Subgroups — C₂⁴×#560 #72295508

Order 1024.

635397 conjugacy classes of subgroups.

The subgroup lattice has 195 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#560	C₂	1	240	yes	maximal, direct product factor
2	C₂⁴×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×#126	C₂	1	2	yes	maximal, complemented
4	C₂⁴×#123	C₂	1	3	yes	maximal, complemented
5	C₂⁴×#121	C₂	1	6	yes	maximal, complemented
6	C₂⁴×#120	C₂	1	3	yes	maximal, complemented
7	C₂⁸	C₂²	1	1	yes	characteristic
8	C₂⁵×D₄	C₂²	1	6	yes	—
9	C₂⁵×D₄	C₂²	1	3	yes	—
10	C₂⁵×D₄	C₂²	1	3	yes	complemented
11	C₂⁶×C₄	C₂²	1	3	yes	complemented
12	C₂²×#560	C₂²	1	8960	yes	direct product factor
13	C₂³×#127	C₂²	1	120	yes	complemented
14	C₂³×#126	C₂²	1	240	yes	complemented
15	C₂³×#123	C₂²	1	360	yes	complemented
16	C₂³×#121	C₂²	1	720	yes	complemented
17	C₂³×#120	C₂²	1	360	yes	complemented
18	C₂⁴×C₄⋊C₄	C₂²	1	6	yes	complemented
19	C₂⁴×C₂²⋊C₄	C₂²	1	6	yes	—
20	C₂⁴×C₂²⋊C₄	C₂²	1	3	yes	—
21	C₂⁴×C₂²⋊C₄	C₂²	1	3	yes	complemented
22	C₂⁴×C₄²	C₂²	1	1	yes	characteristic, complemented
23	C₂⁷	—	4	48	no	—
24	C₂⁷	C₂³	1	3	yes	—
25	C₂⁷	C₂³	1	2	yes	—
26	C₂⁷	C₂³	1	1	yes	characteristic
27	C₂⁷	C₂³	1	60	yes	—
28	C₂⁴×D₄	—	4	96	no	—
29	C₂⁴×D₄	—	4	48	no	—
30	C₂⁴×D₄	—	4	48	no	—
31	C₂⁴×D₄	C₂³	1	360	yes	—
32	C₂⁴×D₄	C₂³	1	180	yes	—
33	C₂⁴×D₄	C₂³	1	180	yes	complemented
34	C₂⁵×C₄	—	4	48	no	—
35	C₂⁵×C₄	C₂³	1	6	yes	—
36	C₂⁵×C₄	C₂³	1	3	yes	—
37	C₂⁵×C₄	C₂³	1	180	yes	complemented
38	C₂×#560	C₂³	1	61440	yes	direct product factor
39	C₂²×#127	C₂³	1	2240	yes	complemented
40	C₂²×#126	C₂³	1	4480	yes	complemented
41	C₂²×#123	C₂³	1	6720	yes	complemented
42	C₂²×#121	C₂³	1	13440	yes	complemented
43	C₂²×#120	C₂³	1	6720	yes	complemented
44	C₂³×C₄⋊C₄	C₂³	1	360	yes	complemented
45	C₂³×C₂²⋊C₄	C₂³	1	360	yes	—
46	C₂³×C₂²⋊C₄	C₂³	1	180	yes	—
47	C₂³×C₂²⋊C₄	C₂³	1	180	yes	complemented
48	C₂³×C₄²	C₂³	1	60	yes	complemented
49	C₂⁶	—	2	48	no	—
50	C₂⁶	—	2	96	no	—
51	C₂⁶	—	2	96	no	—
