Subgroups — C₂⁴×#509 #72286510

Order 1024.

96405 conjugacy classes of subgroups.

The subgroup lattice has 241 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁵×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂⁴×SD₃₂	C₂	1	2	yes	maximal, complemented
4	C₂³×#509	C₂	1	120	yes	maximal, direct product factor
5	C₂⁴×D₁₆	C₂	1	2	yes	maximal, complemented
6	C₂⁴×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁴×#41	—	2	1	no	—
8	C₂⁵×D₄	—	2	1	no	—
9	C₂³×#135	—	2	30	no	—
10	C₂³×#135	C₂²	1	60	yes	complemented
11	C₂⁴×Q₁₆	C₂²	1	1	yes	characteristic, complemented
12	C₂⁴×SD₁₆	—	2	1	no	—
13	C₂⁴×D₈	—	2	1	no	—
14	C₂⁴×D₈	—	2	30	no	—
15	C₂⁴×D₈	C₂²	1	2	yes	—
16	C₂⁴×D₈	C₂²	1	60	yes	complemented
17	C₂⁴×D₈	C₂²	1	1	yes	characteristic, complemented
18	C₂⁵×C₈	C₂²	1	1	yes	characteristic
19	C₂²×#509	C₂²	1	2240	yes	direct product factor
20	C₂³×SD₃₂	C₂²	1	120	yes	complemented
21	C₂³×D₁₆	C₂²	1	120	yes	complemented
22	C₂³×M₃₂	C₂²	1	60	yes	complemented
23	C₂⁴×C₁₆	C₂²	1	2	yes	complemented
24	C₂⁷	—	4	1	no	—
25	C₂³×#41	—	2	60	no	—
26	C₂³×#41	—	4	30	no	—
27	C₂⁴×Q₈	—	2	1	no	—
28	C₂⁴×D₄	—	2	2	no	—
29	C₂⁴×D₄	—	2	60	no	—
30	C₂⁴×D₄	—	2	1	no	—
31	C₂⁴×D₄	—	4	1	no	—
32	C₂⁴×D₄	—	4	30	no	—
33	C₂⁴×D₄	—	4	1	no	—
34	C₂⁵×C₄	—	4	1	no	—
35	C₂⁵×C₄	D₄	1	1	yes	characteristic
36	C₂²×#135	—	2	840	no	—
37	C₂²×#135	C₂³	1	560	yes	complemented
38	C₂³×Q₁₆	—	2	15	no	—
39	C₂³×Q₁₆	C₂³	1	30	yes	complemented
40	C₂³×SD₁₆	—	2	60	no	—
41	C₂³×D₈	—	2	30	no	—
42	C₂³×D₈	—	2	60	no	—
43	C₂³×D₈	—	2	840	no	—
44	C₂³×D₈	—	2	15	no	—
45	C₂³×D₈	C₂³	1	60	yes	—
46	C₂³×D₈	C₂³	1	560	yes	complemented
47	C₂³×D₈	C₂³	1	30	yes	complemented
48	C₂⁴×C₈	D₄	1	1	yes	characteristic
49	C₂⁴×C₈	D₄	1	30	yes	—
50	C₂⁴×C₈	C₂³	1	1	yes	characteristic
51	C₂⁴×C₈	C₂³	1	30	yes	—
52	C₂×#509	C₂³	1	7680	yes	direct product factor
53	C₂²×SD₃₂	C₂³	1	1120	yes	complemented
54	C₂²×D₁₆	C₂³	1	1120	yes	complemented
55	C₂²×M₃₂	C₂³	1	560	yes	complemented
56	C₂³×C₁₆	C₂³	1	60	yes	complemented
57	C₂⁶	—	4	60	no	—
58	C₂⁶	—	4	1	no	—
59	C₂⁶	—	4	2	no	—
60	C₂⁶	—	8	32	no	—
61	C₂⁶	D₈	1	1	yes	characteristic
62	C₂²×#41	—	2	560	no	—
63	C₂²×#41	—	4	840	no	—
64	C₂³×Q₈	—	2	30	no	—
