Subgroups — C₂⁴×#559 #72295510

Order 1024.

554821 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	C₂³×#559	C₂	1	240	yes	maximal, direct product factor
2	C₂⁴×#127	C₂	1	1	yes	maximal, characteristic
3	C₂⁴×#124	C₂	1	4	yes	maximal, complemented
4	C₂⁴×#123	C₂	1	4	yes	maximal, complemented
5	C₂⁴×C₄×Q₈	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₄×D₄	C₂	1	4	yes	maximal, complemented
7	C₂⁵×Q₈	C₂²	1	2	yes	—
8	C₂⁵×D₄	C₂²	1	4	yes	—
9	C₂⁵×D₄	C₂²	1	2	yes	—
10	C₂⁶×C₄	C₂²	1	4	yes	—
11	C₂²×#559	C₂²	1	8960	yes	direct product factor
12	C₂³×#127	C₂²	1	120	yes	—
13	C₂³×#124	C₂²	1	480	yes	complemented
14	C₂³×#123	C₂²	1	480	yes	complemented
15	C₂³×C₄×Q₈	C₂²	1	240	yes	complemented
16	C₂³×C₄×D₄	C₂²	1	480	yes	complemented
17	C₂⁴×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂⁴×C₄⋊C₄	C₂²	1	4	yes	—
19	C₂⁴×C₂²⋊C₄	C₂²	1	8	yes	—
20	C₂⁴×C₂²⋊C₄	C₂²	1	4	yes	—
21	C₂⁴×C₄²	C₂²	1	4	yes	—
22	C₂⁴×C₄²	C₂²	1	1	yes	characteristic
23	C₂⁷	C₂³	1	4	yes	—
24	C₂⁴×Q₈	—	2	64	no	—
25	C₂⁴×Q₈	C₂³	1	120	yes	—
26	C₂⁴×D₄	—	2	128	no	—
27	C₂⁴×D₄	—	4	32	no	—
28	C₂⁴×D₄	C₂³	1	240	yes	—
29	C₂⁴×D₄	C₂³	1	120	yes	—
30	C₂⁵×C₄	—	4	64	no	—
31	C₂⁵×C₄	C₂³	1	4	yes	—
32	C₂⁵×C₄	C₂³	1	1	yes	characteristic
33	C₂⁵×C₄	C₂³	1	4	yes	—
34	C₂⁵×C₄	C₂³	1	240	yes	—
35	C₂⁵×C₄	C₂³	1	2	yes	—
36	C₂×#559	C₂³	1	61440	yes	direct product factor
37	C₂²×#127	C₂³	1	2240	yes	—
38	C₂²×#124	C₂³	1	8960	yes	complemented
39	C₂²×#123	C₂³	1	8960	yes	complemented
40	C₂²×C₄×Q₈	C₂³	1	4480	yes	complemented
41	C₂²×C₄×D₄	C₂³	1	8960	yes	complemented
42	C₂³×C₄⋊C₄	C₂³	1	120	yes	—
43	C₂³×C₄⋊C₄	C₂³	1	240	yes	—
44	C₂³×C₂²⋊C₄	C₂³	1	480	yes	—
45	C₂³×C₂²⋊C₄	C₂³	1	240	yes	—
46	C₂³×C₄²	C₂³	1	240	yes	—
47	C₂³×C₄²	C₂³	1	60	yes	—
48	C₂⁶	—	2	128	no	—
49	C₂⁶	—	2	64	no	—
50	C₂⁶	C₂⁴	1	120	yes	—
51	C₂⁶	C₂⁴	1	1	yes	characteristic
52	C₂³×Q₈	—	2	1920	no	—
53	C₂³×Q₈	C₂⁴	1	1120	yes	—
54	C₂³×D₄	—	2	3840	no	—
55	C₂³×D₄	—	4	960	no	—
56	C₂³×D₄	C₂⁴	1	2240	yes	—
57	C₂³×D₄	C₂⁴	1	1120	yes	—
58	C₂⁴×C₄	—	2	64	no	—
59	C₂⁴×C₄	—	2	64	no	—
60	C₂⁴×C₄	—	2	16	no	—
61	C₂⁴×C₄	—	4	1920	no	—
62	C₂⁴×C₄	#41	1	64	yes	—
63	C₂⁴×C₄	C₂⁴	1	120	yes	—
64	C₂⁴×C₄	C₂⁴	1	30	yes	—
65	C₂⁴×C₄	C₂⁴	1	120	yes	—
