Subgroups — C₂⁴×#2380 #72325851

Order 2048.

2255208 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	C₂⁴×#383	C₂	1	2	yes	maximal, complemented
2	C₂⁵×#127	C₂	1	1	yes	maximal, characteristic
3	C₂⁴×#386	C₂	1	12	yes	maximal, complemented
4	C₂³×#2380	C₂	1	240	yes	maximal, direct product factor
5	C₂⁶×D₄	C₂²	1	6	yes	—
6	C₂⁷×C₄	C₂²	1	4	yes	—
7	C₂⁴×#127	C₄	1	128	yes	complemented
8	C₂⁴×#127	C₂²	1	120	yes	—
9	C₂⁵×C₂²⋊C₄	C₂²	1	12	yes	—
10	C₂⁵×C₄²	C₂²	1	1	yes	characteristic
11	C₂²×#2380	C₂²	1	8960	yes	direct product factor
12	C₂³×#386	C₂²	1	1440	yes	complemented
13	C₂³×#383	C₂²	1	240	yes	complemented
14	C₂⁴×#95	C₂²	1	12	yes	—
15	C₂⁸	C₂³	1	4	yes	—
16	C₂⁵×D₄	—	4	96	no	—
17	C₂⁵×D₄	—	4	96	no	—
18	C₂⁵×D₄	C₂×C₄	1	384	yes	complemented
19	C₂⁵×D₄	C₂³	1	360	yes	—
20	C₂⁶×C₄	—	4	192	no	—
21	C₂⁶×C₄	C₂³	1	8	yes	—
22	C₂⁶×C₄	C₂³	1	3	yes	—
23	C₂⁶×C₄	C₂³	1	240	yes	—
24	C₂³×#127	C₂×C₄	1	7680	yes	complemented
25	C₂³×#127	C₂³	1	2240	yes	—
26	C₂⁴×C₂²⋊C₄	—	4	192	no	—
27	C₂⁴×C₂²⋊C₄	C₂³	1	720	yes	—
28	C₂⁴×C₄²	C₂×C₄	1	64	yes	complemented
29	C₂⁴×C₄²	C₂³	1	60	yes	—
30	C₂×#2380	C₂³	1	61440	yes	direct product factor
31	C₂²×#386	C₂³	1	26880	yes	complemented
32	C₂²×#383	C₂³	1	4480	yes	complemented
33	C₂³×#95	C₂³	1	720	yes	—
34	C₂⁷	—	2	192	no	—
35	C₂⁷	—	2	192	no	—
36	C₂⁷	C₂²×C₄	1	128	yes	—
37	C₂⁷	C₂⁴	1	120	yes	—
38	C₂⁷	C₂⁴	1	1	yes	characteristic
39	C₂⁴×D₄	—	4	6144	no	—
40	C₂⁴×D₄	—	4	2880	no	—
41	C₂⁴×D₄	—	4	2880	no	—
42	C₂⁴×D₄	C₂²×C₄	1	11520	yes	complemented
43	C₂⁴×D₄	C₂⁴	1	3360	yes	—
44	C₂⁵×C₄	—	2	96	no	—
45	C₂⁵×C₄	—	2	384	no	—
46	C₂⁵×C₄	—	4	5760	no	—
47	C₂⁵×C₄	—	4	4096	no	—
48	C₂⁵×C₄	C₂²×C₄	1	96	yes	—
49	C₂⁵×C₄	C₂⁴	1	240	yes	—
50	C₂⁵×C₄	C₂⁴	1	90	yes	—
51	C₂⁵×C₄	C₂⁴	1	2240	yes	—
52	C₂²×#127	C₂²×C₄	1	71680	yes	complemented
53	C₂²×#127	C₂⁴	1	7680	yes	—
54	C₂³×C₂²⋊C₄	—	4	5760	no	—
55	C₂³×C₂²⋊C₄	C₂⁴	1	6720	yes	—
56	C₂³×C₄²	C₂²×C₄	1	1920	yes	complemented
57	C₂³×C₄²	C₂⁴	1	560	yes	—
58	#2380	C₂⁴	1	65536	yes	direct product factor
59	C₂×#386	C₂⁴	1	92160	yes	complemented
60	C₂×#383	C₂⁴	1	15360	yes	complemented
61	C₂²×#95	C₂⁴	1	6720	yes	—
62	C₂⁶	—	2	6144	no	—
63	C₂⁶	—	2	2880	no	—
64	C₂⁶	—	2	2880	no	—
65	C₂⁶	—	4	1024	no	—
66	C₂⁶	—	4	3072	no	—
67	C₂⁶	C₂³×C₄	1	1920	yes	—
68	C₂⁶	C₂³×C₄	1	16	yes	—
69	C₂⁶	H(5,𝔽₂)	1	96	yes	—
70	C₂⁶	C₂⁵	1	560	yes	—
71	C₂⁶	C₂⁵	1	15	yes	—
72	C₂³×D₄	—	4	92160	no	—
73	C₂³×D₄	—	4	13440	no	—
74	C₂³×D₄	—	4	13440	no	—
75	C₂³×D₄	C₂³×C₄	1	53760	yes	complemented
76	C₂³×D₄	C₂⁵	1	5760	yes	—
77	C₂⁴×C₄	—	2	1536	no	—
78	C₂⁴×C₄	—	2	1440	no	—
79	C₂⁴×C₄	—	2	5760	no	—
80	C₂⁴×C₄	—	4	26880	no	—
81	C₂⁴×C₄	—	4	61440	no	—
82	C₂⁴×C₄	—	4	2048	no	—
83	C₂⁴×C₄	C₂³×C₄	1	1440	yes	—
84	C₂⁴×C₄	C₂⁵	1	1120	yes	—
85	C₂⁴×C₄	C₂⁵	1	420	yes	—
86	C₂⁴×C₄	C₂⁵	1	3840	yes	—
87	C₂×#127	C₂³×C₄	1	122880	yes	complemented
88	C₂×#127	C₂⁵	1	4096	yes	—
89	C₂²×C₂²⋊C₄	—	4	26880	no	—
90	C₂²×C₂²⋊C₄	C₂⁵	1	11520	yes	—
91	C₂²×C₄²	C₂³×C₄	1	8960	yes	complemented
