Subgroups — C₂²×#65500 #12257564

Order 1280.

8062 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₂²⋊Dic₅	C₂	1	2	yes	maximal
3	C₂×#65500	C₂	1	24	yes	maximal, direct product factor
4	C₂²×#3963	C₂	1	4	yes	maximal, complemented
5	C₂⁶×C₅	C₂²	1	1	yes	characteristic
6	C₂⁴×C₄×C₅	—	2	16	no	—
7	C₂⁴×C₄×C₅	C₄	1	16	yes	complemented
8	C₂⁴×C₄×C₅	C₂²	1	2	yes	—
9	C₂⁴×C₄×C₅	C₂²	1	12	yes	—
10	C₂⁴×Dic₅	C₂²	1	4	yes	—
11	C₂²×C₂²⋊Dic₅	—	2	16	no	—
12	C₂²×C₂²⋊Dic₅	C₂²	1	24	yes	—
13	#65500	C₂²	1	64	yes	direct product factor
14	C₂×#3963	C₂²	1	48	yes	complemented
15	C₂²×#381	—	5	1	no	maximal, Sylow
16	C₂⁵×C₅	—	2	16	no	—
17	C₂⁵×C₅	C₂×C₄	1	8	yes	—
18	C₂⁵×C₅	D₄	1	16	yes	—
19	C₂⁵×C₅	C₂³	1	6	yes	—
20	C₂⁵×C₅	C₂³	1	1	yes	characteristic
21	C₂³×C₄×C₅	—	2	128	no	—
22	C₂³×C₄×C₅	—	2	16	no	—
23	C₂³×C₄×C₅	—	2	96	no	—
24	C₂³×C₄×C₅	C₂×C₄	1	16	yes	—
25	C₂³×C₄×C₅	C₂×C₄	1	96	yes	complemented
26	C₂³×C₄×C₅	C₂³	1	12	yes	—
27	C₂³×C₄×C₅	C₂³	1	16	yes	—
28	C₂³×Dic₅	—	2	16	no	—
29	C₂³×Dic₅	—	2	32	no	—
30	C₂³×Dic₅	C₂³	1	24	yes	—
31	C₂×C₂²⋊Dic₅	—	2	96	no	—
32	C₂×C₂²⋊Dic₅	C₂³	1	32	yes	—
33	#3963	C₂³	1	64	yes	complemented
34	C₂⁵×C₄	D₅	1	1	yes	characteristic
35	C₂³×C₂²⋊C₄	—	5	2	no	—
36	C₂×#381	—	5	24	no	—
37	C₂²×#95	—	5	4	no	—
38	C₂⁴×C₅	—	2	64	no	—
39	C₂⁴×C₅	—	2	128	no	—
40	C₂⁴×C₅	—	2	48	no	—
41	C₂⁴×C₅	C₂²⋊C₄	1	64	yes	—
42	C₂⁴×C₅	C₂²×C₄	1	24	yes	—
43	C₂⁴×C₅	C₂²×C₄	1	4	yes	—
44	C₂⁴×C₅	C₂×D₄	1	48	yes	—
45	C₂⁴×C₅	C₂×D₄	1	8	yes	—
46	C₂⁴×C₅	#41	1	16	yes	—
47	C₂⁴×C₅	C₂⁴	1	4	yes	—
48	C₂⁴×C₅	C₂⁴	1	3	yes	—
49	C₂²×C₄×C₅	—	2	64	no	—
50	C₂²×C₄×C₅	—	2	384	no	—
51	C₂²×C₄×C₅	—	2	48	no	—
52	C₂²×C₄×C₅	—	2	64	no	—
53	C₂²×C₄×C₅	C₂²×C₄	1	48	yes	—
54	C₂²×C₄×C₅	C₂²×C₄	1	64	yes	complemented
55	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
56	C₂²×Dic₅	—	2	48	no	—
57	C₂²×Dic₅	—	2	96	no	—
58	C₂²×Dic₅	—	4	64	no	—
59	C₂²×Dic₅	C₂⁴	1	16	yes	—
60	C₂²⋊Dic₅	—	2	64	no	—
61	C₂⁶	C₂×D₅	1	1	yes	characteristic
62	C₂⁴×C₄	—	2	16	no	—
63	C₂⁴×C₄	—	5	4	no	—
64	C₂⁴×C₄	Dic₅	1	16	yes	complemented
65	C₂⁴×C₄	C₂×D₅	1	2	yes	—
66	C₂⁴×C₄	C₂×D₅	1	12	yes	—
67	C₂²×C₂²⋊C₄	—	5	24	no	—
68	C₂²×C₂²⋊C₄	—	10	16	no	—
69	#381	—	5	64	no	—
70	C₂×#95	—	5	48	no	—
71	C₂³×C₅	—	2	96	no	—
72	C₂³×C₅	—	2	192	no	—
