Subgroups — C₂²×#64888 #13043371

Order 1280.

862 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#674	C₂	1	2	yes	maximal
2	C₂²×C₈×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂×#64888	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₅×C₈	C₄	1	1	yes	characteristic
5	C₂²×C₄×Dic₅	C₄	1	1	yes	characteristic
6	C₂³×#168	C₂²	1	1	yes	characteristic
7	C₂²×#674	—	2	8	no	—
8	C₂²×#674	C₂²	1	12	yes	—
9	C₂×C₈×Dic₅	C₄	1	6	yes	—
10	C₂×C₈×Dic₅	C₂²	1	6	yes	—
11	#64888	C₂²	1	16	yes	direct product factor
12	C₂²×C₄⋊C₁₆	—	5	1	no	maximal, Sylow
13	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
14	C₂³×Dic₅	C₈	1	1	yes	characteristic
15	C₂³×Dic₅	C₈	1	1	yes	characteristic
16	C₂²×C₅×C₈	—	2	4	no	—
17	C₂²×C₅×C₈	C₂×C₄	1	6	yes	—
18	C₂×C₄×Dic₅	C₈	1	6	yes	—
19	C₂×C₄×Dic₅	C₂×C₄	1	6	yes	—
20	C₂²×#168	C₂×C₄	1	3	yes	—
21	C₂²×#168	D₄	1	4	yes	—
22	C₂²×#168	Q₈	1	4	yes	—
23	C₂²×#168	C₂³	1	3	yes	—
24	C₂×#674	—	2	24	no	—
25	C₂×#674	C₂³	1	8	yes	—
26	C₈×Dic₅	C₂×C₄	1	12	yes	—
27	C₈×Dic₅	C₂³	1	4	yes	—
28	C₂³×C₁₆	—	5	2	no	—
29	C₂²×C₄×C₈	—	5	1	no	—
30	C₂×C₄⋊C₁₆	—	5	12	no	—
31	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
33	C₂²×C₄×C₅	C₂×C₈	1	3	yes	—
34	C₂²×C₄×C₅	M₁₆	1	4	yes	—
35	C₂²×C₄×C₅	C₂²×C₄	1	3	yes	—
36	C₂²×Dic₅	—	2	4	no	—
37	C₂²×Dic₅	C₁₆	1	8	yes	complemented
38	C₂²×Dic₅	C₂×C₈	1	6	yes	—
39	C₂²×Dic₅	C₂×C₈	1	6	yes	—
40	C₂×C₅×C₈	—	2	12	no	—
41	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
42	C₄×Dic₅	C₂×C₈	1	12	yes	—
43	C₄×Dic₅	C₂²×C₄	1	4	yes	—
44	C₂×#168	C₄⋊C₄	1	12	yes	—
45	C₂×#168	C₂²×C₄	1	3	yes	—
46	C₂×#168	C₂×D₄	1	6	yes	—
47	C₂×#168	C₂×Q₈	1	6	yes	—
48	C₂×#168	C₂⁴	1	1	yes	characteristic
49	#674	—	2	8	no	—
50	C₂³×C₈	—	5	1	no	—
51	C₂³×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₂²×C₄²	—	5	1	no	—
53	C₂²×C₁₆	—	5	12	no	—
54	C₂²×C₁₆	—	10	8	no	—
55	C₂×C₄×C₈	—	5	6	no	—
56	C₂×C₄×C₈	—	5	6	no	—
57	C₄⋊C₁₆	—	5	16	no	—
58	C₂³×C₅	C₄⋊C₈	1	4	yes	—
59	C₂³×C₅	C₂×C₁₆	1	4	yes	—
60	C₂³×C₅	M₃₂	1	4	yes	—
61	C₂³×C₅	C₂²×C₈	1	3	yes	—
62	C₂×C₄×C₅	C₄⋊C₈	1	12	yes	—
63	C₂×C₄×C₅	C₂×C₄⋊C₄	1	6	yes	—
64	C₂×C₄×C₅	C₂²×C₈	1	3	yes	—
65	C₂×C₄×C₅	C₂×M₁₆	1	6	yes	—
66	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
67	C₂×Dic₅	—	2	12	no	—
68	C₂×Dic₅	C₂×C₁₆	1	24	yes	complemented
69	C₂×Dic₅	C₂²×C₈	1	4	yes	—
70	C₂×Dic₅	C₂²×C₈	1	4	yes	—
71	C₅×C₈	—	2	4	no	—
72	#168	C₂×C₄⋊C₄	1	6	yes	—
73	#168	C₂²×D₄	1	1	yes	characteristic
74	#168	C₂²×Q₈	1	1	yes	characteristic
75	C₂³×C₄	—	5	1	no	—
76	C₂³×C₄	—	5	1	no	—
77	C₂³×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₂²×C₈	—	2	4	no	—
79	C₂²×C₈	—	5	4	no	—
80	C₂²×C₈	—	5	4	no	—
81	C₂²×C₈	—	5	3	no	—
82	C₂²×C₈	—	5	3	no	—
83	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	6	yes	—
84	C₂×C₄²	—	5	6	no	—
85	C₂×C₄²	—	5	6	no	—
86	C₂×C₁₆	—	5	8	no	—
87	C₂×C₁₆	—	10	24	no	—
88	C₄×C₈	—	5	12	no	—
89	C₄×C₈	—	5	4	no	—
