Subgroups — C₄₁×#22282 #23798483

Order 10496.

810 conjugacy classes of subgroups.

The subgroup lattice has 228 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×#2932	C₂	1	2	yes	maximal, complemented
3	C₄₁×#1963	C₂	1	2	yes	maximal, complemented
4	C₄₁×#1876	C₂	1	8	yes	maximal, complemented
5	C₄₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#135×C₄₁	C₂²	1	4	yes	—
8	C₂×#131×C₄₁	C₂²	1	4	yes	—
9	C₄₁×#532	C₂²	1	2	yes	—
10	C₄₁×#495	C₄	1	4	yes	complemented
11	C₄₁×#494	C₄	1	1	yes	characteristic, complemented
12	C₄₁×#493	C₄	1	1	yes	characteristic, complemented
13	C₄₁×#492	C₄	1	2	yes	complemented
14	C₄₁×#435	—	2	8	no	—
15	C₄₁×#423	C₂²	1	2	yes	—
16	C₂×#104×C₄₁	C₂²	1	8	yes	—
17	C₄₁×#408	C₂²	1	4	yes	—
18	C₂×#97×C₄₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₄₁	C₂²	1	1	yes	characteristic
20	C₄₁×#376	C₂²	1	1	yes	characteristic
21	C₄₁×#341	C₂²	1	4	yes	—
22	C₄₁×#336	C₂²	1	4	yes	—
23	C₂×#41×C₄₁	C₂³	1	4	yes	—
24	#135×C₄₁	—	4	8	no	—
25	C₂×Q₁₆×C₄₁	C₂×C₄	1	4	yes	—
26	C₂×SD₁₆×C₄₁	C₂×C₄	1	8	yes	—
27	C₂×D₈×C₄₁	C₂×C₄	1	4	yes	—
28	#131×C₄₁	—	4	8	no	—
29	C₂×M₁₆×C₄₁	—	2	4	no	—
30	C₂×M₁₆×C₄₁	C₂³	1	4	yes	—
31	C₂²×C₈×C₄₁	—	2	4	no	—
32	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
33	C₂²×C₈×C₄₁	C₂³	1	2	yes	—
34	C₄⋊Q₈×C₄₁	C₂×C₄	1	2	yes	—
35	#127×C₄₁	C₂×C₄	1	2	yes	—
36	#124×C₄₁	C₂×C₄	1	4	yes	—
37	#121×C₄₁	—	2	4	no	—
38	C₄×D₄×C₄₁	—	2	4	no	—
39	C₂×C₄²×C₄₁	C₂³	1	2	yes	—
40	C₂×C₄²×C₄₁	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₄₁	—	2	4	no	—
42	#104×C₄₁	—	4	8	no	—
43	C₂²⋊C₈×C₄₁	—	2	8	no	—
44	#97×C₄₁	D₄	1	4	yes	—
45	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₄₁	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₄₁	C₂×C₄	1	2	yes	—
48	#95×C₄₁	—	2	2	no	—
49	#41×C₄₁	—	2	8	no	—
50	#41×C₄₁	—	2	8	no	—
51	C₂×Q₈×C₄₁	C₂²×C₄	1	4	yes	—
52	C₂×D₄×C₄₁	—	2	4	no	—
53	C₂×D₄×C₄₁	—	2	4	no	—
54	C₂×D₄×C₄₁	C₂²×C₄	1	4	yes	—
55	C₂²×C₄×C₄₁	—	2	4	no	—
56	C₂²×C₄×C₄₁	—	2	2	no	—
57	C₂²×C₄×C₄₁	#41	1	2	yes	—
58	C₂²×C₄×C₄₁	C₂⁴	1	1	yes	characteristic
59	Q₁₆×C₄₁	—	4	4	no	—
60	SD₁₆×C₄₁	—	4	8	no	—
61	D₈×C₄₁	—	4	4	no	—
62	M₁₆×C₄₁	—	2	8	no	—
63	M₁₆×C₄₁	—	4	8	no	—
64	C₂×C₈×C₄₁	—	2	4	no	—
65	C₂×C₈×C₄₁	—	2	4	no	—
66	C₂×C₈×C₄₁	—	2	4	no	—
67	C₂×C₈×C₄₁	—	4	8	no	—
68	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₄₁	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₄₁	C₂×D₄	1	4	yes	—
71	C₄⋊C₄×C₄₁	—	2	4	no	—
72	C₂²⋊C₄×C₄₁	—	2	8	no	—
73	C₄²×C₄₁	—	2	4	no	—
74	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₄₁	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₄₁	C₂×D₄	1	2	yes	—
77	C₂³×C₄₁	—	2	4	no	—
78	C₂³×C₄₁	C₂×#41	1	1	yes	characteristic
79	Q₈×C₄₁	—	2	8	no	—
80	D₄×C₄₁	—	2	8	no	—
