Subgroups — C₂⁴×#534 #72278216

Order 1024.

1140821 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×D₄	C₂	1	2	yes	maximal, complemented
2	C₂³×#534	C₂	1	240	yes	maximal, direct product factor
3	C₂⁴×#123	C₂	1	4	yes	maximal, complemented
4	C₂⁴×#121	C₂	1	4	yes	maximal, complemented
5	C₂⁴×#120	C₂	1	4	yes	maximal, complemented
6	C₂⁵×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁸	C₂²	1	1	yes	characteristic
8	C₂⁸	C₂²	1	2	yes	complemented
9	C₂⁵×D₄	—	2	128	no	—
10	C₂⁵×D₄	C₂²	1	4	yes	—
11	C₂⁵×D₄	C₂²	1	4	yes	complemented
12	C₂⁵×D₄	C₂²	1	4	yes	complemented
13	C₂⁵×D₄	C₂²	1	240	yes	complemented
14	C₂⁶×C₄	C₂²	1	2	yes	complemented
15	C₂⁶×C₄	C₂²	1	2	yes	complemented
16	C₂²×#534	C₂²	1	8960	yes	direct product factor
17	C₂³×#123	C₂²	1	480	yes	complemented
18	C₂³×#121	C₂²	1	480	yes	complemented
19	C₂³×#120	C₂²	1	480	yes	complemented
20	C₂⁴×C₄⋊C₄	C₂²	1	4	yes	complemented
21	C₂⁴×C₂²⋊C₄	C₂²	1	4	yes	complemented
22	C₂⁴×C₂²⋊C₄	C₂²	1	8	yes	complemented
23	C₂⁴×C₂²⋊C₄	C₂²	1	120	yes	complemented
24	C₂⁷	—	2	64	no	—
25	C₂⁷	—	4	32	no	—
26	C₂⁷	—	4	64	no	—
27	C₂⁷	D₄	1	64	yes	complemented
28	C₂⁷	C₂³	1	4	yes	—
29	C₂⁷	C₂³	1	1	yes	characteristic
30	C₂⁷	C₂³	1	2	yes	—
31	C₂⁷	C₂³	1	60	yes	—
32	C₂⁷	C₂³	1	120	yes	complemented
33	C₂⁴×D₄	—	2	128	no	—
34	C₂⁴×D₄	—	2	128	no	—
35	C₂⁴×D₄	—	2	7680	no	—
36	C₂⁴×D₄	—	4	64	no	—
37	C₂⁴×D₄	C₂³	1	240	yes	—
38	C₂⁴×D₄	C₂³	1	240	yes	complemented
39	C₂⁴×D₄	C₂³	1	240	yes	complemented
40	C₂⁴×D₄	C₂³	1	4480	yes	complemented
41	C₂⁵×C₄	—	2	64	no	—
42	C₂⁵×C₄	—	4	32	no	—
43	C₂⁵×C₄	C₂³	1	4	yes	—
44	C₂⁵×C₄	C₂³	1	4	yes	—
45	C₂⁵×C₄	C₂³	1	120	yes	complemented
46	C₂⁵×C₄	C₂³	1	120	yes	complemented
47	C₂×#534	C₂³	1	61440	yes	direct product factor
48	C₂²×#123	C₂³	1	8960	yes	complemented
49	C₂²×#121	C₂³	1	8960	yes	complemented
50	C₂²×#120	C₂³	1	8960	yes	complemented
51	C₂³×C₄⋊C₄	C₂³	1	240	yes	complemented
52	C₂³×C₂²⋊C₄	C₂³	1	240	yes	complemented
