Subgroups — C₄₁×#20816 #23797017

Order 10496.

982 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₄₁	C₂	1	1	yes	maximal, characteristic
2	C₄₁×#2285	C₂	1	2	yes	maximal
3	C₄₁×#1844	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1834	C₂	1	2	yes	maximal, complemented
5	C₄₁×#1807	C₂	1	4	yes	maximal
6	C₄×#106×C₄₁	C₂	1	4	yes	maximal, complemented
7	C₂³×C₈×C₄₁	C₂²	1	2	yes	—
8	C₂×#117×C₄₁	—	2	2	no	—
9	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic
10	C₂×#106×C₄₁	C₂²	1	8	yes	—
11	C₂×C₄×C₈×C₄₁	C₄	1	8	yes	complemented
12	C₂×C₄×C₈×C₄₁	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	C₄₁×#384	C₂²	1	4	yes	—
16	C₄₁×#382	—	2	2	no	—
17	C₄₁×#380	—	2	2	no	—
18	C₄₁×#380	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄×C₄₁	—	2	2	no	—
20	C₄×C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
21	C₄₁×#340	C₂²	1	8	yes	—
22	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₄₁	D₄	1	1	yes	characteristic
24	C₂³×C₄×C₄₁	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₄₁	—	2	8	no	—
26	C₂²×C₈×C₄₁	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
28	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
29	#117×C₄₁	—	4	8	no	—
30	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₄₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₄₁	C₂³	1	8	yes	—
33	C₂×C₂²⋊C₄×C₄₁	—	2	4	no	—
34	C₂×C₄²×C₄₁	—	2	4	no	—
35	C₂×C₄²×C₄₁	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₄₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₄₁	Q₈	1	4	yes	—
39	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
41	#106×C₄₁	—	2	16	no	—
42	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₄₁	C₂×C₄	1	8	yes	—
44	C₄×C₈×C₄₁	C₂×C₄	1	4	yes	—
45	#95×C₄₁	—	2	4	no	—
46	#95×C₄₁	—	2	4	no	—
47	C₂⁴×C₄₁	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₄₁	—	2	4	no	—
49	C₂²×C₄×C₄₁	—	2	4	no	—
50	C₂²×C₄×C₄₁	—	2	8	no	—
51	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₄₁	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₄₁	SD₁₆	1	8	yes	—
54	C₂²×C₄×C₄₁	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₁	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₁	C₂×Q₈	1	2	yes	—
61	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
62	C₂×C₈×C₄₁	—	2	4	no	—
63	C₂×C₈×C₄₁	—	2	16	no	—
64	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
65	C₂×C₈×C₄₁	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₄₁	#41	1	8	yes	—
67	C₄⋊C₄×C₄₁	—	2	16	no	—
68	C₄⋊C₄×C₄₁	—	4	8	no	—
69	C₄⋊C₄×C₄₁	—	4	8	no	—
70	C₂²⋊C₄×C₄₁	—	4	8	no	—
71	C₄²×C₄₁	—	4	8	no	—
72	C₄²×C₄₁	C₄⋊C₄	1	4	yes	—
73	C₄²×C₄₁	C₄⋊C₄	1	4	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	2	no	—
79	C₂³×C₄₁	—	2	2	no	—
80	C₂³×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
81	C₂³×C₄₁	C₂×SD₁₆	1	4	yes	—
82	C₂³×C₄₁	C₂²×D₄	1	1	yes	characteristic
83	C₂×C₄×C₄₁	—	2	2	no	—
84	C₂×C₄×C₄₁	—	2	2	no	—
85	C₂×C₄×C₄₁	—	2	4	no	—
86	C₂×C₄×C₄₁	—	2	4	no	—
87	C₂×C₄×C₄₁	—	2	8	no	—
88	C₂×C₄×C₄₁	—	4	8	no	—
89	C₂×C₄×C₄₁	—	4	8	no	—
90	C₂×C₄×C₄₁	—	4	8	no	—