73	C₂³×C₅	—	2	16	no	—
74	C₂³×C₅	—	2	256	no	—
75	C₂³×C₅	C₂×C₂²⋊C₄	1	96	yes	—
76	C₂³×C₅	C₂×C₂²⋊C₄	1	16	yes	—
77	C₂³×C₅	#117	1	32	yes	—
78	C₂³×C₅	#123	1	64	yes	—
79	C₂³×C₅	C₂³×C₄	1	8	yes	—
80	C₂³×C₅	C₂³×C₄	1	6	yes	—
81	C₂³×C₅	C₂²×D₄	1	16	yes	—
82	C₂³×C₅	C₂²×D₄	1	12	yes	—
83	C₂³×C₅	C₂×#41	1	24	yes	—
84	C₂³×C₅	C₂⁵	1	1	yes	characteristic
85	C₂×C₄×C₅	—	2	96	no	—
86	C₂×C₄×C₅	—	2	128	no	—
87	C₂×C₄×C₅	—	2	16	no	—
88	C₂×C₄×C₅	C₂³×C₄	1	16	yes	—
89	C₂×Dic₅	—	2	16	no	—
90	C₂×Dic₅	—	2	32	no	—
91	C₂×Dic₅	—	4	96	no	—
92	C₂⁵	—	2	16	no	—
93	C₂⁵	C₂×Dic₅	1	8	yes	—
94	C₂⁵	C₂²⋊D₅	1	16	yes	—
95	C₂⁵	C₂²×D₅	1	6	yes	—
96	C₂⁵	C₂²×D₅	1	1	yes	characteristic
97	C₂³×C₄	—	2	128	no	—
98	C₂³×C₄	—	2	16	no	—
99	C₂³×C₄	—	2	96	no	—
100	C₂³×C₄	—	5	24	no	—
101	C₂³×C₄	—	10	32	no	—
102	C₂³×C₄	—	10	16	no	—
103	C₂³×C₄	C₂×Dic₅	1	16	yes	—
104	C₂³×C₄	C₂×Dic₅	1	96	yes	complemented
105	C₂³×C₄	C₂²×D₅	1	12	yes	—
106	C₂³×C₄	C₂²×D₅	1	16	yes	—
107	C₂×C₂²⋊C₄	—	5	32	no	—
108	C₂×C₂²⋊C₄	—	10	96	no	—
109	#95	—	5	64	no	—
110	C₂²×C₅	—	2	16	no	—
111	C₂²×C₅	—	2	32	no	—
112	C₂²×C₅	—	2	192	no	—
113	C₂²×C₅	#381	1	64	yes	complemented
114	C₂²×C₅	C₂²×C₂²⋊C₄	1	16	yes	—
115	C₂²×C₅	C₂²×C₂²⋊C₄	1	12	yes	—
116	C₂²×C₅	C₂×#117	1	24	yes	—
117	C₂²×C₅	C₂×#123	1	48	yes	—
118	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
119	C₂²×C₅	C₂³×D₄	1	2	yes	—
120	C₂²×C₅	C₂²×#41	1	4	yes	—
121	C₄×C₅	—	2	16	no	—
122	Dic₅	—	4	16	no	—
123	C₂⁴	—	2	48	no	—
124	C₂⁴	—	2	64	no	—
125	C₂⁴	—	2	128	no	—
126	C₂⁴	C₂²⋊Dic₅	1	64	yes	—
127	C₂⁴	#709	1	16	yes	—
128	C₂⁴	C₂²×Dic₅	1	24	yes	—
129	C₂⁴	C₂²×Dic₅	1	4	yes	—
130	C₂⁴	C₂×C₂²⋊D₅	1	48	yes	—
131	C₂⁴	C₂×C₂²⋊D₅	1	8	yes	—
132	C₂⁴	C₂³×D₅	1	4	yes	—
133	C₂⁴	C₂³×D₅	1	3	yes	—
134	C₂²×C₄	—	2	64	no	—
135	C₂²×C₄	—	2	384	no	—
136	C₂²×C₄	—	2	48	no	—
137	C₂²×C₄	—	2	64	no	—
138	C₂²×C₄	—	5	16	no	—
139	C₂²×C₄	—	10	96	no	—
140	C₂²×C₄	—	10	48	no	—
141	C₂²×C₄	—	20	64	no	—
142	C₂²×C₄	C₂²×Dic₅	1	48	yes	—
143	C₂²×C₄	C₂²×Dic₅	1	64	yes	complemented
144	C₂²×C₄	C₂³×D₅	1	8	yes	—
145	C₂²⋊C₄	—	10	64	no	—
146	C₂×C₅	—	2	16	no	—
147	C₂×C₅	C₂×#381	1	24	yes	complemented
148	C₂×C₅	C₂³×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₅	C₂²×#117	1	2	yes	—
150	C₂×C₅	C₂²×#123	1	4	yes	—
