Subgroups — C₂×#10600113 #12254776

Order 1280.

558 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	C₂×#65163	C₂	1	2	yes	maximal
2	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×#64724	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆×Dic₅	C₂	1	1	yes	maximal, characteristic
5	C₂×#64709	C₂	1	2	yes	maximal
6	#10600113	C₂	1	8	yes	maximal, direct product factor
7	C₂×#4034	—	2	2	no	—
8	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₅×#107	C₂²	1	2	yes	—
10	C₂×#3950	C₂²	1	2	yes	—
11	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
12	C₂²×#672	C₂²	1	1	yes	characteristic
13	C₂×#3939	—	2	2	no	—
14	#65163	—	2	4	no	—
15	#65163	C₂²	1	8	yes	—
16	C₅×#366	C₂²	1	4	yes	—
17	#64724	C₂²	1	4	yes	—
18	C₁₆×Dic₅	C₂²	1	4	yes	—
19	#64709	C₂²	1	8	yes	—
20	C₂×#2252	—	5	1	no	maximal, Sylow
21	C₂²×Dic₁₀	—	4	2	no	—
22	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₄⋊Dic₅	—	2	2	no	—
25	C₂×#683	—	4	2	no	—
26	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
27	C₂²×#168	D₄	1	1	yes	characteristic
28	#4034	—	2	8	no	—
29	#4034	—	4	4	no	—
30	C₂×C₅×C₁₆	Q₈	1	4	yes	complemented
31	C₂×C₅×C₁₆	C₂³	1	2	yes	—
32	C₅×#107	—	2	2	no	—
33	C₅×#107	C₂³	1	4	yes	—
34	#3950	—	2	2	no	—
35	#3950	C₂³	1	4	yes	—
36	C₈×Dic₅	D₄	1	2	yes	—
37	C₈×Dic₅	C₂³	1	2	yes	—
38	C₂×#672	Q₈	1	4	yes	complemented
39	C₂×#672	C₂³	1	2	yes	—
40	#3939	—	2	8	no	—
41	C₂×#500	—	5	2	no	—
42	#2252	—	5	8	no	—
43	C₂×#366	—	5	1	no	—
44	C₂×#366	—	5	2	no	—
45	C₂×#366	D₅	1	1	yes	characteristic
46	C₂×C₄×C₁₆	—	5	1	no	—
47	C₂²×C₄×C₅	—	4	2	no	—
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×Dic₅	—	4	2	no	—
50	C₂²×Dic₅	D₈	1	1	yes	characteristic
51	C₂²×Dic₅	D₈	1	1	yes	characteristic
52	C₂×Dic₁₀	—	4	8	no	—
53	C₂×Dic₁₀	—	8	8	no	—
54	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
56	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
57	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
58	C₅×C₄⋊C₄	—	2	4	no	—
59	C₅×C₄⋊C₄	—	4	2	no	—
60	C₄⋊Dic₅	—	2	4	no	—
61	C₄⋊Dic₅	—	4	2	no	—
62	#683	—	4	8	no	—
63	C₄×Dic₅	D₈	1	2	yes	—
64	C₄×Dic₅	C₂×D₄	1	2	yes	—
65	C₂×#168	Q₁₆	1	4	yes	—
66	C₂×#168	C₂×D₄	1	2	yes	—
67	C₅×C₁₆	C₂×Q₈	1	4	yes	complemented
68	#672	C₂×Q₈	1	4	yes	complemented
69	C₂×C₄⋊Q₈	—	10	2	no	—
70	C₂²×C₁₆	—	5	1	no	—
71	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
72	#500	—	5	8	no	—
73	#500	—	10	4	no	—
74	C₂×#107	—	5	2	no	—
75	C₂×#107	—	10	2	no	—
76	C₂×#107	C₂×D₅	1	2	yes	—
77	C₂×C₄×C₈	—	5	1	no	—
78	#366	—	5	4	no	—
79	#366	—	5	8	no	—
80	#366	C₂×D₅	1	4	yes	—
81	C₄×C₁₆	—	5	4	no	—
82	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₅	—	4	4	no	—
84	C₂×C₄×C₅	—	8	4	no	—
85	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
89	C₂×Dic₅	—	4	4	no	—
90	C₂×Dic₅	—	8	4	no	—
91	C₂×Dic₅	D₁₆	1	4	yes	—
92	C₂×Dic₅	Q₃₂	1	4	yes	—
93	C₂×Dic₅	C₂×D₈	1	2	yes	—
94	C₂×Dic₅	C₂×D₈	1	2	yes	—
95	Dic₁₀	—	8	16	no	—
96	C₅×C₈	C₄⋊Q₈	1	2	yes	—
97	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
98	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
99	#168	C₂×Q₁₆	1	4	yes	—
100	C₂²×Q₈	—	20	2	no	—
101	C₂²×C₈	—	5	1	no	—
102	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
103	C₄⋊Q₈	—	10	8	no	—
104	C₄⋊Q₈	—	20	4	no	—
105	C₂×C₄⋊C₄	—	2	2	no	—
106	C₂×C₄⋊C₄	—	10	2	no	—
107	C₂×C₄⋊C₄	—	20	2	no	—
108	C₂×C₄²	—	5	1	no	—
