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
218	C₂×C₄	#117×C₄₁	1	8	yes	—
219	C₂×C₄	#124×C₄₁	1	2	yes	—
220	C₂×C₄	#124×C₄₁	1	2	yes	—
221	C₂×C₄	#125×C₄₁	1	8	yes	—
222	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
223	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
224	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
225	C₂×C₄	C₂×#41×C₄₁	1	2	yes	—
226	C₈	—	2	8	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	2	4	no	—
229	C₂²	—	2	4	no	—
230	C₂²	C₄₁×#382	1	4	yes	—
231	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
232	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
233	C₂²	C₂×#124×C₄₁	1	1	yes	characteristic
234	C₂²	C₄₁×#575	1	4	yes	—
235	C₄	—	2	4	no	—
236	C₄	—	2	4	no	—
237	C₄	—	8	4	no	—
238	C₄	—	8	4	no	—
239	C₄	C₄₁×#382	1	4	yes	—
240	C₄	C₂×#117×C₄₁	1	2	yes	—
241	C₄	C₂×#125×C₄₁	1	2	yes	—
242	C₂	—	2	4	no	—
243	C₂	C₂×C₄₁×#382	1	1	yes	characteristic
244	C₂	C₄₁×#2911	1	2	yes	—
245	C₂	C₄₁×#3140	1	2	yes	—
246	C₂	C₄₁×#3142	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
prime radical	C₂⁴×C₄₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ 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
4. C₄₁×#1836	C₂	semidirect
5. C₄₁×#1835	C₂	semidirect
11. C₄×M₁₆×C₄₁	C₄	semidirect
43. #97×C₄₁	C₂×C₄	semidirect
44. C₄×C₈×C₄₁	C₂×C₄	semidirect
103. #20765	C₄₁	direct (N × H)
122. #1836	C₂×C₄₁	semidirect
123. #1835	C₂×C₄₁	semidirect
134. C₄×M₁₆	C₄×C₄₁	semidirect
146. C₄₁	#20765	direct (N × H)
167. #97	C₂×C₄×C₄₁	semidirect
168. C₄×C₈	C₂×C₄×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×M₁₆×C₄₁	2
C₄₁×#2285	2
C₂×C₄₁×#382	2
C₄₁×#1836	2
C₄₁×#1835	2
C₄₁×#1736	2
#20765	41

Subgroups — C41×#20765 #23796966

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20765 #23796966