Subgroups — C₂²×#65225 #12257575

Order 1280.

4286 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊Dic₅	C₂	1	2	yes	maximal
3	C₂²×#3963	C₂	1	4	yes	maximal
4	C₂×#65225	C₂	1	24	yes	maximal, direct product factor
5	C₂⁴×C₄×C₅	C₂²	1	2	yes	—
6	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂⁴×Dic₅	C₂²	1	4	yes	—
8	C₂²×C₄²×C₅	C₄	1	16	yes	complemented
9	C₂²×C₄²×C₅	C₂²	1	12	yes	—
10	C₂²×C₄⋊Dic₅	—	2	16	no	—
11	C₂²×C₄⋊Dic₅	C₂²	1	24	yes	—
12	C₂×#3963	C₂²	1	48	yes	—
13	#65225	C₂²	1	64	yes	direct product factor
14	C₂²×#382	—	5	1	no	maximal, Sylow
15	C₂⁵×C₅	C₂³	1	1	yes	characteristic
16	C₂³×C₄×C₅	—	2	16	no	—
17	C₂³×C₄×C₅	C₂×C₄	1	16	yes	—
18	C₂³×C₄×C₅	C₂×C₄	1	8	yes	—
19	C₂³×C₄×C₅	D₄	1	8	yes	—
20	C₂³×C₄×C₅	Q₈	1	8	yes	—
21	C₂³×C₄×C₅	C₂³	1	12	yes	—
22	C₂³×C₄×C₅	C₂³	1	6	yes	—
23	C₂³×Dic₅	—	2	16	no	—
24	C₂³×Dic₅	—	2	32	no	—
25	C₂³×Dic₅	C₂³	1	24	yes	—
26	C₂×C₄²×C₅	C₂×C₄	1	96	yes	complemented
27	C₂×C₄²×C₅	C₂³	1	16	yes	—
28	C₂×C₄⋊Dic₅	—	2	96	no	—
29	C₂×C₄⋊Dic₅	C₂³	1	32	yes	—
30	#3963	C₂³	1	64	yes	—
31	C₂³×C₄⋊C₄	—	5	2	no	—
32	C₂³×C₄²	D₅	1	1	yes	characteristic
33	C₂×#382	—	5	24	no	—
34	C₂²×#95	—	5	4	no	—
35	C₂⁴×C₅	C₂²×C₄	1	4	yes	—
36	C₂⁴×C₅	C₂×D₄	1	4	yes	—
37	C₂⁴×C₅	C₂×Q₈	1	4	yes	—
38	C₂⁴×C₅	#41	1	16	yes	—
39	C₂⁴×C₅	C₂⁴	1	3	yes	—
40	C₂²×C₄×C₅	—	2	64	no	—
41	C₂²×C₄×C₅	—	2	48	no	—
42	C₂²×C₄×C₅	C₄⋊C₄	1	64	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	48	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	24	yes	—
45	C₂²×C₄×C₅	C₂×D₄	1	24	yes	—
46	C₂²×C₄×C₅	C₂×Q₈	1	24	yes	—
47	C₂²×C₄×C₅	C₂⁴	1	8	yes	—
48	C₂²×C₄×C₅	C₂⁴	1	4	yes	—
49	C₂²×Dic₅	—	2	48	no	—
50	C₂²×Dic₅	—	2	96	no	—
51	C₂²×Dic₅	—	4	64	no	—
52	C₂²×Dic₅	C₂⁴	1	16	yes	—
53	C₄²×C₅	C₂²×C₄	1	64	yes	complemented
54	C₄⋊Dic₅	—	2	64	no	—
55	C₂⁴×C₄	—	5	4	no	—
56	C₂⁴×C₄	C₂×D₅	1	2	yes	—
57	C₂⁴×C₄	C₂×D₅	1	1	yes	characteristic
58	C₂²×C₄⋊C₄	—	5	24	no	—
59	C₂²×C₄⋊C₄	—	10	16	no	—
60	C₂²×C₄²	Dic₅	1	16	yes	complemented
61	C₂²×C₄²	C₂×D₅	1	12	yes	—
62	#382	—	5	64	no	—
63	C₂×#95	—	5	48	no	—
64	C₂³×C₅	C₂×C₄⋊C₄	1	16	yes	—
65	C₂³×C₅	#117	1	32	yes	—