81	D₄×C₄₁	—	4	8	no	—
82	D₄×C₄₁	—	8	4	no	—
83	C₂×C₄×C₄₁	—	2	4	no	—
84	C₂×C₄×C₄₁	—	4	2	no	—
85	C₂×C₄×C₄₁	—	4	4	no	—
86	C₂×C₄×C₄₁	—	4	4	no	—
87	C₂×C₄×C₄₁	—	4	4	no	—
88	C₂×C₄×C₄₁	—	4	2	no	—
89	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₄₁	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₄₁	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₄₁	C₂²×D₄	1	1	yes	characteristic
93	C₂×C₄×C₄₁	H(5,𝔽₂)	1	4	yes	—
94	C₈×C₄₁	—	2	2	no	—
95	C₈×C₄₁	—	2	2	no	—
96	C₈×C₄₁	—	2	4	no	—
97	C₈×C₄₁	—	4	4	no	—
98	#22282	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₄₁	—	2	2	no	—
100	C₂²×C₄₁	—	4	4	no	—
101	C₂²×C₄₁	—	4	8	no	—
102	C₂²×C₄₁	C₂×C₄×D₄	1	1	yes	characteristic
103	C₂²×C₄₁	#540	1	2	yes	—
104	C₄×C₄₁	—	2	2	no	—
105	C₄×C₄₁	—	2	2	no	—
106	C₄×C₄₁	—	4	4	no	—
107	C₄×C₄₁	—	8	2	no	—
108	C₄×C₄₁	#518	1	2	yes	—
109	C₄×C₄₁	#535	1	2	yes	—
110	#3147	C₂×C₄₁	1	1	yes	characteristic
111	#2932	C₂×C₄₁	1	2	yes	complemented
112	#1963	C₂×C₄₁	1	2	yes	complemented
113	#1876	C₂×C₄₁	1	8	yes	complemented
114	#1837	C₂×C₄₁	1	1	yes	characteristic, complemented
115	#1746	C₂×C₄₁	1	1	yes	characteristic, complemented
116	C₂×C₄₁	—	2	2	no	—
117	C₂×C₄₁	—	8	4	no	—
118	C₂×C₄₁	#2328	1	1	yes	characteristic
119	C₂×C₄₁	#2957	1	2	yes	—
120	C₂×#135	C₂²×C₄₁	1	4	yes	—
121	C₂×#131	C₂²×C₄₁	1	4	yes	—
122	#532	C₂²×C₄₁	1	2	yes	—
123	#495	C₄×C₄₁	1	4	yes	complemented
124	#494	C₄×C₄₁	1	1	yes	characteristic, complemented
125	#493	C₄×C₄₁	1	1	yes	characteristic, complemented
126	#492	C₄×C₄₁	1	2	yes	complemented
127	#435	—	2	8	no	—
128	#423	C₂²×C₄₁	1	2	yes	—
129	C₂×#104	C₂²×C₄₁	1	8	yes	—
130	#408	C₂²×C₄₁	1	4	yes	—
131	C₂×#97	C₂²×C₄₁	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₄₁	1	1	yes	characteristic
133	#376	C₂²×C₄₁	1	1	yes	characteristic
134	#341	C₂²×C₄₁	1	4	yes	—
135	#336	C₂²×C₄₁	1	4	yes	—
136	C₄₁	#22282	1	1	yes	characteristic, direct product factor, Sylow
137	C₂×#41	C₂³×C₄₁	1	4	yes	—
138	#135	—	4	8	no	—
139	C₂×Q₁₆	C₂×C₄×C₄₁	1	4	yes	—
140	C₂×SD₁₆	C₂×C₄×C₄₁	1	8	yes	—
141	C₂×D₈	C₂×C₄×C₄₁	1	4	yes	—
142	#131	—	4	8	no	—
143	C₂×M₁₆	—	2	4	no	—
144	C₂×M₁₆	C₂³×C₄₁	1	4	yes	—
145	C₂²×C₈	—	2	4	no	—
146	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
147	C₂²×C₈	C₂³×C₄₁	1	2	yes	—
148	C₄⋊Q₈	C₂×C₄×C₄₁	1	2	yes	—
149	#127	C₂×C₄×C₄₁	1	2	yes	—
150	#124	C₂×C₄×C₄₁	1	4	yes	—
151	#121	—	2	4	no	—
152	C₄×D₄	—	2	4	no	—
153	C₂×C₄²	C₂³×C₄₁	1	2	yes	—
154	C₂×C₄²	C₂³×C₄₁	1	1	yes	characteristic
155	C₄⋊C₈	—	2	4	no	—
156	#104	—	4	8	no	—
157	C₂²⋊C₈	—	2	8	no	—
158	#97	D₄×C₄₁	1	4	yes	—
159	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
160	C₄×C₈	C₂×C₄×C₄₁	1	1	yes	characteristic
161	C₄×C₈	C₂×C₄×C₄₁	1	2	yes	—
162	#95	—	2	2	no	—
163	#41	—	2	8	no	—