91	C₂×C₄×C₄₁	#106	1	16	yes	complemented
92	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₄₁	C₂×C₄⋊C₄	1	2	yes	—
97	C₂×C₄×C₄₁	C₂×SD₁₆	1	4	yes	—
98	C₂×C₄×C₄₁	C₂×SD₁₆	1	4	yes	—
99	C₂×C₄×C₄₁	#135	1	4	yes	—
100	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₁	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₄₁	C₂×#41	1	4	yes	—
103	C₈×C₄₁	—	2	4	no	—
104	C₈×C₄₁	#117	1	8	yes	complemented
105	#20816	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₄₁	—	2	2	no	—
107	C₂²×C₄₁	—	2	8	no	—
108	C₂²×C₄₁	—	2	2	no	—
109	C₂²×C₄₁	C₂×#106	1	4	yes	—
110	C₂²×C₄₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₄₁	#532	1	2	yes	—
112	C₂²×C₄₁	C₂²×SD₁₆	1	2	yes	—
113	C₂²×C₄₁	C₂×#135	1	2	yes	—
114	C₄×C₄₁	—	2	2	no	—
115	C₄×C₄₁	—	2	2	no	—
116	C₄×C₄₁	—	8	8	no	—
117	C₄×C₄₁	C₂×#106	1	4	yes	—
118	C₄×C₄₁	C₂×#106	1	4	yes	—
119	C₄×C₄₁	#427	1	4	yes	—
120	C₄×C₄₁	C₂×#117	1	2	yes	—
121	C₄×C₄₁	#533	1	2	yes	—
122	C₂²×C₄×C₈	C₂×C₄₁	1	1	yes	characteristic
123	#2285	C₂×C₄₁	1	2	yes	—
124	#1844	C₂×C₄₁	1	2	yes	complemented
125	#1834	C₂×C₄₁	1	2	yes	complemented
126	#1807	C₂×C₄₁	1	4	yes	—
127	C₄×#106	C₂×C₄₁	1	4	yes	complemented
128	C₂×C₄₁	—	2	4	no	—
129	C₂×C₄₁	#2285	1	1	yes	characteristic
130	C₂×C₄₁	C₂²×#106	1	1	yes	characteristic
131	C₂×C₄₁	C₂×#427	1	1	yes	characteristic
132	C₂×C₄₁	#3167	1	4	yes	—
133	C₂³×C₈	C₂²×C₄₁	1	2	yes	—
134	C₂×#117	—	2	2	no	—
135	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic
136	C₂×#106	C₂²×C₄₁	1	8	yes	—
137	C₂×C₄×C₈	C₄×C₄₁	1	8	yes	complemented
138	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₄₁	1	2	yes	—
141	#384	C₂²×C₄₁	1	4	yes	—
142	#382	—	2	2	no	—
143	#380	—	2	2	no	—
144	#380	C₂²×C₄₁	1	4	yes	—
145	C₄×C₄⋊C₄	—	2	2	no	—
146	C₄×C₄⋊C₄	C₂²×C₄₁	1	4	yes	—
147	#340	C₂²×C₄₁	1	8	yes	—
148	C₄₁	#20816	1	1	yes	characteristic, direct product factor, Sylow
149	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
150	C₂³×C₄	D₄×C₄₁	1	1	yes	characteristic
151	C₂³×C₄	C₂³×C₄₁	1	1	yes	characteristic
152	C₂²×C₈	—	2	8	no	—
153	C₂²×C₈	C₂×C₄×C₄₁	1	8	yes	—
154	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
155	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
156	#117	—	4	8	no	—
157	C₂×C₄⋊C₄	—	2	4	no	—
158	C₂×C₄⋊C₄	—	2	4	no	—
159	C₂×C₄⋊C₄	C₂³×C₄₁	1	8	yes	—
160	C₂×C₂²⋊C₄	—	2	4	no	—
161	C₂×C₄²	—	2	4	no	—
162	C₂×C₄²	C₂×C₄×C₄₁	1	4	yes	—
163	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
164	C₂×C₄²	D₄×C₄₁	1	1	yes	characteristic
165	C₂×C₄²	Q₈×C₄₁	1	4	yes	—
166	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
167	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
168	#106	—	2	16	no	—
169	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
170	C₄×C₈	C₂×C₄×C₄₁	1	8	yes	—
171	C₄×C₈	C₂×C₄×C₄₁	1	4	yes	—
172	#95	—	2	4	no	—
173	#95	—	2	4	no	—
174	C₂⁴	C₂×D₄×C₄₁	1	1	yes	characteristic
175	C₂²×C₄	—	2	4	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	8	no	—
178	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
179	C₂²×C₄	C₄⋊C₄×C₄₁	1	4	yes	—
180	C₂²×C₄	SD₁₆×C₄₁	1	8	yes	—