52	C₂⁶	—	2	48	no	—
53	C₂⁶	—	4	1440	no	—
54	C₂⁶	—	4	1024	no	—
55	C₂⁶	C₂⁴	1	90	yes	—
56	C₂⁶	C₂⁴	1	60	yes	—
57	C₂⁶	C₂⁴	1	30	yes	—
58	C₂⁶	C₂⁴	1	560	yes	—
59	C₂⁶	C₂⁴	1	1	yes	characteristic
60	C₂³×D₄	—	4	2880	no	—
61	C₂³×D₄	—	4	1440	no	—
62	C₂³×D₄	—	4	1440	no	—
63	C₂³×D₄	C₂⁴	1	3360	yes	—
64	C₂³×D₄	C₂⁴	1	1680	yes	—
65	C₂³×D₄	C₂⁴	1	1680	yes	complemented
66	C₂⁴×C₄	—	2	96	no	—
67	C₂⁴×C₄	—	2	48	no	—
68	C₂⁴×C₄	—	4	1440	no	—
69	C₂⁴×C₄	C₂⁴	1	180	yes	—
70	C₂⁴×C₄	C₂⁴	1	90	yes	—
71	C₂⁴×C₄	C₂⁴	1	1680	yes	complemented
72	#560	C₂⁴	1	65536	yes	direct product factor
73	C₂×#127	C₂⁴	1	7680	yes	complemented
74	C₂×#126	C₂⁴	1	15360	yes	complemented
75	C₂×#123	C₂⁴	1	23040	yes	complemented
76	C₂×#121	C₂⁴	1	46080	yes	complemented
77	C₂×#120	C₂⁴	1	23040	yes	complemented
78	C₂²×C₄⋊C₄	C₂⁴	1	3360	yes	complemented
79	C₂²×C₂²⋊C₄	C₂⁴	1	3360	yes	—
80	C₂²×C₂²⋊C₄	C₂⁴	1	1680	yes	—
81	C₂²×C₂²⋊C₄	C₂⁴	1	1680	yes	complemented
82	C₂²×C₄²	C₂⁴	1	560	yes	complemented
83	C₂⁵	—	2	720	no	—
84	C₂⁵	—	2	1440	no	—
85	C₂⁵	—	2	1440	no	—
86	C₂⁵	—	2	720	no	—
87	C₂⁵	—	4	6720	no	—
88	C₂⁵	—	4	15360	no	—
89	C₂⁵	—	4	768	no	—
90	C₂⁵	—	4	512	no	—
91	C₂⁵	—	4	256	no	—
92	C₂⁵	H(5,𝔽₂)	1	48	yes	—
93	C₂⁵	C₂⁵	1	280	yes	—
94	C₂⁵	C₂⁵	1	140	yes	—
95	C₂⁵	C₂⁵	1	420	yes	—
96	C₂⁵	C₂⁵	1	960	yes	—
97	C₂⁵	C₂⁵	1	15	yes	—
98	C₂²×D₄	—	4	13440	no	—
99	C₂²×D₄	—	4	6720	no	—
100	C₂²×D₄	—	4	6720	no	—
101	C₂²×D₄	C₂⁵	1	5760	yes	—
102	C₂²×D₄	C₂⁵	1	2880	yes	—
103	C₂²×D₄	C₂⁵	1	2880	yes	complemented
104	C₂³×C₄	—	2	1440	no	—
105	C₂³×C₄	—	2	720	no	—
106	C₂³×C₄	—	4	6720	no	—
107	C₂³×C₄	C₂⁵	1	840	yes	—
108	C₂³×C₄	C₂⁵	1	420	yes	—
109	C₂³×C₄	C₂⁵	1	2880	yes	complemented
110	#127	C₂⁵	1	4096	yes	complemented
111	#126	C₂⁵	1	8192	yes	complemented
112	#123	C₂⁵	1	12288	yes	complemented
113	#121	C₂⁵	1	24576	yes	complemented
114	#120	C₂⁵	1	12288	yes	complemented
115	C₂×C₄⋊C₄	C₂⁵	1	5760	yes	complemented
116	C₂×C₂²⋊C₄	C₂⁵	1	5760	yes	—
117	C₂×C₂²⋊C₄	C₂⁵	1	2880	yes	—
118	C₂×C₂²⋊C₄	C₂⁵	1	2880	yes	complemented
