Subgroups — C₂×#10600634 #12099530

Order 1280.

830 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#65213	C₂	1	2	yes	maximal
3	C₂×#64772	C₂	1	4	yes	maximal
4	#10600634	C₂	1	8	yes	maximal, direct product factor
5	C₂³×C₅×C₈	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂²×#681	C₂²	1	4	yes	—
8	C₂×C₅×#97	C₄	1	4	yes	—
9	C₂×C₅×#97	C₄	1	4	yes	—
10	C₂×C₅×#97	C₂²	1	4	yes	—
11	#65213	—	2	4	no	—
12	#65213	C₂²	1	8	yes	—
13	C₂×C₄⋊#168	—	2	4	no	—
14	#64772	C₂²	1	16	yes	—
15	C₂×#1838	—	5	1	no	maximal, Sylow
16	C₂³×C₄×C₅	Q₈	1	2	yes	—
17	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
18	C₂²×C₅×C₈	—	2	8	no	—
19	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₅×C₈	C₂³	1	4	yes	—
22	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅	D₄	1	2	yes	—
25	C₂×C₄²×C₅	D₄	1	2	yes	—
26	C₂×C₄²×C₅	Q₈	1	2	yes	—
27	C₂×C₄²×C₅	C₂³	1	2	yes	—
28	C₂×#681	—	2	4	no	—
29	C₂×#681	—	2	16	no	—
30	C₂×#681	C₂³	1	8	yes	—
31	C₂²×#168	—	2	4	no	—
32	C₅×#97	C₂×C₄	1	16	yes	—
33	C₄⋊#168	—	2	16	no	—
34	C₂×#423	—	5	2	no	—
35	C₂²×#97	D₅	1	1	yes	characteristic
36	#1838	—	5	8	no	—
37	C₂×#341	—	5	4	no	—
38	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅	—	4	4	no	—
40	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
41	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
47	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	#41	1	8	yes	—
49	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
50	C₂×C₅×C₈	—	2	8	no	—
51	C₂×C₅×C₈	—	2	8	no	—
52	C₂×C₅×C₈	—	2	8	no	—
53	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
54	C₄²×C₅	C₄⋊C₄	1	4	yes	—
55	C₄²×C₅	C₄⋊C₄	1	4	yes	—
56	C₄²×C₅	C₂²×C₄	1	4	yes	—
57	C₄²×C₅	C₂×D₄	1	4	yes	—
58	#681	—	2	16	no	—
59	#681	—	4	8	no	—
60	C₂×#168	—	2	8	no	—
61	C₂×#168	—	4	8	no	—
62	C₂²×M₁₆	—	5	4	no	—
63	C₂³×C₈	C₂×D₅	1	2	yes	—
64	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
65	#423	—	5	8	no	—
66	#423	—	10	4	no	—
67	C₂×C₄⋊C₈	—	10	4	no	—
68	C₂×#97	Dic₅	1	4	yes	—
69	C₂×#97	Dic₅	1	4	yes	—
70	C₂×#97	C₂×D₅	1	4	yes	—
71	#341	—	5	16	no	—
72	C₂³×C₅	—	2	4	no	—
73	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂³×C₅	#125	1	4	yes	—
76	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₅	—	2	8	no	—
78	C₂×C₄×C₅	—	4	4	no	—
79	C₂×C₄×C₅	—	4	4	no	—
80	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
81	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅	#117	1	4	yes	—
84	C₂×C₄×C₅	#117	1	4	yes	—
85	C₂×C₄×C₅	#124	1	4	yes	—
86	C₂×C₄×C₅	#124	1	4	yes	—
87	C₂×C₄×C₅	#125	1	4	yes	—
88	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×#41	1	4	yes	—
91	C₅×C₈	—	2	8	no	—
92	#168	—	4	8	no	—
93	C₂³×C₄	Dic₁₀	1	2	yes	—
94	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
95	C₂×M₁₆	—	5	8	no	—
96	C₂×M₁₆	—	10	4	no	—
97	C₂×M₁₆	—	10	16	no	—
98	C₂²×C₈	—	2	8	no	—
99	C₂²×C₈	—	10	4	no	—
100	C₂²×C₈	C₂×Dic₅	1	4	yes	—
101	C₂²×C₈	C₂×Dic₅	1	4	yes	—
102	C₂²×C₈	C₂²×D₅	1	4	yes	—
103	C₂×C₄²	Dic₁₀	1	2	yes	—
104	C₂×C₄²	D₂₀	1	2	yes	—
105	C₂×C₄²	D₂₀	1	2	yes	—
106	C₂×C₄²	C₂×Dic₅	1	2	yes	—
107	C₂×C₄²	C₂×Dic₅	1	2	yes	—
108	C₂×C₄²	C₂²×D₅	1	2	yes	—
109	C₄⋊C₈	—	10	16	no	—