53	C₂³×C₂²⋊C₄	C₂³	1	480	yes	complemented
54	C₂³×C₂²⋊C₄	C₂³	1	2240	yes	complemented
55	C₂⁶	—	2	32	no	—
56	C₂⁶	—	2	64	no	—
57	C₂⁶	—	2	64	no	—
58	C₂⁶	—	2	1920	no	—
59	C₂⁶	—	2	128	no	—
60	C₂⁶	—	4	1024	no	—
61	C₂⁶	—	4	960	no	—
62	C₂⁶	—	4	1920	no	—
63	C₂⁶	—	4	2048	no	—
64	C₂⁶	C₂×D₄	1	64	yes	—
65	C₂⁶	C₂×D₄	1	32	yes	—
66	C₂⁶	C₂×D₄	1	1920	yes	complemented
67	C₂⁶	C₂⁴	1	120	yes	—
68	C₂⁶	C₂⁴	1	30	yes	—
69	C₂⁶	C₂⁴	1	60	yes	—
70	C₂⁶	C₂⁴	1	560	yes	—
71	C₂⁶	C₂⁴	1	1	yes	characteristic
72	C₂⁶	C₂⁴	1	1120	yes	complemented
73	C₂³×D₄	—	2	3840	no	—
74	C₂³×D₄	—	2	3840	no	—
75	C₂³×D₄	—	2	71680	no	—
76	C₂³×D₄	—	4	1920	no	—
77	C₂³×D₄	C₂⁴	1	2240	yes	—
78	C₂³×D₄	C₂⁴	1	2240	yes	complemented
79	C₂³×D₄	C₂⁴	1	2240	yes	complemented
80	C₂³×D₄	C₂⁴	1	15360	yes	complemented
81	C₂⁴×C₄	—	2	64	no	—
82	C₂⁴×C₄	—	2	1920	no	—
83	C₂⁴×C₄	—	2	64	no	—
84	C₂⁴×C₄	—	4	960	no	—
85	C₂⁴×C₄	C₂⁴	1	120	yes	—
86	C₂⁴×C₄	C₂⁴	1	120	yes	—
87	C₂⁴×C₄	C₂⁴	1	1120	yes	complemented
88	C₂⁴×C₄	C₂⁴	1	1120	yes	complemented
89	#534	C₂⁴	1	65536	yes	direct product factor
90	C₂×#123	C₂⁴	1	30720	yes	complemented
91	C₂×#121	C₂⁴	1	30720	yes	complemented
92	C₂×#120	C₂⁴	1	30720	yes	complemented
93	C₂²×C₄⋊C₄	C₂⁴	1	2240	yes	complemented
94	C₂²×C₂²⋊C₄	C₂⁴	1	2240	yes	complemented
95	C₂²×C₂²⋊C₄	C₂⁴	1	4480	yes	complemented
96	C₂²×C₂²⋊C₄	C₂⁴	1	7680	yes	complemented
97	C₂⁵	—	2	1024	no	—
98	C₂⁵	—	2	512	no	—
99	C₂⁵	—	2	480	no	—
100	C₂⁵	—	2	960	no	—
101	C₂⁵	—	2	960	no	—
102	C₂⁵	—	2	8960	no	—
103	C₂⁵	—	2	1920	no	—
104	C₂⁵	—	4	15360	no	—
105	C₂⁵	—	4	4480	no	—
106	C₂⁵	—	4	8960	no	—
107	C₂⁵	—	4	30720	no	—
108	C₂⁵	—	4	1024	no	—
109	C₂⁵	C₂²×D₄	1	960	yes	—
110	C₂⁵	C₂²×D₄	1	480	yes	—
111	C₂⁵	C₂²×D₄	1	16	yes	—
112	C₂⁵	C₂²×D₄	1	8960	yes	complemented
113	C₂⁵	H(5,𝔽₂)	1	32	yes	—
114	C₂⁵	C₂⁵	1	560	yes	—
115	C₂⁵	C₂⁵	1	280	yes	—
116	C₂⁵	C₂⁵	1	140	yes	—
117	C₂⁵	C₂⁵	1	960	yes	—
118	C₂⁵	C₂⁵	1	15	yes	—
119	C₂⁵	C₂⁵	1	1920	yes	complemented
120	C₂²×D₄	—	2	17920	no	—