92	C₂²×C₄²	C₂⁵	1	960	yes	—
93	#386	C₂⁵	1	49152	yes	complemented
94	#383	C₂⁵	1	8192	yes	complemented
95	C₂×#95	C₂⁵	1	11520	yes	—
96	C₂⁵	—	2	6720	no	—
97	C₂⁵	—	2	6720	no	—
98	C₂⁵	—	2	46080	no	—
99	C₂⁵	—	4	32768	no	—
100	C₂⁵	—	4	7680	no	—
101	C₂⁵	—	4	23040	no	—
102	C₂⁵	#518	1	768	yes	—
103	C₂⁵	#560	1	1024	yes	—
104	C₂⁵	C₂⁴×C₄	1	4480	yes	—
105	C₂⁵	C₂⁴×C₄	1	120	yes	—
106	C₂⁵	C₂×H(5,𝔽₂)	1	720	yes	—
107	C₂⁵	C₂⁶	1	480	yes	—
108	C₂⁵	C₂⁶	1	35	yes	—
109	C₂²×D₄	—	4	215040	no	—
110	C₂²×D₄	—	4	11520	no	—
111	C₂²×D₄	—	4	11520	no	—
112	C₂²×D₄	C₂⁴×C₄	1	46080	yes	complemented
113	C₂²×D₄	C₂⁶	1	1536	yes	—
114	C₂³×C₄	—	2	3360	no	—
115	C₂³×C₄	—	2	11520	no	—
116	C₂³×C₄	—	2	13440	no	—
117	C₂³×C₄	—	4	23040	no	—
118	C₂³×C₄	—	4	143360	no	—
119	C₂³×C₄	—	4	15360	no	—
120	C₂³×C₄	C₂⁴×C₄	1	3360	yes	—
121	C₂³×C₄	C₂⁶	1	960	yes	—
122	C₂³×C₄	C₂⁶	1	360	yes	—
123	C₂³×C₄	C₂⁶	1	1024	yes	—
124	#127	C₂⁴×C₄	1	32768	yes	complemented
125	C₂×C₂²⋊C₄	—	4	23040	no	—
126	C₂×C₂²⋊C₄	C₂⁶	1	3072	yes	—
127	C₂×C₄²	C₂⁴×C₄	1	7680	yes	complemented
128	C₂×C₄²	C₂⁶	1	256	yes	—
129	#95	C₂⁶	1	3072	yes	—
130	C₂⁴	—	2	2880	no	—
131	C₂⁴	—	2	2880	no	—
132	C₂⁴	—	2	53760	no	—
133	C₂⁴	—	4	122880	no	—
134	C₂⁴	—	4	8960	no	—
135	C₂⁴	—	4	26880	no	—
136	C₂⁴	#2380	1	4096	yes	direct product factor
137	C₂⁴	C₂×#518	1	2880	yes	—
138	C₂⁴	C₂×#560	1	3840	yes	—
139	C₂⁴	C₂⁵×C₄	1	1920	yes	—
140	C₂⁴	C₂⁵×C₄	1	140	yes	—
141	C₂⁴	C₂²×H(5,𝔽₂)	1	840	yes	—
142	C₂⁴	C₂⁷	1	64	yes	—
143	C₂⁴	C₂⁷	1	15	yes	—
144	C₂×D₄	—	4	92160	no	—
145	C₂×D₄	—	4	1536	no	—
146	C₂×D₄	—	4	1536	no	—
147	C₂×D₄	C₂⁵×C₄	1	6144	yes	complemented
148	C₂²×C₄	—	2	1440	no	—
149	C₂²×C₄	—	2	13440	no	—
150	C₂²×C₄	—	2	5760	no	—
151	C₂²×C₄	—	4	3072	no	—
152	C₂²×C₄	—	4	61440	no	—
153	C₂²×C₄	—	4	17920	no	—
154	C₂²×C₄	C₂⁵×C₄	1	1440	yes	—
155	C₂²×C₄	C₂⁷	1	128	yes	—
156	C₂²×C₄	C₂⁷	1	48	yes	—
157	C₂²⋊C₄	—	4	3072	no	—
158	C₄²	C₂⁵×C₄	1	1024	yes	complemented
159	C₂³	—	2	11520	no	—
160	C₂³	—	2	192	no	—
161	C₂³	—	2	192	no	—
162	C₂³	—	4	71680	no	—
163	C₂³	—	4	1920	no	—
164	C₂³	—	4	5760	no	—
165	C₂³	C₂×#2380	1	7680	yes	direct product factor
166	C₂³	C₂²×#518	1	1680	yes	—
167	C₂³	C₂²×#560	1	2240	yes	—
168	C₂³	C₂⁶×C₄	1	128	yes	—
169	C₂³	C₂⁶×C₄	1	30	yes	—
170	C₂³	C₂³×H(5,𝔽₂)	1	180	yes	—
171	C₂³	C₂⁸	1	1	yes	characteristic
172	D₄	—	4	6144	no	—
173	C₂×C₄	—	2	2880	no	—
174	C₂×C₄	—	2	96	no	—
175	C₂×C₄	—	2	384	no	—
176	C₂×C₄	—	4	4096	no	—
177	C₂×C₄	—	4	3840	no	—
178	C₂×C₄	C₂⁶×C₄	1	96	yes	—
179	C₂²	—	2	384	no	—
180	C₂²	—	4	64	no	—
181	C₂²	—	4	192	no	—
182	C₂²	—	4	7680	no	—
183	C₂²	C₂²×#2380	1	2240	yes	direct product factor
184	C₂²	C₂³×#518	1	180	yes	—
185	C₂²	C₂³×#560	1	240	yes	—
186	C₂²	C₂⁷×C₄	1	1	yes	characteristic
187	C₂²	C₂⁴×H(5,𝔽₂)	1	6	yes	—
188	C₄	—	2	96	no	—
189	C₄	—	4	128	no	—
190	C₂	—	4	128	no	—
191	C₂	C₂³×#2380	1	120	yes	direct product factor
192	C₂	C₂⁴×#518	1	3	yes	—
193	C₂	C₂⁴×#560	1	4	yes	—