151	C₂³	—	2	96	no	—
152	C₂³	—	2	192	no	—
153	C₂³	—	2	256	no	—
154	C₂³	—	2	16	no	—
155	C₂³	#4072	1	32	yes	—
156	C₂³	#4075	1	64	yes	—
157	C₂³	C₂×C₂²⋊Dic₅	1	96	yes	—
158	C₂³	C₂×C₂²⋊Dic₅	1	16	yes	—
159	C₂³	C₂×#709	1	24	yes	—
160	C₂³	C₂³×Dic₅	1	8	yes	—
161	C₂³	C₂³×Dic₅	1	6	yes	—
162	C₂³	C₂²×C₂²⋊D₅	1	16	yes	—
163	C₂³	C₂²×C₂²⋊D₅	1	12	yes	—
164	C₂³	C₂⁴×D₅	1	1	yes	characteristic
165	C₂×C₄	—	2	96	no	—
166	C₂×C₄	—	2	128	no	—
167	C₂×C₄	—	2	16	no	—
168	C₂×C₄	—	10	32	no	—
169	C₂×C₄	—	10	16	no	—
170	C₂×C₄	—	20	96	no	—
171	C₂×C₄	C₂³×Dic₅	1	16	yes	—
172	C₅	C₂²×#381	1	1	yes	characteristic, complemented, Sylow
173	C₂²	—	2	192	no	—
174	C₂²	—	2	16	no	—
175	C₂²	—	2	32	no	—
176	C₂²	#65500	1	64	yes	direct product factor
177	C₂²	C₂×#4072	1	24	yes	—
178	C₂²	C₂×#4075	1	48	yes	—
179	C₂²	C₂²×C₂²⋊Dic₅	1	16	yes	—
180	C₂²	C₂²×C₂²⋊Dic₅	1	12	yes	—
181	C₂²	C₂²×#709	1	4	yes	—
182	C₂²	C₂⁴×Dic₅	1	1	yes	characteristic
183	C₂²	C₂³×C₂²⋊D₅	1	2	yes	—
184	C₄	—	2	16	no	—
185	C₄	—	20	16	no	—
186	C₂	—	2	16	no	—
187	C₂	C₂×#65500	1	24	yes	direct product factor
188	C₂	C₂²×#4072	1	2	yes	—
189	C₂	C₂²×#4075	1	4	yes	—
190	C₂	C₂³×C₂²⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅
commutator	C₂²×C₅	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅
Fitting	C₂⁵×C₄×C₅	C₂
socle	C₂⁵×C₅	C₂³
prime radical	C₂⁶×C₅	C₂²
cosocle	C₂³×C₅	C₂⁵
hypercenter	C₂⁵×C₄	D₅
hypocenter	C₅	C₂²×#381
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁵ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×#65500	C₂	direct (N × H)
4. C₂²×#3963	C₂	semidirect
7. C₂⁴×C₄×C₅	C₄	semidirect
13. #65500	C₂²	direct (N × H)
14. C₂×#3963	C₂²	semidirect
25. C₂³×C₄×C₅	C₂×C₄	semidirect
33. #3963	C₂³	semidirect
54. C₂²×C₄×C₅	C₂²×C₄	semidirect
64. C₂⁴×C₄	Dic₅	semidirect
104. C₂³×C₄	C₂×Dic₅	semidirect
113. C₂²×C₅	#381	semidirect
143. C₂²×C₄	C₂²×Dic₅	semidirect
147. C₂×C₅	C₂×#381	semidirect
172. C₅	C₂²×#381	semidirect
176. C₂²	#65500	direct (N × H)
187. C₂	C₂×#65500	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₅	2
C₂³×C₂²⋊Dic₅	2
C₂×#65500	2
C₂²×#3963	2
C₂²×#381	5

Subgroups — C22×#65500 #12257564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65500 #12257564