Subgroups — C₂⁴×#518 #72289363

Order 1024.

1072213 conjugacy classes of subgroups.

The subgroup lattice has 180 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	1	yes	maximal, characteristic
2	C₂³×#518	C₂	1	240	yes	maximal, direct product factor
3	C₂⁴×C₄×D₄	C₂	1	8	yes	maximal, complemented
4	C₂⁴×#117	C₂	1	2	yes	maximal, complemented
5	C₂⁵×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
6	C₂⁸	C₂²	1	2	yes	—
7	C₂⁵×D₄	C₄	1	128	yes	complemented
8	C₂⁵×D₄	C₂²	1	4	yes	—
9	C₂⁵×D₄	C₂²	1	120	yes	—
10	C₂⁶×C₄	C₂²	1	1	yes	characteristic
11	C₂⁶×C₄	C₂²	1	8	yes	complemented
12	C₂²×#518	C₂²	1	8960	yes	direct product factor
13	C₂³×C₄×D₄	C₂²	1	960	yes	complemented
14	C₂³×#117	C₂²	1	240	yes	complemented
15	C₂⁴×C₄⋊C₄	C₂²	1	4	yes	complemented
16	C₂⁴×C₂²⋊C₄	C₂²	1	4	yes	—
17	C₂⁴×C₂²⋊C₄	C₂²	1	8	yes	complemented
18	C₂⁴×C₂²⋊C₄	C₂²	1	480	yes	complemented
19	C₂⁴×C₄²	C₂²	1	4	yes	complemented
20	C₂⁷	—	4	64	no	—
21	C₂⁷	C₂×C₄	1	128	yes	complemented
22	C₂⁷	C₂³	1	1	yes	characteristic
23	C₂⁷	C₂³	1	4	yes	—
24	C₂⁷	C₂³	1	120	yes	—
25	C₂⁴×D₄	C₂×C₄	1	256	yes	complemented
26	C₂⁴×D₄	C₂×C₄	1	7680	yes	complemented
27	C₂⁴×D₄	C₂³	1	240	yes	—
28	C₂⁴×D₄	C₂³	1	2240	yes	—
29	C₂⁵×C₄	—	4	128	no	—
30	C₂⁵×C₄	C₂×C₄	1	64	yes	complemented
31	C₂⁵×C₄	C₂³	1	8	yes	—
32	C₂⁵×C₄	C₂³	1	2	yes	—
33	C₂⁵×C₄	C₂³	1	60	yes	—
34	C₂⁵×C₄	C₂³	1	480	yes	complemented
35	C₂×#518	C₂³	1	61440	yes	direct product factor
36	C₂²×C₄×D₄	C₂³	1	17920	yes	complemented
37	C₂²×#117	C₂³	1	4480	yes	complemented
38	C₂³×C₄⋊C₄	C₂³	1	240	yes	complemented
39	C₂³×C₂²⋊C₄	C₂³	1	240	yes	—
40	C₂³×C₂²⋊C₄	C₂³	1	8960	yes	complemented
41	C₂³×C₂²⋊C₄	C₂³	1	480	yes	complemented
42	C₂³×C₄²	C₂³	1	240	yes	complemented
43	C₂⁶	—	2	128	no	—
44	C₂⁶	—	2	32	no	—
45	C₂⁶	—	4	2048	no	—
46	C₂⁶	—	4	1920	no	—
47	C₂⁶	C₂²×C₄	1	32	yes	—
48	C₂⁶	C₂²×C₄	1	128	yes	—
49	C₂⁶	C₂²×C₄	1	3840	yes	complemented
50	C₂⁶	C₂⁴	1	30	yes	—
51	C₂⁶	C₂⁴	1	120	yes	—
52	C₂⁶	C₂⁴	1	1120	yes	—
53	C₂⁶	C₂⁴	1	1	yes	characteristic
54	C₂³×D₄	C₂²×C₄	1	7680	yes	complemented