66	C₂⁴×C₄	C₂⁴	1	2240	yes	—
67	C₂⁴×C₄	C₂⁴	1	60	yes	—
68	#559	C₂⁴	1	65536	yes	direct product factor
69	C₂×#127	C₂⁴	1	7680	yes	—
70	C₂×#124	C₂⁴	1	30720	yes	complemented
71	C₂×#123	C₂⁴	1	30720	yes	complemented
72	C₂×C₄×Q₈	C₂⁴	1	15360	yes	complemented
73	C₂×C₄×D₄	C₂⁴	1	30720	yes	complemented
74	C₂²×C₄⋊C₄	C₂⁴	1	1120	yes	—
75	C₂²×C₄⋊C₄	C₂⁴	1	2240	yes	—
76	C₂²×C₂²⋊C₄	C₂⁴	1	4480	yes	—
77	C₂²×C₂²⋊C₄	C₂⁴	1	2240	yes	—
78	C₂²×C₄²	C₂⁴	1	2240	yes	—
79	C₂²×C₄²	C₂⁴	1	560	yes	—
80	C₂⁵	—	2	1920	no	—
81	C₂⁵	—	2	960	no	—
82	C₂⁵	—	4	1024	no	—
83	C₂⁵	C₂×#41	1	32	yes	—
84	C₂⁵	H(5,𝔽₂)	1	16	yes	—
85	C₂⁵	C₂⁵	1	560	yes	—
86	C₂⁵	C₂⁵	1	15	yes	—
87	C₂²×Q₈	—	2	8960	no	—
88	C₂²×Q₈	C₂⁵	1	1920	yes	—
89	C₂²×D₄	—	2	17920	no	—
90	C₂²×D₄	—	4	4480	no	—
91	C₂²×D₄	C₂⁵	1	3840	yes	—
92	C₂²×D₄	C₂⁵	1	1920	yes	—
93	C₂³×C₄	—	2	960	no	—
94	C₂³×C₄	—	2	960	no	—
95	C₂³×C₄	—	2	240	no	—
96	C₂³×C₄	—	4	8960	no	—
97	C₂³×C₄	C₂×#41	1	960	yes	—
98	C₂³×C₄	C₂⁵	1	560	yes	—
99	C₂³×C₄	C₂⁵	1	560	yes	—
100	C₂³×C₄	C₂⁵	1	140	yes	—
101	C₂³×C₄	C₂⁵	1	3840	yes	—
102	C₂³×C₄	C₂⁵	1	280	yes	—
103	#127	C₂⁵	1	4096	yes	—
104	#124	C₂⁵	1	16384	yes	complemented
105	#123	C₂⁵	1	16384	yes	complemented
106	C₄×Q₈	C₂⁵	1	8192	yes	complemented
107	C₄×D₄	C₂⁵	1	16384	yes	complemented
108	C₂×C₄⋊C₄	C₂⁵	1	1920	yes	—
109	C₂×C₄⋊C₄	C₂⁵	1	3840	yes	—
110	C₂×C₂²⋊C₄	C₂⁵	1	7680	yes	—
111	C₂×C₂²⋊C₄	C₂⁵	1	3840	yes	—
112	C₂×C₄²	C₂⁵	1	3840	yes	—
113	C₂×C₄²	C₂⁵	1	960	yes	—
114	C₂⁴	—	2	4480	no	—
115	C₂⁴	—	2	2240	no	—
116	C₂⁴	—	4	7680	no	—
117	C₂⁴	#559	1	256	yes	direct product factor
118	C₂⁴	C₂²×#41	1	240	yes	—
119	C₂⁴	C₂×H(5,𝔽₂)	1	120	yes	—
120	C₂⁴	C₂⁶	1	480	yes	—
121	C₂⁴	C₂⁶	1	35	yes	—
122	C₂×Q₈	—	2	7680	no	—
123	C₂×Q₈	C₂⁶	1	512	yes	—
124	C₂×D₄	—	2	15360	no	—
125	C₂×D₄	—	4	3840	no	—
126	C₂×D₄	C₂⁶	1	1024	yes	—
127	C₂×D₄	C₂⁶	1	512	yes	—
128	C₂²×C₄	—	2	2240	no	—
129	C₂²×C₄	—	2	2240	no	—
130	C₂²×C₄	—	2	560	no	—
131	C₂²×C₄	—	4	7680	no	—
132	C₂²×C₄	C₂²×#41	1	2240	yes	—
133	C₂²×C₄	C₂⁶	1	480	yes	—
134	C₂²×C₄	C₂⁶	1	480	yes	—
135	C₂²×C₄	C₂⁶	1	120	yes	—
136	C₂²×C₄	C₂⁶	1	1024	yes	—
137	C₂²×C₄	C₂⁶	1	240	yes	—
138	C₄⋊C₄	C₂⁶	1	512	yes	—
139	C₄⋊C₄	C₂⁶	1	1024	yes	—
140	C₂²⋊C₄	C₂⁶	1	2048	yes	—