119	C₂×C₄²	C₂⁵	1	960	yes	complemented
120	C₂⁴	—	2	1680	no	—
121	C₂⁴	—	2	3360	no	—
122	C₂⁴	—	2	3360	no	—
123	C₂⁴	—	2	1680	no	—
124	C₂⁴	—	4	5760	no	—
125	C₂⁴	—	4	35840	no	—
126	C₂⁴	—	4	5760	no	—
127	C₂⁴	—	4	3840	no	—
128	C₂⁴	—	4	1920	no	—
129	C₂⁴	#560	1	256	yes	direct product factor
130	C₂⁴	C₂×H(5,𝔽₂)	1	360	yes	—
131	C₂⁴	C₂⁶	1	240	yes	—
132	C₂⁴	C₂⁶	1	120	yes	—
133	C₂⁴	C₂⁶	1	360	yes	—
134	C₂⁴	C₂⁶	1	256	yes	—
135	C₂⁴	C₂⁶	1	35	yes	—
136	C₂×D₄	—	4	11520	no	—
137	C₂×D₄	—	4	5760	no	—
138	C₂×D₄	—	4	5760	no	—
139	C₂×D₄	C₂⁶	1	1536	yes	—
140	C₂×D₄	C₂⁶	1	768	yes	—
141	C₂×D₄	C₂⁶	1	768	yes	complemented
142	C₂²×C₄	—	2	3360	no	—
143	C₂²×C₄	—	2	1680	no	—
144	C₂²×C₄	—	4	5760	no	—
145	C₂²×C₄	C₂⁶	1	360	yes	—
146	C₂²×C₄	C₂⁶	1	720	yes	—
147	C₂²×C₄	C₂⁶	1	768	yes	complemented
148	C₄⋊C₄	C₂⁶	1	1536	yes	complemented
149	C₂²⋊C₄	C₂⁶	1	1536	yes	—
150	C₂²⋊C₄	C₂⁶	1	768	yes	—
151	C₂²⋊C₄	C₂⁶	1	768	yes	complemented
152	C₄²	C₂⁶	1	256	yes	complemented
153	C₂³	—	2	720	no	—
154	C₂³	—	2	1440	no	—
155	C₂³	—	2	1440	no	—
156	C₂³	—	2	720	no	—
157	C₂³	—	4	768	no	—
158	C₂³	—	4	15360	no	—
159	C₂³	—	4	6720	no	—
160	C₂³	—	4	4480	no	—
161	C₂³	—	4	2240	no	—
162	C₂³	C₂×#560	1	960	yes	direct product factor
163	C₂³	C₂²×H(5,𝔽₂)	1	420	yes	—
164	C₂³	C₂⁷	1	32	yes	—
165	C₂³	C₂⁷	1	16	yes	—
166	C₂³	C₂⁷	1	48	yes	—
167	C₂³	C₂⁷	1	15	yes	—
168	D₄	—	4	1536	no	—
169	D₄	—	4	768	no	—
170	D₄	—	4	768	no	—
171	C₂×C₄	—	2	1440	no	—
172	C₂×C₄	—	2	720	no	—
173	C₂×C₄	—	4	768	no	—
174	C₂×C₄	C₂⁷	1	48	yes	—
175	C₂×C₄	C₂⁷	1	96	yes	—
176	C₂²	—	2	48	no	—
177	C₂²	—	2	96	no	—
178	C₂²	—	2	96	no	—
179	C₂²	—	2	48	no	—
180	C₂²	—	4	1024	no	—
181	C₂²	—	4	1440	no	—
182	C₂²	—	4	960	no	—
183	C₂²	—	4	480	no	—
184	C₂²	C₂²×#560	1	560	yes	direct product factor
185	C₂²	C₂³×H(5,𝔽₂)	1	90	yes	—
186	C₂²	C₂⁸	1	1	yes	characteristic
187	C₄	—	2	96	no	—
188	C₄	—	2	48	no	—
189	C₂	—	4	48	no	—
190	C₂	—	4	32	no	—
191	C₂	—	4	16	no	—
192	C₂	C₂³×#560	1	60	yes	direct product factor