55	C₂³×D₄	C₂²×C₄	1	71680	yes	complemented
56	C₂³×D₄	C₂⁴	1	2240	yes	—
57	C₂³×D₄	C₂⁴	1	7680	yes	—
58	C₂⁴×C₄	—	2	128	no	—
59	C₂⁴×C₄	—	4	3840	no	—
60	C₂⁴×C₄	C₂²×C₄	1	64	yes	—
61	C₂⁴×C₄	C₂²×C₄	1	1920	yes	complemented
62	C₂⁴×C₄	C₂⁴	1	240	yes	—
63	C₂⁴×C₄	C₂⁴	1	60	yes	—
64	C₂⁴×C₄	C₂⁴	1	560	yes	—
65	C₂⁴×C₄	C₂⁴	1	4480	yes	complemented
66	#518	C₂⁴	1	65536	yes	direct product factor
67	C₂×C₄×D₄	C₂⁴	1	61440	yes	complemented
68	C₂×#117	C₂⁴	1	15360	yes	complemented
69	C₂²×C₄⋊C₄	C₂⁴	1	2240	yes	complemented
70	C₂²×C₂²⋊C₄	C₂⁴	1	2240	yes	—
71	C₂²×C₂²⋊C₄	C₂⁴	1	30720	yes	complemented
72	C₂²×C₂²⋊C₄	C₂⁴	1	4480	yes	complemented
73	C₂²×C₄²	C₂⁴	1	2240	yes	complemented
74	C₂⁵	—	2	512	no	—
75	C₂⁵	—	2	2048	no	—
76	C₂⁵	—	2	1920	no	—
77	C₂⁵	—	2	480	no	—
78	C₂⁵	—	4	30720	no	—
79	C₂⁵	—	4	8960	no	—
80	C₂⁵	C₂³×C₄	1	480	yes	—
81	C₂⁵	C₂³×C₄	1	1920	yes	—
82	C₂⁵	C₂³×C₄	1	16	yes	—
83	C₂⁵	C₂³×C₄	1	17920	yes	complemented
84	C₂⁵	H(5,𝔽₂)	1	32	yes	—
85	C₂⁵	C₂⁵	1	140	yes	—
86	C₂⁵	C₂⁵	1	560	yes	—
87	C₂⁵	C₂⁵	1	1920	yes	—
88	C₂⁵	C₂⁵	1	15	yes	—
89	C₂²×D₄	C₂³×C₄	1	35840	yes	complemented
90	C₂²×D₄	C₂³×C₄	1	122880	yes	complemented
91	C₂²×D₄	C₂⁵	1	3840	yes	—
92	C₂²×D₄	C₂⁵	1	4096	yes	—
93	C₂³×C₄	—	2	1920	no	—
94	C₂³×C₄	—	4	17920	no	—
95	C₂³×C₄	C₂³×C₄	1	960	yes	—
96	C₂³×C₄	C₂³×C₄	1	8960	yes	complemented
97	C₂³×C₄	C₂⁵	1	1120	yes	—
98	C₂³×C₄	C₂⁵	1	280	yes	—
99	C₂³×C₄	C₂⁵	1	960	yes	—
100	C₂³×C₄	C₂⁵	1	7680	yes	complemented
101	C₄×D₄	C₂⁵	1	32768	yes	complemented
102	#117	C₂⁵	1	8192	yes	complemented
103	C₂×C₄⋊C₄	C₂⁵	1	3840	yes	complemented
104	C₂×C₂²⋊C₄	C₂⁵	1	3840	yes	—
105	C₂×C₂²⋊C₄	C₂⁵	1	16384	yes	complemented
106	C₂×C₂²⋊C₄	C₂⁵	1	7680	yes	complemented
107	C₂×C₄²	C₂⁵	1	3840	yes	complemented
108	C₂⁴	—	2	4480	no	—
109	C₂⁴	—	2	1120	no	—
110	C₂⁴	—	2	3840	no	—
111	C₂⁴	—	2	15360	no	—
112	C₂⁴	—	4	71680	no	—
113	C₂⁴	—	4	7680	no	—
114	C₂⁴	#518	1	256	yes	direct product factor
115	C₂⁴	C₂⁴×C₄	1	1120	yes	—
116	C₂⁴	C₂⁴×C₄	1	4480	yes	—
117	C₂⁴	C₂⁴×C₄	1	120	yes	—