181	C₂²×C₄	C₂²×C₄×C₄₁	1	2	yes	—
182	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
185	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₄₁	1	1	yes	characteristic
187	C₂²×C₄	C₂×Q₈×C₄₁	1	2	yes	—
188	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	16	no	—
191	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
192	C₂×C₈	C₂²×C₄×C₄₁	1	4	yes	—
193	C₂×C₈	#41×C₄₁	1	8	yes	—
194	C₄⋊C₄	—	2	16	no	—
195	C₄⋊C₄	—	4	8	no	—
196	C₄⋊C₄	—	4	8	no	—
197	C₂²⋊C₄	—	4	8	no	—
198	C₄²	—	4	8	no	—
199	C₄²	C₄⋊C₄×C₄₁	1	4	yes	—
200	C₄²	C₄⋊C₄×C₄₁	1	4	yes	—
201	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
202	C₄²	C₂²×C₄×C₄₁	1	2	yes	—
203	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
204	C₄²	C₂×Q₈×C₄₁	1	2	yes	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
208	C₂³	C₂×SD₁₆×C₄₁	1	4	yes	—
209	C₂³	C₂²×D₄×C₄₁	1	1	yes	characteristic
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	—	4	8	no	—
218	C₂×C₄	#106×C₄₁	1	16	yes	complemented
219	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
220	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
221	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
222	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
223	C₂×C₄	C₂×C₄⋊C₄×C₄₁	1	2	yes	—
224	C₂×C₄	C₂×SD₁₆×C₄₁	1	4	yes	—
225	C₂×C₄	C₂×SD₁₆×C₄₁	1	4	yes	—
226	C₂×C₄	#135×C₄₁	1	4	yes	—
227	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
228	C₂×C₄	C₂²×Q₈×C₄₁	1	1	yes	characteristic
229	C₂×C₄	C₂×#41×C₄₁	1	4	yes	—
230	C₈	—	2	4	no	—
231	C₈	#117×C₄₁	1	8	yes	complemented
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	8	no	—
235	C₂²	C₂×#106×C₄₁	1	4	yes	—
236	C₂²	C₂²×C₄⋊C₄×C₄₁	1	1	yes	characteristic
237	C₂²	C₄₁×#532	1	2	yes	—
238	C₂²	C₂²×SD₁₆×C₄₁	1	2	yes	—
239	C₂²	C₂×#135×C₄₁	1	2	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	8	8	no	—
243	C₄	C₂×#106×C₄₁	1	4	yes	—
244	C₄	C₂×#106×C₄₁	1	4	yes	—
245	C₄	C₄₁×#427	1	4	yes	—
246	C₄	C₂×#117×C₄₁	1	2	yes	—
247	C₄	C₄₁×#533	1	2	yes	—
248	C₂	—	2	4	no	—
249	C₂	C₄₁×#2285	1	1	yes	characteristic
250	C₂	C₂²×#106×C₄₁	1	1	yes	characteristic
251	C₂	C₂×C₄₁×#427	1	1	yes	characteristic
252	C₂	C₄₁×#3167	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₄₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂³×C₄₁	C₂²×D₄
prime radical	C₂⁴×C₄₁	C₂×D₄
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
3. C₄₁×#1844	C₂	semidirect
4. C₄₁×#1834	C₂	semidirect
6. C₄×#106×C₄₁	C₂	semidirect
11. C₂×C₄×C₈×C₄₁	C₄	semidirect
91. C₂×C₄×C₄₁	#106	semidirect
104. C₈×C₄₁	#117	semidirect
105. #20816	C₄₁	direct (N × H)
124. #1844	C₂×C₄₁	semidirect
125. #1834	C₂×C₄₁	semidirect
127. C₄×#106	C₂×C₄₁	semidirect
137. C₂×C₄×C₈	C₄×C₄₁	semidirect
148. C₄₁	#20816	direct (N × H)
218. C₂×C₄	#106×C₄₁	semidirect
231. C₈	#117×C₄₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₄₁	2
C₄₁×#2285	2
C₄₁×#1844	2
C₄₁×#1834	2
C₄₁×#1807	2
C₄×#106×C₄₁	2
#20816	41

Subgroups — C41×#20816 #23797017

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#20816 #23797017