65	C₂³×Q₈	—	4	15	no	—
66	C₂³×D₄	—	2	60	no	—
67	C₂³×D₄	—	2	560	no	—
68	C₂³×D₄	—	2	30	no	—
69	C₂³×D₄	—	4	60	no	—
70	C₂³×D₄	—	4	60	no	—
71	C₂³×D₄	—	4	30	no	—
72	C₂³×D₄	—	4	840	no	—
73	C₂³×D₄	—	4	15	no	—
74	C₂⁴×C₄	—	4	1	no	—
75	C₂⁴×C₄	—	4	60	no	—
76	C₂⁴×C₄	D₈	1	30	yes	—
77	C₂⁴×C₄	D₈	1	1	yes	characteristic
78	C₂⁴×C₄	C₂×D₄	1	30	yes	—
79	C₂⁴×C₄	C₂×D₄	1	1	yes	characteristic
80	C₂×#135	—	2	3360	no	—
81	C₂×#135	C₂⁴	1	960	yes	complemented
82	C₂²×Q₁₆	—	2	210	no	—
83	C₂²×Q₁₆	C₂⁴	1	140	yes	complemented
84	C₂²×SD₁₆	—	2	560	no	—
85	C₂²×D₈	—	2	420	no	—
86	C₂²×D₈	—	2	560	no	—
87	C₂²×D₈	—	2	3360	no	—
88	C₂²×D₈	—	2	210	no	—
89	C₂²×D₈	C₂⁴	1	280	yes	—
90	C₂²×D₈	C₂⁴	1	960	yes	complemented
91	C₂²×D₈	C₂⁴	1	140	yes	complemented
92	C₂³×C₈	C₂×D₄	1	15	yes	—
93	C₂³×C₈	C₂×D₄	1	30	yes	—
94	C₂³×C₈	C₂×D₄	1	420	yes	—
95	C₂³×C₈	C₂⁴	1	15	yes	—
96	C₂³×C₈	C₂⁴	1	140	yes	—
97	#509	C₂⁴	1	4096	yes	direct product factor
98	C₂×SD₃₂	C₂⁴	1	1920	yes	complemented
99	C₂×D₁₆	C₂⁴	1	1920	yes	complemented
100	C₂×M₃₂	C₂⁴	1	960	yes	complemented
101	C₂²×C₁₆	C₂⁴	1	280	yes	complemented
102	C₂⁵	—	2	16	no	—
103	C₂⁵	—	4	30	no	—
104	C₂⁵	—	4	560	no	—
105	C₂⁵	—	4	60	no	—
106	C₂⁵	—	8	960	no	—
107	C₂⁵	—	8	16	no	—
108	C₂⁵	—	8	32	no	—
109	C₂⁵	C₂×D₈	1	30	yes	—
110	C₂⁵	C₂×D₈	1	1	yes	characteristic
111	C₂×#41	—	2	960	no	—
112	C₂×#41	—	4	3360	no	—
113	C₂²×Q₈	—	2	140	no	—
114	C₂²×Q₈	—	4	210	no	—
115	C₂²×D₄	—	2	280	no	—
116	C₂²×D₄	—	2	960	no	—
117	C₂²×D₄	—	2	140	no	—
118	C₂²×D₄	—	4	560	no	—
119	C₂²×D₄	—	4	560	no	—
120	C₂²×D₄	—	4	420	no	—
121	C₂²×D₄	—	4	3360	no	—
122	C₂²×D₄	—	4	210	no	—
123	C₂³×C₄	—	4	30	no	—
124	C₂³×C₄	—	4	560	no	—
125	C₂³×C₄	C₂×D₈	1	30	yes	—
126	C₂³×C₄	C₂×D₈	1	15	yes	—
127	C₂³×C₄	C₂×D₈	1	420	yes	—
128	C₂³×C₄	C₂²×D₄	1	15	yes	—
129	C₂³×C₄	C₂²×D₄	1	140	yes	—
130	#135	—	2	1920	no	—
131	#135	C₂⁵	1	256	yes	complemented
132	C₂×Q₁₆	—	2	420	no	—
133	C₂×Q₁₆	C₂⁵	1	120	yes	complemented
134	C₂×SD₁₆	—	2	960	no	—
135	C₂×D₈	—	2	840	no	—
136	C₂×D₈	—	2	960	no	—
137	C₂×D₈	—	2	1920	no	—
138	C₂×D₈	—	2	420	no	—
139	C₂×D₈	C₂⁵	1	240	yes	—
140	C₂×D₈	C₂⁵	1	256	yes	complemented
141	C₂×D₈	C₂⁵	1	120	yes	complemented
142	C₂²×C₈	C₂²×D₄	1	105	yes	—