118	C₂⁴	C₂⁴×C₄	1	15360	yes	complemented
119	C₂⁴	C₂×H(5,𝔽₂)	1	240	yes	—
120	C₂⁴	C₂⁶	1	120	yes	—
121	C₂⁴	C₂⁶	1	480	yes	—
122	C₂⁴	C₂⁶	1	512	yes	—
123	C₂⁴	C₂⁶	1	35	yes	—
124	C₂×D₄	C₂⁴×C₄	1	30720	yes	complemented
125	C₂×D₄	C₂⁴×C₄	1	32768	yes	complemented
126	C₂×D₄	C₂⁶	1	1024	yes	—
127	C₂²×C₄	—	2	4480	no	—
128	C₂²×C₄	—	4	15360	no	—
129	C₂²×C₄	C₂⁴×C₄	1	2240	yes	—
130	C₂²×C₄	C₂⁴×C₄	1	7680	yes	complemented
131	C₂²×C₄	C₂⁶	1	960	yes	—
132	C₂²×C₄	C₂⁶	1	240	yes	—
133	C₂²×C₄	C₂⁶	1	256	yes	—
134	C₂²×C₄	C₂⁶	1	2048	yes	complemented
135	C₄⋊C₄	C₂⁶	1	1024	yes	complemented
136	C₂²⋊C₄	C₂⁶	1	1024	yes	—
137	C₂²⋊C₄	C₂⁶	1	2048	yes	complemented
138	C₄²	C₂⁶	1	1024	yes	complemented
139	C₂³	—	2	1920	no	—
140	C₂³	—	2	480	no	—
141	C₂³	—	2	4480	no	—
142	C₂³	—	2	17920	no	—
143	C₂³	—	4	30720	no	—
144	C₂³	—	4	1024	no	—
145	C₂³	C₂×#518	1	960	yes	direct product factor
146	C₂³	C₂⁵×C₄	1	480	yes	—
147	C₂³	C₂⁵×C₄	1	1920	yes	—
148	C₂³	C₂⁵×C₄	1	140	yes	—
149	C₂³	C₂⁵×C₄	1	2048	yes	complemented
150	C₂³	C₂²×H(5,𝔽₂)	1	280	yes	—
151	C₂³	C₂⁷	1	16	yes	—
152	C₂³	C₂⁷	1	64	yes	—
153	C₂³	C₂⁷	1	15	yes	—
154	D₄	C₂⁵×C₄	1	4096	yes	complemented
155	C₂×C₄	—	2	1920	no	—
156	C₂×C₄	—	4	2048	no	—
157	C₂×C₄	C₂⁵×C₄	1	960	yes	—
158	C₂×C₄	C₂⁵×C₄	1	1024	yes	complemented
159	C₂×C₄	C₂⁷	1	128	yes	—
160	C₂×C₄	C₂⁷	1	32	yes	—
161	C₂²	—	2	960	no	—
162	C₂²	—	2	3840	no	—
163	C₂²	—	2	128	no	—
164	C₂²	—	2	32	no	—
165	C₂²	—	4	2048	no	—
166	C₂²	C₂²×#518	1	560	yes	direct product factor
167	C₂²	C₂⁶×C₄	1	32	yes	—
168	C₂²	C₂⁶×C₄	1	128	yes	—
169	C₂²	C₂⁶×C₄	1	30	yes	—
170	C₂²	C₂³×H(5,𝔽₂)	1	60	yes	—
171	C₂²	C₂⁸	1	1	yes	characteristic
172	C₄	—	2	128	no	—
173	C₄	C₂⁶×C₄	1	64	yes	—
174	C₂	—	2	32	no	—
175	C₂	—	2	128	no	—
176	C₂	C₂³×#518	1	60	yes	direct product factor
177	C₂	C₂⁷×C₄	1	1	yes	characteristic
178	C₂	C₂⁴×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator	C₂	C₂⁷×C₄
Frattini, cosocle	C₂²	C₂⁸
prime radical	C₂⁶×D₄	C₂
LCS	G ▷ C₂ ▷ 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