110	#97	C₂×Dic₅	1	16	yes	—
111	C₂²×C₅	—	2	8	no	—
112	C₂²×C₅	—	2	2	no	—
113	C₂²×C₅	—	2	2	no	—
114	C₂²×C₅	#382	1	2	yes	—
115	C₂²×C₅	#382	1	2	yes	—
116	C₂²×C₅	#430	1	4	yes	—
117	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₅	C₂×#125	1	2	yes	—
119	C₄×C₅	—	2	8	no	—
120	C₄×C₅	#382	1	2	yes	—
121	C₄×C₅	#382	1	2	yes	—
122	C₄×C₅	C₂×#117	1	2	yes	—
123	C₄×C₅	C₂×#124	1	2	yes	—
124	C₂⁴	C₂×Dic₁₀	1	1	yes	characteristic
125	C₂²×C₄	—	4	4	no	—
126	C₂²×C₄	C₄⋊Dic₅	1	4	yes	—
127	C₂²×C₄	C₄⋊Dic₅	1	4	yes	—
128	C₂²×C₄	C₂×Dic₁₀	1	4	yes	—
129	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
130	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
131	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
132	C₂²×C₄	#709	1	8	yes	—
133	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
134	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
135	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
136	M₁₆	—	10	16	no	—
137	M₁₆	—	20	8	no	—
138	C₂×C₈	—	2	8	no	—
139	C₂×C₈	—	2	8	no	—
140	C₂×C₈	—	2	8	no	—
141	C₂×C₈	—	10	8	no	—
142	C₂×C₈	—	20	8	no	—
143	C₂×C₈	C₂²×Dic₅	1	8	yes	—
144	C₄²	C₄⋊Dic₅	1	4	yes	—
145	C₄²	C₄⋊Dic₅	1	4	yes	—
146	C₄²	C₂×D₂₀	1	4	yes	—
147	C₄²	C₂²×Dic₅	1	4	yes	—
148	C₂×C₅	—	2	2	no	—
149	C₂×C₅	—	2	2	no	—
150	C₂×C₅	#1838	1	4	yes	complemented
151	C₂×C₅	C₂×#382	1	1	yes	characteristic
152	C₂×C₅	C₂×#430	1	2	yes	—
153	C₂³	—	2	4	no	—
154	C₂³	#4016	1	4	yes	—
155	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
156	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
157	C₂³	C₂²×Dic₁₀	1	1	yes	characteristic
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	#4016	1	4	yes	—
162	C₂×C₄	#4021	1	4	yes	—
163	C₂×C₄	#4021	1	4	yes	—
164	C₂×C₄	C₂×C₄⋊Dic₅	1	8	yes	—
165	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
167	C₂×C₄	#4072	1	4	yes	—
168	C₂×C₄	#4072	1	4	yes	—
169	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
170	C₂×C₄	C₂×#709	1	4	yes	—
171	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
172	C₈	—	2	8	no	—
173	C₈	—	20	8	no	—
174	C₅	C₂×#1838	1	1	yes	characteristic, complemented, Sylow
175	C₂²	—	2	8	no	—
176	C₂²	—	2	2	no	—
177	C₂²	—	2	2	no	—
178	C₂²	#65225	1	2	yes	—
179	C₂²	#65225	1	2	yes	—
180	C₂²	#65414	1	4	yes	—
181	C₂²	C₂×#4016	1	2	yes	—
182	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
183	C₄	—	2	8	no	—
184	C₄	#65225	1	2	yes	—
185	C₄	#65225	1	2	yes	—
186	C₄	C₂×#4021	1	2	yes	—
187	C₄	C₂×#4072	1	2	yes	—
188	C₂	—	2	2	no	—
189	C₂	—	2	2	no	—
190	C₂	#10600634	1	4	yes	direct product factor
191	C₂	C₂×#65225	1	1	yes	characteristic
192	C₂	C₂×#65414	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂²×C₅×#97	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂²×#97	D₅
hypocenter	C₅	C₂×#1838
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
4. #10600634	C₂	direct (N × H)
150. C₂×C₅	#1838	semidirect
174. C₅	C₂×#1838	semidirect
190. C₂	#10600634	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#97	2
C₂×#65213	2
C₂×#64772	2
#10600634	2
C₂×#1838	5

Subgroups — C2×#10600634 #12099530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10600634 #12099530