141	C₂²⋊C₄	C₂⁶	1	1024	yes	—
142	C₄²	C₂⁶	1	1024	yes	—
143	C₄²	C₂⁶	1	256	yes	—
144	C₂³	—	2	1920	no	—
145	C₂³	—	2	960	no	—
146	C₂³	—	4	8960	no	—
147	C₂³	C₂×#559	1	960	yes	direct product factor
148	C₂³	C₂³×#41	1	280	yes	—
149	C₂³	C₂²×H(5,𝔽₂)	1	140	yes	—
150	C₂³	C₂⁷	1	64	yes	—
151	C₂³	C₂⁷	1	15	yes	—
152	Q₈	—	2	1024	no	—
153	D₄	—	2	2048	no	—
154	D₄	—	4	512	no	—
155	C₂×C₄	—	2	960	no	—
156	C₂×C₄	—	2	960	no	—
157	C₂×C₄	—	2	240	no	—
158	C₂×C₄	—	4	1024	no	—
159	C₂×C₄	C₂³×#41	1	960	yes	—
160	C₂×C₄	C₂⁷	1	64	yes	—
161	C₂×C₄	C₂⁷	1	64	yes	—
162	C₂×C₄	C₂⁷	1	16	yes	—
163	C₂×C₄	C₂⁷	1	32	yes	—
164	C₂²	—	2	128	no	—
165	C₂²	—	2	64	no	—
166	C₂²	—	4	1920	no	—
167	C₂²	C₂²×#559	1	560	yes	direct product factor
168	C₂²	C₂⁴×#41	1	60	yes	—
169	C₂²	C₂³×H(5,𝔽₂)	1	30	yes	—
170	C₂²	C₂⁸	1	1	yes	characteristic
171	C₄	—	2	64	no	—
172	C₄	—	2	64	no	—
173	C₄	—	2	16	no	—
174	C₄	C₂⁴×#41	1	64	yes	—
175	C₂	—	4	64	no	—
176	C₂	C₂³×#559	1	60	yes	direct product factor
177	C₂	C₂⁵×#41	1	2	yes	—
178	C₂	C₂⁴×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁸
prime radical	C₂⁴×#127	C₂
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×#559	C₂	direct (N × H)
3. C₂⁴×#124	C₂	semidirect
4. C₂⁴×#123	C₂	semidirect
5. C₂⁴×C₄×Q₈	C₂	semidirect
6. C₂⁴×C₄×D₄	C₂	semidirect
11. C₂²×#559	C₂²	direct (N × H)
13. C₂³×#124	C₂²	semidirect
14. C₂³×#123	C₂²	semidirect
15. C₂³×C₄×Q₈	C₂²	semidirect
16. C₂³×C₄×D₄	C₂²	semidirect
36. C₂×#559	C₂³	direct (N × H)
38. C₂²×#124	C₂³	semidirect
39. C₂²×#123	C₂³	semidirect
40. C₂²×C₄×Q₈	C₂³	semidirect
41. C₂²×C₄×D₄	C₂³	semidirect
68. #559	C₂⁴	direct (N × H)
70. C₂×#124	C₂⁴	semidirect
71. C₂×#123	C₂⁴	semidirect
72. C₂×C₄×Q₈	C₂⁴	semidirect
73. C₂×C₄×D₄	C₂⁴	semidirect
104. #124	C₂⁵	semidirect
105. #123	C₂⁵	semidirect
106. C₄×Q₈	C₂⁵	semidirect
107. C₄×D₄	C₂⁵	semidirect
117. C₂⁴	#559	direct (N × H)
147. C₂³	C₂×#559	direct (N × H)
167. C₂²	C₂²×#559	direct (N × H)
176. C₂	C₂³×#559	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×#559	2
C₂⁴×#127	2
C₂⁴×#124	2
C₂⁴×#123	2
C₂⁴×C₄×Q₈	2
C₂⁴×C₄×D₄	2

Subgroups — C24×#559 #72295510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#559 #72295510