Subgroups — C₂²×C₄₁×#560 #23838888

Order 10496.

9606 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#560	C₂	1	48	yes	maximal, direct product factor
2	C₂²×#127×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#126×C₄₁	C₂	1	2	yes	maximal, complemented
4	C₂²×#123×C₄₁	C₂	1	3	yes	maximal, complemented
5	C₂²×#121×C₄₁	C₂	1	6	yes	maximal, complemented
6	C₂²×#120×C₄₁	C₂	1	3	yes	maximal, complemented
7	C₂⁶×C₄₁	C₂²	1	1	yes	characteristic
8	C₂³×D₄×C₄₁	C₂²	1	6	yes	—
9	C₂³×D₄×C₄₁	C₂²	1	3	yes	—
10	C₂³×D₄×C₄₁	C₂²	1	3	yes	complemented
11	C₂⁴×C₄×C₄₁	C₂²	1	3	yes	complemented
12	C₄₁×#560	C₂²	1	256	yes	direct product factor
13	C₂×#127×C₄₁	C₂²	1	24	yes	complemented
14	C₂×#126×C₄₁	C₂²	1	48	yes	complemented
15	C₂×#123×C₄₁	C₂²	1	72	yes	complemented
16	C₂×#121×C₄₁	C₂²	1	144	yes	complemented
17	C₂×#120×C₄₁	C₂²	1	72	yes	complemented
18	C₂²×C₄⋊C₄×C₄₁	C₂²	1	6	yes	complemented
19	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	6	yes	—
20	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	3	yes	—
21	C₂²×C₂²⋊C₄×C₄₁	C₂²	1	3	yes	complemented
22	C₂²×C₄²×C₄₁	C₂²	1	1	yes	characteristic, complemented
23	C₂⁵×C₄₁	—	4	12	no	—
24	C₂⁵×C₄₁	C₂³	1	2	yes	—
25	C₂⁵×C₄₁	C₂³	1	1	yes	characteristic
26	C₂⁵×C₄₁	C₂³	1	3	yes	—
27	C₂⁵×C₄₁	C₂³	1	12	yes	—
28	C₂²×D₄×C₄₁	—	4	24	no	—
29	C₂²×D₄×C₄₁	—	4	12	no	—
30	C₂²×D₄×C₄₁	—	4	12	no	—
31	C₂²×D₄×C₄₁	C₂³	1	72	yes	—
32	C₂²×D₄×C₄₁	C₂³	1	36	yes	—
33	C₂²×D₄×C₄₁	C₂³	1	36	yes	complemented
34	C₂³×C₄×C₄₁	—	4	12	no	—
35	C₂³×C₄×C₄₁	C₂³	1	3	yes	—
36	C₂³×C₄×C₄₁	C₂³	1	6	yes	—
37	C₂³×C₄×C₄₁	C₂³	1	36	yes	complemented
38	#127×C₄₁	C₂³	1	64	yes	complemented
39	#126×C₄₁	C₂³	1	128	yes	complemented
40	#123×C₄₁	C₂³	1	192	yes	complemented
41	#121×C₄₁	C₂³	1	384	yes	complemented
42	#120×C₄₁	C₂³	1	192	yes	complemented
43	C₂×C₄⋊C₄×C₄₁	C₂³	1	72	yes	complemented
44	C₂×C₂²⋊C₄×C₄₁	C₂³	1	72	yes	—
45	C₂×C₂²⋊C₄×C₄₁	C₂³	1	36	yes	—
46	C₂×C₂²⋊C₄×C₄₁	C₂³	1	36	yes	complemented
47	C₂×C₄²×C₄₁	C₂³	1	12	yes	complemented
48	C₂⁴×C₄₁	—	2	12	no	—
49	C₂⁴×C₄₁	—	2	24	no	—
50	C₂⁴×C₄₁	—	2	24	no	—