164	#41	—	2	8	no	—
165	C₂×Q₈	C₂²×C₄×C₄₁	1	4	yes	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	2	4	no	—
168	C₂×D₄	C₂²×C₄×C₄₁	1	4	yes	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	#41×C₄₁	1	2	yes	—
172	C₂²×C₄	C₂⁴×C₄₁	1	1	yes	characteristic
173	Q₁₆	—	4	4	no	—
174	SD₁₆	—	4	8	no	—
175	D₈	—	4	4	no	—
176	M₁₆	—	2	8	no	—
177	M₁₆	—	4	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	4	8	no	—
182	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
183	C₂×C₈	C₂²×C₄×C₄₁	1	2	yes	—
184	C₂×C₈	C₂×D₄×C₄₁	1	4	yes	—
185	C₄⋊C₄	—	2	4	no	—
186	C₂²⋊C₄	—	2	8	no	—
187	C₄²	—	2	4	no	—
188	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
189	C₄²	C₂²×C₄×C₄₁	1	1	yes	characteristic
190	C₄²	C₂×D₄×C₄₁	1	2	yes	—
191	C₂³	—	2	4	no	—
192	C₂³	C₂×#41×C₄₁	1	1	yes	characteristic
193	Q₈	—	2	8	no	—
194	D₄	—	2	8	no	—
195	D₄	—	4	8	no	—
196	D₄	—	8	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
204	C₂×C₄	C₄×D₄×C₄₁	1	2	yes	—
205	C₂×C₄	C₂³×C₄×C₄₁	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₄×C₄₁	1	1	yes	characteristic
207	C₂×C₄	H(5,𝔽₂)×C₄₁	1	4	yes	—
208	C₈	—	2	2	no	—
209	C₈	—	2	2	no	—
210	C₈	—	2	4	no	—
211	C₈	—	4	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	4	no	—
214	C₂²	—	4	8	no	—
215	C₂²	C₂×C₄×D₄×C₄₁	1	1	yes	characteristic
216	C₂²	C₄₁×#540	1	2	yes	—
217	C₄	—	2	2	no	—
218	C₄	—	2	2	no	—
219	C₄	—	4	4	no	—
220	C₄	—	8	2	no	—
221	C₄	C₄₁×#518	1	2	yes	—
222	C₄	C₄₁×#535	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	—	8	4	no	—
225	C₂	C₄₁×#2328	1	1	yes	characteristic
226	C₂	C₄₁×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₄₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₄₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₄₁
socle	C₂²×C₄₁	C₂×C₄×D₄
prime radical	C₄₁×#3147	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₁ ◁ C₂×#97×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄₁×#2932	C₂	semidirect
3. C₄₁×#1963	C₂	semidirect
4. C₄₁×#1876	C₂	semidirect
5. C₄₁×#1837	C₂	semidirect
6. C₄₁×#1746	C₂	semidirect
10. C₄₁×#495	C₄	semidirect
11. C₄₁×#494	C₄	semidirect
12. C₄₁×#493	C₄	semidirect
13. C₄₁×#492	C₄	semidirect
98. #22282	C₄₁	direct (N × H)
111. #2932	C₂×C₄₁	semidirect
112. #1963	C₂×C₄₁	semidirect
113. #1876	C₂×C₄₁	semidirect
114. #1837	C₂×C₄₁	semidirect
115. #1746	C₂×C₄₁	semidirect
123. #495	C₄×C₄₁	semidirect
124. #494	C₄×C₄₁	semidirect
125. #493	C₄×C₄₁	semidirect
126. #492	C₄×C₄₁	semidirect
136. C₄₁	#22282	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×#2932	2
C₄₁×#1963	2
C₄₁×#1876	2
C₄₁×#1837	2
C₄₁×#1746	2
#22282	41

Subgroups — C41×#22282 #23798483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₁×#22282 #23798483