Subgroups — C₂×#10597967 #12010583

Order 1280.

350 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#64971	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#335	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic
4	#10597967	C₂	1	4	yes	maximal, direct product factor
5	C₂×#4015	—	2	1	no	—
6	C₂×C₅×C₄⋊C₈	—	2	1	no	—
7	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×#3950	—	2	1	no	—
9	C₂×#3950	C₄	1	2	yes	—
10	C₂²×#672	—	2	1	no	—
11	#64971	—	2	1	no	—
12	#64971	—	2	1	no	—
13	#64971	C₄	1	2	yes	—
14	#64971	C₂²	1	2	yes	—
15	C₅×#335	—	2	1	no	—
16	C₅×#335	C₂²	1	2	yes	—
17	C₄⋊#672	—	2	1	no	—
18	C₄⋊#672	C₂²	1	2	yes	—
19	C₂×#1365	—	5	1	no	maximal, Sylow
20	C₂²×Dic₁₀	—	4	1	no	—
21	C₂²×C₅×C₈	—	4	1	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
25	C₂×C₄⋊Dic₅	—	2	2	no	—
26	C₂×C₄⋊Dic₅	—	4	1	no	—
27	#4015	—	2	2	no	—
28	#4015	—	2	2	no	—
29	#4015	—	4	1	no	—
30	#4015	—	4	1	no	—
31	C₅×C₄⋊C₈	—	2	2	no	—
32	C₅×C₄⋊C₈	—	4	1	no	—
33	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₄×C₅×C₈	D₄	1	1	yes	characteristic
35	C₄×C₅×C₈	D₄	1	1	yes	characteristic
36	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
37	#3950	—	2	2	no	—
38	#3950	—	2	2	no	—
39	#3950	—	2	2	no	—
40	#3950	C₂×C₄	1	4	yes	—
41	C₂×#672	—	2	2	no	—
42	C₂×#672	—	4	2	no	—
43	C₂×#500	—	5	1	no	—
44	C₂×C₄⋊C₁₆	—	5	1	no	—
45	C₂×#335	D₅	1	1	yes	characteristic
46	#1365	—	5	4	no	—
47	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
49	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
50	C₂²×Dic₅	—	4	2	no	—
51	C₂×Dic₁₀	—	4	2	no	—
52	C₂×Dic₁₀	—	4	2	no	—
53	C₂×Dic₁₀	—	8	4	no	—
54	C₂×C₅×C₈	—	2	2	no	—
55	C₂×C₅×C₈	—	4	2	no	—
56	C₂×C₅×C₈	—	8	2	no	—
57	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₅×C₈	SD₁₆	1	2	yes	—
60	C₂×C₅×C₈	Q₁₆	1	2	yes	—
61	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₅	D₈	1	1	yes	characteristic
65	C₄²×C₅	SD₁₆	1	1	yes	characteristic
66	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
67	C₄⋊Dic₅	—	2	4	no	—
68	C₄⋊Dic₅	—	4	2	no	—
69	C₄⋊Dic₅	—	4	2	no	—
70	C₄⋊Dic₅	—	4	2	no	—
71	#672	—	4	2	no	—
72	C₂×C₄⋊Q₈	—	10	1	no	—
73	C₂²×C₁₆	—	10	1	no	—
74	#500	—	5	2	no	—
75	#500	—	5	2	no	—
76	#500	—	10	1	no	—
77	#500	—	10	1	no	—
78	C₂×#107	—	5	2	no	—
79	C₂×#107	—	10	1	no	—
80	C₂×C₄⋊C₈	—	2	1	no	—
81	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
82	C₄⋊C₁₆	—	5	2	no	—
83	C₄⋊C₁₆	—	10	1	no	—
84	#335	—	2	1	no	—
85	#335	C₂×D₅	1	2	yes	—
86	C₂³×C₅	#102	1	1	yes	characteristic
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	#101	1	2	yes	—
89	C₂×C₄×C₅	#102	1	1	yes	characteristic
90	C₂×C₄×C₅	#102	1	1	yes	characteristic
91	C₂×C₄×C₅	#102	1	1	yes	characteristic
92	C₂×C₄×C₅	#103	1	2	yes	—
93	C₂×C₄×C₅	SD₃₂	1	2	yes	—
94	C₂×C₄×C₅	Q₃₂	1	2	yes	—
95	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×Dic₅	—	4	4	no	—
99	C₂×Dic₅	—	8	4	no	—
100	Dic₁₀	—	8	4	no	—
101	Dic₁₀	—	8	4	no	—
102	C₅×C₈	—	2	1	no	—
103	C₅×C₈	—	2	1	no	—
104	C₅×C₈	—	8	2	no	—
105	C₅×C₈	#103	1	2	yes	—
106	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
107	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
108	C₂²×Q₈	—	20	1	no	—
109	C₂²×C₈	—	4	1	no	—
110	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
111	C₂²×C₈	D₂₀	1	1	yes	characteristic
112	C₄⋊Q₈	—	10	2	no	—
113	C₄⋊Q₈	—	10	2	no	—
114	C₄⋊Q₈	—	20	1	no	—
115	C₄⋊Q₈	—	20	1	no	—
116	C₂×C₄⋊C₄	—	10	2	no	—
117	C₂×C₄⋊C₄	—	20	1	no	—
118	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
119	C₂×C₁₆	—	10	2	no	—
120	C₂×C₁₆	—	20	2	no	—
121	#107	—	5	4	no	—
122	#107	—	10	2	no	—
123	#107	—	10	2	no	—
124	#107	—	10	2	no	—
125	C₄⋊C₈	—	2	2	no	—