51	C₂⁴×C₄₁	—	2	12	no	—
52	C₂⁴×C₄₁	—	4	72	no	—
53	C₂⁴×C₄₁	—	4	64	no	—
54	C₂⁴×C₄₁	C₂⁴	1	12	yes	—
55	C₂⁴×C₄₁	C₂⁴	1	6	yes	—
56	C₂⁴×C₄₁	C₂⁴	1	18	yes	—
57	C₂⁴×C₄₁	C₂⁴	1	16	yes	—
58	C₂⁴×C₄₁	C₂⁴	1	1	yes	characteristic
59	C₂×D₄×C₄₁	—	4	144	no	—
60	C₂×D₄×C₄₁	—	4	72	no	—
61	C₂×D₄×C₄₁	—	4	72	no	—
62	C₂×D₄×C₄₁	C₂⁴	1	96	yes	—
63	C₂×D₄×C₄₁	C₂⁴	1	48	yes	—
64	C₂×D₄×C₄₁	C₂⁴	1	48	yes	complemented
65	C₂²×C₄×C₄₁	—	2	24	no	—
66	C₂²×C₄×C₄₁	—	2	12	no	—
67	C₂²×C₄×C₄₁	—	4	72	no	—
68	C₂²×C₄×C₄₁	C₂⁴	1	18	yes	—
69	C₂²×C₄×C₄₁	C₂⁴	1	36	yes	—
70	C₂²×C₄×C₄₁	C₂⁴	1	48	yes	complemented
71	C₄⋊C₄×C₄₁	C₂⁴	1	96	yes	complemented
72	C₂²⋊C₄×C₄₁	C₂⁴	1	96	yes	—
73	C₂²⋊C₄×C₄₁	C₂⁴	1	48	yes	—
74	C₂²⋊C₄×C₄₁	C₂⁴	1	48	yes	complemented
75	C₄²×C₄₁	C₂⁴	1	16	yes	complemented
76	C₂³×C₄₁	—	2	36	no	—
77	C₂³×C₄₁	—	2	72	no	—
78	C₂³×C₄₁	—	2	72	no	—
79	C₂³×C₄₁	—	2	36	no	—
80	C₂³×C₄₁	—	4	48	no	—
81	C₂³×C₄₁	—	4	192	no	—
82	C₂³×C₄₁	—	4	48	no	—
83	C₂³×C₄₁	—	4	32	no	—
84	C₂³×C₄₁	—	4	16	no	—
85	C₂³×C₄₁	H(5,𝔽₂)	1	12	yes	—
86	C₂³×C₄₁	C₂⁵	1	8	yes	—
87	C₂³×C₄₁	C₂⁵	1	4	yes	—
88	C₂³×C₄₁	C₂⁵	1	12	yes	—
89	C₂³×C₄₁	C₂⁵	1	3	yes	—
90	D₄×C₄₁	—	4	96	no	—
91	D₄×C₄₁	—	4	48	no	—
92	D₄×C₄₁	—	4	48	no	—
93	C₂×C₄×C₄₁	—	2	72	no	—
94	C₂×C₄×C₄₁	—	2	36	no	—
95	C₂×C₄×C₄₁	—	4	48	no	—
96	C₂×C₄×C₄₁	C₂⁵	1	12	yes	—
97	C₂×C₄×C₄₁	C₂⁵	1	24	yes	—
98	C₂²×#560	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂²×C₄₁	—	2	12	no	—
100	C₂²×C₄₁	—	2	24	no	—
101	C₂²×C₄₁	—	2	24	no	—
102	C₂²×C₄₁	—	2	12	no	—
103	C₂²×C₄₁	—	4	64	no	—
104	C₂²×C₄₁	—	4	72	no	—
105	C₂²×C₄₁	—	4	48	no	—
106	C₂²×C₄₁	—	4	24	no	—
107	C₂²×C₄₁	#560	1	16	yes	direct product factor
108	C₂²×C₄₁	C₂×H(5,𝔽₂)	1	18	yes	—
109	C₂²×C₄₁	C₂⁶	1	1	yes	characteristic
110	C₄×C₄₁	—	2	24	no	—
111	C₄×C₄₁	—	2	12	no	—
112	C₂×#560	C₂×C₄₁	1	48	yes	direct product factor
113	C₂²×#127	C₂×C₄₁	1	1	yes	characteristic, complemented
114	C₂²×#126	C₂×C₄₁	1	2	yes	complemented
115	C₂²×#123	C₂×C₄₁	1	3	yes	complemented
116	C₂²×#121	C₂×C₄₁	1	6	yes	complemented
117	C₂²×#120	C₂×C₄₁	1	3	yes	complemented
118	C₂×C₄₁	—	4	12	no	—
119	C₂×C₄₁	—	4	8	no	—