66	C₂³×C₅	#124	1	32	yes	—
67	C₂³×C₅	#125	1	32	yes	—
68	C₂³×C₅	C₂³×C₄	1	6	yes	—
69	C₂³×C₅	C₂²×D₄	1	6	yes	—
70	C₂³×C₅	C₂²×Q₈	1	6	yes	—
71	C₂³×C₅	C₂×#41	1	24	yes	—
72	C₂³×C₅	C₂⁵	1	1	yes	characteristic
73	C₂×C₄×C₅	—	2	96	no	—
74	C₂×C₄×C₅	—	2	16	no	—
75	C₂×C₄×C₅	C₂×C₄⋊C₄	1	96	yes	—
76	C₂×C₄×C₅	C₂³×C₄	1	16	yes	—
77	C₂×C₄×C₅	C₂³×C₄	1	8	yes	—
78	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
79	C₂×C₄×C₅	C₂²×Q₈	1	8	yes	—
80	C₂×Dic₅	—	2	16	no	—
81	C₂×Dic₅	—	2	32	no	—
82	C₂×Dic₅	—	4	96	no	—
83	C₂⁵	C₂²×D₅	1	1	yes	characteristic
84	C₂³×C₄	—	2	16	no	—
85	C₂³×C₄	—	5	24	no	—
86	C₂³×C₄	—	10	16	no	—
87	C₂³×C₄	—	10	32	no	—
88	C₂³×C₄	Dic₁₀	1	8	yes	—
89	C₂³×C₄	D₂₀	1	8	yes	—
90	C₂³×C₄	C₂×Dic₅	1	16	yes	—
91	C₂³×C₄	C₂×Dic₅	1	8	yes	—
92	C₂³×C₄	C₂²×D₅	1	12	yes	—
93	C₂³×C₄	C₂²×D₅	1	6	yes	—
94	C₂×C₄⋊C₄	—	5	32	no	—
95	C₂×C₄⋊C₄	—	10	96	no	—
96	C₂×C₄²	C₂×Dic₅	1	96	yes	complemented
97	C₂×C₄²	C₂²×D₅	1	16	yes	—
98	#95	—	5	64	no	—
99	C₂²×C₅	#382	1	64	yes	complemented
100	C₂²×C₅	C₂²×C₄⋊C₄	1	12	yes	—
101	C₂²×C₅	C₂×#117	1	24	yes	—
102	C₂²×C₅	C₂×#124	1	24	yes	—
103	C₂²×C₅	C₂×#125	1	24	yes	—
104	C₂²×C₅	C₂⁴×C₄	1	1	yes	characteristic
105	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
106	C₂²×C₅	C₂³×Q₈	1	1	yes	characteristic
107	C₂²×C₅	C₂²×#41	1	4	yes	—
108	C₄×C₅	—	2	16	no	—
109	C₄×C₅	C₂²×C₄⋊C₄	1	16	yes	—
110	Dic₅	—	4	16	no	—
111	C₂⁴	C₂×Dic₁₀	1	4	yes	—
112	C₂⁴	C₂×D₂₀	1	4	yes	—
113	C₂⁴	#709	1	16	yes	—
114	C₂⁴	C₂²×Dic₅	1	4	yes	—
115	C₂⁴	C₂³×D₅	1	3	yes	—
116	C₂²×C₄	—	2	48	no	—
117	C₂²×C₄	—	2	64	no	—
118	C₂²×C₄	—	5	16	no	—
119	C₂²×C₄	—	10	48	no	—
120	C₂²×C₄	—	10	96	no	—
121	C₂²×C₄	—	20	64	no	—
122	C₂²×C₄	C₄⋊Dic₅	1	64	yes	—
123	C₂²×C₄	C₂×Dic₁₀	1	24	yes	—
124	C₂²×C₄	C₂×D₂₀	1	24	yes	—
125	C₂²×C₄	C₂²×Dic₅	1	48	yes	—
126	C₂²×C₄	C₂²×Dic₅	1	24	yes	—
127	C₂²×C₄	C₂³×D₅	1	8	yes	—
128	C₂²×C₄	C₂³×D₅	1	4	yes	—
129	C₄⋊C₄	—	10	64	no	—
130	C₄²	C₂²×Dic₅	1	64	yes	complemented
131	C₂×C₅	C₂×#382	1	24	yes	complemented
132	C₂×C₅	C₂³×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₅	C₂²×#117	1	2	yes	—
134	C₂×C₅	C₂²×#124	1	2	yes	—