121	C₂²×D₄	—	2	17920	no	—
122	C₂²×D₄	—	2	122880	no	—
123	C₂²×D₄	—	4	8960	no	—
124	C₂²×D₄	C₂⁵	1	3840	yes	—
125	C₂²×D₄	C₂⁵	1	3840	yes	complemented
126	C₂²×D₄	C₂⁵	1	3840	yes	complemented
127	C₂²×D₄	C₂⁵	1	8192	yes	complemented
128	C₂³×C₄	—	2	960	no	—
129	C₂³×C₄	—	2	8960	no	—
130	C₂³×C₄	—	2	960	no	—
131	C₂³×C₄	—	4	4480	no	—
132	C₂³×C₄	C₂⁵	1	560	yes	—
133	C₂³×C₄	C₂⁵	1	560	yes	—
134	C₂³×C₄	C₂⁵	1	1920	yes	complemented
135	C₂³×C₄	C₂⁵	1	1920	yes	complemented
136	#123	C₂⁵	1	16384	yes	complemented
137	#121	C₂⁵	1	16384	yes	complemented
138	#120	C₂⁵	1	16384	yes	complemented
139	C₂×C₄⋊C₄	C₂⁵	1	3840	yes	complemented
140	C₂×C₂²⋊C₄	C₂⁵	1	3840	yes	complemented
141	C₂×C₂²⋊C₄	C₂⁵	1	7680	yes	complemented
142	C₂×C₂²⋊C₄	C₂⁵	1	4096	yes	complemented
143	C₂⁴	—	2	1120	no	—
144	C₂⁴	—	2	2240	no	—
145	C₂⁴	—	2	7680	no	—
146	C₂⁴	—	2	3840	no	—
147	C₂⁴	—	2	2240	no	—
148	C₂⁴	—	2	7680	no	—
149	C₂⁴	—	2	4480	no	—
150	C₂⁴	—	4	35840	no	—
151	C₂⁴	—	4	3840	no	—
152	C₂⁴	—	4	7680	no	—
153	C₂⁴	—	4	71680	no	—
154	C₂⁴	—	4	7680	no	—
155	C₂⁴	#534	1	256	yes	direct product factor
156	C₂⁴	C₂³×D₄	1	2240	yes	—
157	C₂⁴	C₂³×D₄	1	1120	yes	—
158	C₂⁴	C₂³×D₄	1	120	yes	—
159	C₂⁴	C₂³×D₄	1	7680	yes	complemented
160	C₂⁴	C₂×H(5,𝔽₂)	1	240	yes	—
161	C₂⁴	C₂⁶	1	480	yes	—
162	C₂⁴	C₂⁶	1	240	yes	—
163	C₂⁴	C₂⁶	1	120	yes	—
164	C₂⁴	C₂⁶	1	256	yes	—
165	C₂⁴	C₂⁶	1	35	yes	—
166	C₂⁴	C₂⁶	1	512	yes	complemented
167	C₂×D₄	—	2	15360	no	—
168	C₂×D₄	—	2	15360	no	—
169	C₂×D₄	—	2	32768	no	—
170	C₂×D₄	—	4	7680	no	—
171	C₂×D₄	C₂⁶	1	1024	yes	—
172	C₂×D₄	C₂⁶	1	1024	yes	complemented
173	C₂×D₄	C₂⁶	1	1024	yes	complemented
174	C₂²×C₄	—	2	2240	no	—
175	C₂²×C₄	—	2	7680	no	—
176	C₂²×C₄	—	2	2240	no	—
177	C₂²×C₄	—	4	3840	no	—
178	C₂²×C₄	C₂⁶	1	480	yes	—
179	C₂²×C₄	C₂⁶	1	480	yes	—
180	C₂²×C₄	C₂⁶	1	512	yes	complemented
181	C₂²×C₄	C₂⁶	1	512	yes	complemented
182	C₄⋊C₄	C₂⁶	1	1024	yes	complemented
183	C₂²⋊C₄	C₂⁶	1	1024	yes	complemented