120	C₂×C₄₁	—	4	4	no	—
121	C₂×C₄₁	C₂×#560	1	12	yes	direct product factor
122	C₂×C₄₁	C₂²×H(5,𝔽₂)	1	3	yes	—
123	C₂⁶	C₂²×C₄₁	1	1	yes	characteristic
124	C₂³×D₄	C₂²×C₄₁	1	6	yes	—
125	C₂³×D₄	C₂²×C₄₁	1	3	yes	—
126	C₂³×D₄	C₂²×C₄₁	1	3	yes	complemented
127	C₂⁴×C₄	C₂²×C₄₁	1	3	yes	complemented
128	#560	C₂²×C₄₁	1	256	yes	direct product factor
129	C₂×#127	C₂²×C₄₁	1	24	yes	complemented
130	C₂×#126	C₂²×C₄₁	1	48	yes	complemented
131	C₂×#123	C₂²×C₄₁	1	72	yes	complemented
132	C₂×#121	C₂²×C₄₁	1	144	yes	complemented
133	C₂×#120	C₂²×C₄₁	1	72	yes	complemented
134	C₂²×C₄⋊C₄	C₂²×C₄₁	1	6	yes	complemented
135	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	6	yes	—
136	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	3	yes	—
137	C₂²×C₂²⋊C₄	C₂²×C₄₁	1	3	yes	complemented
138	C₂²×C₄²	C₂²×C₄₁	1	1	yes	characteristic, complemented
139	C₄₁	C₂²×#560	1	1	yes	characteristic, direct product factor, Sylow
140	C₂⁵	—	4	12	no	—
141	C₂⁵	C₂³×C₄₁	1	2	yes	—
142	C₂⁵	C₂³×C₄₁	1	1	yes	characteristic
143	C₂⁵	C₂³×C₄₁	1	3	yes	—
144	C₂⁵	C₂³×C₄₁	1	12	yes	—
145	C₂²×D₄	—	4	24	no	—
146	C₂²×D₄	—	4	12	no	—
147	C₂²×D₄	—	4	12	no	—
148	C₂²×D₄	C₂³×C₄₁	1	72	yes	—
149	C₂²×D₄	C₂³×C₄₁	1	36	yes	—
150	C₂²×D₄	C₂³×C₄₁	1	36	yes	complemented
151	C₂³×C₄	—	4	12	no	—
152	C₂³×C₄	C₂³×C₄₁	1	6	yes	—
153	C₂³×C₄	C₂³×C₄₁	1	3	yes	—
154	C₂³×C₄	C₂³×C₄₁	1	36	yes	complemented
155	#127	C₂³×C₄₁	1	64	yes	complemented
156	#126	C₂³×C₄₁	1	128	yes	complemented
157	#123	C₂³×C₄₁	1	192	yes	complemented
158	#121	C₂³×C₄₁	1	384	yes	complemented
159	#120	C₂³×C₄₁	1	192	yes	complemented
160	C₂×C₄⋊C₄	C₂³×C₄₁	1	72	yes	complemented
161	C₂×C₂²⋊C₄	C₂³×C₄₁	1	72	yes	—
162	C₂×C₂²⋊C₄	C₂³×C₄₁	1	36	yes	—
163	C₂×C₂²⋊C₄	C₂³×C₄₁	1	36	yes	complemented
164	C₂×C₄²	C₂³×C₄₁	1	12	yes	complemented
165	C₂⁴	—	2	12	no	—
166	C₂⁴	—	2	24	no	—
167	C₂⁴	—	2	24	no	—
168	C₂⁴	—	2	12	no	—
169	C₂⁴	—	4	72	no	—
170	C₂⁴	—	4	64	no	—
171	C₂⁴	C₂⁴×C₄₁	1	12	yes	—
172	C₂⁴	C₂⁴×C₄₁	1	6	yes	—
173	C₂⁴	C₂⁴×C₄₁	1	18	yes	—
174	C₂⁴	C₂⁴×C₄₁	1	16	yes	—
175	C₂⁴	C₂⁴×C₄₁	1	1	yes	characteristic
176	C₂×D₄	—	4	144	no	—