143	C₂²×C₈	C₂²×D₄	1	140	yes	—
144	C₂²×C₈	C₂²×D₄	1	840	yes	—
145	C₂²×C₈	C₂⁵	1	35	yes	—
146	C₂²×C₈	C₂⁵	1	120	yes	—
147	SD₃₂	C₂⁵	1	512	yes	complemented
148	D₁₆	C₂⁵	1	512	yes	complemented
149	M₃₂	C₂⁵	1	256	yes	complemented
150	C₂×C₁₆	C₂⁵	1	240	yes	complemented
151	C₂⁴	—	2	240	no	—
152	C₂⁴	—	4	140	no	—
153	C₂⁴	—	4	960	no	—
154	C₂⁴	—	4	280	no	—
155	C₂⁴	—	8	4480	no	—
156	C₂⁴	—	8	240	no	—
157	C₂⁴	—	8	480	no	—
158	C₂⁴	#509	1	16	yes	direct product factor
159	C₂⁴	C₂²×D₈	1	140	yes	—
160	C₂⁴	C₂²×D₈	1	15	yes	—
161	#41	—	2	256	no	—
162	#41	—	4	1920	no	—
163	C₂×Q₈	—	2	120	no	—
164	C₂×Q₈	—	4	420	no	—
165	C₂×D₄	—	2	240	no	—
166	C₂×D₄	—	2	256	no	—
167	C₂×D₄	—	2	120	no	—
168	C₂×D₄	—	4	960	no	—
169	C₂×D₄	—	4	960	no	—
170	C₂×D₄	—	4	840	no	—
171	C₂×D₄	—	4	1920	no	—
172	C₂×D₄	—	4	420	no	—
173	C₂²×C₄	—	4	140	no	—
174	C₂²×C₄	—	4	960	no	—
175	C₂²×C₄	C₂²×D₈	1	140	yes	—
176	C₂²×C₄	C₂²×D₈	1	105	yes	—
177	C₂²×C₄	C₂²×D₈	1	840	yes	—
178	C₂²×C₄	C₂³×D₄	1	35	yes	—
179	C₂²×C₄	C₂³×D₄	1	120	yes	—
180	Q₁₆	—	2	120	no	—
181	Q₁₆	C₂⁶	1	16	yes	complemented
182	SD₁₆	—	2	256	no	—
183	D₈	—	2	240	no	—
184	D₈	—	2	256	no	—
185	D₈	—	2	120	no	—
186	D₈	C₂⁶	1	32	yes	—
187	D₈	C₂⁶	1	16	yes	complemented
188	C₂×C₈	C₂³×D₄	1	105	yes	—
189	C₂×C₈	C₂³×D₄	1	120	yes	—
190	C₂×C₈	C₂³×D₄	1	240	yes	—
191	C₂×C₈	C₂⁶	1	15	yes	—
192	C₂×C₈	C₂⁶	1	16	yes	—
193	C₁₆	C₂⁶	1	32	yes	complemented
194	C₂³	—	2	560	no	—
195	C₂³	—	4	120	no	—
196	C₂³	—	4	256	no	—
197	C₂³	—	4	240	no	—
198	C₂³	—	8	3840	no	—
199	C₂³	—	8	560	no	—
200	C₂³	—	8	1120	no	—
201	C₂³	C₂×#509	1	120	yes	direct product factor
202	C₂³	C₂³×D₈	1	120	yes	—
203	C₂³	C₂³×D₈	1	35	yes	—
204	Q₈	—	2	16	no	—
205	Q₈	—	4	120	no	—
206	D₄	—	2	32	no	—
207	D₄	—	2	16	no	—
208	D₄	—	4	256	no	—
209	D₄	—	4	256	no	—
210	D₄	—	4	240	no	—
211	D₄	—	4	120	no	—
212	C₂×C₄	—	4	120	no	—
213	C₂×C₄	—	4	256	no	—
214	C₂×C₄	C₂³×D₈	1	120	yes	—
215	C₂×C₄	C₂³×D₈	1	105	yes	—
216	C₂×C₄	C₂³×D₈	1	240	yes	—
217	C₂×C₄	C₂⁴×D₄	1	15	yes	—
218	C₂×C₄	C₂⁴×D₄	1	16	yes	—
219	C₈	C₂⁴×D₄	1	15	yes	—
220	C₈	C₂⁴×D₄	1	16	yes	—
221	C₈	C₂⁷	1	1	yes	characteristic
222	C₂²	—	2	240	no	—
223	C₂²	—	4	16	no	—
224	C₂²	—	4	32	no	—