	subgroup	G/H
center, socle	C₂⁷	C₂⁴
commutator	C₂²	C₂⁷×C₄
Frattini, cosocle	C₂³	C₂⁸
prime radical	C₂⁵×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂⁴×#383	C₂	semidirect
3. C₂⁴×#386	C₂	semidirect
4. C₂³×#2380	C₂	direct (N × H)
7. C₂⁴×#127	C₄	semidirect
11. C₂²×#2380	C₂²	direct (N × H)
12. C₂³×#386	C₂²	semidirect
13. C₂³×#383	C₂²	semidirect
18. C₂⁵×D₄	C₂×C₄	semidirect
24. C₂³×#127	C₂×C₄	semidirect
28. C₂⁴×C₄²	C₂×C₄	semidirect
30. C₂×#2380	C₂³	direct (N × H)
31. C₂²×#386	C₂³	semidirect
32. C₂²×#383	C₂³	semidirect
42. C₂⁴×D₄	C₂²×C₄	semidirect
52. C₂²×#127	C₂²×C₄	semidirect
56. C₂³×C₄²	C₂²×C₄	semidirect
58. #2380	C₂⁴	direct (N × H)
59. C₂×#386	C₂⁴	semidirect
60. C₂×#383	C₂⁴	semidirect
75. C₂³×D₄	C₂³×C₄	semidirect
87. C₂×#127	C₂³×C₄	semidirect
91. C₂²×C₄²	C₂³×C₄	semidirect
93. #386	C₂⁵	semidirect
94. #383	C₂⁵	semidirect
112. C₂²×D₄	C₂⁴×C₄	semidirect
124. #127	C₂⁴×C₄	semidirect
127. C₂×C₄²	C₂⁴×C₄	semidirect
136. C₂⁴	#2380	direct (N × H)
147. C₂×D₄	C₂⁵×C₄	semidirect
158. C₄²	C₂⁵×C₄	semidirect
165. C₂³	C₂×#2380	direct (N × H)
183. C₂²	C₂²×#2380	direct (N × H)
191. C₂	C₂³×#2380	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×#383	2
C₂⁵×#127	2
C₂⁴×#386	2
C₂³×#2380	2

Subgroups — C24×#2380 #72325851

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#2380 #72325851