177	C₂×D₄	—	4	72	no	—
178	C₂×D₄	—	4	72	no	—
179	C₂×D₄	C₂⁴×C₄₁	1	96	yes	—
180	C₂×D₄	C₂⁴×C₄₁	1	48	yes	—
181	C₂×D₄	C₂⁴×C₄₁	1	48	yes	complemented
182	C₂²×C₄	—	2	24	no	—
183	C₂²×C₄	—	2	12	no	—
184	C₂²×C₄	—	4	72	no	—
185	C₂²×C₄	C₂⁴×C₄₁	1	18	yes	—
186	C₂²×C₄	C₂⁴×C₄₁	1	36	yes	—
187	C₂²×C₄	C₂⁴×C₄₁	1	48	yes	complemented
188	C₄⋊C₄	C₂⁴×C₄₁	1	96	yes	complemented
189	C₂²⋊C₄	C₂⁴×C₄₁	1	96	yes	—
190	C₂²⋊C₄	C₂⁴×C₄₁	1	48	yes	—
191	C₂²⋊C₄	C₂⁴×C₄₁	1	48	yes	complemented
192	C₄²	C₂⁴×C₄₁	1	16	yes	complemented
193	C₂³	—	2	36	no	—
194	C₂³	—	2	72	no	—
195	C₂³	—	2	72	no	—
196	C₂³	—	2	36	no	—
197	C₂³	—	4	48	no	—
198	C₂³	—	4	192	no	—
199	C₂³	—	4	32	no	—
200	C₂³	—	4	16	no	—
201	C₂³	—	4	48	no	—
202	C₂³	H(5,𝔽₂)×C₄₁	1	12	yes	—
203	C₂³	C₂⁵×C₄₁	1	8	yes	—
204	C₂³	C₂⁵×C₄₁	1	4	yes	—
205	C₂³	C₂⁵×C₄₁	1	12	yes	—
206	C₂³	C₂⁵×C₄₁	1	3	yes	—
207	D₄	—	4	96	no	—
208	D₄	—	4	48	no	—
209	D₄	—	4	48	no	—
210	C₂×C₄	—	2	72	no	—
211	C₂×C₄	—	2	36	no	—
212	C₂×C₄	—	4	48	no	—
213	C₂×C₄	C₂⁵×C₄₁	1	12	yes	—
214	C₂×C₄	C₂⁵×C₄₁	1	24	yes	—
215	C₂²	—	2	12	no	—
216	C₂²	—	2	24	no	—
217	C₂²	—	2	24	no	—
218	C₂²	—	2	12	no	—
219	C₂²	—	4	64	no	—
220	C₂²	—	4	48	no	—
221	C₂²	—	4	24	no	—
222	C₂²	—	4	72	no	—
223	C₂²	C₄₁×#560	1	16	yes	direct product factor
224	C₂²	C₂×H(5,𝔽₂)×C₄₁	1	18	yes	—
225	C₂²	C₂⁶×C₄₁	1	1	yes	characteristic
226	C₄	—	2	24	no	—
227	C₄	—	2	12	no	—
228	C₂	—	4	8	no	—
229	C₂	—	4	4	no	—
230	C₂	—	4	12	no	—
231	C₂	C₂×C₄₁×#560	1	12	yes	direct product factor
232	C₂	C₂²×H(5,𝔽₂)×C₄₁	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₄₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₄₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#560	C₂	direct (N × H)
2. C₂²×#127×C₄₁	C₂	semidirect
3. C₂²×#126×C₄₁	C₂	semidirect
4. C₂²×#123×C₄₁	C₂	semidirect
5. C₂²×#121×C₄₁	C₂	semidirect
6. C₂²×#120×C₄₁	C₂	semidirect
10. C₂³×D₄×C₄₁	C₂²	semidirect
11. C₂⁴×C₄×C₄₁	C₂²	semidirect
12. C₄₁×#560	C₂²	direct (N × H)
13. C₂×#127×C₄₁	C₂²	semidirect
14. C₂×#126×C₄₁	C₂²	semidirect
15. C₂×#123×C₄₁	C₂²	semidirect
16. C₂×#121×C₄₁	C₂²	semidirect
17. C₂×#120×C₄₁	C₂²	semidirect