109	C₂×C₁₆	—	5	4	no	—
110	C₂×C₁₆	—	5	2	no	—
111	C₂×C₁₆	Dic₁₀	1	4	yes	complemented
112	C₂×C₁₆	C₂²×D₅	1	2	yes	—
113	#107	—	2	2	no	—
114	#107	—	5	4	no	—
115	#107	—	10	8	no	—
116	#107	—	10	2	no	—
117	#107	C₂²×D₅	1	4	yes	—
118	C₄×C₈	—	5	2	no	—
119	C₄×C₈	—	5	2	no	—
120	C₂²×C₅	#500	1	2	yes	—
121	C₂²×C₅	C₂×D₁₆	1	2	yes	—
122	C₂²×C₅	C₂×Q₃₂	1	2	yes	—
123	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
124	C₄×C₅	—	8	4	no	—
125	C₄×C₅	#500	1	2	yes	—
126	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
127	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
128	Dic₅	—	8	4	no	—
129	Dic₅	C₂×D₁₆	1	4	yes	—
130	Dic₅	C₂×Q₃₂	1	4	yes	—
131	C₂×Q₈	—	20	8	no	—
132	C₂×Q₈	—	40	8	no	—
133	C₂²×C₄	—	4	2	no	—
134	C₂²×C₄	—	5	1	no	—
135	C₂²×C₄	—	5	1	no	—
136	C₂²×C₄	—	20	2	no	—
137	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
138	C₂×C₈	—	5	2	no	—
139	C₂×C₈	—	5	4	no	—
140	C₂×C₈	C₂×Dic₁₀	1	2	yes	—
141	C₂×C₈	D₄×D₅	1	1	yes	characteristic
142	C₂×C₈	Q₈×D₅	1	2	yes	—
143	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
144	C₄⋊C₄	—	2	4	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	10	4	no	—
147	C₄⋊C₄	—	20	8	no	—
148	C₄⋊C₄	—	20	2	no	—
149	C₄²	—	5	2	no	—
150	C₄²	—	5	2	no	—
151	C₁₆	—	5	4	no	—
152	C₁₆	C₂×Dic₁₀	1	4	yes	complemented
153	C₂×C₅	#2252	1	4	yes	complemented
154	C₂×C₅	C₂×#500	1	1	yes	characteristic
155	C₂×C₅	C₂²×D₁₆	1	1	yes	characteristic
156	C₂×C₅	C₂²×Q₃₂	1	1	yes	characteristic
157	C₂³	D₅×D₈	1	1	yes	characteristic
158	Q₈	—	40	16	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	5	2	no	—
161	C₂×C₄	—	5	2	no	—
162	C₂×C₄	—	5	4	no	—
163	C₂×C₄	—	5	4	no	—
164	C₂×C₄	—	8	4	no	—
165	C₂×C₄	—	20	4	no	—
166	C₂×C₄	—	40	4	no	—
167	C₂×C₄	#4034	1	2	yes	—
168	C₂×C₄	D₅×D₈	1	1	yes	characteristic
169	C₂×C₄	D₅×Q₁₆	1	2	yes	—
170	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
171	C₈	—	5	4	no	—
172	C₈	#4034	1	2	yes	—
173	C₈	C₂²×Dic₁₀	1	1	yes	characteristic
174	C₈	C₂×Q₈×D₅	1	1	yes	characteristic
175	C₅	C₂×#2252	1	1	yes	characteristic, complemented, Sylow
176	C₂²	#65163	1	2	yes	—
177	C₂²	D₅×D₁₆	1	2	yes	—
178	C₂²	D₅×Q₃₂	1	2	yes	—
179	C₂²	C₂×D₅×D₈	1	1	yes	characteristic
180	C₄	—	5	4	no	—
181	C₄	—	5	4	no	—
182	C₄	—	8	4	no	—
183	C₄	—	40	4	no	—
184	C₄	#65163	1	2	yes	—
185	C₄	C₂×#4034	1	1	yes	characteristic
186	C₄	C₂×D₅×Q₁₆	1	1	yes	characteristic
187	C₂	#10600113	1	4	yes	direct product factor
188	C₂	C₂×#65163	1	1	yes	characteristic
189	C₂	C₂×D₅×D₁₆	1	1	yes	characteristic
190	C₂	C₂×D₅×Q₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₅×D₈
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#366	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
hypercenter	C₂×#366	D₅
hypocenter	C₅	C₂×#2252
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
6. #10600113	C₂	direct (N × H)
30. C₂×C₅×C₁₆	Q₈	semidirect
38. C₂×#672	Q₈	semidirect
67. C₅×C₁₆	C₂×Q₈	semidirect
68. #672	C₂×Q₈	semidirect
111. C₂×C₁₆	Dic₁₀	semidirect
152. C₁₆	C₂×Dic₁₀	semidirect
153. C₂×C₅	#2252	semidirect
175. C₅	C₂×#2252	semidirect
187. C₂	#10600113	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#65163	2
C₂×C₅×#366	2
C₂×#64724	2
C₂×C₁₆×Dic₅	2
C₂×#64709	2
#10600113	2
C₂×#2252	5

Subgroups — C2×#10600113 #12254776

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600113 #12254776