126	C₄⋊C₈	—	4	1	no	—
127	C₄×C₈	C₄×D₅	1	1	yes	characteristic
128	C₄×C₈	D₂₀	1	1	yes	characteristic
129	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
130	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
131	C₂²×C₅	#332	1	2	yes	—
132	C₂²×C₅	#358	1	2	yes	—
133	C₂²×C₅	#361	1	2	yes	—
134	C₂²×C₅	C₂×#102	1	1	yes	characteristic
135	C₄×C₅	—	2	1	no	—
136	C₄×C₅	—	2	1	no	—
137	C₄×C₅	#332	1	2	yes	—
138	C₄×C₅	#358	1	2	yes	—
139	C₄×C₅	C₂×#101	1	1	yes	characteristic
140	C₄×C₅	C₂×#103	1	1	yes	characteristic
141	C₄×C₅	C₂×SD₃₂	1	1	yes	characteristic
142	C₄×C₅	C₂×Q₃₂	1	1	yes	characteristic
143	Dic₅	—	8	4	no	—
144	C₂×Q₈	—	20	2	no	—
145	C₂×Q₈	—	20	2	no	—
146	C₂×Q₈	—	40	4	no	—
147	C₂²×C₄	—	20	2	no	—
148	C₂²×C₄	#685	1	1	yes	characteristic
149	C₂²×C₄	#686	1	1	yes	characteristic
150	C₂²×C₄	#688	1	1	yes	characteristic
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	8	2	no	—
154	C₂×C₈	#677	1	2	yes	—
155	C₂×C₈	Dic₂₀	1	2	yes	—
156	C₂×C₈	#685	1	1	yes	characteristic
157	C₂×C₈	#685	1	1	yes	characteristic
158	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
159	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
160	C₄⋊C₄	—	10	4	no	—
161	C₄⋊C₄	—	20	2	no	—
162	C₄⋊C₄	—	20	2	no	—
163	C₄⋊C₄	—	20	2	no	—
164	C₄²	#685	1	1	yes	characteristic
165	C₄²	#686	1	1	yes	characteristic
166	C₄²	#688	1	1	yes	characteristic
167	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
168	C₁₆	—	20	2	no	—
169	C₂×C₅	#1365	1	4	yes	complemented
170	C₂×C₅	C₂×#332	1	1	yes	characteristic
171	C₂×C₅	C₂×#358	1	1	yes	characteristic
172	C₂×C₅	C₂×#361	1	1	yes	characteristic
173	C₂³	#3941	1	1	yes	characteristic
174	Q₈	—	40	4	no	—
175	Q₈	—	40	4	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	20	4	no	—
178	C₂×C₄	—	40	4	no	—
179	C₂×C₄	#3941	1	1	yes	characteristic
180	C₂×C₄	#3941	1	1	yes	characteristic
181	C₂×C₄	#3941	1	1	yes	characteristic
182	C₂×C₄	#3948	1	2	yes	—
183	C₂×C₄	#3956	1	2	yes	—
184	C₂×C₄	#3959	1	2	yes	—
185	C₂×C₄	#3961	1	2	yes	—
186	C₂×C₄	C₂×#685	1	1	yes	characteristic
187	C₂×C₄	C₂×#686	1	1	yes	characteristic
188	C₂×C₄	C₂×#688	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	8	2	no	—
192	C₈	#3948	1	2	yes	—
193	C₈	C₂×#677	1	1	yes	characteristic
194	C₈	C₂×Dic₂₀	1	1	yes	characteristic
195	C₅	C₂×#1365	1	1	yes	characteristic, complemented, Sylow
196	C₂²	#64702	1	2	yes	—
197	C₂²	#64712	1	2	yes	—
198	C₂²	#64715	1	2	yes	—
199	C₂²	C₂×#3941	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	40	4	no	—
203	C₄	#64702	1	2	yes	—
204	C₄	#64712	1	2	yes	—
205	C₄	C₂×#3948	1	1	yes	characteristic
206	C₄	C₂×#3956	1	1	yes	characteristic
207	C₄	C₂×#3959	1	1	yes	characteristic
208	C₄	C₂×#3961	1	1	yes	characteristic
209	C₂	#10597967	1	4	yes	direct product factor
210	C₂	C₂×#64702	1	1	yes	characteristic
211	C₂	C₂×#64712	1	1	yes	characteristic
212	C₂	C₂×#64715	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#3941
commutator	C₂×C₅×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₂×C₅×#335	C₂
socle, prime radical	C₂³×C₅	#102
cosocle	C₄×C₅×C₈	C₂³
hypercenter	C₂×#335	D₅
hypocenter	C₅	C₂×#1365
LCS	G ▷ C₂×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#335

Complements

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

N	H ≅ G/N	type
4. #10597967	C₂	direct (N × H)
169. C₂×C₅	#1365	semidirect
195. C₅	C₂×#1365	semidirect
209. C₂	#10597967	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×#64971	2
C₂×C₅×#335	2
C₂×C₄⋊#672	2
#10597967	2
C₂×#1365	5

Subgroups — C2×#10597967 #12010583

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10597967 #12010583