18. C₂²×C₄⋊C₄×C₄₁	C₂²	semidirect
21. C₂²×C₂²⋊C₄×C₄₁	C₂²	semidirect
22. C₂²×C₄²×C₄₁	C₂²	semidirect
33. C₂²×D₄×C₄₁	C₂³	semidirect
37. C₂³×C₄×C₄₁	C₂³	semidirect
38. #127×C₄₁	C₂³	semidirect
39. #126×C₄₁	C₂³	semidirect
40. #123×C₄₁	C₂³	semidirect
41. #121×C₄₁	C₂³	semidirect
42. #120×C₄₁	C₂³	semidirect
43. C₂×C₄⋊C₄×C₄₁	C₂³	semidirect
46. C₂×C₂²⋊C₄×C₄₁	C₂³	semidirect
47. C₂×C₄²×C₄₁	C₂³	semidirect
64. C₂×D₄×C₄₁	C₂⁴	semidirect
70. C₂²×C₄×C₄₁	C₂⁴	semidirect
71. C₄⋊C₄×C₄₁	C₂⁴	semidirect
74. C₂²⋊C₄×C₄₁	C₂⁴	semidirect
75. C₄²×C₄₁	C₂⁴	semidirect
98. C₂²×#560	C₄₁	direct (N × H)
107. C₂²×C₄₁	#560	direct (N × H)
112. C₂×#560	C₂×C₄₁	direct (N × H)
113. C₂²×#127	C₂×C₄₁	semidirect
114. C₂²×#126	C₂×C₄₁	semidirect
115. C₂²×#123	C₂×C₄₁	semidirect
116. C₂²×#121	C₂×C₄₁	semidirect
117. C₂²×#120	C₂×C₄₁	semidirect
121. C₂×C₄₁	C₂×#560	direct (N × H)
126. C₂³×D₄	C₂²×C₄₁	semidirect
127. C₂⁴×C₄	C₂²×C₄₁	semidirect
128. #560	C₂²×C₄₁	direct (N × H)
129. C₂×#127	C₂²×C₄₁	semidirect
130. C₂×#126	C₂²×C₄₁	semidirect
131. C₂×#123	C₂²×C₄₁	semidirect
132. C₂×#121	C₂²×C₄₁	semidirect
133. C₂×#120	C₂²×C₄₁	semidirect
134. C₂²×C₄⋊C₄	C₂²×C₄₁	semidirect
137. C₂²×C₂²⋊C₄	C₂²×C₄₁	semidirect
138. C₂²×C₄²	C₂²×C₄₁	semidirect
139. C₄₁	C₂²×#560	direct (N × H)
150. C₂²×D₄	C₂³×C₄₁	semidirect
154. C₂³×C₄	C₂³×C₄₁	semidirect
155. #127	C₂³×C₄₁	semidirect
156. #126	C₂³×C₄₁	semidirect
157. #123	C₂³×C₄₁	semidirect
158. #121	C₂³×C₄₁	semidirect
159. #120	C₂³×C₄₁	semidirect
160. C₂×C₄⋊C₄	C₂³×C₄₁	semidirect
163. C₂×C₂²⋊C₄	C₂³×C₄₁	semidirect
164. C₂×C₄²	C₂³×C₄₁	semidirect
181. C₂×D₄	C₂⁴×C₄₁	semidirect
187. C₂²×C₄	C₂⁴×C₄₁	semidirect
188. C₄⋊C₄	C₂⁴×C₄₁	semidirect
191. C₂²⋊C₄	C₂⁴×C₄₁	semidirect
192. C₄²	C₂⁴×C₄₁	semidirect
223. C₂²	C₄₁×#560	direct (N × H)
231. C₂	C₂×C₄₁×#560	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#560	2
C₂²×#127×C₄₁	2
C₂²×#126×C₄₁	2
C₂²×#123×C₄₁	2
C₂²×#121×C₄₁	2
C₂²×#120×C₄₁	2
C₂²×#560	41

Subgroups — C22×C41×#560 #23838888

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄₁×#560 #23838888