184	C₂²⋊C₄	C₂⁶	1	2048	yes	complemented
185	C₂³	—	2	480	no	—
186	C₂³	—	2	960	no	—
187	C₂³	—	2	8960	no	—
188	C₂³	—	2	4480	no	—
189	C₂³	—	2	960	no	—
190	C₂³	—	2	1024	no	—
191	C₂³	—	2	1920	no	—
192	C₂³	—	4	15360	no	—
193	C₂³	—	4	512	no	—
194	C₂³	—	4	1024	no	—
195	C₂³	—	4	30720	no	—
196	C₂³	—	4	8960	no	—
197	C₂³	C₂×#534	1	960	yes	direct product factor
198	C₂³	C₂⁴×D₄	1	960	yes	—
199	C₂³	C₂⁴×D₄	1	480	yes	—
200	C₂³	C₂⁴×D₄	1	140	yes	—
201	C₂³	C₂⁴×D₄	1	1024	yes	complemented
202	C₂³	C₂²×H(5,𝔽₂)	1	280	yes	—
203	C₂³	C₂⁷	1	64	yes	—
204	C₂³	C₂⁷	1	32	yes	—
205	C₂³	C₂⁷	1	16	yes	—
206	C₂³	C₂⁷	1	15	yes	—
207	D₄	—	2	2048	no	—
208	D₄	—	2	2048	no	—
209	D₄	—	4	1024	no	—
210	C₂×C₄	—	2	960	no	—
211	C₂×C₄	—	2	1024	no	—
212	C₂×C₄	—	2	960	no	—
213	C₂×C₄	—	4	512	no	—
214	C₂×C₄	C₂⁷	1	64	yes	—
215	C₂×C₄	C₂⁷	1	64	yes	—
216	C₂²	—	2	1920	no	—
217	C₂²	—	2	960	no	—
218	C₂²	—	2	32	no	—
219	C₂²	—	2	64	no	—
220	C₂²	—	2	64	no	—
221	C₂²	—	2	128	no	—
222	C₂²	—	4	1024	no	—
223	C₂²	—	4	2048	no	—
224	C₂²	—	4	1920	no	—
225	C₂²	C₂²×#534	1	560	yes	direct product factor
226	C₂²	C₂⁵×D₄	1	64	yes	—
227	C₂²	C₂⁵×D₄	1	32	yes	—
228	C₂²	C₂⁵×D₄	1	30	yes	—
229	C₂²	C₂³×H(5,𝔽₂)	1	60	yes	—
230	C₂²	C₂⁸	1	1	yes	characteristic
231	C₄	—	2	64	no	—
232	C₄	—	2	64	no	—
233	C₂	—	2	64	no	—
234	C₂	—	2	32	no	—
235	C₂	—	4	64	no	—
236	C₂	C₂³×#534	1	60	yes	direct product factor
237	C₂	C₂⁶×D₄	1	1	yes	characteristic
238	C₂	C₂⁴×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂⁶×D₄	C₂	semidirect
2. C₂³×#534	C₂	direct (N × H)
3. C₂⁴×#123	C₂	semidirect
4. C₂⁴×#121	C₂	semidirect
5. C₂⁴×#120	C₂	semidirect
6. C₂⁵×C₂²⋊C₄	C₂	semidirect
8. C₂⁸	C₂²	semidirect
11. C₂⁵×D₄	C₂²	semidirect
12. C₂⁵×D₄	C₂²	semidirect
13. C₂⁵×D₄	C₂²	semidirect
14. C₂⁶×C₄	C₂²	semidirect
15. C₂⁶×C₄	C₂²	semidirect
16. C₂²×#534	C₂²	direct (N × H)
17. C₂³×#123	C₂²	semidirect
18. C₂³×#121	C₂²	semidirect
19. C₂³×#120	C₂²	semidirect
20. C₂⁴×C₄⋊C₄	C₂²	semidirect