135	C₂×C₅	C₂²×#125	1	2	yes	—
136	C₂³	#4016	1	32	yes	—
137	C₂³	#4021	1	32	yes	—
138	C₂³	C₂×C₄⋊Dic₅	1	16	yes	—
139	C₂³	#4072	1	32	yes	—
140	C₂³	C₂²×Dic₁₀	1	6	yes	—
141	C₂³	C₂²×D₂₀	1	6	yes	—
142	C₂³	C₂×#709	1	24	yes	—
143	C₂³	C₂³×Dic₅	1	6	yes	—
144	C₂³	C₂⁴×D₅	1	1	yes	characteristic
145	C₂×C₄	—	2	96	no	—
146	C₂×C₄	—	2	16	no	—
147	C₂×C₄	—	10	16	no	—
148	C₂×C₄	—	10	32	no	—
149	C₂×C₄	—	20	96	no	—
150	C₂×C₄	C₂×C₄⋊Dic₅	1	96	yes	—
151	C₂×C₄	C₂²×Dic₁₀	1	8	yes	—
152	C₂×C₄	C₂²×D₂₀	1	8	yes	—
153	C₂×C₄	C₂³×Dic₅	1	16	yes	—
154	C₂×C₄	C₂³×Dic₅	1	8	yes	—
155	C₅	C₂²×#382	1	1	yes	characteristic, complemented, Sylow
156	C₂²	#65225	1	64	yes	direct product factor
157	C₂²	C₂×#4016	1	24	yes	—
158	C₂²	C₂×#4021	1	24	yes	—
159	C₂²	C₂²×C₄⋊Dic₅	1	12	yes	—
160	C₂²	C₂×#4072	1	24	yes	—
161	C₂²	C₂³×Dic₁₀	1	1	yes	characteristic
162	C₂²	C₂³×D₂₀	1	1	yes	characteristic
163	C₂²	C₂²×#709	1	4	yes	—
164	C₂²	C₂⁴×Dic₅	1	1	yes	characteristic
165	C₄	—	2	16	no	—
166	C₄	—	20	16	no	—
167	C₄	C₂²×C₄⋊Dic₅	1	16	yes	—
168	C₂	C₂×#65225	1	24	yes	direct product factor
169	C₂	C₂²×#4016	1	2	yes	—
170	C₂	C₂²×#4021	1	2	yes	—
171	C₂	C₂³×C₄⋊Dic₅	1	1	yes	characteristic
172	C₂	C₂²×#4072	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂²×D₅
commutator	C₂²×C₅	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₅
Fitting	C₂³×C₄²×C₅	C₂
socle, prime radical	C₂⁵×C₅	C₂³
cosocle	C₂³×C₅	C₂⁵
hypercenter	C₂³×C₄²	D₅
hypocenter	C₅	C₂²×#382
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
4. C₂×#65225	C₂	direct (N × H)
8. C₂²×C₄²×C₅	C₄	semidirect
13. #65225	C₂²	direct (N × H)
26. C₂×C₄²×C₅	C₂×C₄	semidirect
53. C₄²×C₅	C₂²×C₄	semidirect
60. C₂²×C₄²	Dic₅	semidirect
96. C₂×C₄²	C₂×Dic₅	semidirect
99. C₂²×C₅	#382	semidirect
130. C₄²	C₂²×Dic₅	semidirect
131. C₂×C₅	C₂×#382	semidirect
155. C₅	C₂²×#382	semidirect
156. C₂²	#65225	direct (N × H)
168. C₂	C₂×#65225	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×C₄²×C₅	2
C₂³×C₄⋊Dic₅	2
C₂²×#3963	2
C₂×#65225	2
C₂²×#382	5

Subgroups — C22×#65225 #12257575

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#65225 #12257575