2. C₂³×#518	C₂	direct (N × H)
3. C₂⁴×C₄×D₄	C₂	semidirect
4. C₂⁴×#117	C₂	semidirect
5. C₂⁵×C₂²⋊C₄	C₂	semidirect
7. C₂⁵×D₄	C₄	semidirect
11. C₂⁶×C₄	C₂²	semidirect
12. C₂²×#518	C₂²	direct (N × H)
13. C₂³×C₄×D₄	C₂²	semidirect
14. C₂³×#117	C₂²	semidirect
15. C₂⁴×C₄⋊C₄	C₂²	semidirect
17. C₂⁴×C₂²⋊C₄	C₂²	semidirect
18. C₂⁴×C₂²⋊C₄	C₂²	semidirect
19. C₂⁴×C₄²	C₂²	semidirect
21. C₂⁷	C₂×C₄	semidirect
25. C₂⁴×D₄	C₂×C₄	semidirect
26. C₂⁴×D₄	C₂×C₄	semidirect
30. C₂⁵×C₄	C₂×C₄	semidirect
34. C₂⁵×C₄	C₂³	semidirect
35. C₂×#518	C₂³	direct (N × H)
36. C₂²×C₄×D₄	C₂³	semidirect
37. C₂²×#117	C₂³	semidirect
38. C₂³×C₄⋊C₄	C₂³	semidirect
40. C₂³×C₂²⋊C₄	C₂³	semidirect
41. C₂³×C₂²⋊C₄	C₂³	semidirect
42. C₂³×C₄²	C₂³	semidirect
49. C₂⁶	C₂²×C₄	semidirect
54. C₂³×D₄	C₂²×C₄	semidirect
55. C₂³×D₄	C₂²×C₄	semidirect
61. C₂⁴×C₄	C₂²×C₄	semidirect
65. C₂⁴×C₄	C₂⁴	semidirect
66. #518	C₂⁴	direct (N × H)
67. C₂×C₄×D₄	C₂⁴	semidirect
68. C₂×#117	C₂⁴	semidirect
69. C₂²×C₄⋊C₄	C₂⁴	semidirect
71. C₂²×C₂²⋊C₄	C₂⁴	semidirect
72. C₂²×C₂²⋊C₄	C₂⁴	semidirect
73. C₂²×C₄²	C₂⁴	semidirect
83. C₂⁵	C₂³×C₄	semidirect
89. C₂²×D₄	C₂³×C₄	semidirect
90. C₂²×D₄	C₂³×C₄	semidirect
96. C₂³×C₄	C₂³×C₄	semidirect
100. C₂³×C₄	C₂⁵	semidirect
101. C₄×D₄	C₂⁵	semidirect
102. #117	C₂⁵	semidirect
103. C₂×C₄⋊C₄	C₂⁵	semidirect
105. C₂×C₂²⋊C₄	C₂⁵	semidirect
106. C₂×C₂²⋊C₄	C₂⁵	semidirect
107. C₂×C₄²	C₂⁵	semidirect
114. C₂⁴	#518	direct (N × H)
118. C₂⁴	C₂⁴×C₄	semidirect
124. C₂×D₄	C₂⁴×C₄	semidirect
125. C₂×D₄	C₂⁴×C₄	semidirect
130. C₂²×C₄	C₂⁴×C₄	semidirect
134. C₂²×C₄	C₂⁶	semidirect
135. C₄⋊C₄	C₂⁶	semidirect
137. C₂²⋊C₄	C₂⁶	semidirect
138. C₄²	C₂⁶	semidirect
145. C₂³	C₂×#518	direct (N × H)
149. C₂³	C₂⁵×C₄	semidirect
154. D₄	C₂⁵×C₄	semidirect
158. C₂×C₄	C₂⁵×C₄	semidirect
166. C₂²	C₂²×#518	direct (N × H)
176. C₂	C₂³×#518	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×#518	2
C₂⁴×C₄×D₄	2
C₂⁴×#117	2
C₂⁵×C₂²⋊C₄	2

Subgroups — C24×#518 #72289363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#518 #72289363