21. C₂⁴×C₂²⋊C₄	C₂²	semidirect
22. C₂⁴×C₂²⋊C₄	C₂²	semidirect
23. C₂⁴×C₂²⋊C₄	C₂²	semidirect
27. C₂⁷	D₄	semidirect
32. C₂⁷	C₂³	semidirect
38. C₂⁴×D₄	C₂³	semidirect
39. C₂⁴×D₄	C₂³	semidirect
40. C₂⁴×D₄	C₂³	semidirect
45. C₂⁵×C₄	C₂³	semidirect
46. C₂⁵×C₄	C₂³	semidirect
47. C₂×#534	C₂³	direct (N × H)
48. C₂²×#123	C₂³	semidirect
49. C₂²×#121	C₂³	semidirect
50. C₂²×#120	C₂³	semidirect
51. C₂³×C₄⋊C₄	C₂³	semidirect
52. C₂³×C₂²⋊C₄	C₂³	semidirect
53. C₂³×C₂²⋊C₄	C₂³	semidirect
54. C₂³×C₂²⋊C₄	C₂³	semidirect
66. C₂⁶	C₂×D₄	semidirect
72. C₂⁶	C₂⁴	semidirect
78. C₂³×D₄	C₂⁴	semidirect
79. C₂³×D₄	C₂⁴	semidirect
80. C₂³×D₄	C₂⁴	semidirect
87. C₂⁴×C₄	C₂⁴	semidirect
88. C₂⁴×C₄	C₂⁴	semidirect
89. #534	C₂⁴	direct (N × H)
90. C₂×#123	C₂⁴	semidirect
91. C₂×#121	C₂⁴	semidirect
92. C₂×#120	C₂⁴	semidirect
93. C₂²×C₄⋊C₄	C₂⁴	semidirect
94. C₂²×C₂²⋊C₄	C₂⁴	semidirect
95. C₂²×C₂²⋊C₄	C₂⁴	semidirect
96. C₂²×C₂²⋊C₄	C₂⁴	semidirect
112. C₂⁵	C₂²×D₄	semidirect
119. C₂⁵	C₂⁵	semidirect
125. C₂²×D₄	C₂⁵	semidirect
126. C₂²×D₄	C₂⁵	semidirect
127. C₂²×D₄	C₂⁵	semidirect
134. C₂³×C₄	C₂⁵	semidirect
135. C₂³×C₄	C₂⁵	semidirect
136. #123	C₂⁵	semidirect
137. #121	C₂⁵	semidirect
138. #120	C₂⁵	semidirect
139. C₂×C₄⋊C₄	C₂⁵	semidirect
140. C₂×C₂²⋊C₄	C₂⁵	semidirect
141. C₂×C₂²⋊C₄	C₂⁵	semidirect
142. C₂×C₂²⋊C₄	C₂⁵	semidirect
155. C₂⁴	#534	direct (N × H)
159. C₂⁴	C₂³×D₄	semidirect
166. C₂⁴	C₂⁶	semidirect
172. C₂×D₄	C₂⁶	semidirect
173. C₂×D₄	C₂⁶	semidirect
180. C₂²×C₄	C₂⁶	semidirect
181. C₂²×C₄	C₂⁶	semidirect
182. C₄⋊C₄	C₂⁶	semidirect
183. C₂²⋊C₄	C₂⁶	semidirect
184. C₂²⋊C₄	C₂⁶	semidirect
197. C₂³	C₂×#534	direct (N × H)
201. C₂³	C₂⁴×D₄	semidirect
225. C₂²	C₂²×#534	direct (N × H)
236. C₂	C₂³×#534	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×D₄	2
C₂³×#534	2
C₂⁴×#123	2
C₂⁴×#121	2
C₂⁴×#120	2
C₂⁵×C₂²⋊C₄	2

Subgroups — C24×#534 #72278216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#534 #72278216