90	C₂²×C₅	C₄⋊C₁₆	1	16	yes	complemented
91	C₂²×C₅	C₂×C₄⋊C₈	1	6	yes	—
92	C₂²×C₅	C₂²×C₁₆	1	6	yes	—
93	C₂²×C₅	C₂×M₃₂	1	6	yes	—
94	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
95	C₄×C₅	C₂×C₄⋊C₈	1	6	yes	—
96	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
98	Dic₅	—	2	4	no	—
99	Dic₅	C₂²×C₁₆	1	8	yes	complemented
100	C₂⁴	D₅⋊C₈	1	1	yes	characteristic
101	C₂²×C₄	—	5	6	no	—
102	C₂²×C₄	—	5	8	no	—
103	C₂²×C₄	—	5	6	no	—
104	C₂²×C₄	—	10	4	no	—
105	C₂²×C₄	D₅⋊C₈	1	3	yes	—
106	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	4	yes	—
107	C₂²×C₄	#704	1	4	yes	—
108	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	3	yes	—
109	C₂×C₈	—	2	12	no	—
110	C₂×C₈	—	5	6	no	—
111	C₂×C₈	—	5	12	no	—
112	C₂×C₈	—	5	6	no	—
113	C₂×C₈	—	5	1	no	—
114	C₂×C₈	—	5	3	no	—
115	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	4	yes	—
116	C₄²	—	5	12	no	—
117	C₄²	—	5	4	no	—
118	C₁₆	—	10	8	no	—
119	C₂×C₅	C₂×C₄⋊C₁₆	1	12	yes	complemented
120	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
121	C₂×C₅	C₂³×C₁₆	1	1	yes	characteristic
122	C₂×C₅	C₂²×M₃₂	1	1	yes	characteristic
123	C₂³	D₅⋊C₁₆	1	4	yes	—
124	C₂³	#3990	1	4	yes	—
125	C₂³	#4004	1	4	yes	—
126	C₂³	C₂×D₅⋊C₈	1	3	yes	—
127	C₂×C₄	—	5	24	no	—
128	C₂×C₄	—	5	4	no	—
129	C₂×C₄	—	5	4	no	—
130	C₂×C₄	—	10	12	no	—
131	C₂×C₄	#4004	1	12	yes	—
132	C₂×C₄	C₂×D₅⋊C₈	1	3	yes	—
133	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	6	yes	—
134	C₂×C₄	C₂×#704	1	6	yes	—
135	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
136	C₈	—	2	4	no	—
137	C₈	—	5	1	no	—
138	C₈	—	5	6	no	—
139	C₈	—	5	1	no	—
140	C₅	C₂²×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
141	C₂²	#64888	1	16	yes	direct product factor
142	C₂²	C₂×D₅⋊C₁₆	1	6	yes	—
143	C₂²	C₂×#3990	1	6	yes	—
144	C₂²	C₂×#4004	1	6	yes	—
145	C₂²	C₂²×D₅⋊C₈	1	1	yes	characteristic
146	C₄	—	5	8	no	—
147	C₄	—	10	4	no	—
148	C₄	C₂×#4004	1	6	yes	—
149	C₄	C₂²×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₄	C₂²×#704	1	1	yes	characteristic
151	C₂	C₂×#64888	1	12	yes	direct product factor
152	C₂	C₂²×D₅⋊C₁₆	1	1	yes	characteristic
153	C₂	C₂²×#3990	1	1	yes	characteristic
154	C₂	C₂²×#4004	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₂³×C₅×C₈	C₄
socle, prime radical	C₂⁴×C₅	C₂×C₈
cosocle	C₂×#168	C₂⁴
hypercenter	C₂³×C₈	AGL(1,𝔽₅)
hypocenter	C₅	C₂²×C₄⋊C₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#64888	C₂	direct (N × H)
11. #64888	C₂²	direct (N × H)
37. C₂²×Dic₅	C₁₆	semidirect
68. C₂×Dic₅	C₂×C₁₆	semidirect
90. C₂²×C₅	C₄⋊C₁₆	semidirect
99. Dic₅	C₂²×C₁₆	semidirect
119. C₂×C₅	C₂×C₄⋊C₁₆	semidirect
140. C₅	C₂²×C₄⋊C₁₆	semidirect
141. C₂²	#64888	direct (N × H)
151. C₂	C₂×#64888	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×#674	2
C₂²×C₈×Dic₅	2
C₂×#64888	2
C₂²×C₄⋊C₁₆	5

Subgroups — C22×#64888 #13043371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#64888 #13043371