193	C₂	C₂⁴×H(5,𝔽₂)	1	3	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. C₂³×#560	C₂	direct (N × H)
2. C₂⁴×#127	C₂	semidirect
3. C₂⁴×#126	C₂	semidirect
4. C₂⁴×#123	C₂	semidirect
5. C₂⁴×#121	C₂	semidirect
6. C₂⁴×#120	C₂	semidirect
10. C₂⁵×D₄	C₂²	semidirect
11. C₂⁶×C₄	C₂²	semidirect
12. C₂²×#560	C₂²	direct (N × H)
13. C₂³×#127	C₂²	semidirect
14. C₂³×#126	C₂²	semidirect
15. C₂³×#123	C₂²	semidirect
16. C₂³×#121	C₂²	semidirect
17. C₂³×#120	C₂²	semidirect
18. C₂⁴×C₄⋊C₄	C₂²	semidirect
21. C₂⁴×C₂²⋊C₄	C₂²	semidirect
22. C₂⁴×C₄²	C₂²	semidirect
33. C₂⁴×D₄	C₂³	semidirect
37. C₂⁵×C₄	C₂³	semidirect
38. C₂×#560	C₂³	direct (N × H)
39. C₂²×#127	C₂³	semidirect
40. C₂²×#126	C₂³	semidirect
41. C₂²×#123	C₂³	semidirect
42. C₂²×#121	C₂³	semidirect
43. C₂²×#120	C₂³	semidirect
44. C₂³×C₄⋊C₄	C₂³	semidirect
47. C₂³×C₂²⋊C₄	C₂³	semidirect
48. C₂³×C₄²	C₂³	semidirect
65. C₂³×D₄	C₂⁴	semidirect
71. C₂⁴×C₄	C₂⁴	semidirect
72. #560	C₂⁴	direct (N × H)
73. C₂×#127	C₂⁴	semidirect
74. C₂×#126	C₂⁴	semidirect
75. C₂×#123	C₂⁴	semidirect
76. C₂×#121	C₂⁴	semidirect
77. C₂×#120	C₂⁴	semidirect
78. C₂²×C₄⋊C₄	C₂⁴	semidirect
81. C₂²×C₂²⋊C₄	C₂⁴	semidirect
82. C₂²×C₄²	C₂⁴	semidirect
103. C₂²×D₄	C₂⁵	semidirect
109. C₂³×C₄	C₂⁵	semidirect
110. #127	C₂⁵	semidirect
111. #126	C₂⁵	semidirect
112. #123	C₂⁵	semidirect
113. #121	C₂⁵	semidirect
114. #120	C₂⁵	semidirect
115. C₂×C₄⋊C₄	C₂⁵	semidirect
118. C₂×C₂²⋊C₄	C₂⁵	semidirect
119. C₂×C₄²	C₂⁵	semidirect
129. C₂⁴	#560	direct (N × H)
141. C₂×D₄	C₂⁶	semidirect
147. C₂²×C₄	C₂⁶	semidirect
148. C₄⋊C₄	C₂⁶	semidirect
151. C₂²⋊C₄	C₂⁶	semidirect
152. C₄²	C₂⁶	semidirect
162. C₂³	C₂×#560	direct (N × H)
184. C₂²	C₂²×#560	direct (N × H)
192. C₂	C₂³×#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
C₂³×#560	2
C₂⁴×#127	2
C₂⁴×#126	2
C₂⁴×#123	2
C₂⁴×#121	2
C₂⁴×#120	2

Subgroups — C24×#560 #72295508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#560 #72295508