225	C₂²	—	8	512	no	—
226	C₂²	—	8	240	no	—
227	C₂²	—	8	480	no	—
228	C₂²	C₂²×#509	1	140	yes	direct product factor
229	C₂²	C₂⁴×D₈	1	16	yes	—
230	C₂²	C₂⁴×D₈	1	15	yes	—
231	C₄	—	4	16	no	—
232	C₄	C₂⁴×D₈	1	16	yes	—
233	C₄	C₂⁴×D₈	1	15	yes	—
234	C₄	C₂⁵×D₄	1	1	yes	characteristic
235	C₂	—	2	16	no	—
236	C₂	—	8	16	no	—
237	C₂	—	8	32	no	—
238	C₂	C₂³×#509	1	30	yes	direct product factor
239	C₂	C₂⁵×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵	C₂×D₈
commutator, Frattini, cosocle	C₈	C₂⁷
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄ ◁ C₂⁵×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂⁴×#135	C₂	semidirect
2. C₂⁵×D₈	C₂	semidirect
3. C₂⁴×SD₃₂	C₂	semidirect
4. C₂³×#509	C₂	direct (N × H)
5. C₂⁴×D₁₆	C₂	semidirect
6. C₂⁴×M₃₂	C₂	semidirect
10. C₂³×#135	C₂²	semidirect
11. C₂⁴×Q₁₆	C₂²	semidirect
16. C₂⁴×D₈	C₂²	semidirect
17. C₂⁴×D₈	C₂²	semidirect
19. C₂²×#509	C₂²	direct (N × H)
20. C₂³×SD₃₂	C₂²	semidirect
21. C₂³×D₁₆	C₂²	semidirect
22. C₂³×M₃₂	C₂²	semidirect
23. C₂⁴×C₁₆	C₂²	semidirect
37. C₂²×#135	C₂³	semidirect
39. C₂³×Q₁₆	C₂³	semidirect
46. C₂³×D₈	C₂³	semidirect
47. C₂³×D₈	C₂³	semidirect
52. C₂×#509	C₂³	direct (N × H)
53. C₂²×SD₃₂	C₂³	semidirect
54. C₂²×D₁₆	C₂³	semidirect
55. C₂²×M₃₂	C₂³	semidirect
56. C₂³×C₁₆	C₂³	semidirect
81. C₂×#135	C₂⁴	semidirect
83. C₂²×Q₁₆	C₂⁴	semidirect
90. C₂²×D₈	C₂⁴	semidirect
91. C₂²×D₈	C₂⁴	semidirect
97. #509	C₂⁴	direct (N × H)
98. C₂×SD₃₂	C₂⁴	semidirect
99. C₂×D₁₆	C₂⁴	semidirect
100. C₂×M₃₂	C₂⁴	semidirect
101. C₂²×C₁₆	C₂⁴	semidirect
131. #135	C₂⁵	semidirect
133. C₂×Q₁₆	C₂⁵	semidirect
140. C₂×D₈	C₂⁵	semidirect
141. C₂×D₈	C₂⁵	semidirect
147. SD₃₂	C₂⁵	semidirect
148. D₁₆	C₂⁵	semidirect
149. M₃₂	C₂⁵	semidirect
150. C₂×C₁₆	C₂⁵	semidirect
158. C₂⁴	#509	direct (N × H)
181. Q₁₆	C₂⁶	semidirect
187. D₈	C₂⁶	semidirect
193. C₁₆	C₂⁶	semidirect
201. C₂³	C₂×#509	direct (N × H)
228. C₂²	C₂²×#509	direct (N × H)
238. C₂	C₂³×#509	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×#135	2
C₂⁵×D₈	2
C₂⁴×SD₃₂	2
C₂³×#509	2
C₂⁴×D₁₆	2
C₂⁴×M₃₂	2

Subgroups — C24×#509 #